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Quantum coherent control of laser-kicked molecular rotors Bitter, Martin 2016

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Quantum coherent control oflaser-kicked molecular rotorsbyMartin BitterB.Sc., Technische Universita¨t Darmstadt, 2008M.Sc., Technische Universita¨t Darmstadt, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2016© Martin Bitter 2016AbstractThe objective of this dissertation is the experimental study and control oflaser-kicked molecular rotors. Nonresonant rotational Raman excitation oflinear molecules by periodic sequences of ultra-short laser pulses allows forthe realization of a paradigm system - the periodically kicked rotor. Thisapparently simple physical system has drawn much interest within the lastdecades, especially due to its role in the field of quantum chaos.This thesis presents an experimental apparatus capable of producinglong sequences of high-energy femtosecond pulses. Rotation of diatomicmolecules, the most basic version of quantum rotors, is investigated undermulti-pulse excitation. In the case of periodic kicking, the wave function ofthe quantum rotor dynamically localizes in the angular momentum space,similarly to Anderson localization of the electronic wave function in disor-dered solids. We present the first direct observation of dynamical localizationin a system of true rotors. The suppressed growth of rotational energy isdemonstrated, as well as the noise-induced recovery of diffusion, indicativeof classical dynamics. We examine other distinct features of the quantumkicked rotor and report on quantum resonances, the phenomena of rotationalBloch oscillations and Rabi oscillations. In addition, multi-pulse excitationis investigated in the context of creating broad rotational wave packets.Another goal of the reported study is the coherent control of quantumchaos. We demonstrate that the relative phases in a superposition of rota-tional states can be used to control the process of dynamical localization.We specify the sensitivity to external parameters and illustrate the loss ofcontrol in the classical limit of laser-molecule interaction.Our work advances the general understanding of the dynamics of laserkicked molecules and complements previous studies of the quantum kickedrotor in a system of cold atoms. The results encourage further studies, e.g.of quantum phenomena which are unique to true rotors. The possibility ofcontrol in classically chaotic systems has far reaching implications for theultimate prospect of using coherence to control chemical reactions.iiPrefaceAll the work presented in this thesis was conducted in “The Laboratory forAdvanced Spectroscopy and Imaging Research” (LASIR) at the Universityof British Columbia, Vancouver, Canada. I designed and constructed theapparatus with help of Kamil Krawczyk, Andrej Machnev and JonathanMorrison. All the data taking and analysis of the results was done by me. Forthe theoretical analysis of the δ-kicked rotor I used a Matlab program. Thecore of the code, the interaction of a single kick with a diatomic molecule,was written by Johannes Floß. I adapted the program for multiple kicks.All the simulations in this thesis were done by me.The methods to create long sequences of high-energy femtosecond pulsesare described in chapter 3. The main ideas are published in Applied Op-tics [M. Bitter,V. Milner, “Generating long sequences of high-intensity fem-tosecond pulses”, Appl. Opt. 55, 830 (2016)] [23]. A version of Sec. 5.4including all the figures has been published in Physical Review A [M. Bit-ter, V. Milner, “Rotational excitation of molecules with long sequences ofintense femtosecond pulses”, Phys. Rev. A 93, 013420 (2016)] [24]. The keyresults of chapter 6, described in Sec. 6.4, have been published in PhysicalReview Letters [M. Bitter, V. Milner, “Experimental Observation of Dynam-ical Localization in Laser-Kicked Molecular Rotors”, Phys. Rev. Lett. 117,144104 (2016)] [22]. Further investigations on the same topic, discussedin Sec. 6.4.4, are presented in a follow-up manuscript [M. Bitter, V. Mil-ner, “Control of quantum localization and classical diffusion in laser-kickedmolecular rotors”] [20], which has been submitted for publication. Chap-ter 7 is heavily based on a manuscript [M. Bitter, V. Milner, “Experimentaldemonstration of coherent control in quantum chaotic systems”] [21], whichhas been submitted for publication.The work on Bloch oscillations in Sec. 5.3 and 6.5, and the results onRabi oscillations in Sec. 5.2 are currently being prepared for two separatemanuscripts, to be submitted shortly.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Main research objectives . . . . . . . . . . . . . . . . . . . . 41.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 62 The kicked rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Classical kicked rotor . . . . . . . . . . . . . . . . . . . . . . 92.1.1 The standard map . . . . . . . . . . . . . . . . . . . . 102.1.2 Classical diffusion . . . . . . . . . . . . . . . . . . . . 112.2 Quantum kicked rotor . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Quantum resonance . . . . . . . . . . . . . . . . . . . 122.2.2 Dynamical localization . . . . . . . . . . . . . . . . . 122.2.3 Stochasticity . . . . . . . . . . . . . . . . . . . . . . . 132.2.4 Atom-optics kicked rotor . . . . . . . . . . . . . . . . 132.3 Laser-kicked molecular rotor . . . . . . . . . . . . . . . . . . 142.3.1 Light-molecule interaction . . . . . . . . . . . . . . . 142.3.2 Kick strength . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 172.3.4 Wave function . . . . . . . . . . . . . . . . . . . . . . 182.3.5 Density matrix . . . . . . . . . . . . . . . . . . . . . . 182.3.6 Numerical analysis . . . . . . . . . . . . . . . . . . . 20ivTable of Contents2.3.7 Revival time . . . . . . . . . . . . . . . . . . . . . . . 232.3.8 Resonance map . . . . . . . . . . . . . . . . . . . . . 232.3.9 Choice of molecule . . . . . . . . . . . . . . . . . . . . 262.4 Correspondence to crystalline solids . . . . . . . . . . . . . . 262.4.1 The one-dimensional lattice . . . . . . . . . . . . . . 272.4.2 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . 282.4.3 Floquet’s theorem . . . . . . . . . . . . . . . . . . . . 292.4.4 Tight-binding model . . . . . . . . . . . . . . . . . . 302.4.5 Anderson model . . . . . . . . . . . . . . . . . . . . . 322.4.6 Mapping the kicked-rotor onto the Anderson model . 342.4.7 Anderson model of the laser kicked rotor . . . . . . . 373 Techniques I: Generation of pulse sequences . . . . . . . . . 403.1 Laser pulses and pulse sequences . . . . . . . . . . . . . . . . 423.1.1 Transform-limited pulse . . . . . . . . . . . . . . . . . 423.1.2 Linearly-chirped pulse . . . . . . . . . . . . . . . . . . 443.1.3 Pulse sequences . . . . . . . . . . . . . . . . . . . . . 453.2 Laser source . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.1 Pulse characterization . . . . . . . . . . . . . . . . . . 513.3 Pulse shaping I: The pulse shaper . . . . . . . . . . . . . . . 533.3.1 Optical setup . . . . . . . . . . . . . . . . . . . . . . 543.3.2 Implementation of pulse sequences . . . . . . . . . . . 553.3.3 Demonstration of pulse sequences . . . . . . . . . . . 583.4 Pulse shaping II: The Michelson interferometer . . . . . . . . 583.5 Multi-pass amplification . . . . . . . . . . . . . . . . . . . . . 603.5.1 Compressed versus chirped amplification . . . . . . . 623.5.2 Amplification of pulse sequences . . . . . . . . . . . . 643.5.3 Demonstration of amplified pulse sequences . . . . . . 674 Techniques II: Rotational Raman spectroscopy . . . . . . . 694.1 Raman spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 694.1.1 Raman spectrogram . . . . . . . . . . . . . . . . . . . 724.2 Molecular beam source . . . . . . . . . . . . . . . . . . . . . 754.2.1 Theory of supersonic expansion . . . . . . . . . . . . 754.2.2 Beam setup . . . . . . . . . . . . . . . . . . . . . . . 774.2.3 Beam characterization . . . . . . . . . . . . . . . . . 805 Resonant excitation of molecular rotation . . . . . . . . . . 865.1 Demonstration of the resonance map . . . . . . . . . . . . . 875.1.1 Excitation of single coherences . . . . . . . . . . . . . 89vTable of Contents5.2 Rabi oscillations in molecular rotation . . . . . . . . . . . . . 925.2.1 Theory and simulation . . . . . . . . . . . . . . . . . 935.2.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . 995.2.3 Observation of Rabi oscillations . . . . . . . . . . . . 1005.2.4 Kick strength calibration . . . . . . . . . . . . . . . . 1075.3 Bloch oscillations in molecular rotation . . . . . . . . . . . . 1075.3.1 Theory I: Bloch oscillations in crystalline solids . . . 1095.3.2 Theory II: Bloch oscillations in a molecular rotor . . 1105.3.3 Numerical simulation . . . . . . . . . . . . . . . . . . 1125.3.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . 1135.3.5 Observation of Bloch oscillations . . . . . . . . . . . . 1155.4 Generation of broad rotational wave packets . . . . . . . . . 1215.4.1 Periodic excitation . . . . . . . . . . . . . . . . . . . . 1225.4.2 Non-periodic excitation . . . . . . . . . . . . . . . . . 1255.4.3 Propagation effects . . . . . . . . . . . . . . . . . . . 1295.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 Dynamical localization in molecular rotation . . . . . . . . 1356.1 Experiments on Anderson localization . . . . . . . . . . . . . 1366.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.3.1 Calibration of experimental parameters . . . . . . . . 1396.3.2 Population retrieval from the experiment . . . . . . . 1406.4 Observation of dynamical localization . . . . . . . . . . . . . 1416.4.1 Dependence on the number of kicks . . . . . . . . . . 1456.4.2 Dependence on the kick strength . . . . . . . . . . . . 1476.4.3 Rotational energy . . . . . . . . . . . . . . . . . . . . 1486.4.4 Dependence on the period . . . . . . . . . . . . . . . 1486.5 Transition from Bloch oscillations to dynamical localization . 1536.5.1 Anderson wall . . . . . . . . . . . . . . . . . . . . . . 1536.5.2 Evolution of angular momentum . . . . . . . . . . . . 1576.5.3 Evolution of rotational energy . . . . . . . . . . . . . 1616.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627 Coherent control of quantum chaos . . . . . . . . . . . . . . 1647.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.1.1 Coherent control . . . . . . . . . . . . . . . . . . . . . 1657.1.2 Quantum-to-classical transition . . . . . . . . . . . . 1657.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.3 Demonstration of coherent control in quantum chaotic system 169viTable of Contents7.3.1 Robustness of control . . . . . . . . . . . . . . . . . . 1717.3.2 Quantum-to-classical transition . . . . . . . . . . . . 1727.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178AppendicesA List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . 196B Classical dynamics of the kicked rotor . . . . . . . . . . . . . 198B.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 198C Spectral decomposition of the kicked rotor wave function 200C.1 The kicked rotor . . . . . . . . . . . . . . . . . . . . . . . . . 200C.2 The periodically kicked rotor . . . . . . . . . . . . . . . . . . 201C.2.1 The wave function ψ+ . . . . . . . . . . . . . . . . . . 201C.2.2 Decomposition of ψ+ . . . . . . . . . . . . . . . . . . 201C.2.3 The coupling matrix . . . . . . . . . . . . . . . . . . . 202D Fourier transform of shaped pulses . . . . . . . . . . . . . . . 204D.1 Fourier transform of a TL pulse . . . . . . . . . . . . . . . . 204D.2 Fourier transform of a chirped pulse . . . . . . . . . . . . . . 205D.3 Fourier transform of a pulse train . . . . . . . . . . . . . . . 206D.4 Convolution theorem . . . . . . . . . . . . . . . . . . . . . . 208E Semi-classical model of rotational Bloch oscillations . . . . 209E.1 Non-rigid rotor on quantum resonance . . . . . . . . . . . . . 210E.2 Rigid rotor detuned from quantum resonance . . . . . . . . . 210viiList of Tables2.1 Molecular constants and parameters for N2 and O2. . . . . . 262.2 Anderson localization versus dynamical localization . . . . . . 39viiiList of Figures2.1 Standard map of the classical kicked rotor. . . . . . . . . . . 102.2 Illustration of a diatomic molecule in a laser field. . . . . . . . 142.3 Field-free alignment of linear molecules. . . . . . . . . . . . . 152.4 Rotational Raman transitions in linear molecules. . . . . . . . 162.5 Resonance map. . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6 Dispersion relation of the 1D tight-binding model. . . . . . . 282.7 Extended Bloch wave versus localized wave. . . . . . . . . . . 332.8 Real lattice versus rotational lattice. . . . . . . . . . . . . . . 353.1 Electric field of a TL pulse in the spectral and time domains. 433.2 Electric field of a chirped pulse in the spectral and time domains. 453.3 Schematic of a flat pulse train in the time domain. . . . . . . 463.4 Pulse sequences in the spectral and time domains. . . . . . . 473.5 Pulse sequences in the spectral and time domains (PAP). . . 493.6 (Cross-correlation) frequency-resolved optical gating (X)FROG. 523.7 XFROG spectrograms of single pulses. . . . . . . . . . . . . . 533.8 Pulse shaper in ’4f ’-folded design . . . . . . . . . . . . . . . . 543.9 Discretization of the SLM amplitude mask. . . . . . . . . . . 563.10 Optimization protocol to shape flat pulse sequences. . . . . . 573.11 Demonstration of flexibility in shaping pulse trains. . . . . . . 593.12 Diagram of the Michelson interferometer setup. . . . . . . . . 603.13 Diagram of the home-built multi-pass amplifier. . . . . . . . . 613.14 The effect of amplification on the pulse spectrum. . . . . . . . 633.15 Diagram of the setup to create long, high-energy pulse trains. 653.16 XFROG spectrogram of a high-intensity pulse train. . . . . . 663.17 Temporal profile of long, high-intensity pulse trains. . . . . . 684.1 Scheme of the Raman spectroscopy setup. . . . . . . . . . . . 704.2 Raman spectrum of N2 after a single kick. . . . . . . . . . . . 734.3 Raman spectrum of O2 after a single kick. . . . . . . . . . . . 744.4 Diagram of a supersonic jet expansion. . . . . . . . . . . . . . 764.5 Diagram of the molecular beam setup. . . . . . . . . . . . . . 78ixList of Figures4.6 Temperature and pressure as a function of nozzle distance. . 794.7 Rotational spectroscopy of N2 and O2. . . . . . . . . . . . . . 824.8 Collisional decay of the Raman signal. . . . . . . . . . . . . . 834.9 Rotational distribution as a function of nozzle distance. . . . 855.1 Raman spectrogram after a periodic pulse train (I). . . . . . . 885.2 Raman spectrogram after a periodic pulse train (II). . . . . . 905.3 Selective excitation with periodic pulse trains. . . . . . . . . . 915.4 Energy diagram of a two-level system. . . . . . . . . . . . . . 935.5 Rabi oscillations: Simulation at 0 K . . . . . . . . . . . . . . 965.6 Rabi oscillations: Resonance map . . . . . . . . . . . . . . . . 965.7 Rabi oscillations: Simulation at 25 K . . . . . . . . . . . . . . 985.8 Rabi oscillations: Raman spectrograms. . . . . . . . . . . . . 1015.9 Rabi oscillations: Dependence on detuning. . . . . . . . . . . 1035.10 Rabi oscillations: Dependence on kick strength. . . . . . . . . 1045.11 Rabi oscillations: Effect of bandwidth. . . . . . . . . . . . . . 1065.12 Bloch oscillations: Simulation at 25 K . . . . . . . . . . . . . 1135.13 Bloch oscillations: Dependence on detuning. . . . . . . . . . . 1155.14 Bloch oscillations: Dependence on kick strength. . . . . . . . 1165.15 Bloch oscillations: Dependence on energy & detuning. . . . . 1185.16 Bloch oscillations: Effect of bandwidth. . . . . . . . . . . . . 1205.17 Limited excitation due to centrifugal distortion. . . . . . . . . 1245.18 Scheme of an optimized pulse train. . . . . . . . . . . . . . . 1265.19 Excitation with an optimized pulse train. . . . . . . . . . . . 1275.20 Propagation effects: Cascaded Raman scattering. . . . . . . . 1305.21 Propagation effects: Spectral broadening. . . . . . . . . . . . 1326.1 Dynamical localization: Resonance map. . . . . . . . . . . . . 1406.2 Dynamical localization & timing noise. . . . . . . . . . . . . . 1426.3 Dynamical localization & amplitude noise. . . . . . . . . . . . 1446.4 Dynamical localization: Dependence on pulse number. . . . . 1466.5 Dynamical localization: Dependence on kick strength. . . . . 1476.6 Dynamical localization: Rotational energy. . . . . . . . . . . . 1496.7 Dynamical localization: Dependence on pulse period. . . . . . 1506.8 Diffusive growth: Dependence on pulse period. . . . . . . . . 1526.9 Dynamical behaviour near the quantum resonance (I). . . . . 1586.10 Dynamical behaviour near the quantum resonance (II). . . . . 1596.11 Localized population near the quantum resonance. . . . . . . 1606.12 Rotational energy near the quantum resonance (I). . . . . . . 1616.13 Rotational energy near the quantum resonance (II). . . . . . 162xList of Figures7.1 Control of quantum chaos: Pulse train diagram. . . . . . . . . 1677.2 Control of quantum chaos: Resonance map. . . . . . . . . . . 1677.3 Control of the rotational energy. . . . . . . . . . . . . . . . . 1687.4 Control of the rotational population. . . . . . . . . . . . . . . 1707.5 Sensitivity of the control scheme. . . . . . . . . . . . . . . . . 1717.6 Quantum to classical transition. . . . . . . . . . . . . . . . . . 172xiList of AbbreviationsAOKR Atom-Optics Kicked RotorBBO Barium BorateCCD Charge-Coupled DeviceCGC Clebsch-Gordan CoefficientsFROG Frequency Resolved Optical GatingFT Fourier TransformFWHM Full Width at Half MaximumKAM Komolgorov-Arnold-MoserKR Kicked RotorMIIPS Multiphoton Intrapulse Interference Phase ScanMKR Modified Kicked RotorMPA Multi-Pass AmplifierNd:YAG Neodymium-doped yttrium aluminium garnet (Nd:Y3Al5O12)OPA Optical Parametric AmplifierPAP Piecewise Adiabatic PassageQKR Quantum Kicked RotorSLM Spatial Light ModulatorTi:Sapph Titanium-doped sapphire (Ti3+:sapphire)TL Transform-limitedXFROG Cross-correlated Frequency Resolved Optical GatingxiiChapter 1IntroductionControl of molecular rotation with ultra-short laser pulses is an active re-search area of experimental and theoretical molecular science. Interestsare ranging from studying molecular dynamics to chemical reactivity ofmolecules [164, 126, 56] and from implementing model systems like thekicked rotor to exploring quantum chaotic systems [28, 59, 63, 70, 71].The rotation of molecules in the gas phase can be excited coherently byintense non-resonant laser fields. The manipulation with ultra-short laserpulses in particular allows for time-dependent control and field-free studies[152]. If the duration of the laser pulse is short with respect to the periodof molecular rotation, the excited molecules will exhibit complex dynam-ics. The evolution of the rotational wave packet is characterized by “quan-tum revivals”, a consequence of the discreteness of the angular momentumspectrum[151].Many techniques to control molecular rotation exist that use two time-delayed ultra-short pulses, and several techniques are based on pulse se-quences with more than two pulses, e.g. to enhance molecular alignment [41],to control the alignment of asymmetric molecules [139] or to initiate uni-directional rotation [193]. The reasons behind implementing multi-pulseschemes are often to improve the selectivity of excitation or to increase theefficiency. Both are achieved by tailoring the timing between the pulses inaccordance with the rotational dynamics. A few schemes of using long se-quences of pulses, so-called “pulse trains”, have been proposed theoretically[12, 103, 104, 166, 187], and implemented experimentally with sequences ofup to eight laser pulses [41, 192, 64, 88].The impulsive excitation of linear molecules with a sequence of peri-odic high-intensity ultra-short pulses presents the realization of a paradigmsystem known as the “kicked rotor” [59]. In this theoretically well-studiedmodel a rotor is subject to an external driving field of periodic δ-kicks. Clas-sically, the system is described by a set of two simple equations of motion forthe angle and angular momentum of the rotor, which despite being strictlydeterministic, turn chaotic beyond a certain strength of the kicks [31].1Chapter 1. IntroductionIn his ground-breaking paper from 1969, Boris Chirikov introduced thefamous “standard map” to describe a large number of classically chaoticsystems [35]. Among them is the periodically delta-kicked rotor. Ten yearslater, Chirikov and collaborators discovered a totally new and unexpectedeffect: the stochastic behavior of the classical rotor becomes non-stochasticfor the quantum kicked rotor (QKR) [31]. Two fundamentally differentregimes emerge depending on the period of the external driving field. Ifthe external frequency matches the natural frequency of the quantum rotor,the rotational excitation becomes highly efficient [84]. In this case of anexcitation on “quantum resonance” the angular momentum grows linearlywith time, rather than diffusively (∝ √time) like in the case of a classi-cal rotor. However, if the external frequency is incommensurable with thenatural frequency of the quantum rotor, the surprising result is a completesuppression of angular momentum growth. Despite a continued kicking, therotor does not accept any rotational energy. This effect has later been calledthe “dynamical localization”.In 1982 Shmuel Fishman and collaborators [52] pointed out that themechanism of dynamical localization is related to the phenomenon of “An-derson localization” [6] known from solid state physics. Philip Anderson, in1958, had shown that the propagation of an electron in a one-dimensionaldisordered lattice is completely suppressed 1. Fishman et.al. formulated atheory that proved the deep underlying connection between the two types oflocalization. They considered the rotational quantum number of a quantumrotor as an effective site number in a rotational lattice. The disorder couldbe linked to the periodic kicking with a nonlinear potential. In both cases,the effect of localization stems from the destructive interference of quantumpathways: in the real space of the lattice [6] and in the angular momentumspace of the rotor [31, 84].Research on the effect of Anderson localization in a variety of differentsystems is very active and constantly growing [98]. As it relies on the de-structive interference of waves, it has been observed with classical waves, i.e.photons [68, 42, 183, 33, 165, 150, 99] and sound [178, 80], as well as withmatter waves [39, 50, 18, 143]. Dynamical localization of the QKR presentsan alternative model to study the fundamental aspects of Anderson localiza-tion. In 1992, Graham et.al. suggested the realization of a QKR in a relatedsystem by observing periodically kicked ultracold atoms in the momentumspace [74]. The “atom-optics kicked rotor” (AOKR), first demonstrated in1995 [121], has since evolved to a standard approach for studying many in-1For this and related work, Anderson was awarded the Nobel Prize in Physics in 1977.2Chapter 1. Introductionteresting effects, from dynamical localization and quantum resonances, tothe effects of noise and dissipation, etc. [135]. Yet until now, more than 30years after Chirikovs and Fishmans papers, the fundamental effect of dy-namical localization has never been observed in a system of true quantumrotors, i.e. quantum objects that actually rotate in space.Besides being an ideal testing ground for effects related to Anderson lo-calization, the QKR plays another important role in the field of quantumchaos [71], a synonym for quantum systems whose classical counterparts arechaotic. One question of interest is whether the evolution of a chaotic sys-tem can be steered one way or another by adjusting its initial state. Thevery definition of classical chaos, i.e. the exponential sensitivity to initialconditions, seems to leave no room for such controllability. Yet quantummechanics teaches us that quantum trajectories, leading to the same finalstate, will interfere with one another no matter how stochastic they are.Adjusting the relative phase of these trajectories to make their interferenceeither constructive or destructive should then provide full control [157], con-trary to the classical expectations. The question is not new and has beendiscussed in many theoretical works for more than 20 years (for reviews, seeRef. [142, 71]). Gong and Brumer considered the QKR to study the con-trollability of classically chaotic dynamics in the quantum limit [69, 70, 71].They showed theoretically that the energy of the localized state can indeedbe controlled by modifying the initial quantum state of the rotor. However,the experimental study of the predicted controllability has been lacking.Control in the regime of quantum chaos is of immediate relevance for thegeneral goal of controlling molecular dynamics with external fields, sincemany complex molecules often display chaotic behaviour.Several theoretical works suggested diatomic molecules repeatedly kickedby a pulsed external field (microwave, optical or THz) as a realization of theQKR [28, 70, 59, 114]. In their proposal, Averbukh and coworkers usedultra-short laser pulses to periodically excite linear molecules [59]. Theyinvestigated several QKR phenomena [59, 63, 60, 61] and effects related tothe quantum resonance have been experimentally verified since [41, 192, 64].An onset of dynamical localization in laser-induced molecular alignment wasreported [88], but the direct evidence of the exponentially localized statesand the suppressed growth of the rotational energy have not been shown.Laser-kicked molecules as QKR provide some advantages over the cold-atom analogue. The angular momentum of true rotors is inherently quan-tized, in contrast to the continuous spectrum of the translational momentum31.1. Main research objectivesin the case of the AOKR [135]. To reach sufficiently narrow momentum dis-tributions the atoms need to be cooled down to ultra-cold temperatures,which requires a complicated experimental setup. On the other hand, su-personic molecular jets are easier to construct, and they provide low enoughtemperatures for the molecular QKR experiments.Furthermore, the suggested system enables the study of quantum phe-nomena which are unique to true rotors. The effect of centrifugal distortionof spinning molecules leads to oscillations in the angular momentum distri-bution, similar to Bloch oscillations in solids [25]. This has been predictedby Floss and Averbukh [60, 62] and recently demonstrated [64]. Moleculesexperience an “edge” in the semi-infinite lattice of rotational states, sinceonly positive rotational quantum numbers are allowed. Floss and Averbukhtheoretically showed the existence of localized edge states [61], which hasnot been observed yet.1.1 Main research objectivesThe goal of our research is the study of the quantum kicked rotor withlaser-kicked linear molecules. In the last decade some of the many predictedphenomena, i.e. the quantum resonance [41, 192], the existence of Blochoscillations [64] and some first indirect signs for the presence of localiza-tion [88] have been explored experimentally. In order to investigate otherfundamental effects, in particular the demonstration of a localized angularmomentum distribution, all of the above experiments lacked a few key re-quirements, the most important one being a sufficiently long pulse sequenceof high energy ultra-short pulses.We designed and built a new tool to create pulse sequences of twentyor more pulses; up until now the sequences were limited to eight pulses.The technique is based on the principles of femtosecond pulse shaping andinterferometric multiplexing. While other techniques often suffer from insuf-ficient pulse intensities, we use multi-pass amplification to compensate forthe severe energy losses. This approach excels in almost full controllabilityover the profile of the generated pulse train: The amplitudes as well as thetemporal spacing of all sub-pulses can be adjusted individually.In our experiments we use long pulse sequences to excite linear molecules,i.e. nitrogen or oxygen, at rotational temperatures around 25 K. Prior exper-iments have exclusively been done at room temperature, which prohibits theobservation of many effects due to broad angular momentum distributions.In contrast to previously implemented detection techniques, we employed41.1. Main research objectivesa frequency-resolved method allowing for the resolution of individual rota-tional states.Complimenting previous work with AOKR, we directly observed the ef-fect of dynamical localization, demonstrating the characteristic exponentialdistribution of the localized wave function and the suppression of the ro-tational energy growth. We introduced noise in a sequence of rotationalkicks to show the expected destruction of localization and the recovery ofclassically diffusive behaviour.In addition, we continued the research of molecular excitation on quan-tum resonances. We improved the existing studies on Bloch oscillationsin molecular rotation [64] and documented a new phenomenon of Rabi os-cillations between two rotational states. The prospects of creating broadrotational wave packets by means of molecular interaction with long pulsesequences were explored. The timing of the pulses could be optimized topartially mitigate the limitations stemming from the centrifugal distortion.Another objective was to explore the possibility of coherent control ina system that is classically chaotic. We established the first experimentaldemonstration of “quantum chaos under control”. In the verification of theGong-Brumer control scheme [69, 70], our unique ability to fine-tune theinitial rotational state was used to control the rotation of a molecule atlater times. By varying the relative phases of the initial states we effectivelychanged the localization process of the QKR. We proved the quantum na-ture of the demonstrated controllability by showing its disappearance in theclassical regime of laser-molecule interaction.Many of the demonstrated phenomena, e.g. dynamical localization andBloch oscillations, establish an intimate connection between the two fun-damental aspects of modern physics: the motion of a quantum particle ina disordered solid and the motion of a quantum pendulum under periodickicking [52]. Our work opens new opportunities for investigating quantumphenomena which are unique to true rotors, e.g. edge localization [61] orthe effects of centrifugal distortion and rotational decoherence on QKR dy-namics [60]. We believe that our results are important to the general fieldsof laser technology, quantum control of light-matter interaction, nonlineardynamics and quantum chaos.51.2. Outline of the thesis1.2 Outline of the thesisThe goal of this thesis is to give a complete overview on the subject of“Quantum coherent control of laser-kicked molecular rotors”.Chapter 2 describes the kicked rotor model, classically (Sec. 2.1) as wellas its quantum mechanical counterpart (Sec. 2.2). Important concepts tocharacterize different interaction regimes, the stochasticity of the system andthe transition from a quantum-to-classical rotor are discussed. We becomemore rigorous when we discuss the QKR model of interest - the laser-kickedmolecular rotor (Sec. 2.3). We introduce the physics behind laser-moleculeinteraction and the relevant Hamiltonian. We give details how we modelthe system numerically to analyse our results. The kicked rotor model canbe mapped onto a one-dimensional tight-binding problem (Sec. 2.4). Weintroduce the concepts to describe quantum particles in crystalline solids,before we derive the correspondence between a periodically-kicked quantumrotor and a quantum particle in a disordered lattice.In chapter 3 we give a detailed description of the technique that we de-veloped to create long sequences of high-intensity ultra-short laser pulsesand demonstrate its unique abilities. A second technical chapter 4 intro-duces other techniques that proved to be crucial for the implementation ofmany experiments. Rotational Raman spectroscopy (Sec. 4.1) enabled aquantum-state selective detection with a dynamic range over several ordersof magnitude. This Raman technique was implemented to work with rota-tionally cold molecules produced in a supersonic jet expansion (Sec. 4.2).We analyse the performance and characterize the experimental conditions.Chapter 5 is dedicated to molecular excitation via quantum resonances.At first, an intuitive “resonance map” is recorded and used to explain theexistence of various (fractional) quantum resonances (Sec. 5.1). An immedi-ate implication of the periodic excitation on fractional quantum resonancesare rotational Rabi oscillations in an effective two-level system of two ro-tational states (Sec. 5.2). The excitation on full quantum resonances leadsto the phenomenon of Bloch oscillations in the angular momentum space(Sec. 5.3). The effect is studied and compared to the solid state analogue.Finally, the quantum resonances are explored with the objective of excit-ing broad rotational wave packets (Sec. 5.4), which is desirable for creatingstrongly aligned molecular samples.Dynamical localization in a system of true rotors is the subject of chap-ter 6. Starting with a theoretical picture of dynamical localization (Sec. 6.2)and the intricacies of the experimental realization (Sec. 6.3), we present di-rect evidence of localized molecular rotation and test its dependency on61.2. Outline of the thesismultiple experimental parameters and the transition to classical behaviourunder the influence of noise (Sec. 6.4). In addition, we establish a connectionbetween the phenomenon of dynamical localization and Bloch oscillationsby studying the regime around the quantum resonance (Sec. 6.5).The final experimental chapter 7 discusses the topic of controlling quan-tum chaos. We describe the idea of coherent control and the unique op-portunity to tune the QKR dynamics from quantum mechanical to classical(Sec. 7.1). We detail the scheme that we use to control the dynamical local-ization process (Sec. 7.2) and show the observed controllability (Sec. 7.3).We demonstrate that the control relies on quantum coherences by drivingthe system closer to the classical limit. At last, we outline the future direc-tions of this work in chapter 8.7Chapter 2The kicked rotorThe one-dimensional kicked rotor (KR) is a well-studied paradigm system,classically as well as in quantum mechanics [31, 138, 77]. Although it isa strictly deterministic Hamiltonian system, it exhibits chaotic motion incertain regimes. It’s simplicity and the fact that the transition to chaos canbe controlled by the external driving field make it a popular model system[138].The interaction of a diatomic molecule subject to N periodic kicks vianon-resonant linearly polarized laser pulses, provides an experimental real-ization of the “quantum kicked rotor” (QKR). It is described by the followingHamiltonian:Hˆ =Jˆ22I− ~P cos2(θ)N−1∑n=0δ(t− nT ) . (2.1)The first term, the kinetic energy of the rotor, depends on the angularmomentum operator Jˆ and the moment of inertia I. The second term isdictated by a nonlinear potential with the angle θ between the molecularaxis and the laser polarization axis. The period of the driving field is T , thestrength of the kicks is P and ~ is the reduced Planck constant.In section 2.1 we look at the classical version of this rotor. We explain thefundamental concepts of the kicked rotor model and introduce the “standardmap”. Section 2.2 discusses the similarities and the unique differences of thequantum version of the KR. In Sec. 2.3 we return to the system that we willstudy experimentally, the “laser-kicked molecular rotor”. More details aboutthe light-molecule interaction, the Hamiltonian and the rotational dynamicswill be provided.A second important significance of the KR has been shown by Fishmanand coworkers [52, 76]: the QKR can be mapped onto a tight-binding modelknown from solid state physics. Section 2.4 is dedicated to the correspon-dence between a crystalline solid and the QKR, which will be used in ouranalysis of several observed effects.82.1. Classical kicked rotor2.1 Classical kicked rotorThe classical equations of motion of the Hamiltonian in Eq. 2.1 areθN = θN−1 + J˜NJ˜N = J˜N−1 −K sin(2θN−1) .(2.2)In appendix B we show how the equations of motion can be derived from theHamiltonian of the kicked rotor 2. The two coupled equations describe themotion of the KR at discrete steps after each kick N via the two canonicalvariables, the angle θ and the dimensionless angular momentum J˜ ,J˜ = JTI. (2.3)The dynamics of the KR is determined by the parameter K - the “stochas-ticity parameter” - which is a measure of the amplitude of the kicks. In thecase of a diatomic molecule exposed to non-resonant linearly polarized laserpulses, the parameter can be calculated asK =T ·∆α4I∫E2(t)dt , (2.4)where ∆α is the polarizability anisotropy of the molecule and E(t) is theelectric field envelope. The integral is evaluated over one full period T ofthe pulse sequence. Details about the light-molecule interaction for a laser-kicked rotor will be discussed in Sec. 2.3.The KR dynamics can be distinguished into several regimes dependingon the magnitude ofK. For weak kicks the classical equations (2.2) representperiodic motion. Here, the angle and the angular momentum only changein small quantities from one kick to the next, resulting in a deterministicmotion. However, if the amplitude of the kicks becomes large enough, themotion turns chaotic. The behaviour of the KR is no longer regular butrather stochastic, hence the name of the stochasticity paramter. We canunderstand the chaos as a consequence of the nonlinear angle-dependentpotential [135]. At high K values each kick is effectively quasi-random in itsdirection and its amplitude.2 The traditional equations of motion for the planar kicked rotor have a slightly modifiedform θN = θN−1 + J˜N and J˜N = J˜N−1 + K sin(θN−1) [70]. However, the physicalinterpretation remains the same.92.1. Classical kicked rotor𝜃/𝜋-101ac-0.5 0 0.5 -0.5 0 0.5𝜃/𝜋Angular momentum  ሚ 𝐽-1.501.5-606-20020bdFigure 2.1: Sections of the standard map of the kicked rotor for differentvalues of the stochasticity parameter (a) K = 0.25, (b) K = 0.5, (c) K = 1and (d) K = The standard mapThe recursion equations (2.2) are called the “standard map” [31]. Strobo-scopically one can follow the motion of the rotor on classical trajectories inthe phase space. The map depends solely on the stochasticity parameter K.In Fig. 2.1 we show four different maps obtained from Eq. 2.2 correspond-ing to different K parameters. All maps are constructed starting from thesame initial conditions: seven points in phase space at J˜0 = 0 and θ0 = pip/16with p = 1, 2, ..., 7 (plotted in different colours). Then the system is evolvedin time up to N = 10000, which yields either stable trajectories, visible asclosed ellipsoids, or chaotic trajectories. Due to the periodicity of sin(2θ)the complete dynamics can be portrayed on a cylinder, modulo pi.The KR behaviour is analysed in different regimes, controlled by the102.2. Quantum kicked rotorstochasticity parameter. Small values of K  1 lead to periodic motion,seen in Fig. 2.1(a). Once chaos emerges, for small K, it is first isolated inchaotic regions, Fig. 2.1(b). The area of the chaotic regions grows with K,while the regions itself are separated by Komolgorov-Arnold-Moser (KAM)trajectories [75]. Beyond a critical value Kcr the last KAM trajectory dis-appears and the motion is no longer bounded, as in Fig. 2.1(c). Note thedifferent y-scale, shown is only a section of the occupied phase space. Thecritical value for the planar kicked rotor is cited as Kc = 0.97164... [75]. Thelaser-kicked rotor described in Eq. 2.2, however, turns chaotic twice fasteras a function of K 3. Eventually all stable islands disappear in the chaoticsea, shown in Fig. 2.1(d). Although no critical value exists in the literature,it is commonly assumed that the phase space of the planar kicked rotor isglobally chaotic for K & 5 [83]. For the equations of motion (2.2) of thelaser-kicked rotor, no periodic trajectories should be expected for K & Classical diffusionIn the regime of global chaos, the classical rotor performs a random walkin angular momentum space, despite being perfectly deterministic. Themean-square value of the angular momentum grows as 〈J˜2〉 ∝ DN with acharacterisitic diffusion rate D ≈ K2/2 in the globally chaotic regime [36].The mean energy of the rotor 〈E〉 ∝ 〈J˜2〉 therefore increases linearly withthe kick number.The diffusive growth is unbounded for the idealistic δ-kicks. As long asthe rotor does not rotate much during the duration of the kick, the δ-KR isstill a good approximation [28, 63]. If the rotational period is on the sametime scale as the kicking period, the momentum transfer averages out tozero. As a consequence, the achievable angular momentum is bounded.2.2 Quantum kicked rotorThe main interest in the quantum kicked rotor (QKR) is stimulated by thestudy of the effects of quantization on classically stochastic behaviour - afield of physics known as “quantum chaos” [31, 138, 77]. Unless otherwisenoted, in this work we will be concerned with the globally chaotic regime.For a finite amount of time, the quantum motion resembles the chaoticclassical motion. However, after a critical number of kicks Nb - the “quan-3 This is a consequence of the symmetry of linear molecules and an angle dependenceof 2θ in (2.2).112.2. Quantum kicked rotortum break time” - the quantization of angular momenta becomes noticeableand leads to quasi-periodic dynamics. Interferences in the quantum systemprohibit stochasticity [31, 84]. Unlike in the classical case, one has to distin-guish two regimes: one where the period of the kicks is commensurable withthe periodicity of the quantum rotor, and one where it is incommensurable.The respective excitations result in two different phenomena: quantum res-onance and dynamical localization.2.2.1 Quantum resonanceWe introduce a dimensionless time which relates the period of the kicks Tto the revival period of the rotor Trev = 2piI/~,τ = ~TI= 2piTTrev. (2.5)The time τ acts as the effective Planck constant of the system. The discrete-ness of the rotational spectrum of the QKR results in quantum resonanceswhenever τ = 2pip/q, where p and q are integers [84, 184, 59]. Equiva-lently, this condition can be expressed as T/Trev = p/q. Tuning the periodto match a quantum resonance enables an efficient excitation of multiplerotational states. Owing to constructive interference, the rotational energygrows quadratically from kick to kick. This “ballistic” growth exceeds theclassical diffusive growth.2.2.2 Dynamical localizationOn the other hand, away from quantum resonances, the quantum interfer-ences are destructive and suppress the increase in rotational energy after thequantum break time. The growth is completely halted due to a mechanismcalled dynamical localization. It has been shown that this phenomenon isrelated to the absence of diffusion in a one-dimensional disordered latticedue to Anderson localization [52]. According to the commonly acceptedterminology, Anderson localization always refers to a quantum particle in areal lattice, and dynamical localization to the QKR 4.The angular momentum of the QKR localizes around the initial momen-tum. The resulting distribution of the angular momentum states falls offexponentially with a characteristic length called the “localization length”.(The characteristic length scale of the classical KR was given as the diffusionconstant D.)4In the literature it is also common to find the QKR localization under the synonymof Anderson localization.122.2. Quantum kicked rotor2.2.3 StochasticityLooking back at the classical definition of the stochasticity in Eq. 2.4, weobserve that the motion of the QKR is governed by two separate parame-ters, the effective Planck constant τ and a dimensionless kick strength P .The rigorous definition of P will be given in Sec. 2.3.2. The stochasticityparameter is thus derived by their productK = τP . (2.6)We can reduce the quantum effects by decreasing τ while keeping K con-stant. The underlying classical dynamics stays unchanged, but we approachthe classical limit, i.e. the standard map. For quantum effects to becomenoticeable the period has to be on the order of the revival time T → Trev,see Eq. 2.5. For T  Trev, on the other hand, the discretness of the energyspectrum is not yet noticeable in the system’s dynamics.The quantum-classical correspondence can also be achieved by introduc-ing noise or dissipation into the system. Since all quantum effects cruciallyrely on the interference of coherent pathways, such processes of decoherencewill lead to the destruction of dynamical localization and the recovery ofclassical diffusion.2.2.4 Atom-optics kicked rotorIn 1992 Graham et.al. proposed to study the QKR by kicking ultracoldatoms with a standing wave of far-detuned light. [74]. The momentum ofultracold atoms interacting with a pulsed standing wave will spread diffu-sively in a classical description. However, they demonstrated theoreticallythat the momentum transfer is quantum mechanically limited by dynamicallocalization. The first experimental demonstration of dynamical localizationin this system was reported in 1995 [121] and has in fact been the first obser-vation of one-dimensional Anderson localization with atomic matter waves.Since then this system, which we will refer to as the atom-optics kicked rotor(AOKR), has become the standard setup for studying the QKR. Much re-search has been done to investigate different aspects of the QKR, dynamicallocalization and quantum chaos. More information can be found in multiplereviews [135, 136].132.3. Laser-kicked molecular rotor𝜃ℇ(𝑡)Figure 2.2: Diatomic molecule interacting with linearly polarized electricfield.2.3 Laser-kicked molecular rotorWe implement the QKR by exposing linear molecules to ultra-short laserpulses. The goal of this section is to understand the underlying physicsof the system: We describe the light-molecule interaction and the idea of“kicking” molecules. We introduce the Hamiltonian of a kicked rotor, itswave function and the resulting dynamics. A vital part of our work is thecomparison of the experimental results with the respective simulations, wedescribe the numerical methods to simulate the kicked rotor in Sec. 2.3.6.To finish, we discuss the particular diatomic molecules that were used asquantum rotors. We give relevant molecular constants and useful derivedparameters in Sec. Light-molecule interactionAt first, we need to understand the interaction between linear moleculesand a non-resonant laser field, see Fig. 2.2. We start with a classical de-scription. The interaction potential of a molecule with a permanent dipolemoment µ has the nonlinear formV (θ, t) = −µE(t) cos θ , (2.7)with the angle θ between the molecular axis and the polarization axis ofthe electric field E(t). Since we look exclusively at molecules without apermanent dipole moment, µ(t) = α(t)E(t) is the induced dipole moment inthe presence of the laser field. The resulting dipole will interact with theelectric field itself. After averaging over the fast oscillations, the interactionpotential becomes [12]V (θ, t) = −14E2(t) [(α‖ − α⊥) cos2 θ + α⊥] . (2.8)142.3. Laser-kicked molecular rotorThe full derivation can be found in the “Nonlinear Optics” book by Boyd[29]. The second term of V (θ, t) does not dependent on the angle θ. Hence,it will not affect the rotational dynamics induced by a laser pulse and can bedisregarded [63]. The polarizability anisotropy ∆α = α‖−α⊥ is determinedby the parallel α‖ and the perpendicular α⊥ components of the polarizabilitywith respect to the molecular axis.Most molecules have ∆α > 0 and the potential energy of the system isminimized when the angle is θ = 0. This implies, that a thermal ensembleof randomly oriented molecules will, in the presence of a long laser pulse,feel a torque that will align all molecules. For more details about “adiabaticalignment” see Refs. [66, 164]. Here, we work with ultra-short pulses. Inthis impulsive regime, the molecules merely feel a “kick” towards the fieldpolarization direction [12], shown in Fig. 2.3. For a classical rotor, the gainin the angular velocity is proportional to sin(2θ) [104] - which also appearsin the classical equation of motions in Eq. 2.2. Molecules with initial anglesθ  1 acquire a velocity that is proportional to the angle. Thus, thesemolecules arrive at θ = 0 at the same time [12] resulting in a “field-freealignment”. Different speeds of molecules in the ensemble result in richdynamics, including further molecular alignments. For more details on thetopic of impulsive alignment we refer to Refs. [151, 126, 56].𝑡0 time(a) (b) (c) (d) (e)Figure 2.3: Field-free alignment of an ensemble of molecules after the exci-tation with an ultra-short laser pulse.In the quantum mechanical description we need to include the quan-tization of the angular momentum. In the simplest approximation linearmolecules are viewed as rigid rotors with a moment of inertia I = ~(4picB)−1152.3. Laser-kicked molecular rotor| 𝐽 〉| 𝐽 + 2 〉| 𝐽 − 2 〉| 0 〉| 4 〉| 6 〉| 2 〉(a) (b)Figure 2.4: (a) Non-resonant two-photon Raman transitions in a rotationalsystem. The quantum number changes by ∆J = 0,±2. (b) Rotationalladder climbing via consecutive Raman processes.[63], where B is the rotational constant and c is the speed of light. The ro-tational energy spectrum is expressed asEJ = hcBJ(J + 1) . (2.9)However, once real molecules occupy higher rotational quantum numbersJ , they no longer behave like rigid rotors. Owing to the fast rotation, themolecular bond stretches, which is reflected by a centrifugal distortion termwith the centrifugal constant D,EJ = hc[BJ(J + 1)−DJ2(J + 1)2] . (2.10)An ultra-short, non-resonant laser pulse induces two-photon Raman tran-sitions via an intermediate virtual level. In each transition, the rotationalquantum number J changes by either ∆J = +2 (Stokes Raman scattering),∆J = −2 (anti-Stokes Raman scattering) or ∆J = 0 (Rayleigh scattering),as drawn in Fig. 2.4(a). If the pulse is strong, it is possible to climb upthe “rotational ladder” with many consecutive Raman transitions, providedthat the bandwidth of the pulse is wide enough to support all the requiredfrequencies, see Fig. 2.4(b).Finally, we connect the classical and the quantum picture. If a molecule,whose polarizabilities α‖ and α⊥ are different, is rotating, then the effectivepolarizability of the medium α(t), experienced by a linearly polarized light,changes periodically [112]. The refractive index n(t) of a collection of co-herently rotating molecules will be modulated in time n(t) =√1 +Nα(t),162.3. Laser-kicked molecular rotorwhere N is the number density of molecules [29]. This temporal modula-tion of n results in the frequency modulation of the carrier wave, and thecorresponding Raman sidebands.2.3.2 Kick strengthOur ultra-short laser pulses are of a Gaussian shape. The electric fieldenvelope of a single pulse is described by E(t) = E0 exp(−t2/2σ2), with thepeak value of E0 at the center of the pulse at time t = 0 and a width σ.When a short pulse interacts with a molecule it creates a rotational wavepacket. This interaction can be described as an instantaneous “kick”, if theduration of the pulse is substantially shorter than the time scales of thedynamics in the rotational wave packet.The width of the prepared wave packet is a measure of the strength ofthe pulse. One can define a dimensionless kick strength [57]P =∆α4~∫E2(t)dt . (2.11)The kick strength reflects the typical amount of angular momentum (inunits of ~, the reduced Planck’s constant) transferred from the laser pulseto the molecule [57]. Solving the integral for a Gaussian pulse, simplifies theexpression to P = ∆α/4~ · E20√piσ. On the other hand, if we numericallysimulate the effect of δ-kicks, Eq. 2.11 can be reformulated with Dirac’s deltafunction ∆α4~ E2(t) = Pδ(t).2.3.3 HamiltonianThe Hamiltonian for a three-dimensional rigid rotor is H = Jˆ22I +V (θ, t) . Ithas a kinetic term, with the angular momentum operator Jˆ , and a potentialterm V (θ, t) taken from Eq. 2.8 (without the θ-independent term), such thatH =Jˆ22I− ∆α4cos2(θ) E2(t) . (2.12)The Hamiltonian for a periodically kicked rotor simply follows by introduc-ing a summation over N pulses with the period TH =Jˆ22I− ∆α4cos2(θ)N−1∑n=0E2(t− nT ) . (2.13)172.3. Laser-kicked molecular rotorThe Hamiltonian can be written in terms of the kick strength (Eq. 2.11),for a pulse sequence of Gaussian pulsesH =Jˆ22I− P~√piσcos2(θ)N−1∑n=0exp(−(t− nT )2σ2), (2.14)or similarly in the case of δ-pulsesHˆ =Jˆ22I− ~P cos2(θ)N−1∑n=0δ(t− nT ) . (2.15)2.3.4 Wave functionThe natural choice to describe the rotational wave function of a linearmolecule interacting with a light pulse are the eigenfunctions of a three-dimensional rotor: the spherical harmonics | J,M 〉 = YMJ (θ). The quantumnumbers J and M are the molecular angular momentum and its projec-tion on the quantization axis, respectively. The wave function of a rotor isexpanded in spherical harmonics|ψM (t) 〉 =∞∑J=0cMJ e−iEJ t/~ | J,M 〉 , (2.16)with the complex amplitudes cMJ . In the case of a linearly polarized driv-ing field, the quantum number M remains unchanged. The selection rules∆M = 0 and ∆J = 0,±2 follow from the derivation in appendix C.2. Here,we merely use the fact that the wave functions are independent for each M(we do not need to sum over M). The amplitudes cMJ can be calculated bysolving the Schro¨dinger equationi~∂ψ∂t= Hψ , (2.17)with the Hamiltonian of a kicked rotor given in Eq. 2.14. The mathematicalprocedure is described in appendix C. Density matrixTwo important quantities that we will often refer to are the rotational “pop-ulation” and the rotational “coherence”. Both are conveniently defined inthe density matrix formalism, presented here.182.3. Laser-kicked molecular rotorLet the operator A represent an observable of the quantum mechanicalsystem, described by a wave function |ψ(t) 〉 = ∑J cJ(t) | J 〉. In comparisonto Eq. 2.16, we omitted the magnetic quantum number M and included thephase term in cJ(t). The ensemble average 〈A〉 is given by〈A〉 = 〈ψ |A|ψ 〉 =∑J,J ′c∗JcJ ′〈 J |A| J ′ 〉 =∑J,J ′c∗JcJ ′ AJJ ′ . (2.18)The coefficients cJ contribute to the expectation value of A as quadraticterms c∗JcJ ′ which are the matrix elements of the operator ρ = |ψ 〉〈ψ |c∗JcJ ′ = 〈 J ′ |ψ〉〈ψ| J 〉 = ρJ ′J . (2.19)The operator ρ is known as the ‘density matrix’.Population: The diagonal elements ρJJ = |cJ |2 are the populations. Theyspecify the probability of finding the system in a quantum state | J 〉. Thetotal probability is conserved∑J |cJ |2 = 1.Coherence: The off-diagonal elements ρJ ′J = c∗JcJ ′ for all J 6= J ′ arethe coherences. These crossterms are responsible for the interference effectsand appear whenever the wave function is in a superposition of the states| J 〉 and | J ′ 〉. Coherences satisfy two important properties [147]: First,ρ∗JJ ′ = ρJ ′J a consequence of the Hermitian nature of the density matrix.Second, ρJJ ρJ ′J ′ > |ρJ ′J |2, i.e. coherences only exist, if the population ofthe corresponding levels is non-zero.Statistical mixture: The density operator of a statistical mixture isρ =∑k pkρk where ρk are the density matrices for the pure states |ψk 〉 andpk their statistical weights. The same properties as mentioned above stillapply. It can be shown [147] that the population is now ρJJ =∑k pk |c(k)J |2and similarly that the coherence is ρJ ′J =∑k pk c(k)∗J c(k)J ′ .The observable in our experiment is proportional to the modulus squaredof the coherences |c∗JcJ+2|2. More details will be given in Sec. 2.3.6. Wewill talk about the detection technique of rotational Raman spectroscopy inSec. Laser-kicked molecular rotor2.3.6 Numerical analysisIt has been essential to simulate the dynamics of the QKR, not only toshow the correspondence between experiment and theory but also to testhypotheses or to make predictions. The simulations are done in the pro-gramming language Matlab for a δ-kicked rotor. In this section we describethe numerical procedure.The δ-kicked rotorWe solve the Schro¨dinger equation (Eq. 2.17) with the Hamiltonian for asingle pulse (Eq. 2.12). Owing to the impulsive approximation, meaning thatthe duration of the pulse is much shorter than the relevant rotational timescale, we approximate the laser pulse by a δ-function. The wave functionimmediately before the kick is labelled as |ψ− 〉, whereas the one after theδ-kick is labelled |ψ+ 〉. During the δ-pulse, the potential is so strong, thatthe kinetic energy term in the Hamiltonian can be neglected. Thus, theSchro¨dinger equation simplifies toi~∂|ψ+ 〉∂t= −14E2(t) ∆α cos2 θ |ψ+ 〉 . (2.20)In appendix C.2.1 we prove that an analytic solution of Eq. 2.20 is|ψ+ 〉 = exp(iP cos2 θ) |ψ− 〉 . (2.21)Next, we want to rewrite |ψ+ 〉 in the basis of spherical harmonics,|ψ+ 〉 =∑J,McMJ e−iEJ t/~ | J,M 〉 (2.22)to find its complex amplitudes cMJ . This decomposistion into the eigenstatesof the kicked rotor is a non-trivial exercise. The approach used in our code,taken from Ref. [57] is outlined in appendix C.2.2. Alternative methodsare presented in Ref. [12, 104]. We introduce an artificial time τ , definedsuch that before the kick (τ = 0) and immediately after the kick (τ = 1).Eventually, we obtain a set of coupled ordinary differential equations (ODE)∂∂tcM′J ′ (τ) = iP∑J,McMJ 〈 J ′,M ′ | cos2 θ| J,M 〉 , (2.23)that are solved numerically by means of the ‘ode45 solver’ in matlab. Theeigenstates before the kick are | J,M 〉 and after the kick | J ′,M ′ 〉. The202.3. Laser-kicked molecular rotorresulting amplitude coefficients cM′J ′ (τ) are evaluated at (τ = 1) to get thewave function of the created rotational wave packet immediately after thekick.Selection rules: In appendix C.2.3 we further investigate the expres-sion 〈 J ′,M ′ | cos2 θ| J,M 〉, which contains all transition matrix elements forthe relevant two-photon transitions, the Clebsch-Gordan coefficients (CGC).The selection rules follow directly from the CGC∆J = 0,±2 and ∆M = 0 . (2.24)Statistical averaging: In the experiment, we work with a thermal en-semble of molecules. We calculate the final wave function individuallyfor each initially populated state | J0,M0 〉. The results are then addedwith the proper statistical weights ω = ωBωN . The weight of the ther-mal Boltzmann distribution ωB = Z−1 exp (−hcEJ/kBT0) depends on theinitial temperature T0. Here, kB is the Boltzmann constant and Z =∑J(2J + 1) exp (−hcEJ/kBT0) is the partition function. The weight ofthe appropriate nuclear spin statistic ωN is given in Sec. 2.3.9.Sequence of δ-kicks: So far, we mentioned only the wave function of therotor after a single δ-kick. In order to get the wave function for a sequenceof kicks, we implement the following procedure: (1) Calculate the wavefunction after a single kick individually for each initially populated state| J0,M0 〉. (2) Evolve each wave function for the field-free time period Tto the next δ-kick. A free evolution corresponds to the J-dependent phaseterm e−iEJT/~. (3) Apply the next kick to the new wave functions. Againcalculate it individually for all of the wave functions which originated fromthe different | J0,M0 〉. (4) Repeat step (2-3) iteratively for each pulse inthe sequence. (5) Sum over all wave functions with the proper statisticalweights.Experimental observableOur experimental observable, the coherent Raman signal, depends on thecoherences cM∗J cMJ+2 excited in an ensemble of molecules. More about thedetection technique of Raman spectroscopy will be discussed in section 4.1.Here, we only point out that the J-dependent observable is referred to as“Raman spectrum”.212.3. Laser-kicked molecular rotorConsider a coherent superposition of two rotational states,|ψMJ (t) 〉 = cMJ e−iEJ t/~ | J,M 〉+ cMJ+2 e−iEJ+2t/~ | J + 2,M 〉 , (2.25)created by a laser pulse. The coherent dynamics of such a wave packet is J-dependent. Owing to the selection rules for a two-photon excitation process(Eq. 2.24) the superposition |ψMJ (t) 〉 can originate from any initially pop-ulated thermal state | J0 = J ± 2k,M0 = M 〉, where k is an integer. Theintensity of the observed Raman spectrum will be proportional to the mod-ulus squared of the induced coherence,IJ ∝∑M〈|cM∗J cMJ+2|2〉J0,M0 , (2.26)summed over all M -sublevels and averaged over the initial thermal mixture.It is important to realize that the Raman signal IJ as a function of thequantum number J , does in fact depend on both rotational states J andJ + 2.M-degeneracyEach rotational state J is (2J + 1)-fold degenerate in the field-free case.The degenerate states are described by the magnetic quantum number M =−J,−J + 1, ..., J − 1, J . The axes of the polar coordinates are determinedby the polarization direction of the electric field, which is taken along thez-direction. The quantum number M refers to the projection of the angularmomentum J on the z-axis.One important point about the M -substates is that they interact dif-ferently with the laser pulses - the kick strength P will effectively varyfor different M -substates. For multiply-degenerate states the kicks are feltweaker if |M | approaches its maximal value J . In this case the correspondingangular momentum vector is oriented closer to the z-axis, and the rotatingmolecules are thus confined closer to the xy-plane. One can see from Eq. 2.8that for such polar angles θ the interaction potential decreases.Another difference between M -sublevels has its origins in the “dynamicStark effect”, i.e. the AC Stark effect, which leads to a shift in the energylevels due to the non-resonant laser field. During the interaction of thelaser pulse, the M -degeneracy is lifted, because the energy shift will varyfor different M -substates.It is obvious now, that in the experiments it is important to lower theinitial rotational temperature to have fewer M -sublevels populated. This222.3. Laser-kicked molecular rotorwill result in a more accurate extraction of the molecular wave function (de-tails in Sec. 6.3.2) because of a reduced averaging over states with differentdynamics.2.3.7 Revival timeThe discreteness of the rotational energy spectrum leads to periodic dynam-ics of any wave packet and the so called quantum revivals [13], which havebeen observed in many different contexts, e.g. Refs [144, 108, 45, 141] togive just a few examples.The revival time is inversely proportional to the second derivative of therotational energy E(J) with respect to the angular momentum J [105, 151].In the case of a rigid rotor we obtainTrev =12cB, (2.27)an expression that is independent of the rotational quantum number J .Fast molecular rotation leads to a non-negligible centrifugal distortion andtherefore a J-dependent revival time, which will become important for highquantum numbers.2.3.8 Resonance mapAn ultra-short laser ‘kick’ creates a rotational wave packet. We are lookingat its dynamics and want to develop an understanding, how the wave packetwill change once it is exposed to a series of periodic kicks and how it dependson the periodicity. The following arguments are based on the perturbativeregime of light-molecule interaction.The first laser pulse in a sequence of pulses induces a coherent rota-tional wave packet |ψ(t) 〉 = ∑J cJ e−iEJ t/~ |J〉. Consider a wave packet ofonly two states, |J〉 and |J + 2〉. It can be assigned a classical rotation pe-riod τJ = 2h (EJ+2 − EJ)−1 with Planck’s constant h. Given a symmetricmolecule (e.g N2 or O2), the wave function of such a coherent superposi-tion repeats itself every integer multiple of a half-rotation, i.e. at timesTJ = NJτJ/2 with NJ ∈ N. Figure 2.5 indicates all these time moments TJfor all the (|J〉, |J + 2〉) wave packets. Black and red markers represent thetwo independent rotational progressions of even and odd quantum numbersJ , respectively. Note that due to the nuclear spin statistics some spin iso-mers may not be allowed, e.g. in oxygen 16O2 even values of J are prohibited232.3. Laser-kicked molecular rotor0 0.2 0.4 0.6 0.8 1 1.2Time  ( T/Trev)0246810Rotational quantum numberJFigure 2.5: Resonance map: Markers indicate all time moments at whicha coherent rotational wave packet consisting of two states, |J〉 and |J + 2〉,completes half a classical rotation τJ . Time is expressed in units of themolecular revival time Trev. The dotted, dashed and solid lines indicate thethird, half and full quantum resonance, respectively.(Sec. 2.3.9). We refer to all the individual times TJ as “fractional quantumresonances” and we address the plot itself as the “resonance map”.For a rigid rotor, EJ = hcBJ(J + 1) and the rotational period becomesτJ = Trev(J + 3/2)−1, with the J-independent revival time Trev = (2cB)−1.The map is universal since it is plotted as a function of the dimensionlesstime T/Trev. For a non-rigid rotor, however, the map will depend on thechoice of the molecule and its centrifugal terms. In Fig. 2.5 for the lowquantum numbers J 6 10 the effect of centrifugal distortion is not yetvisible.Every laser pulse in a periodic sequence will interfere with the rotationalwave packet created by the previous pulses. Again, we start by looking atthe excited (|J〉, |J + 2〉) wave packet between two neighbouring states. Ifthe time T between two consecutive laser kicks coincides with a fractionalresonance TJ , then this wave packet has evolved by NJ = T/Trev× (2J + 3)half rotations during its free evolution. This means that the next laser pulse‘kicks’ the molecule in the same direction as the previous pulse, enhancingthe rotational excitation of the corresponding wave packet. At all other timesT 6= TJ the period of the pulse train is out of phase with the (|J〉, |J + 2〉)wave packet and the rotational excitation is suppressed.242.3. Laser-kicked molecular rotorQuantum resonanceAt the timing known as the quantum resonance, when T = Trev, all wavepackets perform an integer number of half-rotations with NJ = 2J + 3. InFig. 2.5, these times TJ = (2J + 3)× τJ/2 lie on a vertical trajectory (solidline). Thus, a resonant pulse train with a period T = Trev is equally efficientin exciting all molecules in the ensemble, regardless of their angular momen-tum [192]. However, this is only true for rigid rotors. For non-rigid rotors,the quantum resonance Trev becomes J-dependent and, therefore, impossi-ble to satisfy for all quantum states simultaneously. In our map, the dotswill no longer line up along a vertical line above T/Trev = 1. Consequencesof the centrifugal distortion are investigated in Sec. 5.3 and 5.4.Fractional quantum resonanceBack to Fig. 2.5, we point out that all the (|J〉, |J+2〉) wave packets completean integer number of half-rotations at different time moments. However,there are times when several wave packets corresponding to different valuesof J are in phase simultaneously. Those are called “fractional resonances”,because a fraction of the total wave function revives to its initial state. Anexample of a lower-order fractional resonance, T/Trev = 1/3, is plotted as thedotted line. Here, the quantum states J = 0, 3, 6, ... are resonantly excited,as they complete NJ = 1, 3, 5, ... half rotations, respectively. All other statesare out of phase. At higher-order fractions with TJ/Trev = NJ/(2J + 3),fewer states are simultaneously resonant.The resonance map, although quite simple, proves to be very helpful inthe explanation of many observations. In section 5.1, we present Ramanspectra after the excitation with different periodic pulse trains. By tuningthe train period and having a detection accuracy of individual rotationalstates, we experimentally verify the resonance map of Fig. 2.5. The studyof the kicked rotor crucially depends on the choice of the pulse train period,in particular whether it coincides with a fractional resonance or not. Wewill return to the resonance map in several instances, e.g. in Sec. 5.2 tostudy periodic excitations on fractional quantum resonances, in Sec. 5.3 toinvestigate periodic excitation around the full quantum resonance, and inchapters 6 & 7 to examine the phenomenon of dynamical localization viaoff-resonant periodic excitation.252.4. Correspondence to crystalline solidsBe αe B D Trev ∆α[cm−1] [cm−1] [cm−1] [cm−1] [ps] [A˚3]14N2 1.9982 0.0173 1.9896 5.76E-6 8.383 0.6916O2 1.4376 0.0159 1.4297 4.84E-6 11.666 1.08Table 2.1: Molecular constants and parameters for N2 and O2.2.3.9 Choice of moleculeAll experiments are conducted with either oxygen or nitrogen moleculesfor several reasons: (1) Our Raman detection technique (see Sec. 4.1) iscapable of resolving individual rotational states, whose spacing is given bythe energy separation of neighbouring rotational states and is bigger forlighter molecules. (2) The revival times Trev are on the order of 10 ps, whichallows the generation of multiple pulses in a periodic sequence via our pulseshaping techniques (see Sec. 3.3). (3) Both molecules are non-toxic gasesthat are easy and safe to handle in the laboratory.Necessary molecular constants are taken from the NIST Chemistry Web-book [81] and shown in the table with other derived parameters. The rota-tional constant B = Be−αe(v+1/2) is calculated for the vibrational groundstate v = 0. Higher order terms, beyond the centrifugal constant D, are notnecessary. And to calculate the kick strength of our laser pulses we requirethe polarizability anisotropy ∆α, whose values are taken from Ref. [8].Molecules can often exist in different states, called nuclear spin isomers.These are found by evaluating the symmetry of the total wave function withrespect to the exchange of two identical nuclei. Oxygen 16O has a zeronuclear spin (I = 0). Oxygen molecules (16O2) can therefore only exist inodd rotational states. The nuclear spin of 14N is I = 1 which yields a 2:1statistical ratio of even to odd rotational states in molecular nitrogen (14N2)[79].2.4 Correspondence to crystalline solidsIn this section we outline the similarities of our kicked rotor system to that ofa seemingly unrelated crystalline solid, which was first analysed by Fishmanand coworkers in 1982 [52].We start with the solid state system of an electron in a periodic crystal,Sec. 2.4.1, to introduce the important concepts of reciprocal lattice, Brillouinzone and Bragg reflection. Note, that in this work we will always refer to a262.4. Correspondence to crystalline solidsone-dimensional lattice, where vectors like ~k, ~r, ... reduce to scalars k, r, ....Bloch’s theorem (Sec. 2.4.2) and Floquet’s theorem (2.4.3) are essential toolsto solve Schro¨dinger equations for periodic problems. The first one is appliedto periodic potentials like the electron in a lattice, while the latter onesimplifies periodic time-dependent problems like the QKR.The physical framework in all of our efforts is the tight-binding model.In Sec. 2.4.4 we go over the simplest case for a one-dimensional periodiclattice with only nearest neighbour interactions. Introducing disorder intothe system yields the Anderson model, subject of Sec. 2.4.5. We explainthe emerging phenomenon of Anderson localization in its original context ofelectron conduction in metals. Finally, we are ready to make the connectionback to the QKR in Sec. 2.4.6. We derive how a quantum rotor in a latticeof angular momentum states can be mapped onto a quantum particle ina one-dimensional lattice. In other words, we show how the QKR can beexpressed in the formalism of the Anderson model. Section 2.4.7 describessome specific details for our system of laser-kicked molecular rotors.2.4.1 The one-dimensional latticeFree electrons are described by plane waves eikr with energies E(k) =~2k2/2m, where k is the wave vector. Once an electron is inside a periodiclattice of atoms, its dynamics becomes ‘more interesting’. The equilibriumdistance between identical atoms is given by the lattice constant a, whichestablishes a periodicity r → r + a. The relationship between the energy ofan electron (or frequency ω = E/~) and its wave vector k is described by adispersion relation, which is also periodic, but in k → k + 2pi/a. We stressthe importance: Any system that is periodic in real space with period a, willalso be periodic in reciprocal space with periodicity 2pi/a [161, 9, 91]. Wedefine a lattice vector R = na and a reciprocal lattice vector G = m2pi/awith integers n,m.The unit cell in the reciprocal lattice is called a Brillouin zone. Theexcitation spectrum of waves in periodic media is given as the dispersionrelation E(k). The entire spectrum can be described in the first Brillouinzone, which spans from −pi/a 6 k 6 pi/a [161]. A quantity that will be ofimportance later is the group velocity vG = dω/dk: it is the speed at whicha wave packet moves. The group velocity vanishes (vG = 0) at the Brillouinzone boundary ±pi/a [161, 9, 91]. Here, the electrons, i.e. plane waves, arescattered back due to a mechanism called Bragg reflection [161, 9, 91]. Thisfeature of wave propagation in periodic lattices leads to band gaps, i.e. thereare energies in the spectrum that do not support any wave-like solutions of272.4. Correspondence to crystalline solidsΤ𝜋 𝑎− Τ𝜋𝑎𝐸(𝑘)4𝑊T𝑘Figure 2.6: Dispersion relation of the one-dimensional tight-binding model.the Schro¨dinger equation. In solid state physics, the band structure is usedto phenomenologically describe the existence of metals, semi-conductors andinsulators. We will be able to connect some of these ideas to the angularmomentum lattice of a kicked rotor.An exemplary dispersion relation, illustrating the concepts of reciprocalspace and Brillouin zone, can be seen in Fig. 2.6. Shown is the solutionto the one-dimensional tight-binding model with only nearest neighbourinteractions, which will be discussed later in the corresponding Sec. Bloch’s theoremA Bloch wave ψk is the solution of the Schro¨dinger equation for a periodicpotential, i.e. an electron in a periodic lattice. Bloch’s theorem states thatall Bloch functions can be expressed as a plane wave multiplied by a periodicfunction uαk (r) with the period a of the lattice [162, 161, 91, 9].ψαk (r) = eikr uαk (r)uαk (r) = uαk (r + a) .(2.28)The subscript k is referred to as “quasi-momentum”; it can always be chosenwithin the first Brillouin zone. Note that ~k is not the momentum of anelectron but rather a crystal momentum. It can be seen as a quantumnumber describing the electron’s state within a band α [162, 9]. Each kcan present different states α that belong to different bands. An important282.4. Correspondence to crystalline solidsrelation follows asψαk (r +R) = eik(r+R) uαk (r +R) = eikReikruαk (r) = eikR ψαk (r) , (2.29)where we used the invariance of uαk towards lattice translations.Although the atomic potentials that each electron sees are strong, theelectron itself still moves like a plane wave through the crystal, modulatedby a periodic function. An important difference however, is that the electronmomentum has been replaced by a quasi-momentum of the lattice [161].2.4.3 Floquet’s theoremFor time-periodic problems, the Floquet formalism [58] can be utilized tosimplify the time-dependent Schro¨dinger equation H(t)ψ(t) = i~∂ψ(t)∂t . Itwill become obvious that it is closely related to the Bloch formalism usedwith periodic lattices. Since the QKR presents a periodic problem, we willmake use of Floquet’s theorem.If the Hamiltonian H(t) = H(t+T ) is invariant under a time translationt → t + T , then according to Floquet’s theorem [78], a solution exists suchthatψα(t) = e−iωαt uα(t)uα(t) = uα(t+ T ) .(2.30)The eigenstates ψα(t) are known as Floquet states with periodic Floquetmodes uα(t) and the period T . In direct analogy to the quasi-momentumk of Bloch states, the Floquet states are characterized by a quasi-energyEα = ~ωα [189]. Looking at Eq. 2.30, we know that the eigenfunctions canbe uniquely defined in a time window of width ω = 2pi/T interval. Wedefine the first Brillouin zone of the reciprocal lattice as −pi/T 6 ωα 6 pi/T .For different Floquet modes uα′(t) = uα(t) exp(−inωt) with integers n wemerely get shifted energies Eα′ = Eα + n~ω, that can be mapped into thefirst Brillouin zone. Therefore, the subscript α corresponds to a whole classof solutions with all α′ for n = 0,±1,±2, ... .In other words, if the Hamiltonian is periodic, we can find solutions ofthe Schro¨dinger equation that are periodic in time up to a phase factor [189]ψα(t+ T ) = e−iωαT ψα(t) . (2.31)All linearly independent states with different quasi-energies (Eα′ 6= Eα+n~ω)form a complete basis set [189]. Therefore, the total wave function is a292.4. Correspondence to crystalline solidslinear combination of all quasi-energy states |Ψ(t) 〉 = ∑α cα|ψα(t) 〉 =∑α cα e−iωαt |uα(t) 〉. The time-independent amplitude coefficients areknown at time zero cα = 〈uα(0) |Ψ(0)〉.Owing to the periodicity of the problem, it is often advantageous to usea stroboscopic description. A one-cycle propagator U is defined such thatU(t+ T, t) ψα(t) = ψα(t+ T ) . (2.32)Substituting the right-hand side by Eq. 2.31 reveals that the Floquet statesare eigenstates of the one-cycle propagatorU(t+ T, t) ψα(t) = e−iωαT ψα(t) . (2.33)This means that the Floquet states remain unchanged after each periodapart from a phase factor. In practice, we numerically calculate the propa-gator U(t+ T, t) and diagonalize it to find the quasi-energies.2.4.4 Tight-binding modelWe have a one-dimensional periodic chain of atoms located at sites r with aspacing a. The potentials are so large that the electrons spend most of theirtime close to the core and only occasionaly jump to a neighbouring atom- they are tightly bound [162]. Here, we go through the deviation of themost basic tight-binding model, following Ref. [161]. This serves to estab-lish the formalism and our notations. At multiple occasions in this thesis wewill map the kicked rotor model onto such a one-dimensional tight-bindingmodel.The atomic wave function (atomic orbital) φm(r) of the m-th atom sat-isfies the Schro¨dinger equation Hat φm(r) = Eat φm(r), where Hat is theHamiltonian for an isolated atom. In a lattice, however, the electron seesnot only the atomic potential but also a contribution V (j) from neighbouringatoms j 6= m. Due to the strong binding these contributions are consideredsmall. The tight-binding Hamiltonian for the m-th atom isH = Hat +∑j 6=mV (j) . (2.34)We assume that the wave function of the one-dimensional lattice can beexpressed as a linear combination of all individual atomic wave functions|Ψ 〉 =∑mcm|φm 〉 . (2.35)302.4. Correspondence to crystalline solidsIn the case of isolated atoms the orbitals are orthogonal 〈φn |φm〉 = δn,m, butwhen the atoms get closer - as it is the case in a lattice - the orthogonalityis lifted. Fortunately, the solution of the problem is not much affectedby whether the wave functions are orthogonal or not. Thus, we assumeorthonormal orbitals for simplicity [161].We try to solve the time-independent Schro¨dinger equation H|Ψ 〉 =E|Ψ 〉. Using the linear combination of states and the orthogonality, wearrive at an effective Schro¨dinger equation∑mHn,m cm = E cn , (2.36)with the matrix elements Hn,m = 〈φn |H|φm 〉.In the simplest model of only nearest-neighbour interactions, we set∑j 6=m V(j)n,m = −W for n = m ± 1. This describes a hopping of the elec-tron from the m-th atom to the n-th via the interaction of all j 6= m atoms.We will refer to W as the hopping term (dimension of energy). Further,∑j 6=m V(j)n,m = T0 for n = m, which presents an on-site energy shift. We willrefer to T = T0 + Eat as the on-site energy. The matrix elements of thisHamiltonian areHn,m = Tδn,m −W (δn+1,m + δn−1,m) . (2.37)This model is in accordance with Bloch’s theorem (Sec. 2.4.2). In fact,one can show that the wave function Ψ(r) is indeed a Bloch wave thatmatches the periodicity of the lattice φ(r) = φ(r + R) for all lattice trans-lations R [162]. The tight-binding approach is often used to approximateelectronic band structures in solids.SolutionThe ansatz cn = exp(−ikna) presents a solution to the effective Schro¨dingerequation (Eq. 2.36 and 2.37) [161]. The obtained energy spectrumE(k) = T − 2W cos(ka) (2.38)is periodic in k-space with the period 2pi/a 5. Figure 2.6 shows the dispersionrelation, with the boundaries of the first Brillouin zone marked by dashedvertical lines. Electrons only have energies in a band spanning a width5We will return to this dispersion relation in Sec. 5.3 when we address the phenomenonof Bloch oscillations.312.4. Correspondence to crystalline solidsof 4W . At a hopping strength W = 0, we are at the atomic limit. Forincreasing W , the atomic orbital spread into bands.The implications of this model become obvious when we compare theenergy Ee =~2k22meof a free electron with mass me, to the dispersion rela-tion. For ka  1 equation 2.38 is parabolic E(k) ∝ Wa2k2. This suggeststhat electrons near the bottom of the band move like free electrons with aneffective mass mW =~22Wa2[161]. The larger the hopping term, the smallerthe effective mass, and the more freely move the electrons.2.4.5 Anderson modelWe showed that for a periodic (infinite) lattice the electrons can move quasi-freely through the entire lattice. The system experiences no attenuation dueto scattering, because the periodic order of scatterers is responsible for aconstructive interference of all scattering events. This will no longer be trueif we introduce disorder into the lattice.The Anderson model describes a quantum particle in a lattice where allthe lattice sites m have random potentials Tm. As in the standard tight-binding model the hopping term Wr is responsible for the transfer of theparticle to the r-th neighbour. The probability amplitude um to find theparticle at site m is given by the Schro¨dinger equationTm um +∑rWr um+r = 0 . (2.39)All solutions um, belonging to different quasi-energies, are localized eigen-states. This means that rather than being extended Bloch states that spreadover the full periodic lattice, illustrated in Fig. 2.7(a), the eigenstates arenow exponentially localized in a disordered lattice with um ∼ e−|n−m|/l,illustrated in Fig. 2.7(b). The probability amplitude to find a particle atsite m away from the localization center at n falls off exponentially with acharacteristic localization length l.A simplistic explanation for this intriguing effect can be given as follows.The solutions of the Anderson model are quasi-energy states as we haveshown above. The initial system is a wave packet comprised of a finitenumber of states. At any point of time, we can only populate quasi-energystates that overlap with the initially populated states. It is known thatnearly-identical quasi-energies localize at different sites, whereas states withsimilar localization centers (compared with the localization length) havedifferent quasi-energies [52, 76]. We conclude that the final wave packet willhave a discrete energy spectrum with a finite amount of peaks. The wave322.4. Correspondence to crystalline solids(a) (b)𝑢𝑚2n+1nn-1… … n+1nn-1… …m mFigure 2.7: (a) Extended Bloch waves (red curve) describe a quantum parti-cle in a periodic lattice (black curve), with lattice sites m. (b) The quantumparticle is localized in a disordered lattice (black curve), with a probability(red curve) to find it at site m, which is decreasing exponentially away fromthe localization center at m = n. The disorder in this so-called bi-chromaticlattice is produced by adding two periodic lattices of incommensurable pe-riods: here, the one (black curve) from (a) and a second weaker one (bluecurve).packet dynamics associated with such a spectrum must be quasi-periodic.An initial diffusive spreading of the wave packet will turn into a quasi-periodic motion after a certain time, approximately when the wave packetspread has reached the localization length [77]. An unbounded growth ofthe wave packet is prohibited by Anderson localization.Anderson localization in one-dimensionThe theory of Anderson localization originated more than 50 years ago todescribe electron conduction in solid states [6]. Nowadays, the effect hasbeen observed in many different materials using classical as well as matterwaves, and the same theory has been applied to other physical systems,for instance the kicked rotor. A brief overview will be given later in themotivation of chapter 6. Here, we discuss the phenomenon in disorderedelectronic systems, for reviews see Ref. [97, 98].Electrons do not scatter on ions of a regular lattice, scattering only occursat random impurities, resulting in a diffusive motion of the electrons, a ran-dom walk. In earlier models it was believed that more impurities lower theconductivity according to Ohm’s law [9], which did not capture the real be-haviour. The conductivity completely vanishes beyond a critical amount ofimpurities [6], an effect that could only be explained with the wave characterof the electron. All scattered waves for each electron self-interfere destruc-332.4. Correspondence to crystalline solidstively, such that the electronic wave function becomes spatially localized,with the probability amplitudes falling off exponentially to the sides. An-derson’s theory describes a disorder-induced phase transition from classicaldiffusion to complete localization which prohibits any charge transport.Anderson localization in higher dimensionsIn a one dimensional disordered system all quantum states are localized[6, 7]. The generalization of Anderson’s theory to higher dimensions tookanother 15 years [1]. The formulation of a scaling theory of localization in1979 [2] was another milestone. A scaling parameter was introduced thatgoverned the dependence of the localization transition on the size of thematerial. It was proven that any one- or two-dimensional system localizesregardless of the disorder strength, provided the system size is large enough,a fact that is not true in three-dimensions. Here, the disorder strengthdetermines a critical energy, called the mobility edge. Energy states caneither be localized (insulator) or extended (conductor) depending on themagnitude of the energy with respect to the mobility edge [153] 6.The observation of Anderson localization in a solid lattice is extremlyhard due to several decoherence mechanisms, e.g. lattice vibrations orelectron-electron interactions. Signs of “weak localization” (reduced con-ductivity) have been demonstrated, but “strong localization” (suppressedconductivity) has not been observed in atomic crystals, read the review inRef. [98].2.4.6 Mapping the kicked-rotor onto the Anderson modelIn the seminal paper from 1982, Fishman et.al. gave a mathematical proofthat the periodically kicked quantum rotor can be mapped onto the Ander-son problem of electronic transport in a one-dimensional disordered lattice[52], drawn in Fig. 2.8(a). The equivalent rotational lattice in Fig. 2.8(b) isbuilt of the “J-sites” with the occupation probabilities |cJ |2 known throughthe rotor wave function |ψ 〉 = ∑J cJ | J 〉. We are going to outline this map-ping for the relevant case of laser-kicked molecules. Our derivation is basedon the original work of a planar rotor [52, 76], also see the book on quan-tum chaos by Haake [77]. The extension to real three-dimensional rotorshas been made for the case of linear molecules exposed to either periodicmicrowave fields [28] or ultra-short laser pulses [63].6In general, each energy band of a disordered solid has two mobility edges: The ex-tended states are around the center of each band with the localized states in the wings.342.4. Correspondence to crystalline solids(a) (b)J+4J+2JJ-2J-4n+2n+1nn-1n-2mFigure 2.8: (a) Quantum particle in a disordered lattice of lattice sites m.(b) Quantum rotor as an angular momentum lattice with rotational sites J .We are investigating linear molecules exposed to periodic laser pulses.The pulses are approximated by δ-functions to simplify the mathematics;we refer to Ref. [28, 63] for a treatment with real pulses. The wave functionimmediately before an instantaneous kick is labelled as ψ− and the one aftera δ-kick is labelled ψ+. Due to the periodicity of the kicking, the solutionsof the time-dependent Schro¨dinger equation ψ± must be Floquet states ψ±α .We look at the evolution over one period T in two independent steps. Inthe first step, the time between kicks is a free evolution when the Floquetstate accumulates a phase. The Floquet state is represented in the basis ofthe free rotor (spherical harmonics | J,M 〉) with the Hamiltonian H0ψ−α (t+ T ) = e−iH0T/~ ψ+α (t) = e−iEαT/~ ψ−α (t) . (2.40)The second part of the equation takes advantage of Floquet’s theorem, seeEq. 2.31, with the quasi-energy Eα. We define the phase φ = (Eα −H0)T/~and reorganize to getψ−α (t) = eiφ ψ+α (t) (2.41)u−α = eiφ u+α . (2.42)In Eq. 2.42 we rewrote the Floquet states via ψ±α (t) = e−iEαt/~ u±α (t), seeEq. 2.30, and note that it is sufficient to study u±α [52, 76]. We dropped thet-dependence in the notation of u±α for simplicity. In the second step, theevolution from immediately before to immediately after the δ-kick is givenasψ+α (θ, t) = eiP cos2 θ ψ−α (θ, t) (2.43)u+α (θ) = eiP cos2 θ u−α (θ) , (2.44)352.4. Correspondence to crystalline solidswhich is written in the angle representation (Eq. 2.21) 7. To show theconnection to the Anderson model we need to use the one-cycle propagatorU , which was introduced earlier in Eq. 2.33. We find [28, 63]U u±α = e−iEαT/~ u±α (2.45)U = e−iH0T/~ eiP cos2 θ . (2.46)Next, we have to transform Eq. 2.42 into the angle representation, which canbe done by means of the propagator. We start with the reverse of Eq. 2.44e−iP cos2 θ u+α (θ) = u−α (θ) and propagate it by one cycle by multiplying Ufrom the left side [77]. The left-hand side collapses to e−iH0T/~ u+α (θ) becauseof Eq. 2.46 and the right-hand side yields e−iEaT/~ u−α (θ) due to Eq. 2.45.The combination yields the desired expression in angle representation:u−α (θ) = eiφ u+α (θ) . (2.47)A new vector is defined uα(θ) =12 [u+α (θ) + u−α (θ)] as well as an Hermitianoperator W (θ) [52, 76, 77], its relevance will be discussed shortlyuα(θ) =u+α (θ)1 + iW (θ)=u−α (θ)1− iW (θ) . (2.48)Using these relations, we substitute u±α (θ) in Eq. 2.47.[1− iW (θ)] uα(θ) = eiφ[1 + iW (θ)] uα(θ) , (2.49)which is followed by a separation into two terms, with and without W (θ)dependencei1− eiφ1 + eiφ|uα 〉+W (θ) |uα 〉 = 0 . (2.50)The states are |uα 〉 ≡ uα(θ) in angle representation and | J,M 〉 in angular-momentum representation, which are the eigenstates of the free rotor withEJ the eigenvalues, H0| J,M 〉 = EJ | J,M 〉. Per definition i1−eiφ1+eiφ = tan(φ2 )and we relate the tangent function to an energy T(α)J ≡ tan(φ2 ). All that isleft to do is a projection onto the angular momentum states〈 J,M |T (α)J |uα 〉+ 〈 J,M |W (θ)|uα 〉 = 0T(α)J 〈J,M |uα 〉+∑J ′〈 J,M |W (θ)| J ′,M 〉〈 J ′,M |uα〉 = 0 . (2.51)7A verification of this equation and details were discussed earlier in Sec. Correspondence to crystalline solidsSince the quantum number M does not change in the interaction (Sec. 2.3.6)it is treated as a parameter.T(α)J u(α,M)J +∑J ′W(M)J,J ′ u(α,M)J ′ = 0 . (2.52)The mapping between Anderson’s tight binding model and the periodicallykicked rotor is complete. The equation for u(α,M)J = 〈J,M |uα 〉 matchesthe Schro¨dinger equation of a quantum particle in a lattice with the on-siteenergy T(α)J and the hopping term W(M)J,J ′ [63]T(α)J = tan[(Eα − EJ)T2~](2.53)W(M)J,J ′ = 〈 J,M |W (θ)| J ′,M 〉 . (2.54)The probability amplitude to find the quantum particle at site J is given byu(α,M)J . Considering Eq. 2.44 and 2.48 we can express the kicking operatorin terms of W (θ) as [77, 28, 63]eiP cos2 θ =1 + iW (θ)1− iW (θ) , (2.55)which tells us more about the ‘hopping operator’ W (θ) = tan[P cos2 θ2].2.4.7 Anderson model of the laser kicked rotorWe evaluate the details of the Anderson model (Eq. 2.52) in the case oflinear molecules that are periodically kicked by ultra-short laser pulses. Theperiodic lattice of this true quantum rotor are the angular momentum statesJ . The effective lattice constant is ∆J = 2 because only states of the sameparity are coupled. The energies of the rotational states are EJ = hc[BJ(J+1)−DJ2(J+1)2], with the rotational constant B and the centrifugal constantD. And the on-site energy term (Eq. 2.53) is given by T (J) and φ(J)T (J) = tan[φ(J)]φ(J) = piT(Eαh− cBJ(J + 1) + cDJ2(J + 1)2)=τ4( EαhcB− J(J + 1) + DBJ2(J + 1)2),(2.56)372.4. Correspondence to crystalline solidsas a function of the period T , or alternatively, as a function of the effec-tive Planck constant τ = 2piT/Trev = 4picBT . The molecular constants Band D are irrational numbers, which means that under most conditions thevalues of φ(J) modulus pi uniformly cover all angles. Consequently, the en-ergies TJ that follow from the nonlinear tangent function are distributed ina pseudo-random fashion. The result is dynamical localization in the angu-lar momentum space [59, 63]. Chapter 6 is dedicated to the experimentalinvestigation of this phenomenon.When we excite the rotor with a period that coincides with the quantumresonance at T = Trev = (2cB)−1, the energies TJ are not pseudo-randomanymore. However, this statement applies only to small rotational states J ,when we can neglect the centrifugal termφrev(J) =pi2( EαhcB− J(J + 1)). (2.57)The phase φrev(J) modulo pi is independent of J and the on-site energyof neighbouring states will be constant. Similar non-random energies areobtained at fractional resonances T = pqTrev with integers p, q. All theseresonance excitation scenarios are studied in chapter 5. In section 5.3 wewill investigate how the localized states under non-resonant excitation turninto extended Bloch states when we tune to the quantum resonance. Thephenomenon of Bloch oscillations will be demonstrated.We summarize the mapping between the one-dimensional Anderson modeland the QKR and emphasize the connections and differences in table Correspondence to crystalline solidsElectron in aone-dimensional disorderedlatticeQuantum kicked rotor(off-resonant)lattice in real space ladder of rotational statesstationary potential time-dependent potentialhopping strength kick strengthrandom disorder of on-siteenergypseudo-random rotationalon-site energyAnderson localization inreal spacedynamical localization inangular momentum spaceinteractions betweenelectrons, phononsnon-interactingTable 2.2: Comparison of the electron in a disordered lattice versus a quan-tum kicked rotor.39Chapter 3Techniques I: Generation ofa long and high-energyfemtosecond pulse sequenceSeries of ultra-short laser pulses, also known as “pulse trains”, have foundwidespread use in the field of quantum coherent control of matter with laserlight (for a recent review of this topic, see [177] and references therein).Numerous applications require multiple pulses of relatively high intensity,on the order of 1013 to 1014 W/cm2, to attain the regime of strong-fieldinteraction with each pulse, just below the damage threshold of the ma-terial system under study. Using coherent control of molecular rotation- an area of our own expertise - as only one representative example: Se-quences of intense ultra-short pulses have been key in enhancing molecu-lar alignment [19, 101, 41] and aligning asymmetric top molecules in threedimensions [102, 139], selective excitation of molecular isotopes and spinisomers [54, 55], initiating uni-directional rotation [57, 90, 193] and control-ling gas hydrodynamics [188]. A series of recent works [192, 64, 88] hasused high-intensity pulse trains to study the quantum δ-kicked rotor. Thegreat utility of pulse trains stems from two main factors. First, by matchingthe timing of pulses in the train to the dynamics of the system of inter-est, e.g. the vibrational or rotational period of a molecule, one can oftensignificantly improve the selectivity of excitation. Second, the ability to re-distribute the energy among multiple pulses without losing the cumulativeexcitation strength enables one to avoid detrimental strong-field effects, suchas molecular ionization and gas filamentation.There are two common techniques to produce a pulse train with vari-able time separation between transform-limited (TL) pulses. In the firsttechnique, the incoming laser pulse is split into 2n pulses using n nestedMichelson interferometers. Even though sequences of up to 16 pulses [160]have been generated using this method, the scheme becomes increasinglymore difficult to implement with the increasing value of n. In addition40Chapter 3. Techniques I: Generation of pulse sequencesthe control over the pulse timing is rather limited in that it cannot bechanged independently for each individual pulse in the train. Similarly lim-ited flexibility is characteristic of a pulse splitting method based on stackinga number of birefringent crystals [195]. The second common approach isbased on the technique of femtosecond pulse shaping where the spatiallydispersed frequency components of the pulse are controlled in phase andamplitude [179, 47], or via the direct space-to-time conversion [100]. Thisoffers much higher flexibility at the expense of being limited to the rela-tively low energy trains. The latter limitation is due to both the damagethreshold of a typical pulse shaper, and also the necessity to block multiplespectral components in order to generate a train of pulses in the time do-main. Phase-only shaping has been often used to create a series of pulseswithout the loss of energy [181, 182, 46, 140, 134], but in this case, the dis-tribution of the pulse amplitudes within the train is uneven and no controlover this distribution is available.Our objective was to establish a technique to generate femtosecond pulsetrains, which simultaneously satisfies the following specific characteristics:(1) consist of a large number of transform-limited pulses; (2) exhibit a rel-atively flat amplitude envelope; (3) can be easily tuned in terms of thetiming of the constituent pulses; and (4) carry energies in excess of 100 µJper pulse. The method that we developed is based on the combination of apulse shaper, which provides the often required flexibility in controlling thetiming and amplitudes of individual pulses on the time scale of 50 ps, and aset of Michelson interferometers, which enables extending the overall lengthof the train to much longer times. Key to this approach is the integration ofa multi-pass amplifier (MPA), which compensates the energy loss during thepulse shaping stage. We note that although amplification of shaped pulsesis commonly used in chirped-pulse amplifiers [48, 131], and has also beenemployed to amplify pulse sequences [109, 47, 49, 196], pulse trains with theabove mentioned specifications have not been demonstrated before.This chapter presents all the optical components of our setup and ex-plains all the details necessary to produce the above motivated pulse se-quences. First, we introduce the representation of ultra-short pulses in thefrequency and time domains, as well as the mathematical concepts of pulseshaping in Sec. 3.1. Experimental details about the laser system are givenin Sec. 3.2, where we also elaborate on a technique to characterize our fem-tosecond pulses. This will be of great importance to confirm the quality andaccuracy of the generated pulse trains. Section 3.3 deals with pulse shap-ing via a femtosecond pulse shaper. We explain the optical setup, which is413.1. Laser pulses and pulse sequencesbased on spatial light modulation, and give details of how the generationof pulse sequences is implemented. Section 3.4 is about pulse shaping bymeans of Michelson interferometers. This is followed by the description ofa multi-pass amplifier in Sec. 3.5 which is crucial to reach high-intensitypulse trains. Besides technical considerations, we discuss the performanceand limitations of all essential building-blocks: pulse shaper, Michelson in-terferometer and MPA. The versatility of our techniques is demonstratedwith various different pulse trains, e.g. a long pulse train consisting of 84equally strong pulses.3.1 Laser pulses and pulse sequencesWe establish the mathematical framework and our notation to work withultra-short pulses. We show the connection between spectral and temporalrepresentation via a series of diagrams for different pulse shapes: transform-limited pulses (Sec. 3.1.1), frequency-chirped pulses (Sec. 3.1.2) and pulsesequences (Sec. 3.1.3).3.1.1 Transform-limited pulseUltra-short pulses are called transform-limited (TL) if their duration is atthe lower limit given by the spectral bandwidth of the pulse. This conditionis met when the time-bandwidth product is at its minimum, in other wordsthe spectral phase is constant across the whole spectrum of the pulse. Theultra-short pulses of our laser system are Gaussian pulses. Here, we givethe electric field E of a Gaussian pulse whose phase φ is independent offrequency ωE(ω) = A0 · exp(−(ω − ω0)22Γ2)≡ ETL(ω) . (3.1)The amplitude of the TL pulse is A0, the constant phase φ = φ0 has beenomitted. The Gaussian is centred around the frequency ω0 and has a band-width of Γ. Fourier transform results in a Gaussian pulse in the time domainE(t) = E0 · exp(− t22τ2)exp(iω0t) , (3.2)with the amplitude E0 = A0Γ/√2pi and the duration τ = 1/Γ. A largerbandwidth will thus yield shorter pulses with higher amplitudes. The exactderivation is given in appendix D.1.423.1. Laser pulses and pulse sequencesElectric field (spectral), Loss:0% Electric field (temporal)780 800 820Wavelength (nm)-0.2 -0.1 0 0.1 0.2Time (ps) Mask-303Phase Mask (rad.) b)-303Phase (rad.)Figure 3.1: (a) Amplitude mask (blue line) and phase mask (red line) ap-plied to the electric field of a TL pulse (black dashed line), shown in thespectral domain. (b). Amplitude (blue line) and phase (red line) of thecorresponding electric field in the temporal domain. Electric fields are nor-malized to unity.The intensity I(t) of a pulse is proportional to |E(t)|2 and can be calcu-lated with the expressionI(t) =c02E(t)2 , (3.3)where 0 is the vacuum permittivity and c the speed of light.To illustrate the transformation of the electric field of any laser pulsefrom the spectral domain to the time domain and vice versa, we introduce adiagram showing both domains in two separate plots. Figure 3.1 illustratesthis for a TL pulse. Panel (a) presents the electric field of a given pulseETL as the black dashed line, normalized to unity. The field is plotted as afunction of the wavelength λ, which is related to the frequency as ω = 2pic/λ.The pulse has a central wavelength of 800 nm with a bandwidth of 9 nm(FWHM), corresponding to the parameters of our Ti:Sapph laser system8. Our approach of pulse shaping allows for the individual control of theamplitude and the phase of all spectral components, which is done with thehelp of a spatial light modulator (SLM). A detailed description of the pulseshaper follows in Sec. 3.3. The SLM consists of two masks that are used toimplement an amplitude function ASLM, plotted as the solid blue line (leftaxis, normalized to unity), and a phase function φSLM, plotted as the solidred line (right axis, in radians). Here, for the trivial case of a TL pulse8 For a Gaussian pulse, the expression xFWHM = 2√2 ln 2 xσ ≈ 2.3548 xσ is used toconvert the 1/e2 width to FWHM.433.1. Laser pulses and pulse sequencesneither amplitude ASLM(ω) = 1 nor phase φSLM(ω) = 0 are modulated.Panel (b) displays the electric field of the shaped pulse in the time domainE(t). It is obtained by calculating the Fourier transform of the final electricfield in the spectral domain which is given by the expressionE(ω) = ETL(ω) ·ASLM(ω) · exp[iφSLM(ω)] . (3.4)The phase φ(t) is plotted as the solid red line (right axis, in radians) andthe amplitude A(t) as the solid blue line (left axis). For the case of a TLpulse, shown here, the amplitude is normalized to unity. For all other shapedpulses, shown later, the amplitude is normalized to the amplitude of the TLpulse. This will allow an easy comparison of the absolute pulse amplitudes ineach individual pulse of different pulse trains. In the following subsections,we will frequently revisit this type of diagram. The color coding and theaxes will always remain the same.3.1.2 Linearly-chirped pulseCutting the bandwidth will lead to longer pulses. Another possibility tostretch pulses is to introduce a quadratic phase φ(ω) = α′2 (ω−ω0)2 with thespectral chirp α′. This concept is used in the chirped-pulse amplification,see Sec. 3.5.1. Now, the electric field in Eq. 3.1 acquires the additional phasetermE(ω) = ETL(ω) · exp[iφ(ω)] (3.5)= A0 · exp(−(ω − ω0)22Γ2+ iα′2(ω − ω0)2). (3.6)A similar Fourier transform shown in appendix D.2 yields the electric fieldof a Gaussian pulse in the time domainE(t) = E0 · exp(− t22τ2)exp(iω0t− iα2t2), (3.7)with the temporal phase φ(t) = ω0t− α2 t2 and the temporal chirp α. Thesepulses are called linearly-chirped because their instantaneous frequency changesin a linear fashion ω(t) = dφ/dt = ω0−αt. The relation between bandwidthand duration is given by τ2 = (1 +α′2Γ4)/Γ2 and the one between temporalchirp α and spectral chirp α′ by α = α′ Γ2/τ2 (for details see appendix D.2).If there is no chirp α′ = α = 0, we obtain the limit of a TL pulse. For anon-zero chirp, the pulse duration increases with larger chirp values α′.443.1. Laser pulses and pulse sequencesElectric field (spectral), Loss:0% Electric field (temporal)780 800 820Wavelength (nm)-0.5 0 0.5Time (ps) Mask00.20.40.6Amplitudea) b)Phase Mask (rad.)0102030Phase (rad.)-20-15-10-50Figure 3.2: (a) Amplitude mask (blue line) and phase mask (red line) for afrequency-chirped pulse with α′ = 10.000 fs2, applied to the electric field of aTL pulse (black dashed line, normalized to unity), shown in the spectral do-main. (b) Amplitude (blue line, normalized to the TL pulse in Fig. 3.1) andphase (red line) of the corresponding electric field in the temporal domain.The interplay between the spectral and temporal domains for a linearly-chirped pulse is shown in Fig. 3.2. To achieve such a pulse with the pulseshaper, phase-only shaping is sufficient. This means, that we do not loseany energy in the process; the amplitude mask ASLM(ω) = 1 does not cutany frequencies. The energy is merely distributed over a longer time, whichexplains the drop in amplitude of the stretched pulse.3.1.3 Pulse sequencesWe are interested in creating pulse sequences where each individual pulse isa replica of the initial TL pulse with a duration of ∆tFWHM = 130 fs. Inthe time domain, we design a pulse train of N pulses; N ∈ N is a naturalnumber. The train is strictly periodic with a period T and all pulses havethe same amplitude. We call this a flat pulse train. A schematic is shownin Fig. 3.3. The electric field of such a sequence is described asE(t) =∑kE0 exp(−(t− tk)22τ2+ iω0t)· exp(iβ2k2). (3.8)We need to sum over all N sub-pulses that are labelled with the indexk = [−N−12 ,−N−12 + 1, ..., N−12 − 1, N−12 ]. Each term in the sum describes aGaussian pulse at the time tk = kT , which is a TL pulse (Eq.3.2). We alsodefine a quadratic piecewise chirp β across all sub-pulses by adding a phaseto each k-th pulse, which depends quadratically on k. This kind of chirp453.1. Laser pulses and pulse sequences-4𝑇 -2𝑇 0 2𝑇 4𝑇Time  𝑡𝑘Amplitude (arb. units)Phase (arb. units)𝑇𝛥𝑡𝐹𝑊𝐻𝑀𝑘 = 0 +1 +2−1−2𝛽2Figure 3.3: Schematic of the amplitude (blue line) and phase (red line) ofa flat pulse train in the temporal domain with N = 5 pulses and a piecewisequadratic chirp.has been developed earlier in our group Ref. [154, 156] to execute a processof piecewise adiabatic passage (PAP). In the PAP scheme, population canbe transferred in a two-level system in a stepwise adiabatic fashion. In ourwork, however, the piecewise chirp β plays an important role for a completelydifferent reason: to reduce energy losses in the shaping process (describedin this section below).In appendix D.3 we derive the analytic expression for the correspondingelectric field in the spectral domainE(ω) = A0 exp(−(ω − ω0)22Γ2)·∑kexp(−i(ω − ω0)tk + iβ2k2). (3.9)The term in front of the sum gives the TL spectrum of a single pulse ETL(ω)as in Eq. 3.1. Evaluating the sum and making use of Eq. 3.4 we can thenretrieve the spectral amplitude ASLM(ω) and the spectral phase φSLM(ω)needed to obtain the input pulse trainASLM(ω) · exp(iφSLM(ω)) =∑kexp(−i(ω − ω0)tk + iβ2k2). (3.10)Sequences with zero phaseFirst, we look at the simpler case of β = 0, when all pulses in the sequencehave a zero phase offset with respect to the carrier oscillations ωt. Figure 3.4plots the spectral and temporal domains of three different pulse trains to463.1. Laser pulses and pulse sequences-303Phase (rad.)00.2AmplitudePhase Mask (rad.)03010100.2AmplitudePhase Mask (rad.)03-303Phase (rad.)800 801 803Wavelength (nm)-5 0 5Time (ps)8020Amplitude0.12-303Phase (rad.)Phase Mask (rad.)0301AmplitudeMaskb)d)a)c)e) f)AmplitudeMaskAmplitudeMaskFigure 3.4: (a,c,e) Section of the amplitude mask (blue line) and phase mask(red line) for three different pulse trains, applied to the electric field of a TLpulse (black dashed line, normalized to unity), shown in the spectral domain.(b,d,f) Amplitude (blue line, normalized to the TL pulse in Fig. 3.1) andphase (red line) of the corresponding electric fields in the temporal domain.The three sequences are characterized by the following pulse numbers andperiods: (a,b) N = 5 & T = 1 ps, (c,d) N = 5 & T = 2 ps and (e,f) N = 9& T = 1 ps.473.1. Laser pulses and pulse sequencesvisualize their dependence on the parameters N and T . From top to bottom,we have a sequence of 5 pulses separated by 1 ps (a,b), followed by 5 pulseswith twice the period of 2 ps (c,d) and at last a train of 9 pulses againspaced out by 1 ps (e,f).To understand the Fourier transformation from one domain to the other,we make use of the convolution theorem, see Appendix D.4. A periodic pulsetrain with a finite number of N pulses that are all of equal amplitude canbe described as the multiplication of an infinite pulse train with period Ttimes a squared envelope of width NT . The Fourier transform of a squarefunction yields the sinc function whose width δω is inversely proportionalto NT . We also know, that the spectrum of an infinite pulse train is afrequency comb where the separation of the comb teeth ∆ω is inverselyproportional to the train period T . Then, according to the convolutiontheorem (Eq. D.21) this point-wise multiplication in the time domain isequal to the convolution in the frequency domain. Hence, the spectrum ofa finite train is a frequency comb convolved with the sinc function, exactlywhat can be observed in Fig. 3.4. When we double the pulse train periodfrom (b) to (d), the frequency separation between the main comb teeth∆ω ∝ 1/T from (a) to (c) is split in half. The period from (b) to (f)remains the same, and so does the spacing ∆ω from (a) to (e). Each maincomb tooth has the shape of the sinc function sin(x)/x, when one considersthe pi phase jumps for its negative values. The width of the sinc functionδω ∝ 1/(NT ) decreases by a factor of two from (a) to (c,e) since the totalduration of the pulse train increases from 4 ps (b) to 8 ps (d,f).The same convolution theorem can be applied a second time in the re-verse direction (Eq. D.20) to derive the minimum duration of each individ-ual pulse in the sequence. The electric field due to the amplitude and phasemasks has to be multiplied with the electric field of a TL pulse. This Gaus-sian TL pulse is given by the laser system (dashed black line in (a,c,e)).The Fourier transform of a Gaussian function remains a Gaussian function.Therefore, we have to convolve the electric field in the time domain witha Gaussian function. This explains that each sub-pulse in (b,d,f)) has aduration equal to the single pulse duration prior to shaping.In conclusion, we apply amplitude and phase shaping to achieve a pe-riodic pulse train. The phase mask is comprised of pi-steps, whereas theamplitude mask cuts out a comb-like structure. The (partial) suppressionof frequencies leads to a reduced energy in the shaped output pulses. Theratio δω/∆ω = 1/N estimates the amount of energy remaining in the pulsetrain (PT) after shaping. We confirm the validity of this approximation bycalculating the energy throughput T . The energy W of each shaped pulse483.1. Laser pulses and pulse sequences0100.2AmplitudePhase Mask (rad.)-202Phase (rad.)800 801 803Wavelength (nm)-5 0 5Time (ps)802AmplitudePhase (rad.)Phase Mask (rad.)01AmplitudeMaskb)a)c) d)AmplitudeMask-202460-2-4-600.30-2-4Figure 3.5: (a,c) Section of the amplitude mask (blue line) and phase mask(red line) for two different pulse trains in the PAP scheme, applied to theelectric field of a TL pulse (black dashed line, normalized to unity), shown inthe spectral domain. (b,d) Amplitude (blue line, normalized to the TL pulsein Fig. 3.1) and quadratic piecewise phase (red line) of the correspondingelectric fields in the temporal domain. The two sequences are characterizedby the following parameters: (a,b) N = 5, T = 1 ps & β = 1.17 and (c,d)N = 9, T = 1 ps & β = 0.79.can be calculated by integrating over its intensity spectrum W ∝ ∫ I(ω)dω,with I(ω) ∝ |E(ω)|2. The throughputs T = WPT/WTL for the three differ-ent pulse trains in Fig. 3.4 are (a) 20%, (b) 20% and (c) 11.1%.Sequences with piecewise quadratic phaseSecond, we look at the case of β 6= 0, when all pulses in the sequence changetheir phase offset from pulse-to-pulse in a stepwise fashion. Originally, thepiecewise quadratic phase was intended to emulate the adiabatic popula-tion transfer by a single frequency-chirped pulse. We are interested in thisshaping scheme because of its higher energy throughput T , compared to thesame pulse train with β = 0.In all our experiments, we are using pulse sequences to impulsively exciterotational states via Raman transitions (Sec. 2.3.1). This two-photon pro-cess with the electric field ERaman(t) = E(t)E∗(t) is phase-independent. The493.2. Laser sourcetwo photons that are absorbed and emitted in the Raman process stem fromthe same field E(t) and as such the phases of both photons are identical andcancel each other. As long as the phase offset of each individual pulse in apulse train is constant, regardless of its absolute value, the Raman field willstay the same. In other words, a piecewise chirp will not alter the rotationalRaman excitation. We benefit from this degree of freedom by optimizingβ to minimize our shaping losses. It could be shown empirically that theenergy throughput will lie around 60% independently of N or T . We do notclaim that a piecewise quadratic chirp yields the optimum energy efficiency.However, we were not able to achieve higher efficiencies, with any other em-pirically chosen phase profiles. Significantly higher throughputs than whatwe already obtained cannot be expected, therefore further optimization wasdeemed unnecessary.Figure 3.5 plots the spectral and temporal domains of two pulse trainsin the PAP scheme. Both sequences have a period of 1 ps, one consists of5 pulses with β = 1.17 (a,b) and the other of 9 pulses with β = 0.79 (c,d).In either case, β has been optimized to a maximum throughput, reaching61.7% (a) and 56.3% (b).Other shaping techniques to create pulse sequences do exist, many ofwhich are based on the phase-only shaping [181, 182, 46, 140, 134]. How-ever, neither of these methods yield the necessary flexibilities needed for ourexperiments and will therefore not be introduced. All our pulse trains werebuilt as discussed in this section.3.2 Laser sourceWe use a Titanium:Sapphire (Ti:Sapph) femtosecond laser system (SpitFirePro, Spectra-Physics) producing uncompressed frequency-chirped pulses withthe spectral bandwidth of 9 nm (FWHM) at the central wavelength of800 nm, 1 KHz repetition rate and 2 mJ per pulse. Part of the beam(60% in energy) is compressed to 130 fs pulses (FWHM) via a grating-basedpulse compressor. It is used as a reference beam in cross-correlation mea-surements (see Sec. 3.2.1) or as a probe beam in spectroscopy measurements(see Sec. 4.1). The second part (40% in energy) is used for the generationof high energy pulse sequences, typically as uncompressed pulses of 150 ps(FWHM), as described in Sec. 3.3, 3.4, 3.5.503.2. Laser source3.2.1 Pulse characterizationUltra-short pulses are most commonly characterized with a technique knownas frequency-resolved optical gating (FROG) [169]. The idea is to do a cor-relation measurement in which the pulse to be characterized is gated withanother pulse. In FROG, the spectrally-resolved auto-correlation of the un-known pulse yields enough information to numerically retrieve the spectralphase and amplitude of the pulse. Rather than using an auto-correlationfunction, it is also possible to use a well-characterized reference pulse to per-form cross-correlation frequency-resolved optical gating (XFROG). Furtherinformation about this technique can be found in Ref. [107].We elaborate on both techniques with our experimental setup shownin Fig. 3.6. Panel (a) is used to characterize a single femtosecond pulse,e.g. as produced by our Ti:Sapphire laser system. The incoming beam issplit into two equal parts on a beamsplitter. After travelling some distance,both beams are focused onto a nonlinear BBO crystal where they spatiallyoverlap. The generated second harmonic light of each individual beam isblocked. We are solely interested in measuring the sum-frequency signalwith a spectrometer. The spectrum is recorded as a function of the temporaldelay between both pulses. This way, the unknown pulse is gated by itselfand an iterative algorithm can be used to retrieve the spectral amplitude andphase of the pulse. Panel (b) shows a slight variation, known as XFROG.Here, the unknown pulse is gated by a well characterized reference pulse,in an otherwise identical procedure. We use XFROG to characterize ourpulse sequences. As the reference pulse we use a TL pulse9, that has beencharacterized via FROG.Figure 3.7(a & b) show two different XFROG spectrograms correspond-ing to two different pulse shapes. The spectrum on the vertical axes isrecorded with a spectrometer (Photon Control Inc., SPM-002-B) with aresolution of 0.094 nm at a wavelength of 400 nm. The horizontal axespresent the delay between the shaped pulse and the reference pulse which iscomputer-controlled by a delay line [Newport: Motion Controler, ESP301;translation stage, UTS-100CC (for XFROG) & MFA-CC (for FROG)] with aminimum step size better than 7 fs. We refer to such a two-dimensional rep-resentation of a pulse in the spectral and time domains as an XFROG trace.Panel (a) shows a transform-limited pulse with a symmetric trace. Panel(b), on the other hand, presents a linearly chirped pulse (α′ = 50.000 fs2);the instantaneous frequency changes linearly over time, while the total band-9For a definition of transform-limit (TL) see Sec. Laser sourceReferenceTime delaya)b)SpectrometerSpectrometerFigure 3.6: Two variations of the experimental setup used for ultra-shortpulse characterization: (a) frequency-resolved optical gating (FROG) and(b) cross-correlation frequency-resolved optical gating (XFROG).width is preserved. To obtain the temporal profile of a pulse, we integrateover the entire spectrum, see Fig. 3.7(c). As expected, frequency chirping(dashed black line) leads to a longer pulse of ≈ 1.4 ps (FWHM) with alower peak intensity in comparison to the unshaped pulse (red solid line)with a 130 fs duration (FWHM). The procedure equally applies to a pulsesequence. Various examples of different pulse trains will be shown laterduring the discussion of the shaping techniques.Compensation of phase distortionsUltra-short pulses that propagate through any kind of dispersive material,e.g. glass optics, air, etc., do accumulate phase distortions. As a conse-quence, the spectral phases of a pulse will become frequency dependent,visible in a distorted XFROG trace and leading to longer pulse durations.For the purpose of our experiments, we want to achieve the shortest possiblepulse duration, i.e. a transform-limited pulse. We use multiphoton intra-pulse interference phase scan (MIIPS) to compensate for all phase distortions[111]. The technique relies on a pulse shaper to apply a spectral phase func-tions to compensate for the distorted phase of the pulse. The compensation523.3. Pulse shaping I: The pulse shaper-2 0 2Time (ps)39340040701XFROG intensity(arb. units)-2 0 2Time (ps)393400407Wavelength (nm)abcFigure 3.7: XFROG spectrograms: (a) transform-limited 130 fs pulse, (b)linearly chirped 1.4 ps pulse. (c) Their respective temporal profiles as solid-red line and dashed-black line.function is retrieved iteratively by analysing the second-harmonic spectrumof the pulse. We use the same setup described in Fig 3.6 but measure thesecond-harmonic signal of the pulse. More details of the MIIPS procedureare given in Ref. [191]. We achieve a 130 fs pulse duration for all pulseshaping scenarios.3.3 Pulse shaping I: The pulse shaperThe femtosecond pulse shaper allows the creation of pulse sequences withgreat flexibility. Pulse trains can be formed within certain limitations,namely a minimum pulse duration of 130 fs given by the transform limitof the laser system and a maximum length of the sequence of 50 ps givenby the spectral resolution of the shaper, derived below. The ‘magic’ of apulse shaper is based on spectral phase and amplitude modulation of thefemtosecond pulses implemented with a spatial light modulator (SLM). Weintroduce the optical setup (Sec. 3.3.1), as well as its experimental imple-mentation (Sec. 3.3.2). In particular, we address the method of choice tocreate pulse sequences and demonstrate the shaping flexibility by showingdifferent pulse trains (Sec. 3.3.3).533.3. Pulse shaping I: The pulse shapergratingsphericalmirrormirror (m)SLMmmmInputOutputgratingmirror (m)spherical mirrorSLMftop viewside viewFigure 3.8: Pulse shaper in ’4f ’-folded design: top view and side view. Thedashed boxes indicate the beam profiles on various optical elements.3.3.1 Optical setupWe send the laser beam to a femtosecond pulse shaper which is used to splita single pulse into a series of pulses. The shaper, shown schematically inFig. 3.8, is built in the standard ’4f ’-geometry [180] and uses a liquid crystalspatial light modulator (SLM, Cambridge Research and Instrumentation,Inc.).A transmissive grating (Kaiser Optical Systems, 1800 g/mm) dispersesthe input beam. A spherical mirror (Edmund Optics, f = 36 inch) at thedistance f from the grating focuses different frequency components at dif-ferent locations in the Fourier plane of the shaper, again at the distancef . Here a mirror sends the beam back the same optical path but offset inheight, such that we have a non-dispersed, collimated beam at the output ofthe shaper. The total optical path length sums up to 4f , hence the name ofthe design. The optical path can be followed step-by-step in Fig. 3.8 througha top and a side view, which also visualizes the dispersion and focusing of all543.3. Pulse shaping I: The pulse shaperspectral components. If no spectral mask is applied in the Fourier plane, theinput and output pulses will be identical, in the ideal case that the shaperdoes not introduce any spatial or spectral distortions due to a misalignmentof any optical component. In order to “shape” the femtosecond pulse weintroduce a spatial light modulator (SLM) in the Fourier plane, where all fre-quencies are dispersed and focused. The SLM is the heart of the shaper andit consists of a double-layer 640-pixel mask that is used to control the phaseand amplitude of all spectral components of the laser pulse. To achieve anydesired waveform of the output pulse, e.g. a pulse train, we first calculatethe required frequency masks via the Fourier transform of the target tempo-ral profile and subsequently apply the masks with the SLM. Mathematicaldetails are found in sections 3.1.1 to 3.1.3 and the implementation procedurein section 3.3.2.The spectral resolution of the pulse shaper of ∆λ = 0.04 nm per pixel setsthe upper limit for the total duration of the pulse train to ∆T = λ2/c∆λ ≈50 ps, where the central wavelength λ = 800 nm and c is the speed of lightin vacuum. Within this window, virtually any pulse shape can be achieved.In this work, we focus on producing pulse sequences.Pulse shaping can be done with uncompressed pulses, in which case weuse a grating compressor at the output of the shaper to compress all pulsesin the train. Alternatively, we can compress the pulse prior to the pulseshaping. The choice depends on the amplification process that will followand is discussed in Sec. 3.5.1. In either case, the minimum duration of eachsub-pulse in the train is around 130 fs set by the transform-limited pulseduration of the laser system.Further details about techical aspects of such a femtosecond pulse shaperhave been described in detail in the dissertation of S. Zhdanovich [191]. Thisincludes more information on selecting the appropriate optical componentsto maximize the shaper resolution, and on aligning and calibrating thisspecific instrument.3.3.2 Implementation of pulse sequencesIn Sec. 3.1.3, we introduced the mathematical foundation to form a pulsetrain via phase and amplitude shaping. Here, we will elaborate on theexperimental implementation.The analytically calculated phase and amplitude masks have to be dis-cretized with a step size that matches the resolution of the SLM. Given inunits of wavelength, the resolution is ∆λ = 0.04 nm per pixel. The peakof the spectrum is placed in the center of the 640-pixel mask. Figure 3.9553.3. Pulse shaping I: The pulse shaper640-pixel SLM Masks:  (N=5, T=2ps)640-pixel SLM Masks:  (N=5, T=11.67ps)Matlab: “PulseTrain_LabView.m”WL: 800-803nmPixel: 320 – 394   (75pixel)0101AmplitudeMaska)b)AmplitudeMask800 801 802 803Wavelength (nm)Figure 3.9: Section of an amplitude mask (red dotted line) and its discretizedversion, matching the experimental SLM resolution, to generate a pulse trainwith (a) N = 5, T = 2 ps; β = 0 or (b) N = 5, T = 11.67 ps; β = 0.shows the discretization of two different pulse sequences. For demonstrationpurposes, we plot only the amplitude masks in the spectral window from800 to 803 nm, corresponding to a width of 75 pixels. The center pixel 320matches the central frequency of λ = 800 nm. The red dotted lines showthe desired amplitude masks, the solid blue lines are the discretized versions.Panel (a) presents the mask to achieve a train of 5 pulses with a 2 ps period.The overall length of the train with 8 ps lies well within the shaper limitof ∆T ≈ 50 ps. Panel (b), however, shows an example where the targetpulse train reaches the shaper limitation. A train of 5 pulses with a 11.67 psperiod spans a total duration of 47 ps. Every comb tooth in the spectrumhas the width of only a single pixel of the SLM. A similar pulse train is usede.g. in Sec. 5.4.1, when the period matches the rotational revival time of16O2 (Sec. 2.3.9).Figure 3.9(b) reveals one source of inaccuracy in the shape of the experi-mental pulse train. The discretized function does not describe the amplitudemask well enough and will lead to deviations in the temporal shape. Ratherthan obtaining a sequence of identical pulses, the amplitudes of the indi-vidual pulses will vary. Other discrepancies that we typically observe close563.3. Pulse shaping I: The pulse shaper“target train”Time t“real train”calculate & apply𝐴SLM 𝜔 ,𝜙SLM(𝜔)measure𝐼XFROG(𝑡)retrievecorrectionTermination criterion met ?YESNOFigure 3.10: Protocol to iteratively compensate for irregularities in the out-put temporal profile with the objective to reach a flat pulse train.to the shaper limit are: unwanted pre- and post-pulses appearing outsidethe 50 ps window, and a spectral chirp of the sub-pulses lying closest to theedge of the 50 ps window.Pulse train corrections: We developed an approach to minimize shapingerrors and inaccuracies in the output pulse sequence. The idea is to pre-compensate for all the discrepancies that occur in the shaping process.Starting with Eq. 3.8, we introduce more control parameters: Ratherthan having a flat pulse train, we can adjust the amplitude of each individ-ual pulse; rather than being comprised of TL pulses, each sub-pulse can geta linear chirp; rather than being a strictly periodic pulse train, the periodbetween all pulses can be chosen separately. We use the protocol illustratedin Fig. 3.10 to achieve a flat PT. First, we design the desired pulse trainaccording to the analytic expression in Eq. 3.9 and apply the calculated am-plitude and phase masks via the SLM. Next, we measure the intensity profileIXFROG(t) of the actual shaped pulse via a cross-correlation measurementin the time domain (the technique of XFROG is explained in Sec. 3.2.1).Deviations of all sub-pulses from the ideal are determined and compensatedfor in the next generation of the pulse train. Via a Fourier transform weobtain the new amplitude and phase masks. The procedure is iterativelyrepeated until the termination criterion is met and the output shape closelyresembles the target pulse train.The method proved to be very effective, which will be seen in Sec. 3.3.3and 3.5.3 in several examples. Besides compensating for shaping errors, theprocedure also serves to deal with nonlinearities due to the amplificationin the MPA, discussed in Sec. 3.5. In addition, the flexibility of the shap-ing technique enables us to design any arbitrary pulse sequence, e.g. non-periodic sequences or sequences with various sub-pulse amplitudes. Such573.4. Pulse shaping II: The Michelson interferometersequences will be of importance in almost all of the presented experiments.3.3.3 Demonstration of pulse sequencesFemtosecond pulse shaping offers the flexibility of creating arbitrary pulsesequences within the limits of the shaper’s temporal and spectral resolution.In Fig. 3.11, we demonstrate this flexibility using the example of a pulse trainwith nine pulses. The overall energy of the train was set to 1 mJ. A flat trainof pulses with almost equal amplitudes (7% flatness) separated by T = 4ps isshown in Fig. 3.11(a). In Fig. 3.11(b), a linear amplitude tilt was applied tothe train’s envelope and its sixth pulse was completely suppressed, while thetotal energy was kept constant at 1 mJ. Figures 3.11(c) and (d) demonstrateour ability to produce high-energy flat-amplitude sequences with multipleperiods and completely random timing of pulses, respectively.3.4 Pulse shaping II: The MichelsoninterferometerFor some experiments the number of pulses that can be fitted within a 50 pstime frame is not sufficient. In contrast to the pulse shaper, an interfero-metric setup [160] has no limitation on the overall length of the final pulsetrain. We built two polarization-based Michelson setups that allow us toquadruple the number of pulses in the sequence and extend its duration toat least four times that produced by the shaper.In Fig. 3.12, an incoming linearly polarized laser beam is split intotwo beams with equal amplitude but opposite polarization axes (s & p-polarization) via the combination of a λ/2-waveplate and a polarizationcube. Each beam is reflected back by a dielectric mirror, passing througha λ/4-waveplate twice to flip the polarization axis. The output behind thecube is now a laser beam that consists of two pulses of opposite polarizationwhose temporal spacing can be adjusted with a computer controlled delayline (Newport: Motion Controler, ESP301; translation stage, MFA-CC).Those optical elements can be put in series n times to split a single pulseinto a sequence of 2n pulses. We implemented this design with two Michel-son interferometers n = 2. In order to get a final pulse sequence whereall pulses share the same polarization we use another λ/2-waveplate and acube to split the pulse train into s & p-polarization. In our setup, we useonly p-polarization. Neglecting losses of optical components, polarizationmultiplexing results in the energy loss of 50%.583.4. Pulse shaping II: The Michelson interferometer-20 -15 -10 -5 0 5 10 15 2001-20 -15 -10 -5 0 5 10 15 2001-20 -15 -10 -5 0 5 10 15 2001-20 -15 -10 -5 0 5 10 15 2001Time (ps)Intensity(arb. units)abcdIntensity(arb. units)Intensity(arb. units)Intensity(arb. units)Figure 3.11: Temporal profiles of four different sequences of nine pulses: (a)Periodic train of equal-amplitude pulses separated by 4 ps. (b) Same trainwith linearly decreasing pulse amplitudes and the sixth pulse suppressed.(c) Flat pulse train with two different time periods of 5 ps and 3 ps. (c)Flat non-periodic pulse sequence with a random timing of pulses.593.5. Multi-pass amplificationൗ𝜆 2ൗ𝜆 4ൗ𝜆 4Time delayPolarizerInputOutputFigure 3.12: Two polarization-based Michelson interferometers quadruplethe number of incoming linearly polarized pulses. Computer-controlled delaylines allow variable time delays between the pulses. At the output, half theenergy is lost to the opposite polarization.3.5 Multi-pass amplificationTo compensate for the losses of femtosecond pulse shaping and interfero-metric splitting, we send the long pulse train through a home-built multi-pass amplifier (MPA), see Fig. 3.13. After passing four times through aTi:Sapph crystal, pumped by a neodymium-doped yttrium aluminium gar-net (Nd:YAG) laser (Powerlite Precision II, Continuum, 800 mJ at 532 nmand 10 Hz repetition rate), the weak pulses are amplified to reach energiesof more than 100 µJ per pulse. The size and divergence of the 800 nm beamare adjusted with a telescope to control the gain. In the same way, thepumped volume in the crystal can be controlled with another telescope onthe 532 nm pump beam. The final gain is set to the desired value by tuningthe power of the 532 nm pump beam. The repetition rate of the amplifiedpulses is limited to 10 Hz, the frequency of the Nd:YAG pump laser. Thesynchronization with the 1 KHz femtosecond laser systems is done with ahome-built pulse generator.In the following, we give some more technical details for the setup andthe alignment of the MPA:(1) The 800 nm beam should propagate through the Ti:Sapph crystalat the Brewster angle. It is very important, that the pulses are set to aperfect p-polarization, first, to minimize losses due to the reflection of thesurface, and second, to achieve a smooth amplified pulse train. In the case603.5. Multi-pass amplificationTi:sapp crystal532 nmInputNd:YAG Laser800 nmOutputtelescopetelescopeFigure 3.13: Diagram of the home-built multi-pass amplifier.of a polarization misalignment the birefringence of the Ti:Sapph crystal willlead to a temporal splitting of the input pulse. Recording the final pulseshape via XFROG (Sec. 3.2.1) will reveal undesired pre- and post-pulses ifthey are present.(2) The telescope on the 800 nm beam serves primarily to adjust thebeam divergence. The actual beam size is chosen as large as possible tominimize the pulse intensities. At the same time, the diameter needs to besmaller than the pump beam diameter to obtain a uniform amplification ofthe entire beam profile.(3) Optimal working conditions of the MPA are checked on a weekly(daily) basis. Since our experiment requires long optical beam paths, mi-nuscule changes in the beam alignment can have rather large effects on thenonlinear amplification of the pulses. A rough alignment is guaranteed bymultiple sets of irises throughout the setup. The fine adjustment is doneby measuring the energy iteratively after each stage of the amplifier witha photo-diode. Tuning the respective optical mirror helps to maximize theamplification of each pass. This procedure is done at low amplifications.(4) To attain a nice pulse train of equally strong sub-pulses, the align-ment of the Michelson interferometers is of crucial importance, too. In orderto reach the same gain in each of the four pulse train copies, the four differentbeams originating in the four arms of the Michelson interferometers must bespatially recombined to travel on an identical path through the MPA. Wealign the optical beam path for a single arm of the Michelson setup first,as described in (2). In a second step we overlap each other Michelson armwith the first arm in the far field without amplifying the pulses. At last,we check the uniform amplification of all pulses by means of an XFROGmeasurement.613.5. Multi-pass amplification(5) A potential danger in the nonlinear amplification process lies in self-phase modulation. Self-focusing of the beam can lead to increasing intensi-ties and to the damage of the crystal. We choose a slightly diverging 800 nmbeam at the MPA input (diameter about 1 mm) which has increased toabout twice the size at the output. Once the MPA is running, we check theamplification at each stage to guarantee an equal amplification throughoutthe MPA. A sudden increase in amplification is an indicator for decreasingbeam sizes and rapidly growing intensities. In this case the beam divergenceshould be increased to counteract the detrimental self-focusing. Under idealworking conditions, we achieve amplification factors of around 5 per stage,however we did occasionally operate with factors of up to ∼ 7 per stage.3.5.1 Compressed versus chirped amplificationOur laser generates frequency-chirped pulses. Compressing them down tothe TL duration and shaping them into a pulse train are two linear-opticsoperations. As such, they commute with one another and can be executedin either order.At first, we investigate how the amplification of a single pulse is affectedby its compression. The input laser pulse into the multi-pass amplifier can ei-ther be an uncompressed frequency-chirped pulse or a compressed TL pulse.Hereby, the compression is done with a standard grating-based pulse com-pressor. We compare the output pulses when the amplification is done withchirped pulses versus compressed pulses and give a list of arguments for oragainst each scenario.Compressed-pulse amplification: Multi-pass amplifiers typically oper-ate with chirped pulses that are recompressed after the amplification. Thiskeeps the peak intensities below the damage threshold of the Ti:Sapph crys-tal. However, we show that it is possible to amplify 130 fs pulses (FWHM)up to energies of at least 200 µJ. In this configuration, we first compressthe chirped pulses down to the Fourier transform limit, before creating andamplifying the pulse train. With beam diameters inside the MPA as smallas 1 mm (FWHM), the amplified 800 nm pulse can reach intensities on theorder of 1011W/cm2. Even though this does not damage the Ti:Sapph crys-tal, it does affect the spectrum. The pulses are propagating through airover a distances of about 7 m inside the MPA and acquire spectral modula-tions. This propagation effect due to self-phase modulation (SPM) has beenstudied in detail by Nibbering et.al. [125]. In Fig. 3.14(a) we show howthe input Gaussian spectrum of a single transform-limited pulse (black solid623.5. Multi-pass amplification790 800 81000. (nm)Intensity (arb. units)790 800 81000. (nm)790 800 81000.51a) b)Figure 3.14: (a) Compressed-pulse amplification: spectrum of a single pulsebefore the MPA (black solid), after the MPA unamplified at 5.4 µJ (bluedotted) and amplified to 144 µJ (red dashed). (b) Chirped-pulse amplifi-cation: spectrum of a single pulse unamplified at 0.4 µJ (black solid) andamplified to 100 µJ (red dashed). The inset shows spectral modulations af-ter the amplified pulse is focused through a cell filled with 6 atm of nitrogen(green dotted) or oxygen (blue dashed) molecules.line) changes as it propagates through the MPA. At the output, the spec-trum is already distorted even if the Nd:YAG laser is turned off and thereis no amplification (blue dotted line). The energy of this pulse is measuredto be 5.4 µJ. When the same pulse is amplified to an energy of 144 µJ themodulations become much more severe (red dashed line).Chirped-pulse amplification: In a second scheme, we use the uncom-pressed 150 ps pulses (FWHM) from our laser system. The amplification isapplied to chirped pulses that are only recompressed to femtosecond pulsesat the output. As a consequence, the intensities within the MPA will bedecreased by about three orders of magnitude. Since the propagation effectsscale with intensity, we can now see a clean output spectrum in Fig. 3.14(b),measured after the pulse compression. The spectrum of a single pulse am-plified 250 times to an energy of 100 µJ (red dashed line) is almost indistin-guishable from the spectrum of an unamplified pulse at 0.4 µJ (black solidline). As a proof that the mentioned SPM effects are due to high intensitypulse propagation in air, we focus (f = 100mm) the same transform-limitedpulse into a gas cell filled with 6.5 atm of nitrogen (green dotted line) orOxygen (blue dashed line) in the inset of Fig. 3.14(b). Spectral modulations633.5. Multi-pass amplificationsimilar to the ones in Fig. 3.14(a) become apparent.In conclusion, chirped pulse amplification ensures negligible spectral dis-tortions of the output pulses due to propagation effects in air. It also presentsthe safer option in terms of damaging optical components because of highintensity pulses. Any experiment that requires high pulse energies shouldtherefore implement this scheme. For all other experiments that do not relyon a smooth spectrum or high intensities it is worth considering to workwith compressed pulses for technical reasons. First, no energy is lost af-ter the amplification process, as we can send the pulses straight to the gassample. We do not have to compensate for optical losses of the gratingcompressor. Second, the optical alignment of the pulse shaper is facilitatedwith compressed pulses. It becomes a non-trivial task with chirped pulsesbecause the shaper itself can act like a grating compressor. Third, the MPAneeds regular (daily) adjustments to maintain optimal amplification. Ev-ery change in the beam path through the MPA, necessitates a re-alignmentthrough the compressor. The optimal compression is sensitive to the opticalpath. Experimentally, we simplified this procedure by setting up severalirises throughout the setup to indicate a reference path.3.5.2 Amplification of pulse sequencesMost of our experiments rely on strong pulses. Thus, we chose the schemeof chirped-pulse amplification. The complete setup is depicted in Fig. 3.15.The chirped pulses from our Ti:Sapph laser system are sent to a pulse shaperfollowed by two Michelson interferometers (dotted box) to form long pulsesequences, that are then amplified in an MPA (dashed box). At the finalstage, a standard grating-based pulse compressor compresses each pulse ofthe sequence to a 130 fs duration (FWHM). The second part of the beamfrom the laser system (60% in energy) is immediately compressed with anidentical grating compressor and utilized as a reference beam.Losses and Amplification: Unavoidable optical energy losses in thecombined pulse shaper and Michelson interferometer setup can be quantifiedas follows. In a typical 4f pulse shaper, about 50% of energy is lost due tothe diffraction efficiency of the gratings. The energy throughput owing to thesplitting of the initial pulse into a sequence of N identical pulses by means ofthe spectral shaping can be estimated as 1/N . This scaling can be derivedfrom the following consideration of a pulse train in the frequency domain.The spectrum of an infinite pulse train is a frequency comb. The separation643.5. Multi-pass amplificationTi:Sapph CrystalNd:YAG LaserMPATime delayMichelson interferometersPumpTi:Sapph LaserCompressorProbePulse shaperCompressorFigure 3.15: Diagram of the optical setup. The output beam of a Ti:Sapphlaser with frequency-chirped 150 ps pulses is split into two parts. The pumpbeam is used to generate long sequences of high-energy pulses; the otherbeam serves as a probe. Both outputs are compressed to a 130 fs duration(FWHM).of the comb teeth is inversely proportional to the train period T , ∆ω ∝ 1/T .The spectrum of a finite train of N pulses is the same comb convolved withthe sinc function whose width is inversely proportional toNT , δω ∝ 1/(NT ).The ratio δω/∆ω determines the amount of energy remaining in the trainafter shaping. A more complete mathematical foundation has been derivedin Sec. 3.3.2. The polarization multiplexing in the interferometer setupaccounts for another energy loss of factor 1/2. In total, each of the 4N pulsesin the train carries 1/(16N2) of the input energy.The amount of available energy is typically limited to less than 1 mJ bythe damage threshold of a pulse shaper, which in our case corresponded to300 µJ. For a train of 4N = 84 pulses demonstrated in Sec. 3.5.3, one endsup with . 0.2 µJ per pulse. Imperfections of various optical componentsbring this number even further down, well below the typical requirements fora strong-field regime of the laser-molecule interaction. In order to compen-sate for all the losses, each pulse gets amplified with the MPA. The largestdemonstrated amplification factor of 2800 has been for the mentioned trainof 84 pulses. Generally, we work with pulse sequences where each pulseexceeds an energy of 100 µJ.653.5. Multi-pass amplification-20 -10 0 10 2001Time (ps)bIntensity(arb. units)Wavelength (nm)-20 -10 0 10 203964004040 Intensity (arb. units) 1aFigure 3.16: (a) XFROG spectrogram of an amplified pulse train with21 equally spaced pulses and (b) its corresponding temporal profile before(dashed black) and after (solid red) the amplification by a factor of 400.663.5. Multi-pass amplification3.5.3 Demonstration of amplified pulse sequencesFigure 3.16(a) shows an XFROG spectrogram for the pulse train of 21 equallyspaced pulses with a period of 2 ps. The spectrogram demonstrates that eachpulse in the train is transform-limited with no residual frequency chirp, con-firming that proper pulse compression is attainable at the end of the com-bined shaping and amplification process. Integrating over the whole spec-trum, one finds the distribution of energy among the pulses in the train,plotted as the red solid line in Fig. 3.16(b). Here, the gain factor of 400was required for reaching an average energy of 124 µJ per pulse. Such highgain levels result in the increased sensitivity of the output pulse sequenceto the energy fluctuations in the seed train entering the MPA, as well asin pulse dependent amplification rates. Equalizing the output amplitudesis achieved through the iterative process shown in Fig. 3.10, in which thecorrection function is fed back to the pulse shaper in order to compensatefor the irregularities in the output temporal profile. In the presented exam-ple, the corrected seed train is shown in Fig. 3.16(b) by the black dashedline. The final degree of flatness is limited by the nonlinearity of the MPAamplification process and is on the order of 10%, determined as the standarddeviation of the pulse-to-pulse energy fluctuations.Adding a set of nested Michelson interferometers enables us to generatepulse sequences longer than the 50 ps time limit set by the spectral resolu-tion of the shaper and with more pulses, while still maintaining the energylevel in excess of 100 µJ per pulse. In Figures 3.17(a) and (b), we show pe-riodic sequences of 20 and 84 pulses stretching over a duration of ≈ 170 ps(pulse separation of 8 ps and 2 ps, respectively) and carrying ≈ 110µJ perpulse. The latter train has been amplified 2800 times to the total energy of9 mJ. One can see that longer pulse trains suffer from a higher amplitudenoise, e.g. standard deviations of 13% and 18% in (a) and (b), respectively.The increasing noise is due to the higher MPA amplification factors andthe correspondingly higher nonlinearity of the amplification process. Thefour colors represent the four different pathways through the Michelson in-terferometers. Without amplitude noise, each of the four copies would beidentical.673.5. Multi-pass amplification-20 0 20 40 60 80 100 120 14001Time (ps)b-20 0 20 40 60 80 100 120 14001Intensity(arb. units)Intensity(arb. units)aFigure 3.17: Temporal profile of two periodic pulse sequences: (a) 20 pulsesseparated by 8 ps and amplified 500 times to 114 µJ per pulse; (b) 84 pulsesseparated by 2 ps and amplified 2800 times to 107 µJ per pulse. The fourcolors represent the four different pathways through the Michelson interfer-ometers.68Chapter 4Techniques II: RotationalRaman spectroscopy in amolecular jetIn chapter 3 we introduced a method to generate high-intensity pulse se-quences, which are necessary to implement the “molecular kicked rotor”.This chapter will present the experimental techniques used to explore andunderstand the dynamics of the molecular kicked rotor.All our experiments share the same detection technique: “RotationalRaman spectroscopy”. In Sec. 4.1 we describe how the method works andhow it is implemented. We discuss two slightly different variations: one,where the molecular sample is contained in a gas cell; and the other, wherethe experiments are conducted in a molecular jet. We explain a typicalRaman spectrogram as it will reoccur in the presentation of all later results.Experiments that rely on a narrow initial distribution of rotational stateshave to be done in a molecular jet, topic of Sec. 4.2, where the rotationalmotion is substantially cooled down. We explain how a molecular beamworks and show our experimental realization. We investigate the rotationaltemperature and the molecular density during a supersonic jet expansion byanalysing the Raman spectrum.4.1 Raman spectroscopyIn order to implement the technique of coherent Raman spectroscopy weneed a weak probe pulse with controlled spectral width, which determinesour frequency resolution. The experimental scheme is shown in Fig. 4.1. The800 nm pump pulse, which in almost all of our experiments is a high-intensitysequence of femtosecond pulses, is produced as discussed in chapter 3. Theprobe, which originates from the Ti:sapphire laser system (for details seeSec. 3.2), is sent through another pulse shaper. This shaper is built in thesame folded geometry as the pump shaper in Fig. 3.8 (Sec. 3.3.1), but with694.1. Raman spectroscopyAnalyzerSpectrometerProbePolarizer ൗ𝜆 2 ൗ𝜆4Gas cellൗ𝜆 2BBO crystalPumpTime delayProbePumpTime delayVacuum chamberSpectrometera)b)Figure 4.1: Scheme of the experimental setup: A weak probe pulse isfrequency-doubled in a nonlinear BBO-crystal and combined with a train ofstrong femtosecond pulses (pump). Both beams are focused (a) into a cellfilled with a molecular gas or (b) onto a supersonic jet of molecules inside avacuum chamber. The change of probe polarization is analysed as a functionof the wavelength by means of two crossed polarizers and a spectrometer,and as a function of the tunable time delay.704.1. Raman spectroscopysome different optical components [reflective grating (1200 g/mm), sphericalmirror (f = 50 cm)]. In the Fourier plane of the probe shaper, we place amechanical slit, rather than a liquid-crystal SLM, to withstand higher laserintensities. The slit is used to narrow down the spectrum. The centralwavelength is then shifted to 400 nm by means of the second harmonicgeneration (SHG) in a nonlinear BBO crystal. Finally, the probe pulses of0.15 nm spectral width (FWHM) are linearly polarized at 45° with respect tothe pump pulses. Both beams are combined on a dichroic beamsplitter andfocused onto the molecular sample of oxygen O2 or nitrogen N2. Special careis taken to minimize detrimental effects of spatial averaging by making theprobe beam significantly smaller than the pump (FWHM beam diametersof 20 µm and 60 µm, respectively).We distinguish two scenarios illustrated in Fig. 4.1: In panel (a) the gasof linear molecules is contained in a cell at variable pressure. The pump andprobe beams are combined prior to entering the cell. This setup is used toconduct experiments at room temperature in dense gases. In panel (b) bothbeams are first focused into a vacuum chamber, where they are then com-bined on a dichroic beamsplitter and intersect a supersonic jet of molecules.We use a 500 µm diameter pulsed nozzle, operating at the repetition rateof 10 Hz and the stagnation pressure of 33 bar. We achieve rotational tem-peratures around T = 25K at a distance of 1.9 mm from the nozzle. Asummary of rotational cooling via a supersonic jet expansion is given in thenext section 4.2.Coherent molecular rotation, produced by a pump pulse, modulates therefractive index of the gas. The mechanism can also be viewed as a two-photon Raman process, introduced earlier in Sec. 2.3. As a result, the spec-trum of a weak probe light acquires Raman sidebands shifted from its centralfrequency and polarized orthogonally to its initial polarization [95, 96]. TheRaman sidebands are analysed in a polarization sensitive measurement bypassing the output probe light through an analyser set at 90° with respectto the initial probe polarization.Some of the experiments rely on measuring the Raman spectrum with adynamic range of three to four orders of magnitude. In chapter 6 for exam-ple, we need to demonstrate an exponential shape of the observed Ramanspectrum, which would not be possible without such a dynamic range. Itis therefore important to suppress the initial probe polarization as muchas possible, which in turn will reduce the noise floor. We use two Glan-Thompson polarizers, one before and one after the vacuum chamber, andtwo additional waveplates. The λ/4- and the λ/2-waveplates compensatefor polarization changes that the probe beam accumulates as it propagates714.1. Raman spectroscopythrough the optical windows of the chamber and the dichroic beamsplitters.This results in a lower level of the residual probe light10.We record the rotational Raman spectrum with a spectrometer (McPher-son Inc., model 2035, with a 1800 grooves/mm diffraction grating) equippedwith a CCD camera (Andor iDus, DV401A-BV). The camera has a dynamicrange greater than four orders of magnitude. The resolution of the spectrom-eter of 0.056 nm per pixel (at a wavelength of 400 nm) is good enough toresolve individual rotational states in lighter diatomic molecules (e.g. N2 orO2). A computer-controlled translation stage allows for time-resolved spec-trograms by scanning the delay between the pump and probe pulses (New-port: Motion Controller, ESP301; translation stage, UTS-100CC; minimumincrement ≈ 7 fs). The temporal resolution will be given by the duration ofthe probe pulse.4.1.1 Raman spectrogramWhen a weak probe pulse follows the pump through a cloud of rotatingmolecules, its spectrum acquires Raman sidebands. Here, we will explainthe interpretation of such a Raman spectrum. The shift of each Raman peak,∆ωJ , is equal to the frequency spacing between the rotational levels |J〉 and|J + 2〉, while its magnitude is proportional to the square of the rotationalcoherence [96], a more rigorous expression is found in section 2.3.6.We use oxygen or nitrogen in our experiments, for several reasons de-scribed in Sec. 2.3.9. In the first approximation these linear molecules behaveas rigid rotors and their rotational energy is given by EJ = hcBJ(J + 1).Hence, the frequency shift of each Raman peak is linearly proportionalto the rotational quantum number J and can be expressed as ∆ωJ =(EJ+2 − EJ)/h = 2Bc(2J + 3).Figure 4.2 displays a Raman spectrum of N2 recorded after the rotationalexcitation by a single pump pulse. Shown are the measurements in a gascell (solid black line) in comparison to the measurement with a molecularjet (red dashed line). We see a progression of Raman peaks as a function ofthe wavelength shifted away from the probe wavelength at 400.6 nm. Thiswavelength shift can be translated to the rotational quantum number J ,shown along the lower horizontal axis. According to the nuclear spin statis-tics of nitrogen, the ratio between even and odd rotational states J mustbe 2:1 (described in Sec. 2.3.9). Our probe pulse with a spectral width of0.15 nm (FWHM) is narrower than the line separation ∆ωJ+2−∆ωJ ∝ 8B10The extinction ratio is greater than five orders of magnitude.724.1. Raman spectroscopy401 402 403Wavelength (nm) 01Raman intensity(arb. units)0 4 8 12 16 20Rotational quantum number JFigure 4.2: Raman spectrum of N2 after the rotational excitation by a singleweak pulse, measured in a gas cell (solid black line) or with a molecular jet(red dashed line).enabling us to resolve individual rotational states and thus to determine theshape of the excited rotational wave packet.The entire spectrum consists of a Rayleigh peak, polarized along theinput probe polarization (selection rule: ∆J = 0), and two progressions ofRaman sidebands shifted up (∆J = +2) or down (∆J = −2) with respectto the Rayleigh frequency and polarized in the orthogonal direction. Wemeasure only the red-shifted sidebands. As a consequence of a non-idealpolarization suppression of the input probe light, we detect a Rayleigh peakof substantial magnitude which can exceed that of the scattered Ramanspectrum. We set our spectrometer to measure all Raman spectra only forvalues J ' 1, to truncate the unwanted Rayleigh peak.In Fig. 4.2 the pump pulse is set to a weak kick strength of P  1which will not lead to much angular momentum transfer. Therefore, thepopulation distribution will hardly change and closely represent the initialthermal distribution. The two plotted lines show the clear difference inthe initial rotational temperature. In the case of the gas cell (black solidline) the distribution follows a Boltzmann distribution at room temperature,whereas in the case of the jet (red dashed line) we retrieve a temperaturearound 25 K (details in Sec. 4.2).Figure 4.3 shows the Raman spectrum of oxygen, recorded under iden-734.1. Raman spectroscopy401 402 403Wavelength (nm) 01Raman intensity(arb. units)0 5 10 15 20 25Rotational quantum number JFigure 4.3: Raman spectrum of O2 after the rotational excitation by a singleweak pulse, measured in a gas cell (solid black line) or with a molecular jet(red dashed line).tical conditions as described above for nitrogen. The main difference withFig. 4.2 stems from the nuclear spin statistics of O2, allowing only odd quan-tum numbers J .The interpretation of state-resolved Raman spectra remains the same inall experiments. When the molecules are excited with sequences of strongpulses, however, the assumption of a rigid rotor is not valid anymore. Thus,the exact position of the Raman peaks is calculated with the energy ex-pression of a non-rigid rotor EJ = hc[BJ(J + 1) −DJ2(J + 1)2]. The lineseparation will decrease for higher J values. This change is insignificant forJ < 30, as they are typically excited in our experiments. All the presentedRaman spectra are taken with the probe pulse arriving immediately afterthe last pulse in the train or after the specific pulse of interest within thetrain. This way we minimize the decrease in Raman signal due to collisionaldecay.744.2. Molecular beam source4.2 Molecular beam sourceA majority of the QKR phenomena that we studied can only be observedin rotationally cold molecular samples. One well established technique tocool molecules is based on a supersonic expansion [115, 5]. A high pressuregas that is passing through a nozzle into a low pressure region will expandadiabatically leading to a drop in temperature. Rotational temperatures ofa few Kelvin are routinely achieved.We implemented a supersonic jet of molecules, which resulted in rota-tional temperatures around 25 K. First, we present a concise summary ofthe theory of molecular beams in Sec. 4.2.1. Second, our specific setup witha pulsed nozzle producing a jet suitable for Raman spectroscopy is describedin Sec. 4.2.2. Finally, we show experimental Raman spectra of N2 and O2in Sec. 4.2.3, investigating temperatures and densities and their dependenceon externally controllable parameters.4.2.1 Theory of supersonic expansionAll physical concepts and thermodynamic equations in this section wereadapted from the books by Miller [115] and Anderson [5].At first, we introduce the thermodynamic quantity γ. All moleculeshave a specific heat ratio γ = Cp/Cv, calculated via the specific heats atconstant pressure Cp and constant volume Cv. Next, we look at a beam ofdiatomic molecules (γ = 7/5), i.e. N2 or O2 entering a vacuum chamberthrough a nozzle. Initially the gas is at stagnation pressure p0 and roomtemperature T0. The pressure inside the vacuum chamber is set to pb. Weapproximate the beam as a steady-state continuous jet from the nozzle asshown in Fig. 4.4 (in the actual experiment we will be using a pulsed jet.) Ifthe pressure ratio p0/pb is greater than a critical value of G =(γ+12)γ/(γ−1),for diatomics G ≈ 1.89, the speed of the molecules exiting the orifice will besupersonic with a Mach number M  1. The Mach number measures thespeed v of the gas past the nozzle in relation to the speed of sound a. Foran ideal gas a =√γRT/m with the molecular weight m and the universalgas constant R. Before the expansion, the molecules at the orifice of thenozzle have a velocity of M = 1. In the subsequent adiabatic expansion ofthe gas, the velocity will increase, leading to a drop in molecular density andtemperature. However, if the pressure ratio p0/pb is not sufficiently large,the gas will exit the nozzle at a pressure of pb with no further expansion.Different empirical equations exist to calculate the Mach number alongthe centerline of the jet, see Ref. [115]. We quote one that is suitable for754.2. Molecular beam source𝒑𝒃Jet boundaryBarrel shockZone of silence𝒑𝟎, 𝑻𝟎𝑴 ≪ 𝟏𝑴 = 𝟏𝑴 ≫ 𝟏 𝑴 < 𝟏Mach disc𝑴 > 𝟏𝑴 > 𝟏Figure 4.4: Diagram of a supersonic jet expansion. Indicated are the dif-ferent regions of the beam with their respective molecular speeds given inMach numbers M .close distances from the nozzle, relative to the nozzle diameter dM =(xd) γ−1j[C1 +C2x/d+C3(x/d)2+C4(x/d)3]. (4.1)The constants j, C1, C2, C3 and C4 are γ-dependent (molecule-dependent)and can be looked up in tables [115, 167]. The expression is valid for x/d >0.5.The supersonic expansion is an isentropic process, under the assumptionthat no friction or other dissipative effects occur, which would lead to achange in the entropy. For an ideal gas we can then use known relationsto estimate some thermodynamic quantities like temperature T , pressure pand density ρ inside the beamTT0=(pp0) γ−1γ=(ρρ0)γ−1. (4.2)Primarily, we are interested in the dependency of those quantities on thedistance from the nozzle. We calculate the (translational) temperature as afunction of the Mach number which is related to the distance via Eq. 4.1TT0=(1 +γ − 12M2)−1. (4.3)The temperature and the pressure continue to decrease as long as collisionsoccur in the flow. This correlates with an asymptotically rising velocityof the molecules. Collisions are also responsible for the rotational cooling.During a free-jet expansion the molecules typically experience around 102to 103 binary collisions. This is sufficient to cool the rotational degree of764.2. Molecular beam sourcefreedom: Most diatomics require between 10 to 100 collisions to equilibratewith the translational motion. Hence, we estimate the rotational tempera-ture with Eq. 4.3. The vibrational degree of freedom hardly relaxes in a jetexpansion as many more collisions are necessary > 104. Fortunately, almostall molecules in our experiment at room temperature are already in the vi-brational ground state. At some point the expansion will transition into acollisionless flow, typically after several nozzle diameters. At this point theterminal velocity is reached and the isentropic description breaks down.Eventually, the gas will overexpand as a result of the supersonic flowwhich is unaware of the downstream boundary conditions. Information, i.e.about the pressure pb inside the chamber, cannot travel faster than the speedof sound (M = 1). This results in shock waves indicated in Fig. 4.4, whichwill recompress the system. For the purpose of our experiment, we have tomake sure that the Mach disk locationxMd= 0.67(p0pb)1/2, (4.4)is further away than the interaction region with the laser pulses. The zoneof silence is confined within the barrel shock wave with an approximatediameter of 0.75xM , the diameter of the Mach disk is about 0.5xM [167].Often, experimental setups place skimmers inside the zone of silence toextract a small solid angle of the beam. This reduces the momentum spread,enables differential pumping to prevent the formation of shock waves, andgenerally leads to colder temperatures. Our experiments are done in closeproximity to the nozzle, since we rely on high molecular densities for theRaman detection (described in Sec. 4.1). The implementation of a skimmeris not feasible.4.2.2 Beam setupFigure 4.5 shows a diagram of our molecular beam setup. We use a pulsedvalve (Parker Hannifin, Series9, 009-0582-900) with an orifice of d = 500 µm.The valve is operated at a repetition rate of 10 Hz (Newport Corporation,BV100 Beam Valve Driver). The pulse width is set to ≈ 200µs. The stag-nation pressure p0 can be regulated up to 33 bar. All our experiments weredone at this maximum pressure. The chamber is pumped by a vacuum pump(Edwards, model: E2M40) down to a pressure of 3·10−2 torr without a load.When the pulsed valve is operated, the pressure settles at pb ≈ 1 torr. Thishighlights the importance of working with a pulsed nozzle; the rotary pumpcould not handle the load in the case of a steady flow.774.2. Molecular beam sourcePulsednozzleDistance xZone of silenceMolecular gasPumpProbe𝒑𝟎, 𝑻𝟎𝒑𝒃Ԧ𝑥Ԧ𝑧Figure 4.5: Diagram of the molecular beam setup. The pump and probepulses are both focused onto the supersonic jet of rotationally cold molecules.The probe pulse is focused down to a smaller beam size to reduce spatialaveraging over the pump profile in the interaction region.Our Raman experiments are done at a small distance x from the nozzle,with the exact location being determined as a trade-off between colder rota-tional temperature and sufficiently high density. Raman spectroscopy relieson the induced birefrigence of the gas, which changes the probe polarization[141] (for a detailed explanation see Sec. 4.1). The Raman signal is propor-tional to the molecular density squared and benefits from a smaller distance.At the same time, we want to maximize the distance to reach lower initialtemperatures and not suffer under a collisional decay of the Raman signal.The ideal distance was found at x = 1.9 mm ∼ 4d. All these considerationswill be verified with experimental data in Sec. 4.2.3.The pump and probe beams both propagate through the molecular jet in~z-direction, see Fig. 4.5. We minimize detrimental effects of spatial averagingby making the probe (400 nm) profile significantly smaller than the pump(800 nm) profile. The FWHM beam diameters D are 20 µm and 60 µm,respectively. This way, the probe beam primarily samples the high-intensitycenter of the pump beam and the spatial averaging of the Raman signalover the laser profiles is reduced. At the same time, we also improve theconfinement in ~z-direction because of the jet. Due to a close proximity tothe nozzle, most molecules are present within the Rayleigh range of the laserbeams.In Gaussian beam optics [120], the Rayleigh range zR = piw20/λ is definedas the distance from the smallest beam waist (i.e. the focus) to the pointwhere the beam radius increased by a factor of√2. This is the point, where784.2. Molecular beam sourceNozzle: D=500 m,  p0=478.6psi, pr=1.0e+00torr, Mach Disc: xM=5.27cm shift = 0.9mm0 0.5 1 1.5 2 2.5 3Distance 𝑥 (mm)100101102103104Pressure 𝑝(torr)050100150200Temperature T(K)Figure 4.6: Simulated temperature T (blue solid line, left axis) and pressurep (red dashed line, right axis) as a function of the distance to the nozzle x,based on our molecular beam setup. Experimental values retrieved from N2measurements are indicated as red crosses (p) and dark-blue crosses (T ), thelight-blue crosses (T ) correspond to O2 measurements.the area of the beam cross-section has doubled. The beam radius w0 iswhere the Gaussian intensity is 1/e2 of its peak value. It can be derivedthat w0 = D/√2 ln2. In our experiment, the Rayleigh range of the probebeam is estimated at 2.3 mm and the one of the pump at 10.2 mm.SimulationWe simulate the expected temperature and pressure as a function of thedistance from the nozzle based on our experimental parameters: p0 = 33 bar(3.3 · 106 N/m2), pb = 1 torr (1.3 · 102 N/m2) and T0 = 293 K. At thesesettings we expect a supersonic flow, because the ratio of p0/pb ≈ 2.5 ·104  G lies well above the threshold value of G ≈ 1.89. We use Eq.4.1to calculate the Mach number at a distance x/d with d = 500 µm. Theempirical constants for the axisymmetric expansion of a diatomic moleculeare j = 1, C1 = 3.606, C2 = −1.742, C3 = 0.9226 and C4 = −0.2069 [115].Next, we get the temperature from Eq. 4.3 and subsequently the pressurefrom Eq.4.2.The result is plotted in Fig. 4.6 with temperature as the solid blue line(left axis) and pressure as the dashed red line (right axis, log scale). Wealso retrieved temperatures and pressures experimentally from Raman mea-794.2. Molecular beam sourcesurements at different nozzle distances; the procedure is discussed in thenext section 4.2.3. The experimental values are added to the graph withcrosses. Note, that only the relative distance between different measure-ments are known with certainty. The absolute distance x from the nozzleis approximated by shifting the pressure values from the measurements inN2 (red crosses) to match the simulation. Owing to the retrieval procedure,the obtained pressures are more accurate than the temperatures. There isa discrepancy in the observed temperature which settles at a lower levelthan expected according to the simulation. The dark blue crosses stemfrom a measurement in N2, the light blue ones from O2. (We did not getpressure estimates from the oxygen measurements because of complicatedspin-rotation dynamics, more details in the next section.) There are twoobvious explanations for inaccuracies: First, our setup works with a pulsednozzle, whereas the simplistic calculations are based on a continuous jet.Other approximations, that might cause deviations, assume an ideal gasand a frictionless flow without any dissipative effects. Second, the Rayleighlength of the probe pulse zR ∼ 2.3 mm is of a similar order than the width ofthe molecular jet, sketched in Fig 4.5. As a consequence, our Raman spectrayield an average over the spatial temperature and density distributions in~z-direction, see Ref. [167].For distances of x > 3 mm the local densities become too low for sensitiveRaman spectroscopy. All our measurements are done at x = 1.9 mm, farbefore the Mach disk location at xM ≈ 53 mm, calculated from Eq. 4.4.At last, we point out some general relations: The residual pressure pbinside the chamber determines the Mach disk location, but does not affectthe cooling rate. The stagnation pressure p0 has no influence on the obtainedtemperature T , but the final pressure depends on it p ∝ p0. The nozzlediameter d determines the rate of the cooling process.4.2.3 Beam characterizationIt is possible to map the temperature and density distribution of a supersonicjet expansion using Raman spectroscopy. Rather than creating a completetwo-dimensional map as done in Ref. [167], we look only along the centeraxis of the jet in the region of interest.PressureWe use a single high-intensity pump pulse to create strong rotational co-herences that we subsequently detect with a weak probe pulse via Raman804.2. Molecular beam sourcespectroscopy (Sec. 4.1). Figure 4.7 shows the mean Raman intensity fornitrogen (a) and a state-resolved spectrum for oxygen (b) as a function oftime. Plotted are the results for two relative distances between the nozzleorifice and the intersecting laser beams, a smaller distance of x = 0.9 mmand a larger one of x = 1.9 mm. Our goal is to use the decay of the Ramansignal due to collisions to estimate the pressure. This is done by compar-ing the decay times to the decay rates measured in a gas cell at differentpressures.In nitrogen, all individual rotational states behave identically. Thus, weplot their integrated signal in panel (a) which decays exponentially withtime. The top solid line (x = 1.9 mm) and the bottom solid line (x =0.9 mm) are both fitted with an exponential function f(t) = a · exp(−t/τ),shown as the red-dotted lines. The decay times are found to be τ = (1.13±0.09) ns and (196 ± 6) ps, respectively with the error given by the 95%confidence bounds of the fit.The same measurements done in oxygen at the distance x = 0.9 mm(b1) and x = 1.9 mm (b2) reveal much more complicated dynamics. Thecollisional decoherence is superimposed on spin-rotational oscillations. Para-magnetic oxygen has a non-zero spin in the electronic ground state (S = 1)which is coupled to the molecular rotation (nuclear rotation quantum num-ber N). This coupling splits each rotational level into three levels with thetotal angular momenta (J = N,N ± 1). It also means that the N → N + 2Raman process actually consists of six separate transitions [79]. Owing tothe spin-rotation coupling, we observe beat notes that depend on the ro-tational quantum number N . We plot the state-dependent dynamics forN = 1 (blue dashed), N = 3 (red dotted) and N = 5 (black solid). Thespin-rotational dynamics of oxygen has been investigated experimentally11 [116, 117]. For quantum numbers N > 5 the dynamics is dominated bybranches that lead to the oscillation periods on the order of 600 ps, which ismuch longer than any time scale that we are concerned with. We are onlyaffected by the spin-rotation coupling at low quantum numbers.In this thesis, we will disregard the effect of spin-rotation coupling, sincewe are operating on time scales of 250 ps or less and since we are usually notinterested in the very low quantum numbers. For simplicity, we will alwayslabel the rotational quantum number with J , for both nitrogen and oxygen.We calculate the collisional decay times of nitrogen in the jet for a series11 The fast beating at a period of ∼ 17 ps belongs to the two lowest lines of the R-branch(∆J = 1) and has been observed before [116].814.2. Molecular beam source0 100 200 300 40001Raman intensity(arb. units)aTime (ps)0 100 200 300 4000101Raman intensity(arb. units)b1b2Figure 4.7: (a) Rotational spectroscopy of nitrogen: Mean Raman intensityfor nozzle distances of x = 1.9 mm (top solid line) and x = 0.9 mm (bottomsolid line) and their respective exponential fits (red dotted lines). (b) State-resolved spectroscopy in oxygen for J = 1 (blue dashed), J = 3 (red dotted)and J = 5 (black solid) for nozzle distances of x = 0.9 mm (b1) and x =1.9 mm (b2). Maximum signals are normalized to unity.824.2. Molecular beam sourceCollisional decay in cell  vs Jet0 100 200 300 400 500 600 700 800Pressure (torr)00.511.52Decay time  (ns)0.90 mm1.15 mm1.40 mm1.65 mm1.90 mmNozzle distanceFigure 4.8: Collisional decay times as a function of gas pressure, recordedvia Raman spectroscopy in a cell of nitrogen. Horizontal dashed lines indi-cate the decay rates from molecular jet experiments at various nozzle dis-tances. Their corresponding pressures can be interpolated.of different nozzle distances (1) 0.9 mm, (2) 1.15 mm, (3) 1.4 mm, (4)1.65 mm, (5) 1.9 mm by fitting the Raman signal as a function of timewith an exponential, analogous to the description above and Fig. 4.7(a).In order to estimate the corresponding pressure values, we use Fig. 4.8, agraph that shows the relation of pressure versus collisional decay in a cellfilled with nitrogen. Based on this set of calibrated data points, we caninterpolate the pressure in the jet. The results are (1) 210 torr at a decaytime of 0.20 ns, (2) 110 torr at 0.38 ns, (3) 65 torr at 0.64 ns, (4) 39 torrat 1.01 ns, (5) 34 torr at 1.13 ns. No values exist below the pressure of∼ 30 torr because the collisional decay times become too long to reliablyfit an exponential function to our experimental data (maximum time delaywas 450 ps). Therefore, the procedure could not be applied to distances ofx > 2 mm. The interpolated pressure values are the ones that have beenadded to Fig. 4.6.TemperatureIn a similar fashion we use the Raman spectrum after a single pump pulse toretrieve the rotational temperature. However, this time we use a weak pumppulse with a kick strength of P  1, which creates weak coherences but834.2. Molecular beam sourcehardly changes the rotational state distribution. The shape of the Ramanspectrum then reflects the initial rotational distribution.Figure 4.9 demonstrates the narrowing of the spectrum with larger nozzledistances, which corresponds to rotational cooling of the molecules: nitrogenin (a) and oxygen in (b). The six lines from the broadest to the narrow-est distribution are measured with distances of x = 0.9, 1.15, 1.4, 1.65, 1.9and 2.15 mm. As a representative example we pick the Raman spectrumat 1.9 mm, shown in the insets. The best matched rotational distribu-tion is marked with crosses and serves to estimate the temperature. Thesetheoretical Raman spectra were calculated numerically based on a thermalBoltzmann distribution of states PJ = (2J + 1) exp(−EJ/kBT ). Details ofour numerical simulation are discussed in section 2.3.6. The rotational tem-peratures for all nozzle distances have been added to Fig. 4.6. We note, thatthe Raman spectra, in particular the ones close to the nozzle, deviate fromthe Boltzmann distribution, leading to large errorbars. One possible expla-nation lies in the averaging over the temperature profile of the molecular jetalong the ~z-direction, the direction in which the laser beams propagate.In conclusion, we set our distance to x = 1.9 mm in all experiments.Here, the pressure is estimated to be around (35 ± 5) torr, sufficient forsensitive measurements, and the rotational temperature is approximated tobe (23± 7) K, where the most populated state is J = 2 in nitrogen or J = 3in oxygen.Our results are in agreement with the ones found in other Raman spec-troscopy methods implemented in beams of N2 molecules, e.g. coherentanti-stokes Raman spectroscopy (CARS) [26] or spontaneous Raman scat-tering [137].844.2. Molecular beam source11.4.2016 Oxygen (TL kick)15.4.2016 Nitrogen (TL kick)1 3 5 7 9 11Rotational quantum number J01Raman intensity(arb. units)1 3 5 7 92 4 602 4 6 8001a bFigure 4.9: Rotational Raman spectra of (a) nitrogen and (b) oxygen, mea-sured in a molecular jet at various distances from the nozzle: x = 0.9, 1.15,1.4, 1.65, 1.9 to 2.15 mm (from broadest to narrowest distribution). Theinsets compare the experimental spectrum recorded at a distance of 1.9 mm(solid lines), with the simulated distribution at 23 K (crosses). The maxi-mum signals are normalized to unity.85Chapter 5Resonant excitation ofmolecular rotationThe periodically kicked rotor exhibits rich dynamics. In the classical limit itis described by the “standard map” (Sec. 2.1.1), which is known as one of thesimplest representations of chaotic behaviour. Here, we study the dynamicsof periodically kicked linear molecules - a system of quantum rotors. Owingto the discreteness of rotational energies, we need to distinguish betweentwo regimes: the one of periodic excitation on quantum resonances and theone of periodic excitation away from quantum resonances. In this chapter 5,we exclusively investigate various phenomena that are all a consequenceof quantum resonances. The following chapter 6 will treat the other case,where the period of the pulse train is chosen to be incommensurable withall quantum resonances of the system.We established the “resonance map” (Sec. 2.3.8) as a helpful means tostudy the QKR. In Sec. 5.1, we verify this map experimentally by expos-ing an ensemble of room temperature molecules to a periodic sequence ofpulses. This knowledge is used to set the train period to different resonancesin order to demonstrate different effects. To achieve cleaner results, thesemeasurements are typically done in rotationally cold molecules. When theperiod is chosen to coincide with a single rotational resonance, in Sec. 5.2,we observe Rabi oscillations between the constituent rotational states. Weobserve that the amplitude and period of the oscillations depend on the de-tuning from the resonance. If instead of a single rotational resonance theperiod matches the full quantum resonance, we observe another type of os-cillations. In Sec. 5.3, we investigate how the angular momentum of therotor oscillates in a fashion that has been connected to Bloch oscillations insolid state physics [60, 64, 62]. Again, we can manipulate the dynamics byadjusting the detuning from the quantum resonance.In Sec. 5.4, our goal is the excitation of broad rotational wave packets,which is commonly done via periodic excitation on quantum resonance [41,192]. We evaluate the efficiency of this process and assess the limitations fora non-rigid rotor. Despite long sequences with extremely high cumulative865.1. Demonstration of the resonance mapkick strengths, we fail to populate rotational levels that are significantlyabove thermally excited states. We propose a scheme of a non-periodicpulse train that makes use of fractional resonances to extend the reach ofthe impulsive excitation. Section 5.5 concludes the topic of impulsive multi-pulse excitation of molecular rotation using fractional and full quantumresonances.5.1 Demonstration of the resonance mapIn this first experimental demonstration, we investigate the rotational co-herences created by a periodic pulse sequence. We excite a room tempera-ture ensemble of oxygen molecules with a large number of rotational statesthermally populated. This simplifies the observation of many fractionalresonances and the study of their ensueing dynamics. We use rotationalRaman spectroscopy in a gas cell filled with oxygen, as described earlier inFig. 4.1(a). The exitation is done via relatively weak femtosecond pulses,in sequences of up to 15 pulses. These sequences were produced by a pulseshaper without interferometric splitting (details of the pulse train generationare given in chapter 3).In Fig. 5.1, two-dimensional plots show the observed Raman peaks (colorcoded from dark to bright red) as a function of the pulse train period T .Each Raman spectrum is plotted as a function of wavelength (left verticalaxis) or converted to J-numbers (right vertical axis). The apparent patternof peaks can be interpreted via the resonance map (Sec. 2.3.8). Even J ’s aremissing due to the oxygen nuclear spin statistics. We set all individual pulsesto weak energies, where the accumulated kick strength is PN = P ·N . 1,so as to look at the dynamics in a perturbative regime.In Fig. 5.1(a), the period of five pulses is varied from 10.8 to 12.6 ps.Oxygen 16O2 has a revival time of Trev = 11.67ps. If we choose the periodto match the revival time T = Trev, then we excite the molecules with atrain tuned to the quantum resonance. The result is the generation of abroad rotational wave packet, when all J-states are excited simultaneously.We will investigate this scenario in more detail in Sec. 5.4. The Ramanspectrogram matches well with the simulated resonance map, indicated bylight blue crosses. Whenever the train period coincides with a fractionalresonance, we excite a coherence between the states |J〉 and |J + 2〉. At allother time periods the rotational coherences are suppressed. This supportsthe provided interpretation of the coherent accumulation in the rotationalmulti-pulse excitation.875.1. Demonstration of the resonance mapPeriod T (ps)1 1.5 2 2.5 3 3.5Wavelength (nm)401.2401.6402.0402.45101520Rotational state  JPeriod T (ps)10.8 11 11.2 11.4 11.6 11.8 12 12.2 12.4Wavelength (nm)401.2401.6402.0402.45101520Rotational state  Ja)b)10 Intensity (arb. units)Figure 5.1: State and time resolved Raman spectrogram of room tempera-ture 16O2 after the excitation with a periodic pulse train. The train period Tis scanned around the revival time for a sequence of 5 pulses (a) and arounda quarter of the revival time for a sequence of 15 pulses (b). Light bluecrosses indicate the time moments when a coherent wave packet consistingof two rotational states, |J〉 and |J + 2〉, accumulates a phase of pi.885.1. Demonstration of the resonance mapScanning the period from 0.7 to 3.5 ps for a sequence of 15 pulses inFig. 5.1(b) covers the dynamics around the quarter revival T ≈ 2.92ps.12 Inthis range, no period exists when all J-states can be excited simultaneously.We point out that we can virtually suppress any coherence completely bychoosing a period around 3 ps. This period overlaps with the point in timewhere the molecules are anti-aligned and thus leads to perfectly destructiveinterference. Read more about the revival dynamics and molecular align-ment in the introduction, Sec. Excitation of single coherencesIf the chosen period of a pulse train matches a fractional resonance TJ , thepopulation of this state J will be transferred to the state J + 2. At thesame time, the coherence ρJ,J+2 will grow in a stepwise fashion, which wemeasure as the intensity of the rotational Raman signal. At weak energies,coherences can only be created at thermally populated states. For demon-stration purposes, we select seven different periods T1, marked with solidvertical lines in the two-dimensional Raman spectrogram of Fig. 5.2. Atthese times, the train excites one single coherence, J = 5, 7, 9, 11, 13, 15 or17 at the fractional resonances of T1/Trev equal to2/13,3/17,4/21,1/5,6/29,7/33or 8/37, respectively. In the case of oxygen this corresponds to T1 = 1.80 psfor J = 5 (purple), T1 = 2.06 ps for J = 7 (blue), T1 = 2.22 ps for J = 9(light blue), T1 = 2.33 ps for J = 11 (turquoise), T1 = 2.41 ps for J = 13(yellow), T1 = 2.48 ps for J = 15 (orange) and T1 = 2.52 ps for J = 17(red).Figure 5.3(a) plots the normalized Raman spectra after these periodic se-quences of 15 pulses. The respective target states clearly show the strongestcoherences. Minor signal can be seen at other J values due to the finitepulse duration of 130 fs leading to a partial overlap with other quantumresonances.The concept of accumulating coherence over several pulses is not re-stricted to periodic trains. Building a sequence that alternates between twodifferent periods T1 and T2 both of which are resonant with the same targetstate J will show the same effect. Furthermore, we expect that the selec-tivity can be enhanced, since the overlap of the finite width pulses withother fractional resonances may be different for T1 and T2. We build three12Both sequences utilized in Fig. 5.1(a) & (b) were created with the pulse shaper,without interferometric splitting, in order to simplify the setup. A pulse train of 5 pulseswith a period of 12.6 ps spans a duration of 50 ps, reaching the pulse shaper limit. Apulse train of 15 pulses with a period of 3.5 ps spans a similar duration of 49 ps.895.1. Demonstration of the resonance mapPeriod T  (ps)1.2 1.4 1.6 1.8 2 2.2 2.4 2.6Rotational state J468101214161820Wavelength (nm)401.2401.6402402.4Figure 5.2: Raman spectrogram of room temperature oxygen 16O2 afterthe excitation with a periodic pulse train of 15 pulses (same color map asFig. 5.1). Highlighted are selected periods to excite single coherences.such trains, for J = 5, 9 or 13 at the fractional resonances of T1/Trev andT2/Trev equal to2/13 and3/13,4/21 and2/21, or6/29 and4/29, respectively.The newly added T2 timings are marked as dotted vertical lines in Fig. 5.2.The observed Raman spectra in Fig. 5.3(b-d) (dashed black lines) prove theincreased selectivity compared to the previous results (solid coloured lines)taken from (a). The measurements were done with the following periods:(b) T1 = 1.80 ps, and T2 = 2.69 ps, (c) T1 = 2.22 ps and T2 = 1.11 ps, (d)T1 = 2.41 ps and T2 = 1.61 ps.A number of our studies required periodic sequences of high intensities(P > 1). Once the accumulated kick strength of the pulse train is sufficientto transfer more than 50% of the population from the initial state J to thestate J + 2, the maximum coherence, or the maximum Raman signal, isreached. In the case of even stronger pulses, we observe oscillations betweenthe mentioned states, similar to Rabi oscillations in a two-level system. Thisphenomenon is discussed in Sec.5.2.When we investigate the phenomenon of dynamical localization in chap-ter 6, we examine the angular momentum distribution after the excitationwith a high-intensity pulse train, whose period is chosen off-resonance. How-ever, each pulse of the train has a finite duration. Thus, the choice of thetrain period might affect the shape of the distribution, because of the ‘par-tial’ overlap of pulses with particular quantum resonances. Peaks in thespectra at certain states J can often be explained by the proximity of a corre-sponding fractional resonance. For that reason, the simple non-perturbative905.1. Demonstration of the resonance map00. (arb. units)1 5 9 13 17b c d1 5 9 13 17Rotational state J1 5 9 13 171 3 5 7 9 11 13 15 17 1900. (arb. units)a Rotational state JFigure 5.3: Normalized Raman spectrum after the excitation by a sequenceof 15 pulses. (a) The seven coloured lines are obtained with seven differentperiodic trains. The periods T1 are chosen to each excite only a single co-herence ρJ,J+2. (b-d) The black-dashed line shows the improved selectivityusing a pulse train with two alternating periods T1 and T2 both optimizedto excite the same single coherence ρJ,J+2 for (b) J = 5, (c) J = 9 and (d)J = 13. The coloured lines are the spectra after the strictly periodic trainwith T = T1, as shown in (a). All periods are given in the text and markedin Fig. 5.2.915.2. Rabi oscillations in molecular rotationpicture of the resonance map is useful in the interpretation of the results.5.2 Rabi oscillations in molecular rotationIn this section, we demonstrate robust and selective population transfer be-tween two isolated rotational states of the QKR. If the goal is to maximizethe population transfer in a simple two-level system, one distinguishes be-tween two approaches: A single pulse that executes half a cycle of a Rabioscillation (the so-called pi-pulse) will invert the population [44, 29], howeverthis method is generally very sensitive to the pulse parameters. A robust al-ternative is an adiabatic technique [176] which excels in complete populationtransfer.Many adiabatic schemes lose their appeal in multi-level systems whenthey become sensitive to exact frequency chirps or pulse intensities [159, 30,113]. In a series of theoretical works, Shapiro et.al. [154, 156, 155] studiedpopulation transfer by a piecewise adiabatic passage and a robust populationtransfer was demonstrated between two quantum states [194]. The piecewiseprocedure is transferable to selective excitation in more complex quantumsystems. The idea of using pulse sequences to optimize population transferhas also been applied to many non-adiabatic schemes, e.g. Refs. [186, 132],or has been found as a solution via a genetic search algorithm [168].We present a technique that uses a pulse train of ultra-short pulses toaddress two-states in a multi-level system. We exploit the fact that a pulsetrain of broadband ultra-short pulses is a comb in the frequency domain,which enables selective excitation, just like with narrowband continuous-wave lasers [171]. The selectivity of a comb is combined with the accumula-tive effect of a pulse sequence. More specifically, we choose the pulse trainperiod to pre-select an effective two-level system of two rotational states,| J 〉 and | J + 2 〉. In this non-adiabatic approach the number of pulses andtheir strength can be used to transfer the system into any arbitrary super-position of two states. We demonstrate the effect of Rabi oscillations andstudy their dependence on the pulse train parameters. The scheme relies ona periodic sequence of many femtosecond pulses and has not been reportedyet.In Sec. 5.2.1 we describe the theory of population transfer in a two-levelsystem and apply it to the QKR. In Sec. 5.2.2 we review the experimentaldetails, before presenting the results in Sec. 5.2.3. The observation of Rabioscillations was used to calibrate the kick strength of our laser pulses in all925.2. Rabi oscillations in molecular rotationother experiments. Details of this method can be found in Sec. Theory and simulationAt first we describe the Rabi formalism in a real two-level system, whichis then applied to the case of a kicked molecule. We will then demonstratethe phenomenon via numerical simulations to clarify certain properties thatcannot be shown in the experiment.Two-level systemMonochromatic light of frequency ω is used to excite a transition in a two-level system with the states | a 〉 and | b 〉, whose energy difference is givenby Eb − Ea = ~ω0.| 𝑎 〉| 𝑏 〉𝜔𝛥𝜔0Figure 5.4: Energy diagram of a two-level system.I.I. Rabi showed that the population in a two-level system will oscillatebetween both states under the influence of a constant electric field. Thepotential energy of the interaction is described as the product of the dipolemoment µ of the atomic transition and E(t) the electric field amplitude,V (t) = µE(t). We define the Rabi-frequency asΩ(t) =µE(t)~, (5.1)and a generalized Rabi frequency Ω′ for non-resonant fields with a detuningof ∆ = ω0 − ωΩ′ =√Ω2 + ∆2 . (5.2)The time-dependent probability to occupy either of the two states is [44, 29]|cb(t)|2 = ΩΩ′sin2(Ω′2t)(5.3)|ca(t)|2 = 1− |cb(t)|2 . (5.4)935.2. Rabi oscillations in molecular rotationwith the total wave function |Ψ(t) 〉 = ca(t)e−iEat/~ | a 〉+ cb(t)e−iEbt/~ | b 〉.The population oscillates with the generalized Rabi frequency. For resonantlight, the maximum transfer is 100 %. Once we detune from resonance theefficiency drops - the amplitude of Rabi oscillations decreases while the oscil-lation frequency increases. For time-dependent electric fields it is convenientto introduce the pulse area A =∫dt Ω′. A complete transfer (at ∆ = 0)is achieved when the pulse area is an odd multiple of pi, referred to as a“pi-pulse”.Effective two-level system in the periodically kicked rotorThe QKR is a multi-level system with many rotational levels. By takinga long pulse sequence and setting the period to match a specific fractionalquantum resonance, the QKR will behave as an effective two-level system.According to the resonance map (Sec. 2.3.8) we can choose a periodwhich is in resonance with a superposition of two states | J 〉, | J + 2 〉. Theexcitation of all other states in the vicinity is efficiently suppressed dueto the repeated kicking with a period that is incommensurable with theirrotational periods. The effective wave function reduces to the form |Ψ(t) 〉 =cJ(t)e−iEJ t/~ | J 〉+ cJ+2(t)e−iEJ+2t/~ | J + 2 〉. We have isolated a two-levelsystem that is coupled with a two-photon transition within a multi-levelsystem. The number of kicks N , which are spaced out with a period T ,represents a unit of time. In the experiment we will demonstrate how thepopulation oscillates between the states | J 〉 and | J + 2 〉 as a function ofN .The potential energy of the QKR has been introduced in Eq. 2.8 asV (θ, t) = −∆α4 E2(t) cos2 θ. Similarly to Eq. 5.1, the Rabi-frequency of theQKR behaves according toΩQKR(t) ∝ ∆α E2(t)4~. (5.5)We discuss the case where the period of the pulse train is chosen on resonancefirst. As we learned in the last section, Rabi oscillations depend on the pulsearea A =∫dt Ω(t), which is re-evaluated for the new system by integratingover the duration of the pulse trainAQKR =∫ NT0dt ΩQKR ∝ ∆α4~∫ NT0E2(t)dt = PN . (5.6)Here, we used the definition of the kick strength (Eq. 2.11). On resonance,we expect Rabi oscillations with a frequency that is proportional to the kickstrength and the number of kicks; complete population transfer is possible.945.2. Rabi oscillations in molecular rotationDetuning from the resonance results in a decrease of the oscillation am-plitude and a simultaneous increase of the oscillation frequency, which is bestunderstood in the frequency domain, where the pulse train is representedby a frequency comb. Changing the pulse train period is equivalent to mod-ifying the comb spacing. Thus, it is obvious that a detuning of the trainperiod from a fractional quantum resonance is nothing else but a detuningof the comb teeth from the resonant two-photon frequency.We stress two points: (1) This discussion is only valid for relatively weakpulses (P . 1), when a single pulse couples only nearest neighbour states.We will show that for larger kick strength values population gets lost to otherrotational states. (2) As a consequence of the degenerate M -sublevels, whichinteract slightly differently with the laser pulses, our system is not a truetwo-level system, but rather a superposition of several two-level systems.(3) The shift of the levels due to the AC Stark effect cannot be neglectedand will become noticeable in an off-resonance excitation.Rotational population versus coherenceBy solving the Schro¨dinger equation, we obtain the complex amplitudes cMJused to calculate the rotational population PJ =∑M 〈|cMJ |2〉J′,M′ and themodulus squared of the coherences C(2)J =∑M 〈|cM∗J cMJ+2|2〉J′,M′ . Both ex-pressions include the summation over the degenerate M -sublevels and thethermal average over initially populated states | J ′,M ′ 〉. Check Sec. 2.3.6for details.Figure 5.5 shows the intrinsic connection between both quantities, PJand C(2)J , for the example of nitrogen molecules exposed to a sequence of60 weak pulses at P = 0.2 per pulse. To simplify the picture, we startwith the molecules initially being in the rotational ground state at 0 K.Due to a non-zero nuclear spin of 14N2, there are two spin isomers with thecorresponding statistical weights of P0 =2/3 and P1 =1/3. Both parities arenot coupled and evolve independently; we look at even rotational states.The resonance condition for the quantum state | J,M 〉 = | 0, 0 〉 is metexactly when we choose a train period of T = 0.3354 Trev, marked with asolid line in the resonance map, Fig. 5.6. Owing to the AC Stark-shifts of thelevels, it deviates from the classical prediction of 1/3 by 0.62% (more detailsfollow in the next subsection). The excitation dynamics of this idealisticcase is plotted in Fig. 5.5(a). We achieve a complete population transferbetween the states J = 0 and 2, with the respective populations P0 = |c00|2(dashed black line) and P2 = |c02|2 (dotted black line). The strength of the955.2. Rabi oscillations in molecular rotation0 10 20 30 40 50 60Number of kicks N00.500.500.500.5Population00. squared of coherenceabcdFigure 5.5: Excitation of N2 molecules, initially in J = 0, with a sequence of60 pulses at P = 0.2 per pulse. The population (left axis) oscillates betweenJ = 0 (dashed line) and J = 2 (dotted line). The quantity of interest is thecorresponding modulus squared of the coherence (red solid line, right axis).Shown are different detunings from the period T/Trev = 0.3354 with (a) 0%,(b) 1%, (c) 2% and (d) 4%.0 0.2 0.4 0.6 0.8 10510Pulse train period   ( T/Trev )Quantum numberJFigure 5.6: Resonance map: A pulse train with period T/Trev =1/3 (solidline) induces Rabi oscillations between J = 0 and 2. For T/Trev =1/7(dashed line) or 6/7 (dotted line) they are between J = 2 and 4.965.2. Rabi oscillations in molecular rotationkicks affects the Rabi period, i.e. the number of pulses required to achieveone full cycle. The coherence and its modulus squared C(2)0 = |c0∗0 c02|2 (redsolid line) is maximized when both states are equally populated and is zerowhen one of the states has no population.Figure 5.5(b-d) explores the oscillation dynamics for steadily increasingdetuning from the particular resonance by 1%, 2% and 4%, respectively.As in the case of a two-level system, we expect faster oscillations with asmaller amplitude for growing detunings. We observe the same behaviourin kicked molecules. The transfer of population T decreases from (a) 100%,(b) 78%, (c) 47% to (d) 18%, while the oscillation frequency increases. Theobservable C(2)0 contains the same information in a less obvious way: (a)On resonance (T = 100%) we see regular oscillations with half the periodof the population oscillations; (b) Off resonance (T > 50%) a double-peakoscillation is visible as two maxima merge, the oscillation amplitude remainsunchanged; (c,d) Further off resonance (T < 50%) the regular oscillationshave the same period as the population oscillations, but with a decreasingamplitude.In the experiment, we do not directly detect rotational populations PJ .We use rotational Raman spectroscopy (Sec. 4.1) that yields a state-resolvedRaman spectrum whose intensity is proportional to C(2)J . We fit our calcu-lations to match the measured Raman intensities IJ = b ·C(2)J with a singlefitting parameter b.Manifold of M-substatesIt is impossible to separate individual M channels experimentally. The ob-served Raman spectra are always an incoherent sum over all initially pop-ulated | J ′,M ′ 〉 states. In Fig. 5.7 we reveal the individual dynamics nu-merically for N2 at a realistic temperature of 25 K. The total population inthe initial | 2,M ′ 〉 manifold is 0.37, of which one fifth is in each degenerateM ′ = 0,±1,±2 substate. We excite these molecular states with a sequenceof 100 weak pulses of P = 0.2 with a period of T = 1/7 Trev (dashed linein Fig. 5.6) and compare the evolution of the population P2 (dashed blackline), the population P4 (dotted black line) and the modulus squared of theircoherence C(2)2 (red solid line).The dynamics differ significantly for each individual initial state | J ′,M ′ 〉under identical excitation conditions. In Fig. 5.7 we compare the dynamics ofmolecules initially occupying (a1) | 2, 0 〉, (b1) | 2, 1 〉 and (c1) | 2, 2 〉. Note,that these three plots describe the entire dynamics since each | 2,±M ′ 〉 in-975.2. Rabi oscillations in molecular rotation00.0500.0100.0500.0500.0100.01Modulus squared of coherencePopulationNumber of kicks N0 20 40 60 80 100a1b1c1a2b2c2|𝟐, 𝟎〉|𝟐, 𝟏〉|𝟐, 𝟐〉|𝑱 = 𝟐〉0 20 40 60 80 10000.200.05d1Number of kicks N0 20 40 60 80 100Figure 5.7: Excitation of N2 molecules at 25 K with a sequence of 100 pulsesat P = 0.2 per pulse. Plotted are the populations (left axis) oscillatingbetween J = 2 (dashed line) and J = 4 (dotted line), and the modulussquared of their coherence (red solid line, right axis). In column (1), weset the period to T = 1/7 Trev and resolve the dynamics of different initialstates | J ′,M ′ 〉: (a) | 2, 0 〉, (b) | 2, 1 〉, (c) | 2, 2 〉. (d) is the average of allM ′-substates in | 2,M ′ 〉. In column (2), we adjust the period individuallyfor each substate to regain complete population transfer, achieved at thedetunings of (a) −0.04%, (b) +0.17% and (c) +0.83%.985.2. Rabi oscillations in molecular rotationteracts with the pulses in the same way. We emphasize two key signatures,that were both discussed in Sec. 2.3.6 under “M -degeneracy”. First, theeffective kick strength decreases with higher magnetic quantum numbers- more pulses are needed to complete one Rabi cycle. Second, the reso-nance condition changes due to the dynamic Stark shift. We quantify thedifference with the max population transfer T , which is a measure of thedetuning from the resonance. We obtain efficiencies of T = 100%, 99%and 61%,respectively. The graph (d1) is the average over all five M ′ sub-states starting in the | 2,M ′ 〉 manifold. Since there is no unique period thatsatisfies the resonance condition for all participating states, we cannot ex-pect pure Rabi oscillations. Instead a quasi-periodic pattern with reducedcontrast emerges 13.Owing to the quadratic Stark-shift, which is lifting the M -degeneracy,one obtains a resonance condition that depends on the magnetic quantumnumber. The resonant period is shifted proportionally to the kick strengthP . It is possible to compensate the phase shift by detuning the train periodaway from T = 1/7 Trev. This is done individually for each | 2,M ′ 〉 regaininga 100% population transfer. We empirically retrieve the following detunings(a2) −0.04% , (b2) +0.17% and (c2) +0.83%.5.2.2 ExperimentWe implement rotational Raman spectroscopy in nitrogen molecules cooledto a rotational temperature around 25 K via a supersonic expansion, asdescribed earlier in Fig. 4.1(b). Our pump is a periodic train of intermediatekick strength around P = 1 or less. We conducted measurements with twodifferent sequences, one of 29 pulses at a period of 1/7 Trev, and the otherone with 24 pulses at a period of 6/7 Trev. Both timings are marked onthe resonance map in Fig. 5.6. The revival time of 14N2 is Trev = 8.38 ps.Owing to the overall length of the pulse train, the generation of the firstsequence is done with only the pulse shaper, whereas the latter one requirestwo additional Michelson interferometers (details are found in chapter 3).The weak probe is a single narrowband pulse with a spectral width smallenough to resolve the individual rotational states of nitrogen.We deliberately choose nitrogen over oxygen as our sample. The phe-nomenon is independent of the type of molecule, but oxygen bears the disad-vantage of spin-rotation coupling (Sec. 4.2.3). In particular at low rotationalquantum numbers, where the resulting dephasing of the Raman signal is on13To reproduce the experimental Raman spectrum, we would need to include all otherJ ′-states that can couple to J = 2 or 4, as well.995.2. Rabi oscillations in molecular rotationa similar time scale as the length of the pulse train, the data interpretationwould be more complicated.5.2.3 Observation of Rabi oscillationsThis section summarizes our experimental investigations regarding Rabi os-cillations in a quantum rotational system. We present the dependence of thephenomenon on various pulse train parameters, i.e. the choice of a particularfractional resonance, the detuning from that resonance, the kick strength,the number of pulses and their bandwidth. The results are shown in oneof two ways: Either we plot the Raman intensity for a selected J-state asa function of the pump-probe delay, or we plot a two-dimensional Ramanspectrogram displaying the entire Raman spectrum along its second axis.The time delay is expressed in the number of pulses N that have interactedwith the molecules.Dependence on the detuningWe study the rotational excitation of nitrogen molecules after a pulse se-quence of 29 pulses with a period of 1/7 Trev. This fractional resonance iscommensurable with the dynamics of the (| 2 〉, | 4 〉) wave packet, see Fig. 5.6.It has been chosen because J = 2 is the most populated state (37%) at atemperature of 25 K. (The state J = 4 holds another 13% of the initialpopulation.) Figure 5.8 analyses the rotational dynamics for different de-tunings from the resonance, changing from −10% (top left) in 1% steps to+1% (bottom right). Plotted are the spectra between J = 0 and J = 6.The only state in the Raman spectrum with a substantial intensity IJ is theone at J = 2, which features multiple oscillations with a high contrast. Wepoint out that barely any population escapes to different rotational states,which otherwise would be seen as additional Raman peaks, e.g. at J = 4.Although 50% of the population is initially not in J = 2 or 4, the coherentaccumulation of a sequence of non-resonant pulses results in virtually noRaman signal away from I2.Looking at the oscillations, we make several key observations. Tuningslowly away from the resonance leads to the merging of two maxima, e.g. at−1%,−2%. The explanation has been given in the theory section. Tuningfar away from the resonance leads to faster oscillations and a simultaneousdrop of the amplitude. All spectrograms in Fig. 5.8 share the same intensityscale, the absolute amplitudes are therefore comparable. At the detuning of−10% we count three complete oscillations.1005.2. Rabi oscillations in molecular rotationNumber of Kicks NRotational quantum numberJ0 10 20 0 10 20 0 10 202424242424242424 -10% -9% -8%-7% -6% -5%-4% -3% -2%-1% 0% +1%(2)Figure 5.8: Rabi oscillations observed in the Raman spectrum of 14N2after the excitation with a periodic sequence of N = 29 pulses with a kickstrength of P = 0.33. Amplitude and period of the oscillations are controlledby adjusting the period of the train around the fractional resonance at T =1/7 Trev in twelve 1% steps from −10% (top left) to +1% (bottom right).Each spectrogram in this table compares numerical simulations at T = 25 K(top) with experimental results (bottom).1015.2. Rabi oscillations in molecular rotationThe experimental Raman spectra (bottom) are compared to the numer-ically calculated ones (top). The only free parameter in the simulation -the kick strength - has been adjusted to reproduce the observed oscilla-tions. We found the best match at P = 0.33. For negative detuning, bothgraphs are very similar within experimental noise. On resonance and towardpositive detuning the experimental contrast decreases with the number ofoscillations, which is not reproduced in the simulation. The origin of thisdiscrepancy lies in the finite duration of the pulses, and will be discussedlater in the respective subsection.Figure 5.9 provides a more quantitative comparison between the numer-ics and the experiment by showing the intensity I2 for selected periods. Thedetunings from the resonance are (a) 0%, (b) −3%, (c) −6% and (d) −9%.The graphs represent cross-sections of the two-dimensional Raman spectro-grams in Fig. 5.8 at J = 2. After determining the kick strength of P = 0.33and rescaling, the simulation (black dots) reproduces the experiment (solidred line) with an error of (a) 44%, (b) 12%, (c) 9% and (d) 32%. We cal-culated the root mean square deviation (RMSD) found from the deviationsfor all integer values of N . The percentages are the RMSD values dividedby the mean experimental signal.The explanation of the non-zero minima of Rabi oscillations, e.g. inpanel (a), is the existence of five individual channels corresponding to thefive magnetic quantum numbers M = 0,±1,±2. As discussed earlier, inthe impulsive multi-pulse excitation, the resonance condition for each non-degenerate M -state is different. Our experiments provide a verification ofthat discussion in the theory section. At larger detunings from the fractionalquantum resonance, e.g. in panel (d), the relative differences in the individ-ual detunings between all M -channels become smaller. Thus, the contrastof the oscillation increases.Dependence on the kick strengthSo far, we have looked at Rabi oscillations at weak kicks of P  1. InFig. 5.10 we examine the dynamics due to a periodic pulse sequence withincreasingly higher pulse energies. At P = 0.6 (a) we resolve two oscilla-tions within 24 pulses. We provide a direct comparison between simulation(1) and experiment (2) with a reasonable agreement. At larger strengthsof P = 0.83 (b) and P = 1.1 (c) we observe faster oscillations but the cor-respondence to the simulation becomes increasingly worse. As the numberof pulses grows, the Raman intensity of I2 diminishes. Owing to the finite1025.2. Rabi oscillations in molecular rotationChange Detuning  (1D)(3)0 5 10 15 20 25Number of kicks N00.100.0400.0200.05Raman signal (J=2)   [arb. units]abcdFigure 5.9: Rabi oscillations in 14N2 observed in the Raman signal after theexcitation with a periodic sequence of N = 29 pulses with a kick strengthof P = 0.33. Amplitude and period of the oscillations are controlled bydetuning the period of the train by (a) 0%, (b) −3%, (c) −6% and (d) −9%with respect to the fractional resonance at T = 1/7 Trev. The intensity ofthe experimental result (solid red line) is normalized to fit the numericalsimulation (black dots, connected with a dashed line).1035.2. Rabi oscillations in molecular rotation(5)0 5 10 15 20246246246246246246Number of Kicks NRotational quantum numberJ4812481248120 5 10 15 20a1a2a3b1b2b3c1c2c3Change Energy• For low energy, trivial, need more kicks to get same result• For high energy, more deviations from simulation• For high energy, more Stark-shifting,more dephasing between m’s• More coupling to higher states, pop runs away• 2D (to see all J’s)• At high energies more goes into higher J’s• plot EXP &SIM • Compare with QR (I can populate MUCH higher J-states)• Use different RR, works as wellFigure 5.10: The effect of pulse energy on Rabi oscillations observed in theRaman spectrum of 14N2 after the excitation with a periodic sequence ofN = 24 pulses tuned to the fractional resonance at T = 6/7 Trev. The periodof the oscillations decreases with the kick strength from (a) P = 0.6, (b)P = 0.83 to (c) P = 1.1; shown are numerical simulations at T = 25 K (1)and experimental results (2). The same pulse trains with a period tuned tomatch the quantum resonance at T = Trev lead to a linear growth in angularmomentum; shown are experimental results (3).1045.2. Rabi oscillations in molecular rotationpulse duration, the coupling to other states is larger in the experiment thanin the idealistic calculation with δ-kicks. Consequently, we detect Ramanpeaks belonging to higher-J coherences, e.g. I4. Further investigations onthe subject of pulse duration are performed in the next section.We want to emphasize that Fig. 5.10 has been recorded for a pulse trainwhose period is set to T = 6/7 Trev, unlike the train with T =1/7 Trev inFig. 5.8. Both periods are tailored to the same coherent wave packet andexemplify the universality of the process.Finally, we put the strength of the pulse train in context. For all threescenarios in Fig. 5.10(a-c) we include an additional spectrogram (3) outlin-ing the linear growth of angular momentum when the period of an otherwiseidentical pulse train is moved from fractional quantum resonance to the fullquantum resonance T = Trev. The total energy in the train is sufficient toreach much higher rotational states. This excitation scenario is discussed indetail in Sec. 5.3.Effect of finite pulse durationThe finite duration of our 130 fs pulses manifests itself even at the excitationof such low rotational quantum numbers as studied here in the context ofRabi oscillations. We demonstrate consequences of a limited bandwidthby pointing out differences between the experimental results and the δ-kicksimulation.We excite nitrogen with a pulse train similar to those used before, i.e.29 pulses of medium strength P = 0.57, adjusted to the 1/7 fractional res-onance. We fine tune the period by −4% below this resonance, indicatedwith the dashed line in Fig. 5.11(1). Here, the period is as far detuned aspossible from the resonances of the next higher Raman peak I4. A secondpulse train is tuned to +4% above the resonance, marked by the dotted line,so as to get it closer to the resonance at J = 4.In Fig. 5.11(2) we discover that these small relative changes in the trainperiod do matter in the rotational excitation. Specifically, we look at theintensity of the Raman peaks I2 (blue lines) and I4 (red lines). According tothe simulation (dots, with dashed lines) the expected oscillatory behaviouris apparent for the Raman peak I2, whereas the peak I4 is of virtuallyzero intensity. This is true for both detunings, −4% (2a) and +4% (2b).Hence, the δ-kick excitation is very selective. Only in the proximity of aquantum resonance will the respective states be populated. Compared tothat are the experimental results (solid lines). We see oscillations in I2regardless of the detuning. However, the behaviour of I4 is sensitive to1055.2. Rabi oscillations in molecular rotation0 5 10 15 20 25Number of kicks N00.040.080.1200.010.020.03Raman signal (J=4)  [arb. units]0 5 10 15 20 25Number of kicks N00.040.080.1200.010.020.03Raman signal (J=2)   [arb. units]2a 2b0 0.1 0.20246Pulse train period   ( T/Trev )Quantum numberJ1Figure 5.11: (1) Resonance map, marked are the train periods matchedexactly with the 1/7 fractional resonance (solid line), and detuned from it by−4% (dashed line) and +4% (dotted line). (2) Compromising effect of thelaser bandwidth on Rabi oscillations in 14N2 seen in the Raman signal IJof the states J = 2 (blue lines, left axes) and J = 4 (red lines, right axes).A periodic sequence of N = 29 pulses with a kick strength of P = 0.57is detuned by (2a) −4% and (2b) +4% from the fractional resonance atT = 1/7 Trev. The experimentally measured intensity I2 (solid blue line) isrescaled to fit the simulation (blue dots, dashed line). The same rescalingis applied to the experimental intensity I4 (red solid line) and compared tothe simulation (red dots, dashed line).1065.3. Bloch oscillations in molecular rotationthe choice of the period: very little intensity is measured for the optimizeddetuning of −4% (2a) but a significantly higher intensity is seen for +4%(2b). Hence, the experiment with kicks of finite duration is more sensitiveto different quantum resonances, and the population “leaks” more easily toother rotational states.It is helpful to keep this conclusion in mind for all other QKR studies. Forexample, when we investigate the phenomenon of dynamical localization inchapter 6, it is important to stay away from all fractional resonances. Owingto the finite pulse width, one will always partially overlap some resonances,which in turn will affect the shape of the Raman spectrum. In order tominimize these effects, we will average over several pulse trains with differentperiods.5.2.4 Kick strength calibrationThe fact that we can identify multiple Rabi oscillations with high precisionis used to calibrate the kick strength of our pulses for all future experiments.We obtain an accurate measure of the kick strength in the interaction regionof the experiment, without relying on the (often inaccurate) measurementsof the pulse parameters, i.e. beam diameter, pulse energy or pulse duration.Regardless of the specific experimental requirements, in terms of thenumber of pulses and the period in a pulse sequence, we can always find asuitable fractional quantum resonance to observe Rabi oscillations nearby.We record oscillations for several different detuning from the chosen res-onance to create a compilation of plots similar to Fig. 5.8. The scans aredone at weak pulse intensities (P  1) where the population is well confinedbetween both target states. The actual kick strength in the interaction re-gion of the experiment is found by matching the simulation to the real data.Knowledge of the input energy of the pulse train, which we measure with anenergy meter prior to the vacuum chamber, suffices to extrapolate the finalkick strength at higher pulse energies.5.3 Bloch oscillations in molecular rotationThis section is dedicated to the impulsive excitation of molecules with se-quences of high-intensity, ultra-short laser pulses whose periods are tunedon full quantum resonance, i.e. around the rotational revival time of themolecule. Resonant excitation schemes have been studied before in thermalensembles [41, 192], demonstrating a growing rotational energy. The popu-lation of increasingly higher angular momentum states, however, is limited1075.3. Bloch oscillations in molecular rotationby the centrifugal distortion. In a series of theoretical works [60, 64, 62]Floss et.al. demonstrated that at a critical value of the angular momentumthe centrifugal distortion will lead to a “back-scattering” of population to-wards lower J ’s, representing Bloch oscillations in the molecular rotation.These were reported for the first time in 2015 [64], measured indirectly viamolecular alignment of room temperature nitrogen exposed to eight periodicpulses.We present a more detailed investigation of Bloch oscillations in a quan-tum rotational system. The results demonstrated here improve the previousstudy in several aspects: (1) The use of rotational Raman spectroscopy al-lows for a state-resolved detection. For the first time, Bloch oscillations aredirectly observed in the angular momentum space. (2) Our experiment withmolecules cooled to rotational temperatures below 30 K greatly decreasesthe width of the rotational wave packet, which increases the contrast ofBloch oscillations. (3) Significantly longer pulse sequences enable us to bet-ter capture the rotational dynamics. The disadvantage of our approach isa smaller spectral bandwidth of the laser pulses, which imposes a limita-tion for reaching higher rotational states. Under certain circumstances thisprevents the observation of Bloch oscillations.The phenomenon of Bloch oscillations in angular momentum is uniqueto real molecules. It does not appear in the case of a kicked rigid rotor. It isalso not present in related physical systems like the atom-optic realizationof a kicked rotor [121, 135]. Further, our experiment with real moleculesoffers great controllability over the oscillations. We demonstrate variableoscillation amplitudes and frequencies by detuning the pulse train periodfrom the quantum resonance.Bloch oscillations were first predicted in 1929 [25] to describe the elec-tron motion in a crystalline solid subject to a DC electric field. The effectis extremely hard to show in real lattices as even small defects will destroythe process. The first experimental realization was done only in 1992 in asemiconductor superlattice [51]. Bloch oscillations have since been shownin a few different systems, i.e. in optical lattices, where ultracold atoms aresubject to standing laser waves [17, 123, 93] or in periodic photonic struc-tures [122, 133, 148, 37, 170]. Periodically kicked molecules offer anotheropportunity to study this effect in a new light.The section has the following structure. First, we present the necessarytheoretical background. In Sec. 5.3.1 we talk about Bloch oscillations incrystalline solids. The example of a one-dimensional crystal serves to showthe analogy with the QKR system in Sec. 5.3.2. Section 5.3.3 gives the1085.3. Bloch oscillations in molecular rotationdetails of our numerical simulations and section 5.3.4 the details of ourexperimental studies. All the results are shown and analysed in Sec. Theory I: Bloch oscillations in crystalline solidsElectrons in a crystalline solid subject to an external DC electric field ex-hibit an oscillatory motion, first discovered by Bloch and Zener [25, 190].Although ’quasi-free’ electrons in a band are exposed to an external forcef = −eE , with the electric charge of an electron e and the electric fieldstrength E , the solid acts as an insulator. The electrons oscillate but no netcharge is carried through the crystal. This model only works in a perfectlyperiodic crystal without impurities.The electron wave function describes a wave packet, i.e. a superpositionof plane waves with the group velocity [162]v =1~∂E∂k. (5.7)An external force applied to the electrons in a crystal is equal to [162]f = ~∂k∂t= −eE . (5.8)We explain the mechanism using the tight-binding example of a latticein one dimension with only nearest neighbour interactions of Sec. 2.4.4.The quasi-momentum of the electron is k(t) = − eE~ t found by integratingEq. 5.8 with the boundary condition k(0) = 0. The dispersion relationE(k) = T −2W cos(ka) from Eq. 2.38 is used to calculate the group velocityand the position of the band electron by integrating v = ∂r/∂t with r(0) = 0v(t) = −2Wa~sin(aeE~t)(5.9)r(t) =2WeE[cos(aeE~t)− 1]. (5.10)The electron starts at the bottom of the band at k = 0 when we turn onthe electric field at t = 0. The electron responds by moving in negativer-direction. As k changes uniformly with time, the electron moves up theband E(k). At k = −pi/2a it has reached the maximum velocity, before thevelocity decreases again and reverses the sign at the Brillouin zone boundary−pi/a. At this time the electron has reached its farthest position in real spacebefore it is Bragg scattered in the opposite direction. The quasi-momentum1095.3. Bloch oscillations in molecular rotationkeeps linearly decreasing but due to the 2pi periodicity of the reciprocallattice, the k value changes to pi/a. The process repeats with the velocityand the position of the electron bounded. The electrons oscillate with aBloch frequency of ωB = aeE/~.5.3.2 Theory II: Bloch oscillations in a molecular rotorBehaviour similar to that of an electron in a periodic lattice subject to a DCelectric field reappears in the angular momentum space of the periodicallykicked quantum rotor. The Schro¨dinger equation governing the dynamics ofthe QKR is derived in appendix E based on Ref. [62]idcJ(n)dn= V (J) c(n)J −P4[c(n)J+2 + c(n)J−2]. (5.11)In this semi-classical model the discrete number of kicks N is approximatedwith a continuous dimensionless time n. The parameters cJ(n) are theamplitude coefficients for different eigenstates of the rotor wave function|Ψ(n) 〉 = ∑J c(n)J | J 〉 belonging to the ‘J-sites’ of a one-dimensional rota-tional lattice. The two terms on the right hand side are the on-site potentialV (J) and the kinetic term, which expresses the hopping between neighbour-ing sites (previously labelled T and W , respectively).For a periodic lattice, if V (J) is constant, we recover the tight-bindingmodel yielding extended Bloch states; the rotational population will ‘hop’over the entire lattice. This condition (here, V (J) = 0) is fulfilled only fora rigid rotor excited on quantum resonance, but not for realistic molecules,i.e. non-rigid rotors.Non-rigid rotor, resonant excitation: The period between the kicksis set to match the full quantum resonance T = Trev. The on-site potentialcalculated in appendix E.1 is approximately [62]V (J) ≈ −piDBJ2(J + 1)2 . (5.12)Compared to Eq. E.9, we neglect the small constant shift P/2. The gradi-ent of the potential energy acts as a force, where the spatial coordinate isrepresented by the J-number f = −dV (J)/dJ . To make a connection withthe solid state picture, we can also express the external force (Eq. 5.8) asf = dk/dn in dimensionless units [62]dkdn= −dV (J)dJ≈ 4piDBJ3 . (5.13)1105.3. Bloch oscillations in molecular rotationIn contrast to a uniform force, the effective force here becomes stronger withincreasing J-numbers.Rigid rotor, non-resonant excitation (δ 6= 0): Let us now assume aperiod T = (1 + δ)Trev with a detuning δ. If it is large enough to dominateover the centrifugal distortion effects, the latter one can be neglected. Theon-site potential calculated in appendix E.2 is [62]V (J) = piδJ(J + 1) , (5.14)resulting in an external forcedkdn= −dV (J)dJ≈ −2piδJ . (5.15)The second quantity of interest is the group velocity (Eq. 5.7) in dimen-sionless units v = dE(k)/dk, which in the lattice frame is v = dJ/dn. Indirect comparison to the solid state case [E(k) = T−2W cos(ka) in Eq. 2.38],the energy dispersion relation becomesE(k) = −P2cos(2k) , (5.16)after substituting T = 0 as the on-site energy14 andW = P/4 as the strengthof the hopping and a = 2 for the rotational lattice spacing. The groupvelocity of the rotational wave packet is thereforedJdn=dE(k)dk= P sin(2k) , (5.17)displaying the oscillatory motion, equivalent to Bloch oscillations.The two coupled equations for dJ/dn and dk/dn are equations of motionwith the conjugate variables J and k. Floss et.al. showed the relation k =−θ between quasi-momentum and polar angle [62]. The same undulatingbehaviour manifests itself in the position on the angular momentum latticeJ(n), calculated by solving the two coupled equations. The turning pointsof Bloch oscillations are the edge of the lattice 15 and at the “Bloch wall”14Strictly speaking, this assumption is only valid on resonance at δ = 0.15The edge of the lattice depends on the M -substate and the parity, but cannot liebelow J = |M |.1115.3. Bloch oscillations in molecular rotationJB, which has been derived semi-classically in Ref. [62]Non-rigid rotor(δ = 0) : JB =4√J40 +BDP2≈ 4√BDP2Rigid rotor(δ 6= 0) : JB =√J20 +Ppi|δ| ≈√Ppi|δ| ,(5.18)where J0 is the initial angular momentum. The approximate expressions arevalid if one starts at cold temperatures, close to the rotational ground state16.We interprete the rotational analogue of Bloch oscillations as follows [64]:At time zero, we start with a quasi-momentum k = pi/4 because the initialgrowth rate dJ/dn = P is determined by the strength of the first kick.The rotor feels an accelerating potential V (J) because of the centrifugaldistortion and/or negative detuning. This corresponds to a weak force atlow J-states and an increasingly stronger force at higher states. As the quasi-momentum k grows, it eventually reaches the Brillouin zone boundary at pi/2and the Bloch wave is Bragg reflected because its length is comparable to thelattice spacing of ∆J = 2. We refer to this location as the “Bloch wall” JB.If the detuning is positive, it will counteract the centrifugal term and pushthe Bloch wall to higher rotational states, but since the centrifugal forcescales as J3, Bloch oscillations will always occur. The Bloch wall locationcan be controlled by the detuning as well as the kick strength.5.3.3 Numerical simulationBloch oscillations in molecular rotation manifest themselves in the rotationalpopulations and coherences. We calculate the rotational population PJ andthe modulus squared of the coherences C(2)J , taking into account all thedegenerate M -sublevels and the thermal mixture of initially populated states(Sec. 2.3.6).Figure 5.12 shows a comparison between the calculated population (a)and the calculated modulus squared of the coherence (b) at a realistic tem-perature of 25 K. Plotted are the distributions after each δ-kick in a sequenceof 30 kicks of P = 3 with a period matched to the quantum resonance in16O2. The identical oscillatory behaviour is evident, and expected by the16The calculation of the Bloch wall are based on the -classics approach [62], whichcannot treat a non-rigid rotor with a detuning. Nonetheless, the qualitative behaviour isclear from the above two equations.1125.3. Bloch oscillations in molecular rotation5 100 20 2515 30051015202500.20.400.040.08Number of kicks NRotational quantum numberJ05101520255 100 20 2515 30abPopulationModulus squaredof coherencesFigure 5.12: Bloch oscillations in molecular rotation: Numerically calcu-lated populations (a) and the modulus of coherences squared (b) for 16O2at a temperature of 25 K after the excitation with a sequence of N = 30pulses. The pulses have a kick strength of P = 3 and are separated by therevival time T = Trev, corresponding to the case of zero detuning from thequantum resonance.relation of both quantities (Sec. 2.3.5). There is no coherence if the cor-responding states are not populated; the coherence is maximal when twoRaman-coupled states | J 〉 and | J + 2 〉 are equally populated. These sim-ple arguments require that the oscillations of the population of the rotationalstates must also be present in the coherences (provided the oscillation am-plitude is larger than ∆J = 2).From now on, all our studies concentrate exclusively on the latter quan-tity C(2)J , because experimentally, the intensity of the observed Raman signalis proportional to it, IJ ∝ C(2)J .5.3.4 ExperimentWe decided to study Bloch oscillations in diatomic oxygen instead of nitro-gen for the following reason: The D/B ratios of 16O2 and14N2 are similar,which puts their respective Bloch walls at similar angular momentum values,according to equation 5.18. However, the density of Raman peaks is higherin oxygen due to a smaller rotational constant B, which means that the ab-solute energy of the Bloch wall will be lower. This is a substantial advantage,1135.3. Bloch oscillations in molecular rotationbecause the maximum reachable rotational energy is restricted by the finitelaser bandwidth. Expressed in the rotational quantum number, this limit isJ(N2)lim ≈ 15 or J (O2)lim ≈ 21. The explanation is written down in the followingsubsection. If one can choose the parameters such that JB 6 Jlim, one shouldbe able to demonstrate Bloch oscillations in molecular rotation of oxygen 17.We implement rotational Raman spectroscopy in oxygen molecules cooledto a rotational temperature around 25 K via a supersonic expansion, as seenearlier in Fig. 4.1(b). The driving field is a periodic train of 20 high-intensityfemtosecond pulses. Each individual pulse is set to the same kick strength,whose value is chosen freely inbetween P = 0.45 and P = 4.4, depending onthe experiment. The exact kick strength is calibrated by measuring Rabioscillations first and fitting them to the numerical simulation. The pulse se-quences with periods tuned around the revival time of 16O2 at T = 11.67 psare produced with the combined setup shown in Fig. 3.15 and discussed inSec. 3.5.2. The weak probe is a single narrowband pulse with a spectralwidth small enough to resolve the individual rotational states.Bandwidth limitationWe are limited in the excitation of high values of angular momentum becauseof the finite duration of our laser pulses, i.e. ∆t = 130 fs (FWHM). If amolecule rotates by & 90° during the length of the pulse, the effective kickstrength will diminish and further rotational excitation will be suppressed.The classical rotation period of a rigid rotor has been derived in Sec. 2.3.8as τJ = Trev(J + 3/2)−1. To estimate the bandwidth limit, one needs to seta quarter period equal to the pulse duration τJ/4 = ∆t and solve for therotational quantum number J . For our laser system, we calculate Jlim ≈ 15for nitrogen or Jlim ≈ 21 for oxygen. If the pulse sequence is strong enoughto populate such high rotational states, one will witness a turn-off in therotational excitation at J > Jlim, where the impulsive approximation is notapplicable anymore. It is referred to as “adiabatic localization” [63]. Ourδ-kick simulations do not suffer from a limited bandwidth, and as such donot show adiabatic localization.17The detrimental effect due to spin-rotation coupling in oxygen (Sec. 4.2.3) is expectedto play a minor role. At most times the population resides in higher rotation states thathave a longer dephasing time.1145.3. Bloch oscillations in molecular rotationChange Detuning0 10 20Number of Kicks N51015200 10 20 0 10 20Rotational quantum numberJa b c0 Raman signal (arb. units) 1Figure 5.13: Bloch oscillations observed in the Raman spectrum of 16O2after the excitation with a sequence of N = 20 pulses with P = 2.2 perpulse. The Raman signal is normalized to unity and colour-coded (colormap on top). The amplitude and period of the oscillations are controlledby setting the train period to match the quantum resonance (a), or to bebelow the quantum resonance by δ = −0.2% (b) and −0.6% (c).5.3.5 Observation of Bloch oscillationsThis section is a comprehensive summary of our experimental investigationsregarding Bloch oscillation in a quantum rotational system. We presentthe dependence of the phenomenon on various parameters, i.e. the detuningfrom resonance, the kick strength, the number of pulses and their bandwidth.The results are presented as two-dimensional Raman spectrograms: Eachspectrum is plotted as a function of the rotational quantum number J fora specific pump-probe delay, which is expressed in the number of pulses Nthat have interacted with the molecules.Dependence on the detuningFigure 5.13 displays the predicted oscillatory behaviour for several pulsesequences with different periods T = (1 + δ)Trev at a fixed kick strength ofP = 2.2. The angular momentum increases, following a distinct trace, beforeits direction is reversed at the Bloch wall and the momentum subsequentlydecreases again. Such a well-defined trace can only be observed at coldtemperatures with a narrow initial distribution. We distinguish a periodic1155.3. Bloch oscillations in molecular rotation0 10 20Number of Kicks N51015200 10 20 0 10 20Rotational quantum numberJa b cChange EnergyFigure 5.14: Bloch oscillations observed in the Raman spectrum of 16O2after the excitation with a periodic sequence of N = 20 pulses tuned belowthe quantum resonance by δ = −0.4% (same color map as Fig. 5.13). Theamplitude and period of the oscillations are controlled by setting the kickstrength of each pulse to (a) P = 4.4, (b) P = 2.2 and (c) P = 1.1.excitation (a) on quantum resonance when δ = 0, (b) with δ = −0.2% and(c) −0.6% below resonance, with the Bloch wall shifting from JB = 17 to 13to 9, respectively. The first case barely resolves the upper turn-around forthe given number of 20 pulses, the second case shows one oscillation, whereasthe last case exhibits two complete oscillations. The first minimum, whereno higher angular momentum states are left excited, is achieved with greatfidelity. However, the next oscillation always suffers from a significant dropin Raman intensity. Possible reasons will be discussed later. All Ramanspectra start at J = 2 to cut off the unwanted Rayleigh peak, see Sec. 4.1.1.Dependence on the kick strengthFigure 5.14 illustrates the same oscillatory behaviour when the energy ofthe pulses is changed but the period of the pulse train is fixed at δ = −0.4%below quantum resonance. Shown are the dynamics for trains with the kickstrength set to (a) P = 4.4, (b) P = 2.2 and (c) P = 1.1, reducing theBloch wall from JB = 15 to 11 to 7, respectively. The oscillation perioddecreases from about two oscillations per 20 kicks down to one oscillationper 20 kicks. Again the Raman intensity drops after the first oscillation.This effect seems to intensify for stronger kicks and is not observed in thesimulations.1165.3. Bloch oscillations in molecular rotationExperiment versus simulationSo far, we have confirmed that Bloch oscillations respond to a change in theperiodicity T or the pulse strength P as expected. Now, we will demon-strate that the dynamics is in agreement with our calculations as well. InFig. 5.15 we explore the two-dimensional parameter space with T (rows) andP (columns), while comparing the experimental Raman spectrum (bottom)with the simulated one (top). For the same pulse train of 20 pulses, theperiod changes from (a) on quantum resonance to (b) −0.2% to (c) −0.6%below quantum resonance, with a kick strength that varies from (1) P = 1.8to (2) P = 0.9 to (3) P = 0.45.The traces in each column have the same slope, which is defined by P .If the kick strength is larger, more angular momentum can be transferredto the molecule per kick. The amplitude of the Bloch oscillations, i.e. theBloch wall, is smallest in the bottom right corner (c3) and increases withboth parameters to its maximum in the top left corner (a1). This trend canbe found in both, the simulations and the experiment: the best match isin the bottom right corner, the largest discrepancy is in the top left corner.The culprit is the limited bandwidth of the experimental pulses, which isresponsible for adiabatic localization at Jlim ≈ 21. Once the impulsiveapproximation breaks down, the effective kick strength declines for higherrotational quantum numbers. This becomes visible in a decreased slope ofthe trace.Decay of the signalWhenever the excited states are close to the bandwidth limit Jlim, the in-tensity of the Raman signal decreases with every consecutive kick. Fastrotating molecules get out of phase with ‘long’ kicks, once the impulsiveapproximation is failing. It becomes evident in a loss of coherence. Furthernumerical investigations are necessary to determine the exact effects of thefinite frequency bandwidth.In addition, there is a number of other experimental reasons that resultin a decrease of the Raman signal and a loss of contrast in Bloch oscilla-tions. Spatial averaging over the Gaussian beam profiles is equivalent to anaveraging over different laser intensities, i.e. different molecules experiencedifferent kick strengths. This will ‘smear out’ Bloch oscillations with time.We tried to minimize this effect by sampling primarily the high-intensitycenter of the pump beam with a small probe beam. Experimental amplitudenoise in the pulse train has similar consequences [63]. In a thermal ensem-1175.3. Bloch oscillations in molecular rotationOxygen  (Change Energy & Detuning)Number of Kicks NRotational quantum numberJ515105151051510515100 10 20 0 10 20 0 10 205151051510a1 a2 a3b1 b2 b3c1 c2 c3Figure 5.15: Bloch oscillations observed in the Raman spectrum of 16O2after the excitation with a periodic sequence of N = 20 pulses (same colormap as Fig. 5.13). Amplitude and period of the oscillations are controlled bytwo parameters: the kick strength (1) P = 1.8, (2) P = 0.9, (3) P = 0.45;and the train period (a) on quantum resonance, (b) −0.2%, (c) −0.6% belowquantum resonance. Each spectrogram in this table compares numericalsimulations (top) with experimental results (bottom).1185.3. Bloch oscillations in molecular rotationble of molecules, a multitude of different (J,M)-states are populated. Thedegenerate M -substates interact differently with the same pulse train, asdiscussed earlier in Sec. 5.2.1, yielding slightly different Bloch oscillations.At the same time, Bloch oscillation amplitudes vary for the different (J,M)-states because the lower turning points are set by the M -values, as discussedearlier. Any initial state, that is inbetween both turning points can followa trace toward higher J-values or lower J-values [63], the rotor is eitheraccelerated or decelerated before it is Bragg-reflected. In conclusion, manyBloch oscillations with different amplitudes and periods will ‘wash out’ aclear picture.We reduced the number of initially populated states by lowering thetemperature to ∼ 25 K, i.e. about 40% of the total population are each inthe J = 1 and J = 3 manifolds. The previous study of Bloch oscillations inroom temperature nitrogen [64] attributed deviations from their simulationto collisional decoherence. At our pressures we expect only minor effectsfrom collisions. More research needs to be done to explain why consecutiveoscillations suffer from a significant drop in signal strength. This observationseems to intensify for stronger kicks.Bloch oscillations on quantum resonanceThe dynamics of a kicked rigid rotor can be described by a tight-bindingmodel. A detuning from the quantum resonance introduces an effectivepotential that will cause Bloch oscillations in the angular momentum space.Recently, this effect has been demonstrated with a periodic sequence of eightpulses [64]. We expanded this study in the discussion above.Real molecules - non-rigid rotors - will feature Bloch oscillations evenwhen the periodic kicks are tuned to the quantum resonance. In this case,the effective potential stems from the centrifugal distortion, which results ina detuning for higher J-states. This has not been demonstrated yet, since itis harder to show: The period of Bloch oscillations is longer which requiresmore pulses.Our Raman spectrogram in Fig. 5.13(a) reveals signs of oscillatory dy-namics for periodic excitation on the quantum resonance. We are unableto measure a complete cycle of a single Bloch oscillation as a consequenceof the following dilemma: At the kick strength of P 6 2.2, when the Blochwall is below our laser bandwidth limit (JB < Jlim), we cannot produceenough pulses, due to a technical limitation of the pulse shaping method(Sec. 3.3 & 3.4). However, if we increase the kick strength such that onefull Bloch oscillation occurs within 20 pulses, the Bloch wall is out of reach1195.3. Bloch oscillations in molecular rotation• Tune above QR to reach even higher• Bandwidth limit prevents BO• Pushes down the bandwidth limit• Destroys Bloch Oscillation, if bandwidth limit is lower than Bloch Wall• Signal Intensity goes down –kick molecules out of phaseNo BO above QR (bc of bandwidth limit)Similarly, No BO when chirped(bc of instantaneous bandwidth)BandwidthComparison with slide 2:• Here is less signal after turn-around; b\c I kick stronger, thus the Bloch wall is higher, closer to the bandwidth limit• However, need stronger P, to demonstrate that I can climb higher with positive detuning0 10 20Number of Kicks N0 10 20 0 10 20Rotational quantum numberJa b c510152025Figure 5.16: Compromising effect of the laser bandwidth on Bloch oscil-lations observed in the Raman spectrum of 16O2 after the excitation witha sequence of N = 20 pulses with P = 3 per pulse (same color map asFig. 5.13). The train period is tuned δ = +0.17% above the quantum res-onance (a) or on quantum resonance (b,c). The pulses in the train aretransform-limited at 130 fs (a,b) or freque cy-chirped to ≈ 250 fs (c).(JB > Jlim) and oscillations are suppressed by adiabatic localization. Theoptimized condition to show the onset of Bloch oscillations have been cho-sen in Fig. 5.13(a). Substituting oxygen for another linear molecule, wouldnot help either: The situation for N2 is similar, but the D/B ratio is lessfavourable. Heavier molecules are not compromised by our laser bandwidth,but the smaller B constants result in longer time scales which cannot becovered by our pulse shaper.Effect of bandwidthWe can verify the role that the laser bandwidth plays in the excitation of therotational states by adjusting the instantaneous bandwidth of the pulses. Asequence of 20 pulses with P = 3 per pulse excites a rotational wave packetin oxygen, shown in Fig. 5.16. The plot in the center (b) fulfils the quantumresonance condition, the Bloch wall is situated at J(b)B = 19. In comparisonwith Fig. 5.13(a), a smaller portion of the Raman signal reflects off theBloch wall. The reason lies in stronger kicks resulting in an elevated Blochwall. Therefore, its position in the angular momentum space is similar to thebandwidth limit of the transform-limited pulses J(b)B ∼ J (TL)lim , which yieldsa loss of signal as discussed earlier.1205.4. Generation of broad rotational wave packetsIn Fig. 5.16(a) we tuned the period by δ = +0.17% above the quan-tum resonance. This pushes the Bloch wall higher (J(a)B > J(b)B ) owing toa cancelling effect between the effective potential caused by the centrifu-gal distortion and the one caused by the positive detuning. The angularmomentum keeps growing and saturates significantly higher at J ≈ 21.Instead of lifting the Bloch wall above the bandwidth limit J(a)B > J(TL)lim ,we can also engineer the reversed scenario. In Fig. 5.16(c), for a periodic ex-citation on resonance, we lowered the bandwidth limit below the Bloch wallJ(chirped)lim < J(b)B . By applying a linear frequency chirp, the instantaneousbandwidth of each individual pulse in the train was reduced, as illustratedin Fig. 3.7(b). With a duration of ≈ 250 fs the chirped pulses correspond toa new J(chirped)lim ≈ 10. Indeed, we observe the saturation level in Fig. 5.16(c)drop to J ∼ 13. All three plots are comparable because the strength ofP = 3 is maintained. 185.4 Generation of broad rotational wave packetsIf our goal is to extend the reach of rotational excitation and to excitebroad rotational wave packets, one may distinguish “adiabatic” and “non-adiabatic” techniques. One of the most successful adiabatic methods, anoptical centrifuge [89, 175] has been used to create coherent superpositionsof more than 50 rotational quantum states [118]. However, owing to theadiabatic mechanism, controlling the phases of the individual states in suchultra-broad wave packets is difficult and the absolute number of excitedmolecules can be low. A second alternative route to generate broad rota-tional wave packets with well defined relative phases uses impulsive excita-tion with long sequences of pulses [41]. By tuning the period of a pulse trainto the quantum resonance we expect a ballistic growth of the wave function(Sec. 2.2.1). Under these conditions, however, we also discussed the lim-ited reach of angular momentum states as a consequence of the centrifugaldistortion (Sec. 5.3). In this section, we explore the excitation efficiencywith non-periodic pulse sequences and determine strategies of exciting thebroadest rotational wave packet.A femtosecond laser pulse creates a wave packet whose width can becharacterized by the dimensionless kick strength P , which is proportional to18For chirped pulses, the energy in each pulse is conserved but distributed over a longerduration σ with a smaller peak intensity I0 ∝ E20 ∝ σ−1. According to Eq. 2.11 for aGaussian pulse, the kick strength (P ∝ E20 σ) is constant.1215.4. Generation of broad rotational wave packetsthe intensity of the laser field (Sec. 2.3.2). Naively, a broader wave packet canthus be generated by increasingly stronger pulses. In reality, the ionizationthreshold of a given molecule only allows the excitation of a few rotationalstates with a single kick. The idea of a pulse train is to remain below theionization limit by dividing the total energy among N pulses. If the pulsesare separated by the quantum revival time Trev, the cumulative kick strengthPN of the whole train is the sum of the kick strengths of its individual pulses[12]. Multiple schemes of using long periodic pulse trains for the controlledrotational excitation of molecules have been proposed theoretically [12, 103,104, 166, 187], and implemented experimentally, as discussed in the previoussection [41, 192, 64, 88].Here, we investigate the rotational excitation of room temperature oxy-gen in a gas cell by long sequences up to N = 28 pulses with cumulativekick strength up to PN ≤ 140. These pulse trains are significantly longerand stronger than previous realizations, most notably N = 8, PN ≤ 40 inRef. [64]. In Sec. 5.4.1, we study the shape of the achieved wave packets andits dependence on the train period with respect to the quantum resonance.As discussed in Sec. 5.3, populating high rotational levels proves impossi-ble due to the centrifugal distortion and despite the high cumulative kickstrengths of the periodic train. For fast rotating molecules, the revival timebecomes dependent on the angular momentum J and as a result, the reso-nant condition and the accumulation of the total kick strength from pulse topulse is inhibited. In Sec. 5.4.2, we exploit fractional revivals and apply fournon-periodic kicks per Trev. This optimization for efficient rotational exci-tation extends our reach to J ≈ 29. The propagation of such an optimizedpulse train through a dense medium is examined in Sec. Periodic excitationTo connect with the previous sections, we start by re-examining the ro-tational coherences created by a periodic pulse sequence on and near thequantum resonance. Here, the goal is the creation of broad rotational wavepackets, which can be anticipated on the quantum resonance, i.e. for a pe-riod T = Trev when the excitation of all angular momentum states shouldbe equally efficient. The necessary pulse sequences are produced with thecombined setup shown in Fig. 3.15 and discussed in Sec. 3.5.2. We usestate-resolved coherent Raman detection in a room temperature ensembleof oxygen molecules, see Fig. 4.1(a), to explicitly demonstrate the effect ofthe centrifugal distortion as the main reason for the inhibited rotational lad-der climbing.1225.4. Generation of broad rotational wave packetsFigure 5.17(a) revisits the 2D Raman spectrogram from Fig. 5.1(a), butthis time we analyse the spectrum for much higher quantum numbers J ≈40. The times TJ = (2J + 3)× τJ/2, indicated by light blue squares, markthe times when all (|J〉, |J + 2〉) wave packets perform an integer numberNJ = 2J + 3 of half-rotations19. As mentioned before, these times makeup an almost vertical “trajectory” originating at the revival time for lowvalues of the rotational quantum number (light blue squares). Away fromthe quantum resonance, the NJ = const. trajectories bend away from thevertical lines. For instance, increasing (decreasing) NJ to the next integervalue 2J + 4 (2J + 2) results in a curved trajectory marked with triangles(circles), along which no two rotational wave packets corresponding to twodifferent values of J are in phase simultaneously.Even when the conditions of the quantum resonance are met for the lowerrotational states, they no longer hold at higher J ’s owing to the centrifugaldistortion of the molecular bond. With increasing J , the centrifugal termin the rotational energy EJ = hc[BJ(J + 1) −DJ2(J + 1)2] becomes non-negligible, making the resonance J-dependent and, therefore, impossible tosatisfy for all quantum states simultaneously. In Fig. 5.17(a), this effectis seen through the apparent curving of the resonant trajectory (light bluesquares) away from the vertical line at T = Trev above J ≈ 15. As a result,the efficiency of the accumulative rotational excitation by a resonant pulsetrain (T = Trev) deteriorates with growing J , resulting in Bloch oscillationsdiscussed in Sec. 5.3. To demonstrate the described centrifugal limit weincreased the number of kicks as well as their strength. Fig. 5.17(b) showsthe detected Raman signal generated by the periodic resonant train of 20strong pulses with the pulse intensity of 3 · 1013W/cm2 per pulse (blacksolid line). Despite the increased cumulative kick strength of P20 ≈ 140, thehighest excited level remains significantly below that value, J ≈ 17  P20.In fact, even though the perturbative analysis used earlier is not applicable inthe case of strong pulses, the measured limit agrees well with the conclusionsof Sec. 5.3.Utilizing the resonance map further, one arrives at a simple methodof extending the reach of rotational excitation by a periodic train of fem-tosecond pulses. As seen in Fig. 5.17(a), by shifting the train period abovethe quantum resonance, its overlap with the time of complete half-rotationsTJ of the rotational wave packets with higher J ’s can be improved. In Sec-tion 5.3, this “trick” was introduced in the context of shifting the Bloch wall19For a detailed description of the resonance map, see Sec. Generation of broad rotational wave packetsa)b)1 5 9 13 17 21 25 2901Rotational state  JIntensity(arb. units)Period T (ps)10.8 11 11.2 11.4 11.6 11.8 12 12.2 12.4Wavelength (nm)4024034043759172125293313Rotational state  J10 Intensity (arb.units)Figure 5.17: (a) State and time resolved Raman spectrogram of 16O2 afterthe excitation by a sequence of five pulses with a period T scanned aroundthe rotational revival time Trev (vertical solid line). The vertical shadedband represents the length of our Gaussian pulses (FWHM), whereas theshaded horizontal band covers the rotational quantum numbers (shown onthe right vertical axis) corresponding to the thermally populated rotationalstates (population higher than 10 %). Light blue markers indicate the timemoments at which a coherent rotational wave packet consisting of two states,|J〉 and |J + 2〉, completes an integer multiple of half rotations, NJ . Thecentral trajectory (squares) corresponds to NJ = 2J + 3, while the twoneighbouring sets of circles and triangles represent the cases of NJ = 2J + 2and NJ = 2J + 4, respectively. (b) Raman spectra after a sequence of20 periodic pulses on quantum resonance (black solid), −0.4% below quan-tum resonance (blue dashed) and +0.4% above quantum resonance (greendotted).1245.4. Generation of broad rotational wave packetshigher up by compensating the centrifugal term with the positive detuningfrom the quantum resonance. However, the degree of such control is ratherlimited, because detuning the pulse train far from the resonance reduces theexcitation efficiency of the initially populated low-J states. The limitationcan be analysed by inspecting the two shaded bands in Fig. 5.17(a): thevertical one represents the temporal width of our pulses (FWHM), whereasthe horizontal one covers the thermally populated rotational states. Thelength of a continuous set of blue markers under the vertical band tells usabout the width of the created rotational wave packet. On the other hand,how many of them are covered by the intersection of the two shaded areasis a qualitative indicator of the thermal fraction of molecules in the wavepacket. As the train period shifts to the right of the quantum resonance,higher J states are reached, but the amount of rotationally excited moleculesdecreases.For the length and strength of our experimental pulse trains, we foundthe detuning of +0.4% to result in the highest enhancement of the wavepacket’s width, while not causing a significant loss of the overall excitationefficiency. As illustrated in Fig. 5.17(b), by increasing the train period0.4% above the quantum resonance (green dotted line), we indeed shift therotational excitation to higher J states. Similarly, setting the train period0.4% below the resonance (blue dashed line) results in the narrower excitedwave packet with the lower “center of mass”.We note, that due to the power broadening of the individual rotationaltransitions, higher pulse intensities should allow farther detunings from thequantum resonance and correspondingly broader rotational wave packets.Stronger pulses effectively push the centrifugal limit higher. Nevertheless, asdiscussed earlier, the ultimate limit for the reach of the rotational excitationis set by the finite width of the laser pulses. Once a molecule rotates too fastfor the pulses to act as instantaneous kicks, it does not climb the rotationalladder any further.5.4.2 Non-periodic excitationThe effect of the centrifugal distortion grows in time as the wave packetspreads out. The dispersion of the revival times with J accumulates, makingevery next kick less and less efficient [41]. Hence, to fully benefit from thelarge number of pulses, one needs to apply them on as short of a timescale as possible. To achieve this goal, we employ tunable non-periodicpulse sequences. Here, we make use of fractional quantum revivals [13] andexpose molecules to four rotational kicks per Trev. Such an optimized train1255.4. Generation of broad rotational wave packets0 100 150 20050 01Time (ps)Intensity(arb. units)𝑇4𝑇4𝑇4𝑇420 ⋅ 𝑇45 ⋅ 𝑇4a)010 20 40 60Time (ps)𝑇1𝑇2𝑇3𝑇45 ⋅ 𝑇4𝑇4b)Intensity(arb. units)Figure 5.18: Temporal profile of a long periodic pulse train (a) and a densenon-periodic pulse train (b), 20 pulses each (note different time scale). Thefour shades of red correspond to one of the four different pathways throughthe two Michelson interferometers.allows us to “pack” more pulses within the limited amount of time beforethe centrifugal distortion suppresses further excitation. By fine tuning thetiming around each fractional and full revival, we extend our reach fromJ ≈ 17 to J ≈ 29, utilizing more efficiently the cumulative strength of along train with 20 pulses.Such a dense non-periodic pulse train optimized for the maximum rota-tional excitation is generated in the following way: Rather than adding theinterferometer copies consecutively one after another to form a long periodicpulse train of 20 pulses as shown in Fig. 5.18(a), we interleave those copieswith variable timings T1, T2 and T3 ≡ T1 + T2 as shown in Fig. 5.18(b).Here, Tk is the time between the beginning of the sequence to its k-th pulse.The procedure to find these timings is described next.To start, we demonstrate that utilizing two pulses per every revival timecan indeed enhance the excitation, if the timing is chosen appropriately. Forthat purpose, a resonant train of five pulses with a period T4 = Trev is pro-duced with the pulse shaper. Using a single Michelson interferometer, the1265.4. Generation of broad rotational wave packets1 5 9 13 17 21 25 2901Rotational state  JIntensity(arb. units)c)b)-1012 3 4 5 6 7 8 9 10Time (ps)Intensity(arb. units)Wavelength (nm)4024032 3 4 5 6 7 8 9 105101520Period T1 (ps)Rotational state  JIntensity(arb. units)01a)Figure 5.19: (a) Raman spectrogram for the rotational excitation by asequence of two identical periodic pulse trains, five pulses each (same colormap as Fig. 5.17(a)). The period of each train is fixed at Trev, while thetime delay T1 between the two trains is scanned. The integrated signalabove reveals the times of maximum total coherence. (b) Alignment-inducedbirefringence signal as a function of the probe delay after a weak transform-limited pump pulse. (c) Raman response after the rotational excitation bya sequence of 20 pulses: a periodic pulse train with T = Trev (black solid)and an optimized non-periodic pulse train (red dashed). The two trains aredrawn schematically in the upper right corner.1275.4. Generation of broad rotational wave packetstrain is then split in two parts, which are overlapped in space with the timedelay T1 between them (see Fig. 5.18(b) for the definition of time intervals T1and T4). Similarly to the Raman spectrogram in Fig. 5.17(a), in Fig. 5.19(a)we plot the observed frequency-resolved Raman signal as a function of thedelay T1 between the two interleaved periodic pulse trains. As before, strongRaman peaks correspond to the time moments of enhanced rotational ex-citation. At some of those time moments marked with solid vertical lines,slightly before or after the (14 ,12 and34) × Trev fractional revivals, the Ra-man response is simultaneously high for the largest number of rotationalstates, in exact analogy to the previously discussed response to the periodicexcitation at the quantum resonance. We highlight this in the upper panelby plotting the same Raman signal integrated over wavelength. The latteris proportional to∑J C(2)J20 and should not be confused with the degreeof transient molecular alignment which is plotted in Fig. 5.19(b) for com-parison21. One can now interpret the optimal T1 values in Fig. 5.19(a) asthe times of the maximum positive derivative of the alignment signal, whenthe majority of molecules move towards the aligned state. A second kick,introduced at this time, accelerates the rotation further. In contrast, dashedvertical lines correspond to the maximum negative derivative of the align-ment signal, when the majority of molecules move away from the alignedstate. A second kick arriving at this time decelerates the rotation, loweringthe degree of excitation, as reflected by the dips in the integrated coherencesignal in panel (a).The above analysis suggests a possibility of enhancing the rotationalexcitation by adding up to six pulses per every revival period [six verticalwhite lines in Fig. 5.19(a)]. Our pulse shaping scheme enables convenientscanning of four pulses per Trev: near one full revival and three fractionalrevivals. We set the four variable delay times to the following optimal valuestaken from the integrated coherence signal in Fig. 5.19(a): T1 = 0.242 ·Trev,T2 = 0.519·Trev, T3 = 0.761·Trev and T4 = 1.004·Trev = 11.72 ps, i.e. slightlyabove the full revival as explained earlier. We note that these optimizeddelays depend on the kick strength of the pulses in the train. The strongerthe pulses, the higher the rotational excitation after each pulse, the largerthe centrifugal distortion, the bigger the required shift from every fractionalrevival.The result of the excitation by an optimized non-periodic train is shownin Fig. 5.19(c). By shortening the duration of the pulse train, while using the20Here, C(2)J is the modulus squared of the coherences, described in Sec. alignment signal was recorded with a 130 fs probe pulse following a single kick.1285.4. Generation of broad rotational wave packetssame number of pulses (N = 20), we extend our reach from J=17 with onepulse per revival (black solid curve) to J ≈ 29 with four pulses per revival(dashed red curve). Although the centrifugal limit of the periodic excitationhas been circumvented, the efficiency of the non-periodic optimized pulsetrain is still well below its accumulated kick strength of PN = 140. Themain limitation is now the finite duration of the laser pulses, prohibitingfurther excitation.5.4.3 Propagation effectsLong sequences of femtosecond pulses drive molecules to a highly coherentstate. In a dense medium, it makes the latter a strong light modulator(for recent reviews of this effect, see [163, 14]). In this section, we studyhow pump and probe pulses are affected when propagating through a gas ofrotationally excited molecules under high pressure. In oxygen at pressuresup to 6.5 atmospheres, we observe the generation of frequency sidebands viathe cascaded rotational Raman scattering [124]. We show that the molecularphase modulation (MPM) imparted on the probe pulse is maximized whenits delay coincides with the full revival of the rotational wave packet, aswell as if its timing is close to that of a fractional revival [15]. Here wedemonstrate that when a non-periodic train is optimized for the efficientrotational excitation, the spectral bandwidth is increasingly broadened frompulse to pulse [128].Cascaded Raman scatteringThe rotational coherence in a molecular ensemble induced by our optimizedpulse train is rather strong. This can be seen through Raman processesof higher orders, i.e. cascaded Raman scattering, which are especially pro-nounced at higher gas pressure. In Fig. 5.20(b), we plot the spectra of aweak probe pulse following a pulse train of 28 pulses at four different pres-sures ranging from 1.7 to 6.5 atm. The timing of the pulses in the train hasbeen optimized at 6.5 atm so as to achieve the highest integrated coherencesignal at J > 17, i.e. for maximizing the sum∑J>17C(2)J , by means of thesame optimization procedure as described in the previous section 5.4.2.With an optimized non-periodic pulse sequence, we count up to 250 Ra-man peaks. Note, that this number does not reflect the rotational quantumnumber reached by the end of the excitation process (indeed, at such high Jvalues, the separation between the consecutive peaks should have decreasedsignificantly due to the centrifugal distortion). Rather, large frequency shifts1295.4. Generation of broad rotational wave packets400 405 410 415 420 425 430Wavelength (nm)231233235higher orders1𝑠𝑡 2𝑛𝑑 3𝑟𝑑 (b)10-4Intensity  (arb. units)10-310-210-1100Rotational states1𝑠𝑡2𝑛𝑑3𝑟𝑑(a). . .ΔΔΔJJ+2Figure 5.20: (a) Simplified illustration of the first three scattering ordersin the process of cascaded Raman scattering. The grey profile representsthe Boltzmann distribution of rotational population. (b) Rotational Ra-man spectra of oxygen gas excited by the optimized non-periodic train of28 pulses. The four curves correspond to the gas pressures of 1.7 atm (lowerblue), 2.4 atm (middle orange), 5.1 atm (higher red) and 6.5 atm (top black).First, second and higher orders of the cascaded Raman scattering are indi-cated with arrows. More than 235 Raman peaks are observed at higherpressure values, as shown in the inset.1305.4. Generation of broad rotational wave packetsare caused by the cascaded Raman scattering giving rise to multiple coher-ent sidebands [124]. The process is illustrated schematically in Fig. 5.20(a).A strong pump pulse (solid orange arrow) creates multiple coherences viaseveral two-photon Raman processes (for clarity, only one Raman transitionwith an average frequency shift ∆ is shown in the figure). The shape of theresulting Raman spectrum, which consists of a series of n red-shifted peaks,reflects the initial thermal distribution of molecules among the rotationalstates (grey profile). In the second-order process, all n emitted photons(dashed red arrow) re-scatter off the induced coherences, giving rise to thesecondary set of n2 Raman lines (dotted brown arrow), centered around 2∆from the input laser frequency. The third-order process results in yet an-other red shift by ∆ (dotted brown to dash-dotted black) and so on. Thispicture explains the repetitive broad spectral features in Fig. 5.20(b), e.g.those marked as “higher orders”, resembling the Raman spectrum observedafter a single weak pulse, plotted earlier in Fig. 4.3. The features are sepa-rated by the frequency shift ∆ corresponding to the peak of the Boltzmanndistribution. The number of Raman processes in each order m of the cas-caded scattering grows as nm. We note, that the output spectrum consistsof uniformly spaced Raman lines, owing to the repetitive scattering off thesame set of coherences, induced by the strongest first-order interaction.We verified experimentally that the conversion efficiency into the higher-order sidebands increases with the intensity of the pulse train and the num-ber of molecules in the interaction region, both leading to the stronger ro-tational coherence in the system. Raising the density of molecules by in-creasing the gas pressure revealed an anticipated Bessel-like dependence ofthe strength of the Raman sidebands on the scattering order number [124].For instance, as can be seen in Fig. 5.20(b), at P = 5.1 and 6.5 atm, thesecond-order coherences exceed those induced by the first-order scatteringprocess.Pulse broadeningUntil now, we have used narrowband probe pulses for the state-resolveddetection of the observed spectral broadening driven by the Raman transi-tions of multiple orders. In the time domain, the process can be described asthe transient molecular phase modulation (MPM) owing to the periodicallymodulated refractive index of the medium. If a femtosecond pulse coincideswith a full or a fractional wave packet revival, when the phase modulationis maximized, its frequency bandwidth is broadened [15]. The broadeningeffect has been theoretically predicted to accumulate from pulse to pulse in1315.4. Generation of broad rotational wave packetsTime (ps)Wavelength (nm)-2 0 2 4 6 8380400420a)01234Wavelength (nm)Intensity(arb. units)b)390 395 400 405 41001415Intensity (arb. units)Figure 5.21: (a) Time-resolved Raman spectrogram (log scale) of oxygengas excited by the optimized non-periodic train of 28 pulses under the pres-sure of 6.5 atm. (b) Raman spectra at the time moments indicated witharrows in the spectrogram: before the first kick (black solid), at the time ofthe first kick (blue dashed), and at the time of the third kick (red dotted).1325.5. Conclusiona periodic sequence of pulses, as long as the train period is equal to therotational revival time [128].To demonstrate this accumulative broadening, we scan the time delayof a 130 fs probe pulse across the optimized non-periodic pulse train de-scribed earlier, while recording the probe spectrum. The results are plottedin Fig. 5.21(a) as a function of the probe delay with respect to the firstpulse in the train. As expected, the spectrum of the probe pulse remainsunchanged unless the latter coincides in time with a full or a fractional re-vival of the wave packet. The further down the pulse train we probe, thebroader and more red-shifted the probe spectrum becomes, as illustrated inFig. 5.21(b).5.5 ConclusionWe investigated the excitation of molecular rotation using long pulse se-quences of femtosecond pulses that are tailored to match the fractional andfull quantum resonances. In the case of a pulse sequence with weak pulsesspaced by a fractional revival time, we observe two-photon Rabi oscillations.The population oscillates in an effective two-level system of two rotationalstates as a function of the number of pulses. Once we change the periodicityto the full quantum resonance, the population efficiently moves up the “ro-tational ladder” until the inevitable centrifugal distortion prohibits furtherexcitation. The consequence is oscillations in the angular momentum space,which can be related to the solid-state phenomenon of Bloch oscillations. Wedemonstrated the difficulties in the impulsive excitation of high rotationalstates and presented a strategy to partially mitigate the detrimental effectof the centrifugal distortion by adjusting the train period and by using non-periodic pulse sequences. The induced rotational coherence was significantlyenhanced.In theory, one could follow the rotational wave packet to higher statesby continuously adjusting the train period. By avoiding the periodicityaltogether, one would break the tight-binding analogy and might expect acontinuous diffusive growth of the molecular angular momentum, limitedonly by the laser bandwidth. Non-periodic pulse sequences of this type areinvestigated in the following chapter 6.We envision that our long pulse trains will be useful for all applicationsthat require high transient molecular alignment at field-free conditions suchas the generation of ultra-short laser pulses [15] or the control of high har-monic generation [174, 82]. It will be beneficial in impulsive gas heating [188]1335.5. Conclusionor large amplitude plasma wave generation [172]. Similar effects have beenstudied in the context of propagation of intense femtosecond laser pulses inatmospheric air [173, 129, 128, 188].134Chapter 6Dynamical localization inmolecular rotationThe quantum kicked rotor is characterized by two qualitatively differentregimes: In the last chapter 5 we examined the one when the kicking occurson quantum resonances. In this chapter, the focus is on off-resonant periodicexcitation, when the wave function of the quantum rotor undergoes dynam-ical localization. The wave function does not grow wider in the angularmomentum space with every consecutive kick, but instead localizes near theinitial rotational state due to the interference of quantum interaction path-ways [31, 84]. An exponential distribution around the localization centeris considered a necessary component and a distinct signature of dynamicallocalization. Although the phenomenon has been studied experimentallyin Rydberg atoms [67, 16, 27, 65] and a cold-atom analogue of the QKR[121, 4, 43, 146, 145, 34], it has never been observed in a system of truequantum rotors.Floss et.al. theoretically demonstrated the possibility to observe dynami-cal localization in laser-kicked molecules [59, 63]. The first step toward thatgoal was reported in 2015, when an onset of localization was observed inlaser-induced molecular alignment [88]. Here, we describe the direct obser-vation of dynamical localization. We are able to show the hallmark featuresof the exponentially localized states and the suppressed growth of the rota-tional energy. An overview of all experimental observations, the underlyingtheory and a numerical analysis constitute this chapter.Section 6.1 puts the work on dynamical localization into the contextof the well-known effect of Anderson localization in disordered solids. Sec-tion 6.2 outlines the theoretical basis for our system. We analyse the tight-binding model, which is mapped onto the QKR, and explain the origin ofthe “disorder” in the rotational “lattice” that is responsible for the local-ization. Our experiment is described in Sec. 6.3. We will elaborate on ourdetection scheme and how we retrieve the angular momentum distributionfrom our measurement. Section 6.4 presents our key results: Cold initial1356.1. Experiments on Anderson localizationconditions and high-sensitivity detection enable us to observe the distribu-tion of the molecular angular momenta evolving into an exponential lineshape, characteristic of a localized state. We study the dependence of thelocalization on the number of kicks (6.4.1) and the strength of the kicks(6.4.2). Owing to the state-resolved detection we identify the suppressedgrowth of rotational energy (6.4.3). Dynamical localization is a coherenteffect that relies on wave interferences. Therefore, we also implement twodifferent types of noise in our studies to destroy the localization and to re-cover classical diffusion. Control over the position of the localization centeris investigated (6.4.4). Section 6.5 builds a bridge between chapters 5 and 6where we examine the transition from the quantum resonance to the regimeof localization. Concluding remarks are found in Sec. Experiments on Anderson localizationIn the last decades, there has been great interest and many achievements instudying Anderson localization in various systems of different dimensional-ity. Since the phenomenon relies on the wave character, it is ubiquitious andshould also appear with classical waves. In fact, in our three-dimensionalworld the transition from diffusion to localization should occur for any typeof classical or matter waves in any disordered media if the wavelength be-comes comparable to the mean-free path between random scattering events[165].After the proposal of localization with electromagnetic waves [86], ithas been demonstrated with light in synthesized strong scattering materi-als [183, 165] and with microwaves [68, 42, 33]. Ultrasound was used toshow localization of acoustic waves [178, 80]. All cited works used fully ran-dom potentials, without the presence of any lattice. With the realization ofAnderson localization in photonic crystals [150, 99] experimentalists madeanother step toward the original Anderson model by superposing fluctua-tions onto a periodic structure.Just as Bloch waves describe a quasi-free electron in an atomic crystal,matter waves can be placed in optical potentials created by laser interferencepatterns. Two common techniques are used to create disordered potentials,via the laser speckle patterns [39, 18] or by adding two optical lattices withincommensurable periods [50, 143] (see a review in Ref. [10]). More recently,much attention has been drawn by the studies of the mobility edge [94, 85,153] and the effect of particle interaction on localization [38, 130, 149].The QKR presents an alternative system to study Anderson localization,1366.2. Theoryand offers an interesting testing ground for new physics.6.2 TheoryIn section 2.4.6 we established the relationship between a quantum particlein a disordered solid and the periodically kicked rotor. The Schro¨dingerequation for the quantum rotor was derived in Eq. 2.52 asTJ uJ +∑J ′WJ,J ′ uJ ′ = 0 , (6.1)with the probability amplitude uJ to find the quantum rotor at the rotationalstate J of the angular momentum lattice. The two terms describe the hop-ping term WJ,J ′ and the on-site energy TJ = tan(φJ) with φJ =(Eα−EJ )T2~ .If the function TJ is independent of the rotational states J , we arrive atthe tight-binding model for a periodic lattice. The solutions are extendedBloch states [62]. This scenario has been discussed in the context of rota-tional Bloch oscillations in Sec. 5.3 and it will be further analysed in thecontext of dynamical localization in Sec. 6.5.1. If the on-site energies TJ arerandom, also referred to as “diagonal disorder”, we arrive at the Andersonmodel. In this case, the solution is given by quasi-energy states (Foquetstates) that are localized in the angular momentum space [52].Next we look at the actual expression of TJ for linear molecules. Weneglect the centrifugal term and use the rotational energy EJ = hcBJ(J+1).If the period between kicks T = pqTrev is a rational fraction (p, q are integers)of the revival time Trev = (2cB)−1φ(J) =pi2pq( EαhcB− J(J + 1)). (6.2)The variation of the on-site energy between neighbouring rotational statesJ and J + 2 of the angular momentum lattice is therefore∆φ(J) = |φ(J + 2)− φ(J)| = pipq(2J + 3) . (6.3)If pq = 1, the lattice is strictly periodic (TJ = const.). For other rational val-ues of p/q, we obtain a quasi-periodic lattice. Note, that the phase difference∆φ(J) has been analysed with the resonance map in Sec. 2.3.8, where eachvalue of pi corresponding to a fractional resonance was marked with a cross.In this chapter, we are interested in irrational values of T/Trev, which areof order unity. Due to the periodicity of the tan-function, only ∆φ modulo1376.3. Experimentpi affects the variation of the on-site energy. Here, any rotational quantumnumber yields a magnitude of ∆φ(J) that exceeds several pi. This makesT (J) a pseudo-random number [52], resulting in dynamical localization.Two points are noteworthy: (1) The values of TJ are not truely ran-dom. They are calculated deterministically but they behave stochastically.It has been shown numerically, that “pseudo-random” energies are sufficientto observe Anderson localization in a one-dimensional system [52, 76]. Forthese reasons, the QKR is sometimes called the “pseudo-Anderson model”[76, 77]. (2) The rigid-rotor approximation is sufficient to describe the An-derson model. Adding the centrifugal term merely adds more randomness.6.3 ExperimentOur approach to demonstrate dynamical localization with molecular rotors[63] bears a number of experimental challenges. First, the need to assess theshape of the rotational distribution calls for a sensitive detection methodcapable of resolving individual rotational states. According to the theo-retical studies [63], the population of a few tens of rotational states mustbe measured with high sensitivity over the range of at least two orders ofmagnitude. Second, for the localized state not to be smeared out due tothe averaging over the initial thermal distribution, the latter must be nar-rowed down to as close to a single rotational state as possible, requiring coldmolecular samples. Finally, an important test of dynamical localization, therecovery of classical diffusion under the influence of noise and decoherence,demonstrated experimentally with atoms [27, 4, 92, 119, 127] and theoret-ically with molecular QKR [63], requires long sequences of more than 20strong kicks.We address all three of the above challenges and study the rotationaldynamics of nitrogen molecules, cooled down to 25 K in a supersonic expan-sion and kicked by a periodic series of 24 laser pulses, whose kick strengthsare as high as P = 3 per pulse. We use state-resolved coherent Ramanspectroscopy, described in detail in Sec. 4.1 [Fig. 4.1(b)] to demonstrate theexponential shape of the created rotational wave packet, indicative of dy-namical localization. The dependence of the rotational distribution on thenumber of pulses and their strength is investigated. Our ability to resolveindividual rotational states allows for a direct extraction of the absorbed en-ergy, whose growth is shown to cease completely after as few as three pulses.To confirm the coherent nature of the observed localization, we study theeffect of both timing noise and amplitude noise, which are shown to yield1386.3. Experimenta non-exponential distribution of angular momenta and revive the diffusivegrowth of energy. Our results are in good agreement with the existing the-oretical analysis [63] and our own numerical simulations.6.3.1 Calibration of experimental parametersThe important experimental quantities in this study are the initial tempera-ture of the molecular ensemble and the pulse train parameters: the numberof pulses N , the train period T and the strength P of the individual pulses.Temperature: The rotational temperature of our molecular sample is de-termined by fitting the Raman spectrum after a very weak kick, which toa good degree of approximation does not change the population distribu-tion. The procedure and its results were discussed earlier in the technicalsection 4.2.3. We estimate an initial temperature around 25 K.Pulse train parameters: The exact shape of the pulse train is set bythe pulse shaper and the Michelson interferometers. We confirm the finalpulse train parameters via an XFROG measurement, which was describedin Sec. 3.2.1. This cross-correlation measurement, which is carried out im-mediately before sending the beams into the vacuum chamber, yields thetemporal profile of each sequence. Besides the pulse number and the timeseparation between the pulses, we can also measure the relative pulse am-plitudes, however their absolute values remain unknown.Kick strength: We determine the exact pulse intensity in the interactionregion with an additional measurement. We tune the period of the pulsetrain to T = 67Trev, indicated by the dashed line in Fig. 6.1. As discussedearlier, this timing coincides with the rotational period of a wave packetconsisting of two rotational states with J = 2 and J = 4. Fitting thefrequency of the ensuing Rabi oscillations between the two states provides anaccurate way of measuring the intensity of the pump pulses. For the physicalpicture and more details, read section 5.2 on Rabi oscillations in molecularrotation. The pulse intensity is expressed in the dimensionless units of kickstrength P , reflecting the typical amount of angular momentum (in unitsof ~) transferred from the laser pulse to the molecule [57]. By amplifyingthe sequence of 24 pulses in the multi-pass amplifier (MPA), we are ableto reach kick strengths of up to P = 3 per pulse (2 × 1013 W/cm2). Thestandard deviation of the pulse energy fluctuations is below 15%.1396.3. Experiment0.5 0.6 0.7 0.8 0.9 1Pulse train period   ( T/Trev )0246810Quantum numberJ1013< 𝜏 <5678< 𝜏 <1314Rotational Resonances𝝉 =𝟔𝟕Figure 6.1: Resonance map with an illustration of the choice of pulse trainperiods. The period T/Trev = 6/7 (dashed line) serves to induce Rabioscillations between the states J = 2 and 4. Ten equidistant periods ineach of the two shaded bands are chosen to observe dynamical localizationin molecular rotation.Pulse Period: Localized states in the quantum kicked rotor are known toexist away from the quantum resonances, i.e. when the time between kicksis not equal to a rational fraction of the revival period [52]. For molecularnitrogen 14N2, the revival time is Trev = 8.38 ps. As long as T 6= pqTrev, wherep and q are integers, the behaviour of localization is universal [63]. To satisfythis condition, we choose 10 evenly spaced pulse train periods T in each ofthe two intervals, 10/13 < T/Trev < 5/6 and 7/8 < T/Trev < 13/14, seen inFig. 6.1. Although we (partially) overlap with some fractional resonances,none of them are of low order, i.e. correspond to small integer values of q like12 ,23 ,34 . Hence, their effect on the rotational excitation is rather negligible.Moreover, by taking the mean over all 20 Raman spectra, we reduce theinfluence of nearby quantum resonances and we achieve a better signal-to-noise ratio.6.3.2 Population retrieval from the experimentDynamical localization of the QKR manifests itself in the exponential distri-bution of angular momentum around the localization center. The initial dis-tribution of molecules in our supersonic jet is close to the rotational groundstate. Therefore, we expect the population to fall off exponentially towardshigher rotational quantum numbers. Our detection technique of Ramanspectroscopy, however, does not provide a direct measure of the rotationalpopulation. Here, we describe our procedure to retrieve populations from1406.4. Observation of dynamical localizationthe measured Raman spectra.In Sec. 2.3.6 we derived the intensity of the observed Raman peaks asIJ ∝∑M 〈|cM∗J cMJ+2|2〉J′,M′ , the modulus squared of the induced coherencesummed over the degenerate M -sublevels and averaged over the mixture ofinitially populated states | J ′,M ′ 〉. If the initial ensemble contained onlyone populated level |J ′ = J0,M ′ = M0〉, the strength of the Raman signalwould reduce to IJ ∝ PJ,M0PJ+2,M0 , where PJ,M = |cMJ |2 is the rotationalpopulation. For localized and non-localized dynamics of the QKR, we expectexponential or Gaussian population distributions, respectively [40, 92]. Ineither case, the Raman spectrum can be further simplified to IJ ∝ (PJ,M0)2,offering the direct measure of the rotational population. As we show below,this proportionality holds even at a non-zero temperature, when the Ramansignal is produced by a number of independent rotational wave packets orig-inating from different initial states. At 25 K most population is initially atJ ′ = 2. Thus, the smallness of M ′ = 0,±1,±2 with respect to the angularmomentum of the majority of states in the final wave packet results in aninteraction Hamiltonian which to a good degree of approximation does notdepend on M ′. Having all molecules in the thermal ensemble respond tothe laser field in an almost identical way enables us to extract rotationalpopulations from the Raman signal as PJ = a√IJ , with the coefficients afound from normalizing the total population to unity.6.4 Observation of dynamical localizationFigure 6.2(a) shows a set of 20 Raman spectra, obtained with the 20 differ-ent periodic pulse trains mentioned above. As before, the Raman frequencyshift (horizontal axis) has been converted to the rotational quantum numberJ . All of the observed Raman signals IJ decay exponentially across 4 ordersof magnitude and 15 rotational states, independent of the train period. Theaverage Raman signal is plotted with the solid red line in Fig. 6.2(c). Ithas a distinctly different shape than the initial thermal distribution, plottedwith the solid grey line 22. The remaining oscillations are a consequence ofthe nuclear spin statistics of nitrogen, which dictates the 2:1 ratio for thetwo independent rotational progressions consisting of only even and onlyodd values of angular momentum. The exact shape of each individual dis-tribution in Fig. 6.2(a) depends on the period of the corresponding trainand is affected by its proximity to fractional quantum resonances of higherorders. In Fig. 6.2(d), the solid red line illustrates the distribution of the22The initial distribution was measured after a single weak pulse.1416.4. Observation of dynamical localizationRotational quantum number J10-410-2100Raman signal(arb. units)10-510-410-310-210-1100Ramansignal(arb. units)5 10 15 20 25010-310-210-1Population5 10 15 20 25010-410-2100a) b)c) d)Figure 6.2: Rotational Raman spectrum of nitrogen molecules excited witha train of N = 24 pulses at a kick strength of P = 2.3 for 20 different (a)periodic and (b) non-periodic sequences. (c) The average experimental dis-tributions (solid lines) are compared to the numerical simulations (dashedlines) for both the periodic (middle red lines) and the non-periodic (up-per black lines) pulse trains. The initial distribution is shown by the lowergrey lines. (d) The exact calculated populations (dashed lines) and the ap-proximate populations (solid lines), retrieved from the experimental Ramansignal. The retrieved populations are fitted with an exponential/Gaussianfunction (thick green lines). The dotted vertical line represents the excita-tion limit due to the finite pulse duration.1426.4. Observation of dynamical localizationrotational population, extracted from the average Raman signal accordingto PJ ∝√IJ . The evident exponential shape, highlighted by an exponentialfit (thick green line) with a localization length (1/e width) Jloc = 3.2, is ahallmark of Anderson localization in this true QKR system.To confirm the coherent nature of the observed localization, we repeatthe same measurement with a set of 20 non-periodic pulse trains. Thekick strength is set to the same value of P = 2.3 per pulse, but the timeintervals between the 24 pulses in each train is randomly distributed aroundthe mean value of 0.85Trev with a standard deviation of 33%. Here, all theindividual Raman spectra, their average and the population distributionretrieved from it (solid black lines in Fig. 6.2(b), (c) and (d), respectively)show a qualitatively different non-exponential shape. As expected for aquantum kicked rotor, dynamical localization is destroyed by the timingnoise. Classical diffusion, with its characteristic Gaussian distribution ofangular momentum (thick green line) with a 1/e width of Jdiff = 7.4, isrecovered.In Figures 6.2(c) and (d), we also compare our experimental data tothe results of numerical simulations, shown with dashed lines. The latterare carried out by solving the Schro¨dinger equation for nitrogen moleculesinteracting with a sequence of δ-kicks, as described in Sec. 2.3.6. We cal-culate the complex amplitudes cJ,M of all rotational states in the wavepacket created from each initially populated state |J ′,M ′〉. Averaging overthe initial thermal mixture, we simulate the expected Raman signals IJ ∝∑M 〈|cM∗J cMJ+2|2〉J′,M′ , and find the exact populations PJ =∑M 〈|cMJ |2〉J′,M′ .In the case of a periodic sequence of kicks, the observed Raman line shape[Fig. 6.2(c)] is in good agreement with the numerical result down to the in-strumental noise floor around IJ ≈ 10−4. Calculated populations [Fig. 6.2(d)]demonstrate the anticipated exponential decay with the rotational quantumnumber, but deviate slightly from the experimentally retrieved distribution.We attribute this discrepancy to the small finite thermal width of the initialrotational distribution, not accounted for in approximating the populationsby√IJ , as discussed earlier.When the timing noise is simulated numerically, both the calculatedRaman response and the population distributions show a non-exponentialshape and match the experimental observations below J ≈ 15 (i.e. to theleft of the dotted vertical line). The disagreement at higher values of angularmomentum is because of the finite duration of our laser pulses, as discussedin Sec. 5.3.4. The dotted line represents this bandwidth limit.Dynamical localization is also susceptible to amplitude noise. Rather1436.4. Observation of dynamical localizationRotational quantum number J10-410-2100Raman signal(arb. units)a)10-410-2100 b)10-510-410-310-210-1100Ramansignal(arb. units)5 10 15 20 250c)10-310-210-1Population5 10 15 20 250d)Figure 6.3: Rotational Raman spectrum of nitrogen molecules excited witha train of N = 24 pulses at a kick strength of P = 2.3 for 20 different periodicsequences (a) without amplitude noise and (b) with amplitude noise. (c)The average experimental distributions (solid lines) are compared to thenumerical simulations (dashed lines) for both pulse trains, without noise(lower red lines) and with noise (upper blue lines). (d) The exact calculatedpopulations (dashed lines) and the approximate populations (solid lines),retrieved from the experimental Raman signal. The retrieved populationsare fitted with an exponential/Gaussian function (thick green lines). Thedotted vertical line represents the excitation limit due to the finite pulseduration.1446.4. Observation of dynamical localizationthan introducing timing noise to our pulse sequence, we now randomly varythe pulse amplitudes within a sequence. The individual amplitudes are dis-tributed around the mean kick strength of P = 2.3 with a standard deviationof 41%. We plot the results in Fig. 6.3. The red lines for the periodic pulsetrain without noise are identical to Fig. 6.2. The results for the pulse trainswith amplitude noise are plotted with blue lines. We observe the same qual-itative behaviour: noise destroys the localization. Again, the populationdistribution acquires a non-exponential shape, albeit not as pronounced asin the case of timing noise. Fitting a Gaussian distribution (thick greenline) yields a smaller 1/e width of Jdiff = 6.9. We conclude that the phe-nomenon of dynamical localization in molecular rotation is more susceptibleto timing noise than to amplitude noise. In the context of demonstratingdynamical localization, this works in our favour since the timing noise of ourexperimental pulse trains can be suppressed much better than the remainingfluctuations in the pulse amplitudes.In the whole section 6.4 we will use the same color coding: ‘red’ - periodicpulse train without noise, ‘blue’ - periodic pulse train with amplitude noise,‘black’ - non-periodic pulse train due to timing noise. The amount of noiseremains unchanged at 41% amplitude noise or 33% timing noise, given instandard deviation.6.4.1 Dependence on the number of kicksFigure 6.4 shows the evolution of the rotational distribution with the numberof kicks N . For the case of a periodic pulse train illustrated in Fig. 6.4(a1),the distribution becomes exponential within a few kicks and hardly changesafter that: Jloc = 3.1, 3.3 and 3.3 for N = 8, 16 and 24, respectively. Insharp contrast, the line shapes in Fig. 6.4(b1&c1) remain non-exponentialand keep broadening with increasing N in the case of periodic kicking withamplitude noise (Jdiff = 6.0, 7.2 and 7.4) and non-periodic kicking due totiming noise (Jdiff = 5.6, 6.2 and 7.9), respectively. This behaviour demon-strates the destruction of dynamical localization by noise and clearly distin-guishes it from other mechanisms of suppressed rotational excitation.We also give a comparison between the experimentally retrieved popula-tion in the left column (1) and the exact calculated population in the rightcolumn (2). Qualitatively, both sets are in agreement. Quantitatively, weobserve two substantial deviations, as expected. First, the calculated local-ization lengths in Fig. 6.4(a2) with Jloc = 2.1, 2.3 and 2.3 for N = 8, 16 and24, respectively, are shorter than the experimentally retrieved ones. Thisdiscrepancy is a result of the single-initial-state approximation. Second, a1456.4. Observation of dynamical localization10-210-1100Population010 15 205081624010 15 20510-210-1100Population010 15 205081624010 15 20510-210-1100Population010 15 205081624010 15 205a1) a2)b1) b2)c1) c2)Figure 6.4: Evolution of the molecular angular momentum distributionwith the number of kicks N for a periodic excitation without noise (a),with amplitude noise (b) and with time noise (c). The mean kick strengthper pulse is P = 2.3. The left column (1) shows the populations retrievedfrom the experiment, the right column (2) are the numerically calculatedones. Exponential and Gaussian fits, respectively, indicate the changes ofthe distributions at N = 8, 16 and 24 (thick green lines).1466.4. Observation of dynamical localization10-210-1Population10-3Rotational quantum number J10 20010 20010 200a) b) c)Figure 6.5: Dependence of the experimental populations on the kickstrength P for an excitation with N = 24 kicks, that are periodic with-out noise (a), periodic with amplitude noise (b) and non-periodic (c). Foreach case, the kick strength is varied from P = 1 (dotted), P = 2 (solid) toP = 3 (dashed). The dotted vertical line represents the excitation limit dueto the finite pulse duration.substantial deviation is caused by the bandwidth limit in the experiment.Due to the final pulse duration, the effective kick strength in nitrogen di-minishes for J & 15. This limit in the excitation of rotational states is notpresent in the calculation with δ-kicks. In Fig. 6.4(b2&c2) the 1/e widthsof the Gaussian distributions grow wider than the experimentally retrievedones. In the case of a pulse train with amplitude noise Jdiff = 5.9, 6.8 and7.9 and for time noise Jdiff = 6.4, 8.0 and Dependence on the kick strengthThe dependence of the rotational distribution on the strength of the kicksis shown in Fig. 6.5. For all three of the discussed scenarios, we show thepopulation after a train of pulses with P = 1 (dotted line), P = 2 (solid line)and P = 3 (dashed line). As expected for a periodically kicked quantumrotor (a), the localization length grows with increasing P : Jloc = 2.2, 2.9and 4.7 for P = 1, 2 and 3, respectively. The line shape remains exponentialbelow the cutoff value of J ≈ 15 discussed earlier. For each kick strength, theGaussian distribution after a noisy pulse sequence lies well above its localizedcounterpart, despite being equally affected by the cutoff, and thus confirming1476.4. Observation of dynamical localizationthe universality of the observed dynamics. The distributions obtained underthe influence of timing noise (c) with Jdiff = 6.7, 7.5 and 10.8 are broaderthan those corresponding to amplitude noise (b) with Jdiff = 5.1, 7.3 and9.0, emphasizing the higher susceptibility to timing noise. For clarity, thefitted lines have not been included in Fig. Rotational energyOwing to our state resolved detection, the total rotational energy of amolecule can be calculated as∑J EJPJ , with populations PJ extractedfrom the observed Raman spectra IJ . The rotational energy is plotted as afunction of the number of kicks for all three excitation scenarios in Fig. 6.6.For periodic kicking, the retrieved energy (red squares) increases duringthe first 3 kicks, after which its further growth is completely suppressed -a prominent feature of dynamical localization in the QKR. The same be-haviour is reproduced in the simulation (red dashed line), with the differenceof a systematic offset in the absolute energy. Numerically we can show thatthis offset is due to the approximations done to retrieve the populations.Random variations in the pulse amplitudes destroy the localization, whichis manifested by the continuously increasing rotational energy of the rotorswith the number of kicks. This is true for the experiment (blue triangles)and the exact simulation (blue dash-dotted line). Breaking the periodicityof the pulse sequence with timing noise results in an even stronger recoveryof the classical diffusion (black circles). A linear growth rate is expectedaccording to the calculations (black dotted line). The observed sub-lineargrowth is due to the finite duration of our laser pulses.6.4.4 Dependence on the periodPreviously, we pointed out the necessary condition for dynamical localiza-tion in the QKR: The period of the kicking field must not coincide with anyfractional quantum resonances T 6= pqTrev [52], where p and q are integers.However, we also stressed the influence of a finite pulse duration in termsof ‘partially’ overlapping with exactly those fractional resonances. In thissection, we investigate how the exact choice of the pulse train period willaffect the localization.We switch our molecular sample to oxygen, where only one spin parityexists and only half of all fractional quantum resonances are of concern.This simplifies the interpretation of the results. We use a pulse sequence1486.4. Observation of dynamical localizationEXPERIMENT&1.)&for&J=1&&use&first&data&point&2.)&for&J=0&&&use&assumpEon&that&everything&is&symmetric&around&Coh^2(J0)&=&&0.75*&Coh^2(J2)&&&&&&&&(taken&from&simulaEon&at&N=24)&&3.)&for&N=0&&use&the&measured&thermal&distribuEon&4.)&Pop&=&sqrt(<coh^2>)&&&&&&T&&normalize&only&states&where&&&<coh^2>&&&>&1eT3&&&(noise&floor)&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&T&normalize&even&to&2/3&&&and&odd&to&1/3&<Pop> SIM <sqrt(coh 2 )> EXP <Pop> SIM <sqrt(coh 2 )> EXP <Pop> SIM <sqrt(coh 2 )> EXP 0 4 8 12 16 20 24 Number of kicks N 20 40 60 Energy (arb. units) 0 80 Figure 6.6: Rotational energy as a function of the number of kicks N witha mean strength of P = 2.3. Compared are the experimentally retrievedenergies (connected symbols) with the numerically calculated ones (lines),for a periodic sequence (red squares, dashed line), and the same sequenceafter the introduction of amplitude noise (blue triangles, dash-dotted line)or timing noise (black circles, dotted line).of 13 pulses with periods on the order of T = 13Trev. Experimentally, thesesequences are produced without any Michelson interferometers. In the ab-sence of polarization multiplexing in the latter, we can reduce the opticallosses by a factor of two. This enables us to achieve stronger pulses with upto P = 8 per pulse. More details regarding the generation of pulse sequencescan be found in chapter 3.Figure 6.7(1a) shows an exponential distribution of angular momentumobtained after the molecules were exposed to such a pulse sequence. Thedistribution localizes after about 4 kicks. In order to smoothen the spec-trum, we measured the distribution after each pulse N . Plotted is the meandistribution averaged over the measurements for N = 6 to 13. We alsoaveraged over 10 pulse trains with equally spaced periods in the interval0.26 6 T/Trev 6 0.29, marked in the resonance map (2a). These periods arechosen to not overlap with any fractional resonances associated with low-lying rotational states. As expected, the rotational population distributionfalls off exponentially away from the initially populated states centred atJ = 1. We compare three scenarios with pulses of different kick strengthsP = 4 (dark-red, dashed line), P = 6 (light-red, dotted line) and P = 8 (red,solid line). The localization length increases with stronger kicks. Once the1496.4. Observation of dynamical localization0.2 0.25 0.3 0.35 0.40510QuantumnumberJPulse train period   ( T/Trev )0.2 0.25 0.3 0.35 0.45 10 15 20 2510-210-1PopulationP=4P=6P=81a 1bRotational quantum number J5 10 15 20 252a 2bFigure 6.7: (1a,1b) Localized angular momentum distribution of oxygenmolecules excited with a periodic train of 13 pulses for three selected kickstrengths of P = 4 (dashed), P = 6 (dotted) and P = 8 (solid). Thedotted vertical line represents the excitation limit due to the finite pulseduration. Shown are the mean distributions obtained from ten pulse trainswith equidistant periods in the interval 0.26 6 T/Trev 6 0.29 (2a) and0.315 6 T/Trev 6 0.325 (2b), marked in the resonance maps at the bottom.1506.4. Observation of dynamical localizationpopulation is getting closer to the bandwidth limit at Jlim ≈ 21 (dotted ver-tical line), we notice a sub-exponential tail due to the additional adiabaticlocalization.Next, we compare the key parameters of the QKR: the kick strength P ,the effective Planck constant τ and the stochasticity K = τP (see Sec. 2.2.3),with respect to our previous study in nitrogen, i.e. Fig. 6.2 and Fig. 6.6. (1)The pulse train periods are smaller by about a factor of three. Thereforethe effective Planck constant is reduced to τ ≈ 2pi 13 ≈ 2, which is still inthe quantum regime of the QKR dynamics (τ > 1). (2) Previously we werelimited to P(N2) = 3. Now we reach P(O2) = 8, larger by about a factorof three. Owing to stronger kicks, we observe longer localization lengths.(3) The stochasticity parameter K of both studies is comparable. For allmeasurements presented in Fig. 6.7(1a) it is well in the classically chaoticregime K & 5.Figure 6.7(1b) shows the angular momentum distribution, obtained withthe same procedure, but using 10 equally spaced periods in a different inter-val 0.315 6 T/Trev 6 0.325, marked in the resonance map (2b). Althoughthese periods are non-resonant as well, the proximity of the resonances forJ = 3 and 5 alter the shape of the distributions notably. The final pop-ulation distribution is achieved similarly after about 4 kicks, but the lo-calization center has shifted to J ∼ 5, 7. Apparently, the vicinity of thefractional quantum resonances can (to some degree) facilitate the transferof population into higher states when one uses laser kicks of non-zero length.Provided strong enough kicks, the center of the distribution initially movesup until it is further off all resonances, at which point the wave functiononce again localizes, but now with a shifted localization center.The same principle applies to the diffusive growth of angular momentumthat we observe under the influence of noise. In Fig. 6.8 we use the same13 pulses as before but eliminate the periodicity with timing noise. Themean period remains at T ∼ 13Trev with a standard deviation of ∼ 40%. Forbetter statistics we again average over 10 different pulse trains. In directanalogy to Fig. 6.7(1a,1b), we compare the final population distribution af-ter one ‘quasi’-random pulse train, which is engineered to avoid all quantumresonances that efficiently excite low-lying rotational states J = 1, 3 or 5,in Fig. 6.8(1a), and after another fully random pulse train in Fig. 6.8(1b).The 120 different periods belonging to each measurement are indicated inthe respective resonance maps (2a,2b).In both scenarios (1a,1b) we observe a non-exponential distribution thatdiffusively grows with the number of kicks, shown is the final distribution af-1516.4. Observation of dynamical localization5 10 15 20 2510-210-1PopulationP=4P=6P=8Rotational quantum number J5 10 15 20 250510QuantumnumberJPulse train period   ( T/Trev )0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.82a 2b1a 1bFigure 6.8: (1a,1b) Angular momentum distribution of oxygen moleculesexcited with a non-periodic train of 13 pulses for three selected kick strengthsof P = 4 (dashed), P = 6 (dotted) and P = 8 (solid). The dotted verticalline represents the excitation limit due to the finite pulse duration. Shownare the average distributions obtained from ten different pulse trains. Therandom periods follow a Gaussian distribution with a mean and standarddeviation of (a) T/Trev = 0.34 and 35%, (b) T/Trev = 0.32 and 43%, re-spectively. All 120 periods are marked in the resonance maps (2). In case(a), we make sure that no period is within 150 fs of any fractional resonanceassociated with J = 1, 3 or 5, whereas no such filtering is applied in case(b). The dashed red line marks the 1/3 resonance.1526.5. Transition from Bloch oscillations to dynamical localizationter 13 pulses. With increasing kick strength from P = 4 (dark-blue, dashedline), P = 6 (light-blue, dotted line) and P = 8 (blue, solid line) the distri-bution grows wider. For the quasi random train the most populated statesremain in the vicinity of the initial distribution. For the fully random trainthe distribution grows faster and the distribution center moves to J ∼ 11. Asthe distribution becomes increasingly broader with more/stronger kicks, thedifference between both cases will diminish. Then, the low-lying resonancesplay a minor role in the whole picture and full randomness is approached.6.5 Transition from Bloch oscillations todynamical localizationHere, we examine the phenomenon of localization in the vicinity of thequantum resonance. So far, we have analysed the excitation with periodsfar from the quantum resonance, leading to the dynamical localization ofangular momentum (Sec. 6.4). We also analysed the excitation on or nearresonance to study Bloch oscillations in angular momentum (Sec. 5.3). Now,we will revisit this regime, but with the intent to investigate the exponen-tially localized population distribution and how it is affected by the detun-ing. We will give an alternative explanation of Bloch oscillations where theresonant excitation is in fact limited by the dynamical localization at highquantum numbers. This transition between dynamical localization and alinear growth in momentum has been studied theoretically by Floss et.al.[184, 59, 63]. There exists no published experimental work with the QKRon this subject.6.5.1 Anderson wallThe rotational tight-binding model makes it evident that the occurrenceof Bloch oscillations in the angular momentum space is closely related tothe phenomenon of dynamical localization. Beyond a critical value of theangular momentum, the “Anderson wall”, the wave function localizes. Asa consequence, Bloch oscillations will appear in the molecular rotation onlybelow the Anderson wall.In this section we are investigating the QKR exposed to periodic resonantexcitation with periods T in the proximity of the quantum resonance Trev =(2cB)−1 withT = (1 + δ) Trev , (6.4)and the detuning |δ|  1. We start our theoretical treatment by revisiting1536.5. Transition from Bloch oscillations to dynamical localizationequation 2.57, the on-site energy term T (J) = tan[φ(J)] of the rotationaltight-binding model (Sec. 2.4.6), under our specific conditionsφ(J) =pi2((1 + δ)EhcB− (1 + δ)J(J + 1) + (1 + δ)DBJ2(J + 1)2)φ(J) ≈ pi2((1 + δ)EhcB− δJ(J + 1) + DBJ2(J + 1)2).(6.5)In the second line, we neglected the term, which is of second-order in bothperturbations |δ|  1 and DB  1, as well as the term pi2J(J + 1) becausemultiples of pi result in the same energy T (J) [62].We will explore four distinct scenarios, distinguishing a rigid rotor versusa non-rigid one, that is excited on quantum resonance or detuned from it.Rigid rotor, resonant excitation (δ = 0): In the rigid rotor approxi-mation, the centrifugal term D in the energy expression is set to zeroφ(J) =pi2EhcB. (6.6)The expression has no J-dependence and will result in a constant energy shiftT (J). Therefore, the tight-binding model presents a completely periodiclattice, which supports unlimited spreading of the rotational wave function,see Sec. 2.4.4. The quasi-energy states are extended over the rotationallattice [62], analogous to the Bloch states in crystalline solids.The realization of this scenario leads to a “ballistic growth” of rotationalenergy. Starting from an initial wave packet that is close to the rotationalground state, the wave packet’s center will increase linearly in angular mo-mentum, which translates into a quadratic growth of energy due to theE(J) ∝ J2 relation. This correlates with our intuitive understanding, thatany rotational wave packet reproduces itself after an integer multiple of therevival time (see the description of the resonance map in Sec. 2.3.8). Thus,N kicks with the kick strength of P will add constructively and excite themolecule equivalently to a single kick with the strength NP .Non-rigid rotor, resonant excitation (δ = 0): In the experiment, wewill observe a linear growth of the rotational population only for a limitednumber of pulses N , due to the non-rigidity of real molecules. Faster spin-ning of the molecule results in a stretching of the bond, i.e. centrifugaldistortion. The increased moment of inertia will in turn modify the reso-nant frequency, making the quantum resonance J-dependent, which becomesmost notable at higher quantum numbers.1546.5. Transition from Bloch oscillations to dynamical localizationIn the picture of the tight-binding model, we gradually transition from aperiodic lattice at low J-states to a pseudo-random lattice at higher J-states.Floss et.al have shown numerically that the phenomenon of dynamical lo-calization will occur beyond a threshold value JA, which they called the“Anderson wall” [60, 62]φ(J) =pi2( EhcB+DBJ2(J + 1)2). (6.7)The quantity of importance is the change of the on-site energy between theneighbouring rotational states J and J+2 of the angular momentum lattice,which is carried through∆φ(J) = |φ(J + 2)− φ(J)| ≈ 4piDBJ3 . (6.8)We regard only the leading term that scales with J3. Once ∆φ(J) reachespi/2, the tan(∆φ) diverges. For larger J-values, the phase difference ∆φgrows increasingly faster. However, only ∆φ modulo pi affects the variationof the on-site energy. The continued wrapping makes T (J) a pseudo-randomnumber. This statement is valid because D/B is irrational. The estimatedposition of the Anderson wall is [60, 62]JA ∼ 123√BD. (6.9)If the initial rotational wave packet consists of low-lying rotational statesJ < JA, we anticipate a spreading of the angular momentum with thenumber of kicks, which is bounded by J = 0 below, and the Bloch wall JB 6JA above. The states at J > JA project onto localized quasi-energy states,which have no overlap with the initial wave packet, and therefore cannotbe populated at any point in time. Instead of an unbounded spreading,one observes Bloch oscillations discussed earlier in Sec. 5.3. For oxygen andnitrogen molecules, the Anderson walls are expected around J(O2)A ∼ 33 andJ(N2)A ∼ 35, respectively, way above our bandwidth limit.Rigid rotor, non-resonant excitation (δ 6= 0): A similar behaviourmay be induced by detuning from the quantum resonance T = (1 + δ)Trev.If the detuning δ is large enough, the centrifugal term can be neglected andφ(J) =pi2((1 + δ)EhcB− δJ(J + 1)). (6.10)1556.5. Transition from Bloch oscillations to dynamical localizationAgain, we can distinguish two regimes, the one of extended states and theone of localized states, depending on∆φ(J) = piδ(2J + 3) . (6.11)Owing to the linear dependence on J , the two regimes will alternate in theangular momentum space [62]. Whenever the rotational states are in thevicinity of J(n)A ∼ 2n+14δ such that ∆φ(J) ∼ 2n+12 pi with integers n, the on-site energies are quasi-random for neighbouring J-states. We consider thosestates as effective Anderson walls. If the rotational states, however, are closeto J ∼ 2n4δ such that ∆φ(J) ∼ npi, adjacent lattice sites have nearly equalenergies, which corresponds to a periodic case with extended eigenstates.In the experiment with the initial wave packet close to J ≈ 0, it projectspredominantly onto extended states (n = 0). The initial linear growth inthe angular momentum is bound by the lowest Anderson wall J(0)A . For ourexperimental conditions with moderate kick strengths, we are not able to“jump” across this Anderson wall to the next set of extended states (n = 1).Qualitatively, we expect to observe the same Bloch oscillations mentionedabove (in the case of a non-rigid rotor at δ = 0), but this time we can controlthe location of the Anderson wall by adjusting the detuning δ.Non-rigid rotor, non-resonant excitation (δ 6= 0): This scenario isa combination of the last two. Using the same arguments, the term thatdominates the expression of ∆φ(J) with increasing J-states will determinethe location of the Anderson wall.The magnitude of the phase difference ∆φ(J) determines whether theparticular rotational states will map onto extended quasi-energy eigenstates(∆φ(J) → 0) or the localized ones (∆φ(J) → pi/2). For a specific diatomicmolecule (D/B = const.), the width and the location of these regions ofextended and localized dynamics can be controlled by the detuning δ fromthe quantum resonance. The overlap of the initial states with either of thetwo regions will determine the dynamics of the QKR, i.e. whether it willdynamically localize or feature Bloch oscillations.We note an important difference between the described behaviour ofthe QKR and the dynamics of a quantum particle in a disordered one-dimensional lattice. In the latter case, the particle will localize no matterhow weak the disorder is. In a rotational lattice, however, introducing asmall detuning from the resonance may not necessarily lead to the expo-nential localization of the wave function. The reason is the dependence of1566.5. Transition from Bloch oscillations to dynamical localizationthe on-site energy on the site number J (cf. Eq. 6.11). Effectively, the “ro-tational disorder” increases with the distance from the edge of the latticeat J = 0. Hence, even though any detuning will result in the restrictedgrowth of angular momentum, the distribution of the wave function will notnecessarily be exponential.6.5.2 Evolution of angular momentumAt first, we look at the angular momentum distribution and how it evolveswith the number of kicks. The general procedure including the populationretrieval from the Raman spectra is equivalent to the one described before.Figure 6.9 and 6.10 are a compilation of six different periodic pulse trains,whose periodicities are continuously tuned further below the quantum res-onance. Shown are plots for the following detunings: (a) 0%, (b) −0.08%,(c) −0.16%, (d) −0.24%, (e) −0.5% and (f) −1.0%. The left column (1) ofexperimentally retrieved populations are compared to the calculated exactpopulations in the right column (2). The kick strength of the pulses in asequence of 24 kicks is set to P = 0.9. For nitrogen molecules this meansthat the bandwidth limit of Jlim ≈ 15 will be reached at the end of the train23.On the quantum resonance in Fig. 6.9(a), the center of the populationdistribution linearly increases to higher J-states, as expected and seen inthe calculations (a2). At the end of the train the population reaches as faras the Anderson wall allows. For higher quantum numbers, the distributionfalls off exponentially, owing to dynamical localization due to the centrifugaldistortion of the non-rigid rotor. This concept was introduced in section 5.3on Bloch oscillations. In the experiment (a1) the linear growth of angularmomentum slows down towards the end of the train. For these higher quan-tum numbers, the effective kick strength decreases due to the finite pulseduration. It results in a stronger localization - a superposition of adiabaticlocalization and dynamical localization.In the following rows (b-f) we detune the period increasingly further be-low the quantum resonance. According to the theory (2), we push down thewall responsible for the turning point of Bloch oscillations. At a detuningbetween −0.16% (c2) and −0.24% (d2) the population distribution com-pletes one full Bloch oscillation within 24 kicks. Eventually, at a detuningof −1% (f2), the population distribution is barely distinguishable from its23For quantum numbers exceeding J = 15 the population is adiabatically localized. Thisparticular value of the kick strength has been chosen to avoid multiple Bloch oscillationsunder the resonance condition.1576.5. Transition from Bloch oscillations to dynamical localization10-210-1100Population010 15 205241680010 15 20510-210-1100Population010 15 205241680010 15 20510-210-1100Population010 15 205241680010 15 205a1) a2)b1) b2)c1) c2)Figure 6.9: Evolution of the molecular angular momentum distributionwith the number of kicks N for a periodic sequence. Each individual pulsecarries the kick strength P = 0.9. Left column (1) contains the experimentalresults; right column (2) shows the corresponding numerical plots. Theperiodicity is continuously tuned further below the quantum resonance (a)with detunings of δ = −0.08% (b) and −0.16% (c). (continued in Fig. 6.10.)1586.5. Transition from Bloch oscillations to dynamical localization10-210-1100Population010 15 205241680010 15 20510-210-1100Population010 15 205241680010 15 20510-210-1100Population010 15 205241680010 15 205d1) d2)e1) e2)f1) f2)Figure 6.10: (Continuation of Fig. 6.9.) Evolution of the molecular angularmomentum distribution with the number of kicks N for a periodic sequence.Each individual pulse carries the kick strength P = 0.9. Left column (1)contains the experimental results; right column (2) shows the correspondingnumerical plots. The periodicity is continuously tuned further below thequantum resonance with detunings of δ = −0.24% (d), −0.5% (e) and−1.0% (f).1596.5. Transition from Bloch oscillations to dynamical localizationinitial distribution, apart from some residual fast Bloch oscillations. For de-tunings > 1% the system is completely governed by dynamical localization.The experimental results (1) demonstrate the same behaviour, i.e the Blochwall location in the angular momentum space is continuously lowered, Blochoscillations become visible (d1), and eventually the population distributionlocalizes and does not grow anymore (f1).For large detunings from the quantum resonance our kick strength ofP = 0.9 is not sufficient to significantly alter the initial thermal distri-bution. All states are localized, but the shape of the observed spectrumremains close to the thermal Boltzmann distribution.0.00%-0.08%-0.16%-0.24%-0.50%-1.00%2 4 6 8 10 12 14 16 18 20Rotational quantum number J10-210-1Population10-31Figure 6.11: Populations, retrieved from the experimental Raman signal,after the excitation by 24 periodic pulses with a kick strength of P = 0.9 perpulse. The detuning from the quantum resonance ranges from 0 to −0.08%,−0.16%, −0.24%, −0.5% and −1.0% (blue to red coloured lines, see legend).In Fig. 6.11 we plot the final population of the angular momentum statesof nitrogen molecules after the excitation with 24 pulses. Shown are thedistributions for all six periods with detunings from the quantum resonanceranging between 0% and −1%. The figure provides a good comparison tothe same theoretical plot presented in Ref. [59, 63]. On resonance (blue line)the distribution is flat for J 6 12 (on a logarithmic scale) before it sharplyfalls off. With increasing detuning, the extend of the plateau decreases,but the sharp drop remains. For the largest detuning (red line) the plateauhas vanished and the population is localized. Owing to the weakness of1606.5. Transition from Bloch oscillations to dynamical localization0204060800 24 0 24 0 24 0 24 0 24 0 24Energy (arb. units)Number of kicks Na) b) c) d) e) f)Figure 6.12: Rotational energy as a function of the number of kicks N witha strength of P = 0.9. Different panels show the dependence on the pulsetrain period: (a) on the quantum resonance, 0%, (b) −0.08%, (c) −0.16%,(d) −0.24%, (e) −0.5% and (f) −1.0% away from the quantum resonance.the pulses at P = 0.9, the distribution resembles the initial Boltzmanndistribution at 25 K.6.5.3 Evolution of rotational energyIt is instructive to look at the total rotational energy in the system. Wehave seen in Fig. 6.6 that in the case of non-resonant periodic kicking, theenergy diffusively grows for several kicks before the growth is suppressedby dynamical localization. In the case of resonant excitation the energy isexpected to grow quadratically at first, before the centrifugal distortion willlead to oscillations in the total energy.For the same six periodic pulse trains, we now calculate the rotationalenergy in the molecular system. In Fig. 6.12 the energy is given as a functionof the number of pulses in the train. On resonance (a) the energy increasesmonotonically throughout all 24 pulses. With larger detunings, the totalenergy that the system accepts becomes smaller. The initial growth rate isthe same, but the reduced Bloch wall forces the energy to oscillate. Finally,for a −1% detuning (f) the energy growth is completely suppressed. Thisis a powerful demonstration of the consequence of dynamical localization:Although in all six cases the energy of the pulse train is identical and theperiod changes by merely 1%, the total amount of energy that one can pumpinto the system differs by a factor of almost eight.Besides the dependence on the detuning, we can also plot the dependence1616.6. Conclusion0 12 24 0 12 24 0 12 24Number of kicks NEnergy (arb. units)020406080100 a) b) c)Figure 6.13: Rotational energy as a function of the number of kicks Nwith a strength of P = 1.1 (dashed lines) or P = 2.3 (solid lines). Differentpanels show the dependence on the pulse train period: (a) on the quantumresonance, 0%, (b) −0.24%, (c) −1.0% away from the quantum resonance.on the kick strength. In Fig. 6.13, for three values of the detunings [(a) 0%,(b) −0.24%, (c) −1.0%], we plot the observed energy growth for two kickstrengths of P = 1.1 (dashed line) and P = 2.3 (solid line). Strongerpulses lead to a more rapid growth of energy and to a higher maximumenergy value. The reason has already been given in the context of Blochoscillations in Sec. 5.3 when we demonstrated that the amplitude of theoscillations in the angular momentum space increases with the kick strength.The initial growth of energy has to be quadratic, which can be observed mostclearly in panel (a). For the largest detuning in panel (c), there are abouttwo complete oscillations. According to the theory, one would expect cleanoscillations with maximum contrast: the energy at the minimum of eachoscillation should equal the initial energy of the molecule. This is not thecase in the experimental data. The most likely suspect for the mismatchis that averaging over the spatial beam profiles washes out the oscillations.Similar contrast in Bloch oscillations were reported by Kamalov et.al. witha different detection technique [88].6.6 ConclusionWe presented the first experimental demonstration of the quantum phe-nomenon of dynamical localization in the angular momentum space of aperiodically kicked rotor.We showed that laser-kicked molecules have the potential of a testing1626.6. Conclusionground for a number of new physical phenomena. Some of them have beenpreviously studied in AOKR, e.g. dynamical localization, the quantum res-onance, the susceptibility to noise. At the same time, the QKR systemoffers new perspectives. The molecular angular momenta are quantized,whereas continuous translational atomic momenta hinder a clear observa-tion of momentum distributions. Cold temperatures achieved in standardsupersonic expansions are sufficient to reach the rotational ground state,while the AOKR requires ultracold atoms, obtained in complex experimen-tal setups. The periodic excitation of real rotors also leads to some uniquephenomena, e.g. the occurence of Bloch oscillations due to centrifugal dis-tortion (Sec. 5.3) or the predicted and yet-to-be-demostrated edge states[61].163Chapter 7Coherent control of quantumchaosControl of molecular dynamics with external fields is a long-standing goalof physics and chemistry research. Great progress has been made by ex-ploiting the coherent nature of light-matter interaction. At the heart ofcoherent control is the interference of quantum pathways leading to the de-sired target state from a well-defined initial state [158, 157]. In this context,an exponential sensitivity to the initial conditions, characteristic for classi-cally chaotic systems, poses an important question about the controllabilityin the quantum limit (for a comprehensive review of this topic, see [71]).As the underlying classical ro-vibrational dynamics of the majority of largepolyatomic molecules is often chaotic, the answer to this question has farreaching implications for the ultimate prospects of using coherence to controlchemical reactions.Success in steering the outcome of chemical reactions by the means offeedback-based adaptive algorithms [11], using the methods of optimal con-trol theory [87], proved that such control is feasible. Theoretical works onquantum controllability in the presence of chaos, both in general [142] andwith regard to specific molecular systems [70, 3], pointed at the importanceof coherent evolution. To investigate the roles of coherence, stochasticity andquantumness further, Gong and Brumer considered the quantum kicked ro-tor to study quantum effects on classically chaotic dynamics [69, 70, 71].They demonstrated that the energy of the localized state can be controlledby modifying the initial wave packet. Quantum coherences, as opposed tothe classical structures in the rotor’s phase space, are indeed responsible forthe achieved control over the chaotic dynamics of the QKR [154].In this chapter, we present an experimental proof of the Gong-Brumercontrol scheme. We prepare oxygen molecules in a coherent rotational wavepacket and control the localization process of the QKR by varying the rela-tive phases of the initial states. In Sec. 7.1, we outline relevant theoreticalconcepts and discuss the transition from quantum to classical regimes of the1647.1. TheoryQKR dynamics. We give a brief overview of the experiment in Sec. 7.2 andpresent all results in Sec. 7.3. The conclusions are in Sec. TheoryThe main experimental approach is based on the effect of dynamical local-ization, described in detail in chapter Coherent controlThe general concept of “coherent control” is based on the availability ofquantum interferences, which can be altered in order to change the proba-bility of reaching the desired target state [157]. Until now, coherent controlhas been used most frequently in atomic or molecular systems exhibitingrelatively simple, e.g. periodic, dynamics [71, 157]. The focus of this workis on the control of quantum objects, whose underlying classical dynamicsis chaotic.7.1.2 Quantum-to-classical transitionThe transition from classical behaviour to quantum behaviour can be definedby comparing the Planck’s constant ~ with the classical action of the system[53]. In the case of the quantum kicked rotor, the latter can be expressedas (I/T ), the moment of inertia divided by the period of kicking [77]. Thus,the effective Planck constant τ = ~(I/T )−1, from Eq. 2.5, provides a simpleestimate of the degree of “quantumness”. For τ → 0, the system approachesthe classical limit, whereas for τ > 1 it is expected to show its quantumnature [77, 138].The fact that the stochasticity K = τP of the QKR is comprised of twoindividual parameters, the effective Planck constant and the kick strength,allows for a unique control knob. One can fix K in the deeply chaoticregime and observe the quantum-to-classical transition by reducing τ fromvalues greater than one towards zero. However, even if a quantum systemcan initially be described by means of the classical equations of motion,quantum effects will still accumulate with time, making the system deviatefrom its classical counterpart after the “quantum break time” [53]. In otherwords, the ability to tune τ to smaller values results in a larger windowwhere the classical behaviour can be observed. The correspondingly largervalues of the kick strength P for a constant stochasticity K, result in alonger localization length. Therefore, it takes more pulses for the QKR1657.2. Experimentto diffuse over a distance comparable to the localization length. But onceit has diffused that far, quantum effects will again lead to the dynamicallocalization.The conjugate variables of the QKR are the angle and the angular mo-mentum, which obey the commutation relation [J, θ] = −i~ [77]. The clas-sical trajectory, described by the standard map, is presented via the scaledangular momentum J˜ = J(I/T )−1 = Jτ/~, which yields the dimensionlesscommutation relation [J˜ , θ] = −iτ .In the following we give an intuitive explanation of the classical to quan-tum transition, which is adapted from Ref. [53]. (1) An initial state of aquantum system is assumed to be a localized wave packet, i.e. a small areain the available phase space. Its minimum size, a “Planck cell”, is limitedby the Heisenberg uncertainty principle and according to the commutationrelation is given by τ . The corresponding classical system consists of severalpoints in the phase space occupying the same area. (2) As a function oftime, the wave packet spreads diffusively, with each initial point followingits classical trajectory. (3) After the quantum break time, the wave packethas spread so much, that its different parts start overlapping, and hence in-terfering, with one another. The resulting interference is purely a quantumphenomenon.The true transition to a classical kicked rotor can be implemented byintroducing noise, as we showed in Sec. 6.4, or other decoherence mecha-nisms, e.g. collisions between the molecules. This will destroy the quantumcoherences in the system irrevocably such that dynamical localization canno longer occur.7.2 ExperimentThe setup and the experimental procedure is mostly identical to the onesused to observe dynamical localization (Ch. 6). The only crucial differenceis the shape of the femtosecond pulse train. A sequence of 15 high-intensitylaser pulses is generated in the combined setup of a pulse shaper and amulti-pass amplifier (MPA). It consists of two independent parts, shown inFig. 7.1. The first three “preparation” pulses are separated in time by Tpre =0.237 Trev, close to a fractional quantum resonance at T =1/4 Trev, andare used to excite a broad rotational wave packet, defined by the resonantprocess studied in Sec. 5.4. The period Tloc of the second “localizing” trainof 12 pulses is chosen between 0.26 Trev and 0.27 Trev, corresponding to the1667.3. Demonstration of coherent control in quantum chaotic systemeffective Planck constant of 1.6 < τ < 1.7. This window is chosen so as toavoid strong fractional quantum resonances of low orders.0.2 0.4 0.6 0.8 1Delay  (Δ𝑇/𝑇rev)4080120Energy (arb. units)𝑇pre 𝑇locΔ𝑇𝑁loc = 12𝑁pre = 32 4 6 8 10 12 14Number of kicks N020406080100120Energy (arb. units)bcaFigure 7.1: Train of fifteen laser pulses, with three variable time constantsindicated by horizontal arrows.The experiments are done with oxygen, whose nuclear spin statistics(Sec. 2.3.9) make only half of all fractional resonances relevant. Figure 7.2visualizes pulse train periods with respect to the position of the fractionalresonances. The time delay ∆T between the two pulse trains, and hence therelative quantum phases of the initial states, serves as a “control knob”. Itwill be used to control the amount of the rotational energy that is absorbedby the molecules before its further growth is suppressed by localization.0 0.1 0.2 0.3 0.405101520Pulse train period   ( T/Trev )Quantum numberJFigure 7.2: Resonance map indicating all relevant times: Tpre = 0.237 Trev(red dotted line) and 0.26 Trev < Tloc < 0.27 Trev (blue area).1677.3. Demonstration of coherent control in quantum chaotic system0.2 0.4 0.6 0.8 1Delay  (Δ𝑇/𝑇rev)4080120Energy (arb. units)2 4 6 8 10 12 14Number of kicks N020406080100120Energy (arb. units)abFigure 7.3: (a) Rotational energy of oxygen molecules as a function of thenumber of kicks N . Shown are thirteen experimental realizations (dottedlines) for each of the two control scenarios corresponding to a maximum(upper green lines, at ∆T1 = 0.243 Trev) and a minimum in the absorbedrotational energy (lower red lines, at ∆T2 = 0.264 Trev). The correspondingaverage values are plotted as the green solid line and the red dashed line,respectively, with error bars representing one standard deviation. In com-parison, the numerical calculations are indicated by connected green circles(∆T1) and red squares (∆T2). (b) Numerically calculated dependence ofthe final rotational energy on the delay ∆T . Two vertical lines mark theexperimental delays ∆T1 (solid green) and ∆T2 (dashed red).1687.3. Demonstration of coherent control in quantum chaotic system7.3 Demonstration of coherent control inquantum chaotic systemOur main result is shown in Fig. 7.3(a), where we plot the rotational energyof oxygen molecules, measured after each of 15 laser pulses for a number ofpulse trains, all with Tloc = 0.267 Trev. The kick strength is set to P = 3.8,which corresponds to a stochasticity parameter K = 6.4 lying deep in theclassically chaotic regime. By design, the first three preparation pulses inall trains lead to a fast growth of molecular energy. The time delay ∆Tbetween the three preparation pulses and the localizing train of 12 pulsesis scanned around the quarter revival time, between ∆T/Trev = 0.223 and0.284, where we anticipate the highest degree of control, as discussed below.When the delay is set to ∆T1 = 0.243 Trev (upper green lines), the en-ergy growth continues for a few more kicks and ceases after that, reflectingdynamical localization of the molecular angular momentum, investigated inchapter 6. Different thin lines correspond to different experimental runs,with their average indicated by the thick green curve. On the other hand,when the very same localizing pulse sequences are separated from the prepa-ration pulses by ∆T2 = 0.264 Trev, the suppression of the energy growthoccurs much earlier and results in a lower (by 40 ± 7%) energy of the finallocalized states (lower red lines).In Fig. 7.3(a) we also show the results of the equivalent numerical cal-culations by connected green circles for the delay ∆T1 and red squares for∆T2. Despite the used approximation of infinitely short δ-kicks, the nu-merical results are in good qualitative agreement with the observations.We further exploit the numerical model for calculating the dependenceof the rotational energy on the single control parameter ∆T , plotted inFig. 7.3(b). The availability of control is apparent around fractional re-vivals, ∆T/Trev =1/4,1/2,3/4 and 1, which suggests an intuitive picture ofits mechanism. The first kick from the localizing pulse train either continuesthe quantum-resonant excitation of the preparation sequence or opposes it,affecting the energy level, at which the rest of the train localizes the system.The dephasing of the rotational states in the prepared wave packet leads toa loss of control between the fractional revivals. The two vertical lines markour experimental values of ∆T in Fig. 7.3(a).The described control mechanism is also evident from the experimentallyretrieved average distributions of the localized angular momentum, shownin Fig. 7.4 by thick lines with no markers. Solid green and dashed red traces1697.3. Demonstration of coherent control in quantum chaotic systemOne PT: optimized delay – average 13x0 5 10 15 20 25Rotational quantum number J10-210-1PopulationFigure 7.4: Localized population distribution of oxygen molecules excitedby a train of 15 pulses with P = 3.8. Plotted is the experimentally retrievedaverage population distribution (thick lines, no markers) and the numeri-cally calculated one (markers, connected by thin lines). The distributionscorrespond to the high (upper green lines, ∆T1 = 0.243 Trev) and low (lowerred lines, ∆T2 = 0.264 Trev) localization energy in Fig. 7.3.correspond to the localized wave packets with higher and lower rotational en-ergies, respectively. As the higher energy clearly correlates with the broaderwave packet, the achieved control can be attributed to populating differentsets of quasienergy (Floquet) states [63]. Because each wave packet containsmore than a single quasienergy state, the distributions are not expected to(and, indeed, do not) exhibit exponential line shapes [69].Numerically calculated population distributions, corresponding to theexperimental parameters for the high and low energy localized wave pack-ets, are shown in Fig. 7.4 with connected green circles and red squares, re-spectively. The simulated and experimental distributions show qualitativeagreement down to the instrumental noise floor around PJ ≈ 5 · 10−3. Thesystematic underestimation of the experimentally extracted population atlow rotational states is attributed to two effects. First, the approximationsin the population retrieval from the measured Raman spectra neglects thedependence on the magnetic quantum number and the thermal populationof multiple rotational states, as discussed in Sec. 6.3.2. Second, the effect ofspin-rotation coupling in oxygen (Sec. 4.2.3) leads to a more rapid dephasingof the Raman signal at low J-states, which is not taken into account in thenumerics.1707.3. Demonstration of coherent control in quantum chaotic system5 different PTs: common vs optimized delays(PLOT4   excha ged by the on  from 1.5.2016)Number of kicks N5 10 5 105 10 5 10 5 10050100050100Energy (arb. units)0.02 ± 0.04 0.20 ± 0.04 −0.06 ± 0.05 0.40 ± 0.07 0.05 ± 0.110.34 ± 0.05 0.39 ± 0.06 0.32 ± 0.04 0.40 ± 0.07 0.36 ± 0.101a 2a 3a 4a 5a1b 2b 3b 4b 5bFigure 7.5: Top row (a): rotational energy for both time delays, ∆T1 =0.243 Trev (sold green line) and ∆T2 = 0.264 Trev (dashed red line) for a setof five different Tloc periods (1)-(5), given in the text. Other parametersof the localizing train remain unchanged. Bottom row (b): for the samefive values of Tloc, delays ∆T1 and ∆T2 are individually adjusted for therespectively highest and lowest energy of the localized state. The degree ofcontrol is given in each plot. Column (4) is equivalent to Fig. Robustness of controlThe stability of the implemented control scheme with respect to the under-lying classically chaotic dynamics is analysed in Fig. 7.5. In the top row(a) we show the dependence of the rotational energy on the period of thelocalizing train Tloc. As earlier, the value of the control parameter is either∆T1 = 0.243 Trev (sold green line) or ∆T2 = 0.264 Trev (dashed red line).Shown is a representative set for five values of Tloc/Trev: (1) 0.260, (2) 0.261,(3) 0.263, (4) 0.267 and (5) 0.270. The respective degree of control, definedas E1−E2(E1+E2)/2 with Ei being the final rotational energy for the delay ∆Ti, isshown at the bottom of each plot. We observe wide fluctuations from a totalloss of control in panels (1a,3a,5a) to the maximum control of about 40%in panel (4a).High sensitivity of the QKR dynamics to the exact train period is well ex-1717.3. Demonstration of coherent control in quantum chaotic system2 4 6 8 10 12 14Number of kicks N2 4 6 8 10 12 14Number of kicks N020406080100120Energy (arb. units)0204060801001205 10 15 20 25 30050100150a bFigure 7.6: Same as Fig.7.3, but for two different values of the effectivePlanck constant: τ = 1.7 (a) and τ = 0.6 (b). The stochasticity parameteris held constant at K = 3.4. The inset gives a numerical comparison ofboth regimes, with lower (top) and higher (bottom) value of τ , for longersequences of infinitely short δ-kicks.pected [154] and can be attributed to the existence of fractional resonances,Tloc/Trev = p/q, where quantum diffusion is accelerated. Yet despite theobserved sensitivity of the control, we found that it can be successfully re-gained by optimizing the control parameter, i.e. the delay time ∆T , foreach individual realization of the localizing train. In the bottom row (b) ofFig. 7.5 we demonstrate this sustained controllability, which supports theassumption of its coherent nature. We note that our numerical calculationsof the molecular response to the localizing train of infinitely short δ-kicks(not plotted) show more stable control, which suggests that the finite ex-perimental pulse width may also contribute to the observed sensitivity.7.3.2 Quantum-to-classical transitionTo distinguish between the quantum and classical mechanisms of the achievedcontrol, we analyse its dependence on the effective Planck constant τ . Smallervalues of τ , realized with shorter periods of the pulse train, take us closerto the classical limit (i.e. the well-known standard map [31]), at which thedynamics is less sensitive to the discreteness of the QKR spectrum. Wekeep the stochasticity parameter constant at K = τP = 3.4, large enough1727.4. Conclusionto remain in the fully chaotic regime, and reduce τ while increasing the kickstrength P proportionally. As demonstrated in Fig. 7.6(a), for τ = 1.7, thelocalized states are reached after about 10 kicks. The quantum break timeis longer than the one in Fig. 7.3(a) due to the lower kick strength (P = 2vs. 3.8). The maximum degree of control (25± 3%) is established between∆T1 = 0.232 Trev (solid green line) and ∆T2 = 0.263 Trev (dashed red line).Figure 7.6(b) shows the result of the same experiment with τ = 0.6and P = 5.6. Owing to much stronger kicks, the quantum break time hasbecome longer than the available number of pulses, such that the kicked rotorbehaves classically within the observable time frame. Although the dynamicsis still sensitive to ∆T , the unbounded growth of rotational energy resultsin the decreasing relative difference between the two cases and, therefore,diminishing degree of coherent control. The apparent energy saturation atlater times is due to the finite duration of our laser pulses. It results in thesuppressed excitation of rotational states with J & 21, as we explained at theend of Sec. 5.3.4. Numerical simulations with a larger number of δ-kicks,shown in the inset, better illustrate the transition between the controlledlocalization at τ = 1.7 (bottom two lines) and the uncontrolled classicaldiffusion at τ = 0.6 (top two lines). Evidently, the latter effective Planckconstant is small enough for the diffusive energy growth to persist. Thisbehaviour is universal if the two periods are both chosen to be in the off-resonance regime.7.4 ConclusionIn summary, we used oxygen molecules exposed to a sequence of stronglaser pulses as true quantum kicked rotors. We demonstrated that despitethe exponential loss of memory about the initial conditions in the classicallychaotic limit, the relative phases in the initial coherent superposition ofrotational states can be used to control the QKR dynamics in the absenceof noise or decoherence. Adjusting a single control parameter results inthe changing rotational distribution of the final localized state: its peak isshifted from a low (here, J = 7) to a high (J = 11) angular momentum.This corresponds to a relative change in the rotational energy, absorbedby the laser-kicked molecules. The coherent quantum nature of the controlmechanism is evident from the demonstrated high sensitivity of the localizedwave packet to the exact period of the pulse train, and the ability to regaincontrol for any value of that parameter. Driving the system closer to theclassical limit, while maintaining the same degree of stochasticity, results in1737.4. Conclusiona gradual loss of control.This proof-of-principle experiment in a simple chaotic system marks afirst step towards the control of more complex systems.174Chapter 8OutlookIn this thesis, we described the recent progress in the experimental stud-ies of the quantum kicked rotor. Linear molecules, i.e. diatomic oxygenor nitrogen, served as quantum rotors and were exposed to periodic se-quences of high-energy femtosecond laser pulses. The main achievementswere the direct observation of dynamical localization and the ability to ex-ecute the methods of coherent control in the regime of quantum chaos. Weanalysed the dependence of the localization phenomenon on various kickingparameters, i.e. kick strength, number of pulses, pulse duration and pulseperiod. We studied the effect of the quantum resonances and investigatedrotational Bloch oscillations and rotational Rabi oscillations. In addition,the quantum nature of the observed phenomena was tested and confirmed.We transitioned from the quantum to the classical limit in two fundamen-tally different ways, by introducing noise or by reducing the time period andwith it, the effective Planck constant of the QKR. Upon this transition, theclassically chaotic behaviour was recovered and the possibility of control waslost. A more detailed summary of each separate topic can be found in theconclusion sections of the experimental chapters 5, 6 and 7.Most of the observations were made possible due to the development ofnew excitation and detection technologies, described in the chapters 3 and4. Our work paves the way to many interesting studies. We outline some ofthem in the next closing pages of this thesis.Edge statesAn interesting theoretical prediction is the existence of the localized edgestates [61]. The rotational quantum number of a three-dimensional rotorhas an effective edge at J = 0 in the rotational lattice. Floss et.al. showedthat under certain conditions, the quantum rotor can be localized at thisedge, similarly to the localization of a particle near the edge of a real lattice.According to the calculations, we can meet all the necessary requirementsto observe this new effect. We did manage to see the glimpse of the edgestate in O2 but the definitive demonstration would require heavier molecules175Chapter 8. Outlooklike N2O or CO2. Such demonstration is of interest because it represents aphenomenon, which is unique to three-dimensional rotors and does not existin the AOKR.Coherent control of quantum chaosWe studied the Gong-Brumer control scheme where the phases of the initialrotational states are used to control the dynamical localization [70, 69].However, changing the phases of the initial states does not affect the Floquetevolution operator. This means that the average localization length, whichis determined by the quasi-energy eigenstates, cannot be changed.Other control scenarios have been studied theoretically, which do allowmore drastic changes of the QKR dynamics. For example, the modifiedkicked rotor (MKR), where the kicking field is reversed every n kicks, cancontrol the localization length and the shape of the localized distribution[73]. The energy absorption of the MKR can be significantly acceleratedcompared to the underlying classical anomalous diffusion [72]. The MKRwould be easy to implement experimentally. The periodic field reversal isequivalent to a pi phase shift, which can be executed by introducing an ap-propriate time delay between the kicks. The pulse train with a field reversalat every second kick (n = 2), could easily be designed with the Michelsoninterferometers by shifting the time delay of one arm of an interferome-ter. One needs to verify, however, whether the limited amount of pulses issufficient to show the predicted effects.DecoherenceWe investigated the effect of noise on the QKR. Timing noise or amplitudenoise in the periodic kicking is essentially destroying the analogy with thetight-binding model of a one-dimensional periodic lattice. Therefore, thesystem no longer corresponds to an Anderson model and dynamical local-ization gives way to classical diffusion.Instead of introducing noise, one could couple the QKR system to theenvironment, which would leave the Anderson model intact. Studying theeffects of true decoherence on quantum chaos could yield important insightsinto the quantum-classical transition and the question of controllability. Apractical way to induce decoherence in the experiment is through collisions.A background gas with tunable pressure can be leaked into the vacuumchamber to create an adjustable decoherence mechanism.176Chapter 8. OutlookDynamical localization in three dimensionsAn interesting future direction is the generalization of the QKR to higherdimensions. It was shown that using several driving fields with incommensu-rable frequencies is equivalent to higher dimensions of the Anderson model[32]. Each frequency corresponds to a spatial coordinate in the Andersonmodel. Such quasi-periodic pulse sequences have been realized in the AOKR[34, 106, 110], enabling for example the study of the metal-insulator phasetransition.Work in this direction will likely require an improvement or modificationof our current pulse shaping setup. Rather than just using the technique ofpulse multiplexing via Michelson interferometers, it might be beneficial tocombine several pulse shapers to create the required multiple frequencies.Multi-pulse excitation schemesOur unique setup could be used to explore other multi-pulse excitationschemes, following goals that are not related to dynamical localization.For instance, one can use long pulse sequences to improve the degree ofmolecular alignment. One could follow the proposal of Averbukh [12] or, asmentioned in Sec. 5.5, adjust the train period from kick to kick. It wouldbe best to try these optimization schemes in heavier molecules where thebandwidth limit is of no concern.More interesting is the alignment or orientation of larger and asymmetricmolecules. In a recent work, three pulses have been used for the three-dimensional alignment of asymmetric top molecules [139]. 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Rotational excitation of molecules withlong sequences of intense femtosecond pulses. Physical Review A, 93,013420, 2016 [24].3. M. Bitter and V. Milner. Experimental Observation of DynamicalLocalization in Laser-Kicked Molecular Rotors. Physical Review Let-ters, 117, 144104, 2016 [22].4. M. Bitter and V. Milner. Experimental demonstration of coherentcontrol in quantum chaotic systems. arXiv :1606.06805, 2016 [21], sub-mitted.5. M. Bitter and V. Milner. Control of quantum localization and classi-cal diffusion in laser-kicked molecular rotors. arXiv :1610.04193, 2016[20], submitted.During my PhD, prior to the work on the quantum kicked rotor, I stud-ied ro-vibrational dynamics of diatomic molecules via coherent anti-StokesRaman spectroscopy (CARS). I also investigated various pulse shaping tech-niques to enhance molecular vibrations in the regime of strong ultra-shortpulses. The results are summarized in three publications. I decided not toinclude this work in this thesis.6. M. Bitter, E. A. Shapiro and V. Milner. Enhancing strong-field-induced molecular vibration with femtosecond pulse shaping. PhysicalReview A, 86, 043421, 2012.7. M. Bitter, E. A. Shapiro and V. Milner. Coherent rovibrational re-vivals in a thermal molecular ensemble. Physical Review A, 85, 043410,2012.196Appendix A. List of Publications8. M. Bitter and V. Milner. Coherent anti-Stokes Raman spectroscopyin the presence of strong resonant signal from background molecules.Optics Letters, 38, 2050, 2013.197Appendix BClassical dynamics of thekicked rotorB.1 Equations of motionWe derive the equations of motion for a classical kicked rotor starting fromthe classical version of the Hamiltonian in Eq. 2.1H =I2θ˙2 + V (θ) , (B.1)with the potential energy term V (θ) = −~P cos2(θ)∑N−1n=0 δ(t− nT ). New-ton’s second law for the rotating KR is of the form I d2θdt2= −dV (θ)dθ , whichresults inId2θdt2= −~P sin(2θ)N−1∑n=0δ(t− nT ) . (B.2)Here, we used the identity 2 sin(θ) cos(θ) = sin(2θ). We define ξ = t/T ,which effectively counts the periods, and a dimensionless angular momentumJ˜ = JTI=(Idθdt)TI=dθdξ. (B.3)Substituting the scaled angular momentum into the differential equationyieldsdJ˜dξ= −P~T2Isin(2θ)N−1∑n=0δ(t− nT ) (B.4)= −Pτ sin(2θ)N−1∑n=0Tδ(t− nT ) , (B.5)where we used the dimensionless time τ = ~T/I. The stochasticity parame-ter is K = τP . Another identity Tδ(t−nT ) = δ(t/T −n) is needed to reach198B.1. Equations of motionthe final form, the two coupled equations of motiondθdξ= J˜dJ˜dξ= −K sin(2θ)N−1∑n=0δ(ξ − n) .(B.6)The equivalent formulation in the form of a discrete mapping gives a stan-dard mapθN = θN−1 + J˜NJ˜N = J˜N−1 −K sin(2θN−1) ,(B.7)with the canonical variables after each kick N .199Appendix CSpectral decomposition ofthe kicked rotor wavefunctionC.1 The kicked rotorThe procedure described here follows the Ref. [63]. We solve the Schro¨dingerequation for a kicked rotor after a single Gaussian pulsei~∂∂t|ψM (t) 〉 =[Jˆ22I− P~√piσcos2 θ e−t2/σ2]|ψM (t) 〉 . (C.1)We insert the wave function of Eq. 2.16 and evaluate the left-hand side (lhs)and the right-hand side (rhs) individually.(lhs) =∑J(i~∂cMJ∂te−iEJ t/~ + EJ cMJ e−iEJ t/~)| J,M 〉(rhs) =∑J[EJ − P~√piσcos2 θ e−t2/σ2]cMJ e−iEJ t/~ | J,M 〉(C.2)The (lhs) was obtained with the product rule, whereas the (rhs) made useof the relation Jˆ2| J,M 〉 = ~2J(J+1)| J,M 〉 and the definition of rotationalenergy EJ . Now, the kinetic energy terms on both sides cancel.∑J∂cMJ∂te−iEJ t/~ | J,M 〉 = iP√piσcos2 θ e−t2/σ2∑JcMJ e−iEJ t/~ | J,M 〉(C.3)We multiply 〈 J ′,M ′ |× from the left to both sides, knowing that the spher-ical harmonics form an orthonormal basis 〈J ′,M ′| J,M 〉 = δJJ ′δMM ′ withthe Kronecker delta δJJ ′ = 1 for J = J′ and zero in all other cases, andδMM ′ = 1 since M = M′.∂∂tcM′J ′ =iP√piσe−t2/σ2∑JcMJ e−i(EJ−E′J )t/~〈 J ′,M ′ | cos2 θ| J,M 〉 (C.4)200C.2. The periodically kicked rotorThis set of coupled differential equations can be solved numerically to getthe complex amplitudes of the wave function.C.2 The periodically kicked rotorThis section is based on Ref. [57] and the quantum nechanics book by Saku-rai [147].C.2.1 The wave function ψ+First, we prove that |ψ+ 〉 = exp(iP cos2 θ) |ψ− 〉 (Eq. 2.21) is indeed asolution to the Schro¨dinger equation of a δ-kicked rotor (Eq. 2.20) by differ-entiating ψ+ with respect to time.i~∂|ψ+ 〉∂t= i~(i cos2 θ∂P∂texp(iP cos2 θ)|ψ− 〉+ exp(iP cos2 θ)∂|ψ− 〉∂t)= −~ cos2 θ∂P∂t|ψ+ 〉= −~ cos2 θ(∆α4~E2(t))|ψ+ 〉= V (θ, t) |ψ+ 〉(C.5)In the first line, we used the product rule of differentiation, whereas thesecond term vanishes, because the initial wave function |ψ− 〉 does not de-pend on time. The second line uses the definition of |ψ+ 〉, the third onethe derivative of the kick strength P defined in Eq. 2.11. As desired, thisdescribes the Schro¨dinger equation (without the kinetic part).C.2.2 Decomposition of ψ+The transformation of |ψ+ 〉 into spherical harmonics is done by introducingan artificial time τ . Immediately before the kick (τ = 0) only the initialstate |ψ− 〉 = | J0,M0 〉 is populated. This translates to cMJ (τ = 0) = 1 for| J,M 〉 = | J0,M0 〉 and zero for all other | J,M 〉. Immediately after the kick(τ = 1) the wave function is |ψ+ 〉. We start with setting the two equationfor |ψ+(τ) 〉 equal, Eq. 2.21 and Eq. 2.22,exp(iP cos2 θ · τ) | J0,M0 〉 =∑J,McMJ (τ) | J,M 〉 . (C.6)201C.2. The periodically kicked rotorNote, that the artificial time has been introduced. We differentiate Eq. C.6with respect to τ∑J,M∂cMJ∂τ| J,M 〉 = iP cos2 θ exp(iP cos2 θ · τ) | J0,M0 〉= iP cos2 θ∑J,McMJ | J,M 〉(C.7)and project both sides onto 〈 J ′,M ′ |×∂∂tcM′J ′ (τ) = iP∑J,McMJ 〈 J ′,M ′ | cos2 θ| J,M 〉 . (C.8)As in appendix C.1 we used the orthonormality of the spherical harmonicsbasis set.C.2.3 The coupling matrixIn this section we study Eq. 2.23 / C.8 and look at the coupling matrix,that couples the quantum numbers before the kick (J,M) with the onesafter the kick (J ′,M ′). At first, however, we will establish some basics forthe addition of angular momenta J = J1 + J2, with the angular momentumoperators J1 and J2. The entire system has two obvious options for a basis.One basis is | J1J2;M1M2 〉 which is simultaneously an eigenfunction for J21,J22, J1z and J2z. The other basis is | J1J2; JM 〉 which is simultaneously aneigenfunction for J21, J22, J2 and Jz. Both bases are connected via a unitarytransformation [147].| J1J2; JM 〉 =∑M1∑M2| J1J2;M1M2 〉〈J1J2;M1M2| J1J2; JM 〉 . (C.9)The elements of the transformation matrix 〈J1J2;M1M2| J1J2; JM 〉 arecalled the Clebsch-Gordan coefficients (CGC). The transformation matrixitself is a unitary matrix, which means that all CGC must be real values.Some important properties of the CGC areM =M1 +M2 (C.10)|J1 − J2| 6J 6 J1 + J2 (C.11)If these conditions are not fulfilled the CGC are zero. For a proof see thebook by Sakurai [147].202C.2. The periodically kicked rotorNow, we switch back to Eq. 2.23 and C.8. We transform the elements ofthe coupling matrix 〈 J ′,M ′ | cos2 θ| J,M 〉 by translating cos2 θ into sphericalharmonics | J,M 〉 = |YMJ (θ, φ) 〉. To do that we use the formal definitionof Y 02 =14√5pi (3 cos2 θ − 1). The new expression reads as〈 J ′,M ′ | cos2 θ| J,M 〉 = 43√pi5〈YM ′J ′ |Y 02 |YMJ 〉+13〈YM ′J ′ |YMJ 〉 (C.12)The second term yields only diagonal matrix elements, i.e. when J ′ = Jand M ′ = M . The more interesting first term contains a product of threespherical harmonics. There exists a helpful relation between such a productand the Clebsch-Gordan coefficients (CGC) [147].∫ 2pi0∫ pi0YM∗J YM1J1YM2J2 sin θ dθdφ =√(2J1 + 1)(2J2 + 1)4pi(2J + 1)〈 J1J2; 00 |J1J2; J0〉〈 J1J2;M1M2 |J1J2; JM〉(C.13)The notation is as introduced above. The square root and the first CGC areindependent of M1 and M2 (independent of the molecular orientation θ).The second CGC describes the angular momentum summation. Matchingthis relation to our product in Eq. C.12 yields〈YM ′J ′ |Y 02 |YMJ 〉 =√5(2J + 1)4pi(2J ′ + 1)〈 2J ; 00 |2J ; J ′0〉〈 2J ; 0M |2J ; J ′M ′〉(C.14)The values of CGC are well known and can be looked up or calculated. Oneimportant revelation from this angular momentum algebra are the selectionrules that apply to our system of a kicked rotor. The CGC properties(Eq. C.10 and C.11) for our system simplify toM ′ = M (C.15)|J − 2| 6J ′ 6 J + 2 . (C.16)It also follows that only J quantum numbers of the same parity are coupled.We summarize: The magnetic quantum number has to be preserved ∆M =M ′ −M = 0, and the angular momenta are only coupled between states ofthe same parity that satisfy the condition ∆J = J ′ − J = 0,±2 .203Appendix DFourier transform of shapedpulsesUltra-short pulses contain a broad frequency spectrum; the shorter thepulses the wider the spectrum. Depending on the circumstances, it canbe beneficial or more intuitive to solve a problem or to understand somephysical concept by looking at the pulses either in the spectral domain oralternatively in the time domain. In general, the mathematical transfor-mation that is applied to switch from one domain to the other is called theFourier transform. It is equally valid for a transformation in both directions.In our case, we will look at the Fourier transform of a function F (ω) in thefrequency domain into the time domain f(t), and vice versaF (ω) =∫ +∞−∞f(t) exp(−iωt) dt (D.1)f(t) =12pi∫ +∞−∞F (ω) exp(iωt) dω. (D.2)We also want to introduce the Fourier transform shift theoremF{f(t− tk)} = exp(−iωtk) · F{f(t)}, (D.3)which is essential in the transformation of pulse sequences with individualpulses at the timings tk (App. D.3).D.1 Fourier transform of a TL pulseWe start with the spectrum of a Gaussian pulse (Eq. 3.1) and do a Fouriertransform (Eq. D.2) to obtain the electric field in the time domainE(t) D.2= 12pi∫ +∞−∞E(ω) exp(iωt) dω3.1=12piA0 ·∫ +∞−∞exp(−(ω − ω0)22Γ2+ iωt)dω .(D.4)204D.2. Fourier transform of a chirped pulseUsing the substitution β = ω−ω0√2Γ− i Γt√2we can rewrite the exponential(ω − ω0)22Γ2− i(ω − ω0)t− iω0t = β2 + Γ2t22− iω0t . (D.5)Using the derivative dβ/dω = 1/√2Γ the expression of the electric field canbe transformed toE(t) = 12piA0 ·∫ +∞−∞exp(−β2 − Γ2t22+ iω0t)·√2Γdβ=Γ√2piA0 · exp(−Γ2t22+ iω0t)·∫ +∞−∞exp(−β2)dβ .(D.6)The solution of the integral is√pi. This yields the analytic expression of theelectric field of a TL Gaussian pulse in the time domainE(t) = E0 · exp(− t22τ2)· exp(iω0t) (D.7)with the amplitude E0 = A0Γ/√2pi and the duration τ = 1/Γ.D.2 Fourier transform of a chirped pulseAgain, we start with the electric field in the spectral domain (Eq. 3.5) andobtain the time domain via the Fourier transform (Eq. D.2)E(t) D.2= 12pi∫ +∞−∞E(ω) exp(iωt) dω3.5=12piA0 ·∫ +∞−∞exp(−(ω − ω0)22Γ2+ iα′(ω − ω0)22+ iωt)dω=12piA0 ·∫ +∞−∞exp(−(ω − ω0)22Γ2(1− iα′Γ2) + iωt)dω .(D.8)We define the constant x2 = (1 − iα′Γ2) and do a similar substitution asbefore β = xω−ω0√2Γ− i Γtx√2such that the exponential can be rewritten asx2(ω − ω0)22Γ2− i(ω − ω0)t− iω0t = β2 + Γ2t22x2− iω0t . (D.9)205D.3. Fourier transform of a pulse trainMaking use of the derivative dβ/dω = x/√2Γ, the altered expression of theelectric field becomesE(t) = 12piA0 ·∫ +∞−∞exp(−β2 − Γ2t22x2+ iω0t)·√2Γxdβ=Γ√2pixA0 · exp(−Γ2t22x2)· exp(iω0t) ·∫ +∞−∞exp(−β2)dβ .(D.10)The solution of the integral is√pi. We still need to re-substitute x. Thefirst exponential in eq. D.10 turns intoΓ2t22x2=Γ2t22(1− iα′Γ2)(1 + iα′Γ2)(1 + iα′Γ2)=Γ2t22(1 + α′2Γ4)+ iα′Γ42(1 + α′2Γ4)t2=t22τ2+ iα2t2(D.11)where we defined the duration τ and the temporal chirp ατ2 =1Γ2(1 + α′2Γ4) (D.12)α = α′Γ41 + α′2Γ4= α′Γ2τ2. (D.13)In the limit of large chirps, we can approximate α ≈ 1/α′. The final versionof eq. D.10 yields the electric field of a linearly chirped Gaussian pulse inthe time domainE(t) = E0 · exp(− t22τ2)exp(iω0t− iα2t2)(D.14)with the complex amplitude E0 = A0Γ/(√2pi · √1− iα′Γ2).D.3 Fourier transform of a pulse trainWe have built the desired pulse train in the time domain with the elec-tric field given in equation 3.8. Now, we will perform a Fourier transform(Eq. D.1) to get the electric field in the spectral domain. Once we have theknowledge of amplitude and phase in the frequency representation we are206D.3. Fourier transform of a pulse trainable to send the corresponding masks to our SLM and create this pulse trainwith the shaper.E(ω) D.1=∫ +∞−∞E(t) exp(−iωt) dt3.8=∑kE0∫ +∞−∞e−(t−tk)22τ2 eiω0t eiβ2k2 exp(−iωt) dt=∑kE0∫ +∞−∞e−(t−tk)22τ2 eiω0(t−tk) e−iωt dt︸ ︷︷ ︸F{f(t−tk)}·eiω0tk eiβ2 k2(D.15)We use the FT shift theorem (Eq. D.3) with the function f(t − tk) =exp(− (t−tk)22τ2+ iω0(t− tk))F{f(t− tk)} = exp (−iωtk)∫ +∞−∞e−t22τ2 eiω0t e−iωt dt . (D.16)Now, we eliminated all k-dependences inside the integralE(ω) =∑ke−iωtk(E0∫ +∞−∞e−t22τ2 eiω0t e−iωt dt)eiω0tk eiβ2k2 . (D.17)We pull everything in brackets in front of the sum and solve the integral.Under closer inspection it can be seen that the integral is merely the Fouriertransform (Eq. D.1) of a TL pulse (Eq. 3.2 or Eq. D.7)∫ +∞−∞E0 e−t22τ2 eiω0t e−iωt dt 3.2=∫ +∞−∞E(t) e−iωt dt D.1= E(ω) (D.18)whose expression we gave earlier in Eq. 3.1. Thus, the analytic expressionfor the electric field of a periodic and flat pulse train in the spectral domainis given asE(ω) = A0 exp(−(ω − ω0)22Γ2)·∑kexp(−i(ω − ω0)tk + iβ2k2)(D.19)with the amplitude A0 = E0√2pi/Γ. The term in front of the sum gives thespectrum of the unshaped, TL pulse as in Eq. 3.1. Once we evaluate thesum, it will tell us how we have to shape the amplitude and phase in orderto obtain the pulse train.207D.4. Convolution theoremD.4 Convolution theoremThe convolution theorem states that the Fourier transform of a convolutionis equal to the point-wise product of Fourier transforms. We denote F asthe Fourier transform operator. Let us use two functions f(t) and g(t) as afunction of time t. Their respective Fourier transforms are F (ω) = F{f(t)}and G(ω) = F{g(t)} as a function of frequency ω, defined in Eq. D.1.F{f(t)⊗ g(t)} = F (ω) ·G(ω) (D.20)F{F (ω)⊗G(ω)} = f(t) · g(t) (D.21)The symbol ⊗ denotes convolution.208Appendix ESemi-classical model ofrotational Bloch oscillationsWe derive a semi-classical model to describe the phenomenon of Bloch os-cillations in rotating molecules. We start with a stroboscopical descriptionof the rotor after each kick n. The wave function is a linear combination offree-rotor eigenstates|Ψ(n) 〉 =∑Jc(n)J | J 〉 , (E.1)where we neglect the magnetic quantum number M since it does not changein the interaction. The wave function right after a δ-kick is governed by|Ψ(n+1) 〉 = eiP cos2 θ e−iEJT/~|Ψ(n) 〉 . (E.2)Here, the one-cycle operator contains the kicking operator, see Sec. C.2.1,and a free-evolution term. Insert Eq. E.1 as a function of J ′ into Eq. E.2and multiply 〈 J | from the left side to get an expression for the amplitudecoefficientsc(n+1)J =∑J ′c(n)J ′ 〈 J |eiP cos2 θ| J ′ 〉 e−iEJT/~ (E.3)The following derivation from Ref. [62] assumes pulses of P  1. However,the same result is obtained in the “-classics” approach [184, 185], which ismore involved but does not rely on the weak pulse approximation. A Taylorexpansion yieldsc(n+1)J =∑J ′c(n)J ′ 〈 J |1 + iP cos2 θ| J ′ 〉 e−iEJ′T/~ . (E.4)Further, we investigate two distinct scenarios.209E.1. Non-rigid rotor on quantum resonanceE.1 Non-rigid rotor on quantum resonanceTaking into account centrifugal distortion the free evolution term with aperiod T = Trev = (2cB)−1 breaks down intoe−iEJ′Trev/~ = e−ipiJ′(J ′+1) eipiDBJ ′2(J ′+1)2 ≈ 1 + ipiDBJ ′2(J ′ + 1)2 (E.5)where the first exponential is equal to unity and the second one is expandedin another Taylor series around DB  1. We insert the expression into Eq. E.4and neglect the (P DB )-termc(n+1)J = c(n)J + ipiDBJ2(J + 1)2c(n)J + iP∑J ′c(n)J ′ 〈 J | cos2 θ| J ′ 〉 . (E.6)The cos2 term is approximated as [62]〈 J | cos2 θ| J ′ 〉 =1/2 for J ′ = J1/4 for J ′ = J ± 20 else(E.7)and we obtain−i[c(n+1)J − c(n)J]=[piDBJ2(J + 1)2 +P2]c(n)J +P4[c(n)J+2 + c(n)J−2]. (E.8)Finally, since each kick is very weak and hardly changes the angular mo-mentum distribution, we replace the difference term c(n+1)J − c(n)J by a dif-ferential dcJ(n)/dn. The result is a Schro¨dinger equation for the rotationaltight-binding model with a continuous dimensionless time nidcJ(n)dn=[−piDBJ2(J + 1)2 − P2]c(n)J −P4[c(n)J+2 + c(n)J−2]. (E.9)E.2 Rigid rotor detuned from quantum resonanceIf we neglect the centrifugal distortion but consider a detuning δ from theresonance T = (1 + δ)Trev the free evolution term has the forme−iEJ′ (1+δ)Trev/~ = e−ipiJ′(J ′+1) e−ipiδJ′(J ′+1) ≈ 1− ipiδJ ′(J ′ + 1) . (E.10)All the other steps are identical to Sec. E.1, and the final differential equationwill beidcJ(n)dn= [piδJ(J + 1)] c(n)J −P4[c(n)J+2 + c(n)J−2]. (E.11)210


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