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Control of molecular rotation with an optical centrifuge Korobenko, Aleksey 2016

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Control of molecular rotation with anoptical centrifugebyAleksey KorobenkoB.Sc., Moscow Institute of Physics and Technology, 2008M.Sc., Moscow Institute of Physics and Technology, 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)July 2016c© Aleksey Korobenko 2016AbstractThe major purpose of this work is the experimental study of the applicabilityof an optical centrifuge – a novel tool, utilizing non-resonant broadband laserradiation to excite molecular rotation – to produce and control moleculesin extremely high rotational states, so called molecular “super rotors”, andto study their optical, magnetic, acoustic, hydrodynamic and quantum me-chanical properties.iiPrefaceAll the work presented in this thesis was conducted using the optical cen-trifuge setup, discussed in section 2.1. I was heavily involved in the designand construction of the centrifuge together with Gilad Hurvitz, AlexanderMilner and Valery Milner. I also actively participated in the developmentof general methods of optical characterization of centrifuge pulses. Amongthem, I conceived and implemented the method of determining an absoluteangle of the centrifuge field, described in subsection 2.4.2. The ultra-highvacuum chamber and the velocity map imaging setup, described in sections2.3.1 and 2.4, respectively, were designed together with Valery Milner andbuilt by me. I also helped to design and construct the gas chamber de-scribed in subsection 2.2.1 and used for the investigation of super rotorsunder ambient conditions.I was heavily involved in designing and interpreting the results of ex-periments on the first direct observation of oxygen and nitrogen super ro-tors, based on coherent Raman spectroscopy, developed and implementedtogether with Alexander Milner and Valery Milner. These results were pub-lished in Paper 1 and discussed in Section 3.1 and Section 4.1.After setting up the tunable dye laser system together with Alice Schmidt-May, I became the primary investigator in the area of resonance-enhancedmulti-photon ionization (REMPI) spectroscopy of centrifuged oxygen. Un-der the guidance of Valery Milner and John Hepburn, I designed the exper-iment, collected data and interpreted the results, which were published inPaper 2 and discussed in Section 3.2.I have designed, implemented and continuously improved the experi-mental setup for the direct imaging of molecular super rotors by means ofvelocity map imaging (VMI). Using the VMI technique, I collected the dataon dispersionless “stopwatch” rotational states, as well as “cogwheel” states,described in Paper 5 and Section 4.2. I also used the same VMI methodto study the properties of centrifuged asymmetric top molecules, collect-ing and analysing the data published in Paper 11 (Section 5.3). Finally, Imodified and applied the ion imaging system to investigate the propertiesof non-magnetic and paramagnetic super rotors in external magnetic fields.iiiPrefaceThe results of this studies, based on my experimental data and numericalanalysis, were published in Paper 8 and discussed in Section 5.2.Magneto-optic properties of molecular super rotors, described in Paper7, were studied in collaboration with the theory group of Ilya Averbukh fromthe Weizmann Institute of Science. The majority of the experimental datafor this work was collected by Alexander Milner, whereas my contributionamounted to the data analysis and to providing an important VMI imagein support of the main claim of the paper, specifically, the visualization ofthe magnetic splitting of the angular distribution of oxygen super rotors inan external magnetic field (Section 5.2).The work on the collisional relaxation of molecular super rotors wasspearheaded by Alexander Milner, who collected the majority of the exper-imental data using the technique of coherent Raman scattering. Rotationaldecoherence of non-magnetic nitrogen and magnetic oxygen was discussedin Papers 3 and 4, respectively (Section 6.1). My contribution to this workconsisted in proposing the experimental configurations and analyzing thecollected data. I provided similar contribution to the work on the colli-sional re-orientation (Paper 9, Section 6.2) and sound generation (Paper6, Section 6.4) in dense ensembles of molecular super rotors. The idea of a“two-dimensional” (2D) centrifuge and its effect on the molecular alignment,described in Section 6.3, was conceived with my direct input. To Paper 10,describing the results of this work, I contributed by supplementing the dataof Alexander Milner, based on the laser-induced optical birefringence, bycollecting and interpreting the VMI images of aligned molecules exposed tothe field of a 2D centrifuge.I participated in writing all the 11 manuscripts, which I co-authored.They are listed below.1. Korobenko A., Milner A. A., Milner V., Complete control, direct ob-servation and study of molecular superrotors. Phys. Rev. Lett., 112,113004 (2014).2. Korobenko A., Milner A. A., Hepburn J.W., Milner V., Rotationalspectroscopy with an optical centrifuge. Phys. Chem. Chem. Phys.,16, 4071 (2014).3. Milner A. A., Korobenko A., Hepburn J.W., Milner V., Effects ofultrafast molecular rotation on collisional decoherence. Phys. Rev.Lett., 113, 043005 (2014).4. Milner A. A., Korobenko A., Milner V., Coherent spin-rotational dy-namics of oxygen superrotors. New J. Phys. 16, 093038 (2014).ivPreface5. Korobenko A., Hepburn J.W., Milner V., Observation of nondispersingclassical-like molecular rotation. Phys. Chem. Chem. Phys. 17, 951(2015).6. Milner A. A., Korobenko A., Milner V., Sound emission from the gasof molecular superrotors. Opt. Express 23, 8603 (2015).7. Milner A. A., Korobenko A., Floss J., Averbukh I. Sh., Milner V.,Magneto-optical properties of paramagnetic superrotors. Phys. Rev.Lett., 115, 033005 (2015).8. Korobenko A., Milner V., Dynamics of molecular superrotors in ex-ternal magnetic field. J. Phys. B. 48, 164004 (2015).9. Milner A. A., Korobenko A., Rezaiezadeh K., Milner V., From gyro-scopic to thermal motion: a crossover in the dynamics of molecularsuperrotors. Phys. Rev. X 5, 031041 (2015).10. Milner A. A., Korobenko A., Milner V., Field-free long-lived alignmentof molecules in extreme rotational states. Phys. Rev. A 93, 053408(2016).11. Korobenko A., Milner V., Adiabatic field-free alignment of asymmetrictop molecules with an optical centrifuge. Phys. Rev. Lett. 116,183001 (2016).vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Rotational motion of molecules . . . . . . . . . . . . . . . . . 31.2 Control of rotational states with intense non-resonant laserfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Optical centrifuge . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Goals of this work . . . . . . . . . . . . . . . . . . . . . . . . 112 Experimental methods . . . . . . . . . . . . . . . . . . . . . . 152.1 Optical centrifuge . . . . . . . . . . . . . . . . . . . . . . . . 162.1.1 Laser system . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Centrifuge shaper . . . . . . . . . . . . . . . . . . . . 172.2 Raman setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Raman chamber . . . . . . . . . . . . . . . . . . . . . 232.2.2 Multi-pass amplifier . . . . . . . . . . . . . . . . . . . 242.2.3 Broadband probe beam . . . . . . . . . . . . . . . . . 262.2.4 Rotational Raman spectroscopy . . . . . . . . . . . . 272.3 Resonance-enhanced multiphoton ionization spectroscopy . . 292.3.1 Utra-high vacuum chamber and molecular source . . 302.3.2 Time-of-flight spectrometer . . . . . . . . . . . . . . . 322.3.3 Dye laser system . . . . . . . . . . . . . . . . . . . . . 35viTable of Contents2.3.4 REMPI spectroscopy of N2 ant O2 . . . . . . . . . . . 362.4 Velocity-map imaging (VMI) . . . . . . . . . . . . . . . . . . 402.4.1 Velocity map imaging reconstruction of the angulardistribution . . . . . . . . . . . . . . . . . . . . . . . 432.4.2 Centrifuge angle measurement . . . . . . . . . . . . . 453 Direct detection of molecular super rotors . . . . . . . . . . 483.1 Raman spectroscopy of molecular super rotors . . . . . . . . 493.2 Rotational spectroscopy with an optical centrifuge . . . . . . 523.3 Mapping out the angular distribution of molecular super ro-tors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 Rotational dynamics of molecular super rotors . . . . . . . 634.1 Rotational revivals in the ensembles of molecular super rotors 634.2 Observation of classical-like molecular rotation . . . . . . . . 654.3 Coherent spin-rotational dynamics in oxygen . . . . . . . . . 734.4 Dynamics of super rotors in external magnetic fields . . . . . 775 Molecular alignment with an optical centrifuge . . . . . . . 855.1 Two-dimensional optical centrifuge . . . . . . . . . . . . . . 865.2 Magneto-optical properties of paramagnetic super rotors . . 935.3 Adiabatic alignment of asymmetric top molecules . . . . . . 996 Collisional decay of rotational excitation in dense media . 1086.1 Effects of ultrafast rotation on collisional decoherence . . . . 1096.2 Crossover from gyroscopic to thermal motion . . . . . . . . . 1176.3 Long-lived permanent molecular alignment . . . . . . . . . . 1266.4 Sound emission from the gas of molecular super rotors . . . . 1277 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.1 Generation of THz radiation . . . . . . . . . . . . . . . . . . 1337.2 Transient magnetization of paramagnetic gases . . . . . . . . 1337.3 Field-free alignment of asymmetric top molecules . . . . . . 1347.4 Potential energy surface reconstruction . . . . . . . . . . . . 1367.5 Harmonic rotational states . . . . . . . . . . . . . . . . . . . 1367.6 Super rotors in He droplets . . . . . . . . . . . . . . . . . . . 1378 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139viiList of Tables3.1 Molecular constants used to fit the data in Fig. 3.6 . . . . . . 58viiiList of Figures1.1 Illustration of (a) adiabatic vs. (b) non-adiabatic rotationalexcitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 (a) Principle of an optical centrifuge excitation. A linearlypolarized intense non-resonant laser field aligns the molecu-lar axes along its polarization direction. This direction thenundergoes accelerated rotation, sweeping the depicted red“corkscrew” shape and dragging the molecular axes along,eventually releasing them to freely rotate with a high angularfrequency. (b) To produce the centrifuge field, one can com-bine two circularly polarized pulses of the opposite chirp andpolarization handedness, shown in the figure. . . . . . . . . . 91.3 Illustration of an adiabatic population transfer in the opticalcentrifuge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 (a) Diagram of our femtosecond laser system. (b) Spectra oflaser pulses before (red line) and after (blue line) amplification. 172.2 (a) Optical centrifuge shaper. IM: input mirror, G0, G1, G2:diffraction gratings, L1, L2, L3: f = 250 mm achromaticlenses, HM: half-cut mirror, FM: folding mirror, RR1, RR2:retroreflectors. Inset: movable shutter. (b) Laser spectrabefore (magenta), and after the shaping for the “blue” (blue)and “red” (red) arms. The “blue” arm spectrum is shownwith the shutter completely out of (dashed), and half way inthe beam (solid) . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Combining the centrifuge arms. HWP: half-wave plate, M:mirror, PBS: polarizing beam splitter cube, QWP: quarter-wave plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Far field profiles of the (a)“blue” arm, (b)“red” arm, (c)bothbeams combined. . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Illustration of the standard cross-correlation setup. . . . . . . 20ixList of Figures2.6 Time- and frequency-resolved characterization of the centrifugebeams. In each panel, the measurements related to the “blue”(“red”) arm are colored blue (red). (a) Time-resolved crosscorrelation measurements. (b) Spectra of the beams. (c)Time-resolved spectrogram of each centrifuge arm. Tiltedlines correspond to a frequency chirp. . . . . . . . . . . . . . 212.7 An illustration of Stokes (left) and anti-Stokes (right) rota-tional Raman scattering. . . . . . . . . . . . . . . . . . . . . . 222.8 An illustration of the rotational Doppler effect. Thick (thin)blue arrows show the circularly polarized probe field of thesame (opposite) direction with respect to the molecular rota-tion (green arrows). . . . . . . . . . . . . . . . . . . . . . . . 232.9 Gas chamber for Raman spectroscopy. . . . . . . . . . . . . . 242.10 (a) Diagram of separate amplification of the two centrifugebeams. (b) Multi-pass amplifier. . . . . . . . . . . . . . . . . 252.11 (a) Probe beam setup. M: mirror, RR: retroreflector. (b)Full centrifuge spectrum (red), full probe spectrum (blue),and truncated probe spectrum (magenta) . . . . . . . . . . . 262.12 Raman setup. PBS: polarizing beam splitter, QWP: quarter-wave plate, DM: dichroic mirror, Filt: dichroic filter, L: lens,M: mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.13 Raman spectrum of laser-kicked oxygen. . . . . . . . . . . . . 282.14 REMPI principle. . . . . . . . . . . . . . . . . . . . . . . . . . 292.15 Ultra-high vacuum chamber with a molecular source. . . . . . 302.16 (a) Magnified view of the UHV chamber. (b) Nozzle controlelectronics. (c) Gas line connections diagram. . . . . . . . . . 312.17 Geometry of the time-of-flight spectrometer (left) and thesimulated electric field distribution (right). . . . . . . . . . . . 332.18 (a) Raw time-of-flight spectrum of photo-dissociated mixtureof gases. (b) Calibrated spectrum. (c) Magnified region ofthe spectrum around mass-to-charge ratio of 8. . . . . . . . . 342.19 Dye laser system. The flashlamps (FL) and Q-switch (Q-sw) of Nd:YAG pump laser were synchronized with the cen-trifuge pulses electronically. The output of the dye laser wasfrequency doubled in a second harmonic generation system(SHG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36xList of Figures2.20 REMPI spectroscopy setup. Centrifuge beam is combinedwith a tunable UV laser pulse and focused inside a vacuumchamber on a supersonically expanded molecular jet betweenthe charged plates of a time-of-flight (TOF) mass spectrom-eter. The ionization rate is measured with a multi-channelplate (MCP). . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.21 REMPI spectra of cold (T = 10 K, blue) and room tempera-ture (red) nitrogen. Black ticks, connected by lines to guidethe eye, show the calculated transition energies for differentbranches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.22 REMPI spectra of cold (T = 10 K, blue) and room temper-ature (red) oxygen. Dashed line demonstrates the effect ofpower broadening. . . . . . . . . . . . . . . . . . . . . . . . . 392.23 Illustration of the C3Πg(v′ = 2) ←← X3Σ−g (v′′ = 0) transi-tion in oxygen. Blue (red) area marks the rotational structureof the ground (excited) electronic state, affected by the spin-spin and spin-rotational (spin-orbit) interaction. . . . . . . . 402.24 Illustration of the process of Coulomb explosion. After beingionized by a fs pulse, the molecules immediately fragment intoions recoiling in the direction of their respective internuclearaxes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.25 Relevant polar variables for the velocity map imaging. . . . . 412.26 (a) Raw ion image after a single laser shot. (b) Processedimage with the determined ion positions, marked by the ma-genta crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.27 N+ ion image distribution measured for 1,000,000 events,taken with probe polarization (a)parallel and (b)normal tothe detector plane. . . . . . . . . . . . . . . . . . . . . . . . . 442.28 Definition of the centrifuge angles. . . . . . . . . . . . . . . . 462.29 (a) Angle measurement setup. Ref: reference femtosecondlaser pulse, BS: 50/50 beam splitter, L: lenses, Ret: glassplate retarder, PD1, PD2: photodiodes. (b) Correlation plotof the two normalized signals from PD1 and PD2, acquiredover 100 laser shots. Red bar charts show the histograms ofthe two signals. Scatter plots of different colors correspond tothree different delays between the reference and the centrifugepulses, with the red trace corresponding to an optimal delay,used for determining θrel. . . . . . . . . . . . . . . . . . . . . 47xiList of Figures3.1 Probe spectra after passing through the counter-clockwise(purple) and clockwise (cyan) centrifuged oxygen molecules.The appearance of only anti-Stokes or only Stokes sidebandsindicate unidirectional rotation. . . . . . . . . . . . . . . . . . 493.2 Time-dependent Raman shifts from the centrifuged N2 (a)and H2 (b) molecules. As the molecules spend longer time inthe centrifuge, the observed Raman frequency shift increasesalong the slopped dashed lines, providing a direct evidenceof accelerated molecular rotation. Some of the molecules“leak” from the accelerating angular trap, leaving the hor-izontal traces at the intermediate Raman shifts. Horizontaldashed lines correspond to the angular velocity of rotationalstates, mostly populated at room temperature. . . . . . . . . 503.3 State-resolved Raman spectra of centrifuged oxygen molecules.Higher curves correspond to longer spinning time inside thecentrifuge. Red vertical arrows mark the rotational quantumnumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Experimental (dots with error bars) and calculated (dashedblue and solid green for the rigid and non-rigid rotor approx-imations, respectively) rotational energy spectrum expressedas a Raman frequency shift. . . . . . . . . . . . . . . . . . . . 523.5 2D REMPI spectrogram for a linearly polarized probe. (a)Experimental spectra of cold (10 K, blue) and centrifugedmolecules (yellow), along with a simulated spectrum of a“hot” thermal ensemble (3000 K, red) calculated with pgo-pher software [134]. (b) Ion signal as a function of the probelaser wavelength and molecular angular momentum definedby the centrifuge final rotation speed. Different areas of the2D plot were measured with different sensitivities and probeintensities and are displayed with different color scales to com-pensate for the broad dynamic range of the data. c Verticalcross-sections of several consecutive peaks from one particu-lar branch, shown in the inset to (b). The peaks are regularlyseparated with a distance of ∆N = 2 reflecting 16O2 nuclearspin statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54xiiList of Figures3.6 2D REMPI spectrogram for a circularly polarized probe. Elec-tric field vector is counter-rotating (a) and co-rotating (b)with the centrifuged molecules. The directionality of laser-induced rotation results in the sensitivity of the measuredsignal to the handedness of probe polarization. The results offitting the data to the theoretical model are shown with col-ored lines and markers for different branches and resonances,respectively. Branch nomenclature is the same as in [122]. . . 563.7 Comparison of the observed REMPI data for the perturbedF1 spin-orbit component with the calculations based on molec-ular constants from our work (red circles), White et al.[135](blue triangles) and Lewis et al.[82] (purple squares) . . . . . 573.8 Ultra-high rotational resonances of O2. The two panels cor-respond to two possible ways of fitting the observed resonantbranches (apparent along white dashed lines) to the calcu-lated Hund’s case (b) structure (labeled with markers). Inpanel (a), the upper branch corresponds to ∆N = −1, andthe lower one to ∆N = 3, resulting in Bv = 1.620 cm−1 andDv = 4.4× 10−6 cm−1. In panel (b), the upper branch over-laps with ∆N = −2, whereas the lower one with ∆N = 2,yielding Bv = 1.664 cm−1 and Dv = 5.7× 10−6 cm−1. . . . . 593.9 Observed linewidths of J ′ = N ′ − 1(triangles) and J ′ =N ′(squares) spin-orbit sublevels of C3Πg (v′ = 2) level asfunctions of rotational quantum number N ′. Inset demon-strates a fit of experimental data (solid red) to a sum oflorentzians in order to extract linewidths. Absolute position,absolute area and the widths of two peaks were fitted for eachdoublet individually, with the areas ratio fixed to a value ex-tracted from the best resolved N ′ = 115 doublet and with adoublet line separation equal to a calculated one. . . . . . . . 603.10 (a) Linear rotor angular distributions. Experimental VMIimages of N2 fragments with the centrifuge turned off (b)and on (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61xiiiList of Figures4.1 (a) Selective centrifuge spinning of oxygen toN ≈ 109. Dashedtilted line shows the increasing angular frequency of the cen-trifuge, terminated at about 90 ps. Dashed horizontal linemarks the most populated rotational state of O2 at roomtemperature, N = 7. Oscillations of the coherent rotationalwave packet are shown in the inset. (b) Truncating the probespectrum to 0.1 nm allowed us to resolve individual rotationalRaman transitions, similar to Fig. 3.2, at the expense of losingtime resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Time evolution of a wide (top) and a narrow (bottom) rota-tional wave packet, consisting of many or only two rotationalstates, respectively. The latter is an example of a quantum“cogwheel state”. Trev is the rotational revival time. . . . . . 654.3 Experimental VMI of O2 taken with the centrifuge turned off(a) and on (b). The apparent alignment of the cold ensem-ble in the plane of probe polarization (PP) in panel (a) isdue to the geometric alignment effect. The enhancement ofthis alignment in panel (b) is caused by the centrifuge. (c)Geometry of the in-plane distribution measurements. . . . . . 694.4 Normalized population of the rotational states with N = 37(blue) and N = 39 (red) as a function of the centrifuge finalangular velocity. The black arrow shows the terminal angu-lar frequency of the centrifuge used in this work for creatingrotational wave packets in O2. . . . . . . . . . . . . . . . . . . 704.5 (a) Probability density as a function of the molecular angleand the free propagation time. Time zero corresponds toapproximately 100 ps since the release from the centrifuge.The blue dashed line marks the calculated trajectory of a“dumbbell” distribution rotating with the terminal angularfrequency of the centrifuge, whose classical period is indicatedwith the white horizontal bar at the lower right corner. Zoom-in to the region near (b) 38Trev and (c)38Trev, taken withbetter averaging and angular resolution. Twice higher densityof the tilted lines in panel (c) (4 per classical period indicatedby the tilted arrows) stems from the emergence of a “cross”-shaped distribution with four lobes along two perpendiculardirections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72xivList of Figures4.6 Probability density as a function of the molecular angle andthe the free propagation time of D2 prepared in the equal-weight superposition of N = 2 and N = 4 states. The ob-served nondispersing behavior illustrates the main propertyof a quantum cogwheel state. The classical period is indicatedwith the white horizontal bar at the lower right corner. . . . . 734.7 (a): Spin-rotational splitting of two rotational levels of oxy-gen, N ′′ and N ′ = N ′′ + 2. Each level is split into threesub-levels with energies Fk, k = 1, 2, 3 for the total angularmomentum J = N+1, N,N−1, respectively. Three strongestRaman transitions (out of the total six allowed by the selec-tion rules) corresponding to the S(N ′) branch are shown andlabeled according to the participating J-states. (b): Depen-dence of the three Raman frequencies (Ωk for Sk line) on therotational quantum number. . . . . . . . . . . . . . . . . . . . 744.8 (a): Experimentally detected Raman spectrogram of cen-trifuged oxygen showing the rotational Raman spectrum as afunction of the time delay between the beginning of the cen-trifuge pulse and the arrival of the probe pulse. Color codingis used to reflect the signal strength in logarithmic scale. (b):Cross-section of the two-dimensional spectrogram at the delayof 200 ps (vertical dashed line in a), showing an ultra-broadrotational wave packet created by the optical centrifuge. (c):Spin-rotation oscillations of the N = 91 Raman line (horizon-tal dashed line in a). Experimental uncertainty is indicatedby the vertical error bars. Note logarithmic scale in all panels. 764.9 The observed data (blue circles, normalized to 1 at t = 100 ps)and the fit to spin-rotation oscillations (red curves, Eq.4.5)for six different Raman lines corresponding to the rotationalquantum numbers N = 5, 7, 9, 21, 61 and 101. Experimen-tal uncertainty (not shown) is similar to Fig. 4.8 (c). Notelogarithmic scale in all panels. . . . . . . . . . . . . . . . . . . 77xvList of Figures4.10 (a) Experimental setup for detecting the plane of molecularrotation. The molecules are excited with a centrifuge pulse,then rotate freely in a magnetic field of a permanent mag-net, and finally are ionized with a femtosecond probe pulse.The resulting atomic ions are extracted and focused with ionoptics of a velocity map imaging (VMI) apparatus onto amicrochannel plate detector (MCP) with a phosphor screen.The images observed without (b) and with (c) the rotationalexcitation by a centrifuge pulse. . . . . . . . . . . . . . . . . . 804.11 Ion images of oxygen super rotors evolving in the externalmagnetic field. Different rows correspond to different degreesof rotational excitation (N value on the left), whereas eachcolumn corresponds to the evolution time indicated at thebottom. The initial disk distribution splits into three disksprecessing with different frequencies according to their spinprojections. . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.12 Experimentally determined precession period as a function ofthe rotational quantum number (blue squares) and the ex-pected theoretical dependence (dashed red line). . . . . . . . 834.13 (a) Ion images of nitrogen super rotors in the external mag-netic field. To improve the visibility of precession, an over-saturated central region is cut off and contour lines (greendashed) are added. (b) Directional REMPI signal as a func-tion of the pump-probe delay. Maximum (minimum) signalis detected when the molecules rotate in the opposite (same)direction with respect to the circularly polarized probe. Solidred and dashed blue lines correspond to the opposite initialdirections of molecular rotation, i.e. opposite handedness ofthe centrifuge pulse. . . . . . . . . . . . . . . . . . . . . . . . 845.1 (a) Illustration of the concept of a “two-dimensional (2D)centrifuge”. The three-dimensional corkscrew-shaped surfacerepresents the field of a conventional “3D centrifuge”, propa-gating from right to left. Shown in blue is the field of a 2Dcentrifuge, created by passing the 3D centrifuge through alinear polarizer. (b, c) Simulated molecular distribution af-ter the rotational excitation by a 2D (b) and a 3D (c) opticalcentrifuge, propagating along the xˆ-axis. . . . . . . . . . . . . 87xviList of Figures5.2 (a) Rotational Raman spectra of the centrifuged oxygen moleculesrecorded with a linearly polarized probe light. From top tobottom, the spectra correspond to the excitation by a sin-gle femtosecond pulse (black), a “truncated” 2D centrifuge(green), a full 2D centrifuge (red), and a full 3D centrifuge(blue) with a terminal rotational frequency around 10 THz.The spectra have been recorded at the delay times of 400ps. (b) Rotational Raman spectra of the centrifuged oxygenmolecules recorded with a circularly polarized probe light.Upper red and lower blue curves correspond to the 2D and3D centrifuge, respectively. The spectra have been recordedat the delay times of 400 ps. . . . . . . . . . . . . . . . . . . . 895.3 Ion images of nitrogen molecules prior to any rotational exci-tation (a), and following the excitation by a 3D (b) and a 2D(c) centrifuge. Calculated molecular distributions are shownabove the respective images. . . . . . . . . . . . . . . . . . . . 905.4 (a) Definition of the spherical coordinates (r, θ, φ) and polarcoordinates (R,Θ), used to describe the full molecular dis-tribution and its two-dimensional projection on the xz planeof the ion detector. (b) R-dependence of the extracted align-ment factor, with the shaded region indicating the experimen-tal error. (c) Ion image of N2 molecules aligned with a 2Doptical centrifuge. The black dotted and white dashed linesshow the circular cross-sections at the radius of a maximumion signal, R = Rmax, and the bigger radius used for estimat-ing the true two-dimensional alignment factor, respectively.The latter is also marked by the red dashed line in plot (b). . 915.5 (a) Setup for the detection of the “magneto-rotational” bire-fringence. BS: beam splitter, DM: dichroic mirror, LP (LA):linear polarizer (analyzer) oriented at angle θp(θp + 90o) withrespect to ~B, DL: delay line, L: lens, M: two magnetic coilsconnected in a Helmholtz configuration. ‘O2’ marks the pres-sure chamber filled with oxygen gas under pressure P at roomtemperature. An optical centrifuge field is illustrated abovethe centrifuge shaper with ~k being the propagation directionand ~E the vector of linear polarization undergoing an accel-erated rotation. (b) Geometry of the magnetic and opticalfields used in this work. The cloud of centrifuged moleculesis depicted as a dark ellipse. . . . . . . . . . . . . . . . . . . . 94xviiList of Figures5.6 Spectrum of probe pulses transmitted through the ensem-ble of centrifuged oxygen molecules as a function of the ap-plied magnetic field. All spectra have been recorded at theprobe delay of t = 1.14 ns and under 0.3 atm of gas pressure.Crossed circular (rather than linear) polarizer and analyzerwere used here to detect a weak magneto-rotational Ramansignal (note the change of vertical scale (×200) at frequencieshigher than 7 THz). . . . . . . . . . . . . . . . . . . . . . . . 955.7 (a,b) Calculated angular distribution of the molecular axesfor the rotational state with N = 59 at time t = 0.9 ns inan external magnetic field of 0 and 0.32 Tesla, respectively.(c) Birefringence signal (scaled to peak at 1) as a functionof angle θp between the polarization of probe pulses and themagnetic field direction. Black circles: data taken at 2 T,t = 1.5 ns, and N = 95. Red curve is a fit to cos2(θp).(d) Experimentally measured distribution of molecular axes,imaged in the direction of the applied field. All parametersare the same as in panel (b). . . . . . . . . . . . . . . . . . . 965.8 Decay of the birefringence signal for different values of therotational quantum number at B = 2 T. All solid curvesare generated by spline-fitting the experimental data (shownwith colored markers) and normalized to peak at 1. Thecorresponding lifetimes are 85 ± 10, 290 ± 20, 660 ± 50 and610± 50 ps·atm for N = 13, 33, 73 and 99, respectively. . . . 985.9 Dependence of the birefringence signal recorded at t =1 ns onthe strength of the applied magnetic field. . . . . . . . . . . . 995.10 Illustration of the main concept of aligning an asymmetrictop molecule (SO2) with an optical centrifuge. Left side ofthe centrifuge pulse (red corkscrew surface) corresponds toits leading edge, linearly polarized along ~E. Behind the trail-ing edge of the centrifuge (right side), the molecular planeis aligned in the plane of the induced rotation. The threeaxes of SO2 (a, b and c) are shown in red, green and blue,respectively. θb is the orientation angle measured in this study.1005.11 (a,b) Images of S3+ and SO2+ fragments originated fromthe rotationally cold and centrifuged SO2 molecules, respec-tively.(c) Setup for the in-plane distribution measurement ofthe molecular b-axis (green dashed line). PP marks the po-larization plane of the lasers. . . . . . . . . . . . . . . . . . . 101xviiiList of Figures5.12 Illustration of the effect of underestimated and overestimatedalignment of S3+ and SO2+, respectively. . . . . . . . . . . . . 1025.13 (a) Periodic revivals of the calculated two-dimensional align-ment factor β2D = 〈cos2 θ〉2D− 12 , determined from (b) Timeevolution of the molecular in-plane angular distribution. (c)High resolution time scan around the alignment peak. . . . . 1045.14 Schematic rotational spectrum of asymmetric top molecules. . 1055.15 Revival period as a function of the rotational frequency ofSO2, with the rotational quantum numbers shown along theupper horizontal axis. Experimental data (blue circles) arecompared with the results of classical calculations, in whichthe potential energy surface is expanded to second and fourthorder in deformation coordinates (dashed and solid lines, re-spectively). The inset shows the dependence of the 2D align-ment β2D on the pump-probe delay, used for calculating therevival time at two angular frequencies, 18.8 rad/ps (ma-genta) and 39.3 rad/ps (green), labeled with the correspond-ingly colored stars in the main plot. Solid lines representexperimental data, while the dashed lines show the fittingfunctions used to extract the revival period. . . . . . . . . . . 1066.1 Experimentally detected Raman spectrogram of nitrogen show-ing the rotational Raman spectrum as a function of the timedelay between the beginning of the centrifuge pulse and thearrival of the probe pulse. Color coding is used to reflectthe signal strength. Tilted white dashed line marks the lin-early increasing Raman shift due to the accelerated rotationof molecules inside the 100 ps long centrifuge pulse. A one-dimensional cross section corresponding to the Raman spec-trum at t = 270 ps is shown in white. . . . . . . . . . . . . . . 1116.2 Logarithm of the intensity of the experimentally measuredRaman lines as a function of time (normalized to 1 at t = 200ps, chosen so as to avoid the effects of the detector saturationat earlier times). Dots of the same color represent experimen-tal data for one particular value of the rotational quantumnumber. Black solid lines show the numerical fit to the corre-sponding exponential decay. Data collected at P = 0.75 atmand T = 294 K. . . . . . . . . . . . . . . . . . . . . . . . . . . 112xixList of Figures6.3 Decay rate of rotational coherence of N2 as a function of therotational quantum number at two different temperature val-ues, T = 294 K (blue triangles) and T = 503 K (red circles),expressed as the Raman linewidth. Solid curves correspondto the prediction of the simplified “energy corrected sudden”(ECS) approximation at N > 12. Black squares represent thedata from Ref.90. . . . . . . . . . . . . . . . . . . . . . . . . 1136.4 The dependence of the adiabaticity factor a on N for thenitrogen gas temperatures of 294 K (blue line), 503 K (redline) and 1100 K (green line). Solid horizontal line marks theadiabaticity threshold, a = pi. Black filled circles mark the ro-tational evels accessed in previous experiments on rotationaldecoherence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.5 The decay rate of rotational coherence in oxygen (blue circles)and nitrogen (green triangles) as a function of the frequencyof molecular rotation. For convenience, rotational quantumnumbers of O2 and N2 are shown below the frequency axis,and the decay times τN are shown on the right vertical axis.Black asterisks and grey squares depict the data from [92] and[90], respectively, where the rotational decay has been stud-ied in thermal oxygen by two different techniques. Solid red(dashed black) curve shows the result of fitting the adiabatic-ity correction factor Ωlc(N) to the data for oxygen (nitrogen)as discussed in text. . . . . . . . . . . . . . . . . . . . . . . . 1166.6 (a) Schematic diagram of the imaging setup. Nanosecondor picosecond probe pulses (green and blue beams, respec-tively) propagated either collinear with, or perpendicular tothe centrifuge, creating an image of the rotationally excitedvolume of gas on a CCD camera either in the longitudinal ortransverse direction, respectively. DM: dichroic mirror, FL:focusing lens, IL: imaging lens, KE: knife edge (shown alongy instead of z axis for clarity), LP: linear polarizers at ±45◦to y axis, F: frequency filter. An example of the longitudinalimage with a circularly expanding sound wave is shown inpanel (b). Images in the transverse geometry were taken atearly (c), intermediate (d) and late (e) time moments. Im-age (c) was recorded with the two linear crossed polarizers inplace. Schlieren image (e) was recorded with the knife edgein place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118xxList of Figures6.7 Radial cross-section of the images, representing the changein the refractive index of the gas, taken along the centrifugebeam, as a function of time, recorded on a long timescale. . . 1206.8 (a) Radial cross-section of the images, taken along the cen-trifuge beam, as a function of time, recorded on a shorttimescale. The arrow at around 7 ns shows the crossoverbetween the rotation-induced and thermal channels. Sampleimages of both refractive channels are shown in the squareinsets. (b) Image contrast as a function of time. The red lineon top of the experimental data points is shown to guide theeye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.9 Radial cross-section of the images, taken along the centrifugebeam, as a function of time recorded in the gas of N2O moleculesat the pressure of 0.5 atm (red line) and 0.25 atm (blue line).Shaded regions around each line represent the statistical error(one standard deviation) in our experimental data. . . . . . . 1226.10 Birefringence signal (red triangles) and image contrast (bluecircles) of the centrifuged gas, retrieved from the transverseimages taken with and without the crossed polarizers (leftand right insets, respectively) as a function of time. Two datapoints in the lower right corner (within the dashed rectangle)indicate the drop in the phase contrast for the slower rotat-ing centrifuge. The solid black line is a fit to an exponentialdecay of the birefringence signal, which gave τb = 3.4 ns asthe decay time. This time constant was used in the hydrody-namic numerical calculations of ∆n(τ), shown by the dashedblack line. The dashed magenta and green lines correspondto the same calculations, performed with the decay constantbeing five times shorter and longer, respectively. The hori-zontal dash-dotted line indicates the noise floor for the phasecontrast measurement. . . . . . . . . . . . . . . . . . . . . . . 123xxiList of Figures6.11 Comparison of the experimental results with the numericalcalculations of gas hydrodynamics. The calculated change inthe gas density, ∆ρ, is shown in panel (a) as a function oftime and distance. Experimentally determined parameters ofthe observed gyroscopic channel are used to simulate the heatsource, which initiates the dynamics at point (y = 0, τ = 0).In panel (b), the derivative of the calculated density pro-file d/dy[ρ(y)] at 1.2 µs is compared with the y cross-sectionof the schlieren image recorded 1.2 µs after the centrifuge(dashed red and solid blue curves, respectively). The depen-dence of the measured and calculated schlieren signals on bothspace and time is shown in panels (c) and (d), respectively,with the two lines indicating the two cross-sections displayedin plot (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.12 Comparison of the birefringence decay for the 2D centrifugewith a full spectral bandwidth of 20 THz (red diamonds) andwith its bandwidth truncated at ≈ 10 THz (purple triangles)and . 5 THz (green squares). Black circles represent thecase of a 1D kick. In all plots, solid lines show the fits byexponential decays. . . . . . . . . . . . . . . . . . . . . . . . . 1266.13 Typical rotational Raman spectrum of oxygen super rotorsmeasured with a probe pulse delayed by 200 ps with respectto the centrifuge pulse. Inset: an example of the acousticsignal recorded in centrifuged ambient air. . . . . . . . . . . . 1286.14 Amplitude of the recorded sound as a function of the energyof centrifuge pulses, plotted on a log-log scale. Each data setconsists of 10,000 points. (a) Typical acoustic signal fromthe centrifuged gas of nitrogen molecules (blue diamonds) iscompared to the sound generated by the centrifuge in pureargon at the same pressure of 95 kPa (green triangles). (b)Acoustic response of the centrifuged oxygen with and with-out a small admixture of SF6 molecules (black dots and redcircles, respectively). Black dashed lines in both panels showfits to power-law scaling. . . . . . . . . . . . . . . . . . . . . . 129xxiiList of Figures6.15 Amplitude of the recorded sound as a function of the rota-tional energy deposited in the gas sample. (a) Acoustic re-sponse from the centrifuged gas of nitrogen molecules (bluecircles) is compared to the sound generated by the centrifugein pure argon at the same pressure of 95 kPa (green triangles).(b) Acoustic response from the centrifuged oxygen with andwithout a small admixture of SF6 molecules (black dots andred circles, respectively). All insets show Raman spectra cor-responding to the data points marked with black crosses andplotted as a function of rotational frequency in THz. . . . . . 1307.1 Precession of the angular distributions (blue disks) of param-agnetic spin-1 super rotors in magnetic field. Different rowscorrespond to different spin projections SN . Blue and greenarrows represent the angular momentum and spin vectors,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.2 Calculated distributions of molecular axes a (red), b (green)and c (blue) of an asymmetric top molecule in the alignedrotational state. . . . . . . . . . . . . . . . . . . . . . . . . . . 135xxiiiAcknowledgementsI would like to express my sincere gratitude to my supervisor Valery Milnerfor giving me the opportunity to work in a unique creative atmosphere ofthe Ultrafast Coherent Control Group. This work would not be possiblewithout his thoughtful guidance and his scientific vision.I am very grateful to Alexander Milner for sharing his vast knowledgeof experimental physics and life. Being always open to discussions aboutanything, from the laws of nature to Strugatsky’s books, he made everydaylab routine a truly enjoyable experience.I would like to thank our collaborators John Hepburn, Ilya Averbukh andJohannes Floss for the fruitful discussions, which inspired many interestingideas.I am very thankful to Gilad Hurvitz, Sergey Zhdanovich and Guil-laume Bussiere for their help and valuable advice on the experimental setup,and to the rest of the Ultrafast Coherent Control Group, especially AliceSchmidt-May, Martin Bitter, Koosha Rezaiezadeh, Oleksandr Litvinov, IanMacPhail-Bartley and Alexander Ruf for their contribution to the friendlyand creative research environment.Finally, I am very grateful to my wife Natasha for her support andencouragement of my academic pursuits and her patience during my late-night shifts in the lab and to my parents Olga and Victor for first inspiringthe interest in science in me.xxivChapter 1IntroductionControl of molecular rotation has been long recognized and successfully usedas a powerful tool for controlling a broad range of physical processes. In themost popular and developed scenario, such control is achieved by spatialalignment or orientation of the axes of gas-phase molecules in the labora-tory frame. Here, the direction of molecular axes is set by either a line(alignment), or a vector (orientation). First works on this subject date backto 1960-s, when the groups of Brooks and co-workers and Beuhler and co-workers crossed a beam of methyl iodide CH3I with a beam of alkali (K orRb) atoms. CH3I molecules were oriented by means of a hexapole rotationalfiltering followed by applying a homogeneous electric field. The researchersobserved a substantial increase in the yield of KI or RbI when the alkaliatom was approaching CH3I from the side of the I atom, compared to whenit was approaching from the opposite side[9, 10, 87, 95]. Later, Loesch andStienkemeier [84, 85] showed, that similar control of gas-phase reaction, canbe achieved by alignment, rather than orientation. Observing the reactionsSr+HF→SrF+H, K+HF→KF+H, they demonstrated control of the reac-tion yield, by changing the HF alignment angle relative to the approachdirection. Similarly to gas phase, molecular orientation was used for con-trolling gas collisions with a solid surface. Kuipers et al.[73] used a beamof cold oriented NO and showed that the rotational excitation acquired bythe molecules scattered off a solid flat silver surface, dramatically dependedon whether the molecule was approaching the metal with its O- or N-end.Shreenivas et al. [116] proposed and theoretically calculated the possibilityof controlling the orientation of the absorbed iodine on silicon surface byaligning I2 molecules prior to collision.The works mentioned above utilized alignment and orientation to mod-ify the way molecules interact with matter. A different possibility is tocontrol their interaction with an external electromagnetic field. Itatani andcoworkers used the molecular alignment techniques to pin O2 and N2 tothe laboratory frame [53, 54]. In a tunnel ionization process induced by astrong laser field, an electron was removed from the molecular orbital andrecollided with it, emitting extreme ultraviolet radiation, whose intensity1Chapter 1. Introductiondepended on the angle between the molecular axis and the electron ejectiondirection. This information was used by the authors to extract the proper-ties of the molecular orbitals. The dipole force, produced by the interactionof inhomogeneous laser field with the dipole moment it induced in a par-ticle, became an important tool for controlling the spatial position of theparticles. Purcell et al. showed [101], that the alignment can be an effectivetool for controlling molecular polarizability due to its high anisotropy and,therefore, the dipole force.In 1999, Karczmarek et al. proposed a novel tool, employing an intensenon-resonant laser pulse tailored in a special way, to spin diatomic moleculesto quantum states of extremely high angular momentum [57]. The molec-ular axes were first aligned along the pulse polarization and then followedits accelerated rotation. The shaped laser pulses were dubbed “an opticalcentrifuge”. The technique was later demonstrated to spin Cl2 moleculesso fast, that their bonds couldn’t withstand the centrifugal forces, caus-ing the molecule to dissociate [130]. At the same time, Li et al. proposedtheir own technique of producing similar highly rotationally excited diatomicmolecules of Li2[83]. Although their method was never experimentally real-ized, they predicted that these molecules, which they called “super rotors”,will exhibit unique collisional properties. Since then, several theoreticalworks were published, suggesting a number of interesting applications ofsuper rotors.Hasbani and coworkers proposed to use an optical centrifuge to selec-tively break molecular bonds [43]. Similar to the original work by Villeneuveet al. [130], the idea was to spin HCN molecules up to a dissociation. Due toa much larger centrifugal pull from the heavy N atom, compared to the lightH, it was predicted that the much stronger C-N bond will break first. Ger-shnabel et al. studied the effect of molecular alignment and rotation on thedipole force from a focused laser beam and suggested an optical centrifugeto be a promising tool for creating rotational states, strongly antialigned toa plane of rotation [39]. Numerical simulations of Khodorkovsky and co-workers suggested a method of controlling the scattering direction of superrotors from a solid surface [61]. Steinitz et al. considered a dense gas of rota-tionally excited molecules [120]. As the rotational energy dissipated and themolecules reached thermal equilibrium, their high internal angular momen-tum was transforming to a macroscopic angular momentum of gas motion,causing a vortex to form. Calculations of Hartman et al. [42] showed theincreasing stability of molecular super rotors towards the collisions with athermal ensemble, whereas Forrey[33] has proposed to use this stability forsympathetic cooling of molecules.21.1. Rotational motion of molecules1.1 Rotational motion of moleculesConsider a molecule, consisting of two atoms A and B. Its Hamiltonian isgiven byHˆ = − ~22MA∇2rA −~22MB∇2rB + U(rB − rA), (1.1)where ~ is the reduced Planck’s constant, MA,B are the masses of atoms Aand B, respectively, Laplacians ∇2rA,B are taken with respect to the radiusvectors rA,B of atoms A and B, respectively and U(rB− rA) is the potentialenergy of interaction between the atoms. Introducing the new vectors r =MArA+MBrBMA+MBand R = rA − rB, we can rewrite (1.1) as a sum:Hˆ = − ~22(MA +MB)∇2r −~22µ∇2R + U(R), (1.2)where µ = MAMBMA+MB is a reduced mass of the molecule. The first term cor-responds to the kinetic energy of motion of the molecule as a whole; itdisappears in the frame of reference, moving with the molecular centre ofmass. When the molecule is not rotating too fast and the arising centrifugalforces are too weak to significantly stretch its bond, it is useful to intro-duce the so-called rigid rotor approximation. In this approximation, theinteratomic distance R = |R| is assumed constant, leaving only the secondterm in (1.2). Moreover, since the wave function now depends only on thepolar and azimuthal angles of R (θ and φ, respectively), the Laplacian term− ~22µ∇2R simplifies to:Hˆ = − ~22µR2(1sin θ∂∂θ(sin θ∂∂θ)+1sin2 θ∂2∂φ2)=−~22I(1sin θ∂∂θ(sin θ∂∂θ)+1sin2 θ∂2∂φ2), (1.3)where I = µR2 is the molecular moment of inertia.The eigenfunctions of this Hamiltonian, spherical harmonics YMJJ (θ, φ),are parametrized by the two quantum numbers, J = 0, 1, 2, . . . and MJ =−J,−J + 1, . . . , J , called the total angular momentum quantum numberand the magnetic quantum number. These numbers have a meaning of thevalue of the total angular momentum and its projection onto a polar axisz, expressed in units of ~1. The eigenenergies, corresponding to these states1The exact value of the angular momentum is rather ~√J(J + 1), which becomes closeto ~J at large J .31.2. Control of rotational states with intense non-resonant laser fieldsare given byJ =~22IJ(J + 1) = BJ(J + 1), (1.4)where we introduced constant B, called the rotational constant. The energy,as expected, depends quadratically on the angular momentum. Since itdoesn’t depend on MJ , each J-state is (2J + 1)-fold degenerate over thepossible values ofMJ . Because the molecular bond may stretch, fast rotationwould cause an increase of the moment of inertia, introducing higher orderterms in the so-called Dunham expansion:J = BJ(J + 1) +DJ2(J + 1)2 + . . . (1.5)Rotational spectrum of real diatomic molecules becomes more compli-cated for the states with a non-zero electron angular momentum. The totalangular momentum J is, as before, a conserved quantity, but now it is a sumof the angular momenta of the nuclear and electronic motion. For example,diatomic oxygen O2, frequently used in our experiments, has an electronspin of S = 1 in its ground electronic state. Due to the zero electron orbitalmomentum it follows Hund’s case (b) coupling, and its rotational energydoes not follow Eq.(1.5) anymore [11]. However, if we introduce the angularmomentum excluding electron spin N = J− S, which in the case of oxygencorresponds to the rotation of its nuclei, the spectrum will still be followingEq.(1.5) with J substituted by N . Each N -state is split into three closelyspaced spin substates due to the spin-spin and spin-rotational interaction.For that reason N , rather then J , is often used below as a quantum numberdesignating the degree of rotational excitation in oxygen. Sometimes, forconsistency, it is also used for other molecular species, such as N2, with zeroelectron spin, making N and J equivalent.1.2 Control of rotational states with intensenon-resonant laser fieldsThe appeal of rotational control has stimulated the development of multi-ple techniques utilizing external electromagnetic fields [37, 119]. The mostprominent class of such techniques involves subjecting molecules to strongnon-resonant laser pulses[36, 119] in the visible or near-infrared region.The interaction between the non-resonant laser field and the moleculescould be described as follows. Consider a molecule placed in an oscillat-ing electric field. The characteristic timescale of nuclear motion typically41.2. Control of rotational states with intense non-resonant laser fieldsspans from 100 femtoseconds up to tens and hundreds of picoseconds, muchlonger than the period of laser light of a few femtoseconds. This means thatthe heavy nuclei are essentially frozen on the time scale of the field oscilla-tions. On the contrary, the light electrons are typically moving on a shortertime scale and would adiabatically adapt to the electric field, stretching themolecular orbital in the direction of the field vector and producing an in-duced dipole moment. The expectation value of this dipole moment overthe electronic state, in the first order with respect to the electric field E(t),could be expressed asµindρ =∑ρ′αLFρρ′(r1, r2, . . . , rN)Eρ′(t),where the molecular polarizability tensor αLFρρ′ describes the adiabatic elec-tronic response to the field and parametrically depends on the positions ofN nuclei r1, r2, . . . , rN and indices ρ, ρ′ go over Cartesian laboratory-fixedcoordinates x, y, z. In the rigid rotor approximation the positions of thenuclei can be fully described by the three Euler angles (θ, ϕ, χ), character-izing the orientation of molecule in a laboratory-fixed frame. The potentialenergy of the electric field-induced dipole interaction can then be writtenas:Vint(t) = −12µind(t) ·E(t) = −12∑ρρ′αLFρρ′(θ, ϕ, χ)Eρ(t)Eρ′(t), (1.6)where the factor of 1/2 accounts for the induced nature of the dipole. Dueto the time scale separability, mentioned above, it is natural to break thetime dependence of the laser field into a slowly varying envelope and fastoscillating components,E(t) =12(ε(t)eiωt + c.c.). (1.7)In the equation above, ε(t) = εˆε(t), with εˆ and ε(t) being the complex unitpolarization vector and slowly varying pulse envelope, and ω is the angularfrequency of light. For the purpose of studying rotational dynamics, wecan now average the interaction potential (1.6) over the irrelevantly fastoscillations of the field in Eq.(1.7) to get the effective interaction potential51.2. Control of rotational states with intense non-resonant laser fieldsfor the rotational motion2:V effint (t) = −14∑ρρ′αLFρρ′(θ, ϕ, χ)ερ(t)ε∗ρ′(t). (1.8)The angular dependence of the interaction potential is contained in theanisotropy of the molecular polarizability tensor αˆLF(θ, ϕ, χ) that could beexpressed in terms of the molecular-fixed frame polarizability tensor αˆMF asαLFρρ′(θ, ϕ, χ) =∑kk′〈ρ|k〉αMFkk′〈k′∣∣ρ′〉 . (1.9)Here, indexes k and k′ go over the molecular-fixed coordinates X,Y, Z, and〈ρ|k〉 are the directional cosines between the laboratory-fixed and molecular-fixed frames unit vectors, which contain all the angular dependence of theinteraction potential. The final expression for the effective potential is then:V effint (t) = −14∑ρρ′kk′〈ρ|k〉αMFkk′〈k′∣∣ρ′〉 ερ(t)ε∗ρ′(t). (1.10)An illustrative example of potential (1.10) is the case of a field, linearlypolarized along z. The unit polarization vector εˆ is equal to zˆ in this case,and if we choose the molecular-fixed unit vectors Xˆ, Yˆ , Zˆ to be the principalaxes of the polarizability tensor αˆMF, with Zˆ chosen along the direction ofmaximal polarizability, the interaction potential becomes[1]:V effint (t) = −14ε2(t)(∆αZX cos2 θ + ∆αY X sin2 θ sin2 χ), (1.11)where ∆αkk′ = αMFkk −αMFk′k′ . This potential has a global minimum at θ = 0,independent of φ and χ, which means that it is energetically preferable fora molecule to align its axis of maximal polarizability along the polarizationdirection.Most of the methods that use the interaction potential (1.10) to controlthe molecular rotational state could be categorized into two large classes,namely adiabatic and non-adiabatic methods, depending on the shape ofthe envelope ε(t)[36, 119]. The two approaches are illustrated in Fig. 1.1.2Up until now we assumed that the molecular permanent dipole moment µperm is zero.For a non-zero µperm an additional term −µperm ·E(t) would need to be included in theinteraction potential. This term, however, would disappear when averaged over the fieldoscillations.61.2. Control of rotational states with intense non-resonant laser fields(b)(a)J4Time3210Field-free JField intensity43210Laser spectrumFigure 1.1: Illustration of (a) adiabatic vs. (b) non-adiabatic rotationalexcitation.Adiabatic methods, usually used to achieve alignment[74, 113, 119], em-ploy ε(t) that changes slowly on the time scale of the rotational period,adiabatically modifying the rotational states (Fig. 1.1 (a)). If we take amolecule in a rotational ground state and slowly turn a linearly polarizedlaser field on, the molecule will end up near the bottom of the potentialwell (1.11), having its axis of maximal polarizability aligned along the fielddirection. More complicated schemes involve elliptically polarized light foraligning all three axes of a molecule[77]. The adiabatic methods are suffer-ing from two limitations. First, the control is executed only during the laserpulse, with the molecules adiabatically returning to their initial rotationalstate when the laser field is gone. The presence of a high intensity field isundesirable in many experimental scenarios. Second, the adiabatic methodscan only be used to achieve alignment or orientation, while other kinds ofrotational control remain inaccessible to them.A different, non-adiabatic approach employs fields, whose envelops ε(t)change rapidly on the scale of molecular rotational periods. A famous ex-ample, the non-adiabatic “delta-kick” excitation, is when the molecules areirradiated by a very short intense linearly polarized pulse, usually on a fem-tosecond scale. Unlike the adiabatic case, the molecules are essentially frozenduring the pulse, but experience a “kick” towards the direction of the polar-71.3. Optical centrifugeization vector. Quantum mechanically this corresponds to many rotationalstates being populated via multiple Raman transitions, with both pump andStokes photons coming from the same laser field (Fig. 1.1 (b)). Conventionalsingle-pulse excitation schemes lack selectivity with respect to the final speedof molecular rotation and produce broad rotational distributions[112]. Al-though sequences of pulses have been successfully used for selective[17, 28,103, 143] and directional[29, 66, 142] rotational excitation, the range ofaccessible rotational states has been limited to relatively low quantum num-bers (of order 10 above the initial state) due to the molecular breakdown inintense laser fields.1.3 Optical centrifugeAnother approach to the rotational control of linear molecules, combiningthe advantages of both adiabatic and non-adiabatic techniques, was pro-posed in [57]. A linearly polarized intense laser field slowly turns on, align-ing the molecular axes of maximal polarizablity (typically, the internuclearaxis of a linear molecule) along its direction. The polarization plane thenundergoes an accelerated rotation illustrated in Fig. 1.2 (a), adiabaticallyspinning the molecular axes along and eventually releasing them in highrotational states.Consider two laser fields, circularly polarized in the opposite directions,of the same amplitude E0 and slightly different optical frequencies ω− andω+, ω+ > ω−. Provided both fields are propagating along the z direction,the electric fields that each of them produces in the xy plane, can be writtenasε− = ε0(sinω−tcosω−t), ε+ = ε0(sinω+t− cosω+t). (1.12)The total electric field is then:ε = ε− + ε− = ε0(sinω−t+ sinω+tcosω−t− cosω+t)= 2ε0 sinω0t(cos Ωtsin Ωt), (1.13)with ω0 =ω++ω−2 and Ω =ω+−ω−2 . Provided the fields have close opticalfrequencies, Ω  ω0, this expression describes a linearly polarized field offrequency ω0 with a plane of polarization precessing around the propagationdirection with an angular frequency Ω.To achieve the field configuration of Fig. 1.2 (a), it is therefore enough toadd together two circularly polarized laser pulses, whose optical frequencieschange linearly in time, starting at the same value ω0, ω±(t) = ω0 ± βt81.3. Optical centrifugeE(a) (b)Optical frequencyTimeFigure 1.2: (a) Principle of an optical centrifuge excitation. A linearlypolarized intense non-resonant laser field aligns the molecular axes along itspolarization direction. This direction then undergoes accelerated rotation,sweeping the depicted red “corkscrew” shape and dragging the molecularaxes along, eventually releasing them to freely rotate with a high angularfrequency. (b) To produce the centrifuge field, one can combine two cir-cularly polarized pulses of the opposite chirp and polarization handedness,shown in the figure.(Fig. 1.2 (b)). Such frequency chirped pulses can be easily produced bythe femtosecond pulse shaping techniques, as described in subsection 2.1.2.The highest angular frequency Ωmax, that the molecules acquire during thecentrifugal excitation, corresponds to half the laser bandwidth and reach6 × 1013rad/s for currently available broadband laser systems. In diatomicoxygen, this frequency corresponds to the angular momentum of ∼ 120~, incontrast to 10− 20~ typically achievable with non-adiabatic methods.Quantum mechanical description of the optical centrifuge was given byVitanov and Girard[131]. Let us consider a diatomic molecule initially inthe ground rotational state J = 0,MJ = 0, placed in a field of n− andn+ photons of the positively and negatively chirped centrifuge components,respectively. The molecule can absorb a photon from the lower frequencypulse field, and emit it to the higher frequency one. The system is nowdescribed by a “dressed state” with an angular momentum of 2, and thephoton fields population of (n−+1) and (n+−1). Since each of the photonsbears an angular momentum of ~, the magnetic quantum number would alsoincrease by 2, MJ = 2. The process could then be repeated, leading to theexcitation of J = 4 state, then J = 6 state and so on. The centrifuge fieldtherefore couples the states | J,MJ = J, n− + J/2, n+ − J/2 〉.91.3. Optical centrifuge|0, n -, n +>|2, n--1, n++1>|4, n --2, n++2>|6, n--3, n++3>Centrifuge angular frequency/(B/  )Energy/B8010002040600 2 4 6 8 10 12 14ΩrΩr2/BFigure 1.3: Illustration of an adiabatic population transfer in the opticalcentrifuge.The energies of these states in the absence of the interaction betweenthe molecule and the field are given by a simple sum of the energy of thefield and the rotational energy of the molecule:(J, n− + J/2, n+ − J/2) = BJ(J + 1) + ~(n− + J2)ω− + ~(n+ − J2)ω+= BJ(J + 1)− ~ΩJ + const,where const is a constant independent of the rotational state. These energiesare plotted as a function of the centrifuge rotational frequency Ω in Fig. 1.3with dashed lines.If we turn on the interaction between the field and the molecular rota-tional motion, described by the Raman Rabi frequency Ωr =ε2∆α8 [112], theneighboring terms with ∆J = 2 would repel (solid lines in Fig. 1.3), forminga network of avoided crossings at Ω = Ωn ≡ (4n − 1)B/~, n = 1, 2, . . . ,101.4. Goals of this workemphasized by the grey dashed lines in Fig. 1.3. The level separation atthe crossings is ∆ = Ωr. If Ω grows slowly in time, fulfilling the adiabatic-ity condition Ω˙/Ω  Ωr, the molecule, initially in the ground rotationalstate, would follow the lowest solid line in Fig. 1.3, consecutively passingthe rotational states | J = 0 〉 → |J = 2 〉 → . . . .One can notice from Fig. 1.3, that starting from | J = 2 〉, the moleculecan also adiabatically climb the rotational states ladder, this time via | J = 2 〉 →| J = 0 〉 → |J = 4 〉 → |J = 2 〉 → |J = 6 〉 → . . . . The “forbidden” ∆J = 4Raman transitions are allowed because the dressed states are superpositionsof several field-free J-states. The strength of such transition is much lower,corresponding to a level splitting of ∆ = Ω2r/B. Similar splittings with evensmaller values of ∆ occur for higher J-states. The adiabaticity conditionis harder to satisfy in this case, which means that if the field intensity isnot high enough, or the rotational frequency ramps too quickly, part of theinitially rotationally excited population can hop over the avoided crossing,escaping the process of further acceleration.In the classical picture, when the field is turned on, the field free rota-tional states are adiabatically transfered to the eigenstates of the angularpotential well (1.11), the so-called pendular states[36]. In such states, molec-ular axes are aligned with the field polarization. Different initial rotationalstates would adiabatically correspond to different pendular states, with thependular energy in general increasing with field free J . The effect of a non-adiabatic escape of the molecules from the accelerating centrifuge in thislanguage is attributed to their “spilling” from the levels that lie too close tothe angular potential top, if this potential accelerates too quickly or is notdeep enough.1.4 Goals of this workSince the original theoretical proposal, the technique of an optical centrifugewas experimentally realized by two groups. Villeneuve et al.[130] observedthe dissociation of the centrifuged chlorine molecules, which was attributedto bond breaking due to the centrifugal forces. Yuan et al.[138] observedsubstantial heating of the centrifuged molecular gases as the extremely highrotational energy was distributed to other degrees of freedom. In both cases,indirect secondary processes were used to detect rotational excitation. Tostudy the properties of the centrifuge excitation and to explore the possibil-ity of controlled molecular spinning, one needs to develop direct methods ofdetecting extreme rotational states. Two different approaches of character-111.4. Goals of this workizing molecular rotation are available. The first one is based on either timeor frequency resolved spectroscopy, whereas the second one involves directimaging of the molecular angular distribution.There are many spectroscopic techniques suitable for studying the rota-tional structure. Although purely rotational absorption spectroscopy is pos-sible, for typical molecules it would require microwave or terahertz sources,tunable in wide frequency ranges specific to each molecule of interest. Us-ing non-resonant Raman instead of a resonant absorption process lifts therequirement for a molecule-specific source. Time-resolved coherent Ramanscattering also provides information about the relative phase of the rota-tional states. As a third-order nonlinear optical process, however, it requireshigh molecular densities.An alternative to a purely rotational spectroscopy is vibrational or elec-tronic spectroscopy with rotational resolution. Yuan and coworkers [138]used infra-red (IR) spectroscopy to detect super rotors, but their experi-mental system only allowed to see relatively low energy levels. Resonance-enhanced multiphoton ionization (REMPI) spectroscopy [81], on the otherhand, proved to be a very sensitive tool and is routinely used to measure ro-tational state distribution in molecules [122]. Its applicability to super rotorswas unclear, as the excited electronic states, participating in the ionizationprocess, is often poorly known at high degree of rotational excitation.The above mentioned spectroscopic methods provide limited informationabout the relative phases of the rotational states and the distribution amongdegenerate MJ levels. More complete information could be acquired fromthe direct measurement of molecular angular distributions. Such methods,often used for detecting molecular alignment [27], could be applied to thestudy of molecular super rotors.A method for controlling the magnitude of angular momentum, acquiredby the molecules in the optical centrifuge, was proposed in the first experi-mental work on the subject[130]. The authors demonstrated that truncatingthe centrifuge spectrum resulted in the decrease of the centrifuge-inducedangular momentum, reflected by the disappearance of the spinning-induceddissociation below some threshold value of the laser bandwidth. The em-ployed detection method, however, did not allow them to study the limitsand properties of the centrifuge-based rotational control, which remained anopen area for further investigation.In addition to the magnitude, the methods of controlling the direction ofmolecular rotation are equally desired. Simple reorientation of the J vectoris possible by changing the laser propagation direction in a broad range ofangles, which may be technically challenging in many experimental scenar-121.4. Goals of this workios. Yun and coworkers [140] theoretically proposed a method of controllingthe direction of the angular momentum of non-magnetic molecules throughthe interaction of magnetic field with the rotationally induced magnetiza-tion. The applicability of such magnetic control to both non-magnetic andparamagnetic super rotors required further experimental studies.In general, full rotational control implies redistributing the molecularpopulation over the magnetic quantum numbers MJ , rather than a simpleincreasing of J. As discussed above, MJ = J is dictated by the selection rules∆MJ = 2 for the Raman process with two circularly polarized photons ateach step of the rotational ladder climbing. This suggests, that modify-ing the polarization of the centrifuge could provide yet another means forcontrolling molecular rotation.The importance of the adiabaticity of the centrifuge excitation was real-ized in the original paper of Karczmarek and coworkers [57]. Further theo-retical studies of Vitanov and Girard [131] suggested that the deviation fromthe complete adiabaticity would result in a broadened coherent rotationalstate distribution, leading to the wavepacket time evolution. The dynamicsof the wavepackets consisting of rotational states with MJ = J was theoret-ically studied in the context of so-called “cogwheel” states [76] – the statesthat propagate without dispersion along classical trajectories. Cryan andcoworkers[18] proposed to use such states as a “stopwatch” to synchronizeultrafast laser pulses, with the classical rotation of molecular axes playingthe role of a clock’s hand. One could, however, expect that the uncertaintyprinciple puts a limit on the classical molecular rotation. Applicability of thecentrifuge to producing dispersionless states and the dynamical propertiesof the created wavepackets were yet to be studied experimentally.Recently, much effort and resources has been invested in controlling thedegree of molecular alignment[119]. As discussed above, the two main ap-proaches are based on adiabatic and non-adiabatic excitation schemes. Adi-abatic approach was shown very effective, producing high-degree alignmentin various molecular species[63, 64]. Its main drawback is the presence of theintense external field during the alignment window, which is intolerable inmany cases. Adiabaticity is even more desirable for the alignment of asym-metric top molecules[74]. The three different moments of inertia lead to anirregular rotational spectrum. The wide rotational distributions producedby non-adiabatic methods therefore result in complex aperiodic dynamics,which severely complicates the routes to rotational control. Combining adi-abaticity with the field-free excitation, the methods based on the opticalcentrifuge could open new possibilities for reaching strong and permanentalignment of asymmetric molecules.131.4. Goals of this workCollisional relaxation of super rotors was studied both theoretically [33,42] and experimentally [138]. Three stages of such relaxation – decoherence,reorientation and thermalization, were identified. However, the experimen-tal methods based on the rotational excitation with a short femtosecondpulse or indirect rotational detection, provided limited information aboutthe decay dynamics and were not capable of characterizing these decaystages separately. Different detection techniques were also required to ad-dress other theoretical works which suggested a number of exotic effects in agas of molecular super rotors. These effects include vortices formation [120],explosive thermalization and anisotropic diffusion [62].The main goal of this work was to build an optical centrifuge for pro-ducing molecular super rotors and to experimentally address all the openquestions listed above.14Chapter 2Experimental methodsTo experimentally investigate molecular super rotors, two ingredients wererequired. First, we needed a laser and an optical system to produce thecentrifuge field capable of exciting the molecular ensemble to high rotationalstates. Second, a detection setup was required to detect and analyze thebehavior of molecules in those states.An optical centrifuge setup was the same for all of our experiments and isdescribed in section 2.1. The detection methods we used can be divided intotwo classes, applied to gases under ambient conditions (at room temperature,and pressures of order of 1 bar, Raman setup) and to dilute cold gases (UHVsetup).Rotational relaxation of molecular super rotors in dense room-temperaturegases is of great interest in the context of molecular kinetics. To investi-gate the effects of super rotation on various relaxation processes, a hermeticchamber, filled with a few hundred Torr of gas under study, was used. Themain technique utilized in this setup for the rotation detection was coherentRaman spectroscopy. Being a spectroscopic method, it allowed a sensitivemeasurement of the rotational state distributions in a single laser shot. Withthe use of circularly polarized probe beam, it provided the information onthe directionality of molecular rotation. Finally, since Raman scatteringdoes not directly rely on electronic resonances, it could be applied to almostany molecular system without any modifications. This method is describedin details in section 2.2.In many cases, however, initially cold gas ensembles were desired. Stud-ies of the processes taking place on long time scales required the absenceof collisions. In the case of asymmetric top molecules, where at room tem-perature the initial thermal excitation of a broad rotational wave packet ledto overcomplicated dynamics, low temperature was essential. Finally, iondetectors, utilized for imaging the molecular angular distributions, requirehigh-vacuum conditions to operate. For these experiments an ultra-high vac-uum (UHV) chamber was built. A jet of gas, expanded through a narrownozzle from high pressure to vacuum, served as a molecular source. Duringthis expansion, both the translational and the internal temperature of gas152.1. Optical centrifugedecreased dramatically, typically below 10 K, resulting in the collisionlessdynamics and a very narrow rotational distribution. This was accompaniedby a dramatic (roughly estimated at around 8 orders of magnitude) drop inthe molecular number density. Since the coherent Raman scattering signaldepends quadratically on the molecular density, it becomes inapplicable inthe UHV setup, and other detection techniques were required.One such technique, Resonance Enhanced Multiphoton Ionization (REMPI)spectroscopy (section 2.3), uses a tunable ultraviolet laser to ionize themolecular sample while the produced ion current is measured as a func-tion of the UV wavelength. Positions and intensities of the resonances allowto determine the molecular rotational population prior to ionization. UnlikeRaman spectroscopy, it is molecule specific and requires the precise knowl-edge of the molecular energy spectrum. Moreover, it requires long scans ofthe laser wavelength to determine the population. The main advantage ofthis method is its high sensitivity, down to a few ions with a modern iondetectors, such as micro-channel plates (MCP), which makes it suitable forthe spectroscopy of dilute gas samples in the UHV setup.Both methods described above enabled us to measure the rotationallevel population. The phase information was completely lost in the caseof REMPI, and limited in the case of Raman spectroscopy. Moreover, thedistribution among degenerate states, such as the states with different pro-jections of the molecular angular momentum, cannot be fully determinedby these methods. An alternative approach, allowing for a complete recon-struction of the rotational wave function, is based on detecting the time-dependent instantaneous angular distribution of molecules. In practice, thisis typically achieved by breaking the molecules apart with an ultrashortlaser pulse, and measuring the recoil direction of the fragment ions usingthe so-called Velocity-Map Imaging (VMI) technique, described in details insection 2.4.2.1 Optical centrifuge2.1.1 Laser systemThe main laser system, used in our experiments to produce the field of anoptical centrifuge, was a commercially available femtosecond source, shownin Fig. 2.1. Titanium-Sapphire based oscillator (Coherent Micra) generatedspectrally broad, 90 nm full-width at half maximum (FWHM) laser pulses,centered at 800 nm at 80 MHz repetition rate (Fig. 2.1 (b), red line). Asthe energy of each pulse was a mere 5 nJ, and because the centrifuge re-162.1. Optical centrifugeMicra-5Oscillator5 nJ @ 80 MHzBW limited10 mJ @ 1 kHzchirped to 120 ps80 MHz sync signal(a)(b)1 kHz Pockels celland pump trigger500/50/10 Hzoutput triggersSDG Elite amplifiercontrol boxLegend Elite Duotwo stage amplifierIntensity, arb. u.Wavelength, nm7000.00.20.40.60.81.0750 800 850 900112230 nmFWHM90 nmFWHMFigure 2.1: (a) Diagram of our femtosecond laser system. (b) Spectra oflaser pulses before (red line) and after (blue line) amplification.quired intense laser fields, it was amplified in a Titanium-Sapphire two stagechirped-pulse amplifier (Coherent Legend Elite Duo), operating at 1 kHzrepetition rate. The amplified bandwidth decreased to 30 nm (Fig. 2.1 (b),blue line), which corresponds to a 35 fs long Fourier transform-limited pulse.The energy and duration of each pulse was 10 mJ and 120 ps, respectively.2.1.2 Centrifuge shaperIn the heart of our experimental setup was an optical centrifuge shaper,which was built according to the original proposal of [130] to create anoptical field configuration of Fig. 1.2 (b). The shaper diagram is shown inFig. 2.2 (a). The concept is based on a common 4-f optical shaper setup,where the input diffraction grating G0 is followed by a pair of identicalpositive lenses L0 and L1 and a grating G1. The beam passed throughthe shaper in one direction, and was turned back by a retroreflector RR1,missing the input mirror IM on its way out. The effect of such shaping was172.1. Optical centrifugeG1G2L0IML1L2L2 focal planeL1 focal planFM(a)(b)G0RR2RR1HM"Blue" armInput beam "Red" armHMShutter750 770 790 810 8300.00.40.8Intensity, arb. u.Wavelength, nm0-2 mJ-0.1 THz/ps chirp0-7 mJ0.1 THz/ps chirp0-2 mJ0.1 THz/ps chirp11Shutter2233Figure 2.2: (a) Optical centrifuge shaper. IM: input mirror, G0, G1, G2:diffraction gratings, L1, L2, L3: f = 250 mm achromatic lenses, HM: half-cut mirror, FM: folding mirror, RR1, RR2: retroreflectors. Inset: movableshutter. (b) Laser spectra before (magenta), and after the shaping for the“blue” (blue) and “red” (red) arms. The “blue” arm spectrum is shown withthe shutter completely out of (dashed), and half way in the beam (solid)an addition of a linear chirp, proportional to the distance from the outputgrating to the focal plane of the second lens (L1). Putting a half mirror HMin a common focal plane of the two lenses on the optical axis, where thewavelength was directly mapped to a horizontal position, we reflected thelong-wave half of the initial beam towards its own combination of lens L2,grating G2 and retroreflector RR2. This configuration allowed us to controlthe chirps of the two spectral halves individually: while G1 was placedat L1’s focal plane, keeping the chirp of the “blue” part of the spectrumunchanged at 0.1 THz/ps, G2 was shifted so as to invert the chirp of the“red” part.182.1. Optical centrifugePrior to shaping, the laser beam width was decreased by a factor of 2from 12 mm to 6 mm with a telescope, which was necessary for fitting thebeams through the shaper optics.The output of the centrifuge shaper consisted of two oppositely chirpedbeams, “red” and “blue” arms, whose spectra are shown in Fig. 2.2 (b) to-gether with the input spectrum (purple). By putting a partial beam shutterin the Fourier plane of the shaper and moving it in and out of the beam path,as shown in the inset of Fig. 2.2 (a), we were able to control the bandwidthof the “blue” arm, controlling the centrifuge terminal angular frequency.The shutter was mounted on a 25-mm travel motorized translation stage(Newport MFA-CC), computer controlled for precise positioning.HWPMQWP4 mJ1 kHzPBSFigure 2.3: Combining the centrifuge arms. HWP: half-wave plate, M:mirror, PBS: polarizing beam splitter cube, QWP: quarter-wave plate.To create a rotating field of an optical centrifuge, the two arms had tobe circularly polarized and combined together. This was executed with apolarizing beam splitter cube and a quarter-wave plate, as shown in Fig. 2.3,producing an optical centrifuge beam of 4 mJ per pulse at 1 kHz repetitionrate. For some experiments, however, higher pulse energies were required. Inthis case an additional custom-built multi-pass amplifier (MPA, see subsec-tion 2.2.2) was used to amplify the two centrifuge arms. The amplificationhad to be done individually prior to joining them up, due to the polarizationsensitivity of the amplification process.Shaper adjustmentThe centrifuge beam was carefully characterized and adjusted in the spatial,spectral and time domains.To adjust the spatial characteristics of the beams, their profiles wereobserved both in the near field, right at the output of the shaper, as well192.1. Optical centrifuge(a) (b) (c)Figure 2.4: Far field profiles of the (a)“blue” arm, (b)“red” arm, (c)bothbeams combined.as in the far field. In the near field the beams were observed on a piece ofpaper, and the inspection usually boiled down to verifying that they werenot clipped in the shaper. Adjustment in the far field required much finercontrol, as the beams profiles there were more sensitive to the spherical aber-rations in and misalignment of the shaper lenses, and to the spatial chirpsintroduced by the gratings. To do that, the beams were focussed with af=250 mm lens and inspected with a beam profiler (Coherent LaserCam-HR). Lenses L1 and L2 were then positioned so as to achieve the best circu-larity of the beam profiles and simultaneous focussing of the two arms. Theexamples of the optimal beam profiles are shown in Fig. 2.4.Centrifuge BBOLens LensAperturePDProbeFigure 2.5: Illustration of the standard cross-correlation setup.A quicker way of overlapping the two centrifuge arms, used routinelyon a day-to-day basis, was implemented in a UHV setup. The centrifugepulse energy was high enough to cause a detectable multi-photon ionizationfrom the molecules in the chamber (see section 2.2). The correspondingion current was measured as the centrifuge output mirrors were adjusted tomaximize it. Owing to the high nonlinearity of the ionization process, thehighest current was observed at the best overlap of the two beams.To characterize the temporal profile of the centrifuge pulses, cross-correlationmeasurements were carried out (Fig. 2.5). The centrifuge and probe beamswere aligned collinearly and focussed together on a BaB2O4 (BBO) non-202.1. Optical centrifugelinear crystal. The sum frequency signal, generated through the nonlinearmixing of the two fields, was recorded as a function of the probe delay.(b) (c)(a)Opticalfrequency,103cm-1Wavelength,nmIntensity, arb.u.Intensity,arb.u.Time, ps202040406060800.40.88012.282076078080012.412.612.813.0"Blue" arm"Red" armShutterRR1 delay0.00.00.40.81.2Figure 2.6: Time- and frequency-resolved characterization of the centrifugebeams. In each panel, the measurements related to the “blue” (“red”) armare colored blue (red). (a) Time-resolved cross correlation measurements.(b) Spectra of the beams. (c) Time-resolved spectrogram of each centrifugearm. Tilted lines correspond to a frequency chirp.Two kinds of measurements were carried out using this setup. The firstone was executed with a short 40 fs probe, and the sum frequency wasdetected with a photodiode (Thorlabs). The electrical signal from the pho-todiode was integrated on a gated integrator (SRS SR250) and recordedwith a data acquisition board (National Instruments 6024E). The resultingsignal (Fig. 2.6 (a)) represented the time dependence of the centrifuge armstaken with a high temporal resolution. Having taken the signal from theblue and the red arms separately, we were able to adjust the blue arm delay(RR1 position, see Fig. 2.2 (a)), synchronizing it with the red arm.To characterize the centrifuge in the frequency domain, the probe band-width was narrowed and the sum frequency signal was measured with a spec-trometer (Photon Control). The resulting spectrogram, shown in Fig. 2.6 (c)212.2. Raman setupafter the probe photon energy subtraction, provided a way of measuring thechirps of the two arms.2.2 Raman setupTime-resolved coherent Raman scattering is a common tool of choice foranalyzing the dynamics of molecular rotation. It has been successfully usedfor the precision thermometry of flames[109] and the studies of collisionaldecoherence in dense gas media[60, 90]. We employed this technique todetect and study centrifuged molecules in dense gas samples.J+2J+1JFigure 2.7: An illustration of Stokes (left) and anti-Stokes (right) rotationalRaman scattering.Quantum mechanically, synchronous molecular rotation corresponds toa superposition of a few rotational quantum states - a “rotational wavepacket”, with an average frequency separation matching the frequency ofthe classical rotation. Consider two coherently populated rotational states|J,MJ = J〉 and |J + 2,MJ = J + 2〉 (Fig. 2.7). Laser light can coherentlyscatter off of it, producing either Stokes (left diagram) or anti-Stokes (rightdiagram) radiation, detuned from the initial frequency by the energy sepa-ration of the relevant rotational levels. Owing to the well-defined angularmomentum projection difference (∆MJ = 2) between the two states, a circu-222.2. Raman setuplarly polarized light will acquire Stokes or anti-Stokes frequency sidebands,depending on whether the molecules rotate in the same or opposite directionwith respect to the probe polarization. In both cases, illustrated in Fig. 2.7,the magnitude of the Raman shift equals twice the rotation frequency, whileits sign reflects the direction of molecular rotation.Space-fixedframeωMolecularframeω-ΩSpace-fixedframeω-2ΩFigure 2.8: An illustration of the rotational Doppler effect. Thick (thin)blue arrows show the circularly polarized probe field of the same (opposite)direction with respect to the molecular rotation (green arrows).In classical terms, the frequency shift can also be viewed as a result of therotational Doppler effect[68] (Fig. 2.8). Indeed, consider a classical molecule,co-rotating with the probe field of frequency ω. In the frame of referenceconnected to the molecule, the observed light frequency will be lower by theangular frequency of rotation, Ω, due to the Doppler effect. Anisotropic po-larizability of the molecule will introduce birefringence, adding ellipticity tothe circularly polarized probe. This ellipticity is equivalent to the appear-ance of a weak field of the same frequency ω−Ω, circularly polarized in theopposite direction. Transforming this field back to the laboratory frame willfurther decrease its frequency by Ω (since now it is counter-rotating withthe molecule) to ω − 2Ω, in good agreement with the quantum picture ofcoherent Raman scattering, described above. Similar arguments justify theappearance of a ω+ 2Ω sideband in the case, when the rotation direction ofthe molecule coincides with that of the probe field.2.2.1 Raman chamberTo carry out a number of experiments on molecular super rotors underambient conditions, we built a simple gas chamber shown in Fig. 2.9. Thechamber had a barrel shape, with a large 12′′ opening on top, used to mounta wide acrylic viewport, and four 2.75′′ conflat side flanges.232.2. Raman setupViewportclampAcrylicviewportOpticalwindowWindowclampNippleOpticalbreadboardFigure 2.9: Gas chamber for Raman spectroscopy.The two opposite 2.75′′ viewports, through which the centrifuge beamentered the chamber, were additionally extended with vacuum nipples. Thiswas necessary to avoid damage from the high intensity centrifuge beam,focused by a long focal length lens (f = 1 m). The distance between theinput and output windows reached 70 cm after this extension.The two other ports were used for the vacuum pump, gas line and elec-trical connections. In the experiments requiring optical probing from a side,they were used to feed through the probe laser beam. The chamber wassealed and evacuated through one of the ports with a hermetic scroll pumpdown to a few tens of mTorr, and then filled with a studied gas from a gascylinder.2.2.2 Multi-pass amplifierUnder ambient conditions, when the molecules were initially rotationallyhot, an additional amplification of the centrifuge beam was required to adi-242.2. Raman setupabatically excite high rotational states. To achieve higher energy, we built amultipass amplifier (MPA). As the gain of a typical Ti:Sapphire amplifier ispolarization sensitive, both arms of the centrifuge arms had to be amplifiedseparately prior to being combined into the centrifuge pulse.Ti:Sapph PumpInputOutput1.7 mDouble MPARedarmRed armamplifiedBluearmBlue armamplifiedMPA(b)(a)Nd:YAGLaserFL trigQSw trigMPA 2MPA 1Pulsegenerator10 Hz fromLegend Control10 Hz600 mJ532 nm10 Hz0.2 mJ1 kHz20 mJ10 HzFigure 2.10: (a) Diagram of separate amplification of the two centrifugebeams. (b) Multi-pass amplifier.The amplifier therefore consisted of two independent MPA’s (Fig. 2.10 (a)),pumped by a second harmonic of a Q-switched Nd:YAG laser (SpectraPhysics). This laser generated 600 mJ of 532 nm light at 10 Hz. A pulse gen-erator (Quantum Composers 9514), synced to every 100th centrifuge pulse,triggered the pump laser’s flashlamps (FL) and Q-switch (QSw) at 10 Hz.The output was split on a 50/50 beam splitter, and sent to a pair of identicalMPA’s, each amplifying its own centrifuge arm.Inside each MPA, the pump beam was focused down to a few millimeterson a 10 mm diameter, 35 mm long Ti:Sapphire crystal rod, used as a gainmedium. The crystals were cut at the Brewster’s angle to minimize thereflection, and fixed in a water-cooled mounts, to stabilize their temperatureat 18◦. The input beam, passing through the crystal close to its axis, wasretro-reflected by a set of mirrors three times, yielding a total amplificationof about 100, from 0.2 mJ to 20 mJ.252.2. Raman setup2.2.3 Broadband probe beamCompressor750 770 790 810 8300.00.40.8Intensity, arb. u.Wavelength, nmRRReflectiongratingsDelaylineProbeoutputFromLegend(a)(b)30% beamsplitterTo centrifugeshaperVariableattenuator0-7 mJ30 nm FWHMstretched to 120 ps2 mJ30 nm FWHMBW limited 40 fs0.12-20 nm FWHMBW limitedSlitMirrorMMMProbeshaperTransmissiongratingSphericalmirrorRemovablemirror113322SlitAttenuatorFigure 2.11: (a) Probe beam setup. M: mirror, RR: retroreflector. (b)Full centrifuge spectrum (red), full probe spectrum (blue), and truncatedprobe spectrum (magenta)The output of the Legend broadband amplifier was split on a dielectricbeam splitter (Layertech, Fig. 2.11), with 30% (3 mJ) of the total energyserving as a probe and the rest sent to the centrifuge shaper (see subsec-tion 2.1.2). Variable attenuation of the centrifuge beam was provided bythe two cube polarisers (Thorlabs PBS-202), with the first one mounted ina manual rotary stage and the second mounted in a fixed holder. Changing262.2. Raman setupthe angle between the two polarisers allowed us to gradually change the laserenergy from 0 mJ (crossed configuration) to a maximum of 7 mJ (uncrossedconfiguration). The more conventional way of attenuating the beam with arotating wave plate, followed by a polarizer, failed to provide an achromaticattenuation across the bandwidth of our laser pulses, especially at high levelsof attenuation.The probe beam was sent through an optical delay line, computer con-trolled with a 150-mm travel motorized translation stage (Newport UTS50CC),and compressed in a double-grating compressor (Coherent) to compensatefor a chirp introduced in a Legend amplifier. The resulting Fourier transform-limited laser pulse of 40 fs duration was used as a probe in all the experimentsrequiring high temporal resolution. For the spectrally-resolved detection,on the other hand, the probe was redirected to a standard 4f pulse shaper,where its bandwidth was narrowed with a slit aperture down to 0.12 nm.2.2.4 Rotational Raman spectroscopyQWPFiltMLensSpectrometerTo pumpFromgas lineCentrifuge ProbeLLensLensMDMMMDelaylineMQWPPBSPBSBBOGaschamberFigure 2.12: Raman setup. PBS: polarizing beam splitter, QWP: quarter-wave plate, DM: dichroic mirror, Filt: dichroic filter, L: lens, M: mirrors.272.2. Raman setupProbe pulses, circularly polarized with a combination of a polarizer(PBS) and a half-wave plate (HWP) and delayed with a motorized delayline, were scattered off the centrifuged molecules. To separate the weakRaman signal from the strong centrifuge light, probe pulses were frequencydoubled to shift their central wavelength to around 400 nm with total en-ergy of < 1 µJ/pulse. As shown in Fig. 2.12, the two beams were combinedon a dichroic mirror (DM) and focused into the gas chamber with a singlef = 1000 mm lens to a focal beam waist of ≈ 120µm FWHM. Lose focusingwas used to avoid ionization and plasma breakdown by limiting the peakintensity of the excitation field to below 5× 1012 W/cm2. After the outputstage of polarization and wavelength filtering, probe spectrum was recordedwith a 0.1 nm-resolution spectrometer (McPherson).-8 -4 00.00.20.40.60.81.0Raman signal, arb. unitsRaman shift, THzRayleighpeak Ramanpeaks4 8Figure 2.13: Raman spectrum of laser-kicked oxygen.An example of Raman spectrum of the rotationally excited oxygen gasis shown in Fig. 2.13. The chamber was filled with 0.9 bar of O2, and themolecules were exposed to a single strong femtosecond linearly polarizedpulse. This type of excitation, usually referred to as a femtosecond kick, cre-ated a broad rotational wave packet in molecules forcing their axes to rotatetowards the laser polarization. The fs kick was followed by a 0.375 cm−1 (0.1THz) wide probe. The figure shows the observed spectrum of the probe lightscattered from the kicked molecules. It consists of a single Rayleigh peakat the zero frequency shift, which stems from the laser-induced permanentmolecular alignment. The latter leads to a non-zero optical birefringenceand the corresponding change in the probe polarization.282.3. Resonance-enhanced multiphoton ionization spectroscopyAround the central line, there is a series of narrow Raman peaks. Eachpeak to the right corresponds to anti-Stokes Raman scattering |N 〉 →|N + 2 〉, while each peak to the left – to Stokes Raman scattering |N + 2 〉 →|N 〉. The N -th peak has a detuning N+2 − N = B(N + 2)(N + 3) −BN(N + 1) = 2B(2N + 3). Since N in oxygen takes only odd positivevalues[47], the distance between the two consecutive peaks is expected tobe 8B = 11.4 cm−1 = 0.34 THz in a good agreement with the observation.Since equal amount of molecules are rotating both clockwise and counter-clockwise in the kicked ensemble, the Stokes and anti-Stokes peaks are ofsimilar height.2.3 Resonance-enhanced multiphoton ionizationspectroscopyRotational Raman spectroscopy which works well in dense ensembles be-comes inapplicable in a cold ultrasonic jet, where the gas density is 8 ordersof magnitude lower. To detect the low density products, resonance-enhancedmultiphoton ionization (REMPI) is often employed due to its high sensitiv-ity, spectral resolution and versatility [2, 81].Ionization threshold3vvJ'J''221100Figure 2.14: REMPI principle.292.3. Resonance-enhanced multiphoton ionization spectroscopyThe principle of REMPI spectroscopy is depicted in Fig. 2.14. Considera molecule in some rotational state |n′′ = 0, v′′, J ′′〉 of the ground electronicstate, where n′′ and v′′ are the principal and the vibrational quantum num-bers. Using a narrowband ultra-violet laser, the molecule can be resonantlyexcited via a one- or a few-photon transition to another ro-vibrational state|n′ = 1, v′, J ′〉. Since these states are much closer to the ionization threshold,they can be ionized by absorbing one or more photons from the same laserfield. Measuring the ion current as a function of the laser wavelength, onedetects peaks each time the wavelength matches the multiphoton resonance|n′, v′, J ′〉 ←← · · · ← |n′′, v′′, J ′′〉3. The strength of each peak is proportionalto the population of the initial state |n′′, v′′, J ′′〉 and the probability of thismultiphoton transition. Knowing the latter one can extract the molecularrotational distribution.2.3.1 Utra-high vacuum chamber and molecular sourceGas supply lineMCPOpticalwindowTo pump 1To pump 2To pump 3SkimmerAperturesNozzleTOFspectrometerFigure 2.15: Ultra-high vacuum chamber with a molecular source.High signal-to-noise ratio spectroscopy of molecular super rotors requires3Here we follow the standard spectroscopic convention, where the ground state is placedon the right hand side, the excited – on the left hand side, and the number of arrowscorrespond to the number of absorbed photons. The ground end excited states quantumnumbers are marked with two and a single primes, respectively302.3. Resonance-enhanced multiphoton ionization spectroscopya dense source of rotationally cold molecules. This is commonly achievedwith a supersonic expansion of gas from a high pressure, of order of tens ofbars, to low pressure, usually below 10−4 Torr through a small nozzle[108].In practice, maintaining this pressure difference in a continuous mode ofoperation requires unrealistic pumping speeds, hence the nozzles are oftenequipped with pulsed valves, working in sync with the laser pulses. In ourwork, we used Even-Lavie valve with a nozzle diameter of 150 µm, capableof producing sub-20 µs gas bursts with a back-pressure of up to 100 bar anda repetition rate of up to 1 kHz.SkimmerAperturesConcavemirrorTOF spectrometerChargedplates170 mmMCPNozzleGas supply lineTo gaslineTonozzleTopumpLiquid sample loadNozzleLaserQ-swtrig out50 HzSS cylinder(a) (b)(c)PulsegeneratorNozzledriverFigure 2.16: (a) Magnified view of the UHV chamber. (b) Nozzle controlelectronics. (c) Gas line connections diagram.Even with a duty cycle this low, the pressure in the vacuum chamberreached 10−4 Torr, which was too high for the detection scheme based onmolecular ionization. To overcome this, a custom design UHV chamber(Nor-Cal Products, Inc.) was built (Fig. 2.15 (a)). It consisted of fourdifferentially pumped sections, connected via small cross-section aperturesaligned along the trajectory of the expanding gas. The first section (red inFig. 2.15 (a)) contained the nozzle, while the measurements were carriedout in the last “science” chamber. To lessen the effects of shock waves,the first aperture was a 1.5 mm nickel skimmer (Beam Dynamics, Inc.,312.3. Resonance-enhanced multiphoton ionization spectroscopysee Fig. 2.16 (a)). As the amount of gas in the other two chambers wasconsiderably smaller, simple stainless steel plates with 2 mm holes wereused there.Since the gas density in the beam drops rapidly (∼ 1r2) with the distancefrom the nozzle, we tried to keep the latter as close to the detection regionas possible. This was achieved by nesting the three vacuum sections insideone another and allowed us to have the nozzle only 170 mm away from thedetection region.All four sections were pumped with turbomolecular pumps (Leybold(1000 l/s), KYKY(1300 l/s), Pfeiffer (260 l/s), Edwards (75 l/s)). Thesepumping speeds enabled us to pump out the science chamber below 3 ×10−8 Torr at the highest used backing pressures of 30 bar and repetitionrates of 1 kHz.The nozzle valve was driven by a pulsed current driver (Even-Lavie),triggered externally by a signal from a pulse generator (BNC 555), synced tothe output of the Nd:YAG laser trigger (Fig. 2.16 (b)). The pulse generatorwas employed for synchronizing the laser pulses with the gas bursts, as ittook a few hundreds of microseconds, depending on the gas mixture, forthe molecules to reach the detection region. For pure oxygen, this time wasabout ∼ 320 µs, which means the trigger timing had to be advanced by thisamount (in practice, it was achieved by delaying it by Trep − 320µs, whereTrep is a period of the Nd:YAG pulses).The connection diagram of a high-pressure gas line is shown in Fig. 2.16 (c).Using a few splitters and valves allowed us to connect the nozzle directly tothe gas supply, or to the stainless steel cylinder, prefilled with the requiredgas mixture. The whole system was connected to a hermetic vacuum pump(Agilent IDP-15) for the safe removal of potentially hazardous gas species.2.3.2 Time-of-flight spectrometerIn a low-density molecular jet environment very high sensitivity experimen-tal techniques are required. Modern avalanche-based ion detectors, such asmicro-channel plate detectors (MCP), are extremely sensitive, as they arecapable of detecting single ions. For this reason, all methods used in theUHV chamber were based on photoionization by either a fs or a ns probelaser pulse, followed by the ion collection.To detect the ions, a time-of-flight spectrometer was used (Fig. 2.17).The ions were extracted from the interaction region and directed towards adual microchannel plate detector (MCP, Tectra MCP-050-D-P46-V) with aDC field, created by a set of high voltage electrodes. The MCP detector was322.3. Resonance-enhanced multiphoton ionization spectroscopy12.7 mm290 mmMCP12.7 mm∅49.4 mm∅40.0 mmGNDGND2 kV5 kV+3480 V+4690 V∅19.1 mm4 kV2 kV0 kVFigure 2.17: Geometry of the time-of-flight spectrometer (left) and thesimulated electric field distribution (right).equipped with a fast phosphor screen, mounted next to an optical viewport.A single ion collision with the detector gave rise to an electronic avalanche,that, further accelerated towards the phosphor screen, produced a brightspot on the screen. The time of ion arrival was determined by the ion’svelocity acquired from the electric field, which in turn depended on the par-ticle mass-to-charge ratio. These times were of order of a few microsecondsfor the voltages used. As the phosphorescence lifetime for our phosphor was55 ns, it was usually possible to distinguish between the ions of differentspecies by the timing of the corresponding luminescent peaks. Although themapping function can be calculated numerically, in practice, a calibrationwas required to achieve a reasonable accuracy.The three disk-shaped electrodes with circular holes were made out ofstainless steel and held together by UHV-compatible ceramic insulators(Kimball Physics). When the voltages on the plates were adjusted for proper332.3. Resonance-enhanced multiphoton ionization spectroscopyion focusing, the point at which an ion would hit the MCP was determinedsolely by its horizontal velocity. We carried out the numerical calculationsin Comsol software (Fig. 2.17, right half) to find the optimal electrode volt-ages. For REMPI, where only a light electron leaves the molecule, its velocitychanges insignificantly. Since the molecules in the beam are translationallycold, all of the produced ions would have the same initial velocity (the speedof the supersonic expansion), giving a single spot close to the center of thescreen.The light from the phosphor screen was collected with an aspheric con-denser lens (f=40 mm) and focussed onto either a photo-multiplier tube(PMT, Hamamatsu H10721-01), sensitive to the phosphor emission peak at520 nm, or a low-noise fast CCD camera (Point Grey Grasshopper 3).0.00.0(a)(b)0 20 40 60 80 100 120I+SO2+SO+SO2+F+SO22+O2+O+ O2+N2+S3+S2+N+N22+N2+1400.5 1.0Time of flight, �sSignal,arb.u.Signal,arb.u.Mass to charge ratio, atomic units1.5 2.0 2.5 3.01001011021030.20.40.60.81.0(c)7.0 8.0 9.0Figure 2.18: (a) Raw time-of-flight spectrum of photo-dissociated mixtureof gases. (b) Calibrated spectrum. (c) Magnified region of the spectrumaround mass-to-charge ratio of 8.MCP detector was equipped with a gate (Photek GM-MCP), capable ofswitching the high voltage from 1.5 kV to 2.0 kV and back with a 20 nstime constant. Since the detector is relying on a highly non-linear processof avalanche formation, the amplification levels at these two input voltageswere different by several orders of magnitude. By timing the gate withrespect to the photoionization laser, we could probe only the ions arriving342.3. Resonance-enhanced multiphoton ionization spectroscopywithin a 20 ns time window.A mix of several substances (O2,N2,SO2, C6H3IF2) was introduced intothe chamber and exposed to a 35-fs intense laser pulse. The pulse caused themolecules to fragment into ions of various masses and charges. Measuringthe total signal (either on camera or on PMT) as a function of the gatedelay, we obtained the spectrum shown in Fig. 2.18 (a). In this plot, peakscorrespond to the arrival times of the ions with different mass-to-chargeratios.The velocity, acquired by a resting ion of mass m and charge q, when itcrosses the potential difference V is v =√2qVm . The time of reaching thedetector is therefore t = C√mq , where C is the calibration constant. Wetuned C so that all the observed peaks corresponded to the mass-to-chargeratios of the expected ion fragments from our gas mixture (Fig. 2.18 (b)).Other ions were then readily identified.As can be clearly seen in Fig. 2.18 (c), some of the peaks demonstrateda peculiar doublet structure, symmetric around the expected arrival time ofthe corresponding ions. The underlying effect, known as Coulomb explosion,was crucial for using the velocity map imaging technique for the reconstruc-tion of the molecular angular distribution, discussed in section 2.4.2.3.3 Dye laser systemThe laser setup used for REMPI spectroscopy is shown in Fig. 2.19. A 50Hz Nd:YAG laser with a frequency doubler (Spectra Physics Quanta Ray)generated nanosecond pulses carrying 250 mJ at 532 nm, which were usedto pump a three-stage dye laser (Sirah Cobra-Stretch). The active mediumfor the latter was prepared by dissolving Rhodamine 6G in ethanol witha concentration of 0.09 g/l for the resonator and preamplifier stages and0.01125 g/l for the amplifier. The output of the dye laser was frequencydoubled in the second harmonic generator system (Sirah SHG-250), yielding5 mJ per pulse tunable from 279 nm to 288 nm. Computer control ofthe laser wavelength was carried out by simultaneous adjustment of theresonator and the second harmonic generation BBO angle, to reach theoptimal phase matching.To synchronize the fs and ns pulses, we took a 1 kHz trigger signal fromthe Legend control unit and divided its frequency 20-fold, bringing it downto 50 Hz, in a pulse generator (Quantum Composers 9514). The two outputsof the generator were used as the triggers for the Nd:YAG laser’s flashlampsand Q-switch. The flashlamps trigger was additionally advanced by 130 µs352.3. Resonance-enhanced multiphoton ionization spectroscopyNd:YAGlaserDye laserComputer532 nm225 mJ566 nm60 mJ283 nm5 mJLegendcontrolunitSHG1 kHz50 HzFLtrig inQ-swtrig inQ-swtrig outTo vacuumsetup syncPulsegeneratorFigure 2.19: Dye laser system. The flashlamps (FL) and Q-switch (Q-sw) of Nd:YAG pump laser were synchronized with the centrifuge pulseselectronically. The output of the dye laser was frequency doubled in a secondharmonic generation system (SHG).to achieve an optimal population inversion prior to the Q-switch firing.2.3.4 REMPI spectroscopy of N2 ant O2Experimental setup for REMPI spectroscopy is shown in Fig. 2.20. Nanosec-ond probe pulses were combined with a centrifuge beam by means of adichroic mirror. Oxygen or nitrogen at a pressure of 20 bar was supplied tothe back of the nozzle.The blue line in Fig. 2.21 shows the measured (2+2) a1Πg(v′ = 0)←←X1Σ+g (v′′ = 1) REMPI spectrum of nitrogen. Here, “2+2” means that ittakes two photons of a probing laser to excite a molecule from its initialground electronic and vibrational state Σ+g (v′′ = 0) to an excited electronicstate a1Πg(v′ = 1), and then another two photons to ionize it.Since each photon bears an angular momentum of 1, an absorption oftwo photons can change the molecular angular momentum by not more than362.3. Resonance-enhanced multiphoton ionization spectroscopyMCP detectorPhotodetectorPulsed nozzleConcave AlmirrorTOF platesSkimmerVacuumwindowDichroicmirrorCentrifugebeamUV beamGas pulsesFigure 2.20: REMPI spectroscopy setup. Centrifuge beam is combinedwith a tunable UV laser pulse and focused inside a vacuum chamber on asupersonically expanded molecular jet between the charged plates of a time-of-flight (TOF) mass spectrometer. The ionization rate is measured with amulti-channel plate (MCP).2. It is natural to classify the two photon transitions by the change of thetotal angular momentum ∆J = J ′ − J ′′. The possible changes are: ∆J = 2(S-branch), ∆J = 1 (R-branch), ∆J = 0 (Q-branch), ∆J = −1 (P-branch)and ∆J = −2 (O-branch). The transition energies of these branches areshown in Fig. 2.21 by ticks on the black lines, with J ′′ changing by one fromone tick to the next.In the absence of the electronic spin, the rotational structure of both theinitial and the final states correspond to a rotational structure of a simplesymmetric rotor and is easy to calculate. The largest peak on the blueline could then be identified as a transition from J ′′ = 0 of S-branch, etc.After identifying the peaks and accounting for the line strength, we coulddetermine the initial population of all rotational states and estimate therotational temperature to be around 10 K.In the case of nitrogen, the identification of REMPI peaks is simple evenat higher temperatures. For example, the red curve in Fig. 2.21 shows aREMPI spectrum of nitrogen at 300 K. For this experiment, the gas was372.3. Resonance-enhanced multiphoton ionization spectroscopy70520 70540 70560SRQPO70580Two photon energy, cm-170600 70620 70640 70660Figure 2.21: REMPI spectra of cold (T = 10 K, blue) and room tempera-ture (red) nitrogen. Black ticks, connected by lines to guide the eye, showthe calculated transition energies for different branches.leaked in the chamber through the leak valve. Even though the numberof resonances is very high, almost each one of them could be separatedand attributed to a particular transition, which makes the determination ofrotational population still possible.In oxygen, (2+1) C3Πg(v′ = 2)←← X3Σ−g (v′′ = 0) REMPI is often em-ployed due to the convenience of the laser wavelength, the high ionizationcross-section, and rotationally resolved structure. The line identificationis, however, substantially more complicated compared to nitrogen. First,the intermediate state C3Πg(v′ = 2) pre-dissociates, which leads to sig-nificant line broadening of about 10 cm−1. Second, both the ground andexcited states have a non-zero spin, which leads to a more complicated en-ergy level structure (Fig. 2.23). In the ground state, oxygen molecule hasa zero electron orbital momentum projection Λ on the internuclear axis,and the energy structure follows Hund’s case (b)[11]. The energy states aredetermined by the angular momentum excluding electronic spin, that takes382.3. Resonance-enhanced multiphoton ionization spectroscopyTwo photon energy, cm-1F1F2F369300 69400 69500 69600 69700 69800 69900 70000Figure 2.22: REMPI spectra of cold (T = 10 K, blue) and room tempera-ture (red) oxygen. Dashed line demonstrates the effect of power broadening.values N ′′ = 1, 3, . . . . For each N ′′ there is a triplet of states correspondingto the total angular momentum J ′′ = N ′′− 1, N ′′, N ′′+ 1. The line splittingin the triplet is due to the spin-spin and spin-rotation interaction. The ex-cited state has Λ = 1 and, therefore, follows Hund’s case (a). It consists ofthree different sets of rotational levels, according to the total electron angu-lar momentum projection Ω′ = 0, 1, 2. For each Ω′ the levels are determinedby their total angular momentum J ′ = Ω′,Ω′ + 1, . . . .Due to the small value of spin splitting in the ground state (less, than 1cm−1[47]), it was not resolved in our experiments. It is, therefore, useful todesignate the transition branches by two numbers Ω′ and ∆N = J ′ + Ω′ −N ′′ − 1[122]. For each value of Ω′, there are 7 possible ∆N , giving a totalof 21 branches. As a result, when the number of excited rotational levels inthe ground state increases, the spectrum becomes very congested, as can beseen in Fig. 2.22, where the blue and the red lines correspond to 10 K androom temperature ensembles, respectively.392.4. Velocity-map imaging (VMI)J'3Π0(F1)3Π1(F2)3Π2(F3)J'J'3X3Σg-(v''=0)C3Πg(v'=2)33444222N''32111J''024246135Figure 2.23: Illustration of the C3Πg(v′ = 2)←← X3Σ−g (v′′ = 0) transitionin oxygen. Blue (red) area marks the rotational structure of the ground(excited) electronic state, affected by the spin-spin and spin-rotational (spin-orbit) interaction.2.4 Velocity-map imaging (VMI)When a molecule is placed in a non-resonant laser field of high intensity(>1014W/cm2) it often undergoes a process of multiphoton ionization [34],leaving a highly charged molecular ion. This cation is often unstable andthe smaller ions which it consists of, experience strong Coulomb repulsion.In a process, called Coulomb explosion, they immediately recoil from eachother, releasing large amount of kinetic energy.The direction of the recoil is correlated with the angular coordinates ofthe molecule. If the laser pulse used for the Coulomb explosion is shortenough, so that the molecule being ionized cannot rotate during the ex-plosion [27, 34, 100], the fragment momenta can be used to measure themolecular angular distribution right before the ionization [27]. For exam-ple, diatomic molecules, such as nitrogen, recoil in the direction of their402.4. Velocity-map imaging (VMI)e-e-e-Figure 2.24: Illustration of the process of Coulomb explosion. After beingionized by a fs pulse, the molecules immediately fragment into ions recoilingin the direction of their respective internuclear axes.internuclear axis, as demonstrated in Fig. 2.24. Measuring the full fragmentvelocity distribution would therefore allow one to extract the molecular an-gular distribution.The kinetic energy released in the Coulumb explosion is strongly depen-dent on the channel of dissociation. For example, for the N3+2 → N2+ + N+channel, the ions recoil with the energy around 15 eV and a distributionwidth of 5 eV, while for N4+2 → N2+ + N2+ the distribution is centeredaround 29 eV and has a width of 10 eV [25].Figure 2.25: Relevant polar variables for the velocity map imaging.412.4. Velocity-map imaging (VMI)Advanced techniques, such as cold target recoil-ion momentum spec-troscopy (COLTRIMS), allow to determine the full three-dimensional mo-mentum distribution of ions[126]. Our VMI setup was suitable for measur-ing the projection of the fragments momentum onto the plane of the MCP.Let us consider a fragment with an initial velocity v, and introduce vectorr = vt, where t is the time of reaching the detector. Vector R, defining theposition in the MCP detector plane, where this ion would be observed, isthen a projection of r onto a detector’s plane.It is useful to introduce a coordinate system xyz, with xz plane cor-responding to the plane of the detector, spherical coordinates r = (r, θ, φ)and polar coordinates R = (R,Θ) in the xz plane, as defined in Fig. 2.25.Knowing the fragment velocity distribution f(r, θ, φ) we can find the ionimage distribution F (R,Θ) via [128]:F (R,Θ) =∫ ∞Rrf(r, θ, φ)√r2 −R2 dr, (2.1)with θ = arccos(R cos Θr)and φ = ± arccos tan Θtan θ . This so-called Abel trans-form is just an integral along the line normal to the detector (dashed orangeline in Fig. 2.25) of all r contributing to the same R.In practice, the task is usually the opposite – to find the velocity dis-tribution f(r) knowing the measured ion image F (R). Although its notpossible in general, because r → R correspondence is not unique, methodsimplying an additional symmetry from f(r) [23, 26, 38, 132] or using severalprojections along different directions [118] were used to successfully find theinverse.The most popular class of Abel transform inversion methods uses thebasis of 2D images with known 3D distributions to expand the experimentalion images and reconstruct the underlying recoil velocity distribution. Thisapproach was first proposed by Dribinski and co-workers, who suggestedusing a basis of Gaussian functions [26]. This method, which the authorscalled BASEX, proved inaccurate for our images, predicting unphysical di-vergences along the axis of symmetry. Its modification, utilizing a morenatural basis of Legendre polynomials for the angular dependencies, knownas pBASEX method [38], was computationally hard because of the largenumber of the basis functions required for the case of high rotational har-monics. The algorithm that worked best for us belonged to the family ofMaximum Entropy Methods (MEM), often employed to reconstruct noisydata and adapted to the analysis of velocity maps by Dick [23], who kindlyshared his software with us. The idea of the method is to find the velocity422.4. Velocity-map imaging (VMI)distribution which gives a projection close to the measured one, but doesn’tassume any new information (or, in other words, has the highest entropyamong the other distributions with similar projections). Since it is only ap-plicable to the distributions with an axial symmetry, which wasn’t the casein many interesting scenarios, and since the reconstruction suffered fromthe strong drift effect (see section 3.3), it was not used in our analysis. Insection 5.1, we describe an approximate method we used for estimating thedegree of molecular alignment from the acquired VMI data.(a) (b)Figure 2.26: (a) Raw ion image after a single laser shot. (b) Processedimage with the determined ion positions, marked by the magenta crosses.2.4.1 Velocity map imaging reconstruction of the angulardistributionHaving calibrated a TOF spectrometer, we were able to isolate a particulartype of ions we were interested in. For example, after setting our gate tothe mass-to-charge ratio of 14, a single femtosecond laser shot resulted inan image shown in Fig. 2.26 (a). Each small spot on the image correspondsto a single collision of an atomic N+ ion with a phosphor screen. The ionsare scattered around the center due to the kinetic energy they acquired ina Coulomb explosion of randomly oriented cold nitrogen molecules. Thebright spot in the center corresponds to molecular ions N2+2 . Zero kineticenergy release (since the two removed electrons are too light to considerablychange the momentum of the molecule) results in the narrow distributionwidth.To extract the angular distributions from such images, we had to av-erage these images over many laser shots. One method, involving a low432.4. Velocity-map imaging (VMI)frame rate camera, was to take long exposure pictures, with each pixel ofthe camera registering multiple ions. The disadvantage of this method wasits requirement of high homogeneity of the MCP amplification across itsarea. For this reason, we later moved to a fast camera, working at repe-tition rates from a few hundred Hz to a kHz. Using an algorithm writtenin Python programming language and utilizing OpenCV computer visionlibrary, the images were acquired from the camera in real time, noise fil-tered, threshold-discriminated, and the ion peak positions were determined(Fig. 2.26 (b)). After accumulating a million ion peaks, we plotted a 2Dhistogram, corresponding to the final F (R) distribution (Fig. 2.27).E E(a) (b)Figure 2.27: N+ ion image distribution measured for 1,000,000 events,taken with probe polarization (a)parallel and (b)normal to the detectorplane.The two images in Fig. 2.27 (a) and (b) were taken with a probe lin-early polarized parallel to the plane of the detector and perpendicularly toit, respectively. Picture (a) has a clear circle structure, with each circlecorresponding to the largest R for a given dissociation channel. As wasmentioned above, since each channel has a kinetic energy release narrowlydistributed around its average energy, these circles come mostly from themolecules aligned in the detector plane at the moment of explosion.Another important feature of Fig. 2.27 (a) is its high anisotropy with re-spect to Θ. An isotropic molecular angular distribution is expected to giverise to a centrally symmetric ion image, while the anisotropic one corre-sponds to an aligned ensemble. This apparent alignment along the directionof the probe polarization could be accounted by two different effects [100].First, the so-called geometric alignment comes from the anisotropy of theionization cross sections – molecules which happened to be aligned along442.4. Velocity-map imaging (VMI)the laser polarization direction are more likely to be ionized and explodein the direction of their instantaneous alignment. The second effect, calledthe dynamic alignment, corresponds to the molecules being aligned by theCoulomb explosion field itself.While the geometric alignment can be accounted for in molecular angulardistributions calculations by a simple normalization, the dynamic alignment,typically observed in light molecules such as H2 and N2[100] complicatesthese calculations as it affects the angular distributions. Fortunately, theeffect of the dynamic alignment can be reduced by polarizing the probepulses circularly or lowering their intensity [27, 93].2.4.2 Centrifuge angle measurementDuring the shaping, the two centrifuge arms traveled separately over a dis-tance of a few meters. Mirrors vibration and air movement in the lab leadto the randomization of their relative optical phase, changing from pulse topulse. Owing to this random phase, the centrifuge “corkscrew” envelop wasrandomly rotated along the propagation axis in each laser shot. As a result,the molecules were released at a random angle θrel with respect to horizon-tal axis (see Fig. 2.28), but with the same angular frequency. Hence, theangular distribution of super rotors, determined by the method describedin the previous section, was averaged over the rotations along the beampropagation direction.For a number of experiments, we needed to know the value of θrel foreach centrifuge pulse or, alternatively, the angle of centrifuge polarizationθ(τ) at any fixed delay τ from the rising edge of the centrifuge.As was mentioned above (section 2.1.2), overlapping the centrifuge anda short reference pulse on a doubling crystal allowed us to measure thecross-correlation signal, proportional to the centrifuge field intensity as afunction of time. Due to the polarization sensitivity of the sum-frequencygeneration, this signal reflects the strength of a single projection of thecentrifuge field onto the axis of the BBO crystal. If we assume that theangle of the centrifuge polarization changed in time as βτ2 + θ0, where τ isthe time measured from the centrifuge rising edge, β is the frequency chirp,and θ0 is the (random) initial polarization angle, then the cross correlationsignal would be proportional to cos2(βτ2 + θ0)when the reference pulse isdelayed by τ .Using 5% beam splitters we picked off a small portion of the centrifugeand the femtosecond probe beams to use them for the angle measurement.Both beams were split in two equal parts on a 50/50 beam splitter and452.4. Velocity-map imaging (VMI)Figure 2.28: Definition of the centrifuge angles.focused on a pair of identical BBO crystals (30 µm Newlight Photonics,Fig. 2.29 (a)). A 1 mm glass plate was inserted in the centrifuge path inone of the arms, to slightly delay it relative to the reference beam. Themeasured cross correlation signals behind the first and the second crystalswere therefore proportional to cos2(β(τ + τglass)2 + θ0)and cos2(βτ2 + θ0),respectively, where τglass =dc (nglass − 1) is an additional delay, acquired bythe centrifuge in the first arm due to the glass plate, d = 1 mm is the platethickness, nglass is the refractive index of the glass and c is the speed of light.The correlation plot of the two signals measured over many laser shotsreproduced a Lissajous figure of a ratio one, with a phase equal to 4βττglass+2βτ2glass, as demonstrated in Fig. 2.29 (b), where different colours correspondto different values of the delay τ . With enough points accumulated, we wereable to fit an ellipse to them, and extract the centrifuge angle θrel for everylaser shot from the angular position of a point on this ellipse correspondingto that shot. To achieve the best precision of this extraction, τ was tunedto obtain circular correlation plot (red circles on Fig. 2.29 (b)).462.4. Velocity-map imaging (VMI)0.00.00.20.20.40.40.60.60.80.81.0PD1 signal normalizedPD1PD2BBOBBOLLRetMBSRef(a) (b)CentrifugePD2signalnormalized1.0Figure 2.29: (a) Angle measurement setup. Ref: reference femtosecondlaser pulse, BS: 50/50 beam splitter, L: lenses, Ret: glass plate retarder,PD1, PD2: photodiodes. (b) Correlation plot of the two normalized signalsfrom PD1 and PD2, acquired over 100 laser shots. Red bar charts show thehistograms of the two signals. Scatter plots of different colors correspond tothree different delays between the reference and the centrifuge pulses, withthe red trace corresponding to an optimal delay, used for determining θrel.47Chapter 3Direct detection of molecularsuper rotorsSince the original proposal[57], an optical centrifuge has been implementedby two experimental groups. In the pioneering work by Villeneuve et al.,dissociation of chlorine molecules exposed to the centrifuge field has beenattributed to the breaking of the Cl-Cl bond which could not withstandthe extremely high spinning rates[130]. More recently, Yuan et al. ob-served rotational and translational heating in the ensembles of CO2 andN2O molecules and associated it with the collisional relaxation of the cen-trifuged species[138, 139]. In both cases, an incoherent secondary process(i.e. dissociation and multiple collisions) has been used for indirect iden-tification of the formation of super rotors whose most unique property -their synchronous uni-directional rotation, remained hidden. In this chap-ter, we describe the three detection techniques, that we used to directlydetect molecular super rotors.The first section of this chapter (section 3.1) describes the application ofcoherent Raman spectroscopy to detect extreme rotational states in a densemolecular gas. This method allowed us to see the field-enforced coherentadiabatic excitation inside an optical centrifuge to rotational frequencies ofup to 10 THz, followed by a field-free rotation of molecules, and to demon-strate the flexibility of controlling the directionality and frequency of theinduced rotation.Rotationally resolved resonance-enhanced multiphoton ionization (REMPI)spectroscopy is a very sensitive method, capable of detecting extremely lowgas densities. The interpretation of REMPI spectra in the case of rotation-ally hot molecules is not always straightforward, as it relies on the exactknowledge of the intermediate electronic state. The controllability of thecentrifuge excitation was used in a novel spectroscopy method and allowedus to interpret the congested REMPI spectra of oxygen (section 2.3), com-plicated by the spin-rotational and spin-orbit splittings and pre-dissociationof the excited electronic state.The two methods mentioned above were employed to probe the energy483.1. Raman spectroscopy of molecular super rotorsspectrum of the rotational states. A different approach was developed toprobe the molecular instantaneous angular distributions. In the last sec-tion (section 3.3) we discuss the use of the velocity-map imaging techniqueto visualize the angular distributions and their dynamics, modified by theultrafast rotation of molecules.3.1 Raman spectroscopy of molecular superrotorsProbe wavelength (nm)3900.00.10.20.30.40.5Ramansignal,arb.u.395 400 405 410Figure 3.1: Probe spectra after passing through the counter-clockwise (pur-ple) and clockwise (cyan) centrifuged oxygen molecules. The appearance ofonly anti-Stokes or only Stokes sidebands indicate unidirectional rotation.To carry out the Raman spectroscopy of super rotors we used the setupdiscussed in detail in section 2.2. We excited ensembles of various diatomicgases (O2, N2, H2) with an optical centrifuge in a dense room temperaturegas sample and probed it with pulses, spectrally narrowed to 0.375 cm−1and centered around 400 nm with a total energy of < 1 µJ/pulse.As the probe light passed through the ensemble of the centrifuged molecules,its spectrum developed sidebands with a detuning magnitude equal to twicethe molecular angular frequency Fig. 3.1. Choosing the centrifuge to beeither counter- or co-rotating with the probe polarization, we observed theappearance of only anti-Stokes (purple) or only Stokes (cyan) sideband. Thisproves the unidirectionality of molecular rotation. Notice that in both cases,the sidebands were circularly polarized, oppositely to the initial probe po-larization, as expected from the selection rules. Indeed, the centrifuge isinducing coherence between the states of the rotational ladder with ∆N = 2and ∆MN = 2. To change the angular momentum projection by exactly two493.1. Raman spectroscopy of molecular super rotorsin the Raman scattering process, the absorbed and emitted photons shouldbear the angular momentum projection of 1 and -1, respectively. This cor-responds to the fields, circularly polarized in the opposite directions. Hence,using a circular polarizer allowed us to almost completely suppress the muchstronger initial probe line.0 0 50 100 150200 200400 600 8000 0100100200200300300400400500500600(a)Time Timedelay, delay,t t( (ps ps) )Raman shift (cm-1)Raman shift (cm-1)(b)N2 H2Figure 3.2: Time-dependent Raman shifts from the centrifuged N2 (a) andH2 (b) molecules. As the molecules spend longer time in the centrifuge, theobserved Raman frequency shift increases along the slopped dashed lines,providing a direct evidence of accelerated molecular rotation. Some of themolecules “leak” from the accelerating angular trap, leaving the horizontaltraces at the intermediate Raman shifts. Horizontal dashed lines correspondto the angular velocity of rotational states, mostly populated at room tem-perature.Delaying the arrival time of probe pulses with respect to the beginningof the centrifuge pulse enabled us to observe the spinning molecules beforeand after they leave the centrifuge. In Fig. 3.2 (a,b), the rotational Ra-man spectrum of nitrogen and hydrogen is plotted as a function of time themolecules spent in the centrifuge. The tilted dashed white line on Fig. 3.2 (a)marks the accelerated spinning of the molecules inside the centrifuge. Astheir angular frequency grows linearly with time, so does the Raman shift.While spinning up, the molecules are “leaking out”, producing a set of Ra-man sidebands, with each one corresponding to free molecular rotation withquantized angular velocities. Similar behavior was observed in hydrogen503.1. Raman spectroscopy of molecular super rotorsFig. 3.2 (b). Due to its small moment of inertia, the level separation isaround 30 times larger than in N2 and, as a result, we were only able topopulate a few rotational states.11 213141516171810 100 200 300 400 500Raman shift (cm-1)Raman signal (arb. u.)Figure 3.3: State-resolved Raman spectra of centrifuged oxygen molecules.Higher curves correspond to longer spinning time inside the centrifuge. Redvertical arrows mark the rotational quantum numbers.To identify the excited rotational levels, we analyzed the Raman spec-trum of super rotors. The results, corresponding to different centrifuge du-rations and, therefore, different degrees of rotational excitation, are shownin Fig. 3.3 for the case of O2. Each measured spectrum consists of threeparts: (i) an unshifted probe line at 0 cm−1; (ii) a set of stationary linescentered around 80 cm−1 and corresponding to the molecules lost from thecentrifuge during its spinning; and (iii) a set of moving lines correspondingto the molecules which followed the centrifuge up to its terminal angularfrequency.Well resolved peaks in the spectrum correspond to the individual Ra-man transitions between the states with rotational quantum numbers N −2and N . The discreetness of the observed spectra follows from the angularmomentum quantization. The group of the Raman-shifted lines on the rightside of the spectra in Fig. 3.3 reflects an accelerated rotation of the moleculestrapped in the centrifuge field, and demonstrates our ability to control thedegree of rotational excitation. Quantum numbers N can be easily assignedby counting the peaks. Only odd values of N are allowed for 16O2 molecule513.2. Rotational spectroscopy with an optical centrifugeN2O2Rigid rotorNon-rigid rotor0 10 20 30 40 50 60 8070Rotational quantum number, NRaman shift (cm-1)010020030040050090Figure 3.4: Experimental (dots with error bars) and calculated (dashedblue and solid green for the rigid and non-rigid rotor approximations, respec-tively) rotational energy spectrum expressed as a Raman frequency shift.because of its nuclear spin statistics.In a rigid rotor approximation, the rotational energy is E(N) = BN(N+1), where B is the rotational constant of the molecule. This scaling resultsin a series of equidistant Raman peaks separated by ∆Ω = 4B∆N , with∆N being the smallest possible step in the molecular rotational ladder (e.g.1 for N2 and 2 for O2). However, as one can see from the slope of the curvesin Fig. 3.4, the measured peak separation does not stay constant, but ratherdecreases with increasing angular momentum - a direct consequence of thecentrifugal distortion evident at N > 50. Being able to resolve the energiesof extreme rotational states, we quantified the magnitude of the centrifugaldistortion and verified that it is well described by the Dunham expansion tosecond power in N(N + 1) (green solid lines in Fig. 3.4).3.2 Rotational spectroscopy with an opticalcentrifugeAs was discussed in subsection 2.3.4, extracting the rotational distributionsfrom REMPI spectra requires the knowledge of both the initial and finalstates. This means that the electronic spectroscopy of a highly rotationallyexcited molecule must be understood, both the assignment of resonances andtheir strengths. This task can be complicated by a predissociative coupling523.2. Rotational spectroscopy with an optical centrifugeand decay behavior in the intermediate excited electronic states [3].Assuming that the molecule of interest is reasonably stable, REMPI spec-tra recorded at high temperatures can provide information on the propertiesof highly excited states, but this approach is limited both by the tempera-tures that the molecule can tolerate before it dissociates, and by the diffi-culty of unraveling the complex spectrum of a high temperature molecule,as illustrated by the work done on the spectrum of high temperature water[99]. In the case of oxygen, reaching the high rotational states accessed withan optical centrifuge (N > 100) would require a temperature of ≈50,000K,while the states accessed in a non-adiabatically laser-kicked ensembles [35]are limited to ∼ 40 (in the case of oxygen) due to the rapid growth of ioniza-tion rate with increasing kick strength. Even if a broad thermal distributionof highly excited rotational states could be produced in a diatomic moleculesuch as oxygen, the triplet structure of the ground and excited states coupledwith two-photon selection rules (for the C3Πg(v′ = 2) ←← X3Σ−g (v′′ = 0)transition), would result in as many as 21 overlapping rotational branches(see section 2.3), making spectroscopic assignment challenging [122].Here we utilized the unique capability of the centrifuge to control thelevel of rotational excitation for the purpose of obtaining and interpretingcomplex REMPI spectra of oxygen super rotors (0 < N . 120). We excitedoxygen to a narrow rotational wave packet whose center is accurately tunedacross the broad range of well defined N values. The centrifuge excitationwas then followed by a REMPI measurement. Owing to the narrow N dis-tribution, the detected spectrum became significantly less congested, andidentifying rotational resonances was greatly simplified. As we have demon-strated in section 3.1, truncating the spectrum of the centrifuge in a Fourierplane of the pulse shaper by a movable shutter (inset to Fig. 2.2 (a)) enabledaccurate control of the rotational state of the centrifuged molecules. Charac-terizing the centrifuge field with the method of cross-correlation frequency-resolved optical gating (XFROG, section 2.1.2) allowed us to calibrate thefinal rotation speed of the centrifuge, and hence the corresponding molecularangular momentum, as a function of the shutter position. REMPI detectionwas carried out using narrowband nanosecond probe pulses tunable from279 nm to 288 nm (0.1 cm−1 linewidth, 500 µJ, 50 Hz repetition rate, seesubsection 2.3.4 for details).The two-dimensional REMPI spectrogram of centrifuged oxygen is shownin Fig. 3.5, where the detected ion count is plotted against the probe wave-length (horizontal axis) and the final rotation speed Ω of the truncatedcentrifuge (vertical axis). The latter is expressed in terms of the angularmomentum N of an oxygen molecule rotating with the angular frequency533.2. Rotational spectroscopy with an optical centrifugeΔN=2δN=7O2rotationalquantumnumber80(a)(b) (c)706050403020100 69400 69800 70200Two-photon energy, cm-1706003000 K simulationLLHotLL10 KLeaked molecules71000700506995069850204060Figure 3.5: 2D REMPI spectrogram for a linearly polarized probe. (a)Experimental spectra of cold (10 K, blue) and centrifuged molecules (yel-low), along with a simulated spectrum of a “hot” thermal ensemble (3000 K,red) calculated with pgopher software [134]. (b) Ion signal as a functionof the probe laser wavelength and molecular angular momentum definedby the centrifuge final rotation speed. Different areas of the 2D plot weremeasured with different sensitivities and probe intensities and are displayedwith different color scales to compensate for the broad dynamic range of thedata. c Vertical cross-sections of several consecutive peaks from one partic-ular branch, shown in the inset to (b). The peaks are regularly separatedwith a distance of ∆N = 2 reflecting 16O2 nuclear spin statistics.Ω, according to:Ω = [(N)− (N − 1)] /~,(N) = hcBN(N + 1)− hcDN2(N + 1)2,where  is the energy of state |N〉, c is the speed of light in vacuum, B =1.438 cm−1 and D = 4.839× 10−6 cm−1 [51].Each peak in the two-dimensional REMPI spectrogram of Fig. 3.5 (b)corresponds to a two-photon transition between a rotational level in theelectronic ground state, X3Σ−g , and a rotational level of C3Πg . The fi-nite horizontal width of the observed peaks stems from the predissociationlinewidth (as in conventional “1D REMPI” detection), whereas finite verti-cal spread reflects the narrow width of the excited rotational wave packetcreated by the centrifuge.543.2. Rotational spectroscopy with an optical centrifugeThe complexity of the two-photon absorption line structure in rotation-ally hot oxygen gas is illustrated by red and yellow lines in Fig. 3.5 (a) whichcorrespond to the hot thermal ensemble (simulated numerically) and the en-semble of centrifuged molecules (experimentally observed 2D spectrogramintegrated along its vertical dimension), respectively. In sharp contrast toconventional 1D REMPI spectroscopy, controlled centrifuge spinning offersdirect assignment of rotational quantum numbers to the observed REMPIpeaks, as well as significantly better peak separation due to their distributionalong the added second dimension.Vertical traces originating from bright resonance peaks in Fig. 3.5 (b)(examples are marked with white arrows) correspond to molecules which“leaked out” of the weakened centrifuge potential before reaching the ter-minal angular frequency of the centrifuge. After escaping the centrifuge,these molecules continue their free rotation while the trap is acceleratingfurther. The three bright vertical stripes reproduce the initial cold beamspectrum (blue line in Fig. 3.5 (a)) and correspond to the molecules whichwere not trapped by the centrifuge. The width of the final rotational wavepackets can be readily extracted as δN ≈ 7 (FWHM), from the verticalcross sections, shown in Fig. 3.5 (c). The REMPI spectrogram shows anaccelerated centrifuge excitation to the rotational states with N as high as∼ 80 (dim diagonal trace pointing at the upper right corner in Fig. 3.5 (b)).Rotational line broadening above N ≈ 60 can be attributed to the increas-ing Rydberg-valence interaction (governed by the Franck-Condon overlapwith the continuum wavefunctions) similarly to the previously observed ro-tational broadening in the lower vibrational states (v′ = 0, 1) of the excitedpotential[122].One can see that the peaks in Fig. 3.5 are grouped in regular patterns, re-sembling Fortrat parabolas corresponding to different rotational branches.Within a single branch, the center of each consecutive resonant peak isshifted by ∆N = 2 (Fig. 3.5 (c)), reflecting the smallest step in the rota-tional ladder climbing executed by the centrifuge. Circularly polarized lightwas used to further simplify the spectrum. As shown in Fig. 3.6, the signalstrength of different rotational branches depends on the handedness of probepolarization. This is due to the highly non-uniform population distributionamong the magnetic sub-levels in the centrifuged wave packet, with most ofthe population concentrated at MN ' N (or MN ' −N) [57].To identify different rotational branches, we used three sets of molec-ular constants (for F1(Ω = 0), F2(Ω = 1) and F3(Ω = 2) spin-orbit com-ponents of the excited state, respectively, with the quantum number Ωintroduced in section 2.3) from the previous studies on thermally excited553.2. Rotational spectroscopy with an optical centrifugeΔN=ΔN=-4 -3 -2 -2 -2 0-1 -1142 2 233-369300692000010102020303040405050(a)(a)69400 69600 69800F1F2F369500 69700Two-photon energy, cm-1Two-photon energy, cm-169900 70100O2rotationalquantumnumberFigure 3.6: 2D REMPI spectrogram for a circularly polarized probe. Electricfield vector is counter-rotating (a) and co-rotating (b) with the centrifugedmolecules. The directionality of laser-induced rotation results in the sensi-tivity of the measured signal to the handedness of probe polarization. Theresults of fitting the data to the theoretical model are shown with coloredlines and markers for different branches and resonances, respectively. Branchnomenclature is the same as in [122].ensembles[82, 135]. These constants are listed in Table 3.1. For F2 andF3 components, our results are well described by the constants provided byLewis et al.[82]. On the other hand, the observed F1 peaks do not agree wellwith the suggested numerical values (ν0 = 69366 cm−1 and B0 = 1.6 cm−1),as shown in Fig. 3.7. This can be attributed to the complexity of the broad-ened and highly overlapping structure of F1 lines, which makes it hard tointerpret and fit the data from a thermally populated ensemble. Centrifugespectroscopy enabled us to correct the values of F1 molecular constants (Ta-ble 3.1) by performing the fit of the most pronounced ∆N = −2 branch(Fig. 3.7).The lowest vibrational level of C3Πg electronic state of oxygen whichis known to exhibit well-resolved rotational structure is v′ = 2 [122]. Pre-563.2. Rotational spectroscopy with an optical centrifuge69250 69300 69350 69400Two-photon energy, cm-169450 69500 69550 6960001020304050New constantsWhite et al.Lewis et al.ΔN=-4 ΔN=-3 ΔN=-2O2rotationalquantumnumberFigure 3.7: Comparison of the observed REMPI data for the perturbed F1spin-orbit component with the calculations based on molecular constantsfrom our work (red circles), White et al.[135] (blue triangles) and Lewis etal.[82] (purple squares)dissociation to the closely lying valence state 13Πg broadens the rotationalspectrum of the lower vibrational states v′ = 0, 1, where the molecule hasa high probability of tunnelling to the said continuum state, undergoingdissociation. This broadening is weakened in the case of v′ = 2 becausethe repulsive potential crosses the level near the node of the vibrationalwavefunction, lowering the Franck-Condon overlap [122]. The FC overlap,however, increases with the increasing degree of rotational excitation. Athigh values of N , we observed a significant line broadening which results ina completely unresolvable rotational structure at N & 60(see Fig. 3.5).At even higher centrifuge frequencies, corresponding to the extreme ro-tational levels with 99 ≤ N ≤ 125, we observed the re-appearance of nar-row resonances shown in Fig. 3.8. Their highly non-monotonic line width573.3. Mapping out the angular distribution of molecular super rotorsTable 3.1: Molecular constants used to fit the data in Fig. 3.6Spin-orbit branch ν0, cm−1 B0, cm−1 D, cm−13Π0(F1) 69375 1.585 2.5× 10−73Π1(F2) 69445 1.648 1.0× 10−53Π2(F3) 69550 1.685 1.3× 10−5dropped down to a well-resolved ∼ 7 cm−1(Fig. 3.9). Similar linewidthdependencies were previously observed in OD[21] where they were used toanalyse the repulsive state. Our analysis showed that, unlike the previ-ously discussed branches of C3Πg(v′ = 2)←← X3Σ−g (v′′ = 0), the observedultra-high narrow lines originate from the v′ = 1 state, which displayed noresolvable rotational structure at lower rotational levels, but re-appeared athigher J ’s.Well described by Hund’s case (b) at such high degree of rotational ex-citation, the observed rotational structure consisted of a series of spin-orbitmultiplets. Out of 9 possible ∆N branches, we observed only two (dashedlines in Fig. 3.8). Given this limited amount of information, fitting the databy a single set of molecular constants proved difficult. Our analysis resultedin two possibilities shown in panels a and b of Fig. 3.8. The retrieved molec-ular constants are Bv = 1.620 cm−1 and Dv = 4.4× 10−6 cm−1 for plot (a),and Bv = 1.664 cm−1 and Dv = 5.7 × 10−6 cm−1 for plot (b). To choosebetween the two possibilities, we noted that in Hund’s case (a), an effectiverotational constant B0 for F2 spin-orbit component is equal to the true Bvvalue [45]. This implies that at v′ = 2, Bv = 1.648 cm−1 (see Table 3.1).Since we expect Bv to decrease with v′, Fig. 3.8 (b) should reflect the correctbranch assignment.3.3 Mapping out the angular distribution ofmolecular super rotorsAs discussed in the introduction, the wave functions of a linear rotor’s sta-tionary states are given by the spherical harmonics YMJJ (θ, φ), with J,MJbeing the full angular momentum quantum number, and its projection on alaboratory-fixed quantization axis z. With a proper choice of z, parallel oranti-parallel to the laser propagation direction, the states produced in an op-tical centrifuge correspond to the wave functions Y JJ (θ, φ) = C(J) sinJ θeiJφ,where C(J) is a J-dependent normalization factor. Angular distributionsof these states P (θ, φ) = Y JJ∗(θ, φ)Y JJ (θ, φ) ∼ sin2J θ are shown in Fig. 3.10583.3. Mapping out the angular distribution of molecular super rotors7000010010011011012012013013070500 71000J'=N'+-101Two-photon energy, cm-171500O2 rotational quantum number ΔN = -1ΔN = -2ΔN = 3ΔN = 2(a)(b)Figure 3.8: Ultra-high rotational resonances of O2. The two panels corre-spond to two possible ways of fitting the observed resonant branches (ap-parent along white dashed lines) to the calculated Hund’s case (b) structure(labeled with markers). In panel (a), the upper branch corresponds to∆N = −1, and the lower one to ∆N = 3, resulting in Bv = 1.620 cm−1and Dv = 4.4 × 10−6 cm−1. In panel (b), the upper branch overlaps with∆N = −2, whereas the lower one with ∆N = 2, yielding Bv = 1.664 cm−1and Dv = 5.7× 10−6 cm−1.for different values of J . As J grows, sin2J θ becomes more and more con-centrated around θ = pi/2 in the xy plane, corresponding to the localizationof the molecule in the plane of rotation. Since the angular momentum,corresponding to the motion in this plane, is known exactly (MJ = J),the uncertainty principle forbids any knowledge of the angular position φ,leading to the isotropic “pancake”-like in-plane distribution.Using the velocity map imaging technique (2.4.1), we projected themolecular distribution onto a plane of the detector and saw the effect ofan optical centrifuge. Nitrogen gas was cooled down to 10 K in a super-sonic expansion and Coulomb exploded by a short 35 fs laser pulse. TheMCP gate was set to the arrival of N+ ions. To minimize the effect of thedynamic alignment (discussed in section 2.4), the laser intensity was keptbelow 1015 W/cm2 [93]. Choosing the field polarization normal to the planeof detector resulted in centrally symmetric images, as shown in Fig. 3.10 (b),corresponding to the initial isotropic distribution.593.3. Mapping out the angular distribution of molecular super rotorsLinewidth, cm-10100 105 110N' quantum numberTwo-photon energy, cm-1Ion signal, arb. u.115706000.00.20.40.60.871000 714001205101520253530Figure 3.9: Observed linewidths of J ′ = N ′ − 1(triangles) and J ′ =N ′(squares) spin-orbit sublevels of C3Πg (v′ = 2) level as functions of rota-tional quantum number N ′. Inset demonstrates a fit of experimental data(solid red) to a sum of lorentzians in order to extract linewidths. Absoluteposition, absolute area and the widths of two peaks were fitted for eachdoublet individually, with the areas ratio fixed to a value extracted from thebest resolved N ′ = 115 doublet and with a doublet line separation equal toa calculated one.When the molecules were pre-excited with an optical centrifuge priorto explosion, the images acquired a well-pronounced anisotropic structure,shown Fig. 3.10 (c). The centrifuge was used at the full available bandwidth,bringing the molecules to the rotational level with J ≈ 85.An important limitation of using this method for the reconstruction ofthe exact angular distributions of super rotors stems from the drift effectowing to their high rotational frequencies[70]. The rotational energy of thecentrifuged nitrogen reached 1.8 eV/molecule, becoming comparable withthe Coulomb explosion energy release of 6.6 eV/molecule[48]. As a result,the recoil velocities of the fragment ions deviated from the internuclear axisof the exploding molecule and the measured velocity distribution no longerrepresented the true molecular angular distribution.603.3. Mapping out the angular distribution of molecular super rotorsJ=0, MJ=0zyxJ=5, MJ=5 J=10, MJ=10 J=50, MJ=50Centrifugeoff(b)(a)Centrifugeon(c)Figure 3.10: (a) Linear rotor angular distributions. Experimental VMIimages of N2 fragments with the centrifuge turned off (b) and on (c).As was discussed in the previous sections on Raman and REMPI spec-troscopy, the centrifuge typically populates several consecutive rotationalstates. The VMI method described here, however, can not be directly usedto study their interference. Indeed, let us assume that the centrifuge popu-lates a few rotational states Ψ(θ, φ) =∑J CJYJJ (θ, φ). As was explained insubsection 2.4.2, the angle at which the molecules are released are randomlychanging from one laser pulse to another. Mathematically, it is equivalent tothe application of the operator of finite rotation around z axis by a randomangle α to the wavefunctionΨα(θ, φ) = e−iα Jˆz~ Ψ(θ, φ) =∑JCJe−iαJ~ Y JJ (θ, φ). (3.1)The corresponding angular distributionPα(θ, φ) = Ψα(θ, φ) =∑J,J ′CJC∗J ′e−iαJ−J′~ Y JJ (θ, φ)YJ ′J ′∗(θ, φ) (3.2)should be averaged over α, reflecting an average over many laser shots inthe experiments and yielding Pα(θ, φ) =∑J |CJ |2∣∣Y JJ (θ, φ)∣∣2, i.e. an in-coherent sum of all J-states distributions. To study coherent dynamics of613.3. Mapping out the angular distribution of molecular super rotorsthe centrifuged-induced distributions, averaging over the randomly changingangle α was eliminated. The results of that study are described in detail insection 4.2.62Chapter 4Rotational dynamics ofmolecular super rotorsA truly adiabatic centrifugal excitation would bring the ground state to oneof the excited states |J〉, corresponding to stationary time-independent dis-tributions. However, as follows from the results shown in sections 3.1 and3.2, the real centrifuge excites a wave packet of rotational states. This isattributed to the deviation of centrifugal excitation from the perfect adi-abaticity, which leads to broadening of the rotational state wave packet,giving rise to dynamically changing molecular angular distributions. Someof the aspects of this dynamics could be captured in the rotational Ramanspectroscopy, by trading off the method’s spectral resolution for a finer tem-poral one (section 4.1). A deeper insight into the field-free evolution of linearmolecules was gained using the VMI method. It is described in the contextof quantum-classical correspondence in section 4.2.The initial thermal rotational distribution could further enrich the molec-ular dynamics by overlapping the motion of several incoherently populatedwave packets (section 4.3).Even for an ideally adiabatic excitation, a stationary |J,M zJ = J〉 statecould be represented as a superposition of many statesJ∑MxJ=−JCMxJ |J,MxJ 〉in a different quantization basis (indicated with a superscript in MxJ ), degen-erate for an isotropic free space. As discussed in section 4.4, this degeneracycould be lifted by an external field, giving rise to yet another source of angu-lar dynamics, and providing an additional way of controlling the molecularrotation.4.1 Rotational revivals in the ensembles ofmolecular super rotorsThe spectroscopic approach used in section 3.1 utilized a narrowband probe,that, on one hand, allowed to resolve individual rotational transitions, but634.1. Rotational revivals in the ensembles of molecular super rotorson the other hand, lacked the information about their relative phase. Tostudy the rotational motion of super rotors, we used the same setup witha a spectrally wider (3 nm instead of 0.1 nm) probe pulses, allowing theadjacent Raman lines to interfere with one another, and examined the timedependence of the Raman response. In the time domain, the probe durationwas shortened to 500 fs, increasing the time resolution and making possibleto observe finer wavepacket dynamics.0 0120 12080 8040 400 0Rotational quantum number NRotational quantum number N50(a)50Time, ps Time, ps100 100150 150200 200(b)Figure 4.1: (a) Selective centrifuge spinning of oxygen to N ≈ 109. Dashedtilted line shows the increasing angular frequency of the centrifuge, termi-nated at about 90 ps. Dashed horizontal line marks the most populatedrotational state of O2 at room temperature, N = 7. Oscillations of thecoherent rotational wave packet are shown in the inset. (b) Truncating theprobe spectrum to 0.1 nm allowed us to resolve individual rotational Ramantransitions, similar to Fig. 3.2, at the expense of losing time resolution.As shown by a tilted trace in Fig. 4.1 (a), the rotation of moleculestrapped in the centrifuge follows the angular frequency of the laser field. Ahorizontal trace originating at the end of the centrifuge pulse (at around90 ps) indicates the free rotation of molecular super rotors, released fromthe centrifuge. The observed oscillatory signal, emphasized in the inset,is indicative of coherent rotational dynamics. In agreement with the gen-eral theory of quantum wave packets[80, 111], the evolution of freely rotat-ing molecules exhibits a clear periodicity. The periods are inversely pro-portional to the second derivative of E(N) with respect to N , i.e. T =[8Bc(1− 6N(N + 1))]−1, where c is the speed of light in vacuum and := D/B ≈ 3 × 10−6 is the ratio between the two rotational constantsin the Dunham expansion. For a rigid-rotor model ( = 0) of oxygen, this644.2. Observation of classical-like molecular rotationwould result in the main period of T = (8Bc)−1 ≈ 2.9 ps in agreement withour results for “slow” rotation (N = 7, along the horizontal dashed line inFig. 4.1 (a)). Centrifugal distortion of fast super rotors (N = 109) results inthe stretching of the molecular bond and the correspondingly longer periodof 3.8 ps, in a nice agreement with a prediction. In contrast, similar mea-surements done with a 0.1 nm probing pulse, shown in Fig. 4.1 (b), providedresolution of individual rotational transitions, at the cost of time resolutionof signal oscillations. An in-depth analysis of the periodic field-free dynamicsof linear molecules is provided in the next section.4.2 Observation of classical-like molecularrotationMicroscopic quantum objects behaving in a classical manner are of great in-terest due to the fundamental aspects of quantum-classical correspondence.The well known examples include coherent states of the quantum harmonicoscillator, slowly spreading Rydberg wave packets[136] and nonspreadingTrojan wave packets[86] in highly excited atoms. Classically behaving quan-tum wave packets preserve their minimum uncertainty shape as they movealong the corresponding classical trajectories[14, 56, 96]. In the case of ro-tating molecules, classical-like behavior implies nondispersing rotation of themolecular wave function with narrow distribution of angular momenta andstrong localization around the internuclear axis.Wide N-distributionNarrow N-distributionFigure 4.2: Time evolution of a wide (top) and a narrow (bottom) rota-tional wave packet, consisting of many or only two rotational states, respec-tively. The latter is an example of a quantum “cogwheel state”. Trev is therotational revival time.654.2. Observation of classical-like molecular rotationAlthough the popular non-adiabatic approach to molecular alignmentwith intense non-resonant laser pulses can produce very narrow angulardistributions[17, 25, 79, 103, 104, 110, 127, 143], the width of the createdrotational wave packet is quite large, making it quickly disperse in angle andundergo oscillations between aligned and non-aligned distributions. Quan-tum analogs of classical rotational dynamics, such as the gyroscopic desta-bilization in the rotation of an asymmetric top molecule, has been found inthe absorption spectra of thermal molecular ensembles[19]. The centrifugetechnique enabled us to execute a complete transfer of molecules from theinitial ground rotational level to a coherent superposition of a relativelysmall number of states |N,MN = N〉, where N (either only even or onlyodd) and MN are the rotational and magnetic quantum numbers, respec-tively. If such wave packet is initially localized along a certain spatial axis(upper left picture in Fig. 4.2), it will rotate with an angular frequency ofΩN = (N+2−N )/(2~) (where N is the rotational energy) while simultane-ously undergoing angular dispersion. For a homonuclear molecule with therotational constant B, it will then revive to its initial shape, up to a rotationby a constant angle of pi/4, at one quarter of the revival time Trev = h/(2B)(upper right picture in Fig. 4.2). Indeed, the time-dependent rotationalstate can be then expressed as|Ψ 〉(t) =∑NCN |N 〉 exp(− i~N t),where CN and |N 〉 are the expansion coefficients and rotational eigenstates,corresponding to an angular momentum N . For a homonuclear molecule, Ntakes either only even or only odd values, depending on the electronic andnuclear spin state of the molecule[47]. For example, only odd N values areallowed in 16O2, and the above expression at t = Trev/4 can be rewritten:|Ψ 〉(Trev/4) =∑n,2n−1=NCN |N 〉 exp(− ipi42n(2n− 1))=∑n,2n−1=NCN |N 〉e−ipin(n−1)e− ipi4 e−i(2n−1)pi4 = e−ipi4∑n,2n−1=NCN |N 〉e− ipi4 N ,where in the last expression we used the fact that n(n − 1) is always even.The last exponent can be rewritten as e−ipi4MN , which represents an operatorof rotation by pi/4 along the quantization axis, applied to the initial state|Ψ 〉(0). The so-called fractional revivals occur at intermediate times t =pqTrev2 , withpq being an irreducible fraction[5, 6]. Provided the number of664.2. Observation of classical-like molecular rotationpopulated rotational states is large, at the time of a fractional revival theinitial wave packet splits into q4(3 − (−1)q) sub wave packets, uniformlydistributed along the classical rotational trajectory, which keep rotating withthe same frequency ΩN . Indeed, the phases, accumulated by the rotationalstates at this time are given by:φ(N) = −BN(N + 1)~pqh4B= −pin(2n− 1)pq=−2pin2 pq+pi2Npq+pi2pq, 2n− 1 = N.The two last terms of this sum, as before, correspond to a rotation by afinite angle pi2pq and adding a common phase, respectively. Let us considerthe term φn = −2pin2 pq . It is easy to verify, by inspecting the cases of evenand odd values of q, that the phase factor eiφn is periodic in n:eiφn+l = eiφne−2ipi pql(2n+l)= eiφn , (4.1)where the period l = q4(3−(−1)q). It can therefore be expanded in the basisof periodic sequences:eiφn =l−1∑k=0ake−2ipi kln, (4.2)where ak’s are n-independent coefficients of this expansion. The wavefunc-tion can therefore be represented, up to an irrelevant phase and a finiterotation by pi2pq , as a sum|Ψ 〉(pqTrev2) =l−1∑k=0ake−2ipi klNˆ |Ψ 〉(0). (4.3)For example, around t = Trev/8, the angular distribution exhibits four peaksin the plane of rotation, as shown in the top row of Fig. 4.2, correspondingto the splitting of the initial wave packet into l = 44(3− (−1)4) = 2 sub wavepackets. As a transient effect, such multi-axial alignment has been observedexperimentally[25] and used as a main ingredient in the new scheme forquantum logic gates based on the control of rotational wave packets[78, 115].The nature of the periodic Raman signal oscillation, discussed in theprevious section, can now be understood in terms of the rotational Dopplereffect (section 2.2). Indeed, around the times 0 and Trev/4 in Fig. 4.2 (a),the rotating molecular ensemble has highly anisotropic polarizability in the674.2. Observation of classical-like molecular rotationplane of rotation, leading to the birefringence in the rotating frame of ref-erence and the appearance of a strong Raman signal. Around Trev/8 thepolarizability becomes isotropic, leading to the disappearance of Ramansignal.If the wave packet consists of only two states |N,MN = N〉 and |N +2n,MN = N + 2n〉, where n is an integer, the resulting “cogwheel state”rotates without spreading, preserving its shape at all times[76]. The wavefunction of a quantum cogwheel is symmetric with respect to rotation by pi/nand follows the classical-like nondispersing motion indefinitely[140]. Thesimplest case of a cogwheel state with n = 1 is shown in the lower row ofFig. 4.2. Similarly to the importance of transient cogwheel distributions inquantum information processing[115], it has been speculated that nondis-persing cogwheel states could open new perspectives in the developmentof molecular machines, whereas in metrology, they could serve as “molecu-lar stopwatches” on a femtosecond time scale with immediate applicationsin synchronizing ultrafast laser sources[18]. The latter could be accom-plished by Coulomb exploding two spatially separated ensembles of rotatingmolecules, and measuring the relative angle of recoil between the ions, pro-duced by different sources. Using an optical centrifuge, we investigated thenondispersing n = 1 cogwheel state in centrifuged D2 molecules. In oxygen,we created a coherent superposition of more than two rotational states andobserved the transition between the classical-like rotation of a molecularstopwatch and that of a quantum cogwheel.We tracked the rotation of centrifuged molecules by means of the VMItechnique described in section 2.4. We combined a beam of circularly polar-ized 35 fs ionizing probe pulses with a beam of centrifuge pulses. Circularpolarization was used to eliminate the effect of the dynamic alignment. Bothlaser beams were focused on a supersonically expanded gas of either O2 orD2 molecules between the charged plates of the VMI setup. The rotationaltemperature in the supersonic jet was below 10 K, which in the case ofoxygen meant that the majority of molecules were in the ground rotationalstate, N = 1. The resulting images, averaged over 10000 laser shots, areshown in Fig. 4.3 (a,b). The images consisted of a set of concentric rings,corresponding to different ionization channels. Even in the absence of thecentrifugal excitation (panel a), the intensity was higher in the plane oflaser polarization (PP). This is due to the geometric alignment effect – theionization probability is higher for those molecules which happened to liein the plane of the probe’s circular polarization. When the molecules werecentrifuged prior to the Coulomb explosion (panel b), the image becameeven more squeezed towards the polarization plane, because of the molecu-684.2. Observation of classical-like molecular rotationCentrifuge onCentrifuge offPP(a) (c)(b)r1r2PinholepositionRFigure 4.3: Experimental VMI of O2 taken with the centrifuge turned off(a) and on (b). The apparent alignment of the cold ensemble in the planeof probe polarization (PP) in panel (a) is due to the geometric alignmenteffect. The enhancement of this alignment in panel (b) is caused by thecentrifuge. (c) Geometry of the in-plane distribution measurements.lar localization at that plane.As was discussed in section 2.4, extracting the exact 3D velocity dis-tribution from a 2D image is non-trivial, especially in the case of axially-asymmetric distributions. We developed a new method, that allowed usto estimate the angular distribution in the polarization plane. Assuming,that the kinetic energy release is constant for a given channel, the veloc-ity distribution function from Equation (2.1) can be written as f(r, θ, φ) =∑n cnδ(r − rn)f(θ, φ), where cn and rn is the fraction of molecules dissoci-ating via the n-th channel and their displacement along the recoil directionby the time of arrival at the detector (see Fig. 4.3 (c) for the definition ofthe relevant coordinates). Integral (2.1) then simplifies to:F (R,Θ) =∑n:rn>Rcnrn√r2n −R2(f(θn, φn) + f(θn,−φn)), (4.4)where θn = arccos(R cos Θrn)and φn = arccostan Θntan θn. The physical meaning694.2. Observation of classical-like molecular rotationof this formula is illustrated in Fig. 4.3 (c), where the recoil velocities of thetwo dissociation channels are distributed over the blue and orange spheres.The Abel integral over the projection line (dashed white line) is just a sumover all intersection points with the channel spheres.We introduced an opaque mask with a pinhole, centered at the circum-ference of the largest observable dissociation ring in the polarization planePP, as shown in Fig. 4.3 (c). At this point, Θ = 0 and R = max(rn), whichleaves in (4.4) a single term ∼ f(0, pi/2), resulting from a single molec-ular orientation angle at θ = 0◦. The phosphorescence signal, recordedbehind this pinhole with a photomultiplier tube (PMT), was proportionalto the molecular angular distribution along θ = 0◦, fθrel(θ ≡ 0◦). The de-pendence of this distribution on the release angle θrel, included in fθrel(θ)as a parameter, is equivalent to the dependence on θ at a fixed θrel ≡ 0,i.e. fθrel(0) ≡ f0(−θrel). We investigated the latter as a function of thecentrifuge-to-probe delay time. In practice, instead of manually varyingθrel, we relied on its natural fluctuations due to the centrifuge instability,and measured this angle for every centrifuge pulse, as described in 2.4.2.Centrifuge angular velocity, 1013 rad/sNormalizedstatepopulation 1.20.80.40.01.6 1.8 2.0 2.2 2.4N=37N=39Figure 4.4: Normalized population of the rotational states with N = 37(blue) and N = 39 (red) as a function of the centrifuge final angular velocity.The black arrow shows the terminal angular frequency of the centrifuge usedin this work for creating rotational wave packets in O2.To monitor the free evolution of the rotational wave packets in cen-trifuged oxygen, we truncated the spectral bandwidth of the centrifuge soas to match its terminal rotational frequency with that of an oxygen moleculeoccupying rotational states with N = 37 and N = 39. The population of704.2. Observation of classical-like molecular rotationthe two states, measured with REMPI (section 3.2), is shown by the blueand red curves in Fig. 4.4 as a function of the terminal angular frequencyof the centrifuge. To make them equally populated, we fixed the latter atthe value indicated by the black arrow. Note, however, that since the dis-tance between the two peaks and their half-widths are comparable, the twoneighboring states with N = 35 and N = 41 were also populated by thecentrifuge. The nature of the distribution width could be explained by thenon-adiabatic turn-off of the centrifuge field. Indeed, in the frame of refer-ence rotating with the centrifuge, the molecule is in one of the eigenstatesof the oscillating electric field potential (the so-called pendular states), withits axis aligned along the field polarization. A sudden turn-off of the fieldwould preserve the instantaneous probability density, leaving the moleculein a superposition of several consecutive |N,MN = N 〉 states.Two VMI images of oxygen, one with and one without the centrifuge,are shown in Fig. 4.3 (a,b). Pre-excitation with an optical centrifuge re-sulted in the visible narrowing of the ion distribution in space, owing tothe localization of the rotational |N,MN = N〉 wave functions in the planeperpendicular to the propagation direction x of the centrifuge (plane ‘PP’in Fig. 4.3 (a)). After binning the PMT signal into 11 intervals, uniformlydistributed from θrel = 0 to θrel = pi, we plotted it as a function of the timedelay between the centrifuge and the Coulomb-explosion pulses.The result is shown in Fig. 4.5 (a). Having been aligned at the moment ofthe release from the centrifuge, the molecules first underwent free rotationwith the expected classical frequency of 3.3 full rotations per picosecond,indicated by the blue dashed line at the beginning of the scan and shownwith a better resolution in Fig. 4.5 (b). The wave packet, however, graduallydispersed and, although the rotational dynamics could still be identified bythe overall linear tilt of the signal, its contrast decreased dramatically. Thishappened because of the admixture of the neighboring N = 35, 41 states tothe rotational wave packet created around N = 37 and N = 39.As expected, the alignment reappeared every quarter-revival time, 14Trev(middle section in Fig. 4.5 (a)). In the vicinities of 18Trev and38Trev, anotherfractional revival was observed. After repeating the measurement around38Trev with better averaging and higher angular resolution, we plotted theresult in Fig. 4.5 (c). Here, instead of two tilted traces per one classi-cal period (marked by a horizontal white bar), reflecting a double-peaked“dumbbell”-shaped alignment geometry, four parallel traces per period canbe seen, corresponding to the “cross”-shaped angular distribution peaked infour spatial directions. This observation is well anticipated from the analysisof the rotational dynamics outlined above and illustrated in the top row of714.2. Observation of classical-like molecular rotation000001 2 3Probe delay, psMolecular angle44.22.7 4.42.9(a)(b) (c)4.63.1 4.83.35 6Figure 4.5: (a) Probability density as a function of the molecular angleand the free propagation time. Time zero corresponds to approximately100 ps since the release from the centrifuge. The blue dashed line marks thecalculated trajectory of a “dumbbell” distribution rotating with the terminalangular frequency of the centrifuge, whose classical period is indicated withthe white horizontal bar at the lower right corner. Zoom-in to the region near(b) 38Trev and (c)38Trev, taken with better averaging and angular resolution.Twice higher density of the tilted lines in panel (c) (4 per classical periodindicated by the tilted arrows) stems from the emergence of a “cross”-shapeddistribution with four lobes along two perpendicular directions.Fig. 4.2.To create a truly nondispersing quantum cogwheel state, one needs toexcite a narrower wave packet, truly consisting of only two N -states. Toachieve this, one can use a gas of lighter molecules with a larger energyspacing between the rotational levels. We repeated the experiment withmolecular deuterium, whose moment of inertia is ∼ 20 times lower thanthat of oxygen. D2 has two spin isomers, ortho- and paradeuterium, withonly even or only odd rotational quantum numbers in their rotation spec-trum, respectively. The terminal frequency of the centrifuge was set at halfthe frequency of the N = 2 → N = 4 transition. Under these conditions,orthodeuterium was excited to the coherent superposition of equally popu-lated N = 2 and N = 4 states, while paradeuterium was transferred into a724.3. Coherent spin-rotational dynamics in oxygen0.000 0.1 0.2 0.3 0.4 0.5Probe delay, psMolecular angleFigure 4.6: Probability density as a function of the molecular angle and thethe free propagation time of D2 prepared in the equal-weight superposition ofN = 2 and N = 4 states. The observed nondispersing behavior illustratesthe main property of a quantum cogwheel state. The classical period isindicated with the white horizontal bar at the lower right corner.single state with N = 3. As the probability density of the latter does notdependent on the polar angle θ, it introduces a homogeneous backgroundwhich did not affect the observed dynamics of orthodeuterium, shown inFig. 4.6. One can see that during one revival time, the molecule completesexactly 312 full rotations in agreement with the expected frequency of clas-sical rotation Ω2 =12piω4−ω22 = 7Bh =721Trev. We observed no dispersion inthe angular shape of the created wave packet, as anticipated for the timeevolution of a cogwheel state.4.3 Coherent spin-rotational dynamics in oxygenAmong simple diatomic molecules, O2 stands out because of its nonzeroelectron spin (S = 1) in the ground electronic state, X3Σ−g . The interactionbetween the spins of the two unpaired electrons and the magnetic field ofthe rotating nuclei results in the spin-rotation (SR) coupling on the order of734.3. Coherent spin-rotational dynamics in oxygena few wave numbers, which grows with increasing nuclear rotation quantumnumber N [45]. This coupling of the electron magnetism with molecularrotation, readily controllable with laser light, offers new opportunities forcontrolling molecular dynamics in external magnetic fields[40].0 10 20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.8Rotational quantum number, NRaman splitting (cm-1)0 10 20 30 40 50 60 70 80 90 100-0.04-0.0200.020.040.060.08(b)(a)Figure 4.7: (a): Spin-rotational splitting of two rotational levels of oxygen,N ′′ and N ′ = N ′′ + 2. Each level is split into three sub-levels with energiesFk, k = 1, 2, 3 for the total angular momentum J = N + 1, N,N − 1, respec-tively. Three strongest Raman transitions (out of the total six allowed bythe selection rules) corresponding to the S(N ′) branch are shown and la-beled according to the participating J-states. (b): Dependence of the threeRaman frequencies (Ωk for Sk line) on the rotational quantum number.SR coupling splits each rotational level in three (Fig. 4.7 (a)), withthe total angular momentum J = N,N ± 1 [45]. The energy splitting,originally observed by Dieke and Babcock in 1927[24] and later calculatedby Kramers[72] and Schlapp[107], is currently known with a very high degreeof accuracy[137]. Though routinely observed in the frequency domain withthe methods of microwave spectroscopy[8, 91], the spin-rotation splittingand the associated with it SR dynamics have not been previously studied inthe time domain.We employed the method of time-resolved coherent Raman spectroscopy(section 3.1) to study this dynamics. Because of the spin-rotation coupling,any N → (N + 2) Raman transition in oxygen consists of six separate linesbelonging to one Q, two R and three S branches with ∆J = 0, 1 and 2,respectively. The strength of both Q and R branches drops quickly withincreasing N and becomes negligibly small at N > 5 [7]. The frequency744.3. Coherent spin-rotational dynamics in oxygendifference between the lowest R lines, R(1) and R(3), is about 2 cm−1. Theirinterference in the time domain results in the oscillations with a period of≈ 17 ps, which has been observed experimentally[90]. On the other hand,the three stronger S branches shown in Fig. 4.7 (a) are split by less than 0.05cm−1 (Fig. 4.7 (b)), which corresponds to the oscillation period of about600 ps. This time scale is much longer than the collisional decoherencetime of the thermally populated rotational levels at ambient pressure[7, 92],explaining why no spin-rotational dynamics has been seen for N > 3 in thetime-resolved experiments[90].We employed the technique of an optical centrifuge to excite oxygenmolecules to ultra-high angular momentum states, reaching rotational quan-tum numbers as high as N = 109. Due to the substantially increased rota-tional decoherence time at high N (see section 6.1), the detection of spin-rotational oscillations was possible even at the pressure of 1 atmosphere.By lowering the pressure, we observed SR dynamics in the broad range ofangular momentum, 3 6 N 6 109.For this study, we used the experimental setup developed for the coher-ent Raman spectroscopy of super rotors (section 3.1). Probe pulses werenarrowed down to 3.75 cm−1 (FWHM) to allow the resolution of individualrotational lines.As discussed earlier, the centrifuge-induced coherence between the states|J,MJ = J〉 and |J + 2,MJ = J + 2〉 results in the Raman frequency shiftof the probe field. The Raman spectrum of the probe pulses scattered offthe centrifuged molecules was measured with a spectrometer as a function ofthe probe delay relative to the centrifuge. An example of the experimentallydetected Raman spectrogram is shown in Fig. 4.8 (a). While spinning up,the molecules were “leaking” from the centrifuge. Narrow probe bandwidthenabled us to resolve individual rotational states and make an easy assign-ment of the rotational quantum numbers to the observed spectral lines asshown in Fig. 4.8 (b) for the Raman spectrum taken at t = 200 ps. Thecreated wave packet consisted of a large number of odd N -states, with evenN ’s missing due to the oxygen nuclear spin statistics. Each Raman line un-derwent quasi-periodic oscillations due to the interference between the threefrequency-unresolved components S1,2,3(N) of the S(N) branch split by thespin-rotation interaction. An example of these spin-rotation oscillations forthe N = 91 Raman line is shown in Fig. 4.8 (c). The oscillations start ataround 100 ps, after the super rotors with the rotational angular momentumof 91~ have escaped from the centrifuge.The intensity of a Raman line corresponding to the transition between754.3. Coherent spin-rotational dynamics in oxygenData from May 7.4000 800 1200 1600Delay time, ps390392394396398Wavelength, nm0 10 20 30 40 50 60 70 80 90 10010-210-1100J numberData from May 7. Delay of 200 ps.110-110-20 10 20 30 40 50 60 70 80 90 100Rotational quantum number, NData from May 7. Delay of 200 ps and N=91.4000 800 1200 1600Delay time, ps110-110-210-310-4(a) (b)Data from May 7, N=91, Errorbars calculated in Analysis1.m0 2 4 6 8 10 12 14 16 18x 10-10-4-3-2-10Delay (s)Data from May 7, N=91, Errorbars calculated in Analysis1.m(c)N=91Figure 4.8: (a): Experimentally detected Raman spectrogram of centrifugedoxygen showing the rotational Raman spectrum as a function of the timedelay between the beginning of the centrifuge pulse and the arrival of theprobe pulse. Color coding is used to reflect the signal strength in logarithmicscale. (b): Cross-section of the two-dimensional spectrogram at the delayof 200 ps (vertical dashed line in a), showing an ultra-broad rotational wavepacket created by the optical centrifuge. (c): Spin-rotation oscillations of theN = 91 Raman line (horizontal dashed line in a). Experimental uncertaintyis indicated by the vertical error bars. Note logarithmic scale in all panels.the states N and N − 2 can be described asIN (t) = I0 |ρN,N−2(t)|2 e−t/τN , (4.5)where I0 is determined by a number of time-independent parameters, suchas molecular concentration and probe intensity, τN is the collisional decaytime constant and ρN,N−2(t) is the centrifuge induced coherence between thecorresponding rotational states. As discussed above, at N > 5, the latterconsists of three main frequency components corresponding to the three Sbranch transitions (see Fig. 4.7),ρN,N−2(t) =∑k=1,2,3akeiΩk(N)(t−t0), (4.6)with amplitudes ak and frequencies Ωk(N). Time t0 (0 < t0 < 100 ps)represents the release time of the corresponding rotational state from the764.4. Dynamics of super rotors in external magnetic fieldscentrifuge. For any N , the three frequencies are simplyΩk(N) = [Fk(N)− Fk(N − 2)] /~,where Fk(N) are the well known spin-rotational energies of oxygen[45]. Afternormalizing each measured Raman line to 1 at t = 100 ps (i.e. shortly afterthe end of the centrifuge pulse), we fitted the theoretical expression to the ob-served signals using the following five fitting parameters {a1, a2, a3, t0, τN }.As demonstrated by a few examples in Fig. 4.9, the oscillatory behavior ofour experimental data is well described by Eq.4.5 over the whole range ofangular momentum accessed by the centrifuge, from N = 5 to N = 109.Note that the weaker the line (e.g. N = 5) the smaller the dynamic range,ultimately determined by the sensitivity of our detector.Data from Apr17. N=5. Analysis1.m Data from Apr17. N=7. Analysis1.m Data from Apr17. N=9. Analysis1.m0 0.5 1 1.5 2Data from Apr17. N=21. Analysis1.m Data from Apr17. N=61. Analysis1.m Data from Apr17. N=101. Analysis1.m110-110-210-310-4110-110-210-310-40 0.5 1 1.5 2 0 0.5 1 1.5 2Delay time, ns Delay time, ns Delay time, nsN=5 N=7 N=9N=21 N=61 N=101Figure 4.9: The observed data (blue circles, normalized to 1 at t = 100ps) and the fit to spin-rotation oscillations (red curves, Eq.4.5) for sixdifferent Raman lines corresponding to the rotational quantum numbersN = 5, 7, 9, 21, 61 and 101. Experimental uncertainty (not shown) is similarto Fig. 4.8 (c). Note logarithmic scale in all panels.4.4 Dynamics of super rotors in externalmagnetic fieldsMagnetic field is one of the most powerful tools for controlling atomic andmolecular dynamics. Both translational and rotational degrees of freedom of774.4. Dynamics of super rotors in external magnetic fieldsmolecules have been successfully manipulated via various types of magneticinteraction. Alignment of molecular axes has been accomplished by creatingpendular states in paramagnetic molecules at low temperature [37, 117]. Gy-roscopic precession along the direction of the applied magnetic field has beenpredicted for non-magnetic molecules in dispersionless “cogwheel” states[76, 140].As discussed above, the technique of an optical centrifuge proved capa-ble of producing synchronously rotating molecules with narrow rotationalstate distribution in a wide variety of species up to extremely high angu-lar frequencies. However, the spatial orientation of the induced angularmomentum is defined by the propagation direction of the excitation laserbeam, and is therefore not easy to manipulate. To gain control over the axisof the induced molecular rotation, we applied an external magnetic field tomolecular super rotors and exploited the coupling between the rotation ofa molecule and its magnetic moment. An applied magnetic field causes thismoment to precess, “dragging” the axis of rotation with it. The method isapplicable to the molecules with magnetic moments of different nature, e.g.due to the rotation of the molecular frame or due to the electronic spin.Consider a Hund’s case (b) molecule possessing an electronic spin, suchas molecular oxygen 16O2 in the ground electronic state. Its electronic spinis strongly coupled to the molecular angular momentum N. In an externalmagnetic field, sufficiently weak for not causing the spin to decouple fromN, the angular momentum will precess with frequency [31]Ω = µBgSN |B| 1~N , (4.7)where µB is the Bohr’s magneton, g is the electron g-factor, B is the mag-netic field strength, SN is the projection of an electronic spin on N and ~ isthe reduced Plank’s constant. The 1/N dependence in the above expressioncomes from the gyroscopic effect: at higher angular momenta, it becomesharder and harder for the same torque (determined by the product of themagnetic moment µBgSN and the field strength B) to change the rotationalaxis.If the molecule does not have an electronic spin, its magnetic momentmay stem from the electronic and nuclear currents (equal in magnitude andopposite in direction). The magnitude of both currents can be written asZeν, where Z is the electron (proton) number of the molecule, e is theelementary charge and ν is the rotational frequency. The latter can beexpressed through the molecular angular momentum N as ν = ~N2piI , where Iis the molecular moment of inertia. The circular currents from the electrons784.4. Dynamics of super rotors in external magnetic fieldsand nuclei will produce magnetic moments µelec and µnucl in the oppositedirections, adding to:µtotal = µelec +µnucl = −ZeνAeffelec +ZeνAeffnucl =Ze~N2piI(Aeffnucl−Aeffelec), (4.8)where Aeffnucl and Aeffelec are the effective areas, encircled by the nuclei andelectrons, respectively. Notice, that Aeffnucl ∼ Aeffelec ∼ pir2, and I ∼ nmpr2/2,where r is the internuclear distance of the molecule, n is its mass in theatomic units, n ∼ 2Z, and mp is the mass of a proton. The magnitudes ofboth contributing magnetic moments can be estimated as:|µelec,nucl| ∼ Ze~N4piZmpr2/2pir2 = µNN, (4.9)where µN is the nuclear magneton. This suggests, that the total magneticmoment would be of the same order, µtotal = µNgrN , where gr is the so-called rotational g-factor[32, 52], specific for a given molecule. For typicalmolecular systems it takes both positive and negative value of order of one.For example, for a ground state of N2 it is −0.2681 [20]. In the presence ofan external magnetic field, this magnetic moment will lead to an effectiveinteraction Hamiltonian Hm = µNgrN · B [32, 52], which gives rise to theprecession with frequency [140]Ω = µNgr |B| 1~ . (4.10)As the magnetic moment, and hence the torque, grows together with theangular momentum, no N -dependence is expected in this case. The preces-sion frequency, however, is a few orders of magnitude lower compared to themagnetic molecules, due to the large difference between µN and µB.Using the technique of an optical centrifuge, we excited O2 and N2molecules to high rotational states and observed the dynamics of their an-gular wave function in the magnetic field applied perpendicular to the ro-tation axis. By imaging the plane of molecular rotation with ion imaging,we followed its evolution in time and observed two different mechanisms ofmagnetic interaction, described above.The imaging was carried out using the same VMI setup described insection 2.4, with a modification to include magnetic field. A circular holewas cut in the bottom electrode of the TOF spectrometer (Fig. 4.10 (a)),through which a cylindrical rare-earth permanent magnet was introduced.The magnetic field at the center was roughly measured with a gaussmeter to794.4. Dynamics of super rotors in external magnetic fieldsCentrifugeProbe MagnetMCPIon optics(a) (b)(c)Centrifuge offCentrifuge onFigure 4.10: (a) Experimental setup for detecting the plane of molecularrotation. The molecules are excited with a centrifuge pulse, then rotatefreely in a magnetic field of a permanent magnet, and finally are ionizedwith a femtosecond probe pulse. The resulting atomic ions are extractedand focused with ion optics of a velocity map imaging (VMI) apparatusonto a microchannel plate detector (MCP) with a phosphor screen. Theimages observed without (b) and with (c) the rotational excitation by acentrifuge pulse.be around 350 mT at the molecule-centrifuge interaction region. Nitrogenor oxygen gas was supersonically expanded through a pulsed nozzle , whichcooled most of the molecules down to their ground rotational state. Themolecules were then excited with a centrifuge pulse to high (up to 10 THz)angular frequencies.After evolving in the applied magnetic field for a certain time period t,the molecules were exposed to an intense 35-femtosecond probe pulse, whichlead to Coulomb explosion, discussed in detail earlier. Gating the voltage onthe MCP allowed us to mass-select only doubly-ionized atomic ions, eitherO2+ or N2+.For a given probe pulse energy (∼ 1015 W/cm2), the ions came primar-ily through the single ionization channel X2 −→ X2+ + X+ + 3e−. Dueto the highly anisotropic ionization probability (the geometric alignment),molecules undergoing Coulomb explosion are mostly those aligned vertically804.4. Dynamics of super rotors in external magnetic fieldsalong the linear polarization of the probe field. Hence, in the absence of thecentrifuge, the observed image consisted of a single spot originating fromthe ions which had zero velocity component in the horizontal plane of thedetector (Fig. 4.10 (b)).Rotational excitation by an optical centrifuge resulted in the transversecomponent of the ion recoil momentum due to the high angular velocity ofthe atomic nuclei at the moment of explosion (the drift effect). The energy ofthis transverse motion reached ∼0.86 eV for the highest accessible rotationalstates and was comparable to the Coulomb explosion energy of a few eV. Asa result, the ions spread out horizontally in the plane of molecular rotation,as can be seen in Fig. 4.10 (c). The long axis of the observed ion image(horizontal axis in Fig. 4.10 (c)) can therefore serve as a direct indication ofthe plane of molecular rotation. As expected, the degree of image elongationgrew with the degree of rotational excitation.In the experiment with O2, the delay between the centrifuge and probepulses was controlled with a motorized delay line and changed from 0 to 0.9ns. The final molecular angular momentum was varied by truncating thespectrum of an optical centrifuge as described in section 2.1. The observedion images are shown in Fig. 4.11 for different degrees of rotational excitation(rows) and at different probe delays (columns).For all observed angular momenta, as the pump-probe delay increases,the initial plane of molecular rotation splits into three: one which does notprecess (horizontal axis in Fig. 4.11) and two others, precessing in clockwiseand counter-clockwise direction. The splitting into three planes, clearlyvisible at longer delays and intermediate N values (e.g. N=61), resultsfrom the dependence of the precession frequency on the projection of theelectronic spin on the rotational angular momentum SN (see Eq.(4.7)), whichcan take the values of 0 and ±1.It could be seen from the images, that the precession frequency decreaseswith increasing angular momentum, in agreement with Eq.4.7. The quan-titative comparison is shown in Fig. 4.12, where the observed values (bluesquares) are plotted together with the theoretical prediction (red dashedline) calculated for the magnetic field strength of |B| = 316 mT, determinedfrom the nitrogen experiments (as explained below). The experimental dataare in good agreement with the predicted values, except of the higher angu-lar momentum. We attributed this discrepancy to the poorer performanceof the centrifuge at higher N values. As the centrifuge field becomes weakerat higher angular velocities, some molecules are “spilled over” from the cen-trifuge prior to reaching its terminal rotational frequency.We repeated the experiment with molecular nitrogen. To observe the814.4. Dynamics of super rotors in external magnetic fields21Nt=0 ns 0.15 ns 0.3 ns 0.45 ns 0.6 ns 0.75 ns 0.9 ns416181101Figure 4.11: Ion images of oxygen super rotors evolving in the externalmagnetic field. Different rows correspond to different degrees of rotationalexcitation (N value on the left), whereas each column corresponds to theevolution time indicated at the bottom. The initial disk distribution splitsinto three disks precessing with different frequencies according to their spinprojections.magnetic precession of this non-magnetic molecule, probe pulses were de-layed for up to 100 ns. The resulting images, taken with the maximum avail-able centrifuge excitation (N ≈ 60), are shown in Fig. 4.13 (a). Althoughlong time delays resulted in the reduced image quality, the precession isstill clearly seen (notice dashed contour lines indicating counter-clockwiserotation).To observe this precession at even longer delays, the directional resonance-enhanced multi-photon ionization (directional REMPI) approach was em-ployed [66]. Centrifuged molecules were ionized with a dye laser, resonantwith a1Πg(v′ = 0, J ′ = 31) ←← X1Σ+g (v′′ = 0, J ′′ = 33) two-photon tran-sition (285.813 nm), thus probing the population of the J = 33 rotationallevel in the ground electronic state. The terminal rotational frequency of thecentrifuge field was tuned to populate this state. In a centrifuged ensemble824.4. Dynamics of super rotors in external magnetic fields108642Rotational quantum number N00 20 40 60 80 100Precession period, nsFigure 4.12: Experimentally determined precession period as a function ofthe rotational quantum number (blue squares) and the expected theoreticaldependence (dashed red line).rotating primarily in one direction, the ionization probability is substantiallydifferent when the direction of rotation is the same as, or opposite to, thedirection of the circular polarization of the probe light. Namely, when thedirection of rotation coincides with that of the probe polarization, the two-photon transition is forbidden by the conservation of the angular momentumprojection. Indeed, in this case, the absorption of two photons should in-crease the magnetic quantum number by two to J ′′+ 2. This value is higherthan the highest possible magnetic quantum number for a given angularmomentum M ′max ≡ J ′ = J ′′ − 2, making this process impossible. In theopposite case, i.e. for the molecules rotating against the probe polarization,the two-photon transition is allowed and results in strong ion yield.Directional REMPI signal was measured as a function of the probe delay.At each time delay, the probe beam was moved downstream from the pumpto account for the supersonic drift of the molecular cloud, which becomesrelevant on these time scales. To compensate for the spread of the movingmolecular cloud and the change in the beams overlap, each intensity wasnormalized to the signal obtained with vertical probe polarization at thatdelay time.The results are shown in Fig. 4.13 (b). When the direction of the cen-834.4. Dynamics of super rotors in external magnetic fields0 ns (b)(a)50 ns100 nsRelative REMPI signalProbe delay, ns0.50.40.30.20.10.00 500 1000 1500 2000 2500BFigure 4.13: (a) Ion images of nitrogen super rotors in the external mag-netic field. To improve the visibility of precession, an over-saturated centralregion is cut off and contour lines (green dashed) are added. (b) DirectionalREMPI signal as a function of the pump-probe delay. Maximum (minimum)signal is detected when the molecules rotate in the opposite (same) directionwith respect to the circularly polarized probe. Solid red and dashed bluelines correspond to the opposite initial directions of molecular rotation, i.e.opposite handedness of the centrifuge pulse.trifuge spinning was opposite to the probe polarization handedness, themeasured signal was initially high (solid red line). However, in half the pre-cession period, when the axis of molecular rotation turned by 180 degrees,the direction of rotation coincided with that of the probe polarization andthe signal reached its minimum. These oscillations continued over the wholeobserved time range, with the plane of polarization completing a total ofmore than 1.5 full turns over 2500 ns, giving a precession period of 1550 ns.Given the rotational g-factor of N2 (−0.2681 [20]), this corresponds to themagnetic field strength of 316 mT, comparable with the expected value forour magnet. When the centrifuge direction was reversed (dashed blue linein Fig. 4.13 (b)), oscillations of similar period and amplitude, but shiftedby pi radian, were observed in agreement with the expected behavior.84Chapter 5Molecular alignment with anoptical centrifugeThe two commonly used techniques for molecular alignment are the adia-batic and non-adiabatic alignment with intense non-resonant laser pulses[113,119]. As was discussed in the introduction, the first class is limited to the ap-plications that tolerate the presence of intense aligning laser field, whereasthe second one can produce only transient alignment. Very high degreesof rotational excitation became available through the method of an opti-cal centrifuge. However, in contrast to the dynamics of impulsively kickedmolecules, the rotation of the centrifuged linear molecules is confined to aplane (section 3.3), rather then a line in the laboratory frame, demonstrat-ing a so-called “antialignment” . Section 5.1 discusses how a modificationto the centrifuge allowed us to achieve molecular alignment, similar to thatinduced by a femtosecond kick, simultaneously with extreme levels of rota-tional excitation characteristic of the centrifuge spinning.Another approach to break the planar anisotropy of the centrifuged lin-ear molecules involves the use of static external fields. In section 5.2 we showthat strong planar confinement of paramagnetic molecular super rotors canbe converted into an anisotropic angular distribution by means of an ap-plied magnetic field. The effect is based on spin-rotational coupling, alreadydiscussed in section 4.4, and can be used for fast switching of optical bire-fringence in gases. Such magneto-rotational birefringence, investigated here,also offers a way of exploring the effect of collisions on the directionality ofmolecular rotation in dense media.In the case of asymmetric top molecules, the adiabatic approach is evenmore desirable. In contrast to linear molecules, asymmetric tops have com-plicated rotational spectrum which leads to very complicated aperiodic dy-namics. In section 5.3, we demonstrate a new mechanism of field-free align-ment of asymmetric top molecules with an optical centrifuge. Owing to theadiabaticity of the centrifuge spinning, SO2 super rotors are left rotatingwith their O-S-O plane aligned with the plane of rotation long after the ex-citation pulse is gone. Remarkably, this planar alignment reduces complex855.1. Two-dimensional optical centrifugerotational dynamics of an asymmetric top to that of a simple linear rotor.5.1 Two-dimensional optical centrifugeAs was discussed in the introduction (section 1.3), rotational excitation byan optical centrifuge may be thought of as a series of adiabatic Ramantransitions between the states with increasing rotational quantum number,∆J = 2 [131], and its projection onto the laser propagation axis ∆MJ = 2.If the two chirped pulses, used to produce the centrifuge field, are linearlypolarized along the same direction, the same reasoning can be applied toshow that the molecule under such excitation would still climb the ladder ofrotational levels, changing its angular momentum by ∆J = 2 at each step,but keeping the projection MJ constant.The latter feature stems from theselection rules for a linearly polarized field, ∆MJ = 0 along the field polar-ization. This excitation scheme, which we called 2D centrifuge (as opposedto a traditional 3D centrifuge, where the meaning of the ”dimensionality”will be clear on the next page) will therefore bring a linear molecule, initiallyin the ground rotational state, to a highly excited state |J,MJ = 0〉. Sincethe symmetry between positive and negative values of MJ is not broken (un-like the case of a 3D centrifuge) the directionality of the induced rotationis lost. In return, however, we gained the ability to fix the direction of themolecular alignment along the (now constant, as opposed to rotating) fieldpolarization.The field of a 2D centrifuge, shown in Fig. 5.1 (a), was created by passinga 3D centrifuge beam through a linear polarizer. It consists of a series ofpulses, all linearly polarized along the same direction. Both the individualpulse width and the time interval between the pulses are gradually decreasingfrom the head to the tail of the sequence.To illustrate and compare the effects of the two centrifuge geometries,2D and 3D, on the molecular alignment, we plot the wave function of arigid rotor |N = 20 〉 with MN = N for the 3D and MN = 0 for the 2Dcentrifuge. The relatively low rotational angular momentum of the finalstate was chosen here for illustration purposes. Although we routinely spunnitrogen to N > 70, the calculated distributions, shown in Fig. 5.1 (b,c),become extremely narrow at higher values of N . A 2D centrifuge is seen toproduce a well-aligned ensemble, shown in panel (a), whereas the effect of a3D centrifuge is very different: molecules are confined in the plane of theirrotation, but exhibit no preferential alignment axis, as seen in panel (b).The induced molecular rotation was analyzed by means of coherent Ra-865.1. Two-dimensional optical centrifuge(a)(b) (c)2D 3DFigure 5.1: (a) Illustration of the concept of a “two-dimensional (2D) cen-trifuge”. The three-dimensional corkscrew-shaped surface represents thefield of a conventional “3D centrifuge”, propagating from right to left. Shownin blue is the field of a 2D centrifuge, created by passing the 3D centrifugethrough a linear polarizer. (b, c) Simulated molecular distribution after therotational excitation by a 2D (b) and a 3D (c) optical centrifuge, propagat-ing along the xˆ-axis.man spectroscopy (similar to section 3.1). Oxygen molecules in the gaschamber were rotationally excited by either a single femtosecond pulse (here-875.1. Two-dimensional optical centrifugeafter referred to as a “1D kick”), a conventional (3D) centrifuge or its two-dimensional projection (2D centrifuge). The excitation light was focusedto a spot size of 90 µm (FWHM diameter). For a 1D kick we used a 60 fspulse, whose total energy of 630 µJ resulted in the peak intensity of 5.4×1013W/cm2, close to the ionization threshold of oxygen. Probe pulses were 4 pslong (3.75 cm−1 spectral bandwidth (FWHM) centered around 400 nm).As in our previous Raman experiments, we sent the pump and probebeams collinearly and recorded the Raman spectrum of the probe with aspectrometer. Raman spectrum of oxygen molecules exposed to a singleintense femtosecond kick is plotted at the top of Fig.5.2(a). The unshiftedRayleigh peak at 0 THz is surrounded by a series of lines indicative of thelaser-induced coherent rotation, with each line corresponding to an indi-vidual | J 〉 → | J + 2 〉 (Stokes) or | J + 2 〉 → | J 〉 (anti-Stokes) Ramantransition. The amplitude envelope of these Raman lines reflects the initialthermal rotational distribution, shifted toward higher rotational levels by∆J ≈ 15, in agreement with the calculated amount of the angular momen-tum, transferred to the molecule by a laser pulse with intensity 5.4 × 1013W/cm2.The result of applying a conventional (3D) optical centrifuge, with itstotal frequency bandwidth of about 20 THz, is shown by the blue plot atthe bottom of Fig.5.2(a). Aside from the low-frequency thermal envelopes,corresponding to the molecules too hot to follow the adiabatic spinning[69],the spectrum contains the response from the molecules centrifuged to theangular frequencies between 7.5 and 10 THz. Unlike Raman scattering ofa probe light with circular polarization, discussed in section 2.2, linearlypolarized probe scatters into both Stokes and anti-Stokes Raman sidebands(slightly asymmetric due to the residual ellipticity of the probe polarization).Applying the 2D centrifuge resulted in the Raman spectra shown in themiddle of Fig.5.2(a). Even though the high-frequency Raman sidebands aresomewhat lower than in the case of a 3D excitation (compare the lower redand blue curves), the 2D centrifuge was clearly producing molecular superrotors, spinning as fast as 9 THz (remember that the corresponding Ramanshift is twice that value). The appearance of a localized (in frequency) wavepacket, well separated from the thermal envelope, indicated the adiabaticnature of the rotational excitation. Similar to its 3D prototype, the 2Dcentrifuge offers high degree of control over the frequency of molecular rota-tion. We illustrated this by truncating the spectral bandwidth of the pulsesaround 8 THz, thus moving the centrifuged rotational wave packet to lowerfrequencies, as shown by the green plot in Fig.5.2(a).In contrast to the uni-directional rotation of a 3D centrifuge, the 2D885.1. Two-dimensional optical centrifugeRaman spectra from June 12. Linear probe. Delay = 200 ps. Analysis1.mRaman spectra from June 25. Circular probe. Delay = ? (notebook) Analysis3.m3D centrifuge, matched, 20150625032D centrifuge, full, 201506250500.511.521D kick2D centrifuge(truncated)2D centrifuge(full)3D centrifuge(full)Linear Probe(a)-20 -15 -10 -5 0 5 10 15 20-15 -10 -5 0 5 10 1500.20.40.60.81Raman shift (THz)2D centrifuge3D centrifugeCircular Probe(b)Raman signal (arb.u.)Raman signal (arb.u.)Figure 5.2: (a) Rotational Raman spectra of the centrifuged oxygenmolecules recorded with a linearly polarized probe light. From top to bot-tom, the spectra correspond to the excitation by a single femtosecond pulse(black), a “truncated” 2D centrifuge (green), a full 2D centrifuge (red), anda full 3D centrifuge (blue) with a terminal rotational frequency around 10THz. The spectra have been recorded at the delay times of 400 ps. (b) Ro-tational Raman spectra of the centrifuged oxygen molecules recorded with acircularly polarized probe light. Upper red and lower blue curves correspondto the 2D and 3D centrifuge, respectively. The spectra have been recordedat the delay times of 400 ps.895.1. Two-dimensional optical centrifugeexcitation field has no preferential sense of rotation. Owing to the selectionrule ∆MJ = 0, the vectors of angular momentum of the molecular superrotors created by a 2D centrifuge lie in the plane perpendicular to its lin-ear (and non-rotating) polarization, similarly to the effect of a single 1Dkick. This lack of directionality is demonstrated in Fig.5.2(b), where weplotted Raman spectra observed with a circularly polarized probe. Whilethe missing Stokes lines in the lower (blue) spectrum indicate that the 3D-centrifuged molecules rotate in the direction opposite to the circular probepolarization, the presence of both Stokes and anti-Stokes Raman peaks onthe upper (red) plot confirms no preferential sense of rotation induced bythe 2D centrifuge.Nocentrifuge(a)3Dcentrifuge(b)2Dcentrifuge(c)Figure 5.3: Ion images of nitrogen molecules prior to any rotational excita-tion (a), and following the excitation by a 3D (b) and a 2D (c) centrifuge.Calculated molecular distributions are shown above the respective images.To determine the degree of molecular alignment produced by a 2D opticalcentrifuge, we employed the method of velocity-map imaging (VMI), similarto the setup used in section 3.3. A supersonic expansion cooled the nitrogengas down to the rotational temperature of ∼ 10 K, with most moleculesoccupying the rotational ground state. An intense 35 fs laser pulse, linearlypolarized along the TOF axis, Coulomb-exploded the molecules into the ionfragments. For the peak intensities of < 1015W/cm2, used in this study, themain source of anisotropy in Coulomb explosion of N2 stems from the geo-metric alignment effect[93], with the fragment ions recoiling in the directionof the internuclear axis prior to ionization. The ions, extracted with a DCelectric field along the probe polarization, were imaged on a phosphor screenof a microchannel plate detector. The latter was gated in time to mass-selectN+ ions, originated predominantly from the N2+2 −→ N+ + N+ dissociationchannel. Each image was averaged over 500,000 ionization events.With the absence of rotational excitation, the observed images were cir-905.1. Two-dimensional optical centrifugecularly symmetric, in accordance with the symmetry of the initial ensemble,as shown in Fig. 5.3 (a). Applying either 3D or 2D centrifuge resulted inanisotropic VMI images, plotted in Fig. 5.3 (b,c), respectively, reflectinghighly anisotropic angular distributions of the centrifuged molecules.0.720.680.640.600.56 1.0 1.1 1.2 1.3(c)(a)TOFdirectionCentrifuge(b)Figure 5.4: (a) Definition of the spherical coordinates (r, θ, φ) and polarcoordinates (R,Θ), used to describe the full molecular distribution and itstwo-dimensional projection on the xz plane of the ion detector. (b) R-dependence of the extracted alignment factor, with the shaded region indi-cating the experimental error. (c) Ion image of N2 molecules aligned witha 2D optical centrifuge. The black dotted and white dashed lines show thecircular cross-sections at the radius of a maximum ion signal, R = Rmax,and the bigger radius used for estimating the true two-dimensional align-ment factor, respectively. The latter is also marked by the red dashed linein plot (b).Since the molecular distribution f(r, θ, φ), produced by a 2D centrifuge,is axially symmetric, its two-dimensional cross-section f(r, θ, 0), containingthe axis of symmetry, completely characterizes the degree of the inducedalignment (see Fig. 5.4(a) for the definition of the relevant spherical andpolar coordinates). In this case, the alignment factor is commonly expressedas 〈cos2 θ2D〉 =∫cos2(θ)f(r0, θ, 0)dθ, where r0 = v0t is the radial distancecovered by the ion of interest, released from the Coulomb explosion withvelocity v0, during the time of flight t.The experimentally recorded ion images provided us with the projec-tion F (R,Θ) of the molecular distribution f(r, θ, φ) onto the xz plane. Asdiscussed earlier, two distributions are related to one another through the915.1. Two-dimensional optical centrifugewell-known Abel transform[128]:F (R,Θ) =+∞∫Rrf(r, θ, φ)√r2 −R2 dr, (5.1)with θ = arccos R cos Θr and φ = arccostan Θtan θ . We note that if the veloc-ity distribution falls off quickly away from its average value v0 and if Ris sufficiently large, the integral in Eq. (5.1) will be accumulated mostlyaround its lower limit r ≈ R. In this case, the observed angular distributionF (R,Θ) becomes proportional to f(R,Θ, 0), and the true two-dimensionalalignment factor can be approximated as 〈cos2 θ2D〉 ≈ 〈cos2 Θ〉. Followingthis argument, we extracted the angular distributions at various radii andcalculated the corresponding values of 〈cos2 Θ〉 as a function of R. As shownin Fig. 5.4(b), the latter indeed saturates at large R’s [white dashed circlein Fig. 5.4(c)], reaching the value of 0.69.The retrieved two-dimensional alignment factor of 0.69 is still consider-ably lower than 〈cos2 θ2D〉 =∫Y 0J∗(θ, 0) cos2 θY 0J (θ, 0)dθ = 0.87, anticipatedfor the rotational quantum state with J = 85, accessed by our centrifuge.We attribute this disagreement to the following two reasons. First, as Rincreases, dissociation channels other than N2+2 −→ N+ +N+ may provide anon-negligible contribution to the number of N+ ions, smearing the angulardistribution and lowering the extracted alignment factor.Another important correction to our estimate of 〈cos2 θ2D〉 stems fromthe drift effect owing to the high rotational frequencies of molecular superrotors[70]. The rotational energy of the centrifuged nitrogen, studied here,reached 1.8 eV/molecule, becoming comparable with the Coulomb explosionenergy release of 6.6 eV/molecule[48]. As a result, the recoil velocities of thefragment ions deviated from the internuclear axis of the exploding moleculeand the measured velocity distribution no longer represented the molecularangular distribution.It should be emphasized, that although the angular distribution acquiresan extremely narrow spike-like shape (Fig. 5.3 (c)) at high values of J , the3D alignment factor〈cos2 θ〉=∫ 2pi0 dφ∫ pi0 sin θdθ∣∣Y 0J (θ, φ)∣∣2 cos2 θ will notexceed a modest factor of 0.5. This can be understood in the following way.Molecules in states | J,MxJ = J 〉4 can be thought of as rotating around adefinite axis, x, while being localized in the plane yz. The molecules areuniformly distributed along the circle in this plane, and, therefore, theiralignment factor along the z-axis is given by 0.5. States | J,M zJ = 0 〉 all4Note the new quantization axis, x.925.2. Magneto-optical properties of paramagnetic super rotorshave their angular momentum lying in plane xy, and can be thought of as| J,MxJ = J 〉 “smeared” by the rotation around z. This rotation, howeverdoes not change the angle between the molecular axes and z direction, andtherefore, keeps the alignment factor at 0.5.5.2 Magneto-optical properties of paramagneticsuper rotorsIn the previous section we showed how the strong centrifuge-induced anti-alignment can be transformed to the axial alignment by, effectively, smearingthe distribution around the polarization axis. A somewhat similar smear-ing effect, induced by the magnetic field, was demonstrated with oxygenmolecules in section 4.4, where spin-rotation coupling mediated the inter-action between the external magnetic field and the nuclear motion, leadingto fast precession around the magnetic field direction. Due to the depen-dence of the precession frequency on the projection of the electronic spinon the angular momentum vector, the initial disk distribution was split intothree disks, converting the planar confinement (〈cos θ2y,z〉 = 0.5) of paramag-netic molecular super rotors into an aligned anisotropic angular distribution(〈cos θ2y〉 < 0.5, 〈cos θ2z〉 = 0.5). Here, θα is the angle between the molecularaxis and unit vector αˆ, the molecules initially rotate in the yz plane, thefield is along z, and 〈..〉 represents ensemble averaging. In this section, weshow our ability to use this effect for controlling an optical birefringenceof gases with an optical centrifuge in the presence of an external magneticfield.The experimental system, based on the modified Raman setup (sec-tion 2.2), is shown in Fig. 5.5 (a). Both centrifuge and probe beams werefocused collinearly between the two Helmholtz coils of a pulsed magnet insidethe chamber filled with oxygen at room temperature and variable pressure(Fig. 5.5 (b)). The coils were driven by a 50 µs current pulse and producedmagnetic fields of up to 4 T, which, on the time scale relevant to this study,could be considered static. After scattering off the centrifuged molecules, thepolarization of the probe pulses became elliptical, reflecting linear opticalbirefringence of the molecular ensemble induced by the combination of thecentrifuge and magnetic fields. To characterize this “magneto-rotational”birefringence, we passed probe pulses through a pair of crossed linear polar-izers, LP and LA in Fig. 5.5 (a).In Fig. 5.6, the observed spectrum of probe pulses, passed through theensemble of centrifuged oxygen, is plotted as a function of the magnetic field935.2. Magneto-optical properties of paramagnetic super rotorsfs SourceCentrifugeAmplifierCentrifugeShaperPO2DLBSDMDMLPLAProbe Pulse ShaperSpectrometerLBCentrifuge PulseM(a) (b)Centrifuge beamProbe beamOUTProbe beamINO2MagnetFigure 5.5: (a) Setup for the detection of the “magneto-rotational” birefrin-gence. BS: beam splitter, DM: dichroic mirror, LP (LA): linear polarizer(analyzer) oriented at angle θp(θp + 90o) with respect to ~B, DL: delay line,L: lens, M: two magnetic coils connected in a Helmholtz configuration. ‘O2’marks the pressure chamber filled with oxygen gas under pressure P at roomtemperature. An optical centrifuge field is illustrated above the centrifugeshaper with ~k being the propagation direction and ~E the vector of linearpolarization undergoing an accelerated rotation. (b) Geometry of the mag-netic and optical fields used in this work. The cloud of centrifuged moleculesis depicted as a dark ellipse.strength. At zero field, the probe spectrum consisted of two components:the frequency-unshifted Rayleigh line and the down-shifted (Stokes) Ramanpeak. The former was a result of the depolarization of probe light due tothe non-uniform (and randomly changing from pulse to pulse) distributionof molecular axes in the plane of rotation. The second component reflectscoherent molecular rotation with an angular frequency of about 6 THz (halfthe Raman shift). Oxygen molecules rotating with this frequency occupyrotational quantum states with N = 71.When the magnetic field was turned on, the amplitude of the Rayleighpeak became much higher, corresponding to strong incoherent (hence, frequency-unshifted) magneto-rotational birefringence. Unlike the weak initial depolar-ization which had no preferential axis at zero field, the magnetically induced945.2. Magneto-optical properties of paramagnetic super rotorsData from 20140522Processing with Analysis11.m Probe spectralpower (arb. u.)0123-12012 01234Figure 5.6: Spectrum of probe pulses transmitted through the ensemble ofcentrifuged oxygen molecules as a function of the applied magnetic field.All spectra have been recorded at the probe delay of t = 1.14 ns and under0.3 atm of gas pressure. Crossed circular (rather than linear) polarizer andanalyzer were used here to detect a weak magneto-rotational Raman signal(note the change of vertical scale (×200) at frequencies higher than 7 THz).birefringence exhibited a well defined optical axis. The applied magneticfield also affected the coherent Raman sidebands. While the initial Ramanpeak fell off with increasing B-field, a weak anti-Stokes Raman line grew onthe other side of the spectrum, reflecting molecular rotation in the opposite,with respect to the centrifuge, direction.In section 4.4, the observed dynamics was described by a simple model,in which the electronic spin, coupled to the molecular angular momentum,was precessing under the effect of magnetic field. To understand the natureof the observed magneto-rotational effects in the gas of paramagnetic superrotors, we calculated the angular distribution of O2 molecules subject toconstant magnetic field. As discussed in chapter 4, the interaction betweenthe spins of the two unpaired electrons and the nuclear rotation results inthe spin-rotation coupling, which splits each rotational level in three com-955.2. Magneto-optical properties of paramagnetic super rotorsData from file 'CosSquared.zip'(a)Simulations with Analysis8.mN=59, .9 ns, .316 T(b)t=900 ps, N=61, B=316 mTFrom Aleksey’s paper on magnetic precession0.2.4.6.81Bi re fr in ge nc e si gn al  (a rb . u. )-20 0 20 40 60 80 100Angle, θp (deg)(c) (d)~BFigure 5.7: (a,b) Calculated angular distribution of the molecular axes forthe rotational state with N = 59 at time t = 0.9 ns in an external magneticfield of 0 and 0.32 Tesla, respectively. (c) Birefringence signal (scaled topeak at 1) as a function of angle θp between the polarization of probe pulsesand the magnetic field direction. Black circles: data taken at 2 T, t = 1.5ns, and N = 95. Red curve is a fit to cos2(θp). (d) Experimentally measureddistribution of molecular axes, imaged in the direction of the applied field.All parameters are the same as in panel (b).ponents characterized by the total angular momentum J = N,N ± 1. Anapplied B-field lifts the degeneracy of each level with respect to the projec-tion MJ of ~J on the field axis. We calculated the energies EN,J,MJ (B) of themagnetic sublevels by numerically diagonalizing the electron-spin ZeemanHamiltonian, starting from its matrix elements in Hund’s case (b) basis set|N,S, J,MJ〉, with S = 1 being the total electronic spin.Given the initial thermal ensemble, the centrifuge creates an incoherentmixture of three states |N,S, (J = N,N ± 1),MJ = J〉 with maximumprojections MJ on the centrifuge direction. Using J  S, we approximatedthe angular distribution of the state |N,S, J,MJ〉 by the spherical harmonicsYJ,MJ (θ, φ). This approximation was verified by a more elaborate calculationof the exact angular distributions [31]. Using this approximation, the final965.2. Magneto-optical properties of paramagnetic super rotorsangular distribution at time t is given as:ρN (θ, φ) ≈∑J=N,N±1∣∣∣∑MJcJ,MJ eiEN,J,MJ (B)t/~ YJ,MJ (θ, φ)∣∣∣2,where cJ,MJ are the wave function amplitudes in the coordinate frame withthe quantization axis along the applied magnetic field. The results of ourcalculations for N = 59 are shown in Fig. 5.7. A disk-like distribution atB = 0 (panel (a)) corresponds to the molecular rotation around the cen-trifuge propagation direction ~k (x axis), with the molecular axes isotropicallydistributed in the perpendicular yz plane. As seen in panel (b) of Fig. 5.7,an applied magnetic field splits the disk into three components. This resultagrees well with our VMI measurements (section 4.4), reproduced in panel(d) for the same magnetic field and rotational state. In the majority of casesconsidered here, the coupling between the electronic spin and an ultrafastrotation of super rotors is stronger than its interaction with the externalB-field. The perturbative effect of the latter on the precession of the spinaround ~N results in a clockwise (counter-clockwise) rotation of ~N around~B for J = N − 1 (J = N + 1) states and no rotation for J = N states.This leads to an anisotropic distribution stretched along zˆ. In agreementwith this model, an experimentally measured cos2(θp) dependence shownin Fig. 5.7 (c) indicates the appearance of a well defined anisotropy axisalong the direction of the field, as expected from the geometry of the cal-culated distribution. We note that for stronger magnetic fields, the threespin states decouple from the molecular angular momentum and the phys-ical picture changes. For example, at the highest magnetic field, achievedwith our setup, B = 4 T, the Zeeman energy µBB reached 1.9 cm−1 andcould not be considered small compared to the spin-rotation interaction, oforder of 1 cm−1.Using the same experimental setup, we analyzed the temporal evolutionof the birefringence signal, which we defined as the difference between theamplitude of the Rayleigh peak with and without the applied magneticfield. Fig. 5.8 shows this signal, measured at a fixed magnetic field of 2Tesla, as a function of the probe time delay with respect to the beginningof the centrifuge pulse. After a rapid initial growth, the magneto-rotationalbirefringence reached its maximum and then decayed exponentially due tothe collisional re-orientation of super rotors.Truncating the centrifuge pulse in the centrifuge shaper, so as to stopthe accelerated rotation at different angular frequencies, allowed us to cre-ate rotational wave packets centered at different values of N and study the975.2. Magneto-optical properties of paramagnetic super rotorsBirefringence Signal(arb. u,log scale)0.10.20.40.60.810 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time (ns)N=33N=13N=99N=73Figure 5.8: Decay of the birefringence signal for different values of therotational quantum number at B = 2 T. All solid curves are generatedby spline-fitting the experimental data (shown with colored markers) andnormalized to peak at 1. The corresponding lifetimes are 85 ± 10, 290 ±20, 660± 50 and 610± 50 ps·atm for N = 13, 33, 73 and 99, respectively.dependence of the magnetic effect on the frequency of molecular rotation.The results are plotted in Fig. 5.8 and demonstrate that with the increasingangular momentum, both the growth rate and the decay rate of the inducedbirefringence became lower. This can be explained by a gyroscopic effect:as the rotational inertia of molecules grew, it became harder and harder tochange their angular momenta, either by an external field or by the collisionswith other molecules. The effect of high N on the rate of gyroscopic pre-cession of paramagnetic molecules in magnetic field was already explainedin section 4.4, In chapter 6, we will discuss in greater detail the effect ofcollisions on the directionality and the coherence of super rotation.The dependence of the magneto-rotational birefringence on the strengthof the applied magnetic field is shown in Fig. 5.9. After the initial growthwith increasing B, the signal saturated. For different rotational frequencies,the saturation occurred at different field amplitudes. Slower super rotorsdid not only require shorter time to respond to the external magnetic field,but also needed weaker fields to reach the plateau. Our numerical estimatesof the birefringence signal, based on the calculated angular distributionsof molecular axes (see Fig. 5.7), successfully reproduce our experimentalfindings, showing qualitatively similar behavior.985.3. Adiabatic alignment of asymmetric top molecules0 0.5 1 1.5 2 2.5 300.20.40.60.81Magnetic field (T).Birefringence Signal(arb.u)N=13N=89Figure 5.9: Dependence of the birefringence signal recorded at t =1 ns onthe strength of the applied magnetic field.Angular distributions are also instructive in the interpretation of themagnetic reversal of the coherent Raman scattering. The latter effect isclearly seen in Fig. 5.6 as the appearance of a weak Raman line with afrequency shift corresponding to the molecular rotation in the direction op-posite to the initial, centrifuge-induced rotation. Indeed, the precession ofthe two parts of the angular distribution around the applied magnetic field,as described earlier in the text, resulted in a small negative component inthe projection of the molecular angular momentum on the centrifuge axis.5.3 Adiabatic alignment of asymmetric topmoleculesThe two previous sections dealt with the axial molecular alignment inducedby an optical centrifuge in a gas of linear rotors. In this section we demon-strate that the centrifuge can also be used to achieve a planar alignmentof asymmetric top molecules. Our method combines the robustness of anadiabatic alignment with the much desired field-free conditions of the finalaligned state. Its essence is illustrated in Fig. 5.10. The leading edge of thecentrifuge (left plane) adiabatically aligns the molecular axis of maximumpolarizability (a axis of SO2, parallel to the O-O axis) along the field direc-tion. As the rotational frequency Ω increases, slowly emerging centrifugal995.3. Adiabatic alignment of asymmetric top moleculesRotationalplanecabFigure 5.10: Illustration of the main concept of aligning an asymmetric topmolecule (SO2) with an optical centrifuge. Left side of the centrifuge pulse(red corkscrew surface) corresponds to its leading edge, linearly polarizedalong ~E. Behind the trailing edge of the centrifuge (right side), the molecularplane is aligned in the plane of the induced rotation. The three axes ofSO2 (a, b and c) are shown in red, green and blue, respectively. θb is theorientation angle measured in this study.forces in the rotating frame of reference pull the sulfur atom into the planeof rotation, so as to minimize the effective potential Veff = −12IΩ2 by max-imizing the molecular moment of inertia I around the rotation axis. For acentrifuged SO2 molecule rotating at 4.5× 1013 rad/s, the well depth of thiscentrifugal potential is about 0.9 eV. To produce a comparable potentialwell via the interaction with the molecular polarizability, one would needlaser powers of order of 1016 W/cm2, well above the ionization threshold ofmost molecules.Observation of the field-free rotational dynamics of the centrifuged SO2molecules was carried out in the VMI setup described in section 3.3. Asupersonic jet of helium-seeded sulfur dioxide was exposed to the field ofan optical centrifuge. Rotationally excited molecules were then Coulomb-exploded by an ultrashort intense probe pulse of either linear or circular1005.3. Adiabatic alignment of asymmetric top moleculesPPPinholeposition(a) (c)(b) CentrifugeonCentrifugeoff SO2+SO2+S3+S3+Figure 5.11: (a,b) Images of S3+ and SO2+ fragments originated from therotationally cold and centrifuged SO2 molecules, respectively.(c) Setup forthe in-plane distribution measurement of the molecular b-axis (green dashedline). PP marks the polarization plane of the lasers.polarization. The momenta of the fragment atomic and molecular ions,bearing the information about the molecular orientation at the momentof explosion, were projected onto the microchannel plate detector with aphosphor screen.Gating the MCP voltage for the arrival of either S3+ or SO2+ ions, werecorded the corresponding VMI images obtained with linear probe polar-ization, normal to the plane of the MCP. Due to the symmetry of SO2, therecoil momenta of both species lie in the O-S-O plane of the fragmentedmolecule and, therefore, define unambiguously the molecular plane at thetime of explosion. Without the centrifuge, the observed images exhibited anaxial symmetry, expected for the spherically symmetric molecular distribu-tion (Fig. 5.11 (a)). As we turned the centrifuge on, both images collapsedto the plane of the laser-induced rotation, as shown in Fig. 5.11 (b). Thesimultaneous collapse of the two distributions for both ion fragments indi-cated strong planar alignment of SO2 molecules.At high rotational energies, comparable to the kinetic energy release ofthe Coulomb explosion, extracting the degree of planar alignment from thevelocity map images in Fig. 5.11 (c) is complicated by the additional driftof the ion fragments away from the rotationally unperturbed trajectories1015.3. Adiabatic alignment of asymmetric top moleculesS3+ SO2+Figure 5.12: Illustration of the effect of underestimated and overestimatedalignment of S3+ and SO2+, respectively.(section 3.3). Ultrafast synchronous rotation of the centrifuged moleculesresults in tilting the Newton sphere of the fragment velocities with respectto the sphere of possible molecular orientations. In the case of S3+ ions,preferably ejected orthogonally to the laser field polarization[48], the drifteffect leads to the compression of the ion image towards the center. Onthe other hand, the image of SO2+, preferably ejected along the field polar-ization, is stretched outwards. This amounts to an underestimated planaralignment factor determined from the S3+ image, 〈cos2 θ〉2D = 0.64, andan overestimated value of 0.90 determined from the SO2+ channel. Thiseffect is illustrated in Fig. 5.12, where the anisotropic distributions of S3+and SO2+ recoil velocities are depicted with red ellipsoids, compressed orstretched along the probe polarization direction (axis y), respectively. Thedrift effect due to the fast molecular rotation around x axis correspondsto the effective tilting of the distribution ellipsoids around this axis. As aresult, the observed distribution along the z axis is either underestimatedor overestimated with respect to the x axis, as demonstrated by the com-pression and expansion of the green ellipses in the bottom row of Fig. 5.12,respectively.We studied the rotational dynamics of aligned SO2 within the rota-1025.3. Adiabatic alignment of asymmetric top moleculestion plane using circularly polarized probe pulses to ensure an isotropicin-plane ionization probability, similarly to the experiments with diatomicsuper rotors (section 4.2). The in-plane angular distribution of centrifugedmolecules was measured as a function of the delay time between the end ofthe centrifuge pulse and the probe-induced Coulomb explosion. The distri-bution was determined by recording the ion signal at a single location on theMCP detector as a function of the release angle of the centrifuge, θrel (seeFig. 5.11 (c)). As opposed to O2, the kinetic energy release in the Coulombexplosion of SO2 was distributed over a broader range of energies, causingthe circles of the dissociation channels to have a diffuse edge (Fig. 5.11 (a)).For this reason, the signal was measured at the center of the image, ratherthen at its boundary.With the MCP voltage gated at the arrival of S3+ ions, the detectedsignal was proportional to the amount of molecules with the orientationangle θb = 90◦ between the horizontal axis and the molecular b axis (theone bisecting the O-S-O bond angle, Fig. 5.11 (c)), f bθrel(θb = 90◦). Aswas shown in section 4.2, the distribution f bθrel(θb = 90◦) is equivalent tof b0(θb = 90◦ − θrel). In what follows, we investigate the latter distribution[hereafter referred to as f(θrel)] as a function of the centrifuge-to-probe delaytime.The observed time dependence of f(θrel) is shown Fig. 5.13 (b). Trun-cating the centrifuge pulse in time enables us to control the final angu-lar frequency of the centrifuged molecules, which in this case was set to1013 rad/s. Due to the centrifuge-induced planar alignment, the number ofthe rotational degrees of freedom of SO2 molecules was reduced to one andtheir dynamics became periodic, similar to the dynamics of a linear rotor.This periodicity, though not apparent from the coarse two-dimensional scan,is clearly evident from the extracted alignment factor, β2D = 〈cos2 θ〉2D − 12(with 〈..〉2D being the in-plane average and θ being measured from the axis ofmaximum, for a given delay, alignment), plotted in Fig. 5.13 (a). Each peakcorresponds to the field-free three-dimensional alignment of SO2(FF3DA).We note, that although the observed two-dimensional alignment factor inthe plane of rotation is not much higher than its isotropic value of 0.5, thestrong molecular confinement to this plane results in rather high values ofFF3DA. Using the lower and upper bounds of the achieved planar alignmentextracted from the two VMI channels (as discussed above), we arrive at thecorresponding estimates of 〈cos2 θb〉 of 0.40 and 0.49. These values are com-parable with the theoretical predictions for other methods of non-adiabaticthree-dimensional alignment[105, 127].1035.3. Adiabatic alignment of asymmetric top moleculesThe relatively small degree of the observed FF3DA stems from the adi-abatic mechanism of the centrifuge spinning, which results in a low numberof quantum rotational states in the excited wave packet. As we have shownin section 4.2, a small number of participating states gives rise to long win-dows of classical-like rotation around the time of each revival, with therotational frequency equal to the terminal frequency of the centrifuge. Thesame classical-like rotation was observed here with SO2 molecules when weperformed a fine time scan around any of the alignment peaks, as demon-strated in Fig. 5.13 (c).Trev2 (a)(b)(c)100 50 100 150 200 2500001 2 3 4 5 6 7Delay time, ps 0.95 1.00 1.05Figure 5.13: (a) Periodic revivals of the calculated two-dimensional align-ment factor β2D = 〈cos2 θ〉2D − 12 , determined from (b) Time evolution ofthe molecular in-plane angular distribution. (c) High resolution time scanaround the alignment peak.As expected for the effectively one-dimensional rotation, the alignmentpeaks are separated by half the revival period Trev = 2pi~(d2E/dJ2)−1,where E(J) is the rotational energy for a given rotational quantum number Jand ~ is the reduced Plank’s constant [5, 111]. Using the rigid rotor’s energy1045.3. Adiabatic alignment of asymmetric top moleculesAngular momentum J Rotational energyFigure 5.14: Schematic rotational spectrum of asymmetric top molecules.spectrum E(J) = ~22I J(J+1), with I being the molecular moment of inertia,one finds Trev = 2piI/~. At lower centrifuge frequencies, the experimentallydetected revival period of 57 ps corresponds to I = 58 A˚2 · amu in goodagreement with the known value for the molecule’s largest moment of inertia(Ic) around the axis normal to its plane (c-axis) [46].To better understand the mechanism of revivals, it is useful to first de-scribe the rotational spectrum of an asymmetric top molecule. The spectrumof a classical asymmetric top is shown in Fig. 5.14 by the light blue area.For any angular momentum J , the energy of the top can take any value be-tween the values, corresponding to the rotation around “heavy” (bright blueline) and “light” (purple line) axes. In a quantum mechanical system, theangular momentum is quantized and can only take values of integer numberof ~. For each allowed J , there are now only (2J + 1) allowed energy levels,shown in Fig. 5.14 by thin red lines, that are enumerated by a new quantumnumber τ , called the asymmetric top quantum number. For a given J , τtakes integer values from −J to J , with τ = −J corresponding to the lowestenergy state, and τ = J – to the highest one. Due to the quantum-classicalcorrespondence, the energies of high states | J, τ = −J 〉 are expected to beclose to the classical limit J = CJ(J + 1), corresponding to the blue line.The observed revivals, known as C-type transients[55], can be attributedto the beating between a few such states, populated by the centrifuge and1055.3. Adiabatic alignment of asymmetric top moleculeslabeled with thick red lines in Fig. 5.14.Angular frequency, rad/psRevivaltime,psRotational quantum number JExperimentUp to quadratic terms PESa.u.Up to quartic terms PES00 40Delay, ps800202001010050607080901001101203030040400Figure 5.15: Revival period as a function of the rotational frequency of SO2,with the rotational quantum numbers shown along the upper horizontalaxis. Experimental data (blue circles) are compared with the results ofclassical calculations, in which the potential energy surface is expanded tosecond and fourth order in deformation coordinates (dashed and solid lines,respectively). The inset shows the dependence of the 2D alignment β2D onthe pump-probe delay, used for calculating the revival time at two angularfrequencies, 18.8 rad/ps (magenta) and 39.3 rad/ps (green), labeled withthe correspondingly colored stars in the main plot. Solid lines representexperimental data, while the dashed lines show the fitting functions used toextract the revival period.At higher angular frequencies, the rising centrifugal forces stretch andbend the molecule, distorting the two S-O bonds and the angle betweenthem. This causes the moment of inertia, and hence the revival period, toincrease, as evident from the experimental data in Fig. 5.15 (blue circles).To describe the effect of the centrifugal distortion on the revival period, wecarried out the following classical calculations. For a given rotational state1065.3. Adiabatic alignment of asymmetric top moleculesJ , we minimized the total energy:E(J, r1, r2, α) =~2J22Ic(r1, r2, α)+ V (r1, r2, α),over the S-O bonds lengths r1,2 and O-S-O angle α. The potential energysurface (PES) V of SO2 was expanded in Taylor series up to the fourth orderin deformations δxi = r1− re, r2− re, α−αe, where re and αe are the valuesof r1,2 and α for the molecule at rest:V (r1, r2, α) = V (re, re, αe) +12∑ijfijδxiδxj+16∑ijkfijkδxiδxjδxk +124∑ijklfijklδxiδxjδxkδxl,with the corresponding force constants f taken from Ref.88. We then de-termined the classical angular velocity and the revival period for a given Jasω(J) = [E(J + 1)− E(J)] /~andTrev = 2pi~ [E(J − 1)− 2E(J) + E(J + 1)]−1 ,respectively. The results of these calculations are shown in Fig. 5.15. Ex-panding PES to second order in deformations δxi failed to explain the ex-perimental observations at frequencies higher than 3 × 1013 rad/s (dashedline). The quartic expansion, on the other hand, proved sufficient (solidline). At the highest achieved rotational frequencies of 4.4× 1013 rad/s, thecalculated bending angle and bond stretching reached 10◦ and 10 pm, re-spectively. Both bonds were found to stretch by an equal amount, as couldbe anticipated from the symmetry of the system.107Chapter 6Collisional decay ofrotational excitation in densemediaPrevious chapters concentrated mostly on the implementation of the rota-tional control in molecules, i.e. the generation and characterization of superrotor states with an optical centrifuge. In this chapter we will discuss howthese extreme rotational states decay due to inter-molecular collisions in adense gas medium.Recent developments in controlling molecular rotation with nonresonantlaser pulses[30, 94] have stimulated active research on the exchange of energybetween a rotating molecule and its environment. Transient molecular align-ment has been first proposed[102] and later implemented as a powerful probeof collisional relaxation in a series of pioneering experiments[50, 58, 129].The three relaxation steps, associated with (i) rotational decoherence, (ii)rotational reorientation and (iii) rotation-translation (RT) thermalization,have been identified in the theoretical model[42], yet found to overlap intime too closely to enable an individual experimental study of each processseparately[58].Rotational decoherence in dense gaseous media is important for the fun-damental understanding of the dissipative properties of gases, as well as inthe practical aspects of thermochemistry and combustion research[89, 90,109, 123]. One of the most interesting aspects of the collision-induced ro-tational decoherence is the question about its dependence on the speed ofmolecular rotation and temperature. From the very first experimental workson the topic [13, 98], it was suggested that the rate of rotational relaxationshould drop with increasing rotational quantum number J , i.e. that fastermolecular rotors are more robust with respect to collisions. In thermal en-sembles, both the rotational energy of the accessible states as well as thecollision energy scale equally with temperature. Hence, reaching higher val-ues of J by means of heating up the gas sample does not allow extending1086.1. Effects of ultrafast rotation on collisional decoherencethe study of decoherence rates to a new regime, where the rotational energyexceeds the collisional energy. In section 6.1, we describe how an optical cen-trifuge allowed us to reach this regime, controlling the angular momentumexcitation separately from the gas temperature.Ultrahigh values of the molecular angular momentum J , provided by thecentrifuge, together with the propensity of collisions to conserve its orienta-tion [42] resulted in the “gyroscopic stage”[62], which outlived the rotationalcoherence by a few nanoseconds. We studied the dynamics of this relax-ation step by tracking an optical birefringence of the centrifuged gas in thedirection perpendicular to the direction of the centrifuge. This transversebirefringence stemmed from a strong permanent confinement of molecularsuper rotors in the plane of their rotation, already discussed in the previouschapter. In the longitudinal direction along the centrifuge, the change inrefractive index owing to the fast unidirectional molecular rotation createda refractive “gyroscopic channel”, which is described in section 6.2.Combining the longevity of the gyroscopic stage with the axial alignmentprovided by the 2D centrifuge approach, demonstrated in section 5.1, allowedus to create a long lived permanent alignment in the ensembles of diatomicmolecules, discussed in section 6.3.The final stage of rotational relaxation, the RT thermalization, leads toa local heating of the gas followed by the formation of an acoustic wave[15,59, 65, 106, 141]. Previous studies of rotational photo-acoustics were basedon the impulsive excitation of molecular rotation by a single femtosecondlaser pulse or a short series of up to four pulses. These methods were limitedto relatively low levels of rotational excitation with molecules gaining onlya few units of angular momentum. At this limit, the amount of rotationalenergy deposited by the laser field is rather hard to control. As a result,the direct connection between the rotational energy and the amplitude ofthe sound wave has not been established beyond the perturbative limit. Byutilizing optical centrifuge and a state-resolved Raman detection, we wereable to measure the rotational energy of molecules and correlate it with theindependently measured acoustic wave intensity. This study is described insection 6.4.6.1 Effects of ultrafast rotation on collisionaldecoherenceThe decrease of the collisional decoherence rate with the angular momentumN is expected from the intuitive “exponential-gap law” (EGL) according to1096.1. Effects of ultrafast rotation on collisional decoherencewhich the decay rate decreases as exp[−∆EN/kBT ] with the increasing dis-tance between the rotational levels ∆EN (here kB is the Boltzmann constantand T is the temperature of the gas). The refined version of EGL, knownas the “energy corrected sudden” (ECS) approximation and introduced byDePristo et al.[22], is a popular model which successfully explained a largenumber of experimental observations[12, 67, 71, 89, 90, 123].The ECS theory describes the collisional decay rate in terms of an “adi-abaticity parameter” a ≡ ωNτc = ωN lc/vc, where ωN is the frequency ofmolecular rotation, τc is the collision time, lc is a characteristic interactionlength (usually determined empirically) and vc is the mean relative velocitybetween the collision partners. Since a = 2piτc/TN (with TN being the rota-tion period), it may also be viewed as the angle, by which a molecule rotatesduring the collision process. When a  pi, the collision is sudden and theenergy transfer does not depend on N (the non-adiabatic regime). In thecase of the finite duration of collisions, i.e. a & pi, the ECS model callsfor scaling the decay rate with a N - and temperature-dependent correctionfactorΩlc,vc(N) ≡[(1 + a2/6)]−2, (6.1)with ‘1/6’ being specific to R−6 interaction potentials. In this regime, withincreasing a collisions become more and more adiabatic.We utilized an optical centrifuge for studying collisional decoherence asa function of the degree of rotational excitation. By spinning molecules inextremely broad range of N numbers, we observed more than an order-of-magnitude change in the decoherence rates. Since in our case the controlof the molecular angular momentum is executed separately from changingthe gas temperature, we were able to explore the molecular dynamics at thecross-over between the non-adiabatic and adiabatic regimes of collisionalrelaxation, testing the ECS model beyond the adiabaticity threshold.For this experiment, we used the Raman spectroscopy setup of section 3.1with a narrowband probe light (0.1 nm FWHM). The gas chamber was filledwith nitrogen or oxygen at room temperature, and the Raman spectrum wasrecorded as a function of the delay between the centrifuge and probe pulses.An example of the experimentally detected Raman spectrogram of nitrogenis shown in Fig. 6.1. The created wave packet consisted of a large number ofodd and even N -states, corresponding to para- and ortho-nitrogen, respec-tively, whose 1:2 relative population ratio explains the observed alternationof amplitudes.The intensity of each Raman line is proportional to the square of the ro-tational coherence ρN,N+2, while the frequency shift is equal to the frequency1106.1. Effects of ultrafast rotation on collisional decoherenceD ta file 'mean130920-163634.dat'0 200 400 600 800390392394396398Time delay, t (ps)Wavelength (nm)390 391 392 393 394 395 396 397 39801000200030004000500060007000Wavelength (nm)605040302010Rotational quantum numberFigure 6.1: Experimentally detected Raman spectrogram of nitrogen show-ing the rotational Raman spectrum as a function of the time delay betweenthe beginning of the centrifuge pulse and the arrival of the probe pulse.Color coding is used to reflect the signal strength. Tilted white dashed linemarks the linearly increasing Raman shift due to the accelerated rotation ofmolecules inside the 100 ps long centrifuge pulse. A one-dimensional crosssection corresponding to the Raman spectrum at t = 270 ps is shown inwhite.difference ωN,N+2 between the rotational levels separated by |∆N | = 2. Asone can see in Fig. 6.1, Raman lines corresponding to higher values of an-gular momentum decayed slower than those with lower N ’s. The decay washappening on the time scale of hundreds of picoseconds, much longer thanthe duration of our probe pulses (≈ 4.5 ps). This offered the possibility toanalyze the decay of rotational coherences with both state and time resolu-tion. We plot the time dependence of the intensity of 50 Raman peaks ona logarithmic scale in Fig. 6.2 (colored dots). The observed decay of eachRaman line is well described by a single exponential decay, in agreementwith a simple theory of decoherence due to random binary collisions.The dependence of the exponential decay rate on the rotational quan-tum number at room temperature is shown by blue triangles in Fig. 6.3.Expressed in the often used units of Raman linewidth (cm−1 atm−1), it was1116.1. Effects of ultrafast rotation on collisional decoherenceDecay data from Mar.13, 2013. '20140313-0.75atm-21C.zip' 200500800110014001700 102030405060-2-10Logarithm of Raman signalFigure 6.2: Logarithm of the intensity of the experimentally measured Ra-man lines as a function of time (normalized to 1 at t = 200 ps, chosen soas to avoid the effects of the detector saturation at earlier times). Dotsof the same color represent experimental data for one particular value ofthe rotational quantum number. Black solid lines show the numerical fit tothe corresponding exponential decay. Data collected at P = 0.75 atm andT = 294 K.calculated as[67, 90]:ΓN = (2picτN )−1 , (6.2)where τN is the exponential decay time of the Raman signal correspondingto the transition between states N−2 and N (right vertical scale in Fig. 6.3,and c is the speed of light in cm/s. Note that the units of (cm−1 atm−1)indicate linear dependence of the decay rate on pressure, exactly as expectedfrom a linear dependence of the collision rate on gas density. This linearrelationship has been confirmed in our experiments conducted at variouspressure values ranging from 0.5 to 1.5 atm.The observed decoherence rate dropped by more than an order of mag-nitude with the angular momentum of nitrogen molecules increasing fromN = 8 to N = 66. At the lower end of the scale, N < 20, which can beaccessed at room temperature without the use of the centrifuge, our resultswere in good agreement with the known data from thermal ensembles[90],shown in Fig. 6.3 by black squares. At N = 10, the decay rate corresponded1126.1. Effects of ultrafast rotation on collisional decoherence10 20 30 40 50 60Rotational quantum number, N0102030405060708090100Raman linewidth, (cm-1atm-1) 10-3T = 294 KT = 503 K 5312651771331068876665953Raman decay time, (psatm)Figure 6.3: Decay rate of rotational coherence of N2 as a function of the rota-tional quantum number at two different temperature values, T = 294 K (bluetriangles) and T = 503 K (red circles), expressed as the Raman linewidth.Solid curves correspond to the prediction of the simplified “energy correctedsudden” (ECS) approximation at N > 12. Black squares represent the datafrom Ref.90.to the exponential time constant of 62 ps, whereas it grew to 664 ps atN = 66. Since the intensity of Raman signal scales as the square of therotational coherence, this yielded the coherence lifetime of 1.33 ns, or theequivalent of about 10 collisions. In contrast to “slow rotors” whose dynam-ics are altered by a single collision, super rotors are much more resilient tocollisional relaxation.We start the analysis of the observed behavior by first noting that ourexperimental technique cannot distinguish between the elastic and inelas-tic mechanisms of rotational decoherence. The latter mechanism, how-ever, has been suggested as the main contributor to the decay of rotationalcoherence[42, 121]. It is therefore instructive to examine the utility of theenergy corrected sudden approximation, which gives the recipe for calculat-ing the rate of inelastic N -changing collisions, in describing our experimentalfindings. In ECS, the rate of transition from N to N ′ is given by the follow-1136.1. Effects of ultrafast rotation on collisional decoherenceing expression:γescN,N ′ = (2N′ + 1) exp(EN − EN>kBT)×∑L(N N ′ L0 0 0)2(2L+ 1)Ωlc,vc(N)Ωlc,vc(L)γL0, (6.3)where EN is the rotational energy (with N> denoting the largest value be-tween N and N ′), (:::) is the Wigner 3J symbol, and Ωlc,vc(N) is the cor-rection factor introduced earlier. Given that the basic rate γL0 is known tofall off rather steeply with L [12], we simplify Eq.6.3 by leaving only thesingle dominant term with L = 2 in the sum. Because of the exponentialgap factor in the first line, we also assume that the main contribution tothe decay of rotational coherence comes from the downward transitions withN ′ = N − 2. Noticing that the corresponding 3J symbol scales as 1/√N atlarge N , we arrive at the following simplified decay rate:ΓN1 ≈ γescN,N−2(N1)≈ 15γ208Ωlc,vc(2)Ωlc,vc(N) ≡ Γ0Ωlc,vc(N), (6.4)where all the factors which are independent on N have been included in Γ0.With only two fitting parameters, Γ0 and lc, this simple expression resultsin a reasonably good agreement with our experimental observations in theregion of high N values. This is demonstrated by fitting the data usingEq.6.4 at N > 12, i.e. above the visible bend in the curve predicted bythe full ECS model and observed here similarly to a number of previousreports[67, 89, 90]. The fit (solid blue line in Fig. 6.3) corresponds to thecharacteristic interaction length lc = 0.74 ± 0.03 A˚, in excellent agreementwith the previously reported value of 0.75 A˚[89, 123].To test the proposed simplified scaling law, we repeated the experimentat a higher temperature. A small heater was placed inside the chamber toheat the gas up locally to T = 503 K. The measured decay rates are shownwith red circles in Fig. 6.3. Importantly, fitting the high-temperature datafor N > 12 with Eq.6.4 yields the same characteristic length lc = 0.74 A˚,confirming the validity of the model. In accord with all previous observationsof thermally accessible N levels, increasing the temperature results in aslower decay rate, primarily due to the reduced gas density. Increasing thevalue of N leads to the similar decrease of ΓN , this time due to the growingadiabaticity of collisions. However, as can be seen from Fig. 6.4, the colderthe ensemble the faster the growth of the adiabaticity factor a with N . This1146.1. Effects of ultrafast rotation on collisional decoherence1100 K294 K503 K0 10 20 30 40 50 60 700123456Rotational quantum number, NAdiabaticity factor,aFigure 6.4: The dependence of the adiabaticity factor a onN for the nitrogengas temperatures of 294 K (blue line), 503 K (red line) and 1100 K (greenline). Solid horizontal line marks the adiabaticity threshold, a = pi. Blackfilled circles mark the rotational evels accessed in previous experiments onrotational decoherence.fact explains the observed reversal of the temperature dependence betweenN ≈ 35 and N ≈ 50, where the molecules in a hot (and therefore moredilute) ensemble lose their rotational coherence faster than the denser coldones. In this window of N ’s, the value of a is already above the adiabaticitythreshold at T =294 K, yet still below it at T =503 K.Black circles in Fig. 6.4 mark the highest observed states of nitrogen atT = 300, 500 and 1100 K (also corresponding to the rotational levels con-taining 0.2% of the total rotational population, which seems to represent thesmallest fraction of molecules, detectable in a typical Raman experiment).One can see that, although the range of N values can be extended by in-creasing the temperature, the maximum accessible adiabaticity parameterremains constant and rather low. Controlling molecular rotation with anoptical centrifuge eliminated this limitation and enabled us to cross over toand explore the adiabatic regime of rotational decoherence.Similar measurements were carried out in oxygen. Since the observedtime dependencies was complicated by the spin-rotational dynamics (sec-tion 4.3), the fitting procedure to extract the time constant τN of the1156.1. Effects of ultrafast rotation on collisional decoherencecollision-induced exponential decay was modified to account for the SR oscil-lations. Similar to nitrogen, for the slower rotating molecules, the coherencelife time is shorter than for the faster rotors. This is shown with blue circlesin Fig. 6.5, where the decay rate is plotted as a function of the angular fre-quency of molecular rotation. Black asterisks and grey squares depict thepreviously reported data obtained in a thermal ensemble of oxygen moleculesat room temperature (hence, N 6 25) using frequency-resolved[92] and time-resolved[90] detection techniques, respectively. Our results at low N ’s showsatisfactory agreement with both data sets.0 20 40 60 80 1000 10 20 30 40 50 60 70 8001020304050607080901000 1 2 3 4 5 6 7 8 9 105312651771331068876665953Raman line width, (cm-1atm-1) 10-3Raman decay time, (psatm)(THz)(O2)(N2)Rotation frequency (THz) and rotational quantum numbers of O2 and N2Thermal oxygen, Miller et al.Centrifuged oxygenCentrifuged nitrogenThermal oxygen,  Millot et al.Figure 6.5: The decay rate of rotational coherence in oxygen (blue circles)and nitrogen (green triangles) as a function of the frequency of molecularrotation. For convenience, rotational quantum numbers of O2 and N2 areshown below the frequency axis, and the decay times τN are shown on theright vertical axis. Black asterisks and grey squares depict the data from[92] and [90], respectively, where the rotational decay has been studied inthermal oxygen by two different techniques. Solid red (dashed black) curveshows the result of fitting the adiabaticity correction factor Ωlc(N) to thedata for oxygen (nitrogen) as discussed in text.We repeated the fitting procedure as before, to extract Γ0 and lc param-eters of the ECS model. Unlike nitrogen, an additional fitting parameter1166.2. Crossover from gyroscopic to thermal motionΓ∞ was required to achieve a good agreement with experiment:ΓN = Γ0Ωlc(N) + Γ∞. (6.5)Red solid curve in Fig. 6.5 shows the result of fitting Eq.6.5 to our data in therange of high adiabaticity (aN > 1). Good fit was achieved with the valuesΓ0 = 76.9×10−3 cm−1 atm−1, Γ∞ = 19.6×10−3 cm−1 atm−1 and lc = 0.56A˚. The latter is close to the value found from applying the ECS model inthe limit of low angular momentum[123], which may suggest a possibility ofextending the known scaling law to the limit of high adiabaticity using thesame correction factor.The necessity of adding an additional parameter Γ∞ for oxygen maypoint at a non-negligible contribution of the spin-flipping collisions in the de-cay of coherent Raman scattering in the gas of magnetic O2 molecules. Thecross section for the spin relaxation of oxygen at room temperature is notknown and its exact calculation is a formidable task. However, an insight canbe gained from the recent studies of this process at low temperatures[4, 125].As the temperature exceeds ≈ 0.01 K and approaches 1 K, the ratio betweenthe elastic and spin-flipping collisions has been found to reach a constantof order 10, which shows little dependence on T . When the temperaturerises above 1 K, the energy scale of the spin-rotation splitting becomes lesssignificant and no dramatic changes in the elastic-to-inelastic cross sectionratio are expected. Given this argument, the spin-flipping cross section mayindeed be somewhat below, yet comparable to, the cross section of the ro-tational relaxation at low N values.Alternatively, faster decoherence of oxygen rotation may stem from thehigher concentration of O2 super rotors with respect to the concentrationof the centrifuged N2. This empirically found difference may result in theincreasing local gas temperature and correspondingly higher rates of rota-tional energy transfer.6.2 Crossover from gyroscopic to thermal motionAs the super rotors underwent collisional relaxation, their rotationalenergy was released, driving the temperature of the gas up. The rotation-translation energy transfer was accompanied by a crossover from the non-equilibrium gyroscopic to the thermal phase of molecular dynamics. Hydro-dynamics of a locally heated gas result in the corresponding change in itspressure, followed by the emission of an acoustic wave and the formation of1176.2. Crossover from gyroscopic to thermal motiona low-density thermal channel. In this section, we describe our experimentaland numerical investigation of this process.Optical CentrifugeData 2015022303. Analysis3.m. Delay 2000 nsData 2015022303. Analysis3.m. Delay 1500 ns200 mData 2015021601. Analysis4.m. Delay 1400 nsData 2015040901. Analysis7.m. Delay 120 ps200 mData 2015041804. Analysis7.m. Delay 23 ns120 ps200 m23 ns200 m1.4 s2.0 sgas cell FLP, 45KEFLns or ps probeDM ILLP, ‐45fEDelayCCDCCD~y(b)(c) (d) (e)(a)Figure 6.6: (a) Schematic diagram of the imaging setup. Nanosecond orpicosecond probe pulses (green and blue beams, respectively) propagatedeither collinear with, or perpendicular to the centrifuge, creating an imageof the rotationally excited volume of gas on a CCD camera either in thelongitudinal or transverse direction, respectively. DM: dichroic mirror, FL:focusing lens, IL: imaging lens, KE: knife edge (shown along y instead of zaxis for clarity), LP: linear polarizers at ±45◦ to y axis, F: frequency filter.An example of the longitudinal image with a circularly expanding soundwave is shown in panel (b). Images in the transverse geometry were takenat early (c), intermediate (d) and late (e) time moments. Image (c) wasrecorded with the two linear crossed polarizers in place. Schlieren image (e)was recorded with the knife edge in place.Our experimental configuration is depicted in Fig. 6.6. The centrifugebeam was focused by a weak 1 m focal length lens down to a spot sizeof 90 µm diameter (full width at half maximum) inside the Raman cham-ber filled with 0.9 bar of oxygen. The performance of the centrifuge wasmonitored by means of coherent Raman spectroscopy, described earlier.We imaged the volume of the centrifuged gas both on short and longtime scales, τ < 23 ns and τ < 2 µs, respectively. For a short-time scan,1186.2. Crossover from gyroscopic to thermal motionpicosecond pulses were extracted from the same Ti:Sapphire ultrafast sys-tem as the centrifuge, and delayed with femtosecond precision by meansof computer controlled motorized translation stages. Long-time scans wereexecuted with nanosecond pulses from a separate YAG laser, while theirdelay was controlled electronically. In both cases, single-shot images wererecorded with a CCD camera and typically averaged over 400 laser pulses.Two imaging geometries were implemented: longitudinal and transverse,with probe pulses propagating collinear with, or at 90 degrees to the direc-tion of the centrifuge. A sample longitudinal image, taken 2 µs after thecentrifuge pulse, is shown in the upper right corner of Fig. 6.6. The loca-tion of the central bright spot, surrounded by multiple circular interferencefringes, corresponds to the position of the centrifuge beam. The fringe pat-tern stems from the diffraction of probe pulses inside the centimeter-longdensity depression channel. Owing to the linearity of the weak probe prop-agation and its extended length, the fringe contrast serves as a sensitiveindicator of the centrifuge-induced changes in the refractive index of thegas, ∆n.We analyzed the time dependence of ∆n by recording a series of longitu-dinal images while scanning the delay time between the centrifuge and probepulses. An angle-averaged radial cross-section of the interference pattern,recorded with the nanosecond laser as a probe, was calculated and plottedas a function of the delay in Fig. 6.7. The plot shows a permanent centralcore and an outgoing sound wave, similar to the recently observed dynamicsof plasma filaments[75, 133]. Unlike the case of plasma, however, zoomingin on the first few nanoseconds of the filament formation reveals unique de-tails of its early history, shown in Fig. 6.8. The instantaneous creation of arefractive channel by the centrifuge, reflects its non-thermal origin. Indeed,we detected the signal already in the first picoseconds after the arrival of acentrifuge pulse - too short a time scale to allow collisional thermalization.Instead, the collisions seem to suppress, rather than enhance ∆n, whichdisappears almost completely at around 7 ns. From that moment on, thechannel grows back and eventually emits the sound wave seen in Fig. 6.7, inaccord with the expected thermal dynamics. A clear crossover between thetwo regions is illustrated in panel (b) of Fig. 6.8, where we plot the contrastof the recorded longitudinal images as a function of time. The universalityof the crossover is demonstrated in Fig. 6.9, where it is observed in the gas ofN2O super rotors at two different pressures. Collision-driven dynamics areevident from the increasing lifetime of the gyroscopic stage with decreasingpressure. To identify the nature of the refractive channels before and afterthe crossover, we switched to the transverse imaging geometry.1196.2. Crossover from gyroscopic to thermal motion0 500 1000 1500 2000 2500Time (ns)-800-4000400800Position,r(μm )Figure 6.7: Radial cross-section of the images, representing the change in therefractive index of the gas, taken along the centrifuge beam, as a functionof time, recorded on a long timescale.Three typical transverse images, recorded on three qualitatively differ-ent time scales, are presented at the bottom of Fig. 6.6. The schlierentechnique[114] was employed by inserting a knife edge in the focal plane ofthe imaging lens, to determine the true distribution of the refractive index(more exactly, its d/dy derivative) at long delay times. The side view ofa narrow channel is free of interference fringes, seen in the longitudinal ge-ometry, which greatly simplifies the interpretation of the picture. Schlierenimage (e) on the right shows two high-density waves captured 400 µm aboveand below the low-density depression channel in the center. The snapshot(d) in the middle was taken close to the very origin of the thermal channelaround τ ≈ 20 ns, before the beginning of the hydrodynamic expansion ofthe gas. At even shorter delay times of order of, and below, 5 nanosec-onds, our transverse phase-contrast imaging approached its signal-to-noiselimit. To further explore the molecular dynamics at these early moments, weprobed the centrifuge-induced birefringence of the gas by taking the images1206.2. Crossover from gyroscopic to thermal motion0 5 10 15 20Position,r(μm)-400-2000200400Contrast (arb.u.)00.40.8(a)(b)O2 P = 1.0 atm200 ps 20 nsGyroscopic - ThermalCrossoverTime (ns)Figure 6.8: (a) Radial cross-section of the images, taken along the centrifugebeam, as a function of time, recorded on a short timescale. The arrow ataround 7 ns shows the crossover between the rotation-induced and thermalchannels. Sample images of both refractive channels are shown in the squareinsets. (b) Image contrast as a function of time. The red line on top of theexperimental data points is shown to guide the eye.between two crossed linear polarizers (leftmost picture (c) at the bottom ofFig. 6.6).The difference between the two processes affecting optical properties ofthe centrifuged gas before and after the observed crossover became apparentwhen we compared the time dependence of the detected optical birefringencewith that of the image contrast. The two observables, extracted from theimages taken with and without the crossed polarizers, respectively, are plot-ted in Fig. 6.10. The birefringence signal is expressed in arbitrary unitswith 1 being equivalent to the polarization rotation angle of about 1 mrad.This birefringence, instantaneously induced by the centrifuge, falls off ex-ponentially with a time constant of 3.4 ns (solid black curve). From theseresults, the disappearing channel to the left of the crossover in Fig. 6.8 can1216.2. Crossover from gyroscopic to thermal motionTime (ns)3 5 7 9Contrast (arb.u.)00.40.81 P = 0.50 atmP = 0.25 atmN2OFigure 6.9: Radial cross-section of the images, taken along the centrifugebeam, as a function of time recorded in the gas of N2O molecules at thepressure of 0.5 atm (red line) and 0.25 atm (blue line). Shaded regionsaround each line represent the statistical error (one standard deviation) inour experimental data.be clearly correlated with the decaying birefringence signal, and thereforeassociated with an optical anisotropy induced by the directional molecularrotation. As the molecular super rotors lose their rotational energy to heat,this gyroscopic channel dies off while thermal effects take over, causing anisotropic change in the gas density reflected in the growing image contrastat τ & 7 ns.Note that the longitudinal imaging is more sensitive to the small changesof the refractive index at early times, when it is changing monotonicallyacross the refractive channel, and not as sensitive to its further growth lateron, when the developing fine structure (due to the formation of an acousticwave) results in multiple diffraction fringes. This explains the fact that thelongitudinal image contrast before and after the crossover in Fig. 6.8 (b)is of the similar magnitude, unlike the case of the transverse contrast inFig. 6.10, which disappears below the cross-over time. To make sure thatsuper rotors are responsible for the observed optical changes, we repeatedthe experiments with the centrifuge pulses of the same peak intensity butlower terminal rotation frequency. Two points inside the dashed rectanglein Fig. 6.10, upper square and lower diamond, correspond to the centrifugeproducing only slow rotors and no rotors at all, respectively.To verify the mechanism behind the rotation-translation crossover, wenumerically simulated the hydrodynamics of the ideal gas exposed to a1226.2. Crossover from gyroscopic to thermal motionAnalysis of data from Apr.9 and Apr.18 using HyFit to exponent for t>100 ps: =3.4 ns2015041820 (One arm, same power as full)2015041822 (Misaligned centrifuge, full power)2015040901 2015041801-18Hydrocode calculation. r0=50 m, dT=140 K, =Data 2015041804. Analysis7.m. Delay 23 nsData 2015040901. Analysis7.m. Delay 120 ps0 5 10 15 200.2.4.6.81Birefringence signal (arb. u.)0.2.4.6.81Image contrast,102Time (ns)Figure 6.10: Birefringence signal (red triangles) and image contrast (bluecircles) of the centrifuged gas, retrieved from the transverse images takenwith and without the crossed polarizers (left and right insets, respectively)as a function of time. Two data points in the lower right corner (within thedashed rectangle) indicate the drop in the phase contrast for the slower ro-tating centrifuge. The solid black line is a fit to an exponential decay of thebirefringence signal, which gave τb = 3.4 ns as the decay time. This timeconstant was used in the hydrodynamic numerical calculations of ∆n(τ),shown by the dashed black line. The dashed magenta and green lines cor-respond to the same calculations, performed with the decay constant beingfive times shorter and longer, respectively. The horizontal dash-dotted lineindicates the noise floor for the phase contrast measurement.known heat source, using the equations:∂/∂t(ρ) = −∇ [ρv] ,∂/∂t(p) = −∇ [pv + P ] ,∂/∂t(E) = −∇ [Ev + Pv + q] ,where ρ, p = ρv and E = 12ρv2 + ε are the mass, momentum and energy ofthe unit gas volume; v its velocity and ε = ρcvT its internal energy, withcv being the heat capacity at constant volume and T the gas temperature.P is the pressure of the gas and q = −κ∇T is the heat flux, with κ beingthe heat conductivity. These equations, corresponding to the conservation1236.2. Crossover from gyroscopic to thermal motionof mass, momentum and energy in the compressible gas flow, were solvedfor ρ, v and P , assuming cylindrical symmetry and equilibrium initial andboundary conditions. Rotation-translation energy exchange was modeledby adding an external pressure source, 1P0 (∂P/∂t)RT =∆TT01τbe−t/τb , to thecorresponding differential equation for P , where T0 (P0) is the ambient tem-perature (pressure) and ∆T is the temperature increase at the end of thethermalization process.An exponential increase of the gas temperature (and hence, its pressurein the first few nanoseconds) with the same time constant τb = 3.4 ns asthe decay of the birefringence signal, stems from the proportionality of boththe birefringence signal and the rotational energy to J2. Indeed, the bire-fringence signal is proportional to(1− 3〈cos2 θz〉)2[49], becoming non-zerowhen the molecules are confined to the xy plane (〈cos2 θz〉 < 1/3). For agas of molecules in the quantum state |J,MJ = J〉, one finds: 〈cos2 θz〉 =〈J, J | cos2 θz|J, J〉 = (2J + 3)−1. As a result, the birefringence signal is pro-portional to J2, as well as the rotational energy, and should therefore bedescribed by the same exponential τb.To estimate ∆T , we note that the peak intensity of our centrifuge pulsesis sufficient to spin adiabatically only 2% of O2 molecules occupying thelowest rotational state at room temperature. After these molecules areexcited to J ≈ 91 by the centrifuge (as determined by means of Ramanspectroscopy), each oxygen super rotor carries 1.5 eV of rotational energy.Redistributing this energy among the whole molecular ensemble and all 5degrees of freedom results in ∆T = 140 K.The main result of our hydrodynamic calculations is shown in Fig. 6.11 (a),where the change in the gas density, ∆ρ(y, τ), is plotted as a function of boththe distance from the centrifuge and the time since its arrival. One can seethe density depression channel forming in the center in the first 200 ns andan acoustic density wave spreading radially away from it. Given the lin-ear dependence of the refractive index of a gas on its density, we comparethe experimentally observed image contrast with the numerically calculated|∆ρ(y = 0, τ)| in Fig. 6.10 (blue circles and dashed black line, respectively).The only fitting parameter was the proportionality coefficient between thegas density and the image contrast. An excellent agreement between theexperimental and numerical line shapes supports our interpretation of thegyroscopic channel and its role as the energy source for the thermal gas ex-pansion. If the RT exchange rate was assumed to be very different from thebirefringence decay rate τb, the numerical results did not fit the experimentaldata as well (dashed magenta and green lines in Fig. 6.10).Further comparison between our experimental findings and numerical1246.2. Crossover from gyroscopic to thermal motionData 2015021601. Analysis4.m Hydrocode simulations. Analysis9.m. 50 um, 140K, 3.4 ns.Delay of 1200 ns. Data 2015021601. Analysis4.m‐1‐0.500.5Schlierensignal (arb.u.)(b)Calculations(d)Position, y (m)‐600 ‐400 ‐200 0 200 400 600(a)Hydrocode simulations. Analysis9.m. 50 um, 140K, 3.4 ns.Position, y (m)‐600 ‐400 ‐200 0 200 400 600Time (s)00.40.81.221.6Experiment(c)0 40001‐.1‐.2‐.3‐.40‐4002Figure 6.11: Comparison of the experimental results with the numerical cal-culations of gas hydrodynamics. The calculated change in the gas density,∆ρ, is shown in panel (a) as a function of time and distance. Experimentallydetermined parameters of the observed gyroscopic channel are used to sim-ulate the heat source, which initiates the dynamics at point (y = 0, τ = 0).In panel (b), the derivative of the calculated density profile d/dy[ρ(y)] at1.2 µs is compared with the y cross-section of the schlieren image recorded1.2 µs after the centrifuge (dashed red and solid blue curves, respectively).The dependence of the measured and calculated schlieren signals on bothspace and time is shown in panels (c) and (d), respectively, with the twolines indicating the two cross-sections displayed in plot (b).model is presented in Fig. 6.11 (b). The schlieren signal at 1.2 µs is calcu-lated by taking the derivative of ∆ρ(y, τ = 1.2 µs) with respect to y, andplotted together with the y cross-section of the experimental schlieren imagetaken at that delay time (cf. Fig. 6.6 (e)). The dependence of schlieren sig-nals on both y and τ is shown in Fig. 6.11 (c) and (d) for the experimentaland numerical data, respectively. The experimental scan, carried out withns probe pulses, extends to 2 µs and shows an acoustic wave traveling withthe speed of sound away from the central core. The equivalent numericalscan displays very similar behavior, supporting the validity of the model.1256.3. Long-lived permanent molecular alignment6.3 Long-lived permanent molecular alignmentAs was discussed in section 5.1, a 2D centrifuge can be used to align di-atomic molecules in extreme rotational states. Reaching high N states hasan immediate advantage over the conventional non-adiabatic alignment witha linearly polarized femtosecond pulse (referred to as a 1D kick). Indeed, ashas been demonstrated above, the decay of both the rotational coherence(section 6.1) and the rotational energy (section 6.2) of molecular super ro-tors due to collisions is much slower than that of slowly rotating molecules.Here, we compare the decay of molecular alignment, induced by a 1D kickand a 2D centrifuge in oxygen under ambient conditions, showing signifi-cantly longer decay times in the latter case.00.10.20.30.40.50.60.70.80.91.0500 1000 1500Time (ps)Birefringence signal (arb. units)2D, 20 THz2D, 10 THz2D, 5 THz1D kickFigure 6.12: Comparison of the birefringence decay for the 2D centrifugewith a full spectral bandwidth of 20 THz (red diamonds) and with itsbandwidth truncated at ≈ 10 THz (purple triangles) and . 5 THz (greensquares). Black circles represent the case of a 1D kick. In all plots, solidlines show the fits by exponential decays.We analyzed this decay of molecular alignment by repeating the bire-fringence measurements of section 6.2 with a 1D kick and a 2D centrifuge.In this study we used the transverse geometry, i.e. sent our probe pulsesperpendicularly to the plane of molecular rotation (xz plane in Fig.5.1). Ap-1266.4. Sound emission from the gas of molecular super rotorsplying the 2D centrifuge resulted in the nonzero linear birefringence (red dia-monds in Fig. 6.12), stemming from the rod-shaped molecular distribution.Similarly to the decoherence rate of molecular super rotors (section 6.1),the decay of the permanent molecular alignment becomes slower for fasterrotating molecules due to the increased adiabaticity of collisions[62]. Wedemonstrated this effect by comparing the decay rates of the birefringencesignal observed with the 2D centrifuges which had different spectral band-widths and, therefore, different terminal rotational frequencies. As shown inFig.6.12, increasing the centrifuge bandwidth from . 5 THz (green squares)to ≈ 8 THz (purple triangles) and 20 THz (red diamonds), resulted in therespective increase of the exponential decay time from 269±9 ps to 461±15ps and 1094± 142 ps.Also plotted in Fig.6.12 is the decay of the permanent molecular align-ment induced by a single femtosecond (1D) kick (black circles). One canclearly see the noticeably shorter life time of this alignment (187 ± 8 ps)with respect to that offered by a 2D centrifuge. Similarly to the previouslydiscussed increasing decay rates with a slower centrifuge, the effect stemsfrom the lower adiabaticity of collisions between the molecules exposed to a1D kick in comparison to those excited by a 2D centrifuge.6.4 Sound emission from the gas of molecularsuper rotorsWe utilized the tunability of the centrifuge to vary the amount of rotationalenergy deposited into the gas of N2 or O2 molecules while recording theacoustic signal produced by the super rotors. The latter was clearly audibleto the unaided ear due to the ultra-high rotational energies exceeding 2.5 eVper oxygen super rotor. The acoustic signal was detected by a microphonepositioned perpendicularly to the centrifuge beam 6 cm away from its focalspot. The sensitivity of the microphone was −31 dBV (≈ 28 mV/Pa) at1 KHz under ambient conditions, and its output was connected directly tothe computer’s analog-to-digital converter with no further amplification. Anexample of the recorded acoustic signal, averaged over 100 laser pulses, isshown in the inset to Fig. 6.13. Frequency bandwidth of the microphonewas not sufficiently broad for resolving the effectively instantaneous pressureburst, which resulted in the decaying oscillatory waveform. Hereafter, werefer to the maximum value of this waveform as the amplitude of the acousticsignal S.The acoustic response of various gases to the field of an optical centrifuge1276.4. Sound emission from the gas of molecular super rotors0 20 40 60 80 100 120 140 16000.10.20.30.40.50.60.70.80.91Rotational quantum number, NRaman signal, R(N)(arb. u)0 0.5 1 1.5 2-20-100102030Time (ms)Sound signal(mV)Super  rotorsSound amplitude, SFigure 6.13: Typical rotational Raman spectrum of oxygen super rotorsmeasured with a probe pulse delayed by 200 ps with respect to the centrifugepulse. Inset: an example of the acoustic signal recorded in centrifugedambient air.is shown in Fig. 6.14 as a function of the laser pulse energy. We first noticethe difference between the sound amplitude detected in the ensemble ofnitrogen super rotors and in pure argon gas under the same pressure andtemperature, clear from Fig. 6.14 (a). Even though the ionization energiesof the two gases are almost the same, 15.58 eV for N2 and 15.76 eV forAr[51], the acoustic signal in nitrogen is an order of magnitude stronger.This points at the non-ionizing nature of the sound waves recorded in thegas of super rotors.To examine this hypothesis further, we intentionally introduced ioniza-tion by adding a small amount of SF6 molecules into the gas of O2. Beinga spherical top, SF6 was immune to centrifuge spinning but caused the for-mation of a clearly visible plasma channel. As can be seen in Fig. 6.14 (b),when ionization was the dominant reason for the gas heating, an amplitudeof the generated sound wave showed very different energy dependence thanthat observed in pure oxygen. Note that the stronger sound signal in plasmacan be attributed to the higher number of ionized molecules in comparison tomolecular super rotors. The non-ionizing mechanism of the recorded soundis also evident from the following test. We modified the laser field in such away as to keep the direction of its vector of linear polarization constant, incontrast to the spinning polarization of the centrifuge pulse, while retain-ing the pulse duration and peak intensity. We observed that together withthe forced accelerated molecular rotation, the acoustic signal disappeared1286.4. Sound emission from the gas of molecular super rotorsbeyond our sensitivity level.0.8 0.9 1.0 1.1Pulse energy, W (log scale, arb. u.)12350.8 0.9 1.0 1.1 1.2Pulse energy, W (log scale, arb. u.)0.10.20.51Sound amplitude, (log scale, arb. u.)ArN2O2O2+SF6(a) (b)Figure 6.14: Amplitude of the recorded sound as a function of the energyof centrifuge pulses, plotted on a log-log scale. Each data set consists of10,000 points. (a) Typical acoustic signal from the centrifuged gas of nitro-gen molecules (blue diamonds) is compared to the sound generated by thecentrifuge in pure argon at the same pressure of 95 kPa (green triangles).(b) Acoustic response of the centrifuged oxygen with and without a smalladmixture of SF6 molecules (black dots and red circles, respectively). Blackdashed lines in both panels show fits to power-law scaling.Shown by black dashed lines in Fig. 6.14, an average sound amplitudefrom both nitrogen and oxygen super rotors exhibited power-law scaling withpulse energy. The exact power is not universal and typically varied between4 and 5 in our experiments with nitrogen, and between 3 and 4 with oxygen.According to the theory of laser-induced pressure waves in dense media[44],as well as numerical simulations of hydrodynamic expansion[133], the soundwave amplitude is expected to scale linearly with the amount of laser energydeposited in the sample. In the case of an impulsive rotational excitation,most of the energy is transferred via a single two-photon Raman transition,suggesting a quadratic dependence on the pulse energy[59], which has beenverified experimentally[65, 141]. In contrast, the number of molecules, whichinitially occupied a particular rotational quantum state and were spun bythe centrifuge, is proportional to the centrifuge volume. The latter can beestimated by the distance from the center of the laser beam, at which theoptical potential of the centrifuge field is higher than the initial rotationalenergy. Since the induced-dipole potential is linear with the laser intensity,1296.4. Sound emission from the gas of molecular super rotorsthe volume will grow as its third power. Larger scaling power, observedin our experiments, can be attributed to the fact that together with thegrowing centrifuge volume for each trapped rotational state, the number ofthese states might have also increased with increasing laser intensity.Because the efficiency of the centrifuge spinning depends on the molecu-lar parameters, such as the moment of inertia and the polarizability anisotropy,this scaling is different for different molecules (compare N2 and O2 in panels(a) and (b) of Fig. 6.14, respectively). The scatter of data points within eachset is due to the fluctuations of the spatial and spectral shape of centrifugepulses, which caused shot-to-shot variations in the centrifuge efficiency.0 1 2 3 4 5 6 7 8Rotational energy, Erot (arb. u.)0123450 1 2 3 4 5 6 7 8 9 10Rotational energy, Erot (arb. u.)00.20.40.60.81Sound amplitude, S(arb. u.)O2O2+SF6N2Ar 0 2 4 6 8 10xRaman spectrum0 2 4 6 8 10Raman spectrumwith Plasmax0 2 4 6 8 10Raman spectrumwithout Plasmax(b)(a)Figure 6.15: Amplitude of the recorded sound as a function of the rota-tional energy deposited in the gas sample. (a) Acoustic response from thecentrifuged gas of nitrogen molecules (blue circles) is compared to the soundgenerated by the centrifuge in pure argon at the same pressure of 95 kPa(green triangles). (b) Acoustic response from the centrifuged oxygen withand without a small admixture of SF6 molecules (black dots and red circles,respectively). All insets show Raman spectra corresponding to the datapoints marked with black crosses and plotted as a function of rotationalfrequency in THz.We eliminated the uncertainty in the degree of rotational excitation bysending a weak probe pulse behind each centrifuge pulse and recording therotational Raman spectrum simultaneously with the corresponding photo-acoustic signal. This enabled us to evaluate the amount of total rotationalenergy deposited in the molecular sample as a sum of rotational populationsPN weighted with the corresponding rotational energy EN and taken over1306.4. Sound emission from the gas of molecular super rotorsthe rotational states |N〉 populated by the centrifuge:Erot =∑NPNEN . (6.6)A crude estimate of populations PN can be obtained from the measured Ra-man spectrum. Indeed, the intensity of each peak, RN , is proportional to thesquare of the centrifuge-induced coherence ρN,N+2 between the two resonantstates |N〉 and |N + 2〉. For N ’s extending beyond the thermally populatedstates (i.e. & 20 for N2 and & 30 for O2), an optical centrifuge excites purequantum states of maximum coherence, |ρN,N+2|2 = PNPN+2. If the createdrotational wave packet is rather broad with a smoothly changing envelope,which was typically the case here, one may approximate PN ≈ PN+2 andcorrespondingly |ρN,N+2|2 ≈ P 2N . Hence, the deposited rotational energycan be estimated from the measured Raman spectrum asErot ≈∑N>Nth√RNEN . (6.7)In Fig. 6.15, the dependence of the sound amplitude on the deposited ro-tational energy, extracted from the observed Raman spectra and calculatedwith Eq. (6.7), is shown for the same experimental runs as in Fig. 6.14. Fornitrogen and oxygen super rotors (left and right panels, respectively), thescaling of the average photo-acoustic signal with Erot appears to be linear,in agreement with the hydrodynamic theory of heat transfer in dense me-dia. The smooth envelope shape of the rotational wave packets in nitrogen,shown in the inset to Fig. 6.15 (a), together with the anticipated linear scal-ing of S(Erot), confirms the validity of the approximation we used to arriveat Eq. (6.7). In oxygen, the created wave packets are typically not as broadand smooth as in nitrogen (lower inset in Fig. 6.15 (b)), but the observedscaling is still linear with Erot.To emphasize the difference in the acoustic response in the presenceof plasma, the same plot shows the experimentally detected distributionS(Erot) in the mixture of O2 with SF6. Whenever the plasma channel isformed, the wavefront of the centrifuge beam is perturbed, which leadsto the loss of coherent rotational excitation and the disappearance of su-per rotors, as demonstrated by the Raman spectrum in the upper inset toFig. 6.15 (b). An increase in the sound loudness is reflected by the ob-served anti-correlation between the two quantities. We also verified thatincluding the low-frequency part of the Raman spectra, corresponding tohot molecules lost from the centrifuge, did not affect the observed linear1316.4. Sound emission from the gas of molecular super rotorsscaling, most probably due to the small amount of rotational energy carriedby those low-N rotors.132Chapter 7OutlookOptical centrifuge proved a remarkably powerful tool for controlling molec-ular rotation and studying molecules in previously inaccessible extreme ro-tational states. In this last chapter I will give a brief outlook on furtherdevelopments and applications of the centrifuge technique and propose afew future experiments.7.1 Generation of THz radiationThe flexibility of the centrifuge excitation, which allows one to control therotational frequencies of molecules in a broad range from 0 to 10 THz,suggests a possibility for a novel way of generating tunable THz radiation.Indeed, a classical rotating dipole radiates at the frequency of its rotation.Quantum mechanically, by creating an isolated rotational wave packet (suchas the one reflected by the Raman spectrum in Fig. 6.13), the centrifuge maynot only populate high rotational states, but also create population inversionbetween them. Provided the transitions between the rotational levels areallowed, as they are in polar molecules, the system may serve either as aTHz amplifier, or even a source of THz superradiance. The frequency of thisTHz radiation, determined by the rotational frequency of molecules, will becontrollable with the centrifuge.7.2 Transient magnetization of paramagneticgasesAs was shown in sections 4.4 and 5.2, paramagnetic super rotors precessaround the direction of external magnetic field, provided that the fieldstrength is not too high. In this regime, the electronic spin S stays cou-pled to the angular momentum N, which makes the precession frequencyΩ depend on the sign of the projection SN , as indicated by Eq.(4.7). Itis this peculiar dependence on SN = 0,±1 that explains the splitting of1337.3. Field-free alignment of asymmetric top moleculesNSN=-1SN=0Time0 TLarmor/4TLarmor/8BSN=1SFigure 7.1: Precession of the angular distributions (blue disks) of param-agnetic spin-1 super rotors in magnetic field. Different rows correspond todifferent spin projections SN . Blue and green arrows represent the angularmomentum and spin vectors, respectively.an isotropic molecular distribution into three parts, observed in our ex-periments and discussed in section 4.4. As illustrated in Fig. 7.1, the twocounter-rotating discs, corresponding to SN = ±1, meet at exactly one quar-ter of the Larmor period. At this time, the two initially opposite projectionsof the electronic spin become the same, causing a macroscopic magnetizationof the gas. Notice, that no initial spin polarization of molecules, but rathertheir centrifuge-induced planar alignment, is required for this effect. Be-cause the magnetization is produced by a rather large number of moleculeswith high magnetic moment of µB, its magnitude could be quite high andpotentially detectable by a small pick-up coil.7.3 Field-free alignment of asymmetric topmoleculesAchieving field-free alignment of asymmetric top molecules is a hard taskdue to the complicated dynamics they exhibit. As was shown in section 5.3,strong planar alignment due to the centrifugal forces reduces the dynamicsof asymmetric tops to that of a simple linear rotor. The latter are rou-tinely aligned using non-adiabatic methods, e.g. by a strong femtosecondpulse. Indeed, the low recurring 3D alignment, demonstrated in Fig. 5.13 (a)was a consequence of the weak non-adiabaticity of the centrifuge excitation.1347.3. Field-free alignment of asymmetric top moleculesThis non-adiabatic part can be substantially enhanced by adding a kickexcitation after the centrifuge, producing much higher degrees of transientthree-dimensional alignment.x yzFigure 7.2: Calculated distributions of molecular axes a (red), b (green) andc (blue) of an asymmetric top molecule in the aligned rotational state.A fully adiabatic field-free technique is even more desirable, since itproduces permanent molecular alignment. The two ingredients are vitalto the development of such methods: the stationary rotational states thatposses some degree of 3D alignment, and the adiabatic way of populatingthese states. Our preliminary calculations confirm the existence of such 3Daligned stationary states of asymmetric top. An example of one such state,∑MJ∈{−2,0,2}1√3| J = 16,MJ , τ = −12 〉,is illustrated in Fig. 7.2 by showing the distributions of molecular axes a(red), b (green) and c (blue). As can be seen, it indeed exhibits a small degreeof permanent 3D alignment. As was demonstrated in section 5.1, changingthe behavior of the centrifuge polarization may offer control over the targetstates of the centrifuge excitation, while keeping it adiabatic. Another degreeof control could be introduced by combining several centrifuge pulses oneafter another, potentially allowing reaching the desired 3D aligned state viaa few consecutive adiabatic transfers. The feasibility of this method requiresfurther study.1357.4. Potential energy surface reconstruction7.4 Potential energy surface reconstructionPotential energy surface (PES) is one of the most important characteristicsof the molecule. It determines how the potential energy V of the molecularbonds depends on the intramolecular coordinates r. A conventional way offinding V (r) involves fitting a particular molecule-specific model potential tothe experimentally-measured vibrational spectrum. At large r, such fittingrequires the the knowledge of high vibrational states.An optical centrifuge could provide an additional information on PES. Aswas shown in section 5.3, fast rotation deforms the molecule and modifies itsfield-free dynamics. Experimentally, the revival time Trev can be determinedas a function of the rotational frequency ω. Since Trev =hd2J/dJ2andω = 1~dJdJ , where J is the energy of the state J , we can write Trev(ω) =2pidω/dJand use2piJ =∫Trev(ω)dωto determine ω(J). Integrating this function one more time, we achievethe rotational spectrum J . Our preliminary results show, that the bonddeformations at the highest achievable J-states become compatible to oreven higher than what can be reached through pure vibrational excitation.This means, that fitting J by the model potential at the extreme level ofrotational excitation may improve the knowledge of V (r) at large r.7.5 Harmonic rotational statesAs the angular momentum of a linear molecule increases in the centrifuge, itstarts stretching. This leads to the increase of its moment of inertia, whichin turn may result in a peculiar situation, when the rotational frequencyω(J) = JI(J) stops increasing with angular momentum J . Our preliminarycalculations and experimental results point to the existence of this effect ina number of linear molecules.Since ω = 1~dJdJ , near this angular velocity the spectrum becomes equidis-tant, making the system equivalent to the quantum harmonic oscillator.Among others, one of the interesting expected features of such “harmonic”rotational states is infinitely large revival time. This means, that evena narrowly localized rotational wave packet would undergo dispersionlessclassical-like motion indefinitely.1367.6. Super rotors in He droplets7.6 Super rotors in He dropletsA lot of attention has recently been attracted to the behavior of moleculesembedded in helium nanodroplets [124]. Modern technology allows creationof small (from a few hundred to a few million of atoms) helium clusters withexactly one molecule embedded inside. Due to the low polarizability andhigh ionization threshold of such helium clusters, an embedded moleculeremains accessible to the rotational excitation by laser fields [16, 97]. Asrecent studies showed [124], the rotational states of embedded moleculesexhibit surprisingly low, for a condensed phase, decoherence rate, which wasattributed to the superfluidity of helium clusters below the lambda point.These studies, however, were limited by a relatively low degree of rotationalexcitation and wave packet controllability.It is therefore of great interest to extend our studies of the collisionaldecay of super rotors (chapter 6) to He nanodroplets. Studying the deco-herence rates as a function of the molecular angular momentum inside aquantum environment of a nanodroplet could provide better understandingof the decoherence mechanisms in the many-body quantum interaction limit.Unlike macroscopic gas samples investigated in this thesis, the thermaliza-tion volume of a nanodroplet is rather small, which could offer opportunitiesfor exciting collective angular momentum states, such as quantum vortices[41].137Chapter 8ConclusionsIn this work we implemented the technique of an optical centrifuge to pro-duce and study the unique states of molecules, molecular super rotors. Sev-eral experimental techniques were developed to detect and characterize su-per rotors of various molecular species in the presence of external fields andunder field-free conditions, in dilute gas samples and in dense medium.We showed, that the centrifuge offered high degree of control over therotational state of molecules, allowing an excitation of synchronous unidi-rectional rotation in a very broad range of frequencies. Finite width ofthe produced wave packet allowed us to observe novel rotational and spin-rotational dynamics. Our studies of the evolution of super rotors in externalmagnetic fields provided a new way of controlling their rotational states.We proposed and implemented a number of experimental techniques thatemployed an optical centrifuge to achieve the molecular alignment at highdegrees of rotational excitation. Permanent field-free planar alignment ofasymmetric top molecules was demonstrated for the first time. 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