Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Wang, Ruoxi
Doctor of Philosophy - PhD
University of British Columbia
In many-body systems out-of-equilibrium, strong coupling triggers emergent properties that can act to limit the natural dissipation of energy and matter. The physics of strong coupling plays an important but incompletely understood role in the dynamics of natural plasmas over a wide range of length scales. Particularly remarkable signs of hindered transport appear under laboratory conditions in the self-organized avalanche, bifurcation, and quench of the nitric oxide molecular ultracold plasma. Here, a UV-UV double-resonant excitation of nitric oxide, cooled to 500 mK in a skimmed supersonic molecular beam prepares an ellipsoidal Rydberg gas of known initial density in a single selected state of principal quantum number, n₀. Penning ionization, followed by an avalanche of electron-Rydberg collisions, forms a plasma of NO⁺ ions and weakly bound electrons, in which a residual population of n₀ Rydberg molecules evolves to a state of high orbital angular momentum, ℓ. A 60 MHz radio frequency pulse with a peak-to-peak amplitude as low as 400 mV/cm, applied to a plasma in a state of arrested relaxation, depletes the residual Rydberg signal. Similarly, mm-wave fields in the range from 70 to 100 GHz tuned to resonance with n₀f (2)→(n₀ ±1)d(2) transitions deplete the plasma signal to an amplitude near zero. We associate both effects with Rydberg electronic orbital angular momentum redistribution. New theoretical work has developed a primitive picture of this system using a Lindblad master equation to describe the Markovian dynamics that arise when a closed quantum system of disordered, randomly interacting molecular dipoles couples to a thermal continuum of free atoms.
Relaxation dynamics in a molecular ultracold plasma:control and modelingbyRuoxi WangM.A. Stony Brook University, 2016B.Sc. University of Waterloo, 2015B.Sc. Beijing Institute of Technology, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)October 2023© Ruoxi Wang, 2023The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Relaxation dynamics in a molecular ultracold plasma: control and mod-elingsubmitted by Ruoxi Wang in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Physics.Examining Committee:Edward Grant, Professor, Department of Chemistry/Department of Physics andAstronomy, UBCSupervisorValery Milner, Associate Professor, Department of Physics and Astronomy, UBCSupervisory Committee MemberTakamasa Momose, Professor, Department of Chemistry/Department of Physicsand Astronomy, UBCUniversity ExaminerKeng Chou, Professor, Department of Chemistry, UBCUniversity ExaminerJacob Roberts, Professor, Department of Physics, Colorado State UniversityExternal ExaminerAdditional Supervisory Committee Members:Kirk Madison, Associate Professor, Department of Physics and Astronomy, UBCSupervisory Committee MemberRoman Krems, Professor, Department of Chemistry, UBCSupervisory Committee MemberiiAbstractIn many-body systems out-of-equilibrium, strong coupling triggers emergent prop-erties that can act to limit the natural dissipation of energy and matter. The physicsof strong coupling plays an important but incompletely understood role in thedynamics of natural plasmas over a wide range of length scales. Particularlyremarkable signs of hindered transport appear under laboratory conditions in theself-organized avalanche, bifurcation, and quench of the nitric oxide molecularultracold plasma.Here, a UV-UV double-resonant excitation of nitric oxide, cooled to 500 mKin a skimmed supersonic molecular beam prepares an ellipsoidal Rydberg gasof known initial density in a single selected state of principal quantum number,n0. Penning ionization, followed by an avalanche of electron-Rydberg collisions,forms a plasma of NO+ ions and weakly bound electrons, in which a residualpopulation of n0 Rydberg molecules evolves to a state of high orbital angularmomentum, ℓ.A 60 MHz radio frequency pulse with a peak-to-peak amplitude as low as 400mV/cm, applied to a plasma in a state of arrested relaxation, depletes the residualRydberg signal. Similarly, mm-wave fields in the range from 70 to 100 GHz tunedto resonance with n0 f (2)→ (n0± 1)d(2) transitions deplete the plasma signal toan amplitude near zero. We associate both effects with Rydberg electronic orbitalangular momentum redistribution.New theoretical work has developed a primitive picture of this system usinga Lindblad master equation to describe the Markovian dynamics that arise whena closed quantum system of disordered, randomly interacting molecular dipolescouples to a thermal continuum of free atoms.iiiLay SummaryUltracold plasmas produced by the resonant photoexcitation of atomic and molecu-lar gases form a class of laboratory systems with important relevance to plasma col-lision and transport physics under strongly coupled conditions. Typically, atomicions and low-energy electrons produced by threshold photoionization relax in amanner that is reasonably well-described by collisional rate processes and hy-drodynamics in a quasi-thermal regime of classical mechanics. However, whenformed by avalanche from a state-selected Rydberg gas, a molecular ultracoldplasma behaves distinctively. Our research has focused on the specific case ofnitric oxide ultracold plasma to explore its exotic behaviours within the realms ofboth classical and quantum mechanics.ivPrefaceThis thesis is based on the results of the collaborative efforts of research andpublication at Professor Edward Grant’s lab at the University of British Columbia.Chapter 4 and Chapter 5 of this thesis have been published and are presented herewith minor revisions. The majority of data collection and analysis, simulations, andmanuscript preparation were performed by the author of this thesis. The followinglist provides all publications that were developed during the course of my thesiswork, detailing the contributions of each author accordingly:• M. Aghigh, K. Grant, R. Haenel, K. L. Marroquı´n, F.B.V. Martins, H. Sadegi,M. Schulz-Weiling, J. Sous, R. Wang, J. S. Keller, and E.R. Grant, Dis-sipative dynamics of atomic and molecular Rydberg gases: Avalanche toultracold plasma states of strong coupling. Journal of Physics B: Atomic,Molecular and Optical Physics, 53(7):074003, 2020M. Aghigh contributed to the literature review and manuscript preparation.M. Aghigh, K. Grant, R. Haenel, K. L. Marroquı´n, F.B.V. Martins, R. Wangand J. S. Keller provided technical support, conducted partial data collec-tion and participated in the discussion. H. Sadegi and M. Schulz-Weilingconducted major data collection and simulations. J. Sous participated inthe discussion. E.R. Grant conducted the writing and submitting of themanuscript.• R. Wang, M. Aghigh, K.L. Marroquı´n, K.M. Grant, J. Sous, F.B.V. Martins,J.S. Keller, and E.R. Grant, Radio frequency field-induced electron mobilityin an ultracold plasma state of arrested relaxation. Physical Review A,102(6):063122, 2020vR. Wang conducted the majority of data collection and analysis, modelling,and manuscript preparation. M. Aghigh participated in the data collectionand manuscript preparation. K.L. Marroquı´n, K.M. Grant, F.B.V. Martinsand J.S. Keller provided technical support and participated in discussions. J.Sous participated in the discussion and modelling. E.R. Grant was responsi-ble for the finalization and submission of the manuscript.• R. Wang, J. Sous, M. Aghigh, K.L. Marroquı´n, K.M. Grant, F.B.V. Martins,J.S. Keller, and E.R. Grant, mm-wave Rydberg–Rydberg transitions gaugeintermolecular coupling in a molecular ultracold plasma, The Journal ofChemical Physics, 157(6):064305, 2022R. Wang collected all data, conducted analysis and modelling, and createdthe manuscript with E.R. Grant. J. Sous contributed to modelling and manuscriptpreparation. M. Aghigh. K.L. Marroquı´n, K.M. Grant, F.B.V. Martins andJ.S. Keller provided technical support and participated in discussions. E.R.Grant was responsible for the finalization and submission of the manuscript.• K.L. Marroquı´n, R. Wang, A. Allahverdian, N. Durand-Brousseau, S. Colom-bini, F. Kogel, J.S. Keller, T. Langen, and E.R. Grant, Self-organizationin the avalanche, quench and dissipation of a molecular ultracold plasma,submitted (2023).K.L. Marroquı´n conducted a major collection and analysis of data and con-tributed to manuscript preparation. R. Wang collected and analyzed the se-lective field ionization data and contributed to simulations. A. Allahverdian,N. Durand-Brousseau, and S. Colombini conducted major simulations andmodelling. F. Kogel, J.S. Keller and T. Langen provided technical supportand participated in discussions. E.R. Grant was responsible for the finaliza-tion and submission of the manuscript.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Ultracold neutral plasma . . . . . . . . . . . . . . . . . . . . . . 11.2 Rydberg gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Molecular ultracold plasma . . . . . . . . . . . . . . . . . . . . . 32 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Fundamental principles . . . . . . . . . . . . . . . . . . . . . . . . . 93.1 Nitric Oxide spectroscopy . . . . . . . . . . . . . . . . . . . . . 93.2 Rydberg molecules . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.1 Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . 12vii3.2.2 Lifetimes of Rydberg states . . . . . . . . . . . . . . . . 143.3 Dynamic process in Rydberg gas and plasma . . . . . . . . . . . 153.3.1 Pening ionization . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Charged particle collisions with Rydberg molecules . . . . 163.4 Field ionization of Rydberg molecules . . . . . . . . . . . . . . . 174 Radio frequency field-induced electron mobility in an ultracold plasmastate of arrested relaxation . . . . . . . . . . . . . . . . . . . . . . . 214.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . 244.2.1 Supersonic molecular beam ultracold plasma spectrometer 244.2.2 Selective field ionization . . . . . . . . . . . . . . . . . . 244.2.3 Radio-frequency electric field . . . . . . . . . . . . . . . 254.2.4 Pulse sequence and evolution of n0 Rydberg density asobserved in a typical RF-SFI experiment . . . . . . . . . 254.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.1 Field-free evolution of the nitric oxide molecular ultracoldplasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.2 Molecular ultracold plasma rate processes in the presenceof a continuous wave (CW) radio frequency field . . . . . 324.3.3 Effects of a pulsed radio frequency field . . . . . . . . . . 324.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4.1 Field-free avalanche dynamics and dissociation in the state-selected nitric oxide Rydberg gas . . . . . . . . . . . . . 354.4.2 Nitric oxide Rydberg predissociation in a regime of ℓ-mixing 364.4.3 Effect of a radio-frequency field . . . . . . . . . . . . . . 394.4.4 Kinetics of radio-frequency accelerated predissociation . . 394.4.5 Mechanics of radio frequency induced ℓ-mixing . . . . . . 424.4.6 Electron mobility in the quenched molecular ultracold plasma 455 mm-wave Rydberg-Rydberg transitions gauge intermolecular cou-pling in a molecular ultracold plasma . . . . . . . . . . . . . . . . . 475.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48viii5.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.1 Formation of Rydberg gas and its evolution to ultracoldplasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.2 Selective field ionization as a gauge of short-time dynam-ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.3 Field-free propagation as a long-time gauge of plasma evo-lution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.4 UV-UV double-resonance and UV-UV-mm-wave triple-resonancespectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3.1 The effect of mm-wave radiation on the ω2 spectrum ofultracold plasma NO Rydberg series . . . . . . . . . . . . 545.3.2 Fixed ω2: mm-wave excitation spectra of state-selectedultracold plasmas . . . . . . . . . . . . . . . . . . . . . . 595.3.3 Ultracold plasma composition dynamics as a function of time 635.3.4 Power dependence of mm-wave excitation spectra . . . . 665.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.4.1 Rydberg spectroscopy of the nitric oxide molecular ultra-cold plasma . . . . . . . . . . . . . . . . . . . . . . . . . 695.4.2 Intensity of mm-wave resonances as a probe of molecularultracold plasma evolution dynamics . . . . . . . . . . . . 735.4.3 Lineshapes of Rydberg-Rydberg transitions in the nitricoxide molecular ultracold plasma . . . . . . . . . . . . . 775.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826 Local dissipation drives global relaxation in a molecular ultracoldplasma with on-site disorder and long-range dipolar interactions . . 856.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2 Model desicription . . . . . . . . . . . . . . . . . . . . . . . . . 886.2.1 Open quantum system in Lindblad formalism . . . . . . . 886.2.2 1D spin-1/2 XY chain . . . . . . . . . . . . . . . . . . . 896.3 Simulation and results . . . . . . . . . . . . . . . . . . . . . . . . 906.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96ix6.5 Effort on large-scale simulation with semi-classical approximation 986.5.1 1st-order mean-field (Gutzwiller) approximation . . . . . 986.5.2 Mean-field method including pair correlation . . . . . . . 1006.5.3 Accuracy of mean-field method with second-order correction1036.6 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . 1057 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109A Radio frequency resonance . . . . . . . . . . . . . . . . . . . . . . . 125A.1 Plasma frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.2 Freqeuncy dependence of RF resonance . . . . . . . . . . . . . . 126B Fano resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128C Shell model simulations . . . . . . . . . . . . . . . . . . . . . . . . . 131C.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131C.1.1 Rate processes associated with avalanche to plasma andquench . . . . . . . . . . . . . . . . . . . . . . . . . . . 131C.1.2 Master equations describing avalanche and charge transferin a plasma of uniform plasma density . . . . . . . . . . . 132C.1.3 Time evolution of electron temperature in a plasma of uni-form density . . . . . . . . . . . . . . . . . . . . . . . . 134C.1.4 Plasma evolution in a Gaussian ellipsoidal density distri-bution represented by 100 concentric shells . . . . . . . . 134C.1.5 Hydrodynamic Shell Model . . . . . . . . . . . . . . . . 135C.1.6 Rate constants . . . . . . . . . . . . . . . . . . . . . . . 138C.1.7 Indirect hydrodynamic expansion of the neutralized prod-ucts of charge-exchange . . . . . . . . . . . . . . . . . . 139C.2 Example results . . . . . . . . . . . . . . . . . . . . . . . . . . . 140C.2.1 Expansion and quench without charge transfer . . . . . . 140C.2.2 Expansion and quench with charge transfer . . . . . . . . 141C.2.3 Bifurcation in the effect of charge transfer . . . . . . . . . 141xC.2.4 Matlab code . . . . . . . . . . . . . . . . . . . . . . . . . 143D Lindblad dynamics simulation program . . . . . . . . . . . . . . . . 151xiList of TablesTable 3.1 Predissociation rates of NO molecules in atomic units . . . . . 14Table 4.1 Kinetic parameters used in Eq (4.1) to fit the exponential decayin the n0 = 49 Rydberg molecule SFI signal in Figure 4.3 underfield-free conditions and in the presence of a 60 MHz radiofrequency field of varying peak-to-peak amplitude, Vpp. . . . . 37Table 4.2 Parameters used in Eqs (4.4) and (4.5) to describe the enhance-ment, suppression and recovery of the n0 Rydberg SFI signalby the action of a 250 ns 60 MHz pulsed radio frequency fieldof peak-to-peak amplitude, Vpp, as time of this pulse, ∆tωrf ad-vances from tω2 to pass through the beginning of the SFI rampfield at ∆tramp for ramp field delays of 2.8 and 4.8 µs. . . . . . 43Table 5.1 Assignment of n0 f (2) positions and n0 f (2) to (n0±1)g(2)Rydberg-Rydberg transitions from SFI-detected ω2 and ωmm excitationspectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Table 5.2 Assignment of n0 f (2) to (n0±1)d(2) Rydberg-Rydberg transi-tions from ωmm excitation spectra detected in the amplitude ofthe late-peak plasma signal . . . . . . . . . . . . . . . . . . . 71Table 5.3 Assignments of interloping n0ℓ(N+) positions for ℓ = 1,2,3,and the energies of Rydberg-Rydberg transitions observed inscans of ω2 (Figure 5.3) and ωmm (Figure 5.4). Asterisks belowrefer to transitions for which calculated energies fall below thelowest scanned frequency. . . . . . . . . . . . . . . . . . . . . 74xiiTable 5.4 Fano parameters for n0 f (2) to (n0± 1)g(2) Rydberg-Rydbergtransitions measured in the late-peak excitation spectrum. Weestimate Γ with an uncertainty of about 10 percent. Lineshapesare perceptibly asymmetric, but the error bounds on q are large,perhaps as great as 25 percent. . . . . . . . . . . . . . . . . . 78Table 5.5 Fano parameters for n0ℓ(2) to (n0± 1)d(2) Rydberg-Rydbergtransitions measured in the late-peak excitation spectrum. Theuncertainty in Γ is again about 10 percent. Greater asymmetryreduces the absolute uncertainty in q, but in relative terms itremains about 25 percent. . . . . . . . . . . . . . . . . . . . . 79Table 6.1 Conversion to real dimensions from ℏ= 1, J¯ = 1, r= 1, W = 10,and γ = 0.1, to J¯=75 GHz µm3 with different densities of spins 97xiiiList of FiguresFigure 2.1 Detected image of the plasma bifurcation evolved from nitric oxideRydberg molecules . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 3.1 Hund’s case: J is the total angular momentum, N is the angu-lar momentum of the nuclear rotation. K is the total angularmomentum apart from spin. L is the total electronic orbitalangular momentum with its projection on the internumclearaxis Λ. S is the total electronic spin with its projection on theinternuclear axis Σ. Ω = Λ+ Σ is the total angular momentprojection on the internuclear axis. . . . . . . . . . . . . . . . 10Figure 3.2 Resonant 1+1 ionization spectrum of the X 2Π1/2 K”=1 to theA2Σ+ K’=0 state fitted to simulated spectrum with a rotationaltemperature TR ≈ 5K. The software used to simulate the NOspectrum is PGOPHER. . . . . . . . . . . . . . . . . . . . . 11Figure 3.3 Coulomb potential and centrifugal potential sum to make theeffective potential of a Rydberg electron with angular momen-tum ℓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 3.4 Potential barrier along the direction of the external field toillustrate the classical field ionization of Rydberg system . . . 18xivFigure 4.1 (left) Diagram illustrating the double-resonant excitation ofa molecular Rydberg gas of nitric oxide, and the conditionsleading to Penning ionization and avalanche to an ultracoldplasma. The atomic-like plasma-excitation spectrum consistsexclusively of N = 1 n0 f (2) Rydberg state resonances con-verging to the N+ = 2 rotational limit of NO+. (top right) Co-propagating laser beams, ω1 and ω2 cross a molecular beamof nitric oxide between entrance aperture G1 and grid G2 of adifferentially-pumped vacuum chamber. (bottom right) Schematicdiagram showing the sequence of pluses in in the RF-SFI ex-periment. The second step of laser excitation follows the firstwith a specified delay, ∆tω2 . A radio-frequency field with anadjustable peak-to-peak amplitude interacts with this ensem-ble, either as a CW field or as a pulse with a duration Wrfapplied at a time, ∆tωrf , after ω2. An electric field ramp from0 to 350 V/cm with a rise-time of 1 µs 0.8 V/ns, applied aftera delay, ∆tramp following ω2, ionizes the excited molecularsystem. Shaded regions represent the surviving fractions ofthe n0 Rydberg molecules to a form a residual fraction of long-lived molecules in the absence (blue) and after the presence ofa 60 MHz radio frequency field (here represented by the pulsein green). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 4.2 (upper frames) Typical SFI spectra, formed by 4,000 SFI tracessorted according to the initial density ρ0, for an n f (2) Rydbergwith an initial principal quantum number n0 = 44, followingramp delays, ∆tramp, of 0,150, 450 and 1000 ns. (bottom frame)Single SFI traces obtained under identical conditions of initial49 f (2) Rydberg gas density (1011 cm−3) and ramp-field delay(2 µs) in the absence (upper) and presence (lower) of a low-amplitude CW 60 MHz radio frequency field (Vpp = 0.125 Vcm−1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30xvFigure 4.3 Ultracold plasmas evolving from 49n f (2)Rydberg gases: (top)SFI amplitude integrated over ramp field from 0 to 50 V cm−1(plasma signal); and (bottom) SFI amplitude integrated overramp field from 50 to 200 V cm−1 (Rydberg signal), as a func-tion of ramp delay, ∆tramp under field-free conditions (A) andin the presence of a 60 MHz rf pulse that starts 500 ns after ω2excitation and extends to fill the elapsed time, ∆tramp, until thestart of ramp field. Peak-to-peak pulse amplitudes vary withincreasing effect over the range, Vpp = 0.125 (B), 0.25 (C),0.5 (D), and 1.25 (E) V cm−1. . . . . . . . . . . . . . . . . . 31Figure 4.4 Integrated electron signal obtained by selective field ionizationof an ultracold plasma as it evolves from a 49 f (2) Rydberggas of NO in the presence of a 250 ns 60 MHz pulsed radiofrequency field, plotted as a function of rf delay, ∆tωrf for fiverf amplitudes, Vpp in V cm−1, 0 (A), 0.125 (B), 0.25 (C), 0.5(D) and 1.25 (E), for a 200 ns ω1 −ω2 delay and two fixedvalues of ramp field delay, ∆tramp, of 2.8 µs (top) and 4.8 µs(middle) after ω2 excitation. (bottom) Fixed ramp field delayand Vpp with three values of ω1−ω2 delay, 100 (A), 200 (B)and 300 (C) ns. Each curve is normalized according to thesignal at -1µs rf pulse delay when the rf pulse has no effect. . 33Figure 5.1 (a) Co-propagating laser beams, ω1 and ω2, cross a molecular beamof nitric oxide between entrance aperture G1 and grid G2 of a differentially-pumped vacuum chamber. G2, G3 and a MCP detector assemblymove on a carriage to afford externally adjustable G1 - G2 spacingsfrom 3 to 60 mm (b) Nd:YAG-pumped dye lasers (ω1 and ω2) pumpground state NO first to the excited A 2Σ+ N′ = 0 state and then to aRydberg level with N = 1. . . . . . . . . . . . . . . . . . . . . 50Figure 5.2 Field free Double-resonant ω2 spectra of nitric oxide n0 f (2) Ryd-berg states for n0 from 38 to 49. (top) Electron signal integratedover Rydberg resonances in the SFI spectra. (bottom) Electron signalintegrated over late peak after a flight time of 20 µs . . . . . . . . 55xviFigure 5.3 ω2 spectra as in Figure 5.2, in pairs with (left) SFI and (right)late peak detection, in each case setting mm-wave fields to thefrequencies indicated. . . . . . . . . . . . . . . . . . . . . . . 57Figure 5.4 mm-wave action spectra in pairs with (left) SFI and (right) latepeak detection fit to Fano lineshapes for ultracold plasmas thatevolve from Rydberg gases with n0 from 39 to 45. . . . . . . 61Figure 5.5 a) Time-dependent mm-wave spectra of 42f(2) state integrated overlate-peak signal and fit to four-component Fano lineshapes, recordedfor mm-wave fields triggered to begin at times -1.8 µs, 0.8 µs and1.4 µs relative to ω2 laser excitation respectively. b) Ratio of theintegrated 42 f (2)→ 43g(2) peak, to that of the 42 f (2)→ 43d(2)dip as a function of mm-wave delay. . . . . . . . . . . . . . . . 64Figure 5.6 Eight frames, each showing four thousand selective field ionization(SFI) spectra arranged by decreasing initial Rydberg gas density,ρ0. Contours describe the relative electron signal as a function ofelectrostatic field, ramped from 0 to 250 V cm−1 with ramp delaysof 0, 150, 300, 450 and 600 ns, as well as 1 and 2 µs. At highdensities and low voltage, the SFI signal reflects weakly bound high-Rydberg molecules and electrons bound to the NO+ space charge.Initial 44 f (2) Rydberg molecules form a distinctive pair of bandsat 125 and 180 V cm−1 signalling diabatic field-ionization to formNO+ in rotational states N+ = 0 and 2, respectively. . . . . . . . 67Figure 5.7 Variation of the amplitude and linewidth of mm-wave transitionsoriginating from the ω2-selected 40f(2) recorded as enhancementfeatures detected by SFI. Frame a) shows the amplitude of the sig-nal integrated from 100 to 102 GHz as it approaches saturation inthe mm-wave intensity. Frame b) shows band shapes indicating anabsence of power broadening for mm-wave intensities indicated byfilled triangles in frame a). Solid lines fit Fano lineshapes to en-hancement and depletion features (See Eqs (B.1) and (B.2)). Curvesdrawn through the electron signal data formed by positive-going fto g resonances are fit to a constant linewidth, Γ = 0.8 GHz for allmm-wave intensity. . . . . . . . . . . . . . . . . . . . . . . . 68xviiFigure 5.8 Measured positions of ωmm resonances for n0 f (2)→ (n0± 1)g(2)(♦) and n0 f (2)→ (n0± 1)d(2) (♢) transitions compared with linesconnecting the calculated displacements of the successive sets ofg(2) and d(2) states from the corresponding member of the n0 f (2)series, for n0 from 39 to 45. . . . . . . . . . . . . . . . . . . . 72Figure 6.1 Schematic illustration of the mm-wave driven plasma dissipa-tion represented by a 1D spin-1/2 chain with on-site energydisorder and local dissipation. . . . . . . . . . . . . . . . . . 88Figure 6.2 (a) Schematic diagram of NO+ core ions paried with eletronsto from interacting dipoles di and d j seperated by ri j = ri−r j.The pentential energy of interaction to lowest order is V ddj j =[di · d j − 3(di · ri j)(d j · ri j)]/r3i j. (b) Illustration of measureddistribution of binding energies, W , defined by SFI spectra. (c)Schematic diagram of the spin-exchange interaction Sˆ+i Sˆ−j oftwo Rydberg molecules in the two-level (spin-half) approxi-mation. (Figures are taken from ref.[146]) . . . . . . . . . . . 90Figure 6.3 (left) Evolution of a closed system of 11 spins with unit coupling(J¯ = 1), zero disorder (W = 0) and zero dissipation (γ j = 0). (right)Evolution of the same system initiating a non-zero local dissipatorafter t = 100 on the central site (S−5 ) with γ5 = 0.1. . . . . . . . . 91Figure 6.4 Evolution of Szi in a system of 11 spins with unit coupling (J¯ =1), moderate disorder (W = 10) and moderate dissipation (γ j =0.5) applied at t ∗ J¯ = 100 to the central site. Curves showingthe rate of decay at the site of dissipation (Site 5), and at oneof the most distant sites (Site 10). . . . . . . . . . . . . . . . 92Figure 6.5 Evolution of Szi in a system of 11 spins with unit coupling (J¯ =1), moderate disorder (W = 10) and moderate dissipation (γ j =0.5) applied at t ∗ J¯ = 100 to an edge site. Curves showing therate of decay at the site of dissipation (Site 0), and at the mostdistant site (Site 10). . . . . . . . . . . . . . . . . . . . . . . 93xviiiFigure 6.6 (left) Exponential decay rates of < Szi > in a system of 11 spins withunit coupling (J¯ = 1), moderate disorder (W = 10) as a function ofsite distances from the central lossy site (Site 5) with varied dissi-pation strength γ5. (right) Exponential decay rates of < Szi > in asystem of 11 spins with unit coupling (J¯ = 1), moderate disorder(W = 10) as a function of site distance from edge lossy site(Site 0)with varied dissipation strength γ0. The curves are fitted to powerlow by the least square method. . . . . . . . . . . . . . . . . . . 95Figure 6.7 Evolution of Sz two sites in a system of 7 spins with unit coupling(J¯ = 1), disorder W = 10 and collective dissipation (γ = 1) appliedat t ∗ J¯ = 10 to all sites. Curves represent calculation by first-order(orange)and second-order (green) mean-field method compared tothe exact solution (blue). All spins are initialized to point to the xdirection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 6.8 Example simulations of the 1st-order (orange) and 2nd-order (green)approximation compared to the exact solution. The top-left frame il-lustrates a scenario where the 2nd-order approximation converges tothe exact solution with good agreement, while the 1st-order approx-imation fails to initiate interaction. The top-right frame shows thedivergence can occur under certain conditions, even in the absenceof disorder and dissipation. The bottom-left frame reveals that diver-gence may take place during the free evolution stage when a randomdisorder configuration is present. The bottom-right frame shows thatdivergence can occur during a stable stage after the introduction ofdissipation with a random disorder configuration in place. . . . . . 104Figure A.1 Measured relative amplitude of the radio frequency output withconnection to experiment grids in the vacuum chamber scannedover different frequencies . . . . . . . . . . . . . . . . . . . . 127Figure A.2 Integrated electron signal obtained by selective field ionizationof an ultracold plasma as it evolves from a 44 f (2) Rydberg gasof NO in the presence of a cw radio frequency field, plotted asa function of rf frequencies, for different input rf amplitudes . 127xixFigure B.1 Fano line shapes of different q. Negative q has a reversed shapeabout ε = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Figure B.2 Labview program for fitting the mmWave spectrum to the Fanoprofile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Figure C.1 (left) The long-time evolution of electron temperature in plasmaformed by avalanche from a Rydberg gas with n0=80 withinan ellipsoid with an initial peak density of 0.4× 1011 cm−3 forfixed geometry (dotted line) and a stepped sequence of coupledrate equation evolution and self-similar ambipolar expansion(solid line). (right) Global population fractions of particlesas they evolve in the avalanche of a shell-model plasma with-out expansion (dotted line) and with intervening steps of self-similar expansion (solid line) as above. . . . . . . . . . . . . 140Figure C.2 Evolution of electron temperature (left) and global populationof particles (right) in plasma formed by avalanche from a Ryd-berg gas with n0=80 in 1D self-similar Gaussian shells with aninitial peak density of 1012 cm−3 (top) / 1011 cm−3 (bottom)for a stepped sequence of coupled rate equation evolution tak-ing into account the charge transfer and self-similar ambipolarexpansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Figure C.3 Evolution of high-ℓ Rydberg molecules in plasma formed froma Rydberg gas with n0=80, an initial peak density of 1010 cm−3and initial σx=1mm and σy=0.5mm in effect of charge transferdriving the bifurcation. The frames are taken at t=20ns, 50ns,100ns ,120ns . . . . . . . . . . . . . . . . . . . . . . . . . . 142xxList of AbbreviationsAL Anderson localizationCW continuous waveDR dissociative recombinationEIT electromagnetic induced transparencyFWHM full width at half maximumMBL many-body localizationMCP multichannel plateMOT magneto-optical trapNO nitric oxidePD predissociationRF radio frequencySFI selective field ionizationSOC self-organizing criticalityTBR three-body recombinationUV ultravioletxxiAcknowledgmentsIt has been an enjoyable and unforgettable journey of all my life to spend the pastseveral years at beautiful UBC and conduct wonderful research at Grant Lab. Thisjourney was made possible, and all the more enjoyable, through the support ofnumerous individuals to whom I am deeply grateful.Firstly, I express my sincerest appreciation to my supervisor, Professor EdwardGrant. Working with Ed has been an incredible journey, filled with learning andexploration. He can always bring ideas that encourage me to explore new andexciting territories. His enthusiasm for science has been a constant inspiration.Ed is not only a skilled scientist but also an artist of communication. I have trulyenjoyed our discussion and learned so much from his elegant writing. Ed has beenmore than a research supervisor to me. He is always considerate, supportive, andpatient as a life coach. As I unfurl the sails toward the next chapter of my life, Iwill always remember his blessing: ”The world is your oyster.”My gratitude is extended to Professor James Keller. Jamie is not only skillfulin every detail of the hands-on experimental work but also enriches us with hisprofound knowledge about the world at the atomic scale. His visits to our lab havealways ensured that all experiments ran smoothly, maintaining an environmentthat is clean and organized, thanks to his meticulous care. Standing beside himand learning from him when troubleshooting a laser, all while punctuated by hisdelightful humor, has been a joy.I extend my sincere gratitude to my Supervisory Committee Members: ValeryMilner, Kirk Madison, and Roman Krems; the University Examiners: TakamasaMomose and Keng Chou; and the External Examiner, Jacob Roberts; for theircritical comments and constructive suggestions that have sharpened my thesis.xxiiI am grateful to our collaborators, Dr. Abhinav Prem from Princeton and Dr.John Sous from UC San Diego. Their insightful discussions and mentorship haveguided me from zero to hero through the theoretical projects.I wish to express my sincere appreciation for the support of my research groupmembers. Specifically, I would like to thank Mahyad Aghigh, Kevin Marroquı´n,Fernanda Banic Viana Martins, Kiara Grant, Amin Allahverdian, Rafael Haenel,Dyuman Das, Nathan Durand-Broussseau, and Smilla Colombini, whose invalu-able contributions have directly fueled my research. I extend my gratitude to LukeMelo, Ashton Christy, and Matthew Kowal for their kind technical support. Furtherwarm thanks go to other group members and our welcomed exchange students,including Teresa Seifried, Zahra Poursorkh, Eric Boateng, Ginger Brown, MahsaZarei, Carraugh Brouwer, Hooman Tavakolizadeh Noghabi, Felix Kogel, FabianMunkes, Xixi Qi, Ede`ne Rocheron, Le´a Brunswic, Angus Hui, and among others,for contributing to the friendly and vibrant environment of the Grant Lab. Thejourney has been both educational and fun thanks to these wonderful individuals.I would like to thank the team of the electronic and machine shops of UBCChemistry Department, under the leadership of Tony Mittertreiner. Many of ourexperimental setups would not be realized without their expertise.I gratefully acknowledge the financial support from Natural Sciences and En-gineering Research Council of Canada (NSERC) and the US Air Force Office ofScientific Research (AFOSR).I want to extend my heartfelt thanks to my friends who have created unfor-gettable moments with me in Vancouver. This journey is enriched with cherishedmemories because of all of you.My deepest gratitude goes to my family. I want to thank my parents, especiallymy mother, for her unconditional love, encouragement, and unwavering supportthroughout my life. Without your constant love and support, my journey of seeingthe world and pursuing my dreams would not have been possible.Last but not least, I would like to thank Ruoyu for her constant companionshipand understanding, always standing with me against all odds.xxiiiChapter 1Background1.1 Ultracold neutral plasmaPlasma is often called the fourth state of matter, defined as being neutral or quasineu-tral, meaning the number of positive charge carriers is approximately equal to thenegative charge carriers in the plasma. Distinct from other states of matter, naturalplasmas can span vast range of temperatures and densities. This is particularlyevident in astrophysical systems, where densities can range from a single particleper cm3 in the interstellar medium to 1030 cm−3 inside white dwarf stars. Withinour solar system, plasma temperatures can vary from 200 K in Earth’s ionosphereto 1.5×107 K at the core of the Sun [80].Over the past decades, cooling techniques of atoms and molecules have beengreatly developed to bring atomic or molecular particles to temperatures on theorder of millikelvin or lower [21, 89, 111, 71], which paved the way for the cre-ation of ultracold plasma. In 1999, Rolston and his coworkers reported the firstexperimental realization of an ultracold plasma [64], created by photoionizinglaser-cooled xenon atoms in a magneto optical trap (MOT). A parallel effort cre-ated plasma that forms spontaneously by Penning ionization and electron-impactavalanche in a dense ultracold Rydberg gas [100]. Since then, various ionizationmethods including photoionization and electron impact ionization, have been em-ployed to create ultracold plasmas from different atomic species such as Sr [63],Ca [27], Rb [36, 164] and Cs [118], as well as from NO molecules [96].1The creation of ultracold neutral plasmas extends the boundaries of traditionalneutral plasma physics, making it a fascinating and rapidly inter-discipline researcharea evolving atomic, molecular, and optical physics, quantum chemistry, and con-densed matter physics. Unlike neutral gas or charged particle collections, neutralplasma contains unbound charge carriers that interact with each other throughCoulomb force, leading to collective behaviour in the presence of external per-turbation. The Coulomb coupling parameter quantifies the ratio of the Coulombinteraction of charged particles to averaged electron kinetic energy:Γ= e2/4πε0akbT (1.1)where a = [3/4πρ]1/3 is the Wigner-Seitz radius that characterize separation be-tween particles of density ρ . The ultracold plasma created from ultracold atomsin magneto-optical trap (MOT), or molecules seeded at higher density in a su-personic expansion can approach a strong coupling regime with Γ> 1. These low-temperature systems with strong and long-range Coulomb interaction provide idealplatforms for studying interesting and rich physics in the laboratory. The studies ofthese ultracold plasma connect directly to a diverse range of phenomena from theevolution of stars and planets to low-temperature charge transport in complex mate-rials. So far, experimental and theoretical investigations have scaled to demonstratekinetic and thermodynamic properties of important high-energy-density systemsincluding strongly coupled astrophysical plasmas [155, 19], as well as terrestrialsources of neutrons [58, 90, 56, 112], and x-ray radiation [121, 31]. In addition,quenched to conditions of very low-temperature, ultracold plasmas can evolve withdynamics that are governed by quantum mechanical particle-particle interactions,and in some cases by coherence with an external electromagnetic field [118].1.2 Rydberg gasA Rydberg gas is a collection of atoms or molecules in highly excited Rydbergstates. Rydberg states are characterized by atoms or molecules having one or moreelectrons in a high principal quantum number n, resulting in large average distancesbetween the electron and the nucleus. The high principal quantum number charac-terizes Rydberg gas by unique features such as strong and long-range interaction,2exaggerated response to electric fields, and long lifetimes[39]. Plenty of phenom-ena are observed in Rydberg systems such as interatomic forces [37, 9], dipoleblockade [24, 29, 78], coherent population trapping [142] and electromagneticinduced transparency (EIT) [46, 84, 95]. The strong and controllable interactionsproduce entanglement states that hold promise for quantum gates [128, 127, 44,150, 109] and exotic nonlinear optical material [23, 38].The Rydberg interactions can lead to the formation of exotic many-body statesand collective phenomena, making Rydberg gases an excellent platform for study-ing many-body physics. As engineered to mimic the arrangement of atoms withina solid, the interacting Rydberg systems provide a powerful quantum simulatorfor probing quantum effects in condensed matter [15, 106, 162] that give riseto high-temperature superconductivity and magnetism. They can themselves beengineered as quantum materials with properties of metallic or dipolar crystallinesolids [75, 70]. The many-body phenomena in Rydberg systems extend frompairwise phenomenon [10, 16] to aggregation [41, 153, 134], polaritons [57, 12,117], dissipation [20, 45, 72, 42, 11], nonequilibrium phase transitions[20, 85], andavalanche to plasma [65, 163, 3]. Recent experiments have established evidencefor self-organizing criticality (SOC) in ultracold and room-temperature Rydbergatoms [54, 28].1.3 Molecular ultracold plasmaIn the presence of buffer gas, a supersonic molecular beam attains sub-Kelvintemperature relative to a local frame that moves with high velocity in the labora-tory. The supersonic beam environment presents substantially different conditionscompared to the laser cooling of atomic gas within a magneto-optical trap (MOT),thus offering distinct advantages for studying ultracold plamsas. First, the charged-particle densities in molecular beams can typically reach 1012 cm−3, greatly sur-passing those in a MOT by several orders of magnitude. This brings molecularplasma systems in or near the strong coupling regime. Second, it supports a varietyof molecular species that form an ultracold plasma, which introduces new degreesof freedom such as rotation, vibration, and dissociative dissipation. The high,controlled ion density and neutral channels of dissociation add control points that3may lead to uncovering of new physics.Since 2008, our research group has pioneered generating and investigatingan ultracold molecular plasma [96]. This system starts as Rydberg gas ellipsoidprepared by double-resonant laser excitation of nitric oxide cooled to milli-Kelvintemperatures in a skimmed supersonic molecular beam. Here, electron impactavalanche ionization followed by a shaped hydrodynamic expansion relaxes thesystem to form a strongly coupled ultracold plasma. The emergence of this novelkind of ultracold plasma has sparked numerous intriguing investigations that canlead to a deeper understanding of the complex plasmas dynamics and uncoverpreviously unknown phenomena in the realm of quantum mechanics and many-body physics[3, 123, 139, 136, 52, 50, 146, 144, 159, 160].4Chapter 2IntroductionThe narrative of our ultracold molecular experiment is set into motion 10 timesevery second in the 10Hz triggering system. At these intervals, a supersonic-cooled molecular beam intersects with two co-propagating pulsed laser beams,leading to the creation of an ellipsoid volume of nitric oxide Rydberg molecules.The first pulsed laser beam, labeled ω1, has a diameter that is approximately afifth of the molecular beam, which defines the Gaussian ellipsoid geometry of theresultant Rydberg gas. This anisotropic geometry is subsequently revealed in thedetection images captured at hundreds of microseconds after the laser excitation,as illustrated in Figure 2.1. We have coined these images as ”bifurcations”.During the investigation of this system, crucial questions emerged: What isthe composition of this material? Is it merely a conventional collection of ionsand electrons that constitutes a plasma? Or is it simply just a Rydberg gas? Moreimportantly, what is the state or phase of this material?Previous work has established an initial kinetics of electron impact avalancheionization based on particle-particle interactions [130, 131, 97]. The short-timedynamics of avalanches from Rydberg gas conform well with the properties ofplasmas that evolve from ultracold atoms in a MOT. Yet unique properties havedeveloped in the relaxation of this NO ultracold plasma system, such as the spatialcorrelation of ions[124] and the slow plasma expansion[136, 52]. We refer to thissystem as being in an ”arrested relaxation” state, which is characterized by its lowelectron temperature, strong correlation, and long lifetime.5Figure 2.1: Detected image of the plasma bifurcation evolved from nitric oxideRydberg moleculesAt high enough density, the neutral Rydberg molecules undergo Penning ion-ization in the effect of dipole-dipole interaction between two neighbouring Ryd-berg molecules. This causes one molecule to be ionized producing a free electron,and the other molecules to go to a lower excited state. The released free electronsfurther collide with other bounded Rydberg molecules to produce more ionizationand avalanche. The spontaneous avalanche to plasma splits the core of an ellip-soidal Rydberg gas of nitric oxide. The electron gas formed promptly in the centerof the plasma drives an ambipolar expansion, pulling NO+ ions through the outershells that contain a balanced population of Rydberg molecules. Subsequently,cycles of charge exchange happen between ballistic ions and stationary Rydbergmolecules in the wings of the ellipsoid, which reduces the ion-Rydberg relativevelocity. In this process, the plasma consumes kinetic energy in the initial gas ofhot electrons to separate plasma into bifurcated volumes in the laboratory frame.These dynamics store avalanche energy in a mass transport reservoir, initiatinga process that anneals recoiling volumes to form an ultracold, strongly coupledplasma.The bifurcated plasma volume is stable with respect to recombination andneutral dissociation, exhibiting a long lifetime of 400µs or more. The classical6model is inadequate to explain the component of such a system based on obser-vations: It cannot be simply a classical ultracold plasma as the cold temperaturewould promote three-body recombination that consumes ions and heat electrons,driving a fast expansion which we do not see; Conversely, it cannot be a system ofpurely high-Rydberg states either because it would then be subject to consump-tion by Penning ionization that drives avalanche in a short time scale. Thus,the boundaries between ions and high Rydberg states are indistinct. We believethe bifurcated ultracold plasma represents an unconventional aggregated state ofRydberg molecules, ions and electrons[52].In my thesis, I found that the long-lived plasma contains a persistent resid-ual population of NO Rydberg molecules that retain the initial principal quantumnumber, n0 selected by double-resonant preparation. As I will introduce in Chapter3, Rydberg states with low orbital angular momentum dissociate on a nanosecondtimescale. In a collision-free environment, molecules of low ℓ rapidly predisso-ciate, leaving this long-lived, high-ℓ residue. The long persistence of arrestedrelaxation indicates the absence of the dissociative recombination collisions thatconsume ions, as well as the suppression of angular momentum-mixing (ℓ-mixing)electron-Rydberg collisions that consume Rydberg states. This long-lived stateof attested relaxation signifies a quench of energy and mass transport, hence pre-sumptively termed as ”localized”. In my thesis, I implemented radio-frequencytechniques and mm-wave techniques to perturb this localized system.The persistence of these residual Rydberg molecules offers a sensitive probeof electron mobility. Any process that stimulates electron collisions scramblesℓ restarts predissociation. In Chapter 4, I will demonstrate the utilization of theresidual population of Rydberg molecules in high angular momentum states of theinitial principal quantum number as a quantum probe of electron mobility. By em-ploying a low amplitude radio-frequency field that induces the plasma resonance,the electron mobility can promote collisional ℓ-mixing transitions that lower an-gular momentum barriers to Rydberg predissociation, with dramatic consequenceson the persistence of the long-lived phase in the ultracold plasma.Similarly, as I will show in Chapter 5, a mm-wave field can cause the Rydbergelectronic orbital angular momentum mixing with frequencies tuned to Rydberg-Rydberg transitions. The variation of predissociation lifetime with orbital angular7momentum changes the Rydberg-ion balance thus affecting the developing plasma.We obtain high-resolution mm-wave spectra of Rydberg-Rydberg transitions andfound that the excitation by mm-wave radiation tuned to resonance with adjacentprincipal quantum numbers of lower angular momentum can open a dissipationchannel that substantially depletes the plasma, forming deep negative-going res-onances that signify a large-scale redistribution of energy in the basis of close-coupled molecular Rydberg states.In previous work, we described the quenched ultracold plasma system of nitricoxide Rydberg molecules based on an effective many-body spin Hamiltonian in-cluding dipole-dipole interaction and van der Waals interactions.[144, 146]. Suchsystem occupies a disordered energy landscape that varies randomly from dipoleto dipole, which allows possible slow dynamics or many-body localization (MBL).The quenched ultracold plasma provides an experimental platform for studyingquantum many-body physics. We are motivated by the linewidths and depths ofdepletion resonances created by the mm-wave field. With mm-wave field tunedresonances driving Rydberg Rydberg transitions to the low angular momentumstates, predissociation transfers the population from the low angular momentumRydberg states to a system of free N(4S) + O(3P) atoms, bridging the closed plasmaensemble to a thermal continuum. This in effect creates an open quantum systemthat combines the plasma with the reservoir of free atoms. In Chapter 6, I willapproximate these dynamics using a Markovian approach that applies the Lindbladmaster equation to a one-dimensional model of eleven spins with disorder and onedissipative site.8Chapter 3Fundamental principles3.1 Nitric Oxide spectroscopyThe ground state of the Nitric Oxide molecule has the(σ1s)2(σ1s)2(σ2s)2(σ2s)2(σ2p)2(π2p)4(π2p)1electronic configuration. This state follows Hund’s case (a) as illustrated in Figure3.1, in which the interaction between electronic motion and nuclear rotation isweak. The total orbital angular momentum L= ∑i ℓi is strongly coupled to theinternuclear axis to make its projection a good quantum number. The projectionof L on the internuclear axis is Λ=1. Similarly, the total electron spin angularmomentum S= ∑i si has its projection on the internuclear axis Σ = ±12 . Thetotal projection of angular momentum on the internuclear axis is Ω = Λ+ Σ =3/2,1/2, leading to the two configurations of the ground state: 2Π3/2 (denoted asF2) and 2Π1/2 (denoted as F1). The most population is in 2Π1/2-state (F1) at lowtemperatures.The 225nm ω1 laser promotes the ground state to the first excited state A2Σ+.The corresponding electronic configuration is(σ1s)2(σ1s)2(σ2s)2(σ2s)2(σ2p)2(π2p)4(σ2p)19Figure 3.1: Hund’s case: J is the total angular momentum, N is the angularmomentum of the nuclear rotation. K is the total angular momentum apart fromspin. L is the total electronic orbital angular momentum with its projection onthe internumclear axis Λ. S is the total electronic spin with its projection on theinternuclear axis Σ. Ω = Λ+Σ is the total angular moment projection on theinternuclear axis.The angular momentum projection is Λ=0, and the state follows Hund’s case(b).In this scenario, the electronic spin is weakly coupled to the internuclear axis,therefore, Σ and Ω are not defined. Λ and N combine to define the value oftotal angular momentum K excluding the spin, which can take values K=Λ, Λ+1,Λ+2,...K and S together define the total angular momentum J that can take valuesJ = (K+ S),(K+ S− 1), ...|K−S|. Here, we denote J=K+1/2 and J=K-1/2 statesas F1 and F2, respectively.Selection rules for electric dipole transitions are:∆J =±1 for J ̸= 0, ∆J = 0 for J = 0Parity: u⇌ g∆S = 0The ω1 laser generates the (1+1) ionization spectrum as an extra photon excitesthe A2Σ+-state NO molecules above the ionization potential. This result is shown inFigure 3.2. One method to enhance such a spectrum involves applying the secondlaserω2 with substantially greater power thanω1, tuned above the ionization poten-tial to produce the resonant-enhanced multiphoton ionization (REMPI) spectrum ofω1.In the spectrum, transitions are denoted by P,Q,R for ∆J =−1,0,1, and p,q,r for∆K =−1,0,1. The subscripts 1 (or 21) represent transitions to F1 (or F2) from F1.10Figure 3.2: Resonant 1+1 ionization spectrum of the X 2Π1/2 K”=1 to the A2Σ+K’=0 state fitted to simulated spectrum with a rotational temperature TR ≈ 5K.The software used to simulate the NO spectrum is PGOPHER.The number in parenthesis represents the total angular momentum of the groundstate. In our experiment, we tune the ω1 laser to the pQ1(12) transition, explicitly:J”=1/2, K”=1, F1 ⇒ J’=1/2, K’=0, F1.The second laser, ω2, excites the NO molecules in the A2Σ+ K’=0 state to aX1Σ+ Rydberg state with correpsonding electronic configuration:(σ1s)2(σ1s)2(σ2s)2(σ2s)2(σ2p)2(π2p)4The high Rydberg states follow Hund’s case (d) (Fig. 3.1 (c)), where thecoupling between L and internuclear axis is weak while that between L and nuclearrotation is strong. Here, the angular momentum of nuclear rotation is denoted byR instead of N where R has amplitude√R(R+1)ℏ/2π with possible values ofR=0,1,2,...R and L combine to form the angular momentum K that can take valuesK = (R+L),(R+L−1), ..., |R−L|. K and spin S together define the total angularmomentum J.11For consistency with the conventions used in the subsequent parts of this thesis,I will modify the notation K to N to represent the total orbital angular momentumexcluding spin, and I will use N+ instead of R to represent the nuclear rotationalquantum number. In our experiment, we do not excite any vibrational state of NOmolecules and will denote the vibrational states as v= 0. The entire photoexcitationexcitation process can be represented as X 2Π1/2 N′′ = 1ω1−→ A 2Σ+ N′ = 0 ω2−→n0ℓ(N+). Here, n0 is the initial principal quantum number of Rydberg states, ℓ isthe quantum number of total orbital angular momentum without spin, N+ is therotational quantum number.The selection rule ∆J = 1 applies to theω2 transition, which requires |N+± ℓ|=1. The possible Rydberg series are p(0),s(1),d(1), p(2), f (2). As I will detailin subsequent chapters, only the n f (2) series have a sufficient lifetime to formRydbegr gas and evolve to ultracold plasma.3.2 Rydberg molecules3.2.1 WavefunctionThe highly excited Rydberg molecules can be well approximated by the wavefunc-tion of a hydrogen atom to the first order. Schro¨dinger equation of the atomichydrogen in atomic units is:(−∇22− 1r)Ψ= EΨ (3.1)where r is the radial distance between the electron and the nucleus. The solutionhas eigenenergies:En =Ryn2(3.2)with the Rydberg constant Ry and principal quantum number n. The azimuthalquantum number ℓ = 0,1, ...,n− 1 determines the magnitude of the angular mo-mentum of the Rydberg electron. The projection of angular momentum on thez-axis are quantized values, determined by the magnetic quantum number m =−ℓ, ..., ℓ.The electronic shell structure has minimal impact on Rydberg electrons with12high principal number n and angular momentum ℓ. However, the wavefunctionof low angular momentum Rydberg atom has a higher possibility to penetrateand polarize the electronic shell of the ionic core. The resulting attractive fieldincreases the bonding energy of the Rydberg electron. To account for this effectof the ionic core, the quantum defect δl is introduced to the energy of Rydbergstates in principle quantum number n and angular momentum ℓ. The values weadopt for nitric oxide molecules are δs = 1.2,δp = 0.7,δd =−0.06,δ f = 0.01,δg =0.003,δℓ>3 ≈ 0 [101]. The degree of freedom in the ionic core of Rydberg atomsor molecules shifts the ionization threshold that the Rydberg series converges to.The potential energy of the Rydberg state is therefore adjusted as follows:ERyd = EIP− Ry(n−δℓ)2 (3.3)In our experiment, the ionization thresholds EIP(N+) converge to differentrotational levels of the core differ by B0N+(N++ 1), where B0 = 1.99 cm−1 isthe rotation constant of the NO+ X1Σ+ state corresponding to vibrational levelv = 0 [165].Figure 3.3: Coulomb potential and centrifugal potential sum to make the effectivepotential of a Rydberg electron with angular momentum ℓ13Eq.3.1 is separable by writing the wavefunction Ψ = Y (θ ,φ)R(r) in sphericalcoordinates. By defining by R(r) = ρ(r)/r, the radial equation follows:[12∂ 2∂ r2− 1r+ℓ(ℓ+1)2r2]ρ = Eρ (3.4)The term −1/r represents Coulomb potential of the electron, and ℓ(ℓ+ 1)/2r2represents the centrifugal potential induced by electron angular momentum. Thetwo terms sum to give the effective radial potential as plotted in Figure 3.3The centrifugal potential produces the centrifugal barrier, which plays an im-portant role in resisting the electron wavefunction from penetrating the ionic core.Consequently, Rydberg molecules with higher angular momentum ℓ are more sta-ble against dissociation and exhibit longer lifetimes.3.2.2 Lifetimes of Rydberg statesThe Rydberg states of NO molecules decay primarily by predissociation:NO∗ kPD−−→ N(4S)+O(3P) (3.5)Here, the core penetrating electron cause an internal transition of the moleculeto an unstable state and dissociation to neutral atoms. The predissociation ratestrongly depends on the angular momentum ℓ for lower ℓ. Table 3.1 lists the ℓ-dependent predissociation rates of NO molecules according to a model developedby Gallagher and coworkers [102], patterned on the work of Bixon and Jortner[13].Table 3.1: Predissociation rates of NO molecules in atomic unitsks 1.4×10−2/(2πn3)kp 4.6×10−2/(2πn3)kd 2.9×10−2/(2πn3)k f 1.2×10−3/(2πn3)kℓ>=3 3×10−5/(2πn3)The Rydberg states in high angular momentum ℓ > 3 exhibit a much longerlifetime than low ℓ states. For example, at principal quantum number level n = 45,14the 45d state has a lifetime of about 0.5 ns while the 45g state has a lifetime of500ns.Given large dipole moments, the Rydberg states undergo ℓ-mixing in the per-turbation of either external field or the Coulomb force induced by ion collision,producing an ℓ-mixed manifold similar to the Stark manifold. The overall predis-sociation rate across all angular momenta with a principal quantum number n isstatistically distributed as follows:kPD(n) =∑ℓ (2ℓ+1)kℓn24.13×1016s−12πn3(3.6)where the first term accounts for the weight of state ℓ. This will effectively lengthenthe observed lifetime of Rydberg states in principal quantum number n. Thisstatistical approach predicts decay times for levels near n = 50 of about 200 ns.3.3 Dynamic process in Rydberg gas and plasma3.3.1 Pening ionizationIn high-density Rydberg gas, Pening ionization occurs when the nearest neighbordistance of Rydberg pair falls within a critical distance. The dipole-dipole inter-action between neighboring Rydberg particles causes one partner to ionize and theother to de-excited to a lower state, accompanied by a change of principal quantumnumber and angular momentum:NO∗(n, ℓ)+NO∗(n, ℓ)−→ NO++NO∗(n′, ℓ′)+ e− . (3.7)Modelling the Penning interaction using classical trajectory Monte Carlo method,Robicheaux [119] predicts that 90% of Rydberg pairs within a critical distance ofrc = 1.8× 2n2a0 undergo Penning ionization within 800 Rydberg periods. Here,a0 is the Bohar radius. The resulting de-excited Pening partner follows a leveldistribution n′ ∝ n′5 with n′max = n0/√2.We can estimate the electron density released from Penning ionization by thenearest neighbor distance at a given density of the Rydberg particle. The distribu-15tion function of nearest neighbor distance is given by the results in [151]:Pnn(r) = ρdvD(r)dre−ρvD(r) (3.8)where v3(r) = 43πr3 is the three-dimention volume of sphere and ρ is a uniformdensity of particle. The local density of Pening electrons (ρe) for a given initial lo-cal density ρ0 and initial principal quantum number n0 is calculated by integratingthe Earlang distribution from r = 0 to the critical distance r = rc:ρe(ρ0,n0) =0.92·4πρ20∫ rc0r2e−4π3 ρ0r3dr =0.92ρ0(1− e− 4π3 ρ0r3c ) (3.9)The 0.9/2 factor here reflects 90% ionization and accounts for the fact that everytwo Rydberg particles release one electron.3.3.2 Charged particle collisions with Rydberg moleculesThe plasma forms as the result of an avalanche of ionizing collisions betweenRydberg molecules and electrons described by a second-order rate constant kioni .Three-body-recombination, with rate constant ktbri , reverses this process:NO∗(ni)+ e−kion−−⇀↽−ktbrNO++2e− . (3.10)Both rate constants depend on electron temperature and vary with ni, the principalquantum number of NO∗.Rydberg-electron n-chaging collisions, ki j, scramble the initial state of NO∗,exchanging binding energy with electron thermal motion:NO∗(ni)+ e−ki j−⇀↽−k jiNO∗(n j)+ e− . (3.11)In a molecular Rydberg gas, dissociation affects plasma evolution.16Predissociation of NO∗,NO∗kPDi−−→ N+O , (3.12)diminishes plasma density and consumes binding energy.Direct dissociative recombination,NO++ e− kDR−−→ N+O , (3.13)reduces the balanced population of electrons and ions.Hydrodynamic charge exchange with electron-ion recombination (charge trans-fer) bifurcation accompanied by electron quench) consumes inner-shell ions asthey expand through stationary outer shells of Rydberg gas.NO++ e−+NO∗(ni)kCTi (ni)−→ NO∗∗(ni)+NO∗∗ (3.14)The coupled rate equation simulations of the dynamic process, including rel-evant rate constants and initial conditions, are elaborately detailed in AppendixC.3.4 Field ionization of Rydberg moleculesWe can consider field ionization in a classical model when the hydrogen potentialin zˆ axis is perturbed by a uniform electric field. The electron moving along the zˆaxis experience a potential given by (in atomic units)V (z) =− 1|z| −Fz (3.15)relative to the ionization threshold. The potential has a saddle point at zsp =√1/Fwith corresponding binding energy:Vsp(zsp) =−2√F (3.16)17and is shown in Figure 3.4. Electrons with energies lower than the saddle point areclassically bound, and the field ionizes electrons with energies above the saddlepoint. Setting this energy equal to the energy of the Rydberg state,−Ry/n2, we getthe classical field required to ionize a Rydberg state of principal quantum numbern:F =Ry24n4(3.17)or in atomic units:F =116n4(3.18)It is convenient to convert the ionization field in V/cm to the ionization energy inwavenumber cm−1 by rewriting Eq. 3.16 as:Wionize[cm−1] = 6.12√F [V/cm] (3.19)This expression is the classical expression of the adiabatic ionization thresholdwithout considering the Stark effect.Figure 3.4: Potential barrier along the direction of the external field to illustrate theclassical field ionization of Rydberg system18This classical field ionization threshold is valid for m= 0 state when the angularmomentum projection on the direction of the electric field, zˆ axis, is zero. Whenm ̸= 0 with higher angular momentum, there is a m2/2(r2− z2) centrifugal termin the potential. This centrifugal barrier keeps the electron away from the zˆ axiswhere the potential saddle point forms and effectively raises the field required toionize the m ̸= 0 states. A physical view to understand the increase of ionizationthreshold is in terms of energy conservation: for an electron with nonzero |m| alongthe field (zˆ) axis, as it moves away along zˆ in the effect of the field, a portion ofits kinetic energy is consumed in changing momentum perpendicular to the field.This energy is not useful for escape therefore the electron requires more energy toescape. Cooke and Gallagher estimated the fractional increase in the field requiredto field ionize m ̸= 0 states compared to the ionization threshold of m = 0 statesbased on the classical field ionization model[25]:δFF=|m|√W√2=|m|2n(3.20)In the presence of an external electric field, the Rydberg system generates aStark manifold with energy levels represented by the parabolic states. The down-shifted states, or the red Stark states, have ionization limits close to the classicalionization limit. The parabolic state gives the binding energy of the lowest redStark state of high n with m=0:W =− 12n2− 3n2F2(3.21)Setting this energy equal to Eq. 3.16 we find the threshold field:F =19n4(3.22)This is the diabatic ionization threshold due to the Stark effect. In the conventionalunits, it is written as:Wionize[cm−1] = 4.59√F [V/cm] (3.23)The states that undergo more avoided crossings in the Stark manifold will follow19an adiabatic passage to a lower ionization field than diabatic ionization. States thatexperience a diabatic passage have a higher ionization threshold.20Chapter 4Radio frequency field-inducedelectron mobility in an ultracoldplasma state of arrestedrelaxationPenning ionization releases electrons in a state-selected Rydberg gas of nitric oxideentrained in a supersonic molecular beam. Subsequent processes of electronimpact avalanche, bifurcation and quench form a strongly coupled, spatially cor-related ultracold plasma of NO+ ions and electrons that exhibits characteristicsof self-organized criticality. This plasma contains a residue of nitric oxide Ryd-berg molecules. A conventional fluid dynamics of ion-electron-Rydberg quasi-equilibrium predicts rapid decay to neutral atoms. Instead, the NO plasma enduresfor a millisecond or more, suggesting that quenched disorder creates a state ofsuppressed electron mobility. Supporting this proposition, a 60 MHz radio fre-quency field with a peak-to-peak amplitude less than 1 V cm−1 acts dramatically tomobilize electrons, causing the plasma to dissipate by dissociative recombinationand Rydberg predissociation. An evident density dependence shows that this effectrelies on collisions, giving weight to the idea of arrested relaxation as a cooperativeproperty of the ensemble.214.1 IntroductionAt a given temperature and density, the properties of its most stable phase deter-mine the natural state of a quantum many-body system. For instance, a lattice ofcations in a sea of delocalized electrons naturally affords a metal with ductilityand a band structure that supports electrical conductivity. Covalent bonds andintermolecular forces organize the atoms in a molecular solid to form a crystal. But,out of equilibrium, in a strongly correlated material, cooperative properties canimpose characteristics quite apart from the long-time limit of Boltzmann statisticalmechanics. Such conditions can produce transient states that persist for a range oftimes.In other words, strong correlations in out-of-equilibrium many-body systemsoffer paths to collective behaviour that cannot occur as a natural property of aground state, or an excited state in its statistical limit [149, 17, 94]. Thus, forexample, correlations in a crystal can cause a particular electromagnetic transi-tion to form a transient high-temperature superconducting phase [35, 93]. Moregenerally, an excited system can show signs of prethermalization [47], or undergoa nonequilibrium phase transition in which physical quantities quench to becomenonanalytic functions of time [61, 30, 26].In particular cases, a driving force balanced by dissipation can organize the mi-croscopic behaviour of a strongly coupled dynamical system to exhibit spatial andtemporal scale invariance [8]. This dynamical property, which parallels the criticalpoint of a conventional phase transition, does not depend on external control pa-rameters, but rather approaches a critical attractor in a process of self-organization– a propagating avalanche with large spatiotemporal correlations known as self-organized criticality (SOC) [152, 6]. Atomic-scale self-organized criticality prop-agates much like large-scale macroscopic phenomena such as earthquakes [107]and forest fires [7].Some of the earliest work on the phenomenon of self-organized criticality fo-cused on resilient, non-local properties of cold plasmas [105, 5], and plasma struc-tures confined magnetically or by pressure-gradient turbulence [22], as manifesta-tions of SOC avalanche and transport dynamics. Recent experiments have foundsigns of self-organizing criticality in ultracold and room-temperature atomic Ryd-22berg gases [54, 28]. For some time, we have observed elements of self-organizationin the properties of the nitric oxide molecular ultracold plasma [98, 125, 139, 53,88].In our experiment, double-resonant excitation of ground-state nitric oxide cooledin a supersonic molecular beam forms a state-selected Rydberg gas, which under-goes spontaneous Penning ionization. This triggers an electron impact avalanchethat splits the cross-beam ellipsoid of NO+ ions and Rydberg molecules [137].Ambipolar expansion quenches electron temperature of this ultracold plasma. Long-range resonant charge transfer from ballistic ions to frozen Rydberg moleculesquenches the ion-Rydberg relative velocity [51]. This sequence of steps gives riseto a remarkable mechanics of self-assembly that forms a gas of canonical density,characteristic of self-organized criticality. In this process, the plasma expends thekinetic energy of initially formed hot electrons to separate plasma volumes in thelaboratory frame [4]. These dynamics sequester energy in a reservoir of masstransport, starting a process that anneals recoiling volumes to form a scale-invariantultracold glass of strongly coupled ions and electrons.This ultracold plasma state of arrested relaxation persists for a millisecond orlonger, directly signalling the absence of dissociative recombination collisions ofelectrons and ions. Evident as well is a suppressed state of ℓ-mixing electron-Rydberg collisions. Rydberg states of low orbital angular momentum dissociateon a nanosecond timescale. Here, instead, a long-lived ultracold plasma trapsan adventitious population of Rydberg molecules with principal quantum number,n, distributed about n0, the value selected by double-resonant preparation of theRydberg gas. In a collision-free environment, molecules of low ℓ rapidly predis-sociate, leaving a long-lived, high-ℓ residue. The persistence of this residue actsas responsive sensor of electron mobility. Any process that stimulates electroncollisions scrambles ℓ, restarting predissociation.The sequence of events that leads to self-organized criticality in the bifurcatednitric oxide ultracold plasma forms a quantum mechanical state of ultracold elec-trons and spatially correlated, locally stationary NO+ ions. We have argued thatsuch a system of coupled dipoles supports a strength of dipole-dipole flip-flopinteraction that compares with the global van der Waals energy to a degree thatpredicts a systematic destruction of transport [145, 147].23Here, we offer direct experimental support for this picture. A radio frequencyfield, applied to a plasma in a state of arrested relaxation, acts to mobilize electrons,restarting dissociative recombination and Rydberg predissociation. The fractionof molecules consumed scales with density, proving that this process relies oncollisions: The field must operate on the plasma volume as a whole, as opposed toa perturbation of individual molecules. This offers support for the idea of arrestedrelaxation as a cooperative property of the ensemble in a state with characteristicsof many-body localization.4.2 Experimental methods4.2.1 Supersonic molecular beam ultracold plasma spectrometerA pulsed free-jet of NO seeded 1:10 in helium expands through 0.5 mm nozzlefrom a stagnation pressure of 5 bar and propagates 35 mm to enter an experimentalchamber through a 1 mm diameter skimmer, as diagrammed in Figure 4.1. Thecollimated supersonic molecular beam travels 70 mm to transit the entrance aper-ture of a first field plate (G1). A second grid G2, held at an externally adjustabledistance from G1, defines a flight path of controlled field.Co-propagating unfocussed Q-switched Nd:YAG pumped dye laser pulses, ω1and ω2, cross the molecular beam 6 mm beyond G1. A spatial filter collimates ω1to propagate as a cylindrical Gaussian. The 1 mm (fwhm) diameter of ω1 definesan ellipsoidal illuminated volume in the 3 mm diameter molecular beam. In thisvolume, double resonant excitation creates a gas of state-selected high-Rydbergmolecules with initial principal quantum number, n0.4.2.2 Selective field ionizationDuring ω1+ ω2 laser excitation, with G1 held to ground, an adjustment G2 overa range of ±100 mV serves to define a field-free region between G1 and G2. Ata predetermined time after ω2, a -3 kV square-wave pulse from a Behlke high-voltage switch, coupled to G1 through a 10 kΩ resistor, forms an electron-forward-bias voltage ramp that rises at a rate of ∼ 0.8 V/cm ns. We precisely fit a poly-nomial function to the leading edge of this voltage pulse that transforms the time-24dependent electron signal waveform to an ionization spectrum as a function of fieldin V/cm.4.2.3 Radio-frequency electric fieldThe experiment uses a Tektronix AWG 7102 10 GS/s arbitrary waveform generatorto produce radio frequency (RF) pulses of selected frequency with an amplitudefrom 0.1 to 1.5 V peak-to-peak. A LabVIEW user interface selects the frequencyand amplitude, with controlled delay and pulse duration. Measurements describedbelow use a frequency of 60 MHz (see Appendix A), an amplitude of 400 mVcm−1 and a pulse duration of 250 ns (Wrf), triggered at a time, ∆tωrf between tω2and tramp. The programmed output of the waveform generator connects to G2.4.2.4 Pulse sequence and evolution of n0 Rydberg density asobserved in a typical RF-SFI experimentFigure 4.1 diagrams the sequence of optical and electronic pulses used in carryingout a typical RF-SFI experiment. The horizontal axis indicates the elapsed timeafter ω2. Waveforms ω1 and ω2, represent the first and second laser pulses, and∆tω2 refers to the time delay between them. We apply an RF pulse of width Wrf,with a time delay of ∆tωrf after the second laser pulse. A voltage ramp that begins ata time ∆tramp after ω2 selectively field ionizes the excited molecular system. In thisstudy, Wrf has a duration of 250 ns for all experiments. We vary the intervals, ∆tωrf ,∆tω2 , and ∆tramp, according to the requirements of the measurements describedabove.4.3 Results4.3.1 Field-free evolution of the nitric oxide molecular ultracoldplasmaDensity distribution of the plasma ellipsoidDouble-resonant (ω1 +ω2) excitation of nitric oxide in a molecular beam formsa quantum-state selected ellipsoidal volume of Rydberg gas. Figure 4.1 diagrams25a) b)NO+ X 1Σ+ v+ = 0X 1Σ+ v+ = 0A 2Σ+ v' = 0NO NO+ X 2Π3/2 v' = 0X 2Π1/2 v' = 0ω2nf(2)N = 1N' = 0N" = 1326 nmω1 226 nmJ" = 1/2F1F2NO X 2ΠA 2Σ+NO+ X 1Σ+ v+ = 0ω1ω2nf(2)N = 1e-G1 G2 G3 100 V0 20 40 60 80 100Time (µs)The grid and detector assembly moves longitudinallyunder external motorized control. By this mechanism, we canvary the flight time through the G −1 G2 evolution regionfrom t = 0, when laser excitation forms the Rydberg gas, untilthe the excitation volume transits G2 and forms an electronsignal.To obtain pulsed-field-ionization spectra, we connect theoutput of a Behlke high-voltage switch to G1. A pulse appliedto this electrode creates a rising electric field in the evolutionregion. A tuned RC circuit controls the rise time of this pulsedfield. Figure 2 shows the experimental field pulse created for afixed G −1 G2 span of 3.6 cm, rising from 0 to 838 V cm−1with a time constant of three μs.To probe the excitation volume by means of a low-vol-tage pulsed field, we use a Tektronix AWG2020 arbitrarywaveform generator to create a one μs, −3 V square wavepulse, which we apply to G1.3. Results3.1. Preparation of the Xe Rydberg gasWith ω1 set to excite the transition in Xe from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2, we tune ω2 over a range from 515 nmto 508 nm, which drives absorption from the p6 state toRydberg states in the ns and nd series with an initial principalquantum number ranging from =n 300 to 70. Using a for-ward-bias high-voltage pulse, applied to G1 after a delay ofone μs, we detect the ω2 absorption spectrum by pulsed-fieldionization. Figure 3 plots the Rydberg spectrum from n = 38to n = 53, observed in ω2 transitions from Xe p6 by inte-grating the pulsed-field ionization signal.If we omit the pulsed field and wait for the excitationvolume to travel the distance set between the excitation regionand G2, we observe a characteristic waveform, as displayed infigure 4. The small prompt electron signal at μ≈t 0 s appearswith an intensity that does not vary with the properties of ω2,but does depend sensitively on the ω1 power, indicating thatthis weak feature arises from ω1 three-photon ionization.The arrival time of the second feature depends on thedistance from the excitation region to G2. This distancedivided by this late-peak arrival time conforms precisely withthe measured −385 m s 1 velocity of Xe atoms in our beam.The width of this feature in time transforms to the spatialwidth of the excitation volume in the direction of propagation.Figure 2. RampQ3 pulse used to obtain field-ionization spectra of XeRydberg states. A tuned RC circuit produces a pulsed electrostaticfield that rises to 838 V cm−1 with a time constant of three μs. Figure 3.A portion of the pulsed-field ionization detected absorptionspectrum of high-Rydberg transitions originating from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2. The ladder marks resonances assigned to thend series converging to Xe+ p P5 5 2 3/2.Figure 4. Oscilloscope trace showing the electron signal as afunction of time at the MCP detector. The prompt peak at 0 μs arisesfrom three ω1-photon ionization of xenon atoms together with theinstantaneous evaporation of prompt electrons following Rydber-g–Rydberg Penning ionization. The second, stronger electron signalmarks the transit of the excited volume through G2.3J. Phys. B: At. Mol. Opt. Phys. 00 (2014) 000000 J Hung et al700600500400300200100 00 1 2 Time (µs)Field V/cm2730380 30400 30420 30440 30460 30480 30500 30520 ω2 Energy (cm-1) 502 50a)b) c)ω1+ω2G1 G3G230400 30420 30440 30460 30480 30500 30520ω2 (cm-1)d)c)NOHeFigure 4.1: (left) Diagram illustrating the double-r so an excitation of a olecularRydberg gas of nitric oxide, and the conditions leading to Penning ionizationand avalanche to an ultracold plasma. The atomic-like plasma-excitationspectrum consists exclusively of N = 1 n0 f (2) Rydberg state resonancesconverging to the N+ = 2 rotational limit of NO+. (top right) Co-propagatinglaser beams, ω1 and ω2 cross a molecular beam of nitric oxide between entranceaperture G1 and grid G2 of a differentially-pumped vacuum chamber. (bottomright) Schematic diagram showing the sequence of pluses in in the RF-SFIexperiment. The second step of laser excitation follows the first with a specifieddelay, ∆tω2 . A radio-frequency field with an adjustable peak-to-peak amplitudeinteracts with this ensemble, either as a CW field or as a pulse with a durationWrf applied at a time, ∆tωrf , after ω2. An electric field ramp from 0 to 350 V/cmwith a rise-time of 1 µs 0.8 V/ns, applied after a delay, ∆tramp following ω2,ionizes the excited molecular system. Shaded regions represent the survivingfractions of the n0 Rydberg molecules to a form a residual fraction of long-lived molecules in the absence (blue) and after the presence of a 60 MHz radiofrequency field (here represented by the pulse in green).these steps of double resonant excitation from the electronic ground state X 2Π1/2 (v′′=0, N′′= 1) to the intermediate A 2Σ+ (v′= 0, N′= 0) state, and then to the selectedRydberg state. The choice of A-state N′ = 0 allows only final Rydberg states oftotal angular momentum neglecting spin of N = 1.The present illumination conditions yield Gaussian widths (σx : σy,z) in a ratioof approximately 3:1, where z defines the propagation direction of the molecular26beam and x denotes the laser axis. At t = 0, the peak density of the Rydberggas depends on the intensity of ω1, the laser that drives the first step of doubleresonance from X 2Π1/2 N′′ = 1 to A 2Σ+ N′ = 0 (ω1). At a fixed intensity ofω1, this density falls exponentially with the time delay between the laser pulsespromoting first and second steps. The density of the Rydberg gas varies in spacealong the axes of this illuminated ellipsoid according to the cylindrical Gaussianphoton density of ω1 crossed by the wider cylindrical Gaussian nitric oxide densityof the molecular beam.Even in a Rydberg gas of comparatively low density, some fraction of initiallyexcited n0 molecules populate the leading tail of the nearest-neighbour distancedistribution, separated by an orbital diameter or less. Thus, for an an initial n0 =50 Rydberg gas density of 1010 cm−1, the orbital radius is about 1 µm whilethe average spacing between Rydberg molecules is 3 µm. But, a good portionof the nearest-neighbour distance distribution falls within 1 µm. These closelyspaced pairs interact by Penning ionization to form prompt electrons, which seedthe avalanche to ultracold plasma [125]. Among the N = 1 high-Rydberg statesaccessible to ω2 excitation, only those in the n0 f (2) series converging to the nitricoxide cation state, NO+ X 1Σ+ N+ = 2 have sufficient lifetime to sustain thisavalanche, as illustrated by the excitation spectrum in Figure 4.1.We adjust the pulse energy of the second laser to saturate the ω2 transition.Under such conditions, the initial density of the Rydberg gas depends entirely onthe instantaneous number of intermediate A 2Σ+ molecules. The experiment usestwo means to regulate this quantity. Varying the ω1 pulse energies from 2 to 6µJ increases the intermediate state density linearly to a degree that approachessaturation. This population decays with a radiative lifetime of 192 ns [141], andthus, a delay of the ω2 laser pulse with respect to ω1 offers a precise means to varythe intermediate state density available to form a Rydberg gas at any chosen ω1pulse energy. Control of ω1 pulse energy and ω1−ω2 delay yields initial Rydberggas peak densities from 1010 to 1012 cm−3.To build realistic simulations of the collisional rate processes that give riseto the avalanche, we numerically approximate this ellipsoid by a system of 100concentric shells [140, 51]. For example, in a laser-crossed molecular beam Ry-dberg gas of nitric oxide with an initial peak density of 3× 1011 cm−3, 4 µs of27avalanche, expansion and dissociation to neutral atoms reduce the peak densityto 4× 1010 cm−3. The average ion/Rydberg density in this ellipsoidal volume is1.4× 1010 cm−3, and about 1.5 percent of the molecules in the plasma occupytwo-thirds of its volume at a much lower density less than 109 cm−3.This Rydberg gas undergoes an avalanche to a quasi-neutral plasma of NO+and electrons at rate that rises sigmoidally in a time interval that varies dependingon local density, from nanoseconds in the core of a higher-density ellipsoid to manymicroseconds in the periphery of an ellipse with a lower peak density [4].Selective field ionizationAn electrostatic field ramp with a rise time of ∼ 0.8 V cm−1 ns−1, started imme-diately after ω2 (∆tramp = 0), drives a diabatic evolution of molecules in the n f (2)Rydberg gas through the Stark manifold to cross a saddle point leading to ionsand free electrons when the field F in atomic units exceeds a threshold amplitudeof 1/9n4. In conventional units, this process forms a selective field ionizationresonance beginning in V/cm at a field, F = (En(2)/4.59)2, where En(2) in cm−1is the binding energy of the n f Rydberg state with respect to a nitric oxide cationin rotational state, N+ = 2.This trajectory through the Stark manifold traverses numerous intersectionswith states of matching electronic and rotational parity built on the ground ro-tational state of the ion. By virtue of these crossings, the wavepacket acquiressufficient N+ = 0 character to form free electrons and rotational ground state NO+cations earlier in the ramp, when the rising field passes an amplitude of (En(0)/4.59)2V/cm.A ramp delayed by a few hundred nanoseconds samples the quantum-statedistribution in an evolving Rydberg gas. During the interval of this delay, promptlyformed electrons collide with Rydberg molecules. This causes ℓ-mixing. Moleculesprepared in the initial state, n0 f (2) change orbital angular momentum, populatinga degenerate manifold of states, |N+ = 2⟩ |n, ℓ⟩. These states of higher orbital an-gular momentum field ionize at slightly higher field amplitudes to produce electronwaveforms reflecting the formation of ions in rotational states, N+ = 0 and 2.The integrated electron signal collected at a given ramp delay changes with28the initial density of the Rydberg gas, ρ0. Normally, we saturate the ω2 transitionand use the available density of NO A 2Σ+ to regulate ρ0. As described aboveand diagrammed in Figure 4.1, we systematically control the density of the A-statemolecules present for ω2 excitation by varying the ω1 pulse energy and, for a fixedpulse energy, by varying the ω1−ω2 delay. Using these tools, we have confirmedover the range of the present experiment, that the integrated electron signal in anSFI trace depends in direct proportion on the initial density of the Rydberg gas.Relying on this relationship, we systematically vary the A-state density and sortthe many traces in a typical SFI contour at fixed ramp delay according to ρ0.Selective field ionization spectra such as these provide a measure of the globalspectrum of electron binding energy as a function of time. Figure 4.2 combines4,000 SFI traces sorted by peak density over a range from 1012 to 1010 cm−3.The signal near zero field represents very high Rydberg molecules and electronsloosely-bound by the plasma space charge, The two features that appear at higherfield reflect the field ionization of the n0 = 44 state to NO+ X 1Σ+ cation rotationalstates, N+ = 0 and N+ = 2. Note how after ramp delay of zero (∆tramp = 0), thesefeatures shift to higher field ionization thresholds, reflective of electron collisionalℓ-mixing of initial 44 f (2) Rydberg molecules that occurs early in the electron-impact avalanche.Here, we see that that Rydberg gases with initial density in the higher rangeof our experiment avalanche fully on a 100 ns timescale, while Rydberg gases oflower density evolve to form a mixture of weakly bound NO+ ions and electrons(ultracold plasma) together with Rydberg molecules that retain the initially selectedprincipal quantum number, n0. Note the absence of electron signal in a wedge ofvery low SFI potential on the left-hand edge of these contours. At the highestdensity, field ionization requires a minimum of a few V cm−1, which correspondsto Coulomb binding energy on the order of 300 GHz.On a timescale of a few microseconds, the weakly bound ion-electron plasmapopulation evolves differently from that of the n0 Rydberg molecules. Figure4.3 compares the magnitude of the plasma signal, integrated over SFI ramp-fieldamplitudes from 0 to 50 V cm−1, with that of the residual Rydberg population,observed for a higher range of ramp-field amplitudes from 50 to 200 V cm−1.Note under field-free conditions that the number of electrons weakly bound290 50 100 150 200 250Electric field (V/cm)-0.200.20.40.60.811.21.41.60MHz60MHz0 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-3101210111010450 ns300 ns150 ns0 ns0 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-3101210111010450 ns300 ns150 ns0 ns0 ns 150 ns0 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-3101210111010 4 µs 2 µs 1 µs600 ns0 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 50 100 150 200 250Field V cm-1Initial density ρ0 cm-31012101110100 5 100 150 200 250Field V cm-1Initial density ρ0 cm-3101210111010450 ns300 ns150 ns0 ns450 ns 1 µs50 2001501000 200Electrostatic field (V cm-1)1.61.20.800.4Electron signal (arb units)Figure 4.2: (upper frames) Typical SFI spectra, formed by 4,000 SFI traces sortedaccording to the initial density ρ0, for an n f (2) Rydberg with an initial principalquantum number n0 = 44, following ramp delays, ∆tramp, of 0,150, 450 and1000 ns. (bottom frame) Single SFI traces obtained under identical conditionsof initial 49 f (2) Rydberg gas density (1011 cm−3) and ramp-field delay (2 µs)in the absence (upper) and presence (lower) of a low-amplitude CW 60 MHzradio frequency field (Vpp = 0.125 V cm−1).300.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5ramp delay (us)00.20.40.60.811.2Rydberg signaln=49 RydbergRamp delay (µs)Rydberg field ionization signal (arb.)0 1 2 3 4 5 6ramp delay (us)00.20.40.60.811.21.41.61.82plasma signaln=49 plasmaVpp0.00VVpp0.50VVpp1.00VVpp2.00VVpp5.00VRamp delay (µs)Rydberg field ionization signal (arb.)ABCDEABCDEFigure 4.3: Ultracold plasmas evolving from 49n f (2) Rydberg gases: (top) SFIamplitude integrated over ramp field from 0 to 50 V cm−1 (plasma signal);and (bottom) SFI amplitude integrated over ramp field from 50 to 200 V cm−1(Rydberg signal), as a function of ramp delay, ∆tramp under field-free conditions(A) and in the presence of a 60 MHz rf pulse that starts 500 ns afterω2 excitationand extends to fill the elapsed time, ∆tramp, until the start of ramp field. Peak-to-peak pulse amplitudes vary with increasing effect over the range, Vpp = 0.125(B), 0.25 (C), 0.5 (D), and 1.25 (E) V cm−1.31to NO+ cations and extracted by the leading edge of the SFI field ramp remainsconstant as we step the start of the ramp over a time from 500 ns to 5.5 µs after theω2 excitation pulse that forms the Rydberg gas. During this same time interval, theRydberg signal falls exponentially, in each case to an apparent plateau.4.3.2 Molecular ultracold plasma rate processes in the presence of acontinuous wave (CW) radio frequency fieldFigure 4.3 also includes equivalent measurements made in the presence of a CWradio frequency field with a frequency of 60 MHz and varying peak to peak ampli-tude. Here we see that the presence of an rf field of this frequency diminishes theplasma signal to a degree that depends on the peak-to-peak amplitude, Vpp. The rffield has a more dramatic effect on the decay of the n0 Rydberg signal. Over therange of amplitudes considered, the decay rate increases by a factor of from twoto three and the plateau, which defines the fraction of n0 Rydberg molecules thatsurvive for very long times, decreases by a factor of ten.This distinct effect of a radio frequency field on the SFI signal appears clearlyin a single raw ramped field-ionization trace. Figure 4.2 shows the electron wave-form obtained at intermediate density by the application of a field-ionization rampdelayed by 2 µs after ω2. As in the accompanying contours, the low-voltage partof the ramp collects loosely-bound electrons from the plasma. At higher voltage,we see the field ionization of the residual n0 = 44 Rydberg gas to form NO+ X 1Σ+cation rotational states, N+ = 0 and N+ = 2. Here, we also see the electron signalproduced by the same ramp in the presence of a 60 MHz radio frequency field withan amplitude of 0.125 V cm−1. Note as above, on this timescale, the rf field haslittle effect on the weakly bound plasma waveform, but it substantially reduces theamplitude of the Rydberg signal.4.3.3 Effects of a pulsed radio frequency fieldFor a nominally constant initial Rydberg gas density, set by a fixed ω1−ω2 delayof 200 ns, we find that a 250 ns pulsed 60 MHz radio frequency field applied toG2 affects the n0 Rydberg field ionization yield to a degree that varies with ∆tωrf ,varied between zero (the time of ω2) and ∆tramp.32-1 0 1 2 3 4 5 6RF delay (us)00.511.522.5Rydberg signalRF delay (µs)Rydberg field ionization signal (arb.)-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5RF delay (us)00.511.522.5Rydberg signalVpp0.00VVpp0.50VVpp1.00VVpp2.00VVpp5.00V-1 0 1 2 3 4 5 6RF delay (us)00.511.522.5Rydberg signalVpp0.00VVpp0.50VVpp1.00VVpp2.00VVpp5.00VRF delay (µs)Rydberg field ionization signal (arb.)Rydberg field ionization signal (arb.)RF delay (µs)-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5RF delay (us)00.511.522.5Rydberg signalVpp0.00VVpp0.50VVpp1.00VVpp2.00VVpp5.00V-1 0 1 2 3 4 5 6RF delay (us)00.511.522.5Rydberg signalVpp0.00VVpp0.50VVpp1.00VVpp2.00VVpp5.00VRF delay (µs)Rydberg field ionization signal (arb.)Rydberg field ionization signal (arb.)RF delay (µs)-1 0 1 2 3rf delay (µs)-rf delay (µs)-1 0 1 2 3rf delay (µs)4 5 6Rydberg!Field-ionization signal (arb units)00.51.52.521Rydberg!Field-ionization signal (arb units)00.51.52.521Rydberg!Field-ionization signal (arb units)00.51.52.521ABCDEABCDEABCFigure 4.4: Integrated electron signal obtained by selective field ionization of anultracold plasma as it evolves from a 49 f (2) Rydberg gas of NO in the presenceof a 250 ns 60 MHz pulsed radio frequency field, plotted as a function of rfdelay, ∆tωrf for five rf amplitudes, Vpp in V cm−1, 0 (A), 0.125 (B), 0.25 (C), 0.5(D) and 1.25 (E), for a 200 ns ω1−ω2 delay and two fixed values of ramp fielddelay, ∆tramp, of 2.8 µs (top) and 4.8 µs (middle) after ω2 excitation. (bottom)Fixed ramp field delay and Vpp with three values of ω1−ω2 delay, 100 (A), 200(B) and 300 (C) ns. Each curve is normalized according to the signal at -1µs rfpulse delay when the rf pulse has no effect.33The SFI signal increases for an rf pulse applied during or just after ω2. Beyonda delay of 500 ns, the application of a pulsed radio frequency field depletes the NORydberg molecule signal to a degree that depends on the temporal relation of therf pulse to ω2 and the time at which the SFI ramp field on G1 begins to rise.Figure 4.4 shows this effect for two different intervals of ∆tramp, 2.8 and 4.8 µs.Here we see that the height of the signal increase for ∆tωrf < 500 ns and the depthof depletion for ∆tωrf > 500 ns both grow with increasing rf field amplitude, Vpp,from 0.125 to 1.25 V cm−1.As the delay time of the radio-frequency pulse, ∆tωrf nears the ramp delay,∆tramp, the Rydberg signal recovers. The steepness with which the Rydberg fieldionization signal grows as ∆tωrf approaches ∆tramp measures the rate of rf-acceleratedpredissociation. The rf-field accelerated component of NO Rydberg predissocia-tion clearly increases with Vpp. The degree of acceleration as a function of Vppmatches for SFI ramp delays, ∆tramp = 4.8 and 2.8 µs.The depth of the n0 Rydberg SFI signal depletion for ∆tωrf > 500 ns, as well asthe shape of its recovery as ∆tωrf approaches ∆tramp, also vary with the initial densityof the Rydberg gas. Figure 4.4 shows examples obtained for ω1−ω2 delays of 100,200 and 400 ns. Here, for a constant rf amplitude, Vpp = 0.4 V cm−1, we see theleast depletion at the lowest initial Rydberg gas density (obtained for the longestω1−ω2 delay). Less obviously, the apparent predissociation rate measured by therise in the signal as ∆tωrf approaches ∆tramp lessens in an ultracold plasma of lowerdensity.Note, as seen for the experiment varying Vpp, the n0 Rydberg SFI signal in-creases in the presence of an rf field for ∆tωrf < 500 ns. Here, for constant Vpp = 0.4V cm−1, an early pulsed rf field produces the greatest proportional enhancementby far for the lowest initial Rydberg gas density. When applied in the first 250 nsafter ω2, a radio frequency field with an amplitude of 400 mV cm−1 causes verylittle if any enhancement in the SFI signal obtained for the Rydberg gas with acomparatively high initial-density, formed by using an ω1−ω2 delay of 100 ns.344.4 Discussion4.4.1 Field-free avalanche dynamics and dissociation in thestate-selected nitric oxide Rydberg gasDouble-resonant excitation of nitric oxide at moderate density in a seeded super-sonic molecular beam entrains an ellipsoidal volume of Rydberg gas that undergoesavalanche to form an ultracold plasma. Electron signal contours in sorted selectivefield ionization spectra such as those evident in Figure 4.2 show how the electronbinding dynamics of this avalanche vary with the initial density of the Rydberg gas.Here we see snapshots of the ultracold plasma electron binding energy spectrumformed by 44 f (2) Rydberg gases ranging over two orders of magnitude in initialdensity after evolution times selected by ∆tramp values of 0, 150, 450 and 1000 ns.For initial densities exceeding 5× 1011 cm−3, virtually all of the charge ex-tracted by an applied ramp appears at relatively low field, over a range that extendsfrom about 5 to no more than 80 V cm−1. The collection of these easily extractedelectrons at zero ramp-delay signals a prompt avalanche that promotes the entiresystem to a state composed of high-Rydberg molecules and/or quasi-free electronsbound by the space charge of NO+ ions.A Rydberg gas of moderate density, on the order of 3×1011 cm−3, also avalanchesto yield electrons of low binding energy, mixed here with a residual populationof Rydberg molecules bound with an energy determined by the initial principalquantum number, n0. This signature of n0 Rydberg molecules, which field-ionizeto form NO+ in N+ = 0 and 2, dominates the SFI spectrum of lower-initial-densityRydberg gases when measured with short ramp delays. For all densities, theparticle balance in this ultracold plasma shifts after a longer evolution time tofavour electrons weakly bound to NO+ ions, as high-n Rydberg molecules or aselectrons bound to the plasma space charge.Inspection of SFI spectra between 0 and 150 ns shows direct evidence of initialelectron mobility in a subtle shift of residual n0 features to higher appearance po-tential. This shift to higher field-ionization threshold points to a dynamic process inwhich electron-Rydberg collisions drive ℓ-mixing interactions that cause the photo-selected 44 f (2) molecules to spread over the full distribution of ℓ ∈ (0,1,2...43)35values of Rydberg orbital angular momentum for n0 = 44 [158].From previous work [53], we know that the overall density of the nitric ox-ide molecular ultracold plasma falls with evolution time during the first few mi-croseconds, owing to channels of neutral decay via NO+ ion-electron dissociativerecombination and neutral NO Rydberg predissociation [126, 132, 125]. Figure 4.3details this effect experimentally. Note that the residual n0 Rydberg signal decays,while the plasma signal remains constant on a microsecond timescale. This directlyshows that n0-Rydberg predissociation dominates ultracold plasma decay on thetimescale of this observation.However, after a period of time that varies to some degree with Rydberg gasdensity and initial principal quantum number, the n0 Rydberg molecules sampledby SFI cease to predissociate. Figure 4.2 shows this effect distinctly in the residualn0-Rydberg signal evident after a delay of 1 µs in the SFI spectrum of a 44 f (2)Rydberg gas with an initial density of 1011 cm−3. This appears more evidently inthe exponential decay of the integrated n0 = 49 Rydberg signal as a function oframp field delay in Figure 4.3, which in the absence of a radio frequency field fallsto a residual plateau.Recognizing this, we fit the data in Figure 4.3 to a rate law of the form:[NO∗][NO∗]0= (1− [NO∗]A[NO∗]0)exp(−kPD [NO∗])+ [NO∗]A[NO∗]0(4.1)where [NO∗] describes the density of predissociating n0 Rydberg molecules, [NO∗]Arepresents the residual population in a state of arrested predissociation. Here, kPDrefers to a phenomenological overall rate of predissociation. Table 4.1 gives theseparameters for the fits plotted in Figure 4.3.4.4.2 Nitric oxide Rydberg predissociation in a regime of ℓ-mixingPrevious studies by Vrakking and Lee have established that a nitric oxide Rydbergmolecule in the n0 f (2) series near n0 = 49 predissociates with a field-free lifetimeof 10 ns [81, 82]. A dc field of a few hundred mV cm−1 mixes ℓ sufficiently toincrease this lifetime to a measured 75 ns. Multichannel effective Hamiltonianmodels predict that effective kPD values for fully coupled bright states in this range36Table 4.1: Kinetic parameters used in Eq (4.1) to fit the exponential decay in then0 = 49 Rydberg molecule SFI signal in Figure 4.3 under field-free conditionsand in the presence of a 60 MHz radio frequency field of varying peak-to-peakamplitude, Vpp.Vpp (V cm−1 ) [NO∗]0 kPD (µs−1) [NO∗]A / [NO∗]0Field Free0.000 1 0.82 0.3060 MHz0.125 1 1.39 0.100.250 1 1.51 0.080.500 1 1.90 0.041.250 1 2.30 0.03of the n f series decrease to ∼ 5 µs−1 [13, 116]. Predissociation in the broadermanifold of n0 = 49 Rydberg states, mixed by electron collisions over all values ofℓ, proceeds with a phenomenological kPD determined by sampling ℓ-detailed rates.Referring to work cited above, we can assume that the Rydberg states of nitricoxide predissociate with characteristic rate constants, kPD, that fall systematicallywith increasing n as 1/n3. For a given n, kPD depends very sensitively on orbitalangular momentum ℓ. Only low-ℓ states decay with appreciable rates. For thepurposes of illustration, we can take rates from a model developed by Gallagherand coworkers [102], patterned on the work of Bixon and Jortner [13]. and estimatekPD for a given n from the statistically weighted sum of ℓ-dependent rate constants,kℓ = 0.014, 0.046, 0.029 and 0.0012 in atomic units for ℓ from 0 to 3, and 0.00003for ℓ≥ 4, scaled by n−3:kPD(n) =∑ℓ (2ℓ+1)kn,ℓn24.13×1016s−12πn3(4.2)This simple statistical approach predicts decay times for levels near n= 50 of about200 ns, in accord with observations for bright states of NO in this range, whenprepared by broad-band excitation in the presence of an ℓ-mixing electric field[81, 13, 116].The initial predissociation kinetics observed in the present experiment conform37with this picture. Coupled differential equations describing inelastic electron-Rydberg collisional evolution in principal quantum number, n, and electron-impactionization, together with three body electron-ion recombination accurately accountfor the the first 500 ns in the field-free relaxation of a nitric oxide Rydberg gas toplasma as a function of initial density, ρ0, and initial principal quantum number, n0[132, 51]. For the particular initial density represented in Figure 4.3, we find that n0nitric oxide molecules in this ultracold plasma initially decay to neutral products ona timescale consistent with the state-detailed rate of NO Rydberg predissociationin a collisional regime of ℓ scrambling.However, as evident in Figure 4.3 and the fits to Eq (4.1) parameterized inTable 4.1, the molecular nitric oxide ultracold plasma displays a persistent residualpopulation of n0 Rydberg molecules that survives the avalanche of a state-selectedn0 Rydberg gas to plasma and the quench of this plasma to a state of very lowelectron binding energy.We can explain this apparently cold, arrested state as evidence for the presenceof a long-lived high-ℓ residue of n0 Rydberg molecules that remains from thestatistical distribution over all accessible values of ℓ created by electron-collisionalℓ-mixing during the avalanche. Such a residue can survive only if ℓ-mixing ceases.Its presence here serves as an adventitious sensor of ℓ-mixing under these condi-tions of arrested relaxation.This enduring population of long-lived n0 Rydberg molecules thus suggeststhat the system evolves to a state of quenched predissociation in which the ultra-cold plasma contains too few free electrons to ℓ-mix these residual n0 Rydbergmolecules. We might explain this by assigning all the signal that appears in theprominent contours at low field in the SFI spectrum to electrons bound in veryhigh-n Rydberg states.However, for a initial principal quantum number in the range of n0 ≈ 44,classical rate theory considerations call for strong Rydberg-Rydberg interactions,Penning ionization and electron-ion-Rydberg molecule collisions, giving rise to anℓ-mixed quasi-equilibrium within 100 µs at a density as low as 108 cm−3 [158].Coupled rate-equation simulations predict that such interactions drive Penning ion-ization and avalanche to plasma in Rydberg gas systems with the particular rangeof densities displayed in Figure 4.2 on a microsecond timescale [51].384.4.3 Effect of a radio-frequency fieldThe nitric oxide molecular ultracold plasma evolves in the long-time limit to a statein which most of the electrons bind very weakly to ions, either individually in veryhigh-n Rydberg orbitals or collectively to the NO+ space charge [53]. Either way,the SFI experiment directly measures an ultracold plasma electron binding energyno greater than ∼ 800 GHz. A small portion of this system survives as Rydbergmolecules with the initially selected principal quantum number, n0, in a high-ℓ stateof quenched predissociation, suggesting an absence of ℓ-mixing electron-Rydbergcollisions.The application of a CW 60 MHz field with a peak-to-peak amplitude in therange from 0.125 to 1.25 V cm−1 evidently accelerates the predissociation of theseresidual n0 Rydberg molecules. A radio-frequency field interacts with a conven-tional plasma of electrons and ions to drive collective modes of motion termedplasma oscillations. A neutral plasma of defined density, ρe, supports a plasmafrequency ωrf =√e2ρe/ε0me. A field with a frequency of 60 MHz resonates withan electron-ion plasma at a density of about 106 cm−3. The density of the nitricoxide molecular ultracold plasma volumes sampled by selective field ionizationfalls in a calibrated range from 1010 to 1012 cm−3. The observed conditions ofplasmon resonance therefore classify the carrier of the measured SFI signal as adielectric phase suspended in a rarified conducting background.A radio frequency field of an amplitude in the range from 0.125 to 1.25 V cm−1drives plasma oscillations in this conducting background. This appears to causea resumption of ℓ-mixing that redistributes Rydberg orbital angular momentumfrom states of high-ℓ to low-ℓ states of shorter predissociation lifetime. As detailedabove, it is quite reasonable to attribute this rf-driven ℓ-mixing to renewed electrontransport in the bulk and an avalanche-like effectiveness of electron collisions[158].4.4.4 Kinetics of radio-frequency accelerated predissociationNitric oxide Rydberg molecules with the originally selected principal quantumnumber, n0, predissociate with detailed unimolecular rate constants kn0,ℓ. In astatistical limit, these rate constants combine in accordance with Eq (4.2) to de-39termine a phenomenological rate constant, kPD(n0) for the residual population ofn0 Rydberg molecules.Under field-free conditions, Figure 4.3 shows that the observed predissocia-tion rate constant varies in time, and ultimately falls to zero, leaving an arrestedpopulation of n0 Rydberg molecules, [NO∗]A. This is easily explained. Withoutefficient redistribution in ℓ, kPD(n0) deviates from the value predicted by Eq (4.2)for a statistical distribution of Rydberg angular momentum. As Rydberg moleculesof lower-ℓ predissociate, the distribution of population over ℓ shifts. Coefficientsof the largest terms in the sum of detailed rate constants that determines kPD(n0)fall to zero, and predissociation ceases.The presence of a CW radio-frequency field appears to sustain ℓ-mixing. Phe-nomenologically the presence of the field serves to maintain the weight of low-ℓterms in Eq (4.2), sustaining a low-ℓ contribution to the phenomenological rateconstant δkPD(n0) and diminishing the fraction of arrested molecules. Table 4.1quantifies these trends for the conditions of the experiment represented in Figure4.3, where a contribution to the predissociation rate process owing to low-ℓ pre-dissociation doubles or triples the phenomenological rate constant, kPD(n0), andreduces the arrested fraction, [NO∗]A / [NO∗]0, to nearly zero, depending on Vpp.A pulsed radio frequency field similarly promotes a redistribution of residual n0Rydberg molecules over ℓ, with a comparable effect on the apparent rate of predis-sociation. We see this in a decreased n0 Rydberg contribution to the SFI spectrum.Let us assume that a radio frequency pulse applied anytime after avalanche andelectron-collisional ℓ-mixing produces the same extent of redistribution over ℓ, andas a result, increases kPD(n0) to the same degree. Accelerated predissociation thenacts to diminish the n0 Rydberg signal by an amount that depends on the elapsedtime between the start of the radio frequency pulse, ∆tωrf , and the beginning of theSFI ramp field at ∆tramp.An advancing ∆tωrf , and thus smaller gap, ∆tramp−∆tωrf , gives an rf-acceleratedpredissociation rate constant less time to act and thus causes a smaller suppressionof the n0 Rydberg signal. We represent this effect by developing a phenomeno-logical expression for the density of Rydberg molecules as a function of ∆tωrfnormalized by its value at a time, t = ∆tramp under field-free conditions for anyinitial density, as determined say by the ω1−ω2 delay.40In a limit of instantaneous rf-induced ℓ-mixing and ultrafast NO Rydberg pre-dissociation, the normalized Rydberg field ionization signal depleted by a pulsedradio frequency field would recover to a value of 1 as the convolution of the risingedge of the rf pulse and the SFI voltage ramp. For present purposes, we canarbitrarily approximate this by a logistic function, with arbitrarily chosen onsetand rise time parameters, [tr,kr]:Se = (1− fe)+ fe[11+ ekr(tr−t)], (4.3)Here, fe represents the full fractional extent to which a radio frequency pulseof peak-to-peak amplitude, Vpp depletes the n0-Rydberg SFI signal measured by avoltage ramp that starts at ∆tramp. If the predissociation proceeds at a rate slowerthan instantaneous, then a pulsed radio-frequency field applied late in time, ∆tωrf ,close to ∆tramp, will act with less effectiveness in depleting the n0-Rydberg signalthan an rf field applied earlier.We can account for less-than-instantaneous predissociation if we regulate thedepletion of Se to a degree determined by an amount added to the predissociationrate constant, δkPD and the time difference between t = ∆tωrf and ∆tramp.Se = (1− fe)+ fe[11+ ekr(tr−t)]+ fe[e−δkPD(∆tramp−t)]×(1−[11+ ekr(tr−t)])(4.4)where we moderate the effect of the exponentially recovered n0 Rydberg signal bya step function that falls from 1 to 0. Here, we assign a uniform midpoint, t0, tothe rising and falling logistic functions that best represent the convolution of the rfpulse with the ramp. We then fit fe and δkPD as they vary with density.Just as late ℓ-mixing diminishes the n0-Rydberg SFI signal for ∆tωrf > 500 ns byredistributing high-ℓ to low, early ℓ-mixing at ∆tωrf times less than 500 ns preservesn0-Rydberg population by accelerating the promotion of n0 f (2) states selected bydouble resonant laser excitation to n0 orbitals of higher ℓ. Experiments some timeago in the Gallagher group spectroscopically quantized a similar effect in low-density Rydberg gases of NO in the present range of principal quantum number41[102].For the simple purpose of describing the observed degree of rf enhancementfor ∆tωrf < 500 ns, we fit the data phenomenologically with an exponential rise andlogistic decay:S0 =(fe+ f0e−δk0(t−t0)2)×(1−[11+ ekr(t0−t)])(4.5)Curves drawn through the data in Figure 4.4 sum S0 contributions owing toℓ-mixing Rydberg lifetime enhancement during the avalanche, Eq (4.5), togetherwith Se contributions owing to ℓ-mixing Rydberg lifetime depletion during thesubsequent phase of arrested relaxation, Eq (4.4), as a function of ∆tωrf for fixedramp field delays, ∆tramp, of 2.8 and 4.8 µs.We recognize that arbitrary logistic functions imperfectly describe convolu-tions of varying degrees of ℓ-mixing and changes in kPD as the rf pulse passesthroughω2 and the SFI ramp. Nevertheless, we readily obtain self-consistent repre-sentations of the SFI signal as a function of ∆tωrf at both ramp field delays, changingonly the known fixed value of ∆tramp. Table 4.2 summarizes the parameters of thesefits. Note that we succeed in describing the recovery of the signal as the rf pulsepasses through the ramp by a logistic function with the same offset, ∆tramp− tr atboth ramp-field delays.Table 4.2 quantitatively represents the clear trends displayed in Figure 4.4.A radio frequency field of growing peak-to-peak amplitude increases the early-time amplitude of the n0-Rydberg signal enhancement ( f0), increases the later-time amplitude of the n0-Rydberg signal depletion ( fe), and increases the degree towhich the rate of predissociation is enhanced (δkPD)4.4.5 Mechanics of radio frequency induced ℓ-mixingThe nitric oxide ultracold plasma evolves to a state of suppressed Rydberg pre-dissociation, marked here by the persistence of a residual population of high-ℓRydberg molecules that retain the initially selected principal quantum number,n0. This occurs as the consequence of a process that begins in the avalanchewith ℓ-mixing electron-Rydberg collisions. This randomization in ℓ is followed42Table 4.2: Parameters used in Eqs (4.4) and (4.5) to describe the enhancement,suppression and recovery of the n0 Rydberg SFI signal by the action of a 250ns 60 MHz pulsed radio frequency field of peak-to-peak amplitude, Vpp, as timeof this pulse, ∆tωrf advances from tω2 to pass through the beginning of the SFIramp field at ∆tramp for ramp field delays of 2.8 and 4.8 µs.Vpp ∆tω2 ρ0 kr t0 tr ∆tramp f0 fe δk0 δkPD(V cm−1) (ns) (µm−3) (µs−1) (µs) (µs) (µs) (µs−1) (µs−1)2.8 µs ramp field delay0.125 200 0.35 10 0.38 2.4 2.8 1.10 0.31 20 0.650.250 200 0.35 10 0.38 2.4 2.8 1.82 0.41 20 0.750.500 200 0.35 10 0.38 2.4 2.8 2.10 0.55 20 1.401.250 200 0.35 10 0.38 2.4 2.8 2.10 0.68 20 2.584.8 µs ramp field delay0.125 200 0.35 10 0.4 4.4 4.8 1.10 0.23 20 0.640.250 200 0.35 10 0.4 4.4 4.8 1.83 0.30 20 0.700.500 200 0.35 10 0.4 4.4 4.8 2.10 0.55 20 1.011.250 200 0.35 10 0.4 4.4 4.8 2.30 0.65 20 2.344.8 µs ramp field delay, varying initial density0.400 100 0.59 10 0.3 4.4 4.8 0.33 0.86 20 0.80.400 200 0.35 10 0.3 4.4 4.8 1.24 0.60 20 0.80.400 300 0.21 10 0.3 4.4 4.8 2.38 0.41 20 0.8by evolution in the bulk plasma to a condition both of low electron binding energyand quenched electron mobility. This quenched environment traps a measurablefraction of n0 Rydberg molecules in states of ℓ too high to predissociate.We could interpret sequences of SFI spectra, such as those shown in Figure4.2 as evidence for the classical evolution of the NO Rydberg gas to an ultracoldplasma background consisting entirely of very high-n Rydberg molecules. As-suming such states were stable, this very high-n Rydberg gas background wouldcontain no free electrons. In the absence of Penning ionization and avalanche inthis background, residual n0 Rydberg molecules of low-ℓ would predissociate, anda very long-lived high-ℓ ensemble of Rydberg molecules would remain, consistentwith the observed field-free state of arrested predissociation.The evident perturbation of this arrested system by a 60 MHz radio frequencyfield would appear to require free electrons activated by plasma oscillations, andthus oppose this Rydberg gas scenario. However, Gallagher and coworkers haveshown that a radio frequency field alone can drive transitions that scramble the43distribution over ℓwithin a single Stark manifold [102]. In their experiment on low-ℓ Rydberg states of NO, this effect lengthened predissociation lifetimes. The samemechanism of ℓ-mixing could accelerate predissociation in an arrested distributionof residual n0 Rydberg molecules of high-ℓ.Coupled rate-equation simulations suggest that Penning ionization and avalancheoccur too quickly at our density for collision-free rf excitation to serve as theleading cause of ℓ-mixing in the present case [51]. For greater certainty, we referto an experimental result presented above that tells us directly whether the rf-depletion observed here arises from electron collisions or field-induced ℓ-mixing.Predissociation stimulated by Stark mixing in a radio frequency field of a givenamplitude occurs to the same degree for every molecule in a sample of any density.Predissociation catalyzed by ℓ-mixing electron collisions proceeds as a pseudofirst-order process, and thus occurs to a fractional extent that varies with the densityof electrons. Figure 4.4 shows immediately that the fractional depletion changeswith the initial density of the Rydberg gas, controlled with precision by adjustingtheω1−ω2 delay (∆tω2). As shown in Table 4.2 curves through these data, obtainedfor densities that differ by about a factor of three, fit Eq (4.4) varying only fe.Note that the initial Rydberg gas density affects only the fractional depletion.The additive contribution to kPD does not vary over the range of charged particledensities formed as a consequence of the variation of initial Rydberg gas density.This suggests that the scrambling in ℓ saturates after very few electron-Rydbergcollisions.It is instructive to consider the effect of an rf pulse applied for ∆tωrf < 500ns on the n0-Rydberg SFI signal under conditions of constant density and varyingpeak-to-peak amplitude, Vpp (Figure 4.4) compared with an rf pulse of constantVpp applied under conditions of varying Rydberg gas density (Figure 4.4).In both cases, we can understand the effect of the radio frequency field as oneof facilitating electron-Rydberg collisions. At early times, when most Rydbergmolecules still occupy the optically selected n0 f (2) Rydberg state, electron colli-sions lead to lifetime lengthening, by scattering ℓ= 3 Rydberg molecules to statesof higher ℓ.Preceding conditions match for the systems of constant density (Figure 4.4),and enhancement at early times occurs in an equivalent growing background of44electrons released by Penning ionization and avalanche. The radio frequency fieldof largest amplitude has greatest effect on the Rydberg population both beforeand after collisional ℓ-mixing is complete. The highest Vpp causes the largestenhancement for ∆tωrf < 500 ns and deepest depletion for ∆tωrf > 500 ns, The Table4.2 amplitudes, f0 and fe increase together with increasing Vpp.Under conditions of constant Vpp but varying density, the initial Penning elec-tron density and avalanche rate differ substantially, as can be read directly in Figure4.4. The system of lowest density sustains a comparatively low avalanche electrondensity and so sees a comparatively smaller degree of rf-induced depletion. But,its early lingering population of n0 f (2) Rydberg molecules exhibits a large degreeof rf-accelerated ℓ-mixing and stabilization.The opposite situation applies in the system of highest density. With a largeavalanche electron density, the second-order amplitude of rf-induced ℓ-mixing andpredissociative SFI signal loss is great. But with a rapid avalanche and nearlycomplete degree of early electron-collisional ℓ-mixing, the system shows very littleSFI signal increase as the rf pulse scans through ∆tωrf < 500 ns. In Table 4.2, theamplitudes, f0 and fe vary in opposite directions with density.4.4.6 Electron mobility in the quenched molecular ultracold plasmaThe nitric oxide molecular ultracold plasma contains a persistent residue of NORydberg molecules that retain the initially selected principal quantum number, n0.The application of a weak radio frequency field subtly changes the state of thisplasma in a way that causes the predissociation of this residue to accelerate.The NO Rydberg residue represents a surviving fraction of n0 molecules trappedin states of high ℓ populated during the avalanche and quenched to a regime ofsuppressed ℓ-mixing. Predissociation resumes when the rf field acts to scramblethe orbital angular momentum of those shelved, high-ℓ, n0 Rydberg molecules.The fractional yield of n0 Rydberg molecules undergoing radio frequency field-accelerated predissociation varies substantially with plasma density. This excludesℓ-mixing as a field-induced process in isolated-molecules. We know that chargedparticles in a plasma also respond collectively to a radio frequency field by exe-cuting oscillatory modes of electron and ion motion. Damping via charge coupling45and collisions couples energy from the rf field to the plasma. If this heating exceedsa mobility threshold for localized electrons, we should expect to see an effect ofthe rf field in an increase of the frequency of electron Rydberg collisions.That appears to be the case here. The kinetics tell us that the rf field acts on theplasma to mobilize electrons. These electrons collide with Rydberg molecules andscramble ℓ. Thus, we see an effect in the arrested residue of n0 Rydberg moleculesowing to a process that occurs in the ultracold plasma background. Note that thismobilization, which dramatically accelerates predissociation kinetics, causes littlechange in the ultracold plasma background.We can therefore regard this adventitious population of high-ℓ n0 Rydbergmolecules and its response to an rf field as a quantum-state probe of the dynamicsof avalanche and quench that form the molecular ultacold plasma. The natural stateof arrested predissociation points to an immobility of electrons in the quenchedplasma. A radio frequency field acts to mobilize electrons, which increases therate of predissociation in a fraction of n0 Rydberg population that grows with thedensity of the background plasma.Important questions remain. If we can conclude that a radio frequency fieldpromotes electron-Rydberg collisions, what initial state of the ultracold plasmaserves as the source of electrons mobilized by the rf field? What constrains themobility of these electrons under field-free conditions? How do we describe theinteraction with the rf field that causes this increase in electron mobility?46Chapter 5mm-wave Rydberg-Rydbergtransitions gauge intermolecularcoupling in a molecular ultracoldplasmaOut-of-equilibrium, strong correlation in a many-body system can trigger emergentproperties that act to constrain the natural dissipation of energy and matter. Signsof such self-organization appear in the avalanche, bifurcation, and quench of astate-selected Rydberg gas of nitric oxide to form an ultracold, strongly correlatedultracold plasma. Work reported here focuses on initial stages of avalanche andquench, and uses the mm-wave spectroscopy of an embedded quantum probe tocharacterize the intermolecular interaction dynamics associated with the evolu-tion to plasma. Double-resonance excitation prepares a Rydberg gas of nitricoxide composed of a single selected state of principal quantum number, n0. Pen-ning ionization, followed by an avalanche of electron-Rydberg collisions, forms aplasma of NO+ ions and weakly bound electrons, in which a residual populationof n0 Rydberg molecules evolves to a state of high orbital angular momentum,ℓ. Predissociation depletes the plasma of low-ℓ molecules. Relaxation ceasesand n0ℓ(2) molecules with ℓ ≥ 4 persist for very long times. At short times,47varying excitation spectra of mm-wave Rydberg-Rydberg transitions mark the rateof electron-collisional ℓ-mixing. Deep depletion resonances that persist for longtimes signal energy redistribution in the basis of central-field Rydberg states. Thewidths and asymmetries of Fano lineshapes witness the degree to which coupling inthe arrested bath i) broadens the allowed transition and ii) mixes the local networkof levels in the ensemble.5.1 IntroductionRydberg-Rydberg interactions occur with exaggerated coupling energies in sys-tems of rarified density to form important prototypical out-of-equilibrium stronglycorrelated many-body systems [14]. Atomic Rydberg ensembles exhibit coopera-tive behaviour ranging from precisely defined pairwise and higher-order coherentphenomenon [143, 38, 40, 16] to aggregation [41, 133, 153], dissipation [72, 42, 73,11], non-equilibrium phase transitions [20, 85] and avalanche to plasma [65, 163,4]. Recent experiments have established evidence for self-organizing criticality inultracold and room-temperature atomic gases [54, 28].Work in our laboratory has focused on the quantum mechanical propertiesof the out-of-equilibrium state formed by comparable interactions in a molecularRydberg gas. Here, we explore the dynamics of by which a state-selected Rydberggas of nitric oxide entrained at milli-Kelvin temperature in a skimmed supersonicmolecular beam evolves via a sequence of electron-impact avalanche, bifurcationand quench to form a strongly coupled, ultracold plasma [98, 125, 139, 138, 53,51]. We assign systematic variations in the intensities and lineshapes of mm-waveRydberg-Rydberg transitions that paint a dynamical picture of this state as a self-assembled, spatially correlated quantum system of randomly interacting dipoles ofrandom energies [145, 147]. These dipoles populate a wide distribution of Rydbergand excitonic states concentrated in an energy interval within a few hundred GHzof the ionization threshold, doped by a trace population of more deeply boundRydberg molecules.This residual population of lower-n Rydberg NO∗ molecules includes a fractionthat retains the selected principal quantum number, n0. These molecules initiallypopulate the optically selected total angular momentum state (neglecting spin),48N = 1, n0 f (2). On a timescale of a few-hundred nanoseconds, electron-Rydbergcollisional ℓ-mixing drives this population to non-penetrating states of high-ℓ. Be-fore this happens, mm-wave radiation tuned in the frequency range between 70 and110 GHz forms an excitation spectrum of transitions from n0 f (2) to (n0±1)g(2)for plasmas evolving from NO Rydberg gases selected by ω2 tuned from n0 = 39to 45.A few microseconds later, ℓ-mixing collisions slow down, conserving a long-lived high-ℓ population that exhibits resonant dips at the frequency of n0ℓ(2) to(n0±1)d(2) transitions. Such transitions deplete the plasma signal to an amplitudenear zero, even though delayed selective field ionization spectra show that thedistribution of states evolves by then to contain fewer than one percent in the n0level resonant with the mm-wave field. Thus, the ωmm excitation spectrum itselfshows direct evidence that intermolecular energy transfer must occur to fill the holeburned by mm-wave radiation in the distribution of Rydberg states.The excitation spectrum of Rydberg-Rydberg transitions to states of higherand lower orbital angular momentum displays Fano lineshapes. Rydberg-Rydberginteractions characteristically broaden and displace these asymmetric features, re-distributing spectral weight to a localized network of coupled Rydberg levels thatforms an underlying quasi-continuum of dark states.Thus, the nitric oxide molecular ultracold plasma presents an out-of-equilibriummany-body system in which strong correlation drives self-organization to a canon-ical state of arrested relaxation. In this state, the mm-wave excitation spectrum ofNO∗, serving as an embedded quantum probe, gauges the degree to which variouselectromagnetic transitions either quench or favour ergodicity.5.2 Experimental5.2.1 Formation of Rydberg gas and its evolution to ultracold plasmaNitric oxide gas, seeded at a ratio of 1:10 in a pulsed supersonic jet of helium,expands from a stagnation pressure of 5 bar. This jet travels 35 mm to pass througha 1 mm diameter skimmer separating the source and experimental chambers of adifferentially pumped vacuum system, as diagrammed in Figure 5.1a. A pair of co-49propagating frequency-doubled 5 ns Nd:YAG pumped dye laser pulses (ω1 +ω2)intercepts the NO molecular beam between two grids G1 and G2 to produce a state-selected Rydberg gas. Normally, we hold G1 to ground and adjust G2 over a rangeof±100 mV to create a field-free region between G1 and G2 during laser excitation. NO+ X 1Σ+ v+ = 0X 1Σ+ v+ = 0A 2Σ+ v' = 0NO NO+ X 2Π3/2 v' = 0X 2Π1/2 v' = 0ω2nf(2)N = 1N' = 0N" = 1326 nmω1 226 nmJ" = 1/2F1F2NO X 2ΠA 2Σ+NO+ X 1Σ+ v+ = 0ω1ω2nf(2)N = 1e-G1 G2 G3 100 V0 20 40 60 80 100Time (µs)The grid and detector assembly moves longitudinallyunder external motorized control. By this mechanism, we canvary the flight time through the G −1 G2 evolution regionfrom t = 0, when laser excitation forms the Rydberg gas, untilthe the excitation volume transits G2 and forms an electronsignal.To obtain pulsed-field-ionization spectra, we connect theoutput of a Behlke high-voltage switch to G1. A pulse appliedto this electrode creates a rising electric field in the evolutionregion. A tuned RC circuit controls the rise time of this pulsedfield. Figure 2 shows the experimental field pulse created for afixed G −1 G2 span of 3.6 cm, rising from 0 to 838 V cm−1with a time constant of three μs.To probe the excitation volume by means of a low-vol-tage pulsed field, we use a Tektronix AWG2020 arbitrarywaveform generator to create a one μs, −3 V square wavepulse, which we apply to G1.3. Results3.1. Preparation of the Xe Rydberg gasWith ω1 set to excite the transition in Xe from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2, we tune ω2 over a range from 515 nmto 508 nm, which drives absorption from the p6 state toRydberg states in the ns and nd series with an initial principalquantum number ranging from =n 300 to 70. Using a for-ward-bias high-voltage pulse, applied to G1 after a delay ofone μs, we detect the ω2 absorption spectrum by pulsed-fieldionization. Figure 3 plots the Rydberg spectrum from n = 38to n = 53, observed in ω2 transitions from Xe p6 by inte-grating the pulsed-field ionization signal.If we omit the pulsed field and wait for the excitationvolume to travel the distance set between the excitation regionand G2, we observe a characteristic waveform, as displayed infigure 4. The small prompt electron signal at μ≈t 0 s appearswith an intensity that does not vary with the properties of ω2,but does depend sensitively on the ω1 power, indicating thatthis weak feature arises from ω1 three-photon ionization.The arrival time of the second feature depends on thedistance from the excitation region to G2. This distancedivided by this late-peak arrival time conforms precisely withthe measured −385 m s 1 velocity of Xe atoms in our beam.The width of this feature in time transforms to the spatialwidth of the excitation volume in the direction of propagation.Figure 2. RampQ3 pulse used to obtain field-ionization spectra of XeRydberg states. A tuned RC circuit produces a pulsed electrostaticfield that rises to 838 V cm−1 with a time constant of three μs. Figure 3.A portion of the pulsed-field ionization detected absorptionspectrum of high-Rydberg transitions originating from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2. The ladder marks resonances assigned to thend series converging to Xe+ p P5 5 2 3/2.Figure 4. Oscilloscope trace showing the electron signal as afunction of time at the MCP detector. The prompt peak at 0 μs arisesfrom three ω1-photon ionization of xenon atoms together with theinstantaneous evaporation of prompt electrons following Rydber-g–Rydberg Penning ionization. The second, stronger electron signalmarks the transit of the excited volume through G2.3J. Phys. B: At. Mol. Opt. Phys. 00 (2014) 000000 J Hung et al700600500400300200100 00 1 2 Time (µs)Field V/cm2730380 30400 30420 30440 30460 30480 30500 30520 ω2 Energy (cm-1) 502 50a)b) c)ω1+ω2G1 G3G230400 30420 30440 30460 30480 30500 30520ω2 (cm-1)1 44,319 cm-12 30,345 to 30,489 cm-1NO+ X 1Σ+ v+ = 0X 1Σ+ v+ = 0A 2Σ+ v' = 0NO NO+ X 2Π3/2 v' = 0X 2Π1/2 v' = 0ω2nf(2)N = 1N' = 0N" = 1326 nmω1 226 nmJ" = 1/2F1F2NO X 2ΠA 2Σ+NO+ X 1Σ+ v+ = 0ω1ω2nf(2)N = 1e-G1 G2 G3 100 V0 20 40 60 80 100Time (µs)The grid and detector assembly moves longitudinallyunder external motorized control. By this mechanism, we canvary the flight time through the G −1 G2 evolution regionfrom t = 0, when laser excitation forms the Rydberg gas, untilthe the excitation volume transits G2 and forms an electronsignal.To obtain pulsed-field-ionization spectra, we connect theoutput of a Behlke high-voltage switch to G1. A pulse appliedto this electrode creates a rising electric field in the evolutionregion. A tuned RC circuit controls the rise time of this pulsedfield. Figure 2 shows the experimental field pulse created for afixed G −1 G2 span of 3.6 cm, rising from 0 to 838 V cm−1with a time constant of three μs.To probe the excitation volume by means of a low-vol-tage pulsed field, we use a Tektronix AWG2020 arbitrarywaveform generator to create a one μs, −3 V square wavepulse, which we apply to G1.3. Results3.1. Preparation of the Xe Rydberg gasWith ω1 set to excite the transition in Xe from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2, we tune ω2 over a range from 515 nmto 508 nm, which drives absorption from the p6 state toRydberg states in the ns and nd series with an initial principalquantum number ranging from =n 300 to 70. Using a for-ward-bias high-voltage pulse, applied to G1 after a delay ofone μs, we detect the ω2 absorption spectrum by pulsed-fieldionization. Figure 3 plots the Rydberg spectrum from n = 38to n = 53, observed in ω2 transitions from Xe p6 by inte-grating the pulsed-field ionization signal.If we omit the pulsed field and wait for the excitationvolume to travel the distance set between the excitation regionand G2, we observe a characteristic waveform, as displayed infigure 4. The small prompt electron signal at μ≈t 0 s appearswith an intensity that does not vary with the properties of ω2,but does depend sensitively on the ω1 power, indicating thatthis weak feature arises from ω1 three-photon ionization.The arrival time of the second feature depends on thedistance from the excitation region to G2. This distancedivided by this late-peak arrival time conforms precisely withthe measured −385 m s 1 velocity of Xe atoms in our beam.The width of this feature in time transforms to the spatialwidth of the excitation volume in the direction of propagation.Figure 2. RampQ3 pulse used to obtain field-ionization spectra of XeRydberg states. A tuned RC circuit produces a pulsed electrostaticfield that rises to 838 V cm−1 with a time constant of three μs. Figure 3.A portion of the pulsed-field ionization detected absorptionspectrum of high-Rydberg transitions originating from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2. The ladder marks resonances assigned to thend series converging to Xe+ p P5 5 2 3/2.Figure 4. Oscilloscope trace showing the electron signal as afunction of time at the MCP detector. The prompt peak at 0 μs arisesfrom three ω1-photon ionization of xenon atoms together with theinstantaneous evaporation of prompt electrons following Rydber-g–Rydberg Penning ionization. The second, stronger electron signalmarks the transit of the excited volume through G2.3J. Phys. B: At. Mol. Opt. Phys. 00 (2014) 000000 J Hung et al700600500400300200100 00 1 2 Time (µs)Field V/cm2730380 30400 30420 30440 30460 30480 30500 30520 ω2 Energy (cm-1) 502 50a)b) c)ω1+ω2G1 G3G230400 30420 30440 30460 30480 30500 30520ω2 (cm-1)NO+ X 1Σ+ v+ = 0X 1Σ+ v+ = 0A 2Σ+ v' = 0NO NO+ X 2Π3/2 v' = 0X 2Π1/2 v' = 0ω2nf(2)N = 1N' = 0N" = 1326 nmω1 226 nmJ" = 1/2F1F2NO X 2ΠA 2Σ+NO+ X 1Σ+ v+ = 0ω1ω2nf(2)N = 1e-G1 G2 G3 100 V0 20 40 60 80 100Time (µs)The grid and detector assembly moves longitudinallyunder exter al motorized control. By this mechanism, we canvary the flight time through the G −1 G2 evolution regionfrom t = 0, when laser excitation forms the Rydberg gas, untilthe the excitation volume transits G2 and forms a electronsignal.To obtain pulsed-field-ionization spectra, we connect theoutput of a Behlke high-voltage switch to G1. A pulse appliedto his electrode creates a rising electric field in the evolutionregion. A tuned RC circuit controls the rise time of this pulsedfield. Figure 2 shows the experimental field pulse created for afixed G −1 G2 span of 3.6 cm, rising from 0 to 838 V cm−1with a time constant of three μs.To probe the excitation volume by me ns f a low-vol-tage pulsed field, we use a Tektronix AWG2020 arbitrarywaveform generator to create a one μs, −3 V square wavepulse, which we apply to G1.3. Results3.1. Preparation of the Xe Rydberg gasWith ω1 set to excite the transition in Xe from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2, we tune ω2 over a range from 515 nmto 508 nm, which drives absorption from the p6 state toRydberg states in the ns and nd series with an initial principalquantum number ranging from =n 300 to 70. Using a for-ward-bias high-voltage pulse, applied to G1 after a delay ofone μs, we detect the ω2 absorption spectrum by pulsed-fieldionization. Figure 3 plots the Rydberg spectrum from n = 38to n = 53, observed in ω2 transitions from Xe p6 by inte-grating the pulsed-field ionization signal.If we omit the pulsed field and wait for the excitationvolume to travel the distance set between the excitation regionand G2, we observe a characteristic waveform, as displayed infigure 4. The small prompt electron signal at μ≈t 0 s appearswith an intensity that does not vary with the properties of ω2,but does depend sensitively on the ω1 power, indicating thatthis weak feature arises from ω1 three-photon ionization.The arrival time of the second feature depends on thedistance from the excitation region to G2. This distancedivided by this late-peak arrival time conforms precisely ththe measured −385 m s 1 velocity of Xe atoms in our beam.The width of this feature in time transforms to the spatialwidth of the excitation volume in the direction of propagation.Figure 2. RampQ3 pulse used to obtain field-ionization spectra of XeRydberg states. A tuned RC circuit produces a puls d electrostaticfiel that rises to 838 V cm−1 with a time constant of three μs. Figure 3.A portion of the pulsed-field ionization detected absorptionspectrum of high-Rydberg transitions originating from Xe⎡⎣ ⎤⎦p p5 P 6 [5/2]5 2 3/2 2 2. The ladder marks resonances assigned to thend series converging to Xe+ p P5 5 2 3/2.Fig re 4. Oscilloscope trace showing the electron signal as afunction of time at the MCP detector. The prompt peak at 0 μs arisesfrom three ω1-photon ionization of xenon atoms together with theinstantaneous evaporation of prompt electrons following Rydber-g–Rydberg Penning ionization. The second, stronger electron signalmarks the transit of the excited volume through G2.3J. Phys. B: At. Mol. Opt. Phys. 00 (2014) 000000 J Hung et al700600500400300200100 00 1 2 Time (µs)Field V/cm2730380 30400 30420 30440 30460 30480 30500 30520 ω2 Energy (cm-1) 502 50a)b) c)ω1+ω2G1 G3G230400 30420 30440 30460 30480 30500 30520ω2 (cm-1)G1ω1 + ωmmNO"HeFigure 5.1: (a) Co-propagating laser beams, ω1 and ω2, cross a molecular beamof nitric oxide between entrance aperture G1 and grid G2 of a differentiall -pumped vacuum chamber. G2, G3 and a MCP det ctor as e bly move on acarriage to afford externally adjustable G1 - G2 spacings from 3 to 60 m (b)Nd:YAG-pumped dye lasers (ω1 and ω2) pump ground state NO fir t to theexcited A 2Σ+ N′ = 0 state and then to a Rydberg level with N = 1.The ω1 pulse, tuned to 226 nm and adjusted to an output energy between 1 and10 µJ, excites ground state NO molecules to the rotationally selected electronicallyexcited intermediate state, A 2Σ+ N ′ = 0. A second dye laser pulse, ω2, tuned from327 nm to 330 nm, promotes these selected A 2Σ+ molecules to high Rydbergstates with electron binding energies in the range from 1 to 120 cm−1 below the50N+ = 0 ionization limit of NO+ (Figure 5.1b).Parity and angular momentum selection rules from intermediate N′= 0 confineRydberg states populated by double resonance to those with total angular momen-tum neglecting spin of N = 1. Among optically accessible Rydberg states, onlythose of principal quantum number n in the f series (l = 3) converging to theN+ = 2 limit, n f (2) for n from 30 to 80, have sufficient lifetime to form a Rydberggas that evolves to form an ultracold plasma.The second laser, with a pulse energy in the range of 20 mJ, saturates the ω2transition, so the intensity of ω1 determines the initial density of the Rydberg gas.A ω1 pulse energy that exceeds 8 µJ saturates the X to A transition, which yieldsan upper-limiting initial Rydberg gas density in excess of 1012 cm−3 [98, 139]. Atany fixed ω1 intensity, the radiative lifetime of the A-state affords a precise meansto reduce the initial Rydberg gas density by choosing a nonzero ω1 −ω2 delay[141].For the purposes of the present experiment, we set the ω1 pulse energy to 2 µJand increase the ω1−ω2 delay to 500 ns, which forms a Rydberg gas with a re-duced initial density of about 1011 cm−3. Prompt Penning ionization in the leadingedge of the Rydberg-Rydberg nearest neighbour distribution yields an adventitiouspopulation of free electrons.These electrons, trapped by the NO+ space charge drive an electron-impactavalanche. The illuminated volume evolves on a 100 ns timescale to form a quasi-equilibrium of NO+ ions, electrons and high-Rydberg molecules [53]. Ensembledynamics redistribute the energy released in the avalanche in a hydrodynamicprocess that quenches the electron temperature as well as the relative motion ofions and Rydberg molecules [138, 53].5.2.2 Selective field ionization as a gauge of short-time dynamicsWe measure the distribution of electron binding energies in the Rydberg gas as afunction of delay after ω2 by means of selected field ionization (SFI) spectroscopy[39]. A Bertran high voltage power supply biases a Behlke high voltage push-pull switch to generate a 3 kV high voltage pulse. A 5 kΩ resistor in series withG1 forms an RC circuit that generates a ramped electric field between G1 and G251that rises at ∼ 0.8 V cm−1ns−1. A polynomial function fit to the leading edgeof the ramp voltage as a function of time transforms the detected electron signalwaveform to a quantity that varies with the electric field between G1 and G2.An electric field with this rise time drives the superposition of mℓ states formedby the initial preparation of a particular n0 f (2) Rydberg level to evolve diabaticallythrough the Stark manifold, crossing a saddle point to form ions and free electronswhen the field F rises to a threshold amplitude, 1/9n4 in atomic units. This sep-aration of charge appears as an electron signal that begins for F = (En(2)/4.59)2V cm−1, where En0(2) in cm−1 expresses the binding energy of the Rydberg statewith respect to a nitric oxide cation in rotational state, N+ = 2.As this N+ = 2 wavepacket propagates, it traverses many intersections withstates of matching total angular momentum and electronic × rotational parity builton the ground rotational state of NO+. These crossings confer sufficient N+ = 0character for amplitude to develop in a final state of free electrons and rotationalground state NO+ ions, which can appear as a field-ionization feature with a lowerappearance potential, when the rising field passes an amplitude of (En(0)/4.59)2V/cm.For an initial density of 1011 cm−3, a field ionization ramp begun 300 nsafter the ω2 samples the system in a state composed mainly of n0 f (2) Rydbergmolecules as initially prepared, together with Rydberg molecules in other levels,n and free electrons bound to the space charge of NO+ ions formed early in theavalanche.SFI ramps initiated with longer time delays sample distributions of n0 Rydbergmolecules altered by numerous collisions with avalanche electrons. At a rate fasterthan that of n-changing interactions, these collisions mix ℓ, populating degeneratemanifolds of states at each n, |N+ = 2⟩ |n, ℓ⟩. These states of higher orbital angularmomentum field ionize at slightly higher field amplitudes, again producing electronsignal waveforms that correspond with the formation of ions in rotational states,N+ = 0 and 2.525.2.3 Field-free propagation as a long-time gauge of plasmaevolutionIn the absence of a field-ionization ramp, the photoexcited gas volume travelswith the propagation of the molecular beam to transit the field-free region andcross the far grid, G2. A bellows actuator longitudinally translates a carriageholding perpendicularly mounted grids G2 and G3 together with a microchannelplate detector assembly (Figure 5.1a). Adjusting the flight distance from 3 to 60mm allows field-free evolution times from 2 to 40 µs.As the plasma volume passes through G2, it encounters a static field set to arange from 30 to 200 V cm−1. This field extracts an electron signal that tracesthe width of the plasma in the in the direction of propagation. We integrate thislate-peak electron signal waveform to measure the total plasma and high-Rydbergelectrons that survives for a given evolution time determined by the time of flightof the excited volume.We vary the ω1−ω2 delay to regulate the initial density of the Rydberg gas.For a controlled density, we detect the change in the integrated late-peak signalin the presence of resonant mm-wave radiation to gauge the effect of early timeRydberg-Rydberg transitions on the fraction of the initial Rydberg gas that evolvesto a long-time ultracold plasma state of arrested relaxation.5.2.4 UV-UV double-resonance and UV-UV-mm-wavetriple-resonance spectroscopyA Virginia Diodes 10-400 GHz solid-state multi-band transmitter (VDI-Tx-S129)produces a tunable CW mm-wave field, ωmm, with an intensity as high as 100 µWcm−2. This output originates from an on-board computer controlled Micro LambdaMLSE-1006 yttrium iron garnet (YIG) oscillator based frequency synthesizer, setfor present purposes to a precisely determined primary frequency, ±1 kHz, inthe range from 10 to 20 GHz. Stages of frequency doubling (WR 5.1×2R2)and tripling (WR 9.3×3) produce continuously tunable output from 60 to 120GHz. The multi-band transmitter broadcasts this output through a VDI-WR-1035.5×16.3 mm conical horn. An f/5 teflon lens collects a portion of this mm-wave electromagnetic field and focuses it on the molecular beam with a maximum53estimated intensity of 25 µW cm−2. We regulate the output by means of cardboardattenuators placed between the horn and lens, and measure the mm-wave powerincident on the 2.5 cm diameter × 1 mm thick silica window of the vacuumchamber using a TK Instruments THz absolute power meter.For ω1 fixed on a transition to the A 2Σ+ N′= 0 state, a scan of ω2 from 327nm to 330 nm yields a double-resonant spectrum of n f (2) Rydberg molecules thatappears both in the SFI and late peak ultracold plasma signals. Such scans in thepresence of a CW mm-wave field forms a strong triple-resonance signal when theω2 scan samples an n0 f (2) state for which an appropriate ∆n= 1 Rydberg-Rydbergresonance exists at ωmm (see below).Tuning ω1 and ω2 to a particular n0 f (2) Rydberg state and scaning the mm-wave frequency, we can detect similar spectra both in the SFI and late peak ul-tracold plasma signals when the three radiation fields satisfy the triple-resonancecondition. We have 1 kHz control of the mm-wave frequency, so the observedvariation of the triple-resonance signal with ωmm provides a means to assess thelineshapes of Rydberg-Rydberg transitions in the plasma.For some experiments we pulse the mm-wave field by applying 5 to 0 V half-wave voltage pulse to the TTL modulation input of the VDI-Tx-S129 transmitter.A 5 V DC control voltage sets the transmitter output to zero. Switching this inputto 0 V sets the transmitter to full power. We use this feature to delay the applicationof the mm-wave field with respect to ω2.5.3 Results5.3.1 The effect of mm-wave radiation on the ω2 spectrum ofultracold plasma NO Rydberg seriesUV-UV double-resonanceFigure 5.2 shows double-resonant spectra obtained in transitions originating fromthe A 2Σ+ N+ = 0 state for ω2 wavelengths scanned 0.6 nm in the range of 328nm in the absence of mm-wave radiation. Here, to obtain the upper spectrum,we integrate the SFI signal formed by Rydberg molecules that retain the initial54principal quantum number of the Rydberg gas, n0. The lower spectrum representsthe late peak signal formed as the illuminated volume transits G2 after 20 µs offield-free flight. Both spectra show resonances in the n f (2) series for principalquantum numbers, n0, that extend from 36 to 49, converging to the NO+ rotationallevel N+ = 2 with orbital angular momentum ℓ= 3.30450 30460 30470 30480 30490w2(cm-1)00.20.40.60.81Integrated Rydberg signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 30490w2(cm-1)00.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49fω2 -1Inegrated late peak (arb.)Inegrated Rydberg (arb.)Figure 5.2: Field free Double-resonant ω2 spectra of nitric oxide n0 f (2) Rydbergstates for n0 from 38 to 49. (top) Electron signal integrated over Rydbergresonances in the SFI spectra. (bottom) Electron signal integrated over late peakafter a flight time of 20 µsNote that, collecting either the SFI or late-peak electron signals, scans over55this principal quantum number range yield Rydberg spectra that have comparablevariations in intensity. In particular, resonances in the upper half of these spectraalternate in intensity enhancing structure for n0 = 40, 45, 47 and 49, while sup-pressing n0 = 41, 46 and 48. Interloping structure appears between resonances forn0 = 36 and 37 and n0 = 38 and 39, seen easily in the SFI-detected spectrum, anddetectable as well in the late-peak spectrum.UV-UV-mm-wave triple resonanceThe application of a 25 µW cm−2 CW mm-wave field at particular frequenciesfrom 70 to 110 GHz dramatically affects the intensities of certain particular featuresin the action spectra of n0 f (2) Rydberg resonances for n0 from 36 to 49, observedboth by integrating the SFI signal and the late peak. The ω2 scans presented inthis section show the effect of mm-wave field frequencies chosen to conform with∆n = 1 energy separation of hydrogenic Rydberg levels beginning with n0 = 39.For each selected frequency, the mm-wave field generally enhances the ω2 reso-nance associated with one particular principal quantum number, while oftentimesdiminishing the intensity of a neighbouring resonance in the n0 f (2) spectrum.108.6GHz Figure 5.3 shows SFI and late-peak ω2 spectra as a function of ω2scanned in the presence of a mm-wave field with a frequency of 108.6 GHz. Com-pare these spectra with those in Figure 5.2, recorded in the absence of a mm-wavefield. There, under field-free conditions, the 40 f (2) resonance dominates the seriesof low-n0 resonances. The addition of a 108.6 GHz field enhances the relativeintensity of the 39 f (2) resonance by a factor of five in the SFI action spectrumand by a factor of ten in the late peak action spectrum compared with the field-freeresults. The same 108.6 GHz field suppresses the 40 f (2) feature by about a factorof 1.5 in the SFI spectrum and a factor of two in the late peak spectrum.100.6 GHz A mm-wave field with a frequency of 100.6 GHz causes a similarpattern of enhancement and suppression in the SFI and late-peak ω2 spectra. Notethat a field of this frequency enhances the relative intensity of the 40 f (2) resonanceby factors of approximately two in both the SFI and late peak detected spectra. At5630450 30460 30470 30480 3049000.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.8136f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.8136f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.8136f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.8136f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.8136f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.8136f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.81Integrated latepeak signal (a.u.)36f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f30450 30460 30470 30480 3049000.20.40.60.8136f 37f 38f 39f 40f 41f 42f 43f 44f 45f 46f 47f 48f 49f 42 39 40 41 4643 44 45 47484936 37 38 4239 40 41 4643 44 45 47484936 37 3830450 30460 30470 30480 30490 30450 30460 30470 30480 30490108.6 GHz100.6 GHz93.7 GHz87.0 GHz81.4 GHz75.6 GHz71.1 GHzω2 (cm-1)Figure 5.3: ω2 spectra as in Figure 5.2, in pairs with (left) SFI and (right) late peakdetection, in each case setting mm-wave fields to the frequencies indicated.57the same time,it reduces the relative intensity of the 41 f (2) resonance slightly inthe SFI spectrum, and decreases it to nearly zero in the late-peak spectrum.93.7 GHz A mm-wave field with a frequency of 93.7 GHz increases the relativeintensity of the 41 f (2) resonance increases by a factor of five in the SFI spectrumand a factor of eight in the late peak spectrum. This mm-wave field suppressesthe 42 f (2) feature by two thirds and the 45 f (2) resonance by half in the latepeak spectrum, but causes no obvious suppression in the ω2 spectrum recordedby monitoring the SFI signal. Note as well that this field enhances an interlopingresonance that appears at 30461.46 cm−1 in the SFI spectrum. This resonanceappears weakly as a new feature in the late peak spectrum in the presence of a 93.7GHz field.87.0 GHz A mm-wave field with a frequency of 87.0 GHz increases the relativeintensity of the 42 f (2) resonance by a factor of five in the SFI spectrum and bya factor of three in the late peak spectrum. We find the 43 f (2) feature to besuppressed by half in the late-peak spectrum, but not suppressed at all in the SFIspectrum. Note that a 87.0 GHz mm-wave field again enhances the interlopingresonance that occurs at 30461.65 cm−1 in the SFI-detected ω2 spectrum. It alsocauses this feature to appear again as a new resonance in the late peak spectrum.81.4 GHz In the presence of a mm-wave field with a frequency of 81.4 GHz,the relative intensity of 43 f (2) resonance increases by a factor of two in the SFI-detected ω2 spectrum and increases by a factor of three in the late-peak ω2 spec-trum. This field suppresses the 44 f (2) feature by two thirds in the late peakspectrum, but not at all in the SFI-detected ω2 spectrum. Note that the presence ofan 81.4 GHz field causes a weak new feature to appear at 30464.71 cm−1 between39 f (2) and 40 f (2) in the SFI and late-peak-detected spectra. This feature is absentin both classes of spectra under mm-wave field-free conditions.75.6 GHz A mm-wave field with a frequency of 75.6 GHz increases the intensityof the 44 f (2) resonance by a factor of two in the SFI-detectedω2 spectrum but only58slightly increases the intensity of this feature in the late-peak spectrum. This mm-wave field suppresses the 45 f (2) feature by two thirds in the late peak spectrumand one third in the SFI-detected ω2 spectrum. Note the creation of a weak newfeature at 30467.56cm−1 between 40 f (2) and 41 f (2) in both SFI and late peakspectra. Again, this resonance does not appear in the field-free spectra.71.1GHz In the presence of a mm-wave field with a frequency of 71.1 GHz, therelative intensity of the SFI-detected 45 f (2) resonance grows by a factor of two.This same resonance increases by a factor of three in the late-peak-detected ω2spectrum. Note that this field also increases the intensity of the 43 f (2) resonanceby a factor of two in the late-peak spectrum. The 71.1 GHz field suppressesthe 46 f (2) resonance by a factor of three in the late-peak spectrum and slightlydecreases the intensity of the 47 f (2) resonance in the SFI-detected ω2 spectrumSummary The ω2 spectra detected by integrating both the SFI and late-peak sig-nals show a pattern of increased intensity in successive n0(2) resonances withdecreasing mm-wave frequency. In addition to these systematically resonant en-hancements, we find that mm-wave fields of certain frequencies cause new, inter-loping features to appear at 30461.46, 30464.71 and 30467.56 cm−1.Generally speaking, a field that enhances the strength of a resonance in thelate-peak-detected ω2 spectrum for a principal quantum number, n0, significantlydepresses the intensity of the late-peak feature at n0 + 1. We notice a similar butmuch weaker pattern of suppressed intensity in the ω2 spectra, monitoring theintegrated SFI signal.5.3.2 Fixed ω2: mm-wave excitation spectra of state-selectedultracold plasmasFor fixed ω2, tuned to a particular n0 f (2) resonance, a scanned mm-wave fre-quency, ωmm, produces a response in the electron signal, monitored either byintegrating the SFI waveform collected 300 ns after ω2 or measuring the late peakafter 20 µs. Figure 5.4 shows mm-wave action spectra observed in the SFI signal(left) and late-peak signal (right) for ultracold plasmas that evolve from state-59selected Rydberg gases with n0 from 39 to 45. In each case, the curve drawnthrough the data represents a least-squares fit to the lineshape function determinedby a sum of four Fano profiles (see below).Ultracold plasma that evolves from 39 f (2) In the top left frame of Figure 5.4,we see the mm-wave excitation spectrum of the ultracold plasma that evolves fromthe 39 f (2) Rydberg gas. Here, mm-wave excitation forms a distinctive pair ofresonant features at 109 and 113 GHz with full-width at half-maximum (fwhm) ofapproximately 1 GHz. These appear in almost identical fashion when detected inthe SFI signal 300 ns after ω2 (left) and after 20 µs in the late peak (right). Note aswell a small, sharper satellite resonance at 104 GHz in both spectra. The mm-wavespectrum gauged in the late peak also exhibits prominent, broader dips at 106 GHzand 115 GHz. We see little if any sign of these dips in the spectrum observed after300 ns using SFI detection.Ultracold plasma that evolves from 40 f (2) The ultracold plasma that evolvesfrom the 40 f (2) Rydberg gas exhibits a closer pair of mm-wave resonances ofsimilar fwhm at lower frequencies of 101 and 105 GHz. Over this scan of 40 GHz,we find no satellite peaks. But the late-peak mm-wave spectrum forms broad dipswith centre frequencies of 98 and 107 GHz. A hint of the higher-frequency dipappear in the short-time SFI-detected spectrum.Ultracold plasma that evolves from 41 f (2) The same pattern of 2 GHz fwhmdoublets appears with narrower spacing at lower frequencies of 94 and 97 GHz intransitions from ω2-selected 41 f (2). Here we also see a small satellite peak at 89GHz in both SFI and late-peak spectra. Broader dips occur at 91 GHz and 100 GHzoccur prominently in the late-peak spectrum. The SFI-detected spectrum shows asuggestion of this structure.Ultracold plasma that evolves from 42 f (2) The mm-wave spectra from 42 f (2)again feature prominent doublets, now at 87 GHz and 90 GHz. Here we see a weakadditional peak at 74 GHz in both spectra. Broad dips appear at 85 GHz and 926070 75 80 85 90 95 100 105 110mm frequency (GHz)00.10.20.30.40.50.60.70.80.91Rydberg signalmm-SFI-Fano-4295 105 115 12500.20.40.60.8185 95 105 11500.20.40.60.8180 90 100 11000.20.40.60.8165 75 85 9500.20.40.60.8160 70 80 9000.20.40.60.8155 65 75 8500.20.40.60.8139f (2)40f (2)43f (2)45f (2)42f (2)41f (2)44f (2)95 105 115 12500.20.40.60.8185 95 105 11500.20.40.60.8180 90 100 11000.20.40.60.8175 85 95 10500.20.40.60.8165 75 85 9500.20.40.60.8160 70 80 9000.20.40.60.8155 65 75 8500.20.40.60.81ωmm (GHz) ωmm (GHz)75 85 95 1050.80.60.2010.4Figure 5.4: mm-wave action spectra in pairs with (left) SFI and (right) late peakdetection fit to Fano lineshapes for ultracold plasmas that evolve from Rydberggases with n0 from 39 to 45. 61GHz in the late-peak signal detected after a flight time of 20 µs.Ultracold plasma that evolves from 43 f (2) The plasma prepared from a 43 f (2)Rydberg gas yields a similar doublet of mm-wave excitation features at 82 GHzand 84 GHz in both SFI and late spectra. The spectrum collected monitoring thelate-peak signal also shows broader flanking dips at 80 GHz and 86 GHz. Both theSFI and late-peak spectra show a small satellite peak at 78 GHz and a second pairof resonances at 70 GHz to 73 GHz, flanked by noticeable dips in the late-peakexcitation spectrum.Ultracold plasma that evolves from 44 f (2) The mm-wave excitation spectra ob-served for a plasma that evolves from the 44 f (2) Rydberg gas exhibits a familiardoublet, moved to lower frequencies of 76 GHz and 78 GHz. Here, the spectrumsampled in the late-peak signal shows narrower and weaker dips at 74 GHz and 81GHz. Both spectra contain an isolated peak at 67 GHz, accompanied by a dip at 70GHz in the late-peak spectrum.Ultracold plasma that evolves from 45 f (2) We find the lowest-frequency, narrowest-spaced doublet of mm-wave transitions in this series in the plasma that originatesfrom a Rydberg gas selected to originate from 45 f (2). Excitation spectra col-lected at short time in the SFI signal and after 20 µs in the late peak exhibitresonances at at 71 GHz and 73 GHz. Pronounced dips at 70 GHz and 75 GHzflank these features in the late-peak mm-wave spectrum. Interestingly, notice thatthe frequency positions of the prominent doublet here closely matches the satellitestructure observed in the plasma that evolves from the 43 f (2) Rydberg gas.Summary Ultracold plasmas evolving from n0 f (2) Rydberg gases for n0 from 39to 45 show a consistent pattern of positive going doublets, detected both in the SFIsignal and the integrated late peak. Both the transition energies and the spacing ofthese doublets decrease systematically with increasing n0.Broad dips flank these doublets in the mm-wave spectrum measured by the late-peak signal. The separation of these features also falls uniformly with increasing62n0. Often the position of a positive-going resonance for a given value of n0 occursat the frequency of the upper dip in the excitation spectrum of the plasma thatevolves from n0+1. These dip resonances appear with much smaller amplitude inωmm excitation spectra recording the integrated SFI signal.This appears to explain the suppression of (n0 + 1) f (2) features in late-peakω2 spectra recorded in the presence of mm-wave fields chosen to enhance n0 f (2)resonances. Consistent with the trend observed in the ωmm spectra, resonant dipsprominent in the ω2 signal recorded after an evolution of 20 µs appear to weakereffect in the short-time SFI-detected ω2 spectra.Nearly all of these mm-wave excitation spectra show additional positive-goingsatellite resonances. In some cases these satellite resonances mirror one or bothcomponents of a principal doublet observed for a different n0.5.3.3 Ultracold plasma composition dynamics as a function of timeTime variation of the mm-wave excitation spectrumFigures 5.3 and 5.4 display mm-wave frequency dependent changes in electron-signal intensities associated with UV excitation from A 2Σ+ N+ = 0 to Rydbergstates with n0 from 39 to 45, as well as mm-wave excitation from selected n0 f (2)states. These changes occur owing to the application of a CW mm-wave fieldduring a period of 300 ns between ω2 excitation and the SFI ramp, and to nearlyan equivalent extent in the presence of the same field for the 20 µs between ω2 andpassage through G2 to form the late peak.The similarity of excitation spectra whether monitoring SFI signal or the latepeak raises the question of when during these intervals of time does interactionof the mm-wave field with the ultracold plasma act to produce the observed ωmmresonant effects.To explore this question, we have used the amplitude modulation feature of ourmulti-band transmitter to delay the application of the mm-wave field by variousintervals of time after ω2 excitation to form the nitric oxide Rydberg gas.The top frame of Figure 5.5 shows a set of mm-wave scans over the resonantdoublet and flanking dips observed for the plasma that evolves from a 42 f (2)63Rydberg gas delaying the start of the ωmm square wave by three different delays,∆tmm, with respect to the ω2 excitation pulse, -1.8 µs, 0.8 µs and 1.4 µs. Here, wehave scaled the data in each case to to match the amplitudes of the evident dips.80 85 90 95 100GHz01230 5 10delay(us)00.511.5ratioωmm (GHz)0.80.60.40.20.1.0Late peak signal (arb.)0.0 2.5 5.0 7.5 10.0Dela (µs)1.00.Amplitude ratioField (V/cm)0 50 100 150 200 250Initial Rydberg gas density (cm-3)101010111012Field (V/cm)0 50 100 150 200 250101010111012a)b)d)c)Figure 5.5: a) Time-dependent mm-wave spectra of 42f(2) state integrated over late-peak signal and fit to four-component Fano lineshapes, recorded for mm-wavefields triggered to begin at times -1.8 µs, 0.8 µs and 1.4 µs relative to ω2 laserexcitation respectively. b) Ratio of the integrated 42 f (2)→ 43g(2) peak, to thatof the 42 f (2)→ 43d(2) dip as a function of mm-wave delay.Recognizing the evident asymmetry of these features, we fit the data obtainedfor the mm-wave field started - .8 µs before ω2 to a sum of four Fano profiles, twothat describe the positive-going central features and two negative-going profilesthat describe the flanking dips. A set of 10 scans for ωmm delays varying from∆tmm = −1.8 µs to 3 µs yields best fitting amplitude ratios as plotted in middle64left frame of Figure 5.5. The three curves drawn through the data in the top frameof Figure 5.5 show the best fit simulations for the corresponding three delay timesmarked by filled points in the middle frame.Note that, scaled to a constant relative dip, the positive-going amplitude ob-served in the late-peak ωmm spectrum decays to near zero on the timescale of theavalanche from Rydberg gas to plasma, about 1 µs. For short variations in ∆tmm,the amplitude of the dips is nearly constant, and this fall in the peak-to-dip ratiosimply reflects a time decay in mm-wave enhancement of n0 f (2) resonant features.Time variation of the initial-density-dependent selective field-ionizationspectrumStarting a 0.8 V cm−1 ns−1 electric field ramp at a chosen particular delay after ω2produces a spectrum of electron binding energies sampled by field-ionization. Wetypically collect sets of 4,000 individual SFI spectra for each ramp-field delay. Theselected delay before the ramp defines an interval of field-free ultracold plasmaevolution. For a fixed ramp delay, systematically varying the time between ω1and ω2 enables the experiment to survey a range of nominal initial Rydberg gasdensities, ρ0. We gather the complete set of SFI spectra obtained for a singleselected ramp delay and plot them as a function of calibrated initial density asappearance potential contours. Figure 5.6 show sets of such SFI spectra, reflectinginitial-density dependent electron binding energies in the ultracold plasma formedby a 44 f (2) Rydberg gas after evolution times of 0, 150, 300, 450 and 600 ns, aswell as 1 and 2 µs.The SFI spectrum for initial Rydberg gas densities above 5×1011 cm−3 evolveson the timescale of the ramp to populate states of lower binding energy (higherRydberg states and electrons bound by the NO+ space charge). This redistributionof binding energy marks the prompt avalanche to plasma. After 150 ns, we seevery little transport in this distribution of binding energies.At lower densities and shorter ramp-field delays, note the evolution of thetwo features that dominate the contour of SFI signal. The appearance-potentialdifference between the leading edges of these features matches the N+ = 0 to 2rotational energy spacing of NO+. In the SFI spectrum obtained immediately afterthe ω2 these bands reflect the quasi-diabatic field ionization of a 44 f (2) Rydberg65gas to yield NO+ in the rotational state, N+ = 0, beginning at a field of 125 V/cm,and N+ = 2 beginning at a field of 180 V/cm. Selective field ionization of nitricoxide Rydberg gases with rotational quantum number of 4 and higher confirm theseassignments [62].In the first few hundred nanoseconds, ℓ-mixing electron-Rydberg-moleculecollisions drive initially prepared 44 f (2) molecules to form 44ℓ(2) states. We seethis in Figure 5.6 as a subtle blue shift of the diabatic ionization thresholds sampledfrom n0 = 44. It is important to recognize that each individual SFI spectrum,measured at a calibrated initial peak density, ρ0, samples a Gaussian ellipsoidalexcited volume that ranges from high density in its core to low density in the wings[139]. This gives rise to collisional rate processes (such as avalanche and electron-impact ℓ-mixing) that occur over a broad range of timescales.5.3.4 Power dependence of mm-wave excitation spectraThe Virginia Diodes solid-state multi-band transmitter forms a much narrowersource of pure CW mm-wave frequencies than the evident widths of ωmm Fig-ures 5.4 and 5.5. It thus remains to explain the persistent ∼ 1 GHz, seeminglyasymmetric lineshapes of these resonant features.Rydberg-Rydberg transitions exhibit enormous dipole transition matrix ele-ments [39], which presents the issue of power broadening. The lower frame ofFigure 5.7 integrates the resonant enhancement of the SFI signal observed for theplasma that forms from the 40 f (2) Rydberg gas for ωmm from 100 to 102 GHz atincident intensities from 0 to 25 µW cm−2. Over this range, the absorption risesfrom a regime of linear absorption to approach saturation. The upper frame ofFigure 5.7 shows µW cm−2 spectra recorded at the incident mm-wave intensitiesmarked in the lower frame by filled triangles. We can see that power broadeninghas no effect on the observed linewidths over this full range. For future reference,compare the amplitudes of the prompt flanking depletion features in these SFI-detected ωmm excitation spectra to those in the late-peak excitation spectra plottedin Figure 5.5660 ns 150 ns300 ns150 ns600 ns450 ns2 µs1 µsFigure 5.6: Eight frames, each showing four thousand selective field ionization(SFI) spectra arranged by decreasing initial Rydberg gas density, ρ0. Contoursdescribe the relative electron signal as a function of electrostatic field, rampedfrom 0 to 250 V cm−1 with ramp delays of 0, 150, 300, 450 and 600 ns, aswell as 1 and 2 µs. At high densities and low voltage, the SFI signal reflectsweakly bound high-Rydberg molecules and electrons bound to the NO+ spacecharge. Initial 44 f (2) Rydberg molecules form a distinctive pair of bands at 125and 180 V cm−1 signalling diabatic field-ionization to form NO+ in rotationalstates N+ = 0 and 2, respectively.6795 100 105 110-2024681012140 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1power00.010.020.030.040.050.06Integrated Signal11010510095ωmm (GHz)0 2015105 25 mm-wave intensity (µW cm-2)6420SFI signal (arb.)8SFI enhancement (arb.) 128642010a)b)Figure 5.7: Variation of the amplitude and linewidth of mm-wave transitionsoriginating from the ω2-selected 40f(2) recorded as enhancement featuresdetected by SFI. Frame a) shows the amplitude of the signal integrated from100 to 102 GHz as it approaches saturation in the mm-wave intensity. Frameb) shows band shapes indicating an absence of power broadening for mm-wave intensities indicated by filled triangles in frame a). Solid lines fit Fanolineshapes to enhancement and depletion features (See Eqs (B.1) and (B.2)).Curves drawn through the electron signal data formed by positive-going f tog resonances are fit to a constant linewidth, Γ = 0.8 GHz for all mm-waveintensity.685.4 Discussion5.4.1 Rydberg spectroscopy of the nitric oxide molecular ultracoldplasmaThe signal produced by a durable ultracold plasma forms an excitation spectrummarking the energy positions of long-lived high-Rydberg states of nitric oxideaccessible in UV-UV double resonance from a state of zero angular momentumneglecting spin. The addition of a mm-wave radiation field tunable from 60 to120 GHz drives Rydberg-Rydberg transitions that enhance or diminish the electronsignal associated with this plasma.These results shed light on the high-Rydberg spectroscopy of NO. They alsopoint with quantum state resolution to dynamical factors that shape the evolutionto an ultracold molecular plasma and determine its stability. We begin with ananalysis of the spectroscopy.UV-UV double-resonance and the n0 f (2) Rydberg spectrum of NOAngular momentum selection rules allow UV-UV double resonance electronic tran-sitions via the A 2Σ+ N′ = 0 intermediate state of nitric oxide to reach all Rydbergstates for which, N, the total angular momentum neglecting spin equals 1. Thisoptical excitation process forms neutral Rydberg gases of varying stability withrespect to predissociation to atoms:NO |n0, ℓ⟩ → N(4S)+O(3P) (5.1)Many previous studies have recognized that the rate of predissociation, whichdepends sensitively on Rydberg orbital angular momentum, ℓ, limits transitionsreadily detected by ionization from A 2Σ+ N′ = 0 to those in the N = 1 n f Rydbergseries that converges to the NO+ rotational level, N+ = 2 [157, 13, 116, 102, 110].This hydrogenic series, n0 f (2), defines a particular subset of nitric oxide Ryd-berg gas initial states that evolve to form a durable ultracold plasma. The integratedamplitude of the selective field ionization (SFI) signal, sampled 300 ns after ω2measures the position of each resonance and the effectiveness by which the asso-69ciated short-time dynamics of avalanche ionization forms a persistent distributionof NO+ ions that binds a comparable number of electrons.On a far longer timescale, after 20, 40 or as much a 400 µs [137, 139, 53], anelectron signal late peak forms when the illuminated volume transits a detectionplane. The amplitude of this signal gauges the effectiveness with which eachn0 f (2) Rydberg gas evolves to yield an ultracold plasma state of arrested dissipa-tion. As we can see from Figure 5.2 the long-time, late-peak signal presents muchthe same intensity pattern as the SFI spectrum sampled after only 300 ns. Fromthis we can conclude that dynamical processes occur very early in the avalanche ofthe Rydberg gas to determine the charge density of the plasma sampled at a muchlater time.UV-UV-mm-wave triple-resonance and the assignment ofn0 f (2)→ (n0±1)g(2) Rydberg - Rydberg transitionsA CW mm-wave field, tuned to a succession of specific points in the range offrequency from 120 and 60 GHz, enhances the intensity of n0 f (2) resonances inthe ω2 spectrum for n0 from 39 to 45. The sequence of spectra in Figure 5.3defines the set of mm-wave resonant frequencies that decreases as a slightly sub-linear function of n0 (cf. Figure 5.8). This set of ωmm frequencies conformsalmost exactly with the set of transition energies from each n0 f (2) level to theneighbouring (n0 + 1)g(2) state calculated simply from Rydberg formula withconstant quantum defects, δ f = 0.023 and δg = 0.003 [102].Table 5.1: Assignment of n0 f (2) positions and n0 f (2) to (n0 ± 1)g(2) Rydberg-Rydberg transitions from SFI-detected ω2 and ωmm excitation spectraω2 cm−1 n0 f (2)n0 f (2)→ (n0+1)g(2) n0 f (2)→ (n0−1)g(2)meas(GHz) calc(GHz) meas(GHz) calc(GHz)30462.46 39 f (2) 108.84 109.04 113.22 113.1430466.03 40 f (2) 101.18 101.16 104.91 104.7630469.35 41 f (2) 94.08 94.01 97.25 97.1930472.42 42 f (2) 87.61 87.53 90.22 90.3330475.29 43 f (2) 81.86 81.63 83.92 84.1030477.96 44 f (2) 76.33 76.25 78.26 78.4330480.46 45 f (2) 71.25 71.33 73.23 73.2570For a set of ω2 frequencies fixed on specific n0 f (2) resonances, scans of ωmmshown in Figure 5.4 confirm these assignments. Here we see clear doublet positionsthat fit precisely with the energies calculated for transitions from n0 f (2) to (n0 +1)g(2) and (n0− 1)g(2). Combinations of energies for the mm-wave transitions(n0−1) f (2) to (n0)g(2) and (n0+1) f (2) to (n0)g(2) accurately reflect the bindingenergy differences between (n0− 1) f (2) and (n0 + 1) f (2) measured by resonantpositions in the ultraviolet ω2 spectrum.Table 5.1 summarizes the ω2 positions of the n0 f (2) Rydberg states for n0from 39 to 45, and compares measured mm-wave n0 f (2) to (n0±1)g(2) Rydberg-Rydberg transition energies with simple calculations assuming constant quantumdefects.Dips in the ωmm spectrum mark Rydberg-Rydberg transitions from n0 f (2) todissociative (n0±1)d(2) statesDistinctive dips flank all of the n0 f (2) → (n0 ± 1)g(2) Rydberg-Rydberg reso-nances in the late-peak excitation spectra plotted on the right in Figure 5.4. Thefrequency positions of features fit well in each instance with transitions from theselected n0 f (2) state to the adjacent (n0±1) levels of the dσ series converging toNO+, N+ = 2 with a quantum defect, δd =−0.057 [102].Table 5.2: Assignment of n0 f (2) to (n0±1)d(2) Rydberg-Rydberg transitions fromωmm excitation spectra detected in the amplitude of the late-peak plasma signalω2 cm−1 n0 f (2)n0 f (2)→ (n0+1)d(2) n0 f (2)→ (n0−1)d(2)meas(GHz) calc(GHz) meas(GHz) calc(GHz)30462.46 39 f (2) 116.42 115.19 105.78 105.9630466.03 40 f (2) 108.10 106.87 98.13 98.1230469.35 41 f (2) 100.12 99.33 91.32 91.0330472.42 42 f (2) 93.07 92.49 85.11 84.6130475.29 43 f (2) 86.10 86.25 79.04 78.7830477.96 44 f (2) 80.34 80.57 74.24 73.4730480.46 45 f (2) 75.65 75.37 69.42 68.63Note, however that mm-wave transitions originating from n0ℓ(2) states of anyhigher orbital angular momentum ℓ, formed as a result of electron-Rydberg colli-716575859510511538.8 39 39.2 39.4 39.6 39.8 40 40.2 40.4 40.6 40.8 41 41.2 41.4 41.6 41.8 42 42.2 42.4 42.6 42.8 43 43.2 43.4 43.6 43.8 44 44.2 44.4 44.6 44.8 45 45.2Frequecy GHzPrincipal quantum number n039 40 41 42 43 44 4511510595857565Initial principal quantum number (GHz)8Rydberg-Rydbergtransitionsexhibitenormousdipoletransitionmatrixelements,whichpresentstheissueofpowerbroadening.ThelowerframeofFigure6inte-gratestheresonantenhancementoftheSFIsignalob-servedfortheplasmathatformsfromthe40f(2)Ryd-berggasfor!mmfrom100to102GHzatincidentpowersfrom0to25µWcm2.Overthisrange,theabsorptionrisesfromaregimeoflinearabsorptiontoapproachsat-uration.TheupperframeofFigure6showsµWcm2spectraatincidentmm-waveintensitiesmarkedbyfilledtriangles.Notethatpowerbroadeninghasnoe↵ectontheobservedlinewidthsoverthisfullrange.IV.DISCUSSIONA.RydbergspectroscopyofthenitricoxidemolecularultracoldplasmaTheformationofadurableultracoldplasmamarkstheenergypositionsoflong-livedhighRydbergstatesofni-tricoxideaccessibleinUV-UVdoubleresonancefromastateofzeroangularmomentumneglectingspin.Theadditionofamm-waveradiationfieldtunablefrom60to120GHzdrivesRydberg-Rydbergtransitionsthaten-hanceordiminishtheelectronsignalassociatedwiththisplasma.Theseresultsshedlightonthehigh-RydbergspectroscopyofNO.Theyalsopointwithquantumstateresolutiontodynamicalfactorsthatshapetheevolutiontoanultracoldmolecularplasmaanddetermineitssta-bility.Webeginwithananalysisofthespectroscopy.1.UV-UVdouble-resonanceandthen0f(2)RydbergspectrumofNOAngularmomentumselectionrulesallowUV-UVdou-bleresonanceelectronictransitionsviatheA2⌃+N0 =0intermediatestateofnitricoxidetoreachallRydbergstatesforwhich,N,thetotalangularmomentumne-glectingspinequals1.ThisopticalexcitationprocessformsneutralRydberggasesofvaryingstabilitywithre-specttopredissociationtoatoms:NO|n0,`i!N(4S)+O(3P)(1)Manypreviousstudieshaverecognizedthattherateofpredissociation,whichdependssensitivelyonRydbergorbitalangularmomentum,`,limitstransitionsdetectedbyionizationtothoseintheN=1nfRydbergseriesthatconvergestotheNO+rotationallevel,N+=2.Thishydrogenicseries,n0f(2),definesaparticularsubsetofnitricoxideRydberggasinitialstatesthatevolvetoformadurableultracoldplasma.Theinte-gratedamplitudeoftheselectivefieldionization(SFI)signal,sampled300nsafter!2measurestheresonantpositionofeachresonanceandthee↵ectivenessbywhichtheassociatedshort-timedynamicsofavalancheioniza-tionformsabounddistributionofNO+-echarge.Onafarlongertimescale,after20,40orasmucha400µs,anelectronsignallatepeakformswhentheillumi-natedvolumetransitsadetectionplane.Theamplitudeofthissignalgaugesthee↵ectivenesswithwhicheachn0f(2)Rydberggasevolvestoyieldanultracoldplasmastateofarresteddissipation.AswecanseefromFig-ure2thelong-time,late-peaksignalpresentsmuchthesameintensitypatternastheSFIspectrumsampledafteronly300ns.FromthiswecanconcludethatdynamicalprocessesoccurveryearlyintheavalancheoftheRyd-berggastodeterminethechargedensityoftheplasmasampledatamuchlatertime.2.UV-UV-mm-wavetriple-resonanceandtheassignmentofn0f(2)!(n0±1)g(2)Rydberg-RydbergtransitionsTuningaCWmm-wavefieldtospecificpointsinarangeoffrequencyfrom120and60GHzenhancestheintensityofsuccessiven0f(2)resonancesinthe!2spec-trumforn0from39to45.ThesequenceofspectrainFig-ure3defineasetofmm-waveresonantfrequenciesthatdecreaseswithn0followingaslightlysub-linearfunction.Thissetof!mmfrequenciesconformsalmostexactlywiththesetoftransitionenergiesfromeachn0f(2)leveltotheneighbouring(n0+1)g(2)statecomputedsimplyfromRydbergformulawithconstantquantumdefects, d=0.0232and g=0.003.Forasetof!2frequenciesfixedonspecificn0f(2)res-onances,scansof!mmshowninFigure4confirmtheseassignments.Hereweseecleardoubletpositionsthatfitpreciselywiththeenergiescalculatedfortransitionsfromn0f(2)to(n0+1)g(2)and(n01)g(2).Combina-tionsofenergiesforthemm-wavetransitions(n01)f(2)to(n0)g(2)and(n0+1)f(2)to(n0)g(2)exactlyequalbindingenergydi↵erencesbetween(n01)f(2)and(n0+1)f(2)measuredbyresonantpositionsintheul-traviolet!2spectrum.TABLEI.Assignmentsofn0f(2)positionsandn0f(2)to(n0±1)g(2)Rydberg-Rydbergtransitionsfrom!2and!mmspectra!2cm1n0f(2)n0f(2)!(n1)g(2)n0f(2)!(n+1)g(2)meas(GHz)calc(GHz)meas(GHz)calc(GHz)30462.4639f(2)108.89109.06113.43113.1230466.0340f(2)101.32101.18104.90104.7430469.3541f(2)94.1394.0397.1897.1730472.4242f(2)87.7087.5590.2190.3130475.2943f(2)81.8681.6484.1384.0830477.9644f(2)76.3976.2678.3378.4130480.4645f(2)71.3171.3473.3273.24TableIsummarizesthe!2positionsofthen0f(2)Ryd-bergstatesforn0from39to45,andcomparesmeasuredmm-waven0f(2)to(n0±1)g(2)Rydberg-Rydbergtransi-tionenergieswithsimplecalculationsassumingconstantquantumdefects.Figure 5.8: Measured positions of ωmm resonances for n0 f (2)→ (n0± 1)g(2) (♦)and n0 f (2)→ (n0±1)d(2) (♢) transitions compared with lines connecting thecalculated displacements of the successive sets of g(2) and d(2) states from thecorresponding member of the n0 f (2) series, for n0 from 39 to 45.sional ℓ-mixing also fall within the 2 GHz bandwidth of these resonant dips. Noteas well that ωmm transitions to (n0± 1)d(2) deplete the 20 µs late-peak ultracoldplasma spectrum nearly to zero, while having a barely perceptible effect on theshort-time signal of the avalanching plasma detected by SFI.Table 5.2 lists the mm-wave n0 f (2) to (n0± 1)d(2) Rydberg-Rydberg transi-tion frequencies measured for these band compared with positions estimated fromthe Rydberg formula assuming constant quantum defects. Note the reversal: Forthis range of n0, the calculated energy of the transition from n0 f (2) to the level,(n0+1)d, exceeds that for the transition to (n0−1)dIn summary, we assign the very evident pairs of doublets that appear in theseωmm scans of resonances in the ultracold plasma response to transitions from n0 f (2)states to adjacent Rydberg levels, (n0±1)g(2) and (n0±1)d(2) for n0 from 39 to45. Figure 5.8 confirms this ass gnment by showing the measured position of eachresonance compared to a plot of the calculated displacement of each (n0±1)g(2)and (n0±1)d(2) state from the corresponding selected n0 f (2) state.72Additional resonances evident in the ωmm spectra and isolated regions of theω2 spectraTheω2 spectra in Figure 5.3 show evidence of additional structure between n0 = 39and 40, and in the shoulder of the resonance for n0 = 44 that varies in the presenceof mm-wave fields of particular frequency. Similarly, the ωmm spectra in Figure 5.4exhibit numerous instances of additional resonances for fixed values of ω2 tunedto n0 f (2) states with n0 = 39, 41, 43 and 44. Table 5.3 summarizes the positionsof these additional bands.As made evident by Figure 5.8 together with Tables 5.1 and 5.2, a simpleRydberg model describes the positions of states with n0 in the range from 39 to 45quite well in terms of separable rotational quantum number and constant quantumdefect for d, f , and g series. Considering series that converge to N+ = 0 and1 as well as 2, we can identify intervals of ω1 +ω2 energy at which membersof these interloping series coincide with n0 f (2) states. This occurs particularly instroboscopic regions where Rydberg intervals approach rotational energy spacings.Gauging the calculated energy separation between candidate interloping n0 p(N+)and n0d(N+) and adjacent (n0± 1)g(N+) states, we can account for all the otherresonant features that appear in Figures 5.3 and 5.4. Table 5.3 assigns these tran-sitions to interloping p, d and f states with total angular momentum neglectingspin of N = 1. It seems likely that initial states with ℓ < 3 all borrow lifetimefrom neighbouring n0 f (2) states, and like those principal ω2 resonances, add tothe plasma signal detected in the SFI spectrum or the late peak when promoted toadjacent (n0±1)g states.5.4.2 Intensity of mm-wave resonances as a probe of molecularultracold plasma evolution dynamicsResonant enhancement of the ultracold plasma SFI and late-peak signalThe UV-UV double-resonance spectra plotted in Figure 5.2 show that, amongaccessible prepared N = 1 states of Rydberg gas, the avalanche to ultracold plasmastrongly selects for those with high orbital angular momentum. The resonancesdetected by SFI or as an ultracold plasma late peak belong almost entirely to the73Table 5.3: Assignments of interloping n0ℓ(N+) positions for ℓ = 1,2,3, and theenergies of Rydberg-Rydberg transitions observed in scans of ω2 (Figure 5.3)and ωmm (Figure 5.4). Asterisks below refer to transitions for which calculatedenergies fall below the lowest scanned frequency.ω2 cm−1 n0ℓ(N+) (n0+1)g(N+)n0ℓ(N+)→ (n0+1)g(2) (n0−1)g(N+) n0ℓ(N+)→ (n0−1)g(2)meas(GHz) calc(GHz) meas(GHz) calc(GHz)ω2 scan30447.98 39p(0) 38g(1) 81.40 83.2830458.35 40d(1) 41g(1) 93.70 93.6630461.46 43p(0) 42g(1) 93.70 92.5630461.65 41d(1) 42g(1) 87.00 87.0530464.26 44p(0) 43g(1) 93.70 94.3730464.71 42d(1) 43g(1) 81.40 81.0530467.56 43d(1) 44g(1) 75.60 75.59ωmm scan: ω2 = 30462.46, [39 f (2)]30461.65 41d(1) 42g(1) * 87.05 40g(1) 103.92 104.15ωmm scan: ω2 = 30469.35, [41 f (2)]30467.56 43d(1) 44g(1) * 75.59 42g(1) 89.79 90.13ωmm scan: ω2 = 30475.29, [43 f (2)]30472.53 45 f (1) 46g(1) 69.87 71.34 44g(1) 72.12 73.2430472.70 45d(1) 46g(1) 66.81 66.06 44g(1) 78.15 78.52ωmm scan: ω2 = 30477.96, [44 f (2)]30477.20 47d(1) 48g(1) * 58.03 46g(1) 66.93 68.82single, atom-like ℓ = 3 series, n0 f (2). The only evidence for states of lower ℓ ap-pears at positions where they occur as N = 1 interlopers, that mix with neighbour-ing states in the n0 f (2) series. Here, they borrow lifetime conferred by admixed,non-penetrating ℓ= 3 character.This selectivity clearly occurs as a consequence of predissociative decay toform neutral N(4S) plus O(3P). Among optically accessible states with N = 1, onlythose with highest orbital angular momentum, ℓ = 3, present a barrier sufficientto suppress the e− - NO+ ion core coupling that causes ultrafast predissociation.Long-lived NO high-Rydberg molecules undergo Penning ionization and electron-impact avalanche to form an ultracold plasma.Promotion of surviving, optically accessible n0 f Rydberg molecule from astate of orbital angular momentum |n0, ℓ= 3⟩ to |(n0±1), ℓ= 4⟩ further decreasescore penetration and extends lifetime. We observe this consequence of an increasein ℓ by an evident increase in the plasma yield, detected after 300 ns in the SFIsignal and 20 µs later in the integrated amplitude of the plasma late peak. Separateevidence suggests that plasma success in evolving to a state of arrested relaxationrelies on a density of surviving Rydberg molecules that balances with the NO+74ions [87].Experimental results plotted in Figure 5.5 serve to define a critical time intervalof evolution after ω2, during which mm-wave plasma enhancement occurs in rela-tion to spectroscopically adjacent suppression. We see here that a mm-wave fielddelayed to begin as little as 1 µs after ω2 acts far less effectively in promoting theevolution from Rydberg gas to plasma by means of n0 f (2)→ (n0±1)g(2) resonantexcitation.The falling amplitude ratio of resonant enhancement to resonant depletionsignifies the disappearance of n0 f (2) molecules. These species, initially populatedby double resonant laser excitation, are consumed by ℓ-mixing, which occurs ona 1 µs timescale owing to Penning electron-Rydberg collisions. As electron colli-sions drive the optically selected high-n ℓ= 3 Rydberg orbital angular momentumdistribution to occupy states of higher ℓ, the advantage of a mm-wave resonant fto g transition disappears.The rate at which this mm-wave resonant intensity decreases in Figure 5.5agrees with the timescale on which we start to observe a blue shift in the pair of then0 Rydberg features evident in the time-resolved SFI spectrum, directly signallinga process of ℓ-mixing in the nitric oxide Rydberg gas early in the avalanche toplasma [53]. We observe a very similar short-time plasma enhancement when wedirectly promote electron-Rydberg ℓ-mixing collisions by applying a 60 MHz radiofrequency pulse [159]. The prominence of f to g lifetime-lengthening resonancesin mm-wave action spectra underline the importance of Rydberg dynamics in de-termining plasma stability.Resonant suppression of the ultracold plasma SFI and late-peak signalThe right-hand frames of Figure 5.4 show clear resonant dips in the late-peakωmm spectra of selected n0 f (2) Rydberg molecules. Figure 5.8 establishes that thepositions of these resonances fit well with assignment to the series of transitions,n0 f (2)→ (n0±1)d(2). The lineshapes of the dips in Figure 5.4 amply overlap theenergies of transitions originating from the nearly iso-energetic less-penetratingstates, n0ℓ(2) for ℓ≥ 4 formed by ℓ-mixing.Isolated nd Rydberg states of NO predissociate on a nanosecond timescale for75principal quantum numbers in this range [13, 157, 116, 102], and this dissipationmust account for the fact that ∆n0 =±1 transitions cause the late-peak plasma den-sity to decrease. A Rydberg-Rydberg transition to an nd state from any originatinglevel with ℓ≥ 3 drives population from a long-lived state to a predissociative one.Interestingly, mm-wave excitation drives these transitions from surviving n0ℓ(2)levels to predissociative (n0± 1)d(2) states with far greater effectiveness duringthe 20 µs flight time from ω2 to the appearance of the late peak than than in thefirst few-hundred nanoseconds of the avalanche to plasma.Initiated as late as 3 µs after ω2, we see in Figure 5.5 that mm-wave fields reso-nant with n0ℓ(2)→ (n0±1)d(2) transitions deplete the integrated late peak signalto nearly zero. But, SFI spectra in the same figure show that on a time-scale muchshorter that this, electron-impact avalanche drives a large fraction of moleculespopulating states of any n0ℓ(2) level to states of far lower binding energy, outof resonance with ωmm. In other words, mm-wave excitation in the first 300 nsproduces little if any signature of n0ℓ(2)→ (n0± 1)d(2) resonant absorption. Bycontrast, a mm-wave field applied continuously after a delay of as much as 3 µsquenches the plasma almost entirely to N(4S)+O(3P).Under field-free conditions, we can explain a molecular ultracold plasma signalthat endures for hundreds of microseconds by recognizing a short-time depletionof Rydberg states with low-ℓ, accompanied by a cessation of ℓ-mixing electron-Rydberg collisions in the ensemble of surviving molecules [4]. In previous work[159], we established that the application of a radio frequency field causes electroncollisions resume. This RF promotion of electron collisions evidently redistributesRydberg population in states of all principal quantum number n over all values ofℓ, including penetrating, short-lived states of low orbital angular momentum, andthis causes predissociative dissipation.In contrast with RF heating, it is difficult to imagine a direct effect of mm-wave radiation on electron mobility. Instead, delayed SFI spectra show evidenceof a trace population of surviving n0ℓ(2) molecules, which can serve as a gatewayto mm-wave driven dissipation only if a mechanism exists to fill the n0 hole burnedby n0ℓ(2)→ (n0±1)d(2) excitation and predissociation.The field-free durability of the plasma to predissociation appears to signalthe absence of ℓ-mixing electron-Rydberg collisions. A trace of mm-wave exci-76tation evidently causes avalanche-like state mixing to resume. Recognizing thisas a many-body problem, we can consider radiative coupling to the continuum tochange the state of the isolated ensemble to that of an open system. We suggestthat this bridges the trace population of n0ℓ(2) states to the predominant plasmapopulation situated very near the ionization threshold, thus serving to maintaina sequence of resonant excitation and predissociation that proceeds on a 20 µstimescale to near completion.5.4.3 Lineshapes of Rydberg-Rydberg transitions in the nitric oxidemolecular ultracold plasmaFunctional form of fits to n0 f (2)→ (n0±1)g(2) andn0 f ,g, ℓ(2)→ (n0±1)d(2) resonancesThe positive-going n0 f (2)→ (n0 ± 1)g(2) resonances highlighted in Figure 5.4display distinctive ∼ 1 GHz linewidths, both as features measured after 300 ns inthe total SFI signal and as structure detected 20 µs later in the amplitude of thelate peak. We cannot readily explain these linewidths based on the properties ofisolated molecules. These f and g Rydberg states have intrinsically long lifetimes.Rydberg-Rydberg transitions do occur with very large dipole transition moments,but the present measurements take place in a range of linear absorption that onlyapproaches saturation. Figure 5.7, establishes that even the most intense mm-wavefields fall short of detectable power broadening.All peaks show a small but very evident degree of asymmetry. Each featurealone directly manifests a position and linewidth. We account for asymmetry assimply as possible by reference to the conventional Fano anti-resonant lineshapefunction:σ(ε) =(q+ ε)21+ ε2, (5.2)which adds just one more term, an asymmetry parameter, q. Here, the relativeabsorption cross section, σ(ε), varies with detuning, ε , defined by:ε =ωmm−ω0Γ/2(5.3)77in which ω0 represents the resonant frequency of the transition and Γ describes itslinewidth.Table 5.4: Fano parameters for n0 f (2) to (n0±1)g(2) Rydberg-Rydberg transitionsmeasured in the late-peak excitation spectrum. We estimate Γwith an uncertaintyof about 10 percent. Lineshapes are perceptibly asymmetric, but the error boundson q are large, perhaps as great as 25 percent.n0 f (2)n0 f (2)→ (n±1)g(2)ω0 Γ q ω0 Γ q(GHz) (GHz) (GHz) (GHz)39 f (2) 108.84 0.8 20 113.22 0.8 -2040 f (2) 101.18 0.9 10 104.91 0.9 -1041 f (2) 94.08 0.8 20 97.25 0.8 -2042 f (2) 87.61 0.8 15 90.22 0.8 -1543 f (2) 81.86 0.9 -15 83.92 0.9 1544 f (2) 76.33 1.0 12 78.26 1.0 -1245 f (2) 71.25 0.8 20 73.23 0.8 -20Table 5.4 summarizes parameters that pertain to all of the n0 f (2) → (n0 ±1)g(2) resonances fit by least-squares in Figure 5.4. Notice that most of the lower-energy n0 f (2)→ (n0 + 1)g(2) transitions, described by positive q, shade to theblue, while lineshapes for n0 f (2)→ (n0 − 1)g(2) transitions best fit to negativeq, and shade to the red, except for the resonances that originate from 43 f (2) forwhich this asymmetry reverses. Relatively large values of q cause lineshapes in allof these cases to approach Lorentzians.Resonances assigned to n0ℓ(3)→ (n0±1)d(2) appear as dips in ωmm late-peakexcitation spectra with broader linewidth and greater asymmetry.Table 5.5 collects the parameters that describe the Fano parameters for all thedips fit to lineshape functions plotted in Figure 5.4.The dynamics that govern Rydberg-Rydberg anti-resonant lineshapesEvery resonant feature considered in Figures 5.4, 5.5, and 5.7 fits well with a simpleasymmetric lineshape function, as defined by Eq. (B.1). Tables 5.4 and 5.5 catalogthe the resonant frequency, ω0, linewidth, Γ, and asymmetry parameter, q, in eachcase that minimize the residuals of these fits. Thus, it seems clear that Rydberg-78Table 5.5: Fano parameters for n0ℓ(2) to (n0±1)d(2) Rydberg-Rydberg transitionsmeasured in the late-peak excitation spectrum. The uncertainty in Γ is againabout 10 percent. Greater asymmetry reduces the absolute uncertainty in q, butin relative terms it remains about 25 percent.n0 f (2)n0ℓ(2) → (n±1)d(2)ω0 Γ q ω0 Γ q(GHz) (GHz) (GHz) (GHz)39 f (2) 116.42 2 10 105.78 2 -1040 f (2) 108.10 2 -2 98.13 2 241 f (2) 100.12 2 10 91.32 2 1042 f (2) 93.07 3 -4 85.11 3 -443 f (2) 86.10 2 4 79.04 2 444 f (2) 80.34 2 4 74.24 2 -445 f (2) 75.65 2 -5 69.42 2 5Rydberg resonances observed in the ultracold plasma conform phenomenologicallywith Fano lineshapes. On the face of it, this hardly seems surprising. The simpleFano formula often serves to describe lineshapes observed in electronic spectra.But, their appearance here invites interpretation.The levels involved confound this task. For principal quantum numbers, n0,between 39 and 45, we should expect to find well-isolated individual transitionsbetween non-penetrating f and g Rydberg states of adjacent n that exhibit mm-wave linewidths limited by the sub-MHz resolution of our synthesizer. Instead, wefind broad features, which must arise from some radiationless process in the systemthat acts in each case to spread the oscillator strengths of the n0 f (2) to (n0±1)g(2)transition over an interval of about 1 GHz.For the conventional case of a single discrete state coupled to a continuum,parallel bound-bound and bound-free transitions interfere, easily rationalizing anasymmetric lineshape [33, 129, 83, 120, 66, 91, 18, 156]. While no direct electro-magnetic transition to a continuum exists to affect the mm-wave Rydberg-Rydbergspectrum of individual nitric oxide molecules below the first ionization poten-tial, cases abound in which discrete-discrete dipole transitions interfere in coupledmany-body systems. Wavefunction interference provides a useful framework forinterpreting bound-quasi-continuum interactions in a range of complex materials,79including quantum dots [67] and plasmonic nanostructures [92, 79, 77, 76]. Theconcept of Fano resonance has also proven useful in describing how compactlocalized states scale the effect of disorder in flat-band lattice systems [115, 74],and reflect the degree to which central-site interactions mix the wavefunctions in asurrounding, Anderson localized (AL) chain [55].A dynamics of avalanche and bifurcation appear to quench the present nitricoxide molecular ultracold plasma to a highly disordered, strongly coupled stateof anomalously slow relaxation [53, 4]. This evolution, directly monitored byexperiment, creates conditions in the system that may approach the threshold formany-body localization (MBL) [43]. We have taken information from experimentto rationalize the behaviour of the NO ultracold plasma in terms of a minimalphenomenological model of randomly interacting dipoles of random energies thatgives rise to a system near a state of MBL [145, 147].In addition to the question of MBL as an enduring phase in various dimensions,theorists have raised important questions about how a localized system responds tocontact with a bath or a rare thermal region [99, 104, 60, 1, 2]. Such questionsof imperfect isolation are of great concern to experimentalists [122]. Generallyspeaking, delocalizing interactions act to mobilize transport and mix localizedstates with consequences for the statistics of energy level spacings.The particular simple limit of a single perturbing level acting with a couplingstrength, m on the quasi-continuum of an Anderson localized bath with disorderwidth, W , causes a configuration interaction that maps onto the Fano resonanceproblem [55]. The ratio, µ = |m/W | determines a leading-order self-energy cor-rection that sets the position and spectral weight of the resonance. In the limit ofweak coupling, |m/W | ≪ 1, a bright state retains its spectral weight and position,and its interaction with the AL bath fails to cause a Poisson energy distribution ofeigenvalues to acquire repulsion. A coupling strength that substantially exceedsthe width of the disorder, |m/W | ≫ 1, shifts the resonance by more than µW , andplaces most of the spectral weight in a [−W,W ] continuum regime of repellinglevels and ballistic entanglement.To draw a crude analogy, let us consider the very few n0 f (2)Rydberg moleculesremaining in the nitric oxide ultracold plasma after 1 µs as points of origin for mm-wave transitions to particular selected bright states in a disordered dipolar system80that spans an energy width [−W,W ], which extends over a measured interval ofabout 500 GHz. This disorder width substantially exceeds the 1 GHz averagedipole-dipole coupling strength in the arrested state of the quenched plasma. In thisbright-state picture, each of the pair of n0 f (2)→ (n0± 1)g(2) Rydberg-Rydbergtransitions in Figure 5.4 yields a Fano lineshape consistent with a weak couplingregime, in which, the coupling of the bright state determines the linewidth, Γ≈ m[42], the spectral weight favours the discrete-discrete transition, q> 10, and dipole-dipole interactions fail to drive level repulsions.From this perspective, the spectra of transitions from high-ℓ to low-ℓ appearqualitatively different. Time-resolved experiments show that n0d(2) states predis-sociate on a nanosecond timescale [13, 157, 116, 102]. This is too slow to explainthe observed values of Γ = 2 GHz as an uncertainty bandwidth. However, a spec-troscopically accessed state that connects the bath of NO dipoles via fast predisso-ciation to a continuum of free N(4S)+O(3P) atoms could alter the coupling andlevel statistics of the bath in such a way as to affect the lineshape of the Rydberg-Rydberg transition. In particular, stronger coupling ought to produce a broaderlinewidth and increase the spectral weight of the quasi-continuum, resulting in abroader, more asymmetric lineshapes, as measured experimentally by dips in thelate peak plasma signal.That these resonances appear as dips, and that the associated depletion faroutweighs the long-time residual fraction of n0 Rydberg molecules speaks to atransport property that must arise in the ultracold plasma to fill the hole createdby n0 f ,g, ℓ(2)→ (n0± 1)d(2) excitation and predissociation. We might look toRydberg molecule collisions with free electrons as a means of randomization byRydberg energy and angular-momentum transport. Evidence for dynamics like thisappears in the first few hundred nanoseconds of plasma evolution, when electron-collisional ℓ-mixing, accelerated by an RF field, suppresses rapid predissociationby scattering n0 f (2) Rydberg molecules into states of higher ℓ [159]. But a mi-crosecond later, the observed predissociative lifetime grows to a millisecond ormore, signalling an end to ℓ-mixing on the timescale of the experiment. To confirmthis, we have shown that the application of a 60 MHz RF field restarts electroncollisions and causes predissociation to resume, neutralizing the plasma to a gas ofnitrogen and oxygen atoms [159].81So the question remains as to how the ensemble fills the hole burned in thescant, late-time population of n0ℓ(2) molecules by resonant n0ℓ(2) to (n0±1)d(2)transitions. As just noted, electron collisions would serve this purpose, but can oc-cur under field-free conditions only if enabled by the agency of the predissociativestates formed in the regions of these dips. Whatever the dynamics that drive thishole filling, they appear to require (n0±1)d(2) excitation to operate.The question of dissipation and its effect on energy flow in a disordered many-body system has attracted considerable interest among theorists [1, 2, 60, 122].In the NO ultracold plasma, it seems that the subset of dissipative states createdby mm-wave excitation acts on the ensemble of long-lived dipoles to increasethe range of state-changing interactions. Predissociation appears to bridge the ndexcited state, and by extension, the entire arrested system to the continuum of freeN(4S)+O(3P) atoms.By forging this connection to a thermal continuum via predissociation, mm-wave driving in effect adds a dissipative term to any Hamiltonian that describesdipolar coupling [145, 147]. Even in the simple limit of a minimal subsystem oflocalized pseudospins, the effects of a dissipative bright state present a significantchallenge to describe with precision. Still, one can imagine realms of tuned bright-state-subsystem coupling and predissociation width. Despite simplifying approx-imations, such calculations could point to intriguing spectroscopic measures oflocalization instability.5.5 ConclusionsDouble-resonant excitation of NO in a supersonic molecular beam creates a state-selected Rydberg gas that evolves to form an ultracold plasma. The efficiencyof this process depends sensitively on the dissociative lifetime of the initially se-lected Rydberg quantum state. Among the exclusively N = 1 levels accessiblevia intermediate resonance with A 2Σ+ N′ = 0 (where N refers to total angularmomentum neglecting spin), only states in the ℓ = 3 series converging to N+ = 2(n0 f (2)) evolve to a long-lived plasma state. Higher-ℓ, n0g(2) states live an orderof magnitude longer, and mm-wave radiation tuned to n0 f (2) → (n0 ± 1)g(2)transitions substantially increases the density of plasma that survives the 300 ns82of avalanche and persists to form a late-peak electron signal after 20 µs or more.The mm-wave excitation spectrum also exhibits transitions from states, n0ℓ(2),to (n0± 1)d(2) that flank each pair of non-penetrating resonances. Here, ℓ refersto orbital angular momenta ℓ ≥ 3, for which quantum defects cluster near zero.Photoproduced nd Rydberg states of NO predissociate with nanosecond lifetimes,and these (n0±1)d(2) resonances appear as small dips in ωmm excitation spectra,sampled under avalanche conditions by the SFI signal at 300 ns. By contrast, ndpredissociation decreases the plasma signal to near-zero in resonance dips sampledin the late-peak after a long-time evolution of 20 µs.At early times, the SFI spectrum shows that the evolving plasma retains asubstantial proportion of n0 f (2) Rydberg molecules. On a time-scale of about1 µs, ℓ-mixing electron-Rydberg collisions drive this population to states of higherorbital angular momentum, and the n0 f (2)→ (n0±1)g(2) resonances disappear.These transformations have no effect on the resonance condition for n0ℓ(2) to(n0±1)d(2) dips, and delaying the onset of the mm-wave field by as much as 2 µsdoes not diminish the amplitude of the dip detected in the late-peak signal.The nd-Rydberg states populated later dissociate as the plasma evolves, form-ing a resonant dip in the late-peak signal, in some cases nearly to zero. SFI spectrashow that this arrested state contains only a very small residual population ofresonant n0ℓ(2) molecules.Consuming this trace population, n0ℓ(2)→ (n0±1)d(2) excitation must there-fore burn a hole in the distribution of Rydberg molecules at n0ℓ(2). Predissociationcannot proceed to a state of deep depletion without some mechanism to repopulatethis originating state. We see no evidence for energy redistribution in the absenceof mm-wave transitions that populate predissociative nd states. This suggests thatpredissociation itself may play a role in the hole-filling required to sustain it.We suggest the model of a spectroscopically bright state that bridges the quenchedultracold plasma to a neutral continuum offers a possible way to think about thisprocess. The plasma exists in an arrested state, consisting of NO+ ions, electronsand high-ℓ Rydberg molecules in quasi-equilibrium - perhaps under conditions oflocalization. This plasma reaches its quenched state depleted in nd molecules bypredissociation.Promotion of the trace n0ℓ(2) molecules in this ensemble to a bright state83coupled by predissociation to the N(4S)+O(3P) continuum bridges the ensembleof localized dipoles to this continuum. This coupling mixes states in the localizedensemble, feeding a flux of molecules that can absorb mm-wave radiation reso-nantly within the bandwidth of the n0ℓ(2)→ (n0±1)d(2) transitions.The evidence for this coupling appears in a broadened linewidth and spectralweight transferred to the continuum that gives rise to anti-resonant lineshape. Thisspectroscopic manifestation offers an intriguing possible point of contact betweentheory and quantum state resolved experiment.84Chapter 6Local dissipation drives globalrelaxation in a molecularultracold plasma with on-sitedisorder and long-range dipolarinteractions6.1 IntroductionThe previous chapter demonstrates the use of Rydberg-Rydberg transitions as aprobe of ultracold plasma relaxation dynamics. In zeroth-order, we can considerthe nitric oxide ultracold plasma as a disordered ensemble of dipoles populating awide distribution of Rydberg and excitonic states concentrated in an energy intervalwithin a few hundred GHz of the ionization threshold, doped by a trace populationof more deeply bound Rydberg molecules. This residual population of lower-nmolecules includes a prominent fraction that retains the selected principal quantumnumber, n0. These molecules initially populate the optically selected total angularmomentum state (neglecting spin), N=1, n0 f (2), where 2 refers to the NO+ ion-core rotational state. Electron-Rydberg, ℓ-mixing collisions drive this population85to non-penetrating states of high-ℓ. At early times, mm-wave radiation tuned inthe frequency range between 70 and 110 GHz forms an excitation spectrum oftransitions from n0 f (2) to (n0± 1)g(2) for plasmas evolving from NO Rydberggases selected by ω2 tuned from n0 = 39 to 45.After a few microseconds, ℓ-mixing collisions cease, and a long-lived high-ℓpopulation exhibits resonant dips at the frequency of n0(2) to (n0± 1)d(2) tran-sitions. At center frequencies, such transitions suppress the plasma signal to anamplitude near zero, even though delayed selective field ionization spectra showthat the distribution of states evolves by then to contain fewer than one percent inthe n0 level resonant with the mm-wave field. Thus, the ωmm excitation spectrumitself shows direct evidence that collisionless energy transfer must occur to fill thehole burned by mm-wave radiation in the distribution of Rydberg states.This excitation spectrum of Rydberg-Rydberg transitions to bright states ofboth higher and lower orbital angular momentum displays Fano lineshapes. Theseasymmetric features are characteristically broadened and displaced by couplingthat redistributes spectral weight to the localized network of coupled Rydberglevels that forms an underlying quasi-continuum of dark states. Reading the natureof this coupling directly in the linewidth, Γ and asymmetry parameter, q, thesespectra show direct evidence that predissociation, which transfers population fromthe spectroscopically active (n0 ± 1)d(2) states to the open-system of N(4S) +O(3P) atoms, bridges the closed plasma ensemble to a thermal continuum. Thisevidently strengthens coupling of the bright state to the localized plasma bath andacts to increase energy mobility in the strongly coupled out-of-equilibrium system.The application of mm-wave radiation for a longer time before late-peak detec-tion depletes the plasma signal nearly to zero, despite the fact that the n0 states ofany ℓ constitute a very small fraction of the relaxed ensemble. This indicates boththat no mechanism exists to protect ωmm-populated d-states from predissociating,and some process of intermolecular energy transfer must act to fill the populationhole burned in the scant n0ℓ(2) population by mm-wave excitation.This observation begs the question of repopulation. What dynamical processrefills the hole in the n0ℓ(2) Rydberg population, and enables the excitation of atransition to a predissociative state to drive nearly all of the NO+ ions and electronsin the plasma to dissipate as neutral N(4S) and O(3P) atoms? We cannot explain n086hole filling as an electron collision-driven process. The ℓ-mixing that accompaniesn-changing collisions of this nature would readily cycle the Rydberg populationthrough low ℓ and cause plasma decay on a microsecond timescale in the absenceof a resonant mm-wave field.Building on our conceptual model for arrested relaxation in an ultracold plasmastate of randomly interacting dipoles of random energies [148, 146], we see mm-wave driving as a means to add dissipative states to the collisionless ensemble. Therapid predissociation of these states serves to bridge the closed quantum system toa thermal continuum of free N(4S)+O(3P) atoms, in effect creating an open quan-tum system that is weakly coupled to a reservoir of the free-atom environment. Tomodel these dynamics, we apply the the Lindblad master equation as a Markovianapproach. Figure 6.1 schematically diagrams a mapping from plasma dissipationdriven by an mm-wave filed to a one-dimensional model spin of eleven spins withdisorder and one dissipative site. We can imagine the local dissipation occurs inthe residual n0ℓ(2) Rydberg states that occupying less than ten percent populationof the arrested NO ultracold plasma system. The mm-wave filed populates the(n0±1)d(2) state which immediately decays to the open-system of N(4S) + O(3P)atoms. The long-range dipolar interaction causes the non-resonant states, whichoccupy a large population and span a wide range of binding energy W , to transitioninto accessible dissociative states.87Figure 6.1: Schematic illustration of the mm-wave driven plasma dissipationrepresented by a 1D spin-1/2 chain with on-site energy disorder and localdissipation.6.2 Model desicription6.2.1 Open quantum system in Lindblad formalismIn a closed quantum system, the density matrix ρ evolves according to the vonNeumann equation:ρ˙ =− iℏ[H,ρ] (6.1)which is equivalent to the Schro¨dinger equation in that picture. An open quantumsystem undergoes a non-unitary evolution. Here, we consider the system embeddedin its environment, such that the combination forms a closed total system that fol-lows the von Neumann equation. Tracing out the environment degrees of freedom,we obtain the reduced density matrix ρsys = Trenv(ρtot). A general trace-preservingand completely positive form of the evolution for ρsys is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) or simply the Lindblad master equation [86]:ρ˙ =L (ρ) =− iℏ[H,ρ]+∑jγ j(L jρL+j −12{L+j L j,ρ}) (6.2)88where L j is the jump operator that couples the subsystem and the environmentwith corresponding coupling rate γ j. {A,B} = AB+BA is the anti-commutator.We apply this approach to approximate the open system in a weak coupling limitusing the Born, Markov, and rotating wave approximations [161]. Here, we find ithelpful to map the d-dimensional Hilbert space to the Fock-Liouville space with ad2 dimension, where we represent the density matrix by a vector |ρ⟩with Liouvillesuper-operators. We obtain the density matrix elements in Eq. (6.2) as numericalsolutions of the the d2 coupled rate equations.6.2.2 1D spin-1/2 XY chainWe extend this concept to a primitive representation of the quantum system ofdisordered dipoles by considering a one-dimensional, spin- 12 XY chain. An effec-tive many-body Hamiltonian governs pair-wise dipolar interactions between ion-electron pairs. Mimicking our experimental landscape, we assume that quenchinggives rise to a wide distribution of random potential with L states. In this dipole-dipole coupling limit, we represent pairwise excitations by spins with energy εi andexchange interactions governed by an XY Hamiltonian:H =∑iεiSˆzi +∑i, jJi j(Sˆ+i Sˆ−j +h.c.), (6.3)where Sˆα = ℏσˆα/2 denotes a spin-L operator for which σˆα is the correspondingPauli matrix for α = x,y,z. Sˆ± = Sˆx± iSˆy are the spin raising and lowering opera-tors.The first term in Eq. (6.3) describes the diagonal disorder from random dis-tributed on-site energy of any particular dipole with disorder strength W, for whichεi ∈ [0,W ]. In the quenched ultracold plasma, a representative selective field ion-ization spectrum gauges a W of ∼500 GHz as illustrated in Figure 6.2(b). TheJi j = J¯/r3i j term determines coupling strength of the spin flip-flops that contributesto the off-diagonal disorder amplitude (Figure 6.2 (c)). J¯ varies as J¯ ∝ |di||d j|,where di and d j are the transition induced dipole moments associated with thelocal spins. Over the present range of W, a simple pair of dipoles formed by s andp Rydberg of the same n has coupling strength J¯ of 75 GHz µm3 [166]. Here, we89Figure 6.2: (a) Schematic diagram of NO+ core ions paried with eletrons to frominteracting dipoles di and d j seperated by ri j = ri− r j. The pentential energyof interaction to lowest order is V ddj j = [di · d j − 3(di · ri j)(d j · ri j)]/r3i j. (b)Illustration of measured distribution of binding energies, W , defined by SFIspectra. (c) Schematic diagram of the spin-exchange interaction Sˆ+i Sˆ−j of twoRydberg molecules in the two-level (spin-half) approximation. (Figures aretaken from ref.[146])neglect the effects of van der Waals interactions, added to the Hamiltonian by aterm, ∑i, j Ui jSˆzi Sˆzj. Instead, we focus on the distributed effects of local dissipationin a many-body problem that is simplified for now to consist of a 1D spin chainof spin- 12 two-level systems. We define Li = S−i in Eq. (6.2) as the jump operatorfor dissipation on site i. Despite its apparent simplicity, this model is intrinsicallyinteracting as it does not map to non-interacting fermions under a Jordan-Wignertransformation. In fact, even for Ji j confined to couple nearest-neighbours, a localdissipation induces loss across the system.6.3 Simulation and resultsIn all cases, we consider an eleven-site, one-dimensional spin chain as representedby Eq 6.3, composed of alternating sites of spin-up |↑⟩ and spin-down |↓⟩ dipoles.To initialize, we set the middle site (5) to |↓⟩, configure the end sites (0,10) to have90open boundary conditions. This initial configuration represents a general scenarioof randomly distributed dipoles with about equal population of excited and groundstates. However this initial configuration is not important after a short time ofunitary evolution. We choose dimensions in which ℏ = 1, J¯ = 1, and ri,i±1 = 1,and vary W and γ j in a range from 0 to 10. Calculations solve the Lindblad masterequation, Eq (6.2), in Liouville space to model the evolution of the expectationvalues, < Szi >, in presence of all-neighbour spin-flip interactions, Ji j = J¯/r3i j.The left-hand frame of Figure 6.3 shows the free evolution of a closed systemwithout dissipation, as governed by Eq 6.1. Note that all states oscillate slightlyabout spin levels, Szi ≈ 0, conserving total spin as ∑i Szi = 1/2.Figure 6.3: (left) Evolution of a closed system of 11 spins with unit coupling (J¯ = 1),zero disorder (W = 0) and zero dissipation (γ j = 0). (right) Evolution of thesame system initiating a non-zero local dissipator after t = 100 on the centralsite (S−5 ) with γ5 = 0.1.To study the effect of dissipation, we introduce a local non-zero dissipator S−ion a single site after a time t ∗ J¯ = 100. For a γ j as low as 0.1 without disorder,dissipation quickly spreads over the whole system. The right-hand frame of Figure6.3 shows the evolution of each spin < Sz > for a local dissipation with γ j = 0.1applied to the central. site in absence of onsite disorder. For W=0, spin-flip inter-actions delocalize dissipation. Dissipation applied locally, spreads instantaneouslyto all sites, and all the expectation values of spin, < Sz >, decay to < Sz >=−1/2at the same rate.Figure 6.4 and Figure 6.5 illustrate this condition of spreading dissipation for91Figure 6.4: Evolution of Szi in a system of 11 spins with unit coupling (J¯ = 1),moderate disorder (W = 10) and moderate dissipation (γ j = 0.5) applied att ∗ J¯ = 100 to the central site. Curves showing the rate of decay at the siteof dissipation (Site 5), and at one of the most distant sites (Site 10).a case of moderate disorder (W = 10) and moderate decay (γ j = 0.5). The resultsare averaged over 20 different disordered energy set {εi}, where εi ∈ [0,W ]. Fromt ∗ J¯ = 0 to t ∗ J¯ = 100, the on-site disorder apparently regularizes the dynamics inthe closed system to make it localized. After t ∗ J¯ = 100, dissipation applied to thecentral site spreads equally to both ends. Contrast this with the evolution of thesystem when dissipation is applied to Site 0. Here we find that a decay channel ofmoderate rate (γ j = 0.5) at a single site causes amplitude to dissipate at all sites,despite the localization created by the on-site disorder. Moderate decay at one92Figure 6.5: Evolution of Szi in a system of 11 spins with unit coupling (J¯ = 1),moderate disorder (W = 10) and moderate dissipation (γ j = 0.5) applied att ∗ J¯ = 100 to an edge site. Curves showing the rate of decay at the site ofdissipation (Site 0), and at the most distant site (Site 10).site, initiated after a delay of t ∗ J¯ = 100, represents the local dissipation in an openquantum system. The rate of decay stimulated in remote sites lessens with distance.Site 10 decays far slower when we initiate dissipation on Site 0, as opposed toSite 5. Note that the symmetric redistribution of dissipative character from Site5 lengthens its lifetime compared to the less efficient unidirectional redistributionfrom the Site 0 edge state.Similar effects are observed with initial γ j in between 0.1 to 10. Figure 6.6illustrates the relationship between the fitted exponential decay rates of < Szi > andsite distances from the central lossy site (Site 5) with varied dissipation strength γ593and from edge lossy site (Site 0) with varied dissipation strength γ0, respectively.In the presence of weak dissipation γ = 0.1, the decay rate of non-lossy sites fallswith the distance from the lossy site according to a power law ∝ 1/r. This power-law relationship fails when the dissipation strength exceeds the limit of γ = 1 at thecentral site and γ = 0.1 at the edge site.941 2 3 4 5 6distance10-310-210-1100101decay rate5=0.1y=0.02x-1.05=0.5y=0.05x-1.05=1y=0.10x-1.25=5y=0.73x-2.25=10y=3.34x-3.90 2 4 6 8 10distance10-410-2100102decay rate0=0.1y=0.03x-1.00=0.5y=0.16x-1.90=1y=0.32x-2.40=5y=2.60x-3.60=10y=6.25x-4.7Figure 6.6: (left) Exponential decay rates of < Szi > in a system of 11 spins with unitcoupling (J¯ = 1), moderate disorder (W = 10) as a function of site distancesfrom the central lossy site (Site 5) with varied dissipation strength γ5. (right)Exponential decay rates of < Szi > in a system of 11 spins with unit coupling(J¯ = 1), moderate disorder (W = 10) as a function of site distance from edgelossy site(Site 0) with varied dissipation strength γ0. The curves are fitted topower low by the least square method.956.4 DiscussionFrom the simulation results in the previous section, we discovered that due to thelong-range dipolar interaction, a local dissipation can widely spread and cause stotal dissipation that exceeds the population of the lossy sites. This is just whatthe experiment conveys, as discussed in Chapter 5. In addition, we found from thesimulation that disorder in onsite potential energies help prevent the decay on thesites far from the dissipation source. We can therefore ask: does the local dissi-pation affect the dynamics of the system in a perturbative way or non-perturbativeway, and how are decay rates modulated in each case?First, we can consider a weak dissipation and analytically show the exponentialdecay of < Szi > at all sites in terms of the robustness of the Anderson localizationsystem. Recall the Lindblad equation with one-site dissipation γ:∂tρ =−i[H,ρ]+ γ(c0ρc†0−12{c†0c0})(6.4)withH =∑i̸= jJ|i− j|α c†i c j +∑jε jc†jc j (6.5)where εi ∈ [−ε,ε]. The equation of motion for expectation values of observablesfollows from the master equation is then:∂t⟨O⟩= Tr[O∂tρ] = i⟨[H,O]⟩− γRe⟨c†0[c0,O]⟩ (6.6)Single particle eigenstates of H can be written asbℓ =∑jαℓ, jc j , c j =∑ℓα∗ℓ, jbℓ (6.7)and since the mode occupation numbers nℓ = b†ℓbℓ commute with the Hamiltonian(by definition), we can write the rate equations as∂t⟨nℓ⟩=−γRe⟨c†0[c0,nℓ]⟩=−γRe[α∗ℓ,0∑jα j,0⟨b†jbℓ⟩] (6.8)96For a localized system, we can assume that the modes are uncorrelated:⟨b†jbℓ⟩ ≈ δ j,ℓ⟨nℓ⟩ (6.9)which gives∂t⟨nℓ⟩=−γ|αℓ,0|2⟨nℓ⟩ . (6.10)This results in an exponential decay of population at site ℓ. The rate of decayscales by the coefficients αℓ,0 that describes the wavefunction overlaps betweeneigenstates at site ℓ and the dissipative site. The coefficient αℓ,0 is determined bythe localization length. Evidently, |αℓ,0|2 ∼ 1/rd , where d denotes the dimension.In the case of a 1D spin chain, this aligns with the results obtained in the weakdissipation limit, as shown in Figure 6.6. This suggests that, at least for weakdissipation, only a vanishing density of sites near the lossy site will decay whilethe others will continue to evolve coherently.At the opposite extreme, the strong dissipation may drive an effective quantumZeno dynamics, in which at longer times with respect to 1/γ , the unitary part (non-lossy sites) exhibits effective decay strengths that are much smaller than γ . Thishas been deliberated in ref [114].The dimensions of the simulation result in conventional units scale with thedensity of spins. Table 6.1 shows the conversion of units in which ℏ = 1, J¯ = 1,r = 1, W = 10, and γ = 0.1, to the real dimensions with J¯=75 GHz µm3 underdifferent densities of spins, A density of 0.5 µm−3 corresponds to an averagenearest neighbour distance of 1.26 µm. At this density, a disorder of W = 10spans a width of 375 GHz, Ji j = J¯/r3i j = 0.5 refers to a coupling width of 38 GHz,and γ j = 0.1 describes a decay time of 270 ps. For such conditions, the full scaleof t ∗ J¯ = 500 defines a total elapsed time of about 2 ns.Table 6.1: Conversion to real dimensions from ℏ = 1, J¯ = 1, r = 1, W = 10, andγ = 0.1, to J¯=75 GHz µm3 with different densities of spinsρ(µm−3) nearest-neighbor distance (µm) 1 unit time=1/(2π*75GHz) ns W (GHz) γ (GHz) decay time (ns) 500 unit time (ns)1 1.0 0.002 750 7.5 0.13 1.10.5 1.3 0.004 375 3.75 0.27 2.10.1 2.2 0.021 75 0.75 1.33 10.60.01 4.6 0.212 7.5 0.075 13.33 106.297Thus, it is possible to derive support from this simple model for the physicalideas conveyed by the experiment. A moderate rate of decay introduced at a singlesite in a dipolar system localized by strong diagonal disorder can drive dissipationthat appears instantaneously at all sites, where each dipole exhibits a rate of decaythat varies algebraically with its distance from the most affected site. We mightalso be able to answer how robust our localization system is against the mmWavefield-driven dissipation by measuring the overall decay rate of the system. Futureexperiments to investigate the resonant dips of n0(2) to (n0 ± 1)d(2) transitionsutilizing time-resolved and extending mmWave pulses may pave the way for amore profound understanding of this problem.6.5 Effort on large-scale simulation with semi-classicalapproximationThe d-dimensional Hilbert space of the spin system grows exponentially with thenumber spins: d = 2N for the spin- 12 subsystem. The corresponding Liouvillespace grows with d2 = 22N . Thus the exact analytic and full numeric solution isintractable for a large number of spins with classical computing power. Numericalapproaches with mean-field approximation as well as semi-classical expansionhave been applied to efficiently simulate the many-body quantum systems rangingfrom fermionic lattice [154] and driven dissipative Bose-Hubbard chain [103] toand spin ensembles [68]. The sequential sections will explore the mean-fieldmethod adapted to the open spin chain described by Eqs. 6.2 and 6.3.6.5.1 1st-order mean-field (Gutzwiller) approximationThe time evolution of a density matrix can be approximated using Guitzwilleransatz, which is essentially a mean-field approximation. This approach assumesa factorization of the density matrix by neglecting correlations of subsystems.Consequently, the density matrix can be expressed as the tensor product of allsubsystems at each site:ρ ≈∏l⊗ρl (6.11)98where ρl are site-dependent density matrices. Using tr(ρl) = 1 and tr(∂tρl) = 0, wecan derive the set of nonlinear master equations of the from the Lindblad equationEq.(6.2) by calculating the partial trace corresponding to site l:∂tρl =−iTr̸=l[H,ρ]+Tr̸=lD(ρ)which yields the result for the spin chain open quantum system:∂tρl =−i[εiSzl +∑iJil(〈S−i〉S+l +〈S+i〉S−l),ρl] (6.12)+ γl(S−l ρS+l −12{S+l S−l ,ρl})The time evolution of the whole system is governed by local on-site density matri-ces, and the matrix elements to be solved grow polynomially with the number ofsites.It is not always necessary to solve all density matrix elements for the spinsystem. Instead, it is useful to obtain the equation of motion for the expectationvalue of each spin,〈Szl〉and〈S+l〉(〈S−l〉∗). Recall the commutation relationshipsof the spin operators: [Szi ,S+j]= δi jS+i[Szi ,S−j]=−δi jS−i[S+i ,S−j]= 2δi jSzi[S+i ,S+j]=[S−i ,S−j]=[Szi ,Szj]= 0The spin-1/2 system also obeys the anti-commutation relationships:{S+i S−i }= 0 S+i S+i = S−i S−i = 0To improve the accuracy, we can use the following identities for operators on the99same site without factorizing the double operators:S+i S−i = 1/2+SziS−i S+i = 1/2−SziS±i Szi =∓S±i /2Szi S±i =±S±i /2We can therefore derive a set of rate equations for the spin system from the Lind-blad equation Eq.(6.2) by making use of the commutation relations and calculating⟨S . . .⟩ ≡ Tr(S . . .ρ). Explicitly we can get:∂∂ t〈Szk〉=+ i∑i̸=kJik(〈S+i S−k〉−〈S+k S−i 〉)− γk(〈Szk〉+1/2) (6.13a)∂∂ t〈S+k〉=iεk〈S+k〉−2i∑i̸=kJik〈S+i Szk〉− 12γk〈S+k〉(6.13b)∂∂ t〈S−k〉=− iεk〈S−k〉+2i∑i ̸=kJik〈S−i Szk〉− 12γk〈S−k〉(6.13c)The expectation value of the two-particle operator can be factorized according tothe assumption given by Eq.(6.11),〈SiS j〉≈ ⟨Si⟩〈S j〉 (6.14)from which we can get a complete set of coupled rate equations up to the first-order approximation. This is essentially a semi-classical approximation with thefirst-order expansion. One can equivalently derive the same dynamic equationsfrom the density matrix of each subsystem given by Eq.(6.12)6.5.2 Mean-field method including pair correlationThe higher order correction is added to the mean-field approach by considering thepair correlations up to the second order, i.e. ρ =∏i⊗ρ(i)+∑ j<k(ρ( j,k)⊗∏i ̸= j,k⊗ρ(i)),where ρ( j,k) is the correlation part. The expectation values of the three-particle100operators is approximated by〈SiS jSk〉≈ ⟨Si⟩〈S jSk〉+〈S j〉⟨SiSk⟩+ ⟨Sk⟩〈SiS j〉−2⟨Si⟩〈S j〉⟨Sk⟩ (6.15)for i ̸= j ̸= k.By employing the same methodology used to calculate the first-order expec-tation value and carefully picking out the terms with overlapping indexes, we canobtain additional coupled rate equations of higher order, as expressed by Eq.(6.16).The combination of Eq.(6.13) and Eq.(6.16) forms a complete set of coupledrate equations up to the second-order expansion. The total number of equationsscales quadratically with the number of spins. Note that not all equations are lin-early independent in the complex domain, i.e.,〈S−k〉=〈S+k〉∗ and 〈S−k S−l 〉=〈S+l S+k 〉∗.Previous research has demonstrated the high accuracy of the second-order mean-field method for open spin systems in the presence of collective dissipation in alldimensions, at least for short-time evolution [68]. In the subsequent section, I willassess the accuracy of the mean-field method when applied to the 1D-spin chainwith a local dissipation, considering the desired initial conditions and evolutiontime. The results will be compared to the exact solutions.101∂∂ t〈S+k S−l〉=i(εk− εl)〈S+k S−l〉− iJkl(〈Szk〉−〈Szl〉) (6.16a)−2i ∑i ̸=k,l(Jik〈S+i S−l Szk〉− Jil 〈S+k S−i Szl〉)− 12(γk + γl)〈S+k S−l〉∂∂ t〈S+k S+l〉=i(εk + εl)〈S+k S+l〉(6.16b)−2i ∑i ̸=k,l(Jik〈S+i S+l Szk〉+ Jil〈S+k S+i Szl〉)− 12(γk + γl)〈S+k S+l〉∂∂ t〈S−k S−l〉=− i(εk + εl)〈S−k S−l〉(6.16c)+2i ∑i ̸=k,l(Jik〈S−i S−l Szk〉+ Jil〈S−k S−i Szl〉)− 12(γk + γl)〈S+k S+l〉∂∂ t〈S+k Szl〉=iεk〈S+k Szl〉− 12iJkl〈S+l〉(6.16d)−2i ∑i ̸=k,lJik〈S+i SzkSzl〉+ i ∑i ̸=k,lJil(〈S+i S+k S−l〉−〈S+k S+l S−i 〉)− (12γk + γl)〈S+k Szl〉− 12γl〈S+k〉∂∂ t〈S−k Szl〉=− iεk〈S−k Szl〉+12iJkl〈S−l〉(6.16e)+2i ∑i ̸=k,lJik〈S−i SzkSzl〉+ i ∑i ̸=k,lJil(〈S+i S−k S−l〉−〈S+l S−k S−i 〉)− (12γk + γl)〈S−k Szl〉− 12γl〈S−k〉∂∂ t〈SzkSzl〉=i ∑i ̸=k,lJik(〈S+i S−k Szl〉−〈S+k S−i Szl〉)+ i ∑i ̸=k,liJli(〈S+i S−l Szk〉−〈S+l S−i Szk〉)(6.16f)− (γk + γl)〈SzkSzl〉− 12γk〈Szl〉− 12γl〈Szk〉1026.5.3 Accuracy of mean-field method with second-order correctionFigure 6.7 shows examples of good accuracy achieved using the mean-field methodunder special conditions of collective dissipation and initial states for a short periodof evolution. Under these scenarios, the second-order approximation accuratelycaptures the exact solution of the 1D-spin chain dynamics with a small error. Thefirst-order approximation captures the decay feature accurately while exhibiting aslightly larger error on the coupling feature.Figure 6.7: Evolution of Sz two sites in a system of 7 spins with unit coupling (J¯ =1), disorder W = 10 and collective dissipation (γ = 1) applied at t ∗ J¯ = 10 toall sites. Curves represent calculation by first-order (orange)and second-order(green) mean-field method compared to the exact solution (blue). All spins areinitialized to point to the x direction.The same scenarios as in the previous simulations are analyzed, where the spin-up state (|↑⟩) and spin-down state (|↓⟩) alternate on the sites, and local dissipationoccurs either at the central or edge site. The evolution time is extended to t ∗ J¯ = 500including a t ∗ J¯ = 100 free evolution period. Figure 6.8 illustrates examples of suchsimulations under different conditions, yielding varying convergence outcomes.The accuracy of the first-order mean-field method depends on the initial orien-tations of spins. When the initial spins are all parallel to each other (|↑⟩ or |↓⟩),the first-order approximation fails to initiate the coupling between the dissipativesite and other sites through the long-range interactions. As a result, all other sitesremain in their initial states.The second-order method exhibits chaotic behavior after a prolonged periodof evolution. While the system converges to a stable solution for certain initial103Figure 6.8: Example simulations of the 1st-order (orange) and 2nd-order (green)approximation compared to the exact solution. The top-left frame illustratesa scenario where the 2nd-order approximation converges to the exact solutionwith good agreement, while the 1st-order approximation fails to initiateinteraction. The top-right frame shows the divergence can occur under certainconditions, even in the absence of disorder and dissipation. The bottom-leftframe reveals that divergence may take place during the free evolution stagewhen a random disorder configuration is present. The bottom-right frameshows that divergence can occur during a stable stage after the introduction ofdissipation with a random disorder configuration in place.conditions, in most cases, small errors accumulate over time, leading to divergencein the solution. This chaotic behavior arises even in the absence of disorder anddissipation.In summary, the first-order mean-field approximation exhibits limited accu-racy and fails to capture interaction features under certain initial conditions. Thesecond-order approximation display improved accuracy over short timescales; how-ever, it exhibits chaotic behavior. The coupled rate equations derived from thesecond-order approximation form a complex and chaotic system, which limits104the numerical methods. Further investigation is necessary to refine the initialconditions and improve the stability of the simulations using these approximations.6.6 Conclusion and outlookWe have developed and analyzed a primitive model using a Lindblad master equa-tion that incorporates local dissipation, long-range interactions, and on-site disor-der. We find that a moderate rate of decay introduced at a single site in a dipolarsystem, which is localized by strong on-site disorder, can drive dissipation thatappears instantaneously at all sites. In this scenario, each dipole exhibits a rateof decay that varies algebraically with its distance from the most affected site.This relatively simple model successfully captures much of the phenomenologyobserved experimentally. At the same time, this model has opened up a new the-oretical front: the dynamics of disordered many-body systems subject to localizedloss, an area that has not yet been thoroughly investigatedDealing with open quantum systems presents a particular challenge on a tech-nical level since it is practically impossible to numerically keep track of both thesystem and the environment. The semi-classical approximation is unable to achievethe accuracy for the desired conditions of the Lindblad simulation. Consequently,we aim to develop new analytic approximations and numerical tools that allowus to effectively describe the system of interest. In particular, we plan to exploremore powerful numerical tools utilizing tensor network techniques such as Dirac-Frenkel time-dependent variational principle (TDVP) [48, 49, 69, 59, 108]. Thesetechniques will enable us to investigate systems of much larger size and study howthese effects of driven dissipation scale with system size and dimensionality. Wewill also conduct further experimental investigations into the global dissipation rateof the system, aiming to correlate these findings with large-scale calculations.105Chapter 7ConclusionsA Rydberg gas of nitric oxide entrained at milli-Kelvin temperature in a skimmedsupersonic molecular beam proceeds via electron-impact avalanche, bifurcationand quench to form a strongly coupled, ultracold plasma. We investigate therelaxation dynamics of this out-of-equilibrium state in both classical and quantummechanical domains.The nitric oxide molecular ultracold plasma contains a persistent residue ofNO Rydberg molecules that retain the initially selected principal quantum number,n0. In Chapter 4, we demonstrate that a radio frequency field, applied to a plasmain a state of arrested relaxation, subtly changes the state of this plasma in a waythat causes the predissociation of this residue to accelerate. The NO Rydbergresidue represents a surviving fraction of n0 molecules trapped in states of high ℓpopulated during the avalanche and quenched to a regime of suppressed ℓ-mixing.The fractional yield of n0 Rydberg molecules undergoing radio frequency field-accelerated predissociation varies substantially with plasma density. This excludesℓ-mixing as a field-induced process in isolated-molecules. We know that chargedparticles in a plasma also respond collectively to a radio frequency field by exe-cuting oscillatory modes of electron and ion motion. Damping via charge couplingand collisions couples energy from the rf field to the plasma. If this heating exceedsa mobility threshold for localized electrons, we should expect to see an effect ofthe rf field in an increase of the frequency of electron Rydberg collisions.That106appears to be the case here. The kinetics tell us that the rf field acts on theplasma to mobilize electrons. These electrons collide with Rydberg molecules andscramble ℓ. Thus, we see an effect in the arrested residue of n0 Rydberg moleculesowing to a process that occurs in the ultracold plasma background. Note thatthis mobilization, which dramatically accelerates predissociation kinetics, causeslittle change in the ultracold plasma background. We can therefore regard thisadventitious population of high-ℓ n0 Rydberg molecules and its response to an rffield as a quantum-state probe of the dynamics of avalanche and quench that formthe molecular ultracold plasma. The natural state of arrested predissociation pointsto an immobility of electrons in the quenched plasma. A radio frequency field actsto mobilize electrons, which increases the rate of predissociation in a fraction of n0Rydberg population that grows with the density of the background plasma.In Chapter 5, we find that resonant mm-wave fields in the range from 50 to 100GHz drive n0 f (2) to (n0 ± 1)g(2) transitions in a state-selected Rydberg gas ofNO. This transformation produces a clear signature in the selected field ionization(SFI) spectrum and dramatically increases the early time intensity of high-Rydbergresonances in the SFI-detected high-Rydberg spectrum. We associate these en-hanced features with a decrease in the rate of predissociation owing to an increasein Rydberg orbital angular momentum. This same signature of early-time Rydbergstabilization appears as an enhanced long-time plasma signal, extracted as a latepeak after 20 µs of field-free evolution. Excitation by mm-wave radiation tuned toresonance with n0 f (2) to (n0±1)d(2) transitions depletes the plasma signal to anamplitude near zero, even though delayed selective field ionization spectra showthat the distribution of states evolves by then to contain fewer than one percent inthe n0 level resonant with the mm-wave field.The excitation spectrum of mm-wave Rydberg-Rydberg transitions to brightstates of both higher and lower orbital angular momentum displays Fano line-shapes. Intermolecular interactions redistribute spectral weight to a localized net-work of coupled Rydberg levels, forming an underlying quasi-continuum of darkstates that characteristically broadens and displaces these asymmetric features. Thelinewidth, Γ and asymmetry parameter, q directly reflect the dynamics of thiscoupling. Predissociation transfers population from the spectroscopically active(n0± 1)d(2) states to the open-system of N(4S)+O(3P) atoms. This bridges the107closed plasma ensemble to a thermal continuum. Predissociation of the bright statethus adds continuum character to the localized plasma bath, and acts to turn upenergy mobility and dissipation in the strongly coupled out-of-equilibrium system.In much the same way, rf-induced electron-collisional energy and orbital angu-lar momentum redistribution directly promotes dissipation, similarly suppressingevolution to a long-lived plasma state.In Chapter 6, we developed a primitive picture of this system using a Lindbladmaster equation to describe the Markovian dynamics that arise when a closedquantum system of disordered, randomly interacting molecular dipoles couples to athermal continuum of free atoms. 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This can be approximated by the simple Langmuir oscilla-tions caused by periodic displacement of electrons. The equation of motion of anelectron in 1D is,med2(∆r)dt2=−eE (A.1)The strength of the electric field of an infinite plasma volume in vacuum is givenby Gauss’s law:∇ ·E = eρeε0(A.2)where eρe represents the charge density. Substituting the gradient by ∆r we get theoscillating motion:med2(∆r)dt2=−ω2pe∆r (A.3)with ωpe =√e2ρe/ε0me is the oscillating frequency. The ions in plasma also havea resonance frequency ωpi =√e2ρi/ε0mi. The ion oscillation however falls in aslower freqency regime with heavy ion mass and does not play an important role125in our experiment.We can estimate that a typical 1V peak-to-peak (0.25V cm−1) 60 MHz radiofrequency field accelerates electrons to a velocity of 5.8 × 103 ms−1 at the zero-crossing. This corresponds to a laboratory frame kinetic energy of 0.097 meV or23 GHz. The amplitude of oscillation is 15.5 µm, which is less than 10 percent ofthe plasma radius.A.2 Freqeuncy dependence of RF resonanceAn effort has been made to investigate plasma resonance with radio frequencyfields of varying frequencies. There are two alternative synthesizing sources togenerate the RF pulses. A Rhode and Schwarz signal generator (SMB100B) and aTektronix AWG 7102 10 GS/s arbitrary waveform generator.When directly measuring the voltage at the output port, both synthesizers havebeen observed to exhibit a consistent output amplitude across different frequen-cies. However, when the voltage is applied to the grids of the vacuum chamber,it produces transmission characteristics of the circuit. Figure A.1 illustrates themeasured peak-to-peak voltage as the synthesizer is connected to the experimentalgrid. The large variation of the output RF amplitude resulting from resonanceswith experimental grids at different frequencies imposes limitations on conductingspectroscopy experiments to explore the resonance characteristics of the plasma.We conducted the RF spectroscopy experiment with the selective field ion-ization signal of plasma and observed the signal response to the RF field. Theresults is presented in Figure.A.2. Here, we can observe that at 60MHz, the plasmasystem exhibits a noticeable response to the RF field with a mild peak-to-peakamplitude. The plasma system responds similarly to substantially higher and lowerfrequencies (between 20 and 100MHz). Over this range, we see mainly the trans-mission characteristics of the circuit that applies a potential to our grids. Theflat frequency response reflects the wide distribution of particle densities in theGaussian ellipsoidal plasma.126Figure A.1: Measured relative amplitude of the radio frequency output withconnection to experiment grids in the vacuum chamber scanned over differentfrequenciesFigure A.2: Integrated electron signal obtained by selective field ionization of anultracold plasma as it evolves from a 44 f (2) Rydberg gas of NO in the presenceof a cw radio frequency field, plotted as a function of rf frequencies, fordifferent input rf amplitudes127Appendix BFano resonanceThe conventional Fano resonant [32, 34] lineshape function isσ(ε) =(q+ ε)21+ ε2, (B.1)where σ(ε) is the the relative absorption cross-section, q is the asymmetry param-eter. The detuning, ε , is defined by:ε =E−E0Γ/2(B.2)in which E0 ∼ω0 represents the resonant frequency of the discrete transition and Γdescribes its linewidth. The measured width relates to the coupling strength of eachdiscrete Rydberg state with the underlying ionization or dissociation continuum.The theoretical value of Γ is given by[32]:Γ= 2π|VE |2 (B.3)where |VE | represents the interaction between Rydberg discrete states and the un-derlying continuum.Consider an ensemble system with a number of states, among them |ϕ⟩ has adiscrete resonance level E0 and states |ψE⟩ has energy E that spans a continuousrange. We can write the coupling between the discrete state and the continuum as:128VE =< ψE |H|ϕ > (B.4)where H is the interaction Hamiltonian operator.Let |i⟩ represent the initial state, and T represents the transition operator, in thiscase T∼ u is the transition dipole moment. The q parameter can be calculated asq =⟨Φ|T|i⟩πVE · ⟨ψE |T|i⟩ (B.5)=⟨ϕ|T|i⟩+P∫ dE ′ ⟨ψE ′ |H|ϕ⟩ · ⟨ψE ′ |T|i⟩/(E−E ′)π ⟨ψE |H|ϕ⟩ · ⟨ψE |T|i⟩ (B.6)where P∫denotes the principal part of the integral and can sometimes be ignoredwhen determining q when the transition moment to the discrete state ⟨ϕ|u|i⟩ islarge [156]. The numerator describes the modification of the discrete state wave-function |ϕ⟩ by an admixture of coupled states of the continuum. Therefore,12πq2 =| ⟨Φ|T|i⟩|2| ⟨ψE |T|i⟩|2Γ(B.7)represents the ratio of transition probabilities to the modified discrete state |Φ⟩ andto a bandwidth Γ of unperturbed continuum states |ψE⟩.Figure B.1 plots the lines shapes for different values of q. Smaller |q| showsasymmetric lineshape indicating a stronger continuum-discrete coupling.We fit our experimental data to the Fano profile by adding individual transitionstogether with either positive or negative amplitudes. The Fano parameters aredetermined according to local least squares error using the Labview program asillustrated in Figure B.2.129Figure B.1: Fano line shapes of different q. Negative q has a reversed shape aboutε = 0Figure B.2: Labview program for fitting the mmWave spectrum to the Fano profile130Appendix CShell model simulationsC.1 MechanismC.1.1 Rate processes associated with avalanche to plasma andquenchThe plasma forms as the result of an avalanche of ionizing Rydberg-electron col-lisions described by a second-order rate constant kioni . Three-body-recombination,with rate constant ktbri , reverses this process:NO∗(ni)+ e−kion−−⇀↽−ktbrNO++2e− . (C.1)Both rate constants depend on electron temperature and vary with ni, the principalquantum number of NO∗.Rydberg-electron collisions, ki j, scramble the initial state of NO∗, exchangingbinding energy with electron thermal motion:NO∗(ni)+ e−ki j−⇀↽−k jiNO∗(n j)+ e− . (C.2)In a molecular Rydberg gas, dissociation affects plasma evolution.131Predissociation of NO∗,NO∗kPDi−−→ N+O , (C.3)diminishes plasma density and consumes binding energy.Direct dissociative recombination,NO++ e− kDR−−→ N+O , (C.4)reduces the balanced population of electrons and ions.Hydrodynamic charge exchange with electron-ion recombination (charge-transferingbifurcation accompanied by electron quench) consumes inner-shell ions as theyexpand through stationary outer shells of Rydberg gas.NO++ e−+NO∗(ni)kCTi (ni)−→ NO∗∗(ni)+NO∗∗ (C.5)Note that the magnitude of kCT(ni) depends on ni. And, charge exchange conservesRydberg principal quantum number. But, we regard those Rydberg formed by thisprocess as special. They do not predissociate, nor do they undergo electron impactionization. In effect, charge exchange exhausts the thermal energy of the associatedfree electron, quenching it to form a companion pair of high-ℓ molecules, one thatconserves the reactant principal quantum number, NO∗∗(ni), and a companion insome very high-n, high-ℓ state.C.1.2 Master equations describing avalanche and charge transfer ina plasma of uniform plasma densityWe define ρi = NO∗(ni) as the density of Rydberg molecules with principal quan-tum number ni in a plasma of uniform density. Let ρe = ne represent the densityof free electrons. We also use ρe to designate the density of [NO+] ions, whichremain always in balance by quasi-neutrality, which is to say, ρe = ne = [NO+]. Asindicated in Eq (C.5) hydrodynamic charge exchange forms a subset of quenchedRydberg molecules at each principal quantum number, ni with an evolving densitydefined by ρ∗∗i .132Note, here because both the quenched NO∗∗(ni)molecules and NO+ ions close-coupled electron pairs form in a regime of suppressed electron collisions, we as-sume that no inelastic electron - NO∗∗ collisions of consequence occur, leading ei-ther to Rydberg states of different ni or free, unassociated NO+ ions and electrons.Thus, we drop the principal quantum number label for coupled rate calculationsand simply say that ρ∗∗ = [NO∗∗] = 2∑iρ∗∗i . Note, in this picture, [NO∗∗] forms asink filled by conversion from ions and Rydberg molecules from which no outletexists in the absence of an additional source of dissipation.For a uniform density, ρi, of Rydberg molecules in state ni and electron density,ρe the rate processes represented by Eqs. C.1 to C.5 yield the system of coupledrate equations:dρidt=−ρiρe∑jki j +ρe∑jk jiρ j− kioni ρiρe+ ktbri ρ3e − kPDi ρi− kCTi ρiρedρedt= ρe∑ikioni ρi−ρ3e ∑iktbri − kDRρ2e −ρe∑ikCTi ρidρ∗∗dt= ρe∑ikCTi ρi .(C.6)where we use quasi-neutrality to set ρion equal to ρe. Introducing the density of dis-sociated molecules ρDR, ρPDi owing to recombination and predissociation togetherwith the total density of quenched NO∗∗ Rydberg molecules, defined by a quantityρ∗∗ = 2∑iρ∗∗i , defines a conserved number of NO-associated particles, N (NO+ion - electron pairs, NO∗ and NO∗∗ Rydberg molecules and sets of N(4S) + O(3P)atomic dissociation products):N/V = ρe+ρDR+∑i(ρi+ρPDi)+ρ∗∗ (C.7)In principal, the dynamics extend over an infinite range of quantum numbersni. In practice, however, we define a lower bound of nmin = 10 and set the upperbound to nmax =√R/kBT , which conforms best with plasma scaling. [132].133C.1.3 Time evolution of electron temperature in a plasma of uniformdensityConsidering the global conservation of total energy coupled with the rapid colli-sional equilibration of electron kinetic energy, the electron temperature in a plasmaof uniform density evolves as:0 =ddtEtot =ddt(32kBTe(t)ρe−R∑iρin2i−Eloss), (C.8)where Eloss describes the energy loss per unit volume owing to the dissociation ofNO to form N(4S) + O(3P), and charge transfer sequestering NO molecules in thebifurcated volumes of quenched high-Rydberg NO∗∗ molecules:Eloss =−32kB∫ tt0dt ′Te(t ′)ρ˙DRe (t′)+R∑iρPDin2i+R∑iρ∗∗in2i. (C.9)This expression for energy conservation assumes that no change in the potential en-ergy of the system accompanies the formation of close-coupled NO+ ion - electronpairs via reaction (C.5).C.1.4 Plasma evolution in a Gaussian ellipsoidal density distributionrepresented by 100 concentric shellsWe can account explicitly for the prolate distribution of plasma density by defininga Gaussian ellipsoid with semi-axes rk = 5σk (k= x,y,z), divided into L= 100 con-centric ellipsoidal shells with walls separated by equal widths along the principalaxes.At t = 0, each shell accounts for a defined uniform initial density of RydbergNO molecules. Taken taken together, these shells describe the discretized initialdistribution of the Rydberg gas. A separate set of coupled rate equations (Eq. C.6)describes the evolution of Rydberg gas to plasma in each shell, as labeled by thesuperscript index α = 1, . . . ,L:134dραidt=−ραi ραe ∑jki j +ραe ∑jk jiραj − kioni ραi ραe + ktbri (ραe )3− kPDi ραi − kCTi ρα+1i ραedραedt= ραe ∑ikioni ραi − (ραe )3∑iktbri − kDR (ραe )2−ραe ∑ikCTi ρα+1idρα∗∗dt= ραe ∑ikCTi ρα+1i .(C.10)At the density of the molecular beam, the recoiling neutral N(4S) + O(3P)products of nitric oxide dissociation have a mean free path that is much largerthan the entire plasma volume [137]. The electrons on the other hand are bound bythe space charge of the ions. The volume occupied by the ions produced in a givenshell define the evolving volume of that shell. Thus, we assume that the constraintof quasi-neutrality confines the corresponding electrons to the same volume. Thelow electron mass ensures a fast kinetic energy transfer between electrons in allshells, which we assume instantaneously equilibrates the electron temperature,Tαe = Tα ′e = Te. We assume that the kinetic energy in the thermal continuum ofN(4S) + O(3P) after dissociation does not contribute to the electron temperaturein the plasma ellipsoid volume. Thus we can ignore the Eloss term in C.8. As aconsequence, energy conservation holds for the ellipsoid as a whole is:0 =ddt∑αEαtot(t)Vα =ddt(32kBTe(t)∑αραe Vα −R∑iαραin2iVα). (C.11)Equations C.10-C.11 form a set of (nmax−nmin+2)L+ 1 ordinary differentialequations which we can solve by numerical integration.C.1.5 Hydrodynamic Shell ModelThe Vlasov equations provide an effective means of describing the collisions self-similar expansion of a spherical Gaussian plasma [65]:135σ2(t) = σ2(0)(1+ t2/τ2) (C.12)u(t) = γ(t)r =tτ2+ t2r (C.13)Ti(t) =Ti(0)1+ t2/τ2(C.14)Te(t) =Te(0)1+ t2/τ2(C.15)where τ defines the characteristic plasma expansion time,τ =√miσ20 /kB[Te(0)+Ti(0)] (C.16)and u represents the radial velocity at radius r.We can also extend this approach to capture the effect of electron energy lossowing to expansion on the collisional dynamics of plasma avalanche bifurcationand quench, in other words.Under the conditions of simultaneous avalanche and expansion, the force of theexpanding electron gas acts on the NO+ ions to increase the radial coordinates ofthe plasma, diminishing its density and reduce the electron temperature accordingto:−mi dudt = e∇Φ= kBTe∇ρeρe, (C.17)where mi represents the ion mass, and u the radial velocity. These effects ofelectron temperature and charged particle density have straightforward effects onthe coupled rate processes ongoing in the shells.We begin by neglecting any dynamical consequences of ion-ion and ion-Rydbergcollisions, and model the hydrodynamic evolution of the plasma by the self-similarexpansion of the Gaussian volume along its ellipsoidal axes, k, governed by thecorresponding charge density gradients. The shell model of this ellipsoid defines136these gradients discretely and enforces self-similarity by approximating,−mi d2rαkdt2≈ kBTe〈∇kρeρerk〉rαk = kBTe1L∑β1ρβe rβkρβ+1e −ρβerβ+1k − rβkrαk . (C.18)Here, rαk is the position of shell α along the k-axis and brackets denote the averageover all shells α . The second equality figures the gradients in x, y, and z discretelyby summing the differences in adjacent charge densities over all cells.Ambipolar expansion transfers electron kinetic energy to the radial motion ofthe ions. For simplicity, we assume three uncoupled energy channels as would bethe case for a spherical distribution:Eαkin ≈mi2(uαx )2+(uαy)2+(uαz)23. (C.19)Adding this to the energy conservation Eq. C.11 gives:0 =ddt(32kBTe(t)∑αραe Vα −R∑iαραin2iVα −∑αEαlossVα +∑αEαkinVα). (C.20)Note that Eq. C.17-Eq. C.19 are alternative approximations to the self-similarexpansion mechanism described by the Vlasov equations Eq. C.12-Eq.C.15We can finally account for the rarefaction of the charge and Rydberg moleculedistributions owing to expansion. Modifying Eq. C.10 yieldsdραidt=−ραi ραe ∑jki j +ραe ∑jk jiραj − kioni ραi ραe + ktbri (ραe )3− kPDi ραi− kCTi ρα+1i ραe −ραiVαdVαdtdραedt= ραe ∑ikioni ραi − (ραe )3∑iktbri − kDR (ραe )2−ραe ∑ikCTi ρα+1i −ραeVαdVαdtdρα∗∗dt= ραe ∑ikCTi ρα+1i −ρα∗∗VαdVαdt,(C.21)137where the volume of shell α changes asdVαdt=4π3ddt(∏krαk −∏krα−1k). (C.22)We may now proceed to integrate Eq.C.12-C.22.C.1.6 Rate constantsWe adopt expressions for rate constants, kioni , ktbri , ki j as functions of electron tem-perature developed with reference to Monte Carlo trajectory simulations by Pohland coworkers [113]. We calculate a rate constant for dissociative recombinationkDR based on the model of Schneider et al. [135]. kDR(Te) fits a power-law intemperature with exponent −0.51. Values of kPD vary with n and l according toour previously introduced model [132].We approximate the rate of NO+-NO∗ charge exchange kCT depending pro-portionally on the square of the principal quantum number of the target Rydbergmolecule and varying linearly with the local radial velocity of the NO+ ions. In aplasma undergoing a self-similar expansion, the NO+ radial velocity and charge-transfer rate constant associated with each shell α , increases linearly with the radialdistance rα138C.1.7 Indirect hydrodynamic expansion of the neutralized productsof charge-exchangeWe can assign the NO+, NO∗, NO∗∗ into three different sets of Gaussianellipsoidal shells. The NO+ ellipsoid expands ballistically according to the Vlasovequation. The initial neutral NO∗ Rydberg molecules stay in the stationary shells.The charge transfer cycles between NO+ and NO∗ when the NO+ shells passthrough the stationary NO∗ shells. This creates radially moving Rydberg moleculesNO∗∗ that populate all values of orbital angular momentum, ℓ. The NO∗∗ expandswith approximately half the velocity of the NO+ shells according to the conser-vation of momentum. The charge transfer happens the most at the wings of thelong axis. High-ℓ molecules move to larger radial distance, unencumbered byconstraints of quasi neutrality. The ambipolar expansion of NO+ + e− in concertwith recombination and predissociation should claim all of the NO+/ NO∗ in the139core of the plasma. These effects combine to deplete the NO+/NO* density inthe centre of the NO+ and NO∗ ellipsoid, beginning the evident bifurcation of theultracold plasma.The self-similar expansion of the Gaussian ellipsoid makes it possible to defineproper shell boundaries such that all boundaries expand to the previous positionsof the adjacent boundaries simultaneously. We can initially construct the NO+,NO∗, NO∗∗ shells with shared boundaries following a self-similar rule. Reactionsproceed within each overlapping shell of NO+, NO∗, and NO∗∗ until the NO+boundaries expand to the next overlapping boundary of the stationary NO∗ shellssimultaneously. The NO∗∗ shells start to expand when its leading shell matches aninner shell of NO+, where the radial velocity is exactly half of the radial velocityof most outer shells of NO+. It is convenient to consider individual 1D expansionon each axis of the ellipsoid.C.2 Example resultsC.2.1 Expansion and quench without charge transferFigure C.1: (left) The long-time evolution of electron temperature in plasma formedby avalanche from a Rydberg gas with n0=80 within an ellipsoid with aninitial peak density of 0.4 × 1011 cm−3 for fixed geometry (dotted line) and astepped sequence of coupled rate equation evolution and self-similar ambipolarexpansion (solid line). (right) Global population fractions of particles as theyevolve in the avalanche of a shell-model plasma without expansion (dotted line)and with intervening steps of self-similar expansion (solid line) as above.140C.2.2 Expansion and quench with charge transferFigure C.2: Evolution of electron temperature (left) and global population ofparticles (right) in plasma formed by avalanche from a Rydberg gas with n0=80in 1D self-similar Gaussian shells with an initial peak density of 1012 cm−3(top) / 1011 cm−3 (bottom) for a stepped sequence of coupled rate equationevolution taking into account the charge transfer and self-similar ambipolarexpansion.C.2.3 Bifurcation in the effect of charge transfer141Figure C.3: Evolution of high-ℓ Rydberg molecules in plasma formed from aRydberg gas with n0=80, an initial peak density of 1010 cm−3 and initialσx=1mm and σy=0.5mm in effect of charge transfer driving the bifurcation.The frames are taken at t=20ns, 50ns, 100ns ,120ns142C.2.4 Matlab codeThe following section provides the Matlab function for the shell model coupled-rate equation simulation. The complete program is accessible on the GitHub repos-itory at the following link: https://github.com/westlike666/NewRESMO/tree/main/1DAll rights to the code developed are reserved by the author of this thesis.143% This function solves the coupled rate equations of shell model with self-similar expansion in 1Dfunction [t,rs, edens, n_ions, n_highs, nDens,Ts, Tn, r0, tau, ns]=complete1D(N,k,dp,n0,purpose,all_pqn,sigma,env,tspan,gamma1,gamma2, phi, d_phi)% N: the number of shells% dp: peak density in um^-3% n0: initial principal quantum number% sigma: initial Gaussian width% env: expand width to sigma enviroment% define initial constantkB = 1.3806504e-23; % #m2 kg s-2 K-1;J K-1; 8.62e-5 #eV K-1,Ry = 2.179872e-18; % #J: Rydberg energy in Jm_NO=30.006/1000/6.02214076e23; % kgkBm=kB/m_NO*1e12/1e18; %k_b / m_NO #um2 ns-2 K-1%kBm=0.0002770924685030197; %k_b / m_NO #um2 ns-2 K-1a0=5.2917721092e-5; % Bohr radius in um%Rn0=n.^2*a0; % radius of Rydb. atom by bohr model using semi-classical methodsigma_z=sigma; % Gaussian width convert from mm to um.T_e=50; T_i=0; % electron and ion temperature in Ktau=(1/kBm*sigma^2/(T_e+T_i))^0.5; % characteristic plasma expansion time in nss0=1;t1=(2^(2/k)-1)^0.5*tau; % t1/tau is always constant no matter of taus=zeros(N+1,1)';for n=[1:length(s)] s(n)=s0.*(1+(t1/tau).^2).^((n-1)/2); % defining the boudaries of initial shells according to s(t1)=s(0)*(1+t1^2/tau^2)^0.5, such that they all overlap at the same timeendr_z=s/max(s); % normalizingn=[1:length(s)-1];Tn=tau*((1+(t1/tau)^2).^(n)-1).^0.5; % the overlapping times can be calculated by s_1(t_n)=s_n(t_1). ignoring Tn=0. There are n boudaries and n-1 Tnr_z=r_z*env*sigma; %boundaries of initial shells; N+1 columes% fill shells with inital NO^+, NO^*, NO^** populations1144%dp=1; % peak densityden0=dp*exp(-([0,r_z(2:end-1)].^2)./(2.*sigma_z^2)); % 1*N rowN0=n0*ones(1,N);[PF,eden,n_Ryd]=penningfraction(N0,den0); % density in um^-3 % Redistributes the Penning partners over lower n's:n_ion=eden;% n_ion=penning_fraction/2*den0; % NO^+ ion% n_Ryd=(1-penning_fraction)*den0; % remaining Rydberg that not involved in ionizationn_high=0*n_ion; % long-lived NO^**expand=all_pqn; % wether to expand the NO^* Rydberg into different levelsif expand n_min=10; n_max=80; f=@(x)5.95*x.^5; % This is the penning fraction distribution np=n_min:fix(n0/sqrt(2)); % Array of n states allowed after Penn ion ind=1:length(np); nl=[n_min:n_max]; ns=length(nl); n_dist=0*nl; n_dist(ind)=f(np/n0)/sum(f(np/n0));% dividing by the sum normalizes the function nDen=n_dist'*eden; nDen(nl==n0,:)=n_Ryd; T_penning=sum(-Ry*den0./n0^2 + Ry*n_Ryd./n0^2 + Ry*sum(nDen(ind,:)./(nl(ind).^2)',1))./sum(3/2*kB.*eden); % by energy conservation, initial before penning is zero T_e=T_penning % initial temperature. comment if wish to use preset Temperature.else nl=n0; ns=1; nDen=den0-n_ion;endr0=r_z';u0=0*r0;y0=[r0; u0; eden'; n_ion'; n_high'; reshape(nDen,[ns*N,1]);T_e];[ni,nf,II,minn,maxn,diffsn]=buildns(nl);2145% define ordinary differential equationsfunction dy=ode1D(t,y)r=y(1:N+1); m=N+1;% there are N+1 boudariesU=y(m+1:m+N+1); m=m+N+1; % radial velocityeden=y(m+1:m+N); m=m+N; % Nx1 colume; electronsn_ion=y(m+1:m+N); m=m+N; % Nx1 colume; NO^+ ionn_high=y(m+1:m+N); m=m+N; % Nx1 colume; long-lived NO^** RydbergnDen=y(m+1:m+ns*N); m=m+ns*N; % Nx1 colume; short-lived NO^* RydbergT=y(m+1); m=m+1; % scaler: electron temperaturenDen=reshape(nDen,[ns,N]); % ns x N martrix or rowU=t/(t^2+tau^2).*r; % expanding volecity at r according to Vlasov equation: N+1 rowsu=U(1:end-1);rr=r(1:end-1);dU=mean(-kBm*T*diff(eden)./eden(1:end-1)./diff(rr)./rr(1:end-1))*r; % averaging gradient%dU=0*dU;%infintesmall volume dV=sin(phi) r^2 dr d_phi d_theta, integrate over dr%from 0 to r and d_theta from 0 to 2*pi to find total volume at r(theta),%the diff calculate the volume of slices betwenn shell boundariesV=2*pi/3*d_phi*(diff(r.^3)); % N*1 colume volume of each shell, expand volume to pseudo 3Dif purpose=='soc' dV=2*pi*d_phi*(diff(r.^2.*U));elseif purpose=='geo'% V=diff(r);% dV=diff(U); dV=0; % This is to aviod the artefact of bifurcation in imagesendk_ion=kION(nl,T); % 1* ns row. rate of ionizationk_tbr=kTBR(nl,T); % 1* ns row. rate of three body recombinationk_n_np=knnp(ni,nf,II,minn,maxn,diffsn,T); % ns * ns matrix. rate for transfer from n to n'k_pd=kPD(nl); % 1* ns row. rate of predissociationk_DR=kDR(T); % scaler. rate of dissociative recombination% arbitrary defined charge transfer rate: larger r (speed) causes more charge transfer within that shell at given amount of time;k_CT=gamma1.*diff(r).*(a0*nl.^2)/T; % N*ns matrix.k_CT2=gamma2.*diff(r).*(a0.^2); % N*1 colume, since NO^** is not distinguished by pqn3146%%e + NO^+ +NO^* == 2NO^** with k_CT%e + NO^+ +NO^** == 2NO^** with k_CT2%e + NO^* ==NO^+ + 2e k_ion%2e + NO^+ ==NO^* + e k_tbr% Now consider the reactions happening in the overlapping shells of NO^+ (moving with u), NO^* (stationary) , NO^** (moving with 0.5u and falling k shells behind NO^+ )% This means that the firth kth NO^** shells are alaways empty. and the kth frontier NO^+ should not contribute to the NO^** production since there is no overlap with NO^** frontiers% The overlapping times are determined by Tnstep=sum(t>=Tn)+1;d_ionize=k_ion.*eden.*[nDen(:,step:end),zeros(ns,step-1)]'; % N*ns matrix: the outer (step) shells are zero% d_ionize(1:end-step+1,:) is the overlapping (non-zero) regiond_tbr=k_tbr.*eden.^4; % N*1 colume. one eden power is aborbed in k_tbr% d_tbr(1:end-step+1,:) is the overlapping (non-zero) regiond_n_np=sum(k_n_np,2).*nDen.*[zeros(1,step-1),eden(1:end-step+1)']; % ns*N matrix%d_n_np=sum(k_n_np,2).*nden.*eden'; % ns*N matrixk_np_n=k_n_np';d_np_n=k_np_n*nDen.*[zeros(1,step-1),eden(1:end-step+1)']; % ns*N matrixd_CT=k_CT.*n_ion.*[nDen(:,step:end),zeros(ns,step-1)]'; % N*ns matrix: the outer (step) shells are zero% d_CT(1:end-step+1,:) is the overlapping (non-zero) regiond_eden=sum(-d_CT+d_ionize-d_tbr,2); % N*1 columsd_ion=d_eden;d_nDen=[zeros(step-1,ns);-d_CT(1:end-step+1,:)-d_ionize(1:end-step+1,:)+d_tbr(1:end-step+1,:)]; % N*ns matrix: the inner (step) shells are zerogate=1; % whether to consider the population from the first few movementsif step<k+1 % let the NO^** wait for k shells before starting move together with NO^+ shells. Within this time the NO^** align with NO^* d_high1=gate*[zeros(step-1,ns);2*d_CT(1:end-step+1,:)]; % N*ns matrix: the inner (step) shells are zero; or set all to zero4147 d_CT2= k_CT2.*n_ion.*[n_high(step:end);zeros(step-1,1)]; % N*1 colume % d_CT2(1:end-step+1) is the overlapping (non-zero) region d_high2=gate*[zeros(step-1,1);d_CT2(1:end-step+1)]; % N*1 columeelse d_high1=[zeros(k,ns);2*d_CT(1:end-step+1,:);zeros(step-1-k,ns)]; % N*ns matrix; NO^** start moving, falling behind of NO^+ by k shells. so the overall velocity is u/2. d_CT2= k_CT2.*n_ion.*[n_high(k+1:end);zeros(k,1)]; % N*1 colume % d_CT2(1:end-k) is the overlapping (non-zero) region d_high2=[zeros(k,1);d_CT2(1:end-k)]; % N*1 columeendd_eden2=-d_CT2;D_eden_old=sum(d_ionize-d_tbr,2); D_eden=d_eden+d_eden2-k_DR*n_ion.*eden-eden.*dV./V; % consider also the expansion of volume D_ion=D_eden; D_nDen_old=[zeros(step-1,ns);-d_ionize(1:end-step+1,:)+d_tbr(1:end-step+1,:)]-d_n_np'+d_np_n'; % N*ns matrix: the inner (step) shells are zero %D_nDen=d_nDen-d_n_np'+d_np_n'-k_pd.*nDen'-nDen'.*dV./V; % assume Rydberg shells expand, D_nDen=d_nDen-d_n_np'+d_np_n'-k_pd.*nDen'; % assume Rydberg not expand, d_high=sum(d_high1,2)+d_high2; % N*1 colums, D_high=d_high-n_high.*dV./V; % the 1/8 factor cancels in dV/V dT1=(Ry*sum(sum(D_nDen_old./nl.^2.*V))-1.5*kB*T*sum(D_eden_old.*V))./(1.5*kB*sum(eden.*V)); % the total energy conserve before the dissipation. Ry and kB both in S.I. unit dT2=-2*t/(t^2+tau^2)*T; % temperature change due to expansion% if isreal(dT1)==0% dT1=0;% end dT=dT1+dT2;dy=[U; dU; D_eden; D_ion; D_high; reshape(D_nDen',[N*ns,1]);dT];5148end% ouput the ode results% t_final=max(Tn); [t,y]=ode45(@(t,y)ode1D(t,y),tspan,y0);rs=y(:,1:N+1); m=N+1; % N+1 boundaries of the shellUs=y(:,m+1:m+N+1); m=m+N+1; %Velocitiesedens=y(:,m+1:m+N); m=m+N; % Nx1 colume; electronsn_ions=y(:,m+1:m+N); m=m+N; % Nx1 colume; NO^+ ionn_highs=y(:,m+1:m+N); m=m+N; % Nx1 colume; long-lived NO^** RydbergnDens=y(:,m+1:m+ns*N); m=m+ns*N; % Nx1 colume; short-lived NO^* RydbergTs=y(:,m+1); m=m+1;% visualize(optional)%visualize=0;% if visualize% y_ana=@(t,y0,tau) y0.*(1+(t/tau).^2).^0.5;%% figure()% plot(r0,'.')% xlabel('shells')% ylabel('initial r')%% figure()% plot(t,y_ana(t,r_z,tau))% hold on% %plot(t,rs,'bx')% xlabel('t')% ylabel('radius(t)')%%%% figure()% plot(t,n_ions)% xlabel('t')% ylabel('NO^+ ')%% figure()% plot(t,nDens)% xlabel('t')% ylabel('NO^* ')%% figure()% plot(t,n_highs)% xlabel('t')% ylabel('NO^{**} ')% if visualize6149% y_ana=@(t,y0,tau) y0.*(1+(t/tau).^2).^0.5;% figure()% hold on% dt=0.1*tau;% t_final=10*tau;% for t=[0:dt:t_final]% s=y_ana(t,r_z,tau);% clf% % ylim([0 5])% t_n=Tn(abs(t-Tn)<dt);% for i=[1:length(s)]% yline(s(i))% yline(0.5*(s(i)),'red') % start with k shells off% % yline(0.5*(Y(i)+Y0(i)),'red') % start together% yline(r_z(i),'green')% end% if isempty(t_n)==0% title(['t=',num2str(t_n)])% end% pause(0.1)%% end% hold off% endendPublished with MATLAB® R2020b7150Appendix DLindblad dynamics simulationprogramThe following section provides the supplementary material for Chapter 6 withsample codes for simulating the dynamics of the 1D spin chain open quantumsystem.This program has been developed using Python programming language andutilized the QuTip package during the development process. More informationabout the Qutip package can be found here: https://qutip.org/To access the complete simulation program, please visit the GitHub repositoryat the following link: https://github.com/westlike666/LindbladAll rights to the code developed are reserved by the author of this thesis.151"""This code defines the ode functions of the mean-field approximation for both1st-order and 2nd-order in vectorization approaches"""import numpy as npfrom qutip import*class ode_funs(): # This is a vectorized version of ode operation, which boosts calcualtion def __init__(self, N, eps, J, U, gamma, Diss='depha'): """ Parameters ---------- N : int. number of sites. eps: list of onsite energies J: matrix of flip-flop term J U: matrix of Ising term index: Dictionary Choose the type of dissipator. 'depha' S_m for dephasing 'diss' S_z for disspipation. Returns ------- a function fun(t,Y,arg) to feed into the ode solver. """ self.N=N self.eps=np.reshape(eps,(N,1)) #column (N,1) self.J=J self.U=U self.gamma=np.reshape(gamma,(N,1)) #column (N,1) self.Diss=Diss def fun_1st_vec(self, t,Y, index, pbar, state): """ Parameters ---------- t : required variable for ode Y : shape of (3N,) for Sz list and Sp (and Sm) 1152 Returns ------- D: shape of (3N,) Derivative of Sz and Sp (and Sm) """ #this is just for progress bar last_t, dt = state n = int((t - last_t)/dt) pbar.update(n) state[0] = last_t + dt * n D=0*Y+0j #coupled rate equations of 1st-order expansion D[index['z']]=1j*(sum1(self.J,Y,index,'-+')-sum1(self.J,Y,index,'+-'))\ -self.gamma*(Y[index['z']]+0.5) D[index['+']]=1j*self.eps*Y[index['+']]-2j*sum1(self.J,Y,index,'z+')\ -0.5*self.gamma*(Y[index['+']])# D[index['-']]=-1j*self.eps*Y[index['-']]+2j*sum1(self.J,Y,index,'z-')\# -0.5*self.gamma*(Y[index['-']]) D[index['-']]=np.conjugate(D[index['+']]) return D def fun_2nd_vec(self, t, Y, index): """ Parameters ---------- t : required variable for ode Y : shape of (3N+9N^2,) for list of all operators Returns ------- D: shape of (3N+9N^2,) for list of deravatives of all operators """ # for key in index: # if len(key)>1: # Y[np.diag(index[key])]=0 2153 D=0*Y #coupled rate equations of 2nd-order expansion D[index['z']]=1j*(sum2(self.J,Y,index,'-+')-sum2(self.J,Y,index,'+-'))\ -self.gamma*(Y[index['z']]+0.5) D[index['+']]=1j*self.eps*Y[index['+']]-2j*sum2(self.J,Y,index,'z+')\ -0.5*self.gamma*(Y[index['+']]) D[index['-']]=-1j*self.eps*Y[index['-']]+2j*sum2(self.J,Y,index,'z-')\ -0.5*self.gamma*(Y[index['-']]) # D[index['-']]=np.conjugate(D[index['+']]) D[index['+-']]=1j*(self.eps-self.eps.T)*Y[index['+-']]\ -1j*self.J*(Y[index['z']]-Y[index['z']].T)\ -2j*(sum_ki(self.J, Y, index, 'z-+')\ -sum_li(self.J, Y, index, 'z+-'))\ -0.5*(self.gamma+self.gamma.T)*Y[index['+-']] D[index['-+']]= -1j*(self.eps-self.eps.T)*Y[index['-+']]+\ 1j*self.J*(Y[index['z']]-Y[index['z']].T)\ +2j*(sum_ki(self.J, Y, index, 'z+-')\ -sum_li(self.J, Y, index, 'z-+'))\ -0.5*(self.gamma+self.gamma.T)*Y[index['-+']] D[index['++']]=1j*(self.eps+self.eps.T)*Y[index['++']]\ -2j*(sum_ki(self.J, Y, index, 'z++')\ +sum_li(self.J, Y, index, 'z++'))\ -0.5*(self.gamma+self.gamma.T)*Y[index['++']] D[index['--']]=-1j*(self.eps+self.eps.T)*Y[index['--']]\ +2j*(sum_ki(self.J, Y, index, 'z--')\ +sum_li(self.J, Y, index, 'z--'))\ -0.5*(self.gamma+self.gamma.T)*Y[index['--']] D[index['zz']]=1j*(sum_ki(self.J, Y, index, '-z+')\ -sum_ki(self.J, Y, index, '+z-'))\ +1j*(sum_li(self.J, Y, index, '-z+')\ -sum_li(self.J, Y, index, '+z-'))\ -(self.gamma+self.gamma.T)*Y[index['zz']]\ -0.5*self.gamma*Y[index['z']].T\3154 -0.5*self.gamma.T*Y[index['z']] D[index['+z']]=1j*self.eps*Y[index['+z']]-0.5j*self.J*Y[index['+']].T\ -2j*sum_ki(self.J,Y,index,'zz+')\ +1j*(sum_li(self.J, Y, index, '-++')\ -sum_li(self.J, Y, index, '++-'))\ -(0.5*self.gamma+self.gamma.T)*Y[index['+z']]\ -0.5*Y[index['+']]*self.gamma.T D[index['-z']]=-1j*self.eps*Y[index['-z']]+0.5j*self.J*Y[index['-']].T\ +2j*sum_ki(self.J,Y,index,'zz-')\ +1j*(sum_li(self.J, Y, index, '--+')\ -sum_li(self.J, Y, index, '+--'))\ -(0.5*self.gamma+self.gamma.T)*Y[index['-z']]\ -0.5*Y[index['-']]*self.gamma.T D[index['z+']]=1j*self.eps.T*Y[index['z+']]-0.5j*self.J*Y[index['+']]\ -2j*sum_li(self.J,Y,index,'zz+')\ +1j*(sum_ki(self.J, Y, index, '-++')\ -sum_ki(self.J, Y, index, '++-'))\ -(0.5*self.gamma.T+self.gamma)*Y[index['z+']]\ -0.5*Y[index['+']].T*self.gamma D[index['z-']]=-1j*self.eps.T*Y[index['z-']]+0.5j*self.J*Y[index['-']]\ +2j*sum_li(self.J,Y,index,'zz-')\ +1j*(sum_ki(self.J, Y, index, '--+')\ -sum_ki(self.J, Y, index, '+--'))\ -(0.5*self.gamma.T+self.gamma)*Y[index['z-']]\ -0.5*Y[index['-']].T*self.gamma # D[index['--']]=D[index['++']].conjugate() # D[index['-z']]=D[index['+z']].conjugate() # D[index['-+']]=D[index['+-']].conjugate() for key in index: if len(key)>1: D[np.diag(index[key])]=0 return D def generate_full_op_list(self): single_ops=['z','+','-']4155 double_ops=[] for s1 in single_ops: for s2 in single_ops: double_ops.append(s1+s2) return single_ops, double_ops def flat_index(self, single_ops, double_ops, index): """ Parameters ---------- single_ops : list of str e.g. ['z', '+'] double_ops : list of str. e.g. ['z+', '+-', ...] index : dictionary DESCRIPTION. Initiate it to empty {}. Returns ------- index : a dictionary contain the index in the flatten vector Y """ N=self.N pre=0 while single_ops: key=single_ops.pop(0) value=np.arange(pre, pre+N) value_vec=np.reshape(value,(N,1)) pre += N index[key]=value_vec # return colum vector (N,1) while double_ops: key=double_ops.pop(0) #print(pre) value=np.array([range(N*i+pre, N*(i+1)+pre) for i in range(N)]) pre += N**2 index[key]=value self.index=index return index 5156 def sum1(J,Y,index,pmz): #sum by vectorization """ 1st order double operator sum pmz='+-' (k,i) the second is free index ! """ vec1=Y[index[pmz[0]]] vec2=Y[index[pmz[1]]] s=vec1*J@vec2 # element wise and matrix product return s # keep (N,1) dimensiondef sum2(J,Y,index,pmz): #sum by vectorization """ 2nd order double operator sum pmz='+-' (k,i) the second is free index ! """ mat=Y[index[pmz]] s=np.sum(J*mat,axis=1,keepdims=True) return s def sum2_loop(J,Y,index,pmz): #sum by loop for check mat=Y[index[pmz]] N=mat.shape[0] s=np.zeros((N,1)) for k in range(N): for i in range(N): s[k] += J[k,i]*mat[k,i] return s def sum_ki(J,Y, index, pmz): #sum by vectorization """ 2nd order triple operator sum pmz='+-z' (k,l,i) assume the third index is always the free index i, sum over i 6157 also assume the first index k goes into the the coefficient J_ki """ vec1=Y[index[pmz[0]]] vec2=Y[index[pmz[1]]] vec3=Y[index[pmz[2]]] #single (N,1) vectors mat12=Y[index[(pmz[0]+pmz[1])]] mat13=Y[index[(pmz[0]+pmz[2])]] mat23=Y[index[(pmz[1]+pmz[2])]] #double operator matrix s_kl_i=mat12*(J@vec3)-J*mat12*vec3.T s_ki_l=np.sum(mat13*J, axis=1,keepdims=True)*vec2.T-J*mat13*vec2.T s_li_k=J@mat23.T*vec1-J*(np.expand_dims(np.diag(mat23),0))*vec1 #expand the dialgonal element to (1,N) row before boradcasting s_kli=(J@vec3)*(vec1*vec2.T)-J*(vec1*vec2.T)*vec3.T return s_kl_i+s_ki_l+s_li_k-2*s_kli def sum_li(J,Y, index, pmz): #sum by vectorization r=sum_ki(J, Y, index, pmz) return r.T def sum_ki_loop(J,Y,index,pmz): #sum by loop for check # pmz order is (k,l,i) vec1=Y[index[pmz[0]]] vec2=Y[index[pmz[1]]] vec3=Y[index[pmz[2]]] #single (N,1) vectors mat12=Y[index[(pmz[0]+pmz[1])]] mat13=Y[index[(pmz[0]+pmz[2])]] mat23=Y[index[(pmz[1]+pmz[2])]] #double operator matrix N=mat12.shape[0] s1=np.zeros((N,N))7158 s2=np.zeros((N,N)) s3=np.zeros((N,N)) s4=np.zeros((N,N)) for k in range(N): for l in range(N): for i in range(N): if i==k or i==l: continue s1[k,l] += J[k,i]*mat12[k,l]*vec3[i][0] s2[k,l] += J[k,i]*mat13[k,i]*vec2[l][0] s3[k,l] += J[k,i]*mat23[l,i]*vec1[k][0] s4[k,l] += J[k,i]*vec1[k][0]*vec2[l][0]*vec3[i][0] s=s1+s2+s3-2*s4 return s def sum_li_loop(J,Y,index,pmz): #sum by loop for check # pmz order is (l,k,i) vec1=Y[index[pmz[0]]] vec2=Y[index[pmz[1]]] vec3=Y[index[pmz[2]]] #single (N,1) vectors mat12=Y[index[(pmz[0]+pmz[1])]] mat13=Y[index[(pmz[0]+pmz[2])]] mat23=Y[index[(pmz[1]+pmz[2])]] #double operator matrix N=mat12.shape[0] s1=np.zeros((N,N)) s2=np.zeros((N,N)) s3=np.zeros((N,N)) s4=np.zeros((N,N)) for k in range(N): for l in range(N): for i in range(N): if i==k or i==l: continue 8159 s1[k,l] += J[l,i]*mat12[l,k]*vec3[i][0] s2[k,l] += J[l,i]*mat13[l,i]*vec2[k][0] s3[k,l] += J[l,i]*mat23[k,i]*vec1[l][0] s4[k,l] += J[l,i]*vec1[l][0]*vec2[k][0]*vec3[i][0] s=s1+s2+s3-2*s4 return s 9160"""This code provides an example of comparing the results of 1st-order and 2nd-order approximation with the exact solution"""import scipyimport numpy as npimport osimport utilsfrom XY_class import*from ode_funs_vec import ode_funsimport matplotlib.pyplot as pltfrom scipy.integrate import solve_ivp, odeintimport time from datetime import datetime from tqdm import tqdmfrom qutip import*import randomfrom numpy.random import default_rngfrom energy_paras import Energycomputer, Jcomputer, Ucomputer, Gammacomputersave=Falseif save: path='results/'+utils.get_run_time() os.mkdir(path)L=2N=7W=0t=1u=0G=1seed=1show_type='z'model=XY(L,N)eps=Energycomputer(N,seed).uniformrandom_e(W)J=Jcomputer(N, nn_only=False, scaled=True, seed=seed).constant_j(t)U=Ucomputer(N, nn_only=False, scaled=True, seed=seed).uniformrandom_u(u)1161#gamma=Gammacomputer(N).central_g(G)gamma=Gammacomputer(N).boundary_g(G)#gamma=Gammacomputer(N).site_g(G,[0,6])#gamma=Gammacomputer(N).constant_g(G)H=model.get_Hamiltonian_MPC(eps, J)#rho0=model.generate_whole_random_density(seed=seed, pure=True) # not separateble state#states,rho0=model.generate_random_density(seed=None, pure=True) #states,rho0=model.generate_coherent_density(alpha=np.pi/4) #states,rho0=model.generate_random_ket()#rng=default_rng(seed=1)#up_sites=rng.choice(N, N//2, replace=False)up_sites=[i for i in range(0,N,2)]states, rho0=model.generate_up(up_sites)#states, rho0=model.generate_pseudo_up(up_sites, portion=0.01)#print(states)print(up_sites)print(eps)#print(rho0)Sz=model.SzSp=model.SpSm=model.Sm#Diss='dephasing' Diss='dissipation't_0=0t_1=100t_span1=(t_0,t_1)times1=np.linspace(t_0, t_1, 1000)t_2=500t_span2=(t_1, t_2)times2=np.linspace(t_1, t_2, 1000)#%%"""sloving by Qutip Lindblad solver2162"""# phase1e_ops=Sz+Sp+Sm # list of expectation values to evaluateresult1_exact=mesolve(H, rho0, times1, progress_bar=True) y1_exact=expect(e_ops, result1_exact.states)#phase2rho1=result1_exact.states[-1]if Diss=='dephasing': diss=Szelse: diss=Sm c_ops=[] for i in range(N): c_ops.append(np.sqrt(gamma[i])*diss[i]) # constant c_opsresult2_exact=mesolve(H, rho1, times2, c_ops, args={},progress_bar=True)y2_exact=expect(e_ops, result2_exact.states) #=============================================================================#%%"""sloving by 1st order approximation: <S1*S2>=<S1>*<S2> """#phase1 y0=expect(e_ops, rho0)ode_class=ode_funs(N, eps, J, U, gamma=np.zeros(N), Diss=Diss) fun=ode_class.fun_1st_vecindex=ode_class.flat_index(single_ops=['z', '+', '-'], double_ops=[], index={}) with tqdm(total=100, unit="‰") as pbar: result1_semi=solve_ivp(fun, t_span=t_span1, y0=y0, t_eval=times1,3163 args=[index, pbar, [t_0, (t_1-t_0)/100]]) t1_semi=result1_semi['t']y1_semi=result1_semi['y']#phase2 ode_class=ode_funs(N, eps, J, U, gamma=gamma, Diss=Diss) fun=ode_class.fun_1st_vec with tqdm(total=100, unit="‰") as pbar: result2_semi=solve_ivp(fun, t_span=t_span2, y0=y1_semi[:,-1], t_eval= times2, args=[index, pbar, [t_1, (t_2-t_1)/100]]) t2_semi=result2_semi['t']y2_semi=result2_semi['y']#=============================================================================#%%"""sloving by 2nd order approximation: <abc>=<ab><c>+<ac>+<bc>-2<c> """#phase1# s=['z', '+', '-'] # keep the same order as the 1st-order does # ss=['+-', '++', '--', '+z', '-z', 'zz']s, ss = ode_class.generate_full_op_list() e_ops=model.generate_single_ops(s)+model.generate_double_ops(ss)index=ode_class.flat_index(s, ss, index={})ode_class=ode_funs(N, eps, J, U, gamma=np.zeros(N), Diss=Diss)fun=ode_class.fun_2nd_vecy0_MPC=expect(e_ops, rho0)for key in index: if len(key)>1: y0_MPC[np.diag(index[key])]=0 4164result1_MPC=solve_ivp(fun, t_span=t_span1, y0=y0_MPC, method='BDF', dense_output=False, args=[index]) y1_MPC=result1_MPC['y']# #phase2ode_class=ode_funs(N, eps, J, U, gamma=gamma, Diss=Diss)fun=ode_class.fun_2nd_vecresult2_MPC=solve_ivp(fun, t_span=t_span2, y0=y1_MPC[:,-1], method='BDF', args=[index]) y2_MPC=result2_MPC['y']#=============================================================================#%%"""plotting"""def plot_evolution(show_type='z', show_ind=0): t_total=np.append(result1_exact.times, result2_exact.times) y_total=np.append(y1_exact[index[show_type][show_ind][0]].real, y2_exact[index[show_type][show_ind][0]].real) y_total_semi=np.append(y1_semi[index[show_type][show_ind][0]].real, y2_semi[index[show_type][show_ind][0]].real) y_total_MPC=np.append(y1_MPC[index[show_type][show_ind][0]], y2_MPC[index[show_type][show_ind][0]]) plt.figure(show_ind) #plt.subplot(211) plt.plot(t_total, y_total, label="Qutip exact $Re <S^{}_{}>$".format(show_type, show_ind)) plt.plot(np.append(t1_semi,t2_semi), y_total_semi, linestyle='-', label="1st order $Re <S^{}_{}>$".format(show_type, show_ind)) plt.plot(np.append(t1_MPC,t2_MPC), y_total_MPC, linestyle='-',label="2nd order $Re <S^{}_{}>$".format(show_type, show_ind)) plt.ylabel("site {}".format(show_ind)) plt.ylim(-5,5) plt.axhline(y=-0.5, color='grey', linestyle='--')5165 plt.legend() plt.xlabel('t') plt.suptitle('XY model L=%d, N=%d eps=%.2f t=%.2f J g=%.1f J' % (L,N, eps[show_ind],t,G)) if save: plt.savefig(path+"/site {}.png".format(show_ind)) for show_ind in range(N): plot_evolution(show_type, show_ind) 6166"""This code provides an example of solving the exact Lindblad equation with different random configurations in parallel using clusters """import scipyimport numpy as npimport osimport utilsfrom XY_class import*from XY_ode_funs import ode_funsimport matplotlib.pyplot as pltfrom scipy.integrate import solve_ivpimport time from datetime import datetime #from tqdm import tqdmfrom qutip import*import randomfrom numpy.random import default_rng#from Lindblad_solver import Lindblad_solvefrom energy_paras import Energycomputer, Jcomputer, Ucomputer, Gammacomputerimport argparseimport sysparser = argparse.ArgumentParser(description='Run the model as set up')parser.add_argument('--N', type=int, nargs='?', help='Number of site', action='store')parser.add_argument('--W', type=float, nargs='?', help='The degree of disorder', action='store')parser.add_argument('--t', type=float, nargs='?', help='The J bar', action='store')parser.add_argument('--u', type=float, nargs='?', help='The u term', action='store')parser.add_argument('--G', type=float, nargs='?', help='dissipation rate', action='store')parser.add_argument('--bc', type=str, nargs='?', help = 'type of start decaying: center/boundary',1167 action='store', default="center")parser.add_argument('--seed', type=int, nargs='?', help='seed for random generator', action='store', default=None)parser.add_argument('--save', type=bool, nargs='?', help = 'Whether to save results to file', action='store', default=False)args=parser.parse_args()save=args.saveL=2N=args.NW=args.Wt=args.tu=args.uG=args.Gbc=args.bcseed=args.seedname='N='+str(N)+' W='+ str(W)+' t='+str(t) \ + ' u='+ str(u) + ' g='+ str(G) +' seed=' +str(seed)if save:# path='results/'+utils.get_run_time() path='results/'+bc+'/'+name os.mkdir(path) show_type='z'model=XY(L,N)eps=Energycomputer(N,seed).uniformrandom_e(W)J=Jcomputer(N, nn_only=False, scaled=True, seed=seed).constant_j(t)U=Ucomputer(N, nn_only=False, scaled=True, seed=seed).constant_u(u)if bc=='center': gamma=Gammacomputer(N).central_g(G)elif bc=='boundary':2168 gamma=Gammacomputer(N).boundary_g(G)H=model.get_Hamiltonian2(eps, J, U)rng=default_rng(seed=seed)up_sites=[i for i in range(0,N,2)]states, rho0=model.generate_up(up_sites)#print(states)print(up_sites)print(eps)#print(rho0)Sz=model.SzSp=model.SpSm=model.Sm#Diss='dephasing' Diss='dissipation'# chose the jump opperator for 'dephasing' or 'dissipation'ode_funs=ode_funs(N, eps, J, U, gamma, Diss=Diss) index=ode_funs.flat_index(single_ops=['z','+'], double_ops=[], index={}) t_0=0t_1=100t_span=(t_0,t_1)t_eval=np.linspace(t_0, t_1, 1000)e_ops=Sz+Sp # list of expectation values to evaluatetimes1=t_eval"""sloving by Qutip Lindblad solver"""# free evolution result1=mesolve(H, rho0, times1, progress_bar=True) y1=expect(e_ops, result1.states)3169rho1=result1.states[-1]if Diss=='dephasing': diss=Szelse: diss=Sm c_ops=[] t_2=500t_span=(t_1, t_2)times2=np.linspace(t_1, t_2, 1000)# applying dissipationfor i in range(N): c_ops.append(np.sqrt(gamma[i])*diss[i]) # constant c_ops result2=mesolve(H, rho1, times2, c_ops, args={},progress_bar=True)y2=expect(e_ops, result2.states) def plot_evolution(show_type='z', show_ind=0): t_total=np.append(result1.times, result2.times) y_total=np.append(y1[index[show_type][show_ind]].real, y2[index[show_type][show_ind]].real) plt.figure(show_ind) #plt.subplot(211) plt.plot(t_total*t, y_total, label="$Re <S^{}_{}>$".format(show_type, show_ind)) plt.ylabel("site {}".format(show_ind)) plt.ylim(-0.5,0.5) plt.axhline(y=-0.5, color='grey', linestyle='--') plt.legend() plt.xlabel('t * $\overline{J}$') plt.suptitle('XY model, N=%d, W=%d, eps=%.2f, $\overline{J}$=%.2f \ u=%d \ $\gamma$=%.1f' % (N,W, eps[show_ind],t,u,gamma[show_ind])) if save: plt.savefig(path+"/site {}.png".format(show_ind)) for show_ind in range(N): plot_evolution('+',show_ind)4170 plot_evolution('z', show_ind) if save: var_to_save={'N':N, 'J':J, 'U':U, 'eps':eps, 'gamma':gamma, 'states': states, 'y1':y1, 'y2':y2, 't1':result1.times, 't2':result2.times , 'index': index, 'up_sites':up_sites } utils.store_vars(path, var_to_save)5171
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Grant, Edward
Relaxation dynamics in a molecular ultracold plasma : control and modeling
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