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Ultracold nitric oxide molecular plasma : characteristic response to time-varying electric fields Aghigh, Seyed Mahyad 2021

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Ultracold Nitric Oxide Molecular Plasma:Characteristic Response to Time-varying ElectricFieldsbySeyed Mahyad AghighB.Sc., Azad University-Tehran Science & Research Branch, 2011M.Sc., The University of British Columbia, 2014A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Chemistry)The University of British Columbia(Vancouver)March 2021© Seyed Mahyad Aghigh, 2021The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the dissertationentitled:Ultracold Nitric Oxide Molecular Plasma: Characteristic Re-sponse to Time-varying Electric Fieldssubmitted by Seyed Mahyad Aghigh in partial fulfillment of the requirements forthe degree of DOCTOR OF PHILOSOPHY in Chemistry.Examining Committee:Edward Grant, Chemistry & Physics and Astronomy, UBCSupervisorValery Milner, Physics and Astronomy, UBCSupervisory Committee MemberAndrew MacFarlane, Chemistry, UBCUniversity ExaminerKeng Chou, Chemistry, UBCUniversity ExaminerTimothy Wright, Chemistry, University of NottinghamExternal ExaminerAdditional Supervisory Committee Members:Takamasa Momose, Chemistry, UBCSupervisory Committee MemberDavid Chen, Chemistry, UBCSupervisory Committee MemberiiAbstractNot long after metastable xenon was photoionized in a magneto-optical trap,groups in Europe and North America found that similar states of ionized gasevolved spontaneously from state-selected Rydberg gases of high principal quan-tum number. Studies of atomic xenon and molecular nitric oxide entrained ina supersonically cooled molecular beam subsequently showed much the samefinal state evolved from a sequence of prompt Penning ionization and electron-impact avalanche to plasma, well-described by coupled rate-equation simulations.However, measured over longer times, the molecular ultracold plasma (UCP) wasfound to exhibit an anomalous combination of very long lifetime and very lowapparent electron temperature. In this thesis I summarize early developments inthe study of UCP formed by atomic and molecular Rydberg gases, and then I detailobservations as they combine to characterize properties of the nitric oxide molec-ular UCP that appear to call for an explanation beyond the realm of conventionalplasma physics. I also explain how I leveraged a radio frequency electric fieldto understand the causes of classically-unexplainable behaviour of our molecularsystem.iiiLay SummaryAtomic and molecular gases driven to produce ultracold plasmas yield isolatedsystems with properties that can provide an important gauge of collision andtransport under strongly coupled conditions. Strong coupling occurs in a plasmawhen the average inter-particle potential energy exceeds the average kinetic en-ergy. This condition causes the formation of extremely non-ideal charged-particlesystems that have fundamentally altered fluid-like properties, which give rise tostates of structural and dynamical order. The physics of strong coupling playimportant roles governing the dynamics of natural plasmas over a wide range oflength scales. However, conditions of strong coupling are difficult to create. Thisthesis explores the regime of strong coupling that occurs naturally in the relaxationof a molecular ultracold plasma.ivPrefaceThis dissertation is based on the experimental setup and data of the ultracoldmolecular plasma experiment in Professor Grant’s research laboratory at The Uni-versity of British Columbia. Some of data collection and theoretical modelingperformed for this work was carried out by different visiting, undergraduate, andgraduate students. However, the majority of experiments, data analysis, interpre-tation and writings were performed by the author of this dissertation. Chapter 1and 4 have been published previously, and they are attached with little revision.The following is a list of publications that the author has, contributed in, duringthe course of this dissertation, along with the contribution from each member.• R. Haenel, M. Schulz-Weiling, J. Sous, H. Sadeghi, M. Aghigh, L. Melo, J.S. Keller, and E. R. GrantArrested relaxation in an isolated molecular ultracold plasma, Phys. Rev.A 96, 023613 2017I contributed in technical lab support and took part in discussions as well asthe writing process. M. Schulz-Weiling and J. Sous, H. Sadeghi modifiedthe experimental setup and collected the initial data. R. Haenel performedmajority of complimentary data collection and data analysis. L. Melo andJ. S. Keller took part in discussions and manuscript review. E. R. Grant isthe principal investigator helping with finalizing the manuscript and sub-mission for publication. Figures 2.1, 2.3, and 3.1, which are presented inthis dissertation, are used with permission from this source.v• M. Aghigh, K. Grant, R. Haenel, K. L. Marroquín, F. B. V. Martins, H.Sadegi, M. Schulz-Weiling, J. Sous, R. Wang, J. S. Keller, E. R. GrantDissipative dynamics of atomic and molecular Rydberg gases: Avalancheto ultracold plasma states of strong coupling, J. Phys. B: At. Mol. Opt.Phys. 53 074003 2020This article summarizes over a decade of research in our laboratory at UBC,which highlights the unpredictable behaviour of our molecular system, serv-ing as the first chapter of my dissertation. I contributed in literature reviewand organizing the flow of background information. K. Grant, R. Haenel,K. L. Marroquín, F. B. V. Martins had contributed in technical lab supportand participated in discussions. M. Schulz-Weiling and H. Sadeghi hadcollected the majority of data presented in this publication during theirPhD programs. J. Sous and R. Wang contributed in theoretical modelingsand discussion. Creation of the manuscript was a combined effort of theauthors. E. R. Grant is the principal investigator helping with finalizing themanuscript and submission for publication.• R. Wang, M. Aghigh, K. L. Marroquín, K. Grant, J. Sous, F. B. V. Martins,J. S. Keller, E. R. GrantRadio frequency field-induced electron mobility in an ultracold plasma stateof arrested relaxation , Phys. Rev. A 102, 063122 2020I collected all the experimental data published in this article, and also con-tributed key points of the interpretation. The numerical analysis and mathe-matical modelings were conducted by Ruoxi Wang and J Sous. K. L. Mar-roquín, K. Grant, F. B. V. Martins, and J. S. Keller contributed in discussionsand review of the manuscript. Creation of the manuscript was a combinedeffort of the authors. E. R. Grant is the principal investigator helping withfinalizing the manuscript and submission for publication. This publication,with little revision, constructs the forth chapter of this dissertation.vi• R. Wang, J. Sous, M. Aghigh, K. L. Marroquín, K. M. Grant, F. B. V.Martins, J. S. Keller, and E. R. Grantmm-wave Rydberg-Rydberg resonances as a witness of intermolecular cou-pling in the arrested relaxation of a molecular ultracold plasma, arXiv:2005.10088K. L. Marroquín and I assisted R. Wang in experimental data collection,contributed to data interpretation, and provided editing. Majority of theexperimental data collection and data analysis was performed by R. Wang.K. M. Grant, F. B. V. Martins, and J. S. Keller contributed in discussion andmanuscript review. Creation of manuscript by Ruoxi Wang and Ed Grant.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The molecular beam ultracold plasma compared with a MOT . . . 51.3 Supersonic molecular beam temperature and particle density . . . 61.4 Penning ionization . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Spontaneous electron-impact avalanche . . . . . . . . . . . . . . 81.6 Evolution to plasma in a Rydberg gas Gaussian ellipsoid . . . . . 11viii1.7 Plasma expansion and NO+ - NO∗ charge exchange as an avenueof quench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Bifurcation and arrested relaxation . . . . . . . . . . . . . . . . . 161.9 A molecular ultracold plasma state of arrested relaxation . . . . . 192 Calibration of the Initial Density of the Nitric Oxide MolecularRydberg Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Sequence of events in a typical SFI experiment . . . . . . 242.2.2 UV-UV resonance production of a cold NO Rydberg gas . 252.2.3 Ramped electric field to extract the electrons . . . . . . . . 272.2.4 Two ways to control the Rydberg gas density . . . . . . . 282.3 Calculating the maximum possible density . . . . . . . . . . . . . 292.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Long-time Dynamics of the Nitric-oxide Molecular System . . . . . 403.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Physics of selective field ionization . . . . . . . . . . . . . . . . . 423.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3.1 Selective field-ionization spectrum as a probe of the relax-ation from Rydberg gas to plasma . . . . . . . . . . . . . 483.3.2 Long-time dynamics of the NO system probed with SFI . . 503.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 Manipulating the Quenched Plasma Using a Radio Frequency Elec-tric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2.1 Supersonic molecular beam ultracold plasma spectrometer 66ix4.2.2 Selective field ionization . . . . . . . . . . . . . . . . . . 704.2.3 Radio-frequency electric field . . . . . . . . . . . . . . . 714.2.4 Pulse sequence and evolution of n0 Rydberg density asobserved in a typical RF-SFI experiment . . . . . . . . . . 724.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.1 Field-free evolution of the nitric oxide molecular ultracoldplasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.2 Molecular ultracold plasma rate processes in the presenceof a continuous wave (CW) radio frequency field . . . . . 754.3.3 Effects of a pulsed radio frequency field . . . . . . . . . . 764.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4.1 Field-free avalanche dynamics and dissociation in the state-selected nitric oxide Rydberg gas . . . . . . . . . . . . . . 784.4.2 Nitric oxide Rydberg predissociation in a regime of `-mixing 814.4.3 Effect of a radio-frequency field . . . . . . . . . . . . . . 834.4.4 Kinetics of radio-frequency accelerated predissociation . . 844.4.5 Mechanics of radio frequency induced `-mixing . . . . . . 864.4.6 Electron mobility in the quenched molecular ultracold plasma 885 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Appendix A Nitric Oxide Spectroscopy . . . . . . . . . . . . . . . . . . 102Appendix B Supplementary Materials for Chapter 2 . . . . . . . . . . 109Appendix C MATLAB Codes and Functions . . . . . . . . . . . . . . . 118xList of TablesTable 4.1 Distribution of ions in an idealized Gaussian ellipsoid shellmodel of a quenched ultracold plasma of NO with a peak den-sity of 4× 1010 cm−3, σx = 1.0 mm, σy = 0.55 mm and σz =0.70 mm. This model quasi-neutral plasma contains a total of1.9×108 NO+ ions. Its average density is 1.4×1010 cm−3 andthe mean distance between ions is 3.32 µm. . . . . . . . . . . 69Table 4.2 Kinetic parameters used in Eq (4.1) to fit the exponential decayin the n0 Rydberg molecule SFI signal in Figure 4.5. . . . . . . 81Table 4.3 Parameters used in Eq (4.4) to describe the recovery of then0 Rydberg SFI signal depleted by a 250 ns 400 V cm−1 60MHz pulsed radio frequency field as time of this pulse, ∆tωrfadvances to pass through the beginning of the SFI ramp field.∆tramp for two different ramp field delays. . . . . . . . . . . . 86Table 4.4 Parameters used in Eq (4.4) to describe the recovery of then0 Rydberg SFI signal depleted by a 250 ns 400 V cm−1 60MHz pulsed radio frequency field as time of this pulse, ∆tωrfadvances to pass through the beginning of the SFI ramp field.∆tramp for two different initial Rydberg gas densities, as deter-mined by ∆tω2 . . . . . . . . . . . . . . . . . . . . . . . . . . 88xiList of FiguresFigure 1.1 Distributions of ion-ion nearest neighbours following Penningionization and electron-impact avalanche simulated for a pre-dissociating molecular Rydberg gas of initial principal quan-tum number, n0, from 30 to 80, and density of 1012 cm−3.Dashed lines mark corresponding values of aws. Calculatedby counting ion distances after relaxation to plasma in 106-particle stochastic simulations. Integrated areas proportionalto populations surviving neutral dissociation. . . . . . . . . . 9Figure 1.2 Rise in fractional electron density as a function of time scaledby the plasma frequency, ωe and fraction, ρe(t = 0)/ρ0 = Pf ,of prompt Penning electrons. Simulation results shown forn0 = 30, 50 and 70 with initial densities, ρ0 = 109, 1010, 1011,and 1012 cm−3. . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 1.3 (top frame) Cross-sectional contour diagram in the x,y planefor z = 0 describing the distribution of ion plus electron den-sity over 100 shells of Gaussian ellipsoid with initial dimen-sions, σx = 0.75 mm and σy = σz = 0.42 mm and an initialn0 = 50 Rydberg gas density, ρ0 = 2× 1011 cm−3 after anevolution time of 100 ns. (bottom frame) Curves describingthe (dashed) ascending ion and (solid) descending Rydberggas densities of each shell as functions of evolution time, fort = 20, 40, 60, 80 and 100 ns. . . . . . . . . . . . . . . . . . . 13xiiFigure 1.4 Global population fractions of particles as they evolve in theavalanche of a shell-model ellipsoidal Rydberg gas with theinitial principal quantum number and density distribution ofFigure 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 1.5 Double-resonant spectra of nitric oxide Rydberg states in then f series converging to NO+ v= 0, N+= 2 (designated, n f (2)),derived from the late-peak signal obtained after a flight timeof 400 µs by scanning ω2 with ω1 tuned to NO A 2Σ+ v = 0,N′ = 0 for initial n f (2) densities from top to bottom of 0.07,0.10, 0.13, 0.19, 0.27, 0.30, 0.32 and 3×1012 cm−3. . . . . . 17Figure 1.6 x,y detector images of ultracold plasma volumes produced by2:1 aspect ratio ellipsoidal Rydberg gases with selected initialstate, 40 f (2) after a flight time of 402 µs over a distanceof 575 mm. Lower frame displays the distribution in x ofthe charge integrated in y and z. Both images represent theunadjusted raw signal acquired in each case after 250 shots. . 18Figure 2.1 Schematics of moving grid apparatus. Co-propagating laserbeams, ω1 and ω2 cross a molecular beam of nitric oxidebetween entrance aperture G1 and grid G2 of a differentially-pumped vacuum chamber to form a dense gas of Rydberg stateNO molecules. . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.2 Sequence of events in a typical SFI experiment. The laserbeams, ω1 and ω2, cross the molecular beam of NO with aspecified delay, ∆tω2 . An electric field ramp from 0 to 350V/cm with a rise-time of 1 µs (0.8 V/ns), applied after time∆tRamp following ω2, ionizes the excited molecular system. . . 25Figure 2.3 Double UV resonance mechanism. Nd:YAG-pumped, frequency-doubled dye laser beams at, ω1 andω2, pump ground state NOfirst to the excited A 2Σ+ (ν ′ = 0, N′ = 0) state and then to aRydberg level with N = 1. . . . . . . . . . . . . . . . . . . . 26xiiiFigure 2.4 Resonant two-photon ionization spectrum for the ω1 laser. Wetune ω1 to the pQ11 transition. . . . . . . . . . . . . . . . . . 27Figure 2.5 Integrated detector signal as a function of second laser wavenum-ber. Principal quantum numbers of some transitions related tothis range of wavenumbers are shown on top of their corre-sponding peaks. . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.6 Histogram of 1000 SFI traces, with the initial principal quan-tum number of n = 44, the first laser pulse energy of 6 µJ andno delay between the two lasers. The second laser was set at8 mJ, which is above the saturation limit for second electrontransition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 2.7 SFI false color plot consisting of 5000 SFI traces collectedwhile changing the delay between the two lasers. On theleft the histograms related to different delays between the twolasers are shown. As we delay the second laser, the meanvalue and the width of the distribution decrease. . . . . . . . . 32Figure 2.8 Relationship between the SFI integrated signal and the delaybetween the two laser pulses. The SFI integrated signal un-dergoes an exponential decay as the delay between the twolasers increases. The time constant of this exponential decaymatches the radiative lifetime of A-state of Nitric oxide. Thedata were collected with ω1 pulse energy of 6 µJ. . . . . . . . 33Figure 2.9 Relationship between the SFI integrated signal and the delaybetween the two laser pulses. The data were collected with ω1pulse energy of 20 µJ. For such a high laser energy, the datado not follow the exponential decay when the delay betweenthe two laser pulses is less than 150 ns. . . . . . . . . . . . . . 34xivFigure 2.10 Set of all histograms related to density calibration experimentby the SFI technique. Each row shows the distribution of SFIintegrated signal for a fixed ω1 pulse energy and different ω2delays. Each column, on the other hand, shows that distri-bution for a fixed ω2 delay, and different ω1 pulse energies.Notice the change in the means and standard deviations of thedistributions as the initial Rydberg density is changed througheither first laser energy or the second laser delay. . . . . . . . 35Figure 2.11 A subset of histograms showing the change in the total SFIintegrated signal distribution as the initial Rydberg density ischanged. On the top row, the density is changed through thefirst laser energy, while the delay between the two laser pulsesis zero for all frames. At the bottom, the density is changedthrough the delay between the two lasers, while the first laserenergy is kept the same. Both these methods of changing den-sities result in the same trend of change in the distributions;as we can see in each row the density is decreased from left toright, resulting in the mean of the distribution to move towardslower integrated signal values. . . . . . . . . . . . . . . . . . 36Figure 2.12 Relationship between the SFI integrated signal and the delaybetween the two laser pulses for the case of non-zero ∆tRamp.The data were collected with ω1 pulse energy of 6 µJ and∆tRamp = 500 ns. The data shows that the exponential decaywith a time constant which matches the radiative lifetime ofA-state of Nitric oxide also holds for non-zero fixed ramp delays. 37xvFigure 3.1 (upper frame) Set of traces showing the charge density andwidth in z of a plasma formed by avalanche from a 50 f (2)Rydberg gas as a function of flight time for a sequence of flightdistances. (lower frame) Integrated plasma signal obtained asthe areas of the traces in the upper frame, plotted as a functionof arrival time at G2. Electron loss in the traveling excitedvolume seems to cease after 7 µs. . . . . . . . . . . . . . . . 41Figure 3.2 Schematic demonstration of classical field ionization of theRydberg state. The external electric field perturbs the fieldfree potential such that there is a saddle point (SP) at zsp =√e/4piε0E. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 3.3 Ramped time-dependent electric field, applied to grid G1 in atypical SFI experiment. The voltage ramp is formed throughpulsing a 3 kV DC voltage into a well-defined RC circuit. Itrises at a rate of 0.8 V/ns, and almost linearly with time forthe first 800 ns. . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.4 A rising electrostatic field collects the extravalent electrons ina Rydberg-plasma volume, yielding a selective field ioniza-tion (SFI) spectrum. Loosely-bound electrons from plasmaappear at low field, until about 80 V/cm. The two peaksnear 125 V/cm and 190 V/cm correspond to the dominantionization pathways for a Rydberg population prepared in the44f(2) state, the N+ = 0 and N+ = 2 states of the ion. . . . . . 45xviFigure 3.5 (left) Selective field ionization spectra of NO: Contour plots show-ing SFI signal as a function the applied field for an n f (2) Rydberggas with an initial principal quantum number, n0 = 44. Each framerepresents 4,000 SFI traces, sorted by initial Rydberg gas density.Ramp field potential, beginning 0, 150, 300 and 450 ns after the ω2laser pulse for the top left, top right, bottom left, and bottom rightrespectively. The two bars of signal most evident at early ramp fielddelay times represent the field ionization of the 44 f (2) Rydbergstate respectively to NO+ X 1Σ+ cation rotational states, N+ = 0and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.6 SFI spectra for principal quantum numbers n0 = 40,44,49 atdifferent ramp delays ∆tRamp = 0,150,300,450 ns. Compar-ing the plots in each row shows that as the initial principalquantum number increases, the electric field required to ionizethe Rydberg molecules is reduced. . . . . . . . . . . . . . . . 49Figure 3.7 Normalized SFI plots for principal quantum number n = 51and Pω1 = 4µJ at long ramp-delays ∆tRamp. . . . . . . . . . . 52Figure 3.8 Normalized SFI plots for principal quantum number n = 51and Pω1 = 6µJ at long ramp-delays ∆tRamp. . . . . . . . . . . 53Figure 3.9 Normalized SFI plots for principal quantum number n = 51and Pω1 = 8µJ at long ramp-delays ∆tRamp. . . . . . . . . . . 54Figure 3.10 Normalized SFI plots for principal quantum number n = 51and Pω1 = 10µJ at long ramp-delays ∆tRamp. . . . . . . . . . . 55Figure 3.11 Classification of scope traces as plasma signal (red) and Rydbergsignal (blue). Each plotted trace corresponds to the average of3000 recorded traces at ∆tω2 = 0. Classification is based onbinding energy. Signal below 50 V/cm is treated as plasmasignal whereas signal in the range of 50− 220 V/cm is clas-sified as Rydberg signal. . . . . . . . . . . . . . . . . . . . . 57xviiFigure 3.12 Total plasma and Rydberg signal (left column) and the decom-position of total signal into these two components (right panel)as a function of ramp delay for different ω2 pulse energies. . . 58Figure 3.13 Average Rydberg signal as a function of ω1 power for variousramp delays. The Rydberg signal is obtained by integrating anexperimentally measured trace (signal as a function of rampedfield) from 50V/cm to 220V/cm. For the case of zero rampdelay (the blue curve), the signal increases with ω1 pulse en-ergy approaching saturation. If we allow the system to evolvefor 1µs before we extract the electrons with the ramped field(i.e. the red curve), the average Rydberg signal initially rises.However, increasing the laser pulse energy further reducesthe Rydberg signal. If we delay the ramp more than 5 µs,however, the Rydberg signal is always reduced, as we increasethe laser pulse energy. . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.14 Average Rydberg signal as a function of ramp delay for var-ious ω1 laser pulse energies. The pulse energies of 1.5, 4,6, 8, and 10 µJ correspond to initial Rydberg state moleculedensities of 9×1011, 2×1012, 3×1012, 4×1012, and 5×1012cm−3, respectively. The Rydberg signal is obtained by inte-grating each experimentally measured trace from 50V/cm to220V/cm. It was normalized such that all signals are 1 at∆tRamp = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 4.1 Co-propagating laser beams, ω1 and ω2 cross a molecularbeam of nitric oxide between entrance aperture G1 and gridG2 of a differentially-pumped vacuum chamber. . . . . . . . . 66xviiiFigure 4.2 Diagram illustrating the double-resonant excitation of a molec-ular Rydberg gas of nitric oxide, and the conditions leadingto Penning ionization and avalanche to an ultracold plasma.The atomic-like plasma-excitation spectrum consists exclu-sively of N = 1 n0 f (2) Rydberg state resonances convergingto the N+ = 2 rotational limit of NO+. For an initial n0 = 50Rydberg gas density of 1010 cm−1, the orbital radius is about1 µm while the average spacing between Rydberg moleculesis 3 µm. However, a good portion of the nearest-neighbourdistance distribution falls within 1 µm. These closely spacedpairs interact by Penning ionization to form prompt electrons,which seed the avalanche to ultracold plasma. . . . . . . . . . 68Figure 4.3 Schematic diagram showing the sequence of pluses in the RF-SFI experiment. The laser beams, ω1 and ω2, cross the molec-ular beam of NO with a specified delay, ∆tω2 . A radio-frequencyfield with an adjustable peak-to-peak amplitude as high as 1 Vcm−1 interacts with this ensemble, either as a CW field or as apulse with a duration Wrf applied at a time, ∆trf, after ω2. Anelectric field ramp from 0 to 350 V/cm with a rise-time of 1µs, applied ∆tramp followingω2, ionizes the excited molecularsystem. Shaded regions represent the dissociative decay of then0 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). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73xixFigure 4.4 Typical SFI spectra, formed by 4,000 SFI traces sorted ac-cording to the initial density ρ0, for an n f (2) Rydberg gaswith an initial principal quantum number n0 = 44. Individ-ually normalized contours from left to right show electronbinding energy spectra for electric field ramp applied 0,150,450 and 1000 ns after the ω2 laser pulse. The signal near zerofield represents very high Rydberg molecules and electronsloosely-bound by the plasma space charge. The two featuresthat appear at higher field reflect the field ionization of then0 = 44 state to NO+ X 1Σ+ cation rotational states, N+ = 0and N+ = 2. Note how after ramp delay of zero (∆tramp = 0),these features shift to higher field ionization thresholds, reflec-tive of electron collisional `-mixing of initial 44 f (2) Rydbergmolecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 4.5 For ultracold plasmas evolving from (left) 45 f (2) and (right)49 f (2) Rydberg gases: (top) SFI amplitude integrated overramp field from 0 to 50 V cm−1; and (bottom) SFI amplitudeintegrated over ramp field from 50 to 200 V cm−1, as a func-tion of ramp delay, ∆tramp in the presence (blue) and absence(orange) of a 400 mV cm−1 CW 60 MHz radio frequencyfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 4.6 Single SFI traces obtained under identical conditions of ini-tial 49 f (2) Rydberg gas density and ramp-field delay in theabsence (upper) and presence (lower) of a CW 60 MHz radiofrequency field. Here, as above, the low-voltage part of theramp collects loosely-bound electrons from the plasma. Athigher 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. . . . . . . . . . . . . . . . . . . . . . . 77xxFigure 4.7 Integrated electron signal from the selective field ionization ofn0 = 49 Rydberg molecules in the presence (blue) and absence(orange) of a 250 ns 400 V cm−1 60 MHz pulsed radio fre-quency field as a function of rf delay, ∆tωrf for two fixed valuesof ramp field delay, ∆tramp, of 2 µs (left) and 4 µs (right). . . 78Figure 4.8 Integrated electron signal from the selective field ionization ofn0 = 49 Rydberg molecules in the presence of a 250 ns 400V cm−1 60 MHz pulsed radio frequency field as a function ofrf delay, ∆tωrf for a fixed ramp field delay, ∆tramp = 5 µs, andtwo values of ω1 - ω2 delay, 0 and 200 ns. With a radiativelifetime of 192 ns, a delay of 200 ns reduces the density of2Σ+ intermediate state NO molecules by a factor of 2.72. . . . 79Figure A.1 Hund’s cases: Coupling of rotation and electronic motion - Jis the total angular momentum, N is the angular momentum ofthe nuclear rotation. K is the total angular momentum apartfrom spin. L is the total electronic orbital angular momentumand Λ its projection on the molecular axis. S is the totalelectronic spin and Σ it’s projection on the molecular axis.In Hund’s case (d), N becomes R, a good quantum number.(Figure (c) does not show the coupling of S and K to J.)Credit: J.P. Morrison . . . . . . . . . . . . . . . . . . . . . . 104Figure A.2 NO molecular orbital diagram - Only the open shell electronlevels are displayed. The excitation pathway for ω1 photonsis indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure A.3 NO level diagram for X, A and Ry-states and transitions A←X,Ry←A. - Labeling information for energy levels and transi-tions are found in the text. . . . . . . . . . . . . . . . . . . . 108xxiFigure B.1 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 2 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 110Figure B.2 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 4 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 111Figure B.3 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 6 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 112Figure B.4 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 8 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 113Figure B.5 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 10 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 114Figure B.6 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 12 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 115Figure B.7 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 15 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 116Figure B.8 Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 20 µJ, for differentdelays of the second laser. . . . . . . . . . . . . . . . . . . . 117xxiiGlossaryMD Molecular DynamicsMOT Magneto Optical TrapNO Nitric OxideRF Radio FrequencySFI Selective Field IonizationUCP Ultracold PlasmaxxiiiAcknowledgmentsMy five years and a bit in this PhD program is coming to an end and the past year,in particular, has by far been the strangest year I have ever experienced. At thetime I am writing these words, COVID-19 has taken over two million and a halflives worldwide and has forced everyone to adapt to the new norms. Many peoplelost loved ones due to this pandemic. I deeply feel sorry for those who had toexperience this. If we have not been one of those suffering loss of loved ones,we must be grateful. Being grateful to have all my friends and family either notaffected or recovered from this situation, in what follows, I would like to thankthose who supported me during the course of my PhD in many different ways, andhelped me become a better version of myself.First and foremost, my deepest gratitude goes to Professor Edward Grant, mysupervisor. Ed not only guided me in the research contained in this dissertation,but also taught me many interpersonal skills. Ed was extremely patient and sup-portive of me while I was altering customary practices in a typical PhD program,to find my way in- and outside academia. Ed was always there for me whenI needed him, always a welcoming supervisor who created such a welcomingenvironment that I could share my deepest concerns both related to my PhDprogram or other aspects of life. He never stopped me from exploring new thingsand that alone was extremely important in turning me into an unbiased researcher.I met with Ed in the last months of my Physics Master’s program at UBC, when Iwas not sure if PhD is the right path for me. Ed let me join his lab as a non-degreeresearch assistant, to evaluate if I enjoy the type of work being done in the Ultra-xxivcold Plasma side of the lab. Those few months was enough to convince me that Iwant to spend years in this lab. I never forget this statement from Ed: "A real PhDgraduate can create knowledge even in an empty room". Now that I’m living ina 4-BR house, I should allocate one of the rooms to test if I have become a ’real’one.Dr. James Keller has been one of my most influential mentors inside thelab. Majority of the technical skills and practices that I utilized throughout thisdissertation was taught to me by Jamie. I was extremely blessed that my first yearof PhD coincided with Jamie’s visit to our lab, as I learned how to align laserssafely, how to perform data collection and how to interpret the selective fieldionization data from him. These constructed the foundation of what I contributedfurther in this research area. I also benefited a lot from Jamie’s feedback whiledrafting this dissertation. I also owe Jamie a few full cups of coffee, as I tippedhis mug over several times while discussing our data (I think they each were half-filled).I am also thankful to different members of the lab for their supports, eitherdirectly or indirectly related to my research work. Hossein Sadeghi, and MarkusSchulz-Weiling taught me many things about Rydberg Physics and our experi-ments during the relatively short overlap between their PhD program and mine.Luke Melo and Ashton Christy always helped me with my computer/softwarerelated (and sometimes other unrelated) questions. Ruoxi Wang joined the lab ayear after I started my PhD, and proved to be exceptionally talented. I am veryhappy to spend many hours with him during the first months of his arrival goingthrough different steps of the experiment, and grateful to him for helping me withsome data collection and discussion over the data in many occasions.Although I can talk a lot about each and every colleague (and friend) of minein and out-side the lab, I just acknowledge them by mentioning their names hereto prevent doubling the size of this thesis. Matt Kowal, Kevin Marroquín Madera(for bringing the warm and joyful Latin American culture to the lab), NajmehTavassoli (for introducing me to Ed), Kiara Grant, Rafael Haenel, Maggie Duanxxv(for all the laughter we had during stressful moments), Fernanda Banic V. Martins(for all the kindness and wisdom you brought to the lab during your internship),Sara Poursorkh (for pushing me in the last moments to finish my thesis) andeveryone else whose name I can’t remember at this moment.This research wouldn’t have been possible without the financial support ofNatural Sciences and Engineering Research Council of Canada (NSERC), theUniversity of British Columbia Faculty of Graduate and Postdoctoral Studies,UBC Department of Chemistry (Teaching Assistantship), and I express my grati-tude to these parties.I am also grateful of the support I received from specific people outside UBC.My managers at work, Mr. Roland Heere, Mr. Cengiz Guldemet, and Mr. RobertGilbert were extremely flexible with and supportive of my work hours. I appreci-ate their trust in me, allowing me to experiment if I can pursue my graduate schoolwhile working at Metro Testing + Engineering company. I’m glad it all workedout, and I could bring some value to the company!The members of my family from the other side of the globe have alwayssupported me in every possible way and in every stage of life. Even though wehave been away for the majority of the past ten years, knowing that I have a lovingfamily that I can always count on, has tremendously helped me get where I amtoday. Special thanks to my Mom and Dad for their extraordinary supports. Istarted the PhD journey mainly to make my Mom happy (a typical wish of almostevery Iranian Mom is to see her children become a ’Doctor’, and I still wonderwhy!). I hope I have made you proud enough, Mom. I, myself, am happy thatI went through this journey, so thank you. Mahdad, Mohanna, Sahar, Minoo,Nikoo, and Mahan Thank you all for the positive energy you gave me through ourvideo chats throughout the years. I would also like to thank my in-law family,Mr. and Mrs. Kavianipour, Maryam and Kimia for always being supportive andencouraging during my studies.Last but not least, the only person without whom I wouldn’t have been whereI am today is my lovely wife. Thank you Mona for being the strong, lovely, smart,xxviand beautiful person that you are. Without you covering me at work when I neededto study, and without you listening to me vent, I would not have been able to finishmy PhD anytime soon! Thank you truly, and I will always owe you.xxviiTo my lovely wife and parentsChapter 1Introduction1.1 BackgroundUltracold neutral plasmas studied in the laboratory offer access to a regime ofplasma physics that scales to describe thermodynamic aspects of important high-energy-density systems, including strongly coupled astrophysical plasmas [1, 2],as well as terrestrial sources of neutrons [3–6] and x-ray radiation [7, 8]. Yet,under certain conditions, low-temperature laboratory plasmas evolve with dynam-ics that are governed by the quantum mechanical properties of their constituentparticles, and in some cases by coherence with an external electromagnetic field.The relevance of ultracold plasmas to such a broad scope of problems in classi-cal and quantum many-body physics has given rise to a great deal of experimentaland theoretical research on these systems since their discovery in the late 90s. Aseries of reviews affords a good overview of progress in the last twenty years [9–12]. Here, we focus on the subset of ultracold neutral plasmas that form via kineticrate processes from state-selected Rydberg gases, and emphasize in particular thedistinctive dynamics found in the evolution of molecular ultracold plasmas.While molecular beam investigations of threshold photoionization spectroscopyhad uncovered relevant effects a few years earlier [13, 14], the field of ultracoldplasma physics began in earnest with the 1999 experiment of Rolston and cowork-1ers on metastable xenon atoms cooled in a magneto optical trap (MOT) [15].This work and many subsequent efforts tuned the photoionization energy asa means to form a plasma of very low electron temperature built on a stronglycoupled cloud of ultracold ions. Experiment and theory soon established that fastprocesses associated with disorder-induced heating and longer-time electron-ioncollisional rate processes act to elevate the ion temperatures to around one degreeKelvin, and constrain the effective initial electron temperature to a range above 30K [16–18].This apparent limit on the thermal energy of the electrons can be more univer-sally expressed for an expanding plasma by saying that the electron correlationparameter, Γe, does not exceed 0.25, where,Γe =e24piε0aws1kBTe(1.1)defines the ratio of the average unscreened electron-electron potential energy tothe electron kinetic energy. aws is the Wigner-Seitz radius, related to the electrondensity by, ρe = 1/(43pia3ws). These plasmas of weakly coupled electrons andstrongly coupled ions have provided an important testing ground for ion transporttheory and the study of electron-ion collision physics [19].Soon after the initial reports of ultracold plasmas formed by direct photoion-ization, a parallel effort began with emphasis on the plasma that forms sponta-neously by Penning ionization and electron-impact avalanche in a dense ultracoldRydberg gas [20]. This process affords less apparent control of the initial electrontemperature. But, pulsed field-ionization measurements soon established that thephotoionized plasma and that formed by the avalanche of a Rydberg gas bothevolve to quasi-equilibria of electrons, ions and high-Rydberg neutrals [9, 21].Early efforts to understand plasmas formed by Rydberg gas avalanche paidparticular attention to the process of initiation. Evolution to plasma in effusiveatomic beams was long known for high-Rydberg gases of caesium and well ex-plained by coupled rate equations [22]. But, low densities and ultracold velocity2distributions were thought to exclude Rydberg-Rydberg collisional mechanismsin a MOT.In work on ultracold Rydberg gases of Rb and Cs, Gallagher, Pillet and cowork-ers describe the initial growth of electron signal by a model that includes ion-ization by blackbody radiation and collisions with a background of uncooledRydberg atoms [9, 20, 23–25]. This picture was subsequently refined to includemany-body excitation and autoionization, as well as attractive dipole-dipole inter-actions [26, 27], later confirmed by experiments at Rice [28].The Orsay group also studied the effect of adding Rydberg atoms to an estab-lished ultracold plasma. They found that electron collisions in this environmentcompletely ionize added atoms, even when selected to have deep binding energies[29]. They concluded from estimates of electron trapping efficiency that theaddition of Rydberg atoms does not significantly alter the electron temperatureof the plasma.Tuning pair distributions by varying the wavelength of the excitation laser,Weidemüller and coworkers confirmed the mechanical effects of van der Waalsinteractions on the rates of Penning ionization in ultracold 87Rb Rydberg gases[30]. They recognized blackbody radiation as a possible means of final-stateredistribution, and extended this mechanical picture to include long-range repul-sive interactions [31]. This group later studied the effects of spatial correlationsin the spontaneous avalanche of Rydberg gases in a regime of strong blockade,suggesting a persistence of initial spatial correlations [32].Robicheaux and coworkers have recently investigated the question of promptmany-body ionization from the point of view of Monte Carlo classical trajectorycalculations [33]. For atoms on a regular or random grid driven classically byan electromagnetic field, they find that many-body excitation enhances promptionization by about twenty percent for densities greater than 5.6×10−3/(n20a0)3,where n0 is the principal quantum number of the Rydberg gas and a0 is the Bohrradius. They observed that density fluctuations (sampled from the distributionof nearest neighbour distances) have a greater effect, and point to the possible3additional influence of secondary electron-Rydberg collisions and the Penningproduction of fast atoms not considered by the model, but already observed byRaithel and coworkers [34].The Raithel group also found direct evidence for electron collisional `-mixingin a Rb MOT [35], and used selective field ionization to monitor evolution toplasma on a microsecond timescale in ultracold 85Rb 65d Rydberg gases withdensities as low as 108 cm−3 [36]. Research by our group at UBC has observedvery much the same dynamics in the relaxation of Xe Rydberg gases of similardensity prepared in a molecular beam [37]. In both cases, the time evolutionto avalanche is well-described by coupled rate equations (see below), assumingan initializing density of Penning electrons determined by Robicheaux’s criterion[38], applied to an Erlang distribution of Rydberg-Rydberg nearest neighbours.Theoretical investigations of ultracold plasma physics have focused for themost part on the long- and short-time dynamics of plasmas formed by direct pho-toionization [11, 12]. In addition to studies mentioned above, key insights on theevolution dynamics of Rydberg gases have been provided by studies of Pohl andcoworkers exploring the effects of ion correlations and recombination-reionizationon the hydrodynamics of plasma expansion [39, 40]. Further research has drawnupon molecular dynamics (MD) simulations to reformulate rate coefficients forthe transitions driven by electron impact between highly excited Rydberg states[41], and describe an effect of strong coupling as it suppresses three-body re-combination [42]. MD simulations confirm the accuracy of coupled rate equationdescriptions for systems with Γe as large as 0.3. Newer calculations suggest astrong connection between the order created by dipole blockade in Rydberg gasesand the most favourable correlated distribution of ions in a corresponding stronglycoupled ultracold plasma [43].Tate and coworkers have studied ultracold plasma avalanche and expansiontheoretically as well as experimentally. Modelling observed expansion rates, theyrecently found that 85Rb atoms in a MOT form plasmas with effective initial elec-tron temperatures determined by initial Rydberg density and the selected initial4binding energy, to the extent that these parameters determine the fraction of theexcited atoms that ionize by electron impact in the avalanche to plasma [44]. Thisgroup also returned to the question of added Rydberg atoms, and managed toidentify a crossover in n0, depending on the initial electron temperature, thatdetermines whether added Rydberg atoms of a particular initial binding energyact to heat or cool the electron temperature [45].Our group has focused on the plasma that evolves from a Rydberg gas underthe low-temperature conditions of a skimmed, seeded supersonic molecular beam.In work on nitric oxide starting in 2008 [46–49], we established an initial kineticsof electron impact avalanche ionization that conforms with coupled rate equationmodels [50–53] and agrees at early times with the properties of ultracold plas-mas that evolve from ultracold atoms in a MOT. We have also observed uniqueproperties of the NO ultracold plasma owing to the fact that its Rydberg statesdissociate [54], and identified relaxation pathways that may give rise to quantumeffects [55, 56]. The remainder of this chapter focuses on the nitric oxide ultracoldplasma and the unique characteristics conferred by its evolution from a Rydberggas in a laser-crossed molecular beam.1.2 The molecular beam ultracold plasmacompared with a MOTWhen formed with sufficient density, a Rydberg gas of principal quantum numbern0 > 30 undergoes a spontaneous avalanche to form an ultracold plasma [23,32, 46]. Collisional rate processes combine with ambipolar hydrodynamics togovern the properties of the evolving plasma. For a molecular Rydberg gas,neutral fragmentation occurs in concert with electron-impact ionization, three-body recombination and electron-Rydberg inelastic scattering. Neutral dissocia-tion combined with radial expansion in a shaped distribution of charged particles,can give rise to striking effects of self-assembly and spatial correlation [54, 57].The formation of a molecular ultracold plasma requires the conditions of localtemperature and density afforded by a high Mach-number skimmed supersonic5molecular beam. Such a beam propagates at high velocity in the laboratory,with exceedingly well-defined hydrodynamic properties, including a propagation-distance-dependent density and sub-Kelvin temperature in the moving frame [58].The low-temperature gas in a supersonic molecular beam differs in three importantways from the atomic gas laser-cooled in a magneto-optical trap (MOT).The milli-Kelvin temperature of the gas of ground-state NO molecules en-trained in a beam substantially exceeds the sub-100 micro-Kelvin temperature oflaser-cooled atoms in a MOT. However, the evolution to plasma tends to erasethis distinction, and the two further characteristics that distinguish a beam offerimportant advantages for ultracold plasma physics: Charged-particle densities ina molecular beam can exceed those attainable in a MOT by orders of magnitude.A great many different chemical substances can be seeded in a free-jet expansion,and the possibility this affords to form other molecular ultracold plasmas, intro-duces interesting and potentially important new degrees of freedom governing thedynamics of their evolution.1.3 Supersonic molecular beam temperature andparticle densitySeeded in a skimmed supersonic molecular beam of helium, nitric oxide forms dif-ferent phase-space distributions in the longitudinal (propagation) and transversecoordinate dimensions. As it propagates in z, the NO molecules reach a terminallaboratory velocity, u‖, of about 1400 ms−1, which varies with the precise seedingratio.The distribution of v‖, narrows to define a local temperature, T‖, of approx-imately 0.5 K. The beam forms a Gaussian spatial distribution in the transversecoordinates, x and y. In this plane, the local velocity, v⊥(r) is defined for anyradial distance almost entirely by the divergence velocity of the beam, u⊥(r).Phase-space sorting cools the temperature in the transverse coordinates, T⊥ to avalue as low as ∼ 5 mK [58].The stagnation pressure and seeding ratio determine the local density distribu-6tion as a function of z. For example, expanding from a stagnation pressure of 500kPa with a 1:10 seeding ratio, a molecular beam propagates 2.5 cm to a skimmerand then 7.5 cm to a point of laser interaction, where it contains NO at a peakdensity of 1.6×1014 cm−3.Here, crossing the molecular beam with a laser beam tuned to the transitionsequence, X 2Π1/2 N′′ = 1ω1−→ A 2Σ+ N′ = 0 ω2−→ n0 f (2) forms a Gaussian ellip-soidal volume of Rydberg gas in a single selected principal quantum number, n0,orbital angular momentum, `= 3, NO+ core rotational quantum number, N+ = 2and total angular momentum neglecting spin, N = 1.A typical ω1 pulse energy of 2 µJ and a Gaussian width of 0.2 mm serves todrive the first step of this sequence in a regime of linear absorption. Overlappingthis volume by an ω2 pulse with sufficient fluence to saturate the second stepforms a Rydberg gas ellipsoid with a nominal peak density of 5× 1012 cm−3[46, 58]. Fluctuations in the pulse energy and longitudinal mode of ω1 cause thereal density to vary. For certain experiments, we find it convenient to saturate theω1 transition, and vary the density of Rydberg gas by delaying ω2. An ω1-ω2delay, ∆t, reduces the Rydberg gas density by a precise factor, e−∆t/τ , where τ isthe 192 ns radiative lifetime of NO A 2Σ+ N′ = 0 [59, 60].1.4 Penning ionizationThe density distribution of a Rydberg gas defines a local mean nearest neighbourdistance, or Wigner-Seitz radius of aws = (3/4piρ)1/3, where ρ refers to the localRydberg gas density. For example, a Rydberg gas with a density of ρ0 = 0.5×1012cm−3 forms an Erlang distribution [61] of nearest neighbour separations with amean value of 2aws = 1.6 µm.A semi-classical model [38] suggests that 90 percent of Rydberg moleculepairs separated by a critical distance, rc = 1.8 · 2n20a0 or less undergo Penningionization within 800 Rydberg periods. We can integrate the Erlang distributionfrom r = 0 to the critical distance r = rc for a Rydberg gas of given n0, to definethe local density of Penning electrons (ρe at t = 0) produced by this prompt7interaction, for any given initial local density, ρ0 by the expression:ρe(ρ0,n0) =0.92·4piρ20∫ rc0r2e−4pi3 ρ0r3dr . (1.2)Evaluating this definite integral yields an equation in closed form that pre-dicts the Penning electron density for any particular initial Rydberg density andprincipal quantum number.ρe(ρ0,n0) =0.9ρ02(1− e− 4pi3 ρ0r3c ) . (1.3)Prompt Penning ionization acts on the portion of the initial nearest-neighbourdistribution in the Rydberg gas that lies within rc. When a molecule ionizes,its collision partner relaxes to a lower principal quantum number, n′ < n0/√2.This close-coupled interaction disrupts the separability of Rydberg orbital config-urations in the Penning partner. This causes mixing with core penetrating statesthat are strongly dissociative. Penning partners are thus very likely to dissociate,leaving a spatially isolated distribution of ions. We refer to the spatial correlationthat results as a Penning lattice [62]. The extent of this effect varies dependingon the local density and the selected initial principal quantum number. Figure 1.1shows the degree to which Rydberg gases with initial principal quantum numbersfrom 30 to 80 form a Penning lattice for an initial density of 1×1012 cm−3.1.5 Spontaneous electron-impact avalancheThe electrons produced by prompt Penning ionization start an electron impactavalanche. The kinetics of this process are well described by a set of coupledrate equations that account for state-to-state electron-Rydberg inelastic scattering,electron-impact ionization and three-body ion-electron recombination [40, 50–52]using detailed rate coefficients, ki j, ki,ion and ki,tbr validated by MD simulations80 2 4300 1350 0.5 1400 0.5 1population density (arb.)450 0.5 1500 0.5 1550 0.5 1600 0.5 1650 0.5 1700 0.5 1750 0.5 180Ion−Ion nearest neighbour distance (µm)Ion-ion nearest neighbour distance (µm)Population density (arbitrary units)Figure 1.1: Distributions of ion-ion nearest neighbours following Penningionization and electron-impact avalanche simulated for a predissoci-ating molecular Rydberg gas of initial principal quantum number, n0,from 30 to 80, and density of 1012 cm−3. Dashed lines mark corre-sponding values of aws. Calculated by counting ion distances afterrelaxation to plasma in 106-particle stochastic simulations. Integratedareas proportional to populations surviving neutral dissociation.[41].−dρidt= ∑jki jρeρi−∑jk jiρeρ j+ki,ionρeρi− ki,tbrρ3e (1.4)9and,dρedt=∑iki,ionρ2e −∑iki,tbrρ3e (1.5)The relaxation of Rydberg molecules balances with collisional ionization todetermine an evolving temperature of avalanche electrons to conserve total energyper unit volume.Etot =32kBTe(t)ρe(t)−R∑iρi(t)n2i, (1.6)Here, for simplicity, we neglect the longer-time effects of Rydberg predissociationand electron-ion dissociative recombination [51].Such calculations show that the conversion from Rydberg gas to plasma oc-curs on a timescale determined largely by the local Penning electron density, orPenning fraction, Pf = ρe/ρ0, which depends on the local density of Rydbergmolecules and their initial principal quantum number.Avalanche times predicted by coupled rate equation calculations range widely.For example, in a model developed for experiments on xenon, simulations predictthat a Rydberg gas with n0 = 42 at a density of 8.8× 108 cm−3 (Pf = 6× 10−5)avalanches with a half life of 40 µs [37]. At an opposite extreme, rate equationsestimate that a Rydberg gas of NO with n0 = 60 at a density of 1× 1012 cm−3(Pf = 0.3) forms a plasma in about 2 ns [51].Selective field ionization (SFI) probes the spectrum of binding energies in aRydberg gas. Applied as a function of time after photoexcitation, SFI maps theevolution from a state of selected initial principal quantum number, n0, to plasma[54]. Specifically, SFI measurement can be performed in a time-resolved way,allowing the study of plasma evolution as a function of initial Rydberg density ρ0.In general, the measured dynamics agree well with the predictions of coupledrate-equation calculations. We can understand this variation in relaxation dynam-ics with ρ0 and n0 quite simply in terms of the corresponding density of promptPenning electrons these conditions afford to initiate the avalanche to plasma.Figure 1.2 illustrates this, showing how rise times predicted by coupled rate-equation simulations for a large range of initial densities and principal quantum10number match when plotted as a function of time scaled by the ultimate plasmafrequency and fraction of prompt Penning electrons. The dashed line gives an ap-proximate account of the scaled rate of avalanche under all conditions of Rydberggas density and initial principal quantum number in terms of the simple sigmoidalfunction:ρeρ0=ab+ e−cτ, (1.7)where,τ = tωeP3/4f , (1.8)in which ωe is the plasma frequency after avalanche, Pf is the fraction of promptPenning electrons, and a = 0.00062, b = 0.00082 and c = 0.075 are empiricalcoefficients.1.6 Evolution to plasma in a Rydberg gas GaussianellipsoidAs outlined above, the local density and principal quantum number together de-termine the rate at which a Rydberg gas avalanches to plasma. Our experimentcrosses a 2 mm wide cylindrically Gaussian molecular beam with a 1 mm diameterTEM00 ω1 laser beam to produce a Gaussian ellipsoidal distribution of moleculesexcited to the A 2Σ+ v = 0, N′ = 0 intermediate state. A larger diameter ω2 pulsethen drives a second step that forms a Rydberg gas in a single n0 f (2) state withthe spatial distribution of the intermediate state.We model this shaped Rydberg gas as a system of 100 concentric ellipsoidalshells of varying density [53]. Coupled rate equations within each shell describethe avalanche to plasma. This rate process proceeds from shell to shell withsuccessively longer induction periods, determined by the local density as detailedabove. The rising conversion of Rydberg molecules to ions plus neutral dissoci-ation products conserves the particle number in each shell. We assume that local11t we pf3/40 100 200 300 400 500 600scaled density00.10.20.30.40.50.60.70.80.9n=30n=50n=702 3 4 5 600.10.20.30.40.50.60.70.80.9Evaluating this definite integral yields an expression in closed form for the density of pairsthat fall within this critical distance for a given initial density and principal quantum number,⇢e(⇢0, n0) =0.9⇢02(1 e 4⇡3 ⇢0r3c ) . (2)This defines the Penning electron density with which we initialize coupled rate-equationsimulations, and ⇢e(t = 0)/⇢0 determines the Penning fraction. Energy conservationdemands relaxation of the partner Rydberg to a quantum number lower than n0/p2.The semi-classical model predicts a level population proportional to n5 [? ].⇢e⇢0t!eP3/4f3Evaluating this definite integral yields an expression in closed form for the density of pairsthat fall within this critical distance for a given initial density and principal quantum number,⇢e(⇢0, n0) =0.9⇢02(1 e 4⇡3 ⇢0r3c ) . (2)This defines the Penning electron density with which we initialize coupled rate-equationsimulations, and ⇢e(t = 0)/⇢0 determines the Penning fraction. Energy conservationdemands relaxation of the partner Rydberg to a quantum number lower than n0/p2.The semi-classical model predicts a level population proportional to n5 [? ].⇢e⇢0t!eP3/4f3Figure 1.2: Rise in fractional electron density as a function of time scaled bythe plasma frequency, ωe and fraction, ρe(t = 0)/ρ0 = Pf , of promptPenning electrons. Simulation results shown for n0 = 30, 50 and 70with initial densities, ρ0 = 109, 1010, 1011, and 1012 cm−3.space charge confines electrons to shells, conserving quasi-neutrality. Electronsexchange kinetic energy at the boundaries of each shell, which determines a singleplasma electron temperature.The upper frame of Figure 1.3 shows contours of NO+ ion density after 100 nsobtained from a shell-model coupled rate-equation simulation of the avalanche ofa Gaussian ellipsoidal Rydberg gas of nitric oxide with a selected initial state,50 f (2) and a density of 2× 1011 cm−3. Here, we simulate a relaxation thatincludes channels of predissociation at every Rydberg level and redistributes the1200.10.20.30.40.50.60.70.80.9-6 -4 -2 0 2 4 6ρ(r)/ρ(0)x in units of σ00.10.20.30.40.50.60.70.8-6 -4 -2 2 4 6ρ(e)/ρ(0)x in units of σ-3 -2 -1 0 1 2 3-2-1.5-1-0.500.511.52 00.020.040.060.080.10.120.14-3 -2 -1 0 1 2 3-2-1.5-1-0.500.511.52 00.050.10.150.20.250.30.350.40.45-3 -2 -1 0 1 2 3x (mm)x (mm)210-2-10.500.400.300.1000.20y (mm)ρ (cm-3)ρ (cm-3)x 011x 10110.60-3 -2 -1 0 1 2 30.81.01.20.60.40.201.4Figure 1.3: (top frame) Cross-sectional contour diagram in the x,y plane forz = 0 describing the distribution of ion plus electron density over 100shells of Gaussian ellipsoid with initial dimensions, σx = 0.75 mmand σy = σz = 0.42 mm and an initial n0 = 50 Rydberg gas density,ρ0 = 2×1011 cm−3 after an evolution time of 100 ns. (bottom frame)Curves describing the (dashed) ascending ion and (solid) descendingRydberg gas densities of each shell as functions of evolution time, fort = 20, 40, 60, 80 and 100 ns.13energy released to electrons, which determines a uniform rising electron temper-ature for all shells.For comparison, the lower frame plots curves describing the ion density ofeach shell as a function of time from 20 to 100 ns, as determined by applyingEq 1.7 for the local conditions of initial Rydberg gas density. This numericalapproximation contains no provision for predissociation. Coupled rate-equationsimulations for uniform volumes show that predissociation depresses yield tosome degree, but has less effect on the avalanche kinetics [51]. Therefore, wecan expect sets of numerically estimated shell densities, scaled to agree withthe simulated ion density at the elapsed time of 100 ns, to provide a reasonableaccount of the earlier NO+ density profiles as a function of time.00.10.20.30.40.50.60.70.80.910 20 40 60 80 100 120 140 160 180 2000 40 80 120 160 200time (ns)0.600.2Fractional number1.00.40.8NO+ + e-NO* N(4S) + O(3P)Figure 1.4: Global population fractions of particles as they evolve in theavalanche of a shell-model ellipsoidal Rydberg gas with the initialprincipal quantum number and density distribution of Figure 1.3For each time step, the difference, ρ0− ρe defines the neutral population ofeach shell. We assign a fraction of this population to surviving Rydberg molecules,such that the total population of NO∗ as a function of time agrees with the pre-diction of the shell-model simulation, as shown in Figure 1.4. We consider the14balance of this neutral population to reflect NO∗ molecules that have dissociatedto form N(4S) + O(3P). Figure 1.3 plots these surviving Rydberg densities as afunction of radial distance for each evolution time. At the initial density of thissimulation, note at each time step that a higher density of Rydberg moleculesencloses the tail of the ion density distribution in x.1.7 Plasma expansion and NO+ - NO∗ chargeexchange as an avenue of quenchWe regard the ions as initially stationary. The release of electrons creates a radialelectric potential gradient, which gives rise to a force,−e∇φk, j(t), that acceleratesthe ions in shell j in direction k according to [63]:−em′∇φk, j(t) =∂uk, j(t)∂ t=kBTe(t)m′ρ j(t)ρ j+1(t)−ρ j(t)rk, j+1(t)− rk, j(t) , (1.9)where ρ j(t) represents the density of ions in shell j.The instantaneous velocity, uk, j(t) determines the change in the radial coordi-nates of each shell, rk, j(t),∂ rk, j(t)∂ t= uk, j(t) = γk, j(t)rk, j(t), (1.10)which in turn determines shell volume and thus its density, ρ j(t). The electrontemperature supplies the thermal energy that drives this ambipolar expansion. Ionsaccelerate and Te falls according to:3kB2∂Te(t)∂ t=− m′∑ j N j∑k, jN juk, j(t)∂uk, j(t)∂ t, (1.11)where we define an effective ion mass, m′, that recognizes the redistribution of the15electron expansion force over all the NO+ charge centres by resonant ion-Rydbergcharge exchange, which occurs with a very large cross section [40].m′ =(1+ρ∗j (t)ρ j(t))m, (1.12)in which ρ∗j (t) represents the instantaneous Rydberg density in shell j.The initial avalanche in the high-density core of the ellipsoid leaves few Rydbergmolecules, so this term has little initial effect. Rydberg molecules predominate inthe lower-density wings. There, momentum sharing by charge exchange assumesa greater importance.We see this most directly in the ω2 absorption spectrum of transitions to statesin the n0 f (2) Rydberg series, detected as the long-lived signal that survives aflight time of 400 µs to reach the imaging detector. The balance between therising density of ions and the falling density of Rydberg molecules depends onthe initial density of electrons produced by prompt Penning ionization. As clearfrom Eq 1.3, this Penning fraction depends sensitively on the principal quantumnumber, and for all principal quantum numbers, on the initial Rydberg gas density.Figure 1.5 shows a series of ω2 late-signal excitation spectra for a set ofinitial densities. Here, we see a clear consequence of the higher-order dependenceof Penning fraction - and thus the NO+ ion - NO∗ Rydberg molecule balance- on n0, the ω2-selected Rydberg gas initial principal quantum number. ThisPenning-regulated NO+ ion - NO∗ Rydberg molecule balance appears necessaryas a critical factor in achieving the long ultracold plasma lifetime required toproduce this signal. We are progressing in theoretical work that explains thestability apparently conferred by this balance.1.8 Bifurcation and arrested relaxationAmbipolar expansion quenches electron kinetic energy as the initially formedplasma expands. Core ions follow electrons into the wings of the Rydberg gas.There, recurring charge exchange between NO+ ions and NO∗ Rydberg molecules16PhD Comprehensive Examination Report Kevin Marroqu´ınthe power, the bifurcation increases with power but the intensity increases untilreaching a maximum and then decreases. For the highest PQN, one can observea much lower intensity than for the other two PQNs, however, the bifurcationbehaviour is still the same, but in this case the lowest power yields the highestintensity and increasing it makes the intensity go lower. By changing the powerof !1, one is changing the initial density of excited molecules (⇢0), so to furtherexplore this e↵ects, it is worthy to investigate how the initial density of theplasma a↵ects the Rydberg spectrum presented in the previous section.3.5 Initial density dependence of the Rydberg spectrumIt is possible to study this e↵ects by scanning !2 not with di↵erent !1 powers,but with di↵erent laser beam diameters at a constant power. The spectrasobtained are presented in Fig.(18).Figure 18: Rydberg spectra for di↵erent ⇢0. From top to bottom in[µm3]=0.07, 0.10, 0.13, 0.19, 0.27, 0.30, 0.32 and 32330,300 30400 30,500ω2 frequency (cm-1)Figure 1.5: Double-resonant spectra of nitric oxide Rydberg states in the n fseries converging to NO+ v = 0, N+ = 2 (designated, n f (2)), derivedfrom the late-peak signal obtained after a flight time of 400 µs byscanning ω2 with ω1 tuned to NO A 2Σ+ v= 0, N′= 0 for initial n f (2)densities from top to bottom of 0.07, 0.10, 0.13, 0.19, 0.27, 0.30, 0.32and 3×1012 cm−3.redistributes the ambipolar force of the expanding electron gas, equalizing ionand Rydberg velocities. This momentum matching effectively channels electronenergy through ion motion into the overall ±x motion of gas volumes in thelaboratory. The internal kinetic energy of the plasma, which at this point is definedalmost entirely by the ion-Rydberg relative motion, falls. Spatial correlationdevelops, and over a period of 500 ns, the system forms the plasma/high-Rydbergquasi-equilibrium evidenced by the results to be discussed in Chapter 3.17050010001500200025003000350040000 100 200 300 400 5001501530 30x (mm)y-z Integrated electron signal (arb.)Figure 1.6: x,y detector images of ultracold plasma volumes produced by2:1 aspect ratio ellipsoidal Rydberg gases with selected initial state,40 f (2) after a flight time of 402 µs over a distance of 575 mm. Lowerframe displays the distribution in x of the charge integrated in y and z.Both images represent the unadjusted raw signal acquired in each caseafter 250 shots.In the wings, momentum redistribution owing to cycles of ion-Rydberg chargetransfer retards radial expansion [40, 64]. By redirecting electron energy fromambipolar acceleration to ±x plasma motion, NO+ to NO∗ charge exchange dis-sipates electron thermal energy. This redistribution of energy released in the18avalanche of the Rydberg gas to plasma, causes the ellipsoidal Rydberg gas tobifurcate [54, 57], forming very long-lived, separating charged-particle distribu-tions. We capture the electron signal from these recoiling volumes on an imagingdetector as pictured in Figure 1.6. Here, momentum matching preserves densityand enables ions and Rydberg molecules to relax to positions that minimize po-tential energy, building spatial correlation.The semi-classical description of avalanche and relaxation outlined above formsan important point of reference from which to interpret our experimental ob-servations. The laser crossed molecular beam illumination geometry creates aRydberg gas with a distinctively shaped high-density spatial distribution. This ini-tial condition has an evident effect on the evolution dynamics. We have developedsemi-classical models that explicitly consider the coupled rate and hydrodynamicprocesses governing the evolution from Rydberg gas to plasma using a realistic,ellipsoidal representation of the ion/electron and Rydberg densities [53]. No com-bination of initial conditions can produce a simulation that conforms classicallywith the state of arrested relaxation we observe experimentally.1.9 A molecular ultracold plasma state of arrestedrelaxationThus, we find that spontaneous avalanche to plasma splits the core of an el-lipsoidal Rydberg gas of nitric oxide. As ambipolar expansion quenches theelectron temperature of this core plasma, long-range, resonant charge transferfrom ballistic ions to frozen Rydberg molecules in the wings of the ellipsoidquenches the ion-Rydberg molecule relative velocity distribution. This sequenceof steps gives rise to a remarkable mechanics of self-assembly, in which the kineticenergy of initially formed hot electrons and ions drives an observed separation ofplasma volumes. These dynamics redistribute ion momentum, efficiently chan-neling electron energy into a reservoir of mass-transport. This starts a process thatevidently anneals separating volumes to a state of cold, correlated ions, electronsand Rydberg molecules.19We have devised a three-dimensional spin model to describe this arrested stateof the ultracold plasma in terms of two, three and four-level dipole-dipole energytransfer interactions (spin flip-flops), together with Ising interactions that arisefrom the concerted pairwise coupling of resonant pairs of dipoles [55, 56].The Hamiltonian includes the effects of onsite disorder owing to the broadspectrum of states populated in the ensemble and the unique electrostatic environ-ment of every dipole. Extending ideas developed for simpler systems [65, 66], onecan make a case for slow dynamics, including an arrest in the relaxation of NORydberg molecules to predissociating states of lower principal quantum number.Systems of higher dimension ought to thermalize by energy transfer that spreadsfrom rare but inevitable ergodic volumes (Griffiths regions) [67–70]. However,a feature in the self-assembly of the molecular ultracold plasma may precludedestabilization by rare thermal domains: Whenever the quenched plasma developsa delocalizing Griffiths region, the local predissociation of relaxing NO moleculespromptly proceeds to deplete that region to a void of no consequence.In summary, the classical dynamics of avalanche and bifurcation appear tocreate a quenched condition of low temperature and high disorder in which dipole-dipole interactions drive self-assembly to a localized state purified by the predis-sociation of thermal regions. We suggest that this state of the quenched ultracoldplasma offers an experimental platform for studying quantum many-body physicsof disordered systems.In the rest of this thesis we will explain the research I performed during thecourse of my PhD; Chapter 2 discusses the methodology we have created tocalibrate the initial Rydberg density at different sets of experimental parameters.In Chapter 3 we talk about the implementation of selective field ionization tech-nique to investigate the long-time dynamics of our molecular system (up to 20microseconds). Chapter 4 explains the effect of specific radio frequency electricfields on the quenched state of plasma. These are followed by a Conclusionschapter which summarizes the findings.20Chapter 2Calibration of the Initial Density ofthe Nitric Oxide Molecular RydbergGas2.1 IntroductionAt fixed energy the fundamental properties of a gas can vary profoundly withincreasing density, ranging from the appearance of non-ideality to a change inphase. For a plasma at fixed temperature of T the charged particle density, ρ ,determines the degree of correlation of ions and electrons, Γ, as given in Eq. 2.1.Γ=q24piε0awskBT(2.1)where q is the electric charge and aws is the Wigner-Seitz radius, related to theparticle density by:43pia3ws =1ρ(2.2)In the nitric oxide system, the initial density of Rydberg molecules directly21affects the avalanche dynamics through which the system evolves into molecularplasma. Density also affects the rate of some secondary processes such as `-mixing in a mixture of Rydberg state molecules and charged particles. Therefore,determination of the electron density proves critical for an understanding of thesystem behavior under varying experimental conditions.This chapter explains (1) the determination of the absolute electron densityand (2) classification of every Selective Field Ionization (SFI) spectrum for theinitial density of every shot.2.2 ExperimentalThe experiment uses the moving grid apparatus, shown in Figure 2.1, to performthe experiments presented in this thesis. A pulsed molecular beam, consisting ofNO premixed with He at a ratio 1:10, enters the vacuum chamber through a 0.5mm-diameter nozzle from a reservoir at stagnation pressure of 5 bars. Two PfeifferTMU 520 turbo pumps effect a differential pumping of two chambers separatedby a skimmer; we call the chamber before the skimmer the source chamber, andthe one after the skimmer the experimental chamber.In an ideal isentropic beam expansion, the centerline intensity proportionallyrelates to the nozzle flow rate by [46]:I0 =κpiF(γ)n0√2kT0m(pid24) (2.3)where κ and F(γ) are the peaking factor and heat capacity function. For ourcondition these have values of 1.98 and 0.51, respectively [71]. n0 and T0 representthe stagnation density and temperature, and d is the diameter of the gas nozzle. Ifwe consider a pure He gas beam under our experimental condition, this formulapredicts I0 = 9×1021 particles s−1sr−1. The gas load is limited by: (a) a GeneralValve solenoid that delivers 400 µs gas pulses to match the 10-Hz laser frequencyof the experiment; and (b) the skimmer, which is located 35 mm from the nozzleexit. Allowing for the attenuation by the skimmer, this flux yields an instantaneous22Figure 2.1: Schematics of moving grid apparatus. Co-propagating laserbeams, ω1 and ω2 cross a molecular beam of nitric oxide betweenentrance aperture G1 and grid G2 of a differentially-pumped vacuumchamber to form a dense gas of Rydberg state NO molecules.density of approximately 5×1014 particles per cm3 at the laser interaction region.Inside the experimental chamber, the molecular beam travels in the z-directionpassing through the entrance aperture of three grids (G1, G2, and G3) each heldat a desired potential. The transit time of the plasma after the laser pulse to thedetector is determined by the position of the moving grid assembly. It can bevaried from 0.5 to 40 microseconds.Two co-propagating Gaussian laser pulses, set as desired frequencies ω1 andω2, intercept this molecular beam in the x-direction. Photo-excitation results inthe formation of a dense Rydberg gas of NO between G1 and G2 as detailed inSec. 2.2.2.If not externally disturbed, Penning-ionization and collisional processes causethe ionization of a fraction of Rydberg molecules on a timescale of a few hundrednanoseconds, resulting in the formation of an ultracold molecular plasma. Thefraction of ionized Rydberg molecules is sensitive to the initial density of theexcited NO gas. The plasma co-exists with Rydberg molecules which have not23been ionized.This excited volume of Rydberg molecules and loosely bound ions and elec-trons travels down the chamber to reach a second grid, G2, which is set to groundpotential. The beam next encounters a third grid (G3), which in a typical exper-iment is set to 200 V. Three ceramic insulator rods keep G2 and G3 at a fixeddistance of 1 cm. This assembly results in a 200 V/cm DC electric field betweenG2 and G3, which liberates the loosely bound plasma electrons and field-ionizesthe remaining Rydberg molecules in the excited volume after it passes G2.Alternatively, we can apply a ramped electric field between G1 and G2 toextract electrons from the plasma/Rydberg packet during their transit. An impor-tant characteristic of this detection scheme is that states of lower electron bindingenergy are ionized earlier in the transit than states of higher electron bindingenergy. This technique is known as Selective Field Ionization (SFI). In effect,it allows the measurement of the binding energy distribution of Rydberg electronsand plasma electrons by analyzing the time-dependent field ionization signal. Iwill discuss the mechanism of SFI in Sec. 3.2.Electrons extracted by either the 200 V/cm potential between G2 and G3or by the ramp potential are accelerated towards a Multichannel Plate (MCP),manufactured by Jordan TOF Products Inc., that amplifies the electron countby a factor of 107. These electrons continue their path to an anode (set at 2kV), which is capacitively coupled to an Agilent DSO7052B, 500 MHz, 4 GSa/soscilloscope. A LabView data acquisition routine records the signal captured bythe oscilloscope. We refer to datasets corresponding to a single shot of the pulsedmolecular beam as a ‘trace’, and I will use this term throughout this thesis.2.2.1 Sequence of events in a typical SFI experimentFigure 2.2 displays the time sequence of events for a typical SFI experiment. Forsimplicity let us denote the first and second laser pulses byω1 andω2, respectively.∆tω2 represents the time delay between the two laser pulses. Then, a voltage rampselectively field ionizes the excited molecular system. ∆tRamp is the elapsed time24Figure 2.2: Sequence of events in a typical SFI experiment. The laser beams,ω1 and ω2, cross the molecular beam of NO with a specified delay,∆tω2 . An electric field ramp from 0 to 350 V/cm with a rise-time of1 µs (0.8 V/ns), applied after time ∆tRamp following ω2, ionizes theexcited molecular system.between the second laser and the onset of the ramp. The time variables ∆tω2 ,and ∆tRamp are set by a BNC Model 575 pulse/delay generator depending on therequirements of a respective experiment.2.2.2 UV-UV resonance production of a cold NO Rydberg gasThe excitation of NO to a Rydberg state of principal quantum number n0 occursin two steps shown schematically in Figure 2.3. The first laser of frequency ω1excites the NO molecules from the electronic ground state X 2Π1/2 (ν ′′ = 0, N′′ =1) to the intermediate A 2Σ+ (ν ′ = 0, N′ = 0) state.At the beginning of every experiment, we tune ω1 using a resonant two-photonionization process. When ω1 is in resonance with an allowed transition, theabsorption of two photons of frequency ω1 results in the ionization of electrons25Figure 2.3: Double UV resonance mechanism. Nd:YAG-pumped,frequency-doubled dye laser beams at, ω1 and ω2, pump groundstate NO first to the excited A 2Σ+ (ν ′ = 0, N′ = 0) state and then to aRydberg level with N = 1.that appears as a signal on the detector. Scanning ω1 over a range of frequenciesand plotting the corresponding two-photon ionization signal yields the blue spec-trum in Figure 2.4. This spectrum is well-understood as shown by the excellentagreement with a simulation by the software PGOPHER [72] for a temperatureof T = 2.7± 1K. We select the A 2Σ+ (ν ′ = 0, N′ = 0) state transition by tuningω1 to the Q11(1/2) peak. The three dominant peaks are due to transitions fromthe rotationless ground state of the NO molecule. Two of the four transitions lienearly on top of each other. From left to right (high to low frequency) the peaksare labeled as rR21, (qQ21 + qR11), and pQ11, as discussed in Appendix A.The second laser, ω2, drives a transition from this intermediate state to aselected Rydberg state of principal quantum number n0 dictated by selection rules.This particular N = 0 intermediate A-state assures that the higher excited Rydbergstates have a total angular momentum, neglecting spin, of N = 1. Among accessi-ble excited states, only those in the f series converging to NO+, X 1Σ+, N+ = 2have sufficient lifetime to form a Rydberg gas that can evolve to plasma [73, 74].26225.56 225.58 225.6 225.62 225.64 225.6600.20.40.60.811.2 Exp. dataSimulationFigure 2.4: Resonant two-photon ionization spectrum for the ω1 laser. Wetune ω1 to the pQ11 transition.Our two lasers originate from pumped dye laser and are frequency doubled. Theintensity of ω1 laser beam is attenuated through a set of Glan-Thompson polarizer.The shape of this laser beam is optimized by means of a 50-micrometer spatialfilter.2.2.3 Ramped electric field to extract the electronsTo select the correct frequency ω2, we first perform SFI experiments for a rangeof frequencies of the second laser. The resulting integrated detector signal ina typical experiment is plotted in Figure 2.5 as a function of wavenumber ofthe second laser. Principal quantum numbers of some transitions related to thisrange of wavenumbers are shown on top of their corresponding peaks. Each peakcorresponds to a Rydberg state of different quantum number n0. The frequencies273.048 3.049 3.05 3.051 3.052 3.053 3.05410400.511.5Signal (arbitr.)Figure 2.5: Integrated detector signal as a function of second laserwavenumber. Principal quantum numbers of some transitions relatedto this range of wavenumbers are shown on top of their correspondingpeaks.of each peak show the 1/n2 behavior to good approximation where we haveneglected the quantum defect; an expected characteristic of molecular systemsexcited to high Rydberg states. Before each experiment, the data in Figure 2.5 isrecorded and analyzed in real-time using a LabVIEW routine to find the principalquantum number, n0, of the Rydberg state that is populated by the second laser.2.2.4 Two ways to control the Rydberg gas densityThe intermediate A-state, has a radiative lifetime of τ = 192 ns [75]. Therefore,the population of the A-state, PNO, decays asPNO(t) = P0e−t/τ . (2.4)28where P0 is the initial population of the NO molecules excited to the A-state,and t is the elapsed time after the first laser pulse. We use the A-state lifetimeto experimentally control the density of Rydberg state molecules: Since ω2 isonly resonant with the transition from the A-state to a selected Rydberg state ofprincipal quantum number n0, the exponential decay of the A-state will be directlyreflected in the Rydberg population density.We can also control the density of Rydberg molecules by the power of thelaser tuned to ω1. We vary the pulse energy of the laser tuned to ω1 to controlthe number of molecules in the A-state and therefore the number of singly-excitedmolecules available for the second transition to the Rydberg states. For the limitof saturated second transition, if the ω1 pulse energy is high enough to saturatethe first transition, the density of the Rydberg states is maximum. If we reduce ω1pulse energy, the density of final Rydberg state molecules also decreases.2.3 Calculating the maximum possible densityOur current experimental setup does not provide any direct means to measure theabsolute density of excited NO molecules in the excited volume. However, it ispossible to make a good estimate for the maximum density of Rydberg moleculesand plasma electrons.As discussed in Section 2.2, the thermodynamic model of the supersonic molec-ular beam predicts the centerline particle density in the molecular beam to beρbeam = 5× 1014 cm−3. The fraction of NO is the ratio of NO in He, whichin our experiment is f = 0.1. At a rotational temperature of 3 K, pg = 87% ofthese molecules populate the two parity components of the rotational ground state.Two saturated steps of laser excitation transfer pr = 12.5% of this populationto a parity selected high-Rydberg state (We only access one of the two paritycomponents in a given experiment, and each saturated step of laser excitationonly transfers 50 percent of the population, hence pr = 12.5%). In combination,these factors predict the maximum density of excited NO molecules available for29plasma formation to beρRydberg = ρbeam f pg pr ≈ 5×1012cm−3 .ρRydberg estimates the peak density at the center of the beam. Away from thecore, the density falls off according to the Gaussian profile of the molecular beam.Together with the Gaussian profile of the laser, this yield the distribution in allthree spatial dimensions asρRydberg(x,y,z) = ρbeam f pg pr exp[− x22σ2x− y22σ2y− z22σ2z]with σx = 0.75mm, σy = 0.425mm, and σz = 0.425mm.Now in order to obtain the density of the system at any given experimental con-dition (i.e. not necessarily having maximum density), let us make the hypothesisthat at a given ramp delay the detector signal at a specific ramp voltage is directlyproportional to the number of electrons which are extracted by the correspondingelectric field. This means that for a constant ramp delay, the total signal collectedby a single SFI ramped field pulse, is directly proportional with the initial densityof the Rydberg gas, ρRydberg. In forming this hypothesis, we assume that all theelectrons excited to Rydberg orbitals at any given flight time remain extractablefrom the excited volume, either as plasma electrons (bound by the space charge),high-n Rydberg states, or residual n0 Rydberg state electrons. In what follows, weare going to investigate the validity of this hypothesis.2.4 ResultsIn a typical SFI experiment the integrated signal varies substantially after eachdouble resonant photoexcitation, even if the experimental parameters are not changed.As an example, figure 2.6 depicts a histogram of the total integrated SFI signal inan experiment where the state with Rydberg principal quantum number n0 = 44has been selected. The horizontal axis shows the upper limit of each bin and the30Figure 2.6: Histogram of 1000 SFI traces, with the initial principal quantumnumber of n = 44, the first laser pulse energy of 6 µJ and no delaybetween the two lasers. The second laser was set at 8 mJ, which isabove the saturation limit for second electron transition.vertical axis shows the number of traces whose integrated values fall within eachbin. The energy of the ω1 pulse was measured at 6 µJ, and ∆tω2 = ∆tRamp = 0. Atotal of 1000 SFI traces were collected.As we delay the second laser, the mean value and the width of the distributiondecreases as shown on the left side in Figure 2.7. For this figure, we collected5000 traces of SFI and sorted them based on their integrated signal values fromthe top (the highest value) to the bottom (the lowest value). The principal quantumnumber for all data in Figure 2.7 is n0 = 44. The energies of the laser pulses are6 µJ and 8 mJ for ω1 and ω2, respectively. The ramp delay is ∆tRamp = 0 and31Δtω20 ns50 ns100 ns150 ns250 ns0 50 100 150 200 250Electric Field  V/cmFigure 2.7: SFI false color plot consisting of 5000 SFI traces collected whilechanging the delay between the two lasers. On the left the histogramsrelated to different delays between the two lasers are shown. As wedelay the second laser, the mean value and the width of the distributiondecrease.the delay time ∆tω2 between the two lasers is varied from 0 to 250 ns, where each1000 traces correspond to a different ∆tω2 . We have shown the histograms relatedto ∆tω2 = 0, 50 ns, 100 ns, 150 ns and 250 ns on the left side of the false colorplot. One can see that the density distributions shift to lower densities and becomenarrower for larger ∆tω2 delay.Now let us compute the mean of the distribution of total integrated signalfor different ∆tω2 . The density of excited molecules follows the well-definedexponential relation Eq. 2.4. If the relationship between integrated signal anddensity is in fact linear, we should see this specific exponential decay as a functionof ∆tω2 in the total integrated signal as well.Figure 2.8 shows this relationship for the case of ω1 pulse energy of 6 µJ.32Figure 2.8: Relationship between the SFI integrated signal and the delaybetween the two laser pulses. The SFI integrated signal undergoes anexponential decay as the delay between the two lasers increases. Thetime constant of this exponential decay matches the radiative lifetimeof A-state of Nitric oxide. The data were collected with ω1 pulseenergy of 6 µJ.In this figure, each point shows the arithmetic mean of the SFI integrated signalin the 1000 SFI traces collected at each ∆tω2 , and error bars indicate standarddeviations. The dashed line shows an exponential decay with timescale of 192ns which is the radiative lifetime of the A-state. Notice that the data is consistentwith this exponential behavior.We repeated the same experiment for different values of ω1 pulse energy, from2 to 12 µJ, and obtained similar results. In all cases an exponential decay of thesignal with lifetime of 192 ns is in agreement with the data.However, for ω1 pulse energies exceeding 12 µJ, the data deviates from theexponential fit. Figure 2.9 shows a representative data set for ω1 pulse energy of20 µJ. We have performed additional experiments at 15, and 20 µJ. For these ω1pulse energies, it is not possible to describe the data with an exponential decay33Figure 2.9: Relationship between the SFI integrated signal and the delaybetween the two laser pulses. The data were collected with ω1 pulseenergy of 20 µJ. For such a high laser energy, the data do not followthe exponential decay when the delay between the two laser pulses isless than 150 ns.consistent with 192 ns radiative lifetime of the A-state. We have shown the setof all histograms related to these measurements in Figure 2.10. The axes limitsand the bin boundaries for all these histograms are the same (with the exceptionof top right frame). Notice for the same value of ∆tω2 , increasing the first laserpulse energy moves the mean value of the histogram towards higher values, andalso increases the width of the distribution. On the other hand, for any given laserpulse energy increasing ∆tω2 results in the shift of the histogram towards lowervalues and narrower distributions. This plot summarizes a total of 40,000 tracesat different laser pulse energies and second laser delays. For easier comparison2.11 shows a subset of these histograms. Appendix B presents version of thesehistograms enlarged to show greater detail.For the results shown so far, the ramped voltage was applied immediately afterthe second laser, i.e. ∆tRamp = 0. One may ask what would happen to the total34ω2 Delay ω1 Energy 0 ns 50 ns 100 ns 150 ns 250 ns4 uJ6 uJ8 uJ2 uJ10 uJ12 uJ15 uJ20 uJFigure 2.10: Set of all histograms related to density calibration experimentby the SFI technique. Each row shows the distribution of SFIintegrated signal for a fixed ω1 pulse energy and different ω2 delays.Each column, on the other hand, shows that distribution for a fixedω2 delay, and different ω1 pulse energies. Notice the change inthe means and standard deviations of the distributions as the initialRydberg density is changed through either first laser energy or thesecond laser delay. 3512 μJ 4 μ J6 μ J8 μ J10 μ Jω1 Energy 15 μ J0 ns 50 ns 100 ns 150 ns 250 nsω2 Delay 0 nsFigure 2.11: A subset of histograms showing the change in the total SFI in-tegrated signal distribution as the initial Rydberg density is changed.On the top row, the density is changed through the first laser energy,while the delay between the two laser pulses is zero for all frames.At the bottom, the density is changed through the delay between thetwo lasers, while the first laser energy is kept the same. Both thesemethods of changing densities result in the same trend of change inthe distributions; as we can see in each row the density is decreasedfrom left to right, resulting in the mean of the distribution to movetowards lower integrated signal values.36Figure 2.12: Relationship between the SFI integrated signal and the delaybetween the two laser pulses for the case of non-zero ∆tRamp. Thedata were collected with ω1 pulse energy of 6 µJ and ∆tRamp = 500ns. The data shows that the exponential decay with a time constantwhich matches the radiative lifetime of A-state of Nitric oxide alsoholds for non-zero fixed ramp delays..integrated SFI signal if we use a non-zero ∆tRamp, and whether or not this causesthe signal to decay differently with ∆tω2 , compared to the case of ∆tRamp = 0. Toanswer these questions, we repeated the SFI experiments for the case of ∆tRamp =500 ns. The results are shown in Figure 2.12.Figure 2.12 shows that the exponential decay of the integrated SFI signalwith a time constant of 192 ns also holds for the case of non-zero ramp delays.Moreover, comparing Figure 2.8 with Figure 2.12, notice that the absolute totalintegrated signal drops when we increase the ramp delay.372.5 DiscussionDue to various experimental factors, the calculated maximum density of excitedmolecules is not always achieved. Incomplete opening of the pulsed nozzle couldlimit flow, reducing the density of the molecular beam. Laser beam intensityinhomogeneities could give rise to regions of the excited volume in which the firstor second steps in the excitation sequence fail to saturate. Moreover, possible fluc-tuations in the wavelength of the first laser as well as imperfect overlap betweenthe laser and the molecular beam could be among these experimental causes whichtend to reduce the initial Rydberg density.These effects lead to shot-to-shot fluctuations of the SFI signal, which generatea distribution of SFI integrated signal as shown in the histograms of Figures 2.6and 2.7. This results in some overlap between the data collected at different∆tω2 , however Figures 2.8 and 2.9 show that the mean value of SFI integratedsignal changes with ∆tω2 . The results in Figure 2.8, shows that the SFI integratedsignal decreases with ∆tω2 along with the lifetime limited decrease in A-state NOmolecules, that is, the integrated signal decreases exponentially in ∆tω2 . Thismeans that the SFI integrated signal is linearly proportional to the number ofmolecules excited to the A-state.The power of the second laser (which excites A-state molecules to high Rydbergstates) was kept constant throughout these experiments. Therefore, we can say thatthe SFI integrated signal is linearly proportional to the number of Rydberg statemolecules in our system, and in turn to the density of our excited system. In ourexperiment, this is the case for values of ω1 pulse energies up to 12 µJ.Thus knowing the upper limit of density in our system, ρ0 = 5× 1012 cm−3,when both lasers saturate the two transitions, we can use SFI to determine thedensity at different delays between the two lasers. For example, if we perform anSFI experiment in which we scan the delay of ω2 from zero to about 1 µs (i.e. fivetimes the A-state radiative lifetime), we have in fact scanned the initial Rydbergdensity from ρ0 to ρ0/e5 (from 5×1012cm−3 to ∼ 1010 cm−3).However, if the energy of the first laser pulse is higher than 12 µJ, and for short38delays between the two lasers, the SFI integrated signal deviates from the expectedexponential behaviour. We believe this is due to the high space charge densityproduced by ω1 two-photon ionization when ω1 energy is high and ω1−ω2 delaytime is short; The trapped electrons interact with orderly avalanche if Rydbergmolecules are formed right after ω1 laser pulse. After approximately 100 ns, theeffect of these space charge electrons dissipates, hence the agreement of data withthe expected exponential decay curve.The histograms shown in Figure 2.10, shows the change in the density of ourmolecular system both as a function of laser pulse energy and the delay betweenthe two lasers. looking at each column on this plot (i.e. same delay, and differentlaser energies) one can see that increased laser pulsed energy causes the his-tograms to shift towards higher densities. On the other hand, looking at each row(fixing the laser pulse energy and changing the second laser delay), it is obviousthat higher delays cause the histograms to shift towards lower densities. Thesebehaviours are expected and consistent with the assumption that the integratedsignal is directly proportional to charged particle density in our system.Now, let us return to the assumptions of Section 2.3. Because the mean ofintegrated SFI integrated signal follows the A-state radiative lifetime for differentω1 laser pulse energies (up to 12 µJ), we can say the first electron loss mechanism(i.e. escape of electrons in the field-free interval) does not vary strongly withinitial Rydberg gas density.Also, we have observed that the integrated signal (and therefore the electrondensity) decreases when the system evolves for some time (i.e. non-zero ∆tRamp)before field ionization. This directly reflects the second electron loss mechanism,predissociation. Nevertheless, Figure 2.12 shows that the density calibration holdsfor non-zero ramp delays. Therefore, we can conclude that the rate of predissoci-ation does not depend on the initial Rydberg gas density, as expected for a simpleunimolecular decay.39Chapter 3Long-time Dynamics of theNitric-oxide Molecular System3.1 IntroductionIn the previous chapter, we discussed the details of how we use a double UVresonance to produce cold NO Rydberg gas, how we control the density of thisgas, and how we extract and detect the electrons with a ramped electric field.There we showed how ∆tω2 enables the precise characterisation and calibration ofthe initial Rydberg density (Section 2.3).In this chapter we study ∆tRamp which determines the evolution time of theRydberg gas until ionization by the ramped electric field. The use of selective fieldionization delay probes the evolution of the excited system over long times. Be-fore discussing the experimental results, this chapter considers previous findingsthat motivate this study. We also discuss the physics of selective field ionizationin more detail.An explicit demonstration of the exceedingly long plasma lifetime is shownin Fig. 3.1. This experiment was performed without a ramped ionizing electricfield. Instead, the volume of excited gas travels with the laboratory velocity of themolecular beam to transit grid G2, where it encounters a static potential applied to400"0.2"0.4"0.6"0.8"1"1.2"0 5 10 15 20 25 30 35 40 Integrated electron signal (arb.) time (µs)  50!40!30!20!10!0Electron signal (arb.)0                   10                   20                   30                   40                time (µs) Integrated electron signal (arb.)0                   10                   20                   30                   40                time (µs)Distance to G2 (mm)Figure 3.1: (upper frame) Set of traces showing the charge density and widthin z of a plasma formed by avalanche from a 50 f (2) Rydberg gas as afunction of flight time for a sequence of flight distances. (lower frame)Integrated plasma signal obtained as the areas of the traces in the upperframe, plotted as a function of arrival time at G2. Electron loss in thetraveling excited volume seems to cease after 7 µs.41G3, set typically to a value between 30 and 200 V cm−1, see Fig. 2.1. The electronsignal produced as the excited volume transits G2 forms a late peak that traces thewidth of the evolving plasma in the z coordinate direction of propagation.The top panel of Fig. 3.1 shows late-peak traces of electron signal for variousflight times. Integrating these late-peak traces gives the integrated electron signalplotted, as a function of evolution time, in the bottom panel which is directlyproportional to the number of plasma electrons. Over the first 10 µs, neutraldissociation processes combined with expansion reduce electron temperature andthe ion-electron density. Thereafter, plasma decay apparently ceases, yielding anintegrated signal of constant area. Extended measurements confirm that this signalpersists undiminished for evolution times as long as 40 µs as shown in Fig. 3.1.Electron signal measurements conducted on a different, longer flight pathimaging instrument yield strong waveforms of equal area after flight times of 200and 400 µs, suggesting a plasma lifetime substantially in excess of 1 millisecond[54]. These results can not be explained using classical rate equation models.The extremely long lifetime of the plasma as discussed above motivates ourinterest in the state of the ultracold plasma: a dense, highly correlated molecularplasma that ceases to evolve in a state far from thermal equilibrium [54]. Whatare the events that lead to the formation of this state? What can we learn about itsarrested dynamics? To answer these questions, this Chapter considers SelectiveField Ionization experiments that detail the early-time dynamics of plasma forma-tion as well as system evolution up to 20µs. For the most part, the experimentalmethod for the results presented in this chapter is identical to that explained in2.2. Modifications are discussed at the beginning of the relevant sub-section.3.2 Physics of selective field ionizationDue to relatively low electron binding energies, atoms and molecules excited inhigh Rydberg states are susceptible to external electric fields. In a completely clas-sical approach (i.e. no consideration of the Stark effect), ionization occurs whenan external electric field lowers the potential barrier of a bound electron enough42Figure 3.2: Schematic demonstration of classical field ionization of theRydberg state. The external electric field perturbs the field freepotential such that there is a saddle point (SP) at zsp =√e/4piε0E.for the electron to escape. This is schematically shown in figure 3.2. Assumingthe potential difference between grids G1 and G2 create a uniform electric fieldalong the z-axis, the hydrogen potential along this axis will be perturbed accordingtoW (z) =− e24piε01|z| − eEz , (3.1)where E = −∆V/(1 mm) and ∆V is the potential difference between two neigh-boring grids. W (z) has a saddle point (SP) at zsp =√e/4piε0E which is calcu-lated through solving the equation ∂W/∂ z = 0. At this saddle point the loweredpotential is W (zsp) =−2√e3E/4piε0. If this reduction is larger than the classicalbinding energy of the electron in a Rydberg state −Ry/n2, the system is ionized.The field strength required to ionize the electron in a Rydberg state with principalquantum number n can then be calculated by setting these two energies equal to43Figure 3.3: Ramped time-dependent electric field, applied to grid G1 in atypical SFI experiment. The voltage ramp is formed through pulsing a3 kV DC voltage into a well-defined RC circuit. It rises at a rate of 0.8V/ns, and almost linearly with time for the first 800 ns.each other:EFI(n) =(Ry)2piε0e3n4(3.2)In an SFI experiment we apply an electron-forward-bias (negative) voltage rampto G1, with a specified delay after the second laser pulse ∆tRamp. In experimentsdescribed in this chapter the ramp delay will be consistently ∆tRamp = 0. Thevoltage ramp, formed by pulsing a 3 kV DC voltage into a well-defined RC circuit,rises at a rate of 0.8 V/ns, and almost linearly with time for the first 800 ns. Themeasured time dependence is shown in Figure 3.3.Rydberg states of principal quantum number n are ionized at a time t∗ whenthe ramped field ERamp(t) between G1 and G2 reaches the threshold defined inEq. 3.2:ERamp(t∗) = EFI(n) . (3.3)The time-dependent character of this electric field therefore allows to measure thebinding energy of electrons in the time domain.44Figure 3.4: A rising electrostatic field collects the extravalent electrons ina Rydberg-plasma volume, yielding a selective field ionization (SFI)spectrum. Loosely-bound electrons from plasma appear at low field,until about 80 V/cm. The two peaks near 125 V/cm and 190V/cm correspond to the dominant ionization pathways for a Rydbergpopulation prepared in the 44f(2) state, the N+ = 0 and N+ = 2 statesof the ion.A typical SFI spectrum of the detected electron signal as a function of rampedelectric field is shown in Figure 3.4. The spectrum shows a large peak at 0 toaround 80 V/cm. The free electrons bound by the space charge of a low-densityplasma separate from the ions at an early point in a typical field ramp [36],affording a field ionization trace that differs very little from the SFI spectrumof a Rydberg gas of very high principal quantum number. The peak at small fieldsmay therefore represent weakly bound electrons in either of two states: loosely45bound plasma electrons or highly excited Rydberg states. For simplicity, we willrefer to this low-field peak as the plasma signal.The binding energy resolution of SFI spectra allows us to calculate the densityof plasma electrons (or electrons is high-n Rydberg states) and the density of un-ionized Rydberg states, separately. For this, we define a threshold of ∼ 80 V/cm.Integrating the detector signal from 0 to ∼ 80 V/cm gives the plasma density,integrating the remaining contribution for fields higher than ∼ 80 V/cm gives thedensity of Rydberg molecules that do not take part in plasma formation.The ionization of Rydberg molecules of principal quantum number n0 occursat much higher fields, in the range 80-275 V/cm. They give rise to two separatepeaks at high fields in Figure 3.4.The question may arise here that how one selected Rydberg state gives twoSFI features? This brings up the topic of adiabatic vs diabatic field ionizationthresholds. When we expose our system to an external electric field, Stark effectcauses different quantum states to split into the so-called Stark manifold andtherefore at certain field strengths their energy levels cross unless intramolecularcoupling causes these states to form avoided crossings. If the ramp rise time (i.e.the slew rate) is not high enough for diabatic ionization, the electrons have time totravel onto different states at these avoided crossings and find their way to a lowerionization potential (i.e. adiabatic field ionization).Under the influence of an electric field applied at our slew rates, these statesundergo diabatic passage [76] to produce an electron and an ion core with angularmomentum N+ = 2. Field ionization can also leave the ion in the N+ = 0 state,again following a dominantly diabatic path (however, there must be at least oneadiabatic crossing to get to the N+ = 0 ion core rotational threshold). The energydifference in these two thresholds causes electrons to appear at two differentamplitudes of the ramped electrostatic field.460                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 20                50            100           150            200           250 Field  (V cm-1)0  (cm-3)101010111012N+ = 0 N+ = 2Figure 3.5: (left) Selective field ionization spectra of NO: Contour plots showingSFI signal as a function the applied field for an n f (2) Rydberg gas with aninitial principal quantum number, n0 = 44. Each frame represents 4,000 SFItraces, sorted by initial Rydberg gas density. Ramp field potential, beginning0, 150, 300 and 450 ns after the ω2 laser pulse for the top left, top right,bottom left, and bottom right respectively. The two bars of signal mostevident at early ramp field delay times represent the field ionization of the44 f (2) Rydberg state respectively to NO+ X 1Σ+ cation rotational states,N+ = 0 and 2.473.3 Results3.3.1 Selective field-ionization spectrum as a probe of therelaxation from Rydberg gas to plasmaFigure 3.5 shows contour plots representing scaled SFI signal as a function ofapplied electric field for an n0 f (2) Rydberg gas with an initial principal quantumnumber, n0 = 44. In order to obtain each of these frames, one needs to change ∆tω2stepwise from zero to 1 µs while recording all the oscilloscope traces. Then thetraces are sorted based on the integrated signal of each trace (in other words basedon the initial densities). Each of the frames in Fig. 3.5 consists of 4000 SFI traces.The four frames shown in this figure investigate the effect of ∆tRamp, starting fromzero, then 150, 300, and 450 ns after the second laser pulse. The two white lineson each plot represents the field ionization threshold of the 44 f (2) Rydberg staterespectively to NO+ cation rotational states, N+ = 0 and N+ = 2. Notice that aswe increase the ramp delay three changes can be observed: the Rydberg signalreduces, the plasma signal increases, and the two Rydberg features are movedtowards higher fields.Now, let us see how SFI plots of Rydberg molecules with different initial prin-cipal quantum numbers compare to one another. Figure 3.6 shows the SFI contourplots for three different initial principal quantum numbers (n0 = 40, 44, 49), andfour different ramp delays (∆tRamp = 0, 150, 300, and 450 ns). Comparing theplots in each row, we can see that for a given ramp delay, the only major differencebetween the experiments initiated in different principal quantum numbers is theelectric field threshold at which the Rydberg molecules are field ionized. As theinitial principal quantum number increases, the electric field required to ionize theRydberg molecules is reduced.48Figure 3.6: SFI spectra for principal quantum numbers n0 = 40,44,49 atdifferent ramp delays ∆tRamp = 0,150,300,450 ns. Comparing theplots in each row shows that as the initial principal quantum numberincreases, the electric field required to ionize the Rydberg moleculesis reduced.493.3.2 Long-time dynamics of the NO system probed with SFINow that we have discussed the SFI technique and its application in our exper-iment, let us investigate the evolution of our molecular system at even longertimes. To perform this experiment, we changed the carrier gas from helium toargon. Since a molecular beam seeded in argon reaches a lower terminal velocitythan one seeded in helium, it allows our excited molecular system to travel for alonger time before it reaches the end of flight path, which is the second grid in theapparatus. Explicitly, in a field free experiment argon affords flight times of up to40µs in our moving grid apparatus. However, in an SFI experiment for a systeminitially excited to principal quantum numbers in the range of 40 to 60, we cannotmove the second grid too far from the first one. This is because the electric fieldwhich is produced by the ramped voltage is inversely proportional to the spacingbetween the two grids, G1 and G2. If we separate the two grids too much, themaximum electric field produced by the maximum ramped voltage will be belowthe ionization threshold of the Rydberg states in this range. Another point whichis worth mentioning is that the peak density achieved with a given ω1 pulse energyand second laser delay, ∆tω2 , in argon is less than that in helium. Therefore, forthe results presented in this section, we only used ∆tω2 = 0, which produces themaximum possible density at a given ω1 pulse energy.Figure 3.7 shows a series of SFI plots with ∆tRamp = 0,1,5,10,15, and 20µsfor a selected principal quantum number of n0 = 51. We chose ω1 laser pulseenergy of 4µJ in this set of experiments. Each plot corresponds to 3000 scopetraces recorded at a specific ramp delay ∆tRamp and ω1 pulse energy Pω1 displayedin each frame’s upper right corner. The delay between ω1 and ω2 pulses was heldas ∆tω2 = 0, as mentioned before. Different densities are therefore caused onlyby the fluctuating factors discussed in Section 2.5. The traces were sorted basedon their total integrated signal and then divided into 100 bins such that each bincontained an equal number of traces. The average trace of each 30 traces in a binwas computed and then divided by its integral. These averaged and scaled traceswere then plotted as rows in false-color plots where the y-coordinate of a row was50given by the total integrated signal (before normalization), on a logarithmic scale.This method of scaling (i.e. dividing each averaged trace by its integral) magnifiesthe features at lower densities, helping with visualization purposes. Therefore, oneshould be careful not to compare the absolute values of features on different partsof each frame which are vertically apart, as the scale factors would be different.From Fig. 3.7 one can see that the Rydberg signal falls as the Ramp delay isincreased. However, notice that this Figure shows measurable Rydberg signalseven after 20µs, the longest recorded ∆tRamp in our experiments. This extremelylong lifetime narrows the possibility that conventional dynamics is responsible forthe evolution of our excited system.We repeated the same set of experiments for other values of ω1 pulse energyranging from 1.5µJ to 10µJ. For comparison, Figures 3.7 through 3.10 demon-strate the results for Pω1 = 4µJ, 8µJ, and 10µJ, respectively (the results forPω1 = 1.5µJ was similar, but with a lower signal to noise ratio. Since it wouldhave not added any extra value to the present discussion, it was decided to notinclude it in this set of figures). From these plots one can observe that the numberof Rydberg state molecules that survive at long times decreases as ω1 pulse energyis increased. Namely, at ω1 = 10µJ almost no Rydberg molecules remain.To investigate this in more detail, first consider Figure 3.11. For each figure,we collected 3000 traces at zero delay between the two laser pulses (i.e. ∆tω2 = 0),and at a specific ramp delay ∆tRamp and then averaged these traces into a singleone. Here, we do not scale the traces (not dividing the averaged trace by itsintegral), which means the absolute values are comparable. We also defined athreshold at 50 V/cm that separates loosely bound plasma electron signal fromthe Rydberg signal at higher binding energies. The detector signal in the bindingenergy range of 50 to 220 V/cm is treated as Rydberg signal. Therefore, asdiscussed before, integrating the signal up to or from 50 V/cm gives a quantitythat is proportional to the number of plasma electrons or Rydberg molecules,respectively. In Fig. 3.11, the plasma signal is shown as the red-colored areaand the Rydberg signal is colored in blue.51Figure 3.7: Normalized SFI plots for principal quantum number n = 51 andPω1 = 4µJ at long ramp-delays ∆tRamp.52Figure 3.8: Normalized SFI plots for principal quantum number n = 51 andPω1 = 6µJ at long ramp-delays ∆tRamp.53Figure 3.9: Normalized SFI plots for principal quantum number n = 51 andPω1 = 8µJ at long ramp-delays ∆tRamp.54Figure 3.10: Normalized SFI plots for principal quantum number n= 51 andPω1 = 10µJ at long ramp-delays ∆tRamp.55Following this method of signal integration, Figure 3.12 shows the composi-tion of our excited volume as a function of ramp delay. In this figure the black linesshow the total integrated signal (including both plasma electrons and Rydbergelectrons). The red curves show the plasma portion of the excited volume and bluecurves show the electrons extracted from the Rydberg molecules in the plasma.The set of frames on the left hand side show the absolute values of the integratedsignals at each ramp delay, while the set of frames on the right hand side showthe normalized plasma and Rydberg fractions, obtained by dividing each curve bythe value of black curve at any given ramp delay. These frames show that as theω1 pulse energy is increased, the initial total signal increases (the black curve).However, notice that the initial total Rydberg signal (the blue curve) increaseswhen we change the first laser pulse energy from 1.5 to 4 µJ (from 2000 to 4000on the arbitrary units of the plots), and then stays constant regardless of how higha laser pulse energy we use. Instead the plasma portion (the red curve) of signalincreases constantly at higher laser energies.Now consider Figure 3.13, in which we have plotted the integrated Rydbergsignal as a function of ω1 laser pulse energy, for different values of ramp de-lays, ∆tRamp. When Rydberg molecules are collected at zero ramp delay (theblue curve), the signal increases with ω1 pulse energy approaching a saturation.Looking at the curve related to ∆tRamp = 1µs (the red curve), we can see that thecurve initially rises. However, increasing the laser pulse energy further leads to anoverall reduced Rydberg signal. This is in contrast with the case of ∆tRamp = 0.Keep in mind we are now allowing the system to evolve for 1µs before we extractthe electrons with the ramped field. Under this experimental condition and thisramp delay, 4µJ proves to yield the maximum number of Rydberg molecules.The system behaves similarly for ramp delay of 5 microseconds. However, if wedelay the ramp further the Rydberg signal always falls, as we increase the laserpulse energy.Finally let us consider the change of Rydberg signal as a function of rampdelay. Figure 3.14 shows the Rydberg signal, normalized to 1 at zero ramp delay,560 50 100 150 200 250Ramp Field (V/cm)00.511.52Detector signal (arb. units)tRamp= 0 sP1= 4 JDetector signalPlasmaRydbergs0 50 100 150 200 250Ramp Field (V/cm)00.511.52Detector signal (arb. units)tRamp= 1 sP1= 4 JDetector signalPlasmaRydbergs0 50 100 150 200 250Ramp Field (V/cm)00.511.52Detector signal (arb. units)tRamp= 5 sP1= 4 JDetector signalPlasmaRydbergs0 50 100 150 200 250Ramp Field (V/cm)00.511.52Detector signal (arb. units)tRamp= 10 sP1= 4 JDetector signalPlasmaRydbergs0 50 100 150 200 250Ramp Field (V/cm)00.511.52Detector signal (arb. units)tRamp= 15 sP1= 4 JDetector signalPlasmaRydbergs0 50 100 150 200 250Ramp Field (V/cm)00.511.52Detector signal (arb. units)tRamp= 20 sP1= 4 JDetector signalPlasmaRydbergsFigure 3.11: Classification of scope traces as plasma signal (red) andRydberg signal (blue). Each plotted trace corresponds to the averageof 3000 recorded traces at ∆tω2 = 0. Classification is based on bindingenergy. Signal below 50 V/cm is treated as plasma signal whereassignal in the range of 50−220 V/cm is classified as Rydberg signal.570 5 10 15 20Ramp delay in s02000400060008000Signal (arb. units)P=1.5 JPlasma signalRydberg signalTotal signal0 5 10 15 20Ramp delay in s00.51Normalized signalP=1.5 JPlasma fractionRydberg FractionTotal0 5 10 15 20Ramp delay in s02000400060008000Signal (arb. units)P=4 J0 5 10 15 20Ramp delay in s00.51Normalized signalP=4 J0 5 10 15 20Ramp delay in s02000400060008000Signal (arb. units)P=8 J0 5 10 15 20Ramp delay in s00.51Normalized signalP=8 J0 5 10 15 20Ramp delay in s02000400060008000Signal (arb. units)P=10 J0 5 10 15 20Ramp delay in s00.51Normalized signalP=10 JFigure 3.12: Total plasma and Rydberg signal (left column) and the decom-position of total signal into these two components (right panel) as afunction of ramp delay for different ω2 pulse energies.581 2 3 4 5 6 7 8 9 101 power in  J050010001500200025003000350040004500Rydberg signal0 s1 s5 s10 s15 s20 s tRampFigure 3.13: Average Rydberg signal as a function of ω1 power for variousramp delays. The Rydberg signal is obtained by integrating anexperimentally measured trace (signal as a function of ramped field)from 50V/cm to 220V/cm. For the case of zero ramp delay (theblue curve), the signal increases with ω1 pulse energy approachingsaturation. If we allow the system to evolve for 1µs before we extractthe electrons with the ramped field (i.e. the red curve), the averageRydberg signal initially rises. However, increasing the laser pulseenergy further reduces the Rydberg signal. If we delay the rampmore than 5 µs, however, the Rydberg signal is always reduced, aswe increase the laser pulse energy.590 5 10 15 20Ramp delay in s10-210-1100Normalized Rydberg signal1.5 J4 J6 J8 J10 JFigure 3.14: Average Rydberg signal as a function of ramp delay for variousω1 laser pulse energies. The pulse energies of 1.5, 4, 6, 8, and10 µJ correspond to initial Rydberg state molecule densities of9×1011, 2×1012, 3×1012, 4×1012, and 5×1012 cm−3, respectively.The Rydberg signal is obtained by integrating each experimentallymeasured trace from 50V/cm to 220V/cm. It was normalized suchthat all signals are 1 at ∆tRamp = 0.on a logarithmic scale, versus the ramp delay. This figure shows that for thisexperimental condition, and for ω1 pulse energies of equal or less than 8µJ,there is a residual Rydberg signal even at ramp delays as long as 20µs. At veryhigh initial densities achieved by Pω1 = 10µJ almost all Rydberg molecules decaywithin the first 15µs.603.4 DiscussionLooking at one of the frames in Figure 3.5 one can easily observe the effectof density on SFI signal; for an experiment with initial densities in the orderof 1012 cm−3, the largest component of the signal comes from free or looselybound electrons. This shows that at high densities, the initially excited Rydbergmolecules undergo Penning ionization and, on a nanosecond timescale, a largefraction of Rydbergs is ionized to form a plasma. However, as the density isreduced to 1011 cm−3, the largest component of the signal is from the field ioniza-tion of Rydberg molecules, suggesting that the initial density is too low to yieldprompt Penning ionization and electron cascade.As we delay the ramp to longer times after the second laser pulse, the intensityof the Rydberg signal is reduced and instead we observe a rise in the plasma signalintensity as shown in Figure 3.5. This shows the avalanche dynamics and theevolution of the initial molecular Rydberg gas to plasma. Figure 3.5 also showsthat the two Rydberg signals shift towards higher electric fields if we increase theramp delay to 300 ns. This signal shifts to higher field with increasing ramp delayis due to `-mixing [36], that relaxes the Rydberg population to lower energy levels.The presence of `-mixing is a consequence of free-moving charged particles.Comparing the bottom two frames of Figure 3.5 (i.e. ∆tRamp=300, and 450 ns),the Rydberg signals do not move significantly to higher fields. That must showthat the plasma electrons, in these time scales, are not quite free to interact withRydberg molecules causing further `-mixing or initiating n-changing processes.Figures 3.7 through 3.10, however, show that if we allow the system to evolvefor much longer times (i.e. tens of microseconds), we start to see some processeswhich change the principal quantum numbers. More specifically, for the caseof n0 = 51 shown in these figures, the ionization threshold for N+ = 2 signalshifts from 105 V/cm to approximately 145 V/cm after 20 µs. This translatesinto changing n0 from 51 to 47. This could be due to expansion of the excitedvolume, allowing some of immobilized electrons to be free-moving again andstart interacting with the residual Rydberg molecules.61This is not all we can learn from the results shown in Figures 3.7 through 3.10.For example, let us take a closer look at Figures 3.8; comparing the first and lastframes of this figure (i.e.∆tRamp = 0 and ∆tRamp = 20µs), one can see that theratio of the two Rydberg signals is different. When ∆tRamp = 0 N+ = 2 signal isweaker that N+ = 0 signal. However, when ∆tRamp = 20µs, the N+ = 2 signalis stronger that N+ = 0 signal. The same observation can be made for any otherof Figures 3.7 through 3.10. This is because of the fact that the coupling at theavoided crossings depends on the density of free charged particles (i.e. density ofplasma). When the ramp is applied immediately after the second laser, the densityis at its maximum and therefore more crossings from N+ = 2 to N+ = 0 state takeplace (in other words, more adiabatic ionization). On the other hand, if we wait fora long time, the density of free-moving charged particles is reduced. Hence fewercrossings from N+ = 2 to N+ = 0 state and therefore more diabatic ionization isobserved.Another way to look at this phenomenon is to compare the first frames ofFigures 3.7 through 3.10 with one another. That is to examine the effect of plasmadensity on the efficiency of the crossings not by waiting for the density to drop,but rather by changing the initial density through the first laser pulse energy. Wecan see as we increase the laser pulse energy (therefore increasing the density),intensity of the N+ = 0 signal is increased and N+ = 0 signal is weakened. Anexpected behaviour if we agree that the coupling at the avoided crossings of aStark manifold depends on the free-moving charged particles density.If we want to learn about the interplay between the plasma and Rydbergportion of our excited volume, Figure 3.12 may be the best representation of this.As mentioned in the previous section, regardless of the utilized first laser pulseenergy (and therefore initial density), the initial Rydberg signal has an upper limit;once the density is higher than 4 µJ, the only component which is increased is theplasma. We attribute this to the increased rate of Penning ionization which ionizesthe Rydberg state molecules and turns them into plasma. This happens on a scaleof nanoseconds, until the density of Rydberg molecules reaches to just below the62critical value needed for Penning ionization to happen. That is why no matterwhat initial density we start with, the ramp always collects the same number ofRydberg electrons when it is applied with no delay after the second laser (keep inmind that it takes the ramp more time than the time scale of Penning ionization toact on the Rydberg molecules).Moreover, this figure shows the avalanche dynamics in our molecular system.When the first laser power is set at 1.5 µJ, we cannot see any sign of avalanche;the fractional Rydberg signal does not decay and the fractional plasma signal doesnot rise. At 4 µJ, the avalanche occurs and Rydberg and plasma signals reach anequilibrium (50 percent of the mix is Rydberg and 50 percent plasma) within thefirst 10 us. This equilibrium holds for the rest of the flight time. As we increase thelaser pulse energy to 8 and 10 µJ, the behavior is what we expect (fast decrease offractional Rydberg signal and quick increase of fractional plasma signal. Out ofall frames in Figure 3.12, only those related to the two lowest laser pulse energies,show a total signal decay which is similar to the late-peak experiment shown inFigure 3.1. This could mean that the late-peak signal is either mainly the result offield ionization of the Rydberg state molecules when they are between G2 and G3grids (for the case of P = 1.5µJ), or the plasma in its arrested state behaves muchlike a long-lived Rydberg gas (for the case of P = 4µJ).The effect of initial density on the long time dynamics of our molecular systemcan be observed for different ramp delays in Figure 3.13 too. When Rydbergmolecules are collected at zero ramp delay (blue curve), the signal increaseswith ω1 pulse energy approaching saturation. This is expected, since higherlaser energy excites more molecules to the A-state and therefore the second laserproduces a higher Rydberg yield. This happens until the A-state is saturated (asdiscussed before), and that is why the curve starts to flatten at higher laser pulseenergies. This behaviour is not observed if we allow the system to evolve for sometime before we apply the ramped field. for example, in the case of ∆tRamp= 1 or 5µs we observe competing components related to the number of surviving Rydbergmolecules; one being the initial density set by the first laser pulse energy (the63higher the initial density the higher number of Rydberg molecules), and the otherbeing collisional decay (the higher the density, the more collisions which resultsin decay of Rydberg molecules). Higher evolution times (i.e. higher ramp delays),afford enough time for collisional decays to outweigh the density component,hence the observed constant decrease in the Rydberg signal for ∆tRamp= 10, 15,and 20 µs curves.Finally, let us discuss the effect of long evolution times on the Rydberg signalobtained at different initial densities, as depicted in Figure 3.14. As we discussedin the previous chapter, pre-dissociation happens in the system independent ofthe initial density. Therefore, the difference between the five curves shown inthis figure must be related to collisional mechanisms. Figure 3.14 reveals that forlow densities collisional mechanisms are almost stopped. In the next chapter wedescribe how we managed to manipulate this state of quenched plasma using aradio frequency electric field.64Chapter 4Manipulating the Quenched PlasmaUsing a Radio Frequency ElectricField4.1 IntroductionIn the previous two chapters, we explained the selective field ionization (SFI)technique, and how I utilized it to first calibrate the absolute density in our system,and second to energetically study the evolution of our excited molecular systemover time.In this chapter, I am going to discuss an important extension of the SFI ex-periment that I conducted in order to further investigate the `-mixing mechanismin our system, allowing for deeper understanding of whether or not many-bodylocalization is happening in our system. To this end, I introduced a radio frequencypulse into the sequence of SFI experiment events.In what follows, I quickly review the steps we take to excite our molecularsystem and will shortly move onto the results of RF-SFI experiments and finallythe conclusion of the results.654.2 Experimental4.2.1 Supersonic molecular beam ultracold plasmaspectrometerA 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.Figure 4.1: Co-propagating laser beams, ω1 and ω2 cross a molecularbeam of nitric oxide between entrance aperture G1 and grid G2 of adifferentially-pumped vacuum chamber.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.Figure 4.2 diagrams these steps of double resonant excitation from the elec-tronic ground state X 2Π1/2 (v′′ = 0, N′′ = 0) to the intermediate A 2Σ+ (v′ =0, N′ = 0) state, and then to the selected Rydberg state. The choice of A-state66N′ = 0 allows only final Rydberg states of total angular momentum neglectingspin of N = 1.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. These pairs of moleculesundergo a prompt Penning interaction that releases electrons and seeds an electronimpact avalanche to plasma. Among the N = 1 high-Rydberg states accessible toω2 excitation, only those in the n0 f (2) series converging to the nitric oxide cationstate, NO+ X 1Σ+ N+ = 2 have sufficient lifetime to sustain this avalanche, asillustrated by the excitation spectrum in Figure 4.2.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 methods to regulate this quantity. Varying the ω1 pulse energies from 2 to6 µJ increases the intermediate state density linearly to a degree that approachessaturation. This population decays with a radiative lifetime of 192 ns [77], 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−3.67Figure 4.2: Diagram illustrating the double-resonant excitation of a molecu-lar Rydberg gas of nitric oxide, and the conditions leading to Penningionization and avalanche to an ultracold plasma. The atomic-likeplasma-excitation spectrum consists exclusively of N = 1 n0 f (2)Rydberg state resonances converging to the N+ = 2 rotational limitof NO+. For an initial n0 = 50 Rydberg gas density of 1010 cm−1,the orbital radius is about 1 µm while the average spacing betweenRydberg molecules is 3 µm. However, a good portion of the nearest-neighbour distance distribution falls within 1 µm. These closelyspaced pairs interact by Penning ionization to form prompt electrons,which seed the avalanche to ultracold plasma.68Table 4.1: Distribution of ions in an idealized Gaussian ellipsoid shell model of a quenched ultracold plasmaof NO with a peak density of 4×1010 cm−3, σx = 1.0 mm, σy = 0.55 mm and σz = 0.70 mm. This modelquasi-neutral plasma contains a total of 1.9×108 NO+ ions. Its average density is 1.4×1010 cm−3 andthe mean distance between ions is 3.32 µm.Shell Radial Coordinates Density Volume Particle Fraction awsNumber rx ry rz cm−3 cm3 Number ×100 µm1 0.10 0.06 0.07 4.0×1010 1.8×10−6 7.0×104 0.04 1.812 0.20 0.11 0.14 3.9×1010 1.1×10−5 4.4×105 0.23 1.833 0.40 0.22 0.28 3.7×1010 9.0×10−5 3.3×106 1.75 1.864 0.62 0.34 0.43 3.3×1010 2.8×10−4 9.3×106 4.87 1.935 0.91 0.50 0.64 2.6×1010 8.3×10−4 2.2×107 11.52 2.086 1.15 0.63 0.81 2.1×1010 1.2×10−3 2.6×107 13.40 2.267 1.38 0.76 0.97 1.5×1010 1.8×10−3 2.8×107 14.46 2.498 1.60 0.88 1.12 1.1×1010 2.4×10−3 2.6×107 13.81 2.789 1.84 1.01 1.29 7.4×109 3.4×10−3 2.5×107 13.28 3.1910 2.11 1.16 1.48 4.3×109 5.1×10−3 2.2×107 11.56 3.8111 2.44 1.34 1.71 2.0×109 8.3×10−3 1.7×107 8.85 4.8912 2.92 1.61 2.04 5.6×108 1.7×10−2 9.4×106 4.94 7.5113 3.53 1.94 2.47 7.9×107 3.1×10−2 2.4×106 1.27 14.4714 4.29 2.36 3.00 4.0×106 5.6×10−2 2.3×105 0.12 38.9615 5.24 2.88 3.67 4.4×104 1.0×10−1 4.6×103 0.00 176.2269The peak density of the Rydberg gas falls off as a Gaussian ellipsoid de-fined by the intersection of the cylindrical Gaussian photon density of ω1 withthe wider cylindrical Gaussian nitric oxide density of the molecular beam. Weapproximate this ellipsoid computationally by a system of 100 concentric shellsto build realistic simulation of the collisional rate processes that give rise to theavalanche [53, 58]. For example, Table 4.1 gives a coarse-grained representationof the density distribution of the ultracold plasma that evolves from a laser-crossedmolecular beam Rydberg gas of nitric oxide with an initial peak density of 3×1011cm−3. In this case, 4 µs of avalanche, expansion and dissociation to neutral atomsreduce the peak density to 4× 1010 cm−3. The average ion/Rydberg density inthis ellipsoidal volume is 1.4×1010 cm−3, and about 1.5 percent of the moleculesin the plasma occupy two-thirds of its volume at a much lower density less than109 cm−3.4.2.2 Selective field ionizationDuring ω1+ ω2 laser excitation, with G1 held to ground, an adjustment of G2over a range of ±100 mV serves to define a field-free region between G1 andG2. At a predetermined time after ω2, a -3 kV square-wave pulse from a Behlkehigh-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 apolynomial function to the leading edge of this voltage pulse that transforms thetime-dependent electron signal waveform to an ionization spectrum as a functionof field in V/cm.A ramp with this rise time, started immediately after ω2 (∆tRamp = 0), drivesa diabatic evolution of molecules in the n f (2) Rydberg gas through the Starkmanifold to cross a saddle point leading to ions and free electrons when the fieldF in atomic units exceeds a threshold amplitude of 1/9n4. In conventional units,this process forms a selective field ionization resonance beginning in V/cm at afield, F = (En(2)/4.59)2, where En(2) in cm−1 is the binding energy of the n fRydberg state with respect to a nitric oxide cation in rotational state, N+ = 2.70This 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 stateNO+ cations earlier in the ramp, when the rising field passes an amplitude of(En(0)/4.59)2 V/cm.A ramp delayed by a few hundred nanoseconds samples the quantum-state dis-tribution in an evolving Rydberg gas. During the interval of this delay, promptlyformed electrons collide with Rydberg molecules. This causes `-mixing. Moleculesprepared in the initial state, n0 f (2) change orbital angular momentum, populatinga degenerate manifold of states, |N+ = 2〉 |n, `〉. These states of higher orbitalangular momentum field ionize at slightly higher field amplitudes to produceelectron waveforms reflecting the formation of ions in rotational states, N+ = 0and 2.The integrated electron signal collected at a given ramp delay changes withthe 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 above anddiagrammed in Figure 4.3, 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 4,000 traces in a typical SFI contour at fixed ramp delay according to ρ0.4.2.3 Radio-frequency electric fieldThe experiment uses a Tektronix AWG 7102 10 GS/s arbitrary waveform gener-ator to produce radio frequency (RF) pulses of selected frequency with an am-plitude from 0.1 to 1.5 V peak-to-peak. A LabVIEW user interface selects thefrequency and amplitude, with controlled delay and pulse duration. Measurements71described below use a frequency of 60 MHz, an amplitude of 400 mV cm−1 anda pulse duration of 250 ns (Wrf), triggered at a time, ∆tωrf between tω2 and tRamp.The programmed output of the waveform generator connects to the grid G2.4.2.4 Pulse sequence and evolution of n0 Rydberg density asobserved in a typical RF-SFI experimentFigure 4.3 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 ∆trf after the second laser pulse. A voltage ramp that begins ata time ∆tramp afterω2 selectively field ionizes the excited molecular system. In thisstudy, Wrf has a duration of 250 ns for all experiments. We vary the intervals, ∆trf,∆tω2 , and ∆tramp, according to the requirements of the measurements describedbelow.4.3 Results4.3.1 Field-free evolution of the nitric oxide molecularultracold plasmaAs discussed above, double-resonant excitation of nitric oxide in a molecularbeam forms a quantum-state selected ellipsoidal volume of Rydberg gas. Thepresent illumination conditions yield Gaussian widths (σx : σy,z) in a ratio ofapproximately 3:1, where z defines the propagation direction of the molecularbeam and x denotes the laser axis. At t = 0, the peak density of the Rydberg gasdepends on the intensity of the laser that drives the first step of double resonancefrom X 2Π1/2 N′′ = 1 to A 2Σ+ N′ = 0 (ω1), as well as the delay between the laserpulses promoting first and second steps.This Rydberg gas undergoes an avalanche to a quasi-neutral plasma of NO+72Figure 4.3: Schematic diagram showing the sequence of pluses in the RF-SFI experiment. The laser beams, ω1 and ω2, cross the molecularbeam of NO with a specified delay, ∆tω2 . A radio-frequency field withan adjustable peak-to-peak amplitude as high as 1 V cm−1 interactswith this ensemble, either as a CW field or as a pulse with a durationWrf applied at a time, ∆trf, after ω2. An electric field ramp from 0to 350 V/cm with a rise-time of 1 µs, applied ∆tramp following ω2,ionizes the excited molecular system. Shaded regions represent thedissociative decay of the n0 Rydberg molecules to a form a residualfraction of long-lived molecules in the absence (blue) and after thepresence of a 60 MHz radio frequency field (here represented by thepulse in green).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 tomany microseconds in the periphery of an ellipsoid with a lower peak density.Selective field ionization provides a measure of the global spectrum of electronbinding energy as a function of time. Figure 4.4 combines 4,000 SFI traces sorted730 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 µsFigure 4.4: Typical SFI spectra, formed by 4,000 SFI traces sorted accordingto the initial density ρ0, for an n f (2) Rydberg gas with an initialprincipal quantum number n0 = 44. Individually normalized contoursfrom left to right show electron binding energy spectra for electricfield ramp applied 0,150, 450 and 1000 ns after the ω2 laser pulse.The signal near zero field represents very high Rydberg molecules andelectrons loosely-bound by the plasma space charge. The two featuresthat appear at higher field reflect the field ionization of the n0 = 44 stateto NO+ X 1Σ+ cation rotational states, N+ = 0 and N+ = 2. Note howafter ramp delay of zero (∆tramp = 0), these features shift to higherfield ionization thresholds, reflective of electron collisional `-mixingof initial 44 f (2) Rydberg molecules.by peak density over a range from 1012 to 1010 cm−3. Here, for a ramp field risingfrom 0 V cm−1 150 ns after ω2, we see that that Rydberg gases in the higher74density range of our experiment avalanche fully on a 100 ns timescale, whileRydberg gases of lower peak density evolve to a mixture of weakly bound NO+ions and electrons (ultracold plasma) together with Rydberg molecules retainingthe initially selected principal quantum number, n0. Note the absence of electronsignal in a wedge of very low SFI potential on the left-hand edge of these contours.At the highest density, field ionization requires a minimum of a few V cm−1,which corresponds to 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. Figure4.5 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 that the number of electrons weakly bound to NO+ cations and extractedby the leading edge of the SFI field ramp remains constant as we step the startof the ramp over a time from 500 ns to 3.5 µs after the ω2 excitation pulse thatforms the Rydberg gas. During this same time interval, the Rydberg signal fallsexponentially, in each case to an apparent plateau.4.3.2 Molecular ultracold plasma rate processes in thepresence of a continuous wave (CW) radio frequencyfieldFigure 4.5 also includes equivalent measurements made in the presence of a CWradio frequency field with a frequency of 60 MHz and a peak to peak amplitudeof 400 mV cm−1. Here we see that the presence of an rf field of this frequencyhas no effect on the weakly bound electrons that form the plasma signal, but itaccelerates the decay of the Rydberg signal in all cases.This distinct effect of a radio frequency field on the SFI signal appears clearlyin a single raw ramped field-ionization trace. Figure 4.6 shows the electron wave-form obtained at intermediate density by the application of a field-ionization rampdelayed by 2 µs after ω2. Here, we also see the electron signal produced by the750.40.60.811.20.5 1 1.5 2 2.5 3 3.5Normalized electron signal Ramp Delay (us)0.40.60.811.20.5 1 1.5 2 2.5 3 3.5Normalized electron signal Ramp Delay (us)00.20.40.60.810.5 1 1.5 2 2.5 3 3.5Normalized electron signal Ramp Delay (us)00.20.40.60.810.5 1 1.5 2 2.5 3 3.5Normalized electron signal Ramp Delay (us)Figure 4.5: For ultracold plasmas evolving from (left) 45 f (2) and (right)49 f (2) Rydberg gases: (top) SFI amplitude integrated over ramp fieldfrom 0 to 50 V cm−1; and (bottom) SFI amplitude integrated over rampfield from 50 to 200 V cm−1, as a function of ramp delay, ∆tramp in thepresence (blue) and absence (orange) of a 400 mV cm−1 CW 60 MHzradio frequency field.same ramp in the presence of a 60 MHz radio frequency field. Note again that therf field has no effect on the weakly bound plasma waveform, but it substantiallyreduces the amplitude of the Rydberg signal.4.3.3 Effects of a pulsed radio frequency fieldThe application of a pulsed radio frequency field depletes NO Rydberg moleculesto a degree that depends on the temporal relation of the rf pulse to ω2 and thetime at which the SFI ramp field begins to rise. Figure 4.7 shows two examples.Here we see that a 250 ns pulsed 60 MHz radio-frequency potential applied to760 50 100 150 200 250Electric field (V/cm)-0.200.20.40.60.811.21.41.60MHz60MHzFigure 4.6: Single SFI traces obtained under identical conditions of initial49 f (2) Rydberg gas density and ramp-field delay in the absence(upper) and presence (lower) of a CW 60 MHz radio frequency field.Here, as above, the low-voltage part of the ramp collects loosely-bound electrons from the plasma. At higher voltage, we see the fieldionization of the residual n0 = 44 Rydberg gas to form NO+ X 1Σ+cation rotational states, N+ = 0 and N+ = 2.G2 before the start of the SFI ramp applied to G1 substantially depletes the n0Rydberg signal. As the delay time of the radio-frequency pulse, ∆tωrf approachesthe ramp delay, ∆tramp, the Rydberg signal recovers. Measured by the increase inRydberg field ionization signal as ∆tωrf approaches ∆tramp, the rate of rf acceleratedpredissociation for a ramp delay of 2 µs equals that observed for ∆tramp = 4 µs.The depth of the depletion and the shape its rise as ∆tωrf approaches ∆trampvaries with the density of the plasma, as shown in Figure 4.8.Note that a reduced initial Rydberg gas density decreases the fractional extentto which 60 MHz radio frequency field depletes the n0 = 49 NO Rydberg signal.Less obviously, the apparent predissociation rate measured by this experiment7700.20.40.60.810 0.5 1 1.5 2Rydberg field ionizaton signal  (arb.)rf delay (µs)00.20.40.60.810 1 2 3 4Rydberg field ionizaton signal  (arb.)rf delay (µs)Figure 4.7: Integrated electron signal from the selective field ionizationof n0 = 49 Rydberg molecules in the presence (blue) and absence(orange) of a 250 ns 400 V cm−1 60 MHz pulsed radio frequency fieldas a function of rf delay, ∆tωrf for two fixed values of ramp field delay,∆tramp, of 2 µs (left) and 4 µs (right).lessens in an ultracold plasma of lower density.4.4 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 under-goes avalanche to form an ultracold plasma. Electron signal contours in sortedselective field ionization spectra such as those evident in Figure 4.4 show howthe electron binding dynamics of this avalanche vary with the initial density ofthe Rydberg gas. Here we see snapshots of the ultracold plasma electron bindingenergy spectrum formed by Rydberg gases ranging over two orders of magnitudein initial density after evolution times selected by ∆tramp values of 0, 150, 450 and1000 ns.For initial densities exceeding 5× 1011 cm−3, virtually all of the charge ex-7800.20.40.60.810 1 2 3 4 5Rydberg field ionizaton signal  (arb.)rf delay (µs)Figure 4.8: Integrated electron signal from the selective field ionization ofn0 = 49 Rydberg molecules in the presence of a 250 ns 400 V cm−160 MHz pulsed radio frequency field as a function of rf delay, ∆tωrffor a fixed ramp field delay, ∆tramp = 5 µs, and two values of ω1 - ω2delay, 0 and 200 ns. With a radiative lifetime of 192 ns, a delay of 200ns reduces the density of 2Σ+ intermediate state NO molecules by afactor of 2.72.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, alsoavalanches to yield electrons of low binding energy, mixed here with a residualpopulation of Rydberg molecules bound with an energy determined by the initialprincipal quantum number, n0. This signature of n0 Rydberg molecules, whichfield-ionize to form NO+ in N+ = 0 and 2, dominates the SFI spectrum of lower-initial-density Rydberg gases when measured with short ramp delays. For all den-79sities, the particle balance in this ultracold plasma shifts after a longer evolutiontime to favour electrons weakly bound to NO+ ions, as high-n Rydberg moleculesor as electrons bound to the plasma space charge.Inspection of SFI spectra between 0 and 150 ns shows direct evidence ofinitial electron mobility in a subtle shift of residual n0 features to higher ap-pearance potential. This shift to higher field-ionization threshold points to adynamic process in which electron-Rydberg collisions drive `-mixing interactionsthat cause the photo-selected 44 f (2) molecules to spread over the full distributionof ` ∈ 0,1,2...44 values of Rydberg orbital angular momentum for n0 = 44 [36].From previous work, we know that the overall density of the nitric oxidemolecular ultracold plasma falls with evolution time during the first few microsec-onds, owing to channels of neutral decay via NO+ ion-electron dissociative re-combination and neutral NO Rydberg predissociation [49, 51, 62]. Figure 4.5details this effect experimentally. Note that the residual n0 Rydberg signal de-cays, while the plasma signal remains constant on a microsecond timescale. Thisdirectly shows that n0-Rydberg predissociation dominates ultracold plasma decayon the timescale 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 4.4 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 evidentlyin the exponential decays of the integrated n0 = 45 and 49 Rydberg signals as afunction of ramp field delay in Figure 4.5, which in every case fall to a residualplateau.Recognizing this, we fit the data in Figure 4.5 to a rate law of the form:d [NO∗]dt=(1− [NO∗]A[NO∗]0)kPD [NO∗]+[NO∗]A[NO∗]0(4.1)where [NO∗] describes the density of predissociating n0 Rydberg molecules,[NO∗]0 shows the initial (i.e. at t = 0) density of Rydberg-state molecules, and80[NO∗]A represents the residual population in a state of arrested predissociation.Here, kPD refers to a phenomenological overall rate of predissociation. Table 4.2gives these parameters for the fits plotted in Figure 4.5.Table 4.2: Kinetic parameters used in Eq (4.1) to fit the exponential decay inthe n0 Rydberg molecule SFI signal in Figure 4.5.[NO∗]0 kPD (µs−1) [NO∗]A / [NO∗]0n0 = 45Field Free 1 0.75 0.2860 MHz 1 1.60 0.22n0 = 49Field Free 1 0.95 0.2760 MHz 1 1.95 0.164.4.2 Nitric oxide Rydberg predissociation in a regime of`-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 [73, 74]. A dc field of a few hundred mV mixes ` sufficiently to increasethis lifetime to a measured 75 ns. Multichannel effective Hamiltonian modelspredict that effective kPD values for fully coupled bright states in this range of then f series decrease to∼ 5 µs−1 [78, 79]. Predissociation in the broader manifold ofn0 = 49 Rydberg states, mixed by electron collisions over all values of `, proceedswith a phenomenological kPD determined by sampling `-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 `. Only low-` states decay with appreciable rates. For thepurposes of illustration, we can take rates from a model developed by Gallagherand coworkers [80], patterned on the work of Bixon and Jortner [78]. and estimatekPD for a given n from the statistically weighted sum of `-dependent rate constants,81k` = 0.014, 0.046, 0.029 and 0.0012 in atomic units for ` from 0 to 3, and 0.00003for `≥ 4, scaled by n−3:kPD(n) =∑` (2`+1)kn,`n24.13×1016s−12pin3(4.2)This simple statistical approach predicts decay times for levels near n = 50 ofabout 200 ns, in accord with observations for bright states of NO in this range,when prepared by broad-band excitation in the presence of an `-mixing electricfield [73, 78, 79].The initial predissociation kinetics observed in the present experiment con-form with 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 ac-count for 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 [51, 53]. For the particular initial density represented in Figure 4.5, we findthat n0 nitric oxide molecules in this ultracold plasma initially decay to neutralproducts on a timescale consistent with the state-detailed rate of NO Rydbergpredissociation in a collisional regime of ` scrambling.However, as evident from Figure 4.5 and the fits to Eq (4.1) parameterized inTable 4.2, 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-` residue of n0 Rydberg molecules that remains from thestatistical distribution over all accessible ` by created electron-collisional `-mixingduring the avalanche. Such a residue can survive only if `-mixing ceases. Itspresence here serves as an adventitious sensor of `-mixing under these conditionsof arrested relaxation.82The presence here of long-lived n0 Rydberg molecules thus suggests that thesystem evolves to a state of quenched predissociation in which the ultracold plasmacontains too few free electrons to `-mix these residual n0 Rydberg molecules. Wemight explain this by assigning all the signal that appears in the prominent con-tours at low field in the SFI spectrum to electrons bound in very high-n Rydbergstates.However, for the range of initial densities and principal quantum number ex-hibited in Figure 4.4 classical rate theory considerations call for strong Rydberg-Rydberg interactions, Penning ionization and electron-ion-Rydberg molecule col-lisions in an `-mixed quasi-equilibrium with no more than a 100 µs lifetime [36].In particular, coupled rate-equation simulations predict that such interactions drivePenning ionization and avalanche to plasma quasi-equilibrium in Rydberg gassystems with densities as displayed in Figure 4.4 on a microsecond timescale [53].4.4.3 Effect of a radio-frequency fieldThe nitric oxide molecular ultracold plasma evolves in the long-time limit to astate in which most of the electrons bind very weakly to ions, either individually invery high-n Rydberg orbitals or collectively to the NO+ space charge [54]. Eitherway, the SFI experiment directly measures an ultracold plasma electron bindingenergy no greater than ∼ 800 GHz. A small portion of this system survives asRydberg molecules with the initially selected principal quantum number, n0, ina high-` state of quenched predissociation, suggesting an absence of `-mixingelectron-Rydberg collisions.A 60 MHz field, applied with a peak-to-peak amplitude of 400 mV cm−1,evidently accelerates the predissociation of these residual n0 Rydberg molecules.A radio-frequency field interacts with a conventional plasma of electrons and ionsto drive collective modes of motion termed plasma oscillations. A neutral plasmaof defined density, ρe, supports a plasma frequency ωrf = 1/2√e2ρe/ε0me. Afield with a frequency of 60 MHz resonates with an electron-ion plasma at adensity of about 106 cm−3.83In the nitric oxide molecular ultracold plasma, a 400 mV cm−1 radio frequencyfield seems to release electrons and cause a resumption of `-mixing that redis-tributes Rydberg orbital angular momentum from states of high-` to low-` statesof shorter predissociation lifetime. As detailed above, it is quite reasonable toattribute this rf-driven `-mixing to a resumed effectiveness of electron collisions[36].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,`. In thestatistical limit, these rate constants combine in accordance with Eq (4.2) to deter-mine a phenomenological rate constant, kPD(n0) for the entire residual populationof n0 Rydberg molecules. Under field-free conditions, Figure 4.5 shows that theeffective predissociation rate constant varies in time, and appears ultimately tofall to zero, leaving an arrested population of n0 Rydberg molecules, [NO∗]A.Without efficient redistribution in `, terms in the sum of kn0,` over ` have coeffi-cients smaller than (2`+1) that decrease over time as predissociation depletes thepopulation of n0 Rydberg molecules with low `.The presence of an rf field serves to maintain the weight of low-` terms, dimin-ishing the fraction of arrested molecules and creating an additive contribution tothe phenomenological rate constant δkPD(n0). Table 4.2 quantifies these trends forthe conditions of the experiment represented in Figure 4.5, where contributions tokPD(n0) owing to low-` predissociation add about 1 µs−1 to the phenomenologicalrate constant and reduce the arrested fraction by an amount in the range of 30percent.A pulsed radio frequency field similarly promotes a redistribution of residualn0 Rydberg molecules over `, with a comparable effect on the apparent rate ofpredissociation. We see this in a decreased n0 Rydberg contribution to the SFIspectrum. Assuming that a pulse applied anytime produces the same degreeof redistribution over `, and as a result, increases kPD(n0) to the same degree,84accelerated predissociation diminishes the n0 Rydberg signal to a varying extentthat depends on the elapsed time between the application of the radio frequencypulse, ∆tωrf , and the beginning of the SFI ramp field at ∆tramp.A smaller value of ∆tramp−∆tωrf gives an rf-accelerated predissociation rateconstant less time to act and thus causes a smaller suppression of the n0 Rydbergsignal. We illustrate this effect by developing an expression for the density ofRydberg molecules as a function of ∆tωrf normalized by its value at a time, t =∆tramp under field-free conditions for any initial density, as determined say by theω1 - ω2 delay.In a limit of instantaneous rf-induced `-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 canapproximate this by a logistic function:Se = (1− fe)+ fe[11+ ekr(t0−t)], (4.3)Here, fe represents the full fractional depletion of the Rydberg density mea-sured by SFI by a voltage ramp that starts at ∆tramp. If the predissociation time isfinite, then a pulsed radio-frequency fields applied at times, ∆tωrf , close to ∆tramp,will be less effective in depleting the n0 Rydberg signal measured by SFI at ∆tramp.We can account for less than instantaneous predissociation if we moderate 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(t0−t)]+ fe[e−δkPD(∆tramp−t)]×(1−[11+ ekr(t0−t)])(4.4)where we moderate the effect of the exponentially recovered n0 Rydberg signalby a step function that falls from 1 to 0. Here, we assign a uniform midpoint, t0,to the rising and falling logistic functions to best represent the convolution of the85rf pulse with the ramp. We then fit fe and δkPD as they vary with density.Curves drawn through the data in Figure 4.7 fit Eq. (4.4) to the n0 Rydbergsignal as a function of ∆tωrf for fixed ramp field delays, ∆tramp, of 2.68 and 4.70 µs.Here, we neglect the evident effect of an rf field when some portion of the pulseoverlaps with ω2 excitation and avalanche. We also recognize that the logisticfunction imperfectly describes the convolution of a varying degree of `-mixingand kPD increase as the rf pulse passes through the SFI ramp. With these provisos,we readily obtain self-consistent descriptions of the measurement at both rampfield delays, changing only the known fixed value of ∆tramp. Table 4.3 summarizesthe parameters of these fits. Note that recovery of the signal as the rf pulse passesthrough the ramp is described by a logistic function with the same offset, ∆tramp−t0.Table 4.3: Parameters used in Eq (4.4) to describe the recovery of the n0Rydberg SFI signal depleted by a 250 ns 400 V cm−1 60 MHz pulsedradio frequency field as time of this pulse, ∆tωrf advances to passthrough the beginning of the SFI ramp field. ∆tramp for two differentramp field delays.∆tω2 (µs) ρ0 kr (µs−1) t0 (µs) ∆tramp (µs) fe δkPD (µs−1)0 1 8.0 1.65 2.70 0.73 0.760 1 8.0 3.65 4.70 0.73 0.764.4.5 Mechanics of radio frequency induced `-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-`Rydberg molecules that retain the initially selected principal quantum number,n0. This occurs as the consequence of a process that begins in the avalanchewith `-mixing electron-Rydberg collisions. This randomization in ` is followedby evolution in the bulk plasma to a condition both of low electron binding energy86and quenched electron mobility. This quenched environment traps a measurablefraction of n0 Rydberg molecules in states of ` too high to predissociate.We could interpret sequences of SFI spectra, such as those shown in Figure4.4 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-`would predissociate, anda very long-lived high-` 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 thedistribution over ` within a single Stark manifold [80]. In their experiment onlow-` Rydberg states of NO, this effect lengthened predissociation lifetimes. Thesame mechanism of `-mixing would accelerate predissociation in an arrested dis-tribution of residual n0 Rydberg molecules of high-`.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 `-mixing in the present case [53]. 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 `-mixing.Predissociation stimulated by Stark mixing in a radio frequency field of agiven amplitude occurs to the same degree for every molecule in a sample ofany density. Predissociation catalyzed by `-mixing electron collisions proceedsas a pseudo first-order process, and thus occurs to a fractional extent that varieswith the density of electrons. Figure 4.8 shows immediately that the fractionaldepletion changes with the initial density of the Rydberg gas, controlled withprecision by adjusting the ω1 - ω2 delay (∆tω2). As shown in Table 4.4 curves87through these data, obtained for densities that differ by about a factor of three, fitEq (4.4) varying only fe. Note that the initial Rydberg gas density affects only theTable 4.4: Parameters used in Eq (4.4) to describe the recovery of the n0Rydberg SFI signal depleted by a 250 ns 400 V cm−1 60 MHz pulsedradio frequency field as time of this pulse, ∆tωrf advances to passthrough the beginning of the SFI ramp field. ∆tramp for two differentinitial Rydberg gas densities, as determined by ∆tω2 .∆tω2 (µs) ρ0 kr (µs−1) t0 (µs) ∆tramp (µs) fe δkPD (µs−1)0 1 8.0 4.85 5.90 0.59 0.76200 0.37 8.0 4.85 5.90 0.32 0.76fractional depletion. The additive contribution to kPD does not vary over the rangeof charged particle densities formed as a consequence of the variation of initialRydberg gas density. This suggests that the scrambling in ` saturates after veryfew electron-Rydberg collisions.4.4.6 Electron mobility in the quenched molecular ultracoldplasmaThe 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 (400 mV cm−1) 60 MHz radio frequency field subtlychanges the state of this plasma in a way that causes the predissociation of thisresidue to accelerate.The NO Rydberg residue represents a surviving fraction of n0 molecules trappedin states of high ` populated during the avalanche and quenched to a regime ofsuppressed `-mixing. Predissociation resumes when the rf field acts to scramblethe orbital angular momentum of those high-`, n0 Rydberg molecules.The fractional yield of accelerated predissociation of n0 Rydberg moleculesvaries substantially with plasma density. This excludes `-mixing in isolated molecules.88We know that charged particles in a plasma also respond collectively to a ra-dio frequency field by executing oscillatory modes of electron and ion motion.Damping via charge coupling and collisions couples energy from the rf field tothe plasma. If this heating exceeds a mobility threshold for electrons trappedeither by localization or as a spin glass, we should expect to see an effect of the rffield 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 `. 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 nodetectable change in the ultracold plasma background.We can therefore regard this adventitious population of high-` n0 Rydbergmolecules and its response to an rf field as a quantum-state probe of the dy-namics of avalanche and quench that form the molecular ultra-cold plasma. Thenatural state of arrested predissociation points to an immobility of electrons inthe quenched plasma. A radio frequency field acts to mobilize electrons, whichincreases the rate of predissociation in a fraction of n0 Rydberg population thatgrows with the density 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?89Chapter 5ConclusionThrough the course of this thesis, we explained the formation of an ultracoldmolecular plasma from a cold gas of nitric oxide molecules which are initiallyexcited to Rydberg states. We have used the selective field ionization method toinvestigate the process of plasma formation energetically. The results show thatunder certain experimental circumstances, plasma co-exists with residual Rydbergstate molecules which survive for much longer times than classically expected.The first chapter provides an introduction to the field of ultracold plasmaincluding a comparison of MOT with supersonic expansion techniques. We alsodiscussed how exactly the Rydberg gas undergos Penning ionization and electron-impact avalanche to form the plasma, and how this plasma bifurcates and takes ona state of arrested relaxation.In the second chapter we discussed the details of our experimental apparatusand the selective field ionization technique. The chapter reviewed how we canchange the Rydberg gas density in our system. It also discussed how we utilize aspecific ramped electric field and the knowledge of nitric oxide A-state lifetime tocalibrate the initial Rydberg density in our system. Combined with the maximumdensity which is calculated through a thermodynamic model of the supersonicmolecular beam, this allows us to exactly determine the density of our Rydbergsystem at any given set of experimental parameters.90The third chapter presented long-time dynamics of our excited molecular sys-tem investigated through selective field ionization. We reviewed physics of se-lective field ionization first and some of the previous experiments in our researchgroup which acted as the motivation for long-time selective field ionization inves-tigation of our system. The results showed that after the initial electron avalanchewhich produces free electrons and NO+ ions, these free electrons cause collisional`-mixing in the first 300 ns. Then such collisions stop for some few hundrednanoseconds. If we wait for much longer times (in the order of several to a fewtens of microseconds), however, we observe surviving Rydberg states in lowerprincipal quantum numbers than initially excited by our double-UV resonances.One of the interesting observations of selective field ionization experiments,was the extremely long-lived Rydberg state molecules in the excited volume. Inthe fourth chapter, we reviewed how we added a radio frequency electric fieldpulse to the sequence of events in a regular selective field ionization experimentto check the electron mobility in the excited system. Conventional fluid dynamicsof ion-electron-Rydberg quasi-equilibrium predicts rapid decay to neutral atoms.Instead, the NO plasma endures for much longer times, suggesting that quencheddisorder creates a state of suppressed electron mobility. Supporting this propo-sition, in this chapter we showed how a 60 MHz radio frequency field with apeak-to-peak amplitude less than 1 V/cm acts dramatically to mobilize electrons,causing the Rydberg molecules to predissociate. An evident density dependenceshows that this effect relies on collisions, giving weight to the idea of arrestedrelaxation as a cooperative property of the ensemble.Although our current collective understanding of our molecular plasma andthe state of arrested relaxation has been greatly advanced by the experimentsreported in this thesis, there is still a great deal left to discover about this many-body system. Our group is aiming to further investigate the system using mm-wave radiation to directly interact with the Rydberg state molecules. Moreover,using new set of laser pulses one can excite a secondary step of double resonancethat precisely selects a particular high-Rydberg state of nitric oxide for use as91a quantum reporter. Selective field ionization then can be utilized to determinethe timescale on which electron collisions cause a precisely selected Rydbergreporter to undergo `-mixing and n-level relaxation. 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Bichsel, M. Morrison, N. Shafer-Ray, and E. Abraham, “Experimentaland theoretical investigation of the stark effect for manipulating coldmolecules: Application to nitric oxide,” Phys. Rev. A, vol. 75, p. 023410,Feb 2007. → page 102[84] M. Schulz-Weiling, Ultracold Molecular Plasma. PhD thesis, 2017. →page 102101Appendix ANitric Oxide SpectroscopyIn this Appendix we briefly discuss spectroscopic properties of nitric oxide.1. Thisshort note only discusses the quantum states which are relevant to our experimen-tal method, as discussed in the previous chapters. Since vibrations do not play arole in our experiments we do not discuss this topic, and we follow the Herzberg[81] notation throughout this Appendix.The motion of electrons in a diatomic molecule takes place within the cylin-drical field of force set by the intermolecular axis. A precession of total electronicorbital angular momentum (without spin), L = ∑i `i, takes place about this axis.Only the projection of L on the molecular axis, with components ML(}/2pi), is aconstant of motion. ML can take values ML = L,L− 1,L− 2, ...,−L. Projectionstates differing only in the sign of ML have the same energy. Since L is not a goodquantum number for molecules, the following notation has been established:Λ= |ML|= 0,1,2, ...,L = Σ,Π,∆,Φ, ... (A.1)1Excerpted from Herzberg’s Spectra of Diatomic Molecules [81], Brown et. al. Chemistry:The Central Science [82] and Abraham’s review paper on nitric oxide spectroscopy [83]. ThisAppendix is to a great extent repeating what has been explained in thesis of previous member ofour research group, Dr. Markus Schulz-Weiling’s thesis [84], and I claim no authorship. My goalfrom including this here is to facilitate understanding of this topic for any future scientist who mayread my thesis.102The representative numbering of Λ through Greek capital letters is analogous tothe mode of designation for atoms. Π,∆,Φ, ... states are doubly degenerate asΛ= |±ML|. States Λ= Σ are non-degenerate.The spin states of the individual electrons form a total spin S=∑i si. In Σ statesand in absence of external fields or rotation, S is uncoupled from the molecularaxis. For states Λ 6= 0, precession of L about the molecular axis causes an internalmagnetic field in the same direction. This in turn causes S to precess about themolecular axis with constant components MS(}/2pi). The notation,Σ= MS = S,S−1,S−2, ...,−S (A.2)has been established. The total electronic angular momentum about the inter-molecular axis is called Ω. It is obtained by addition of Λ and Σ.Ω= |Λ+Σ| (A.3)Angular momentum caused by nuclear rotation of the molecular core is rep-resented through quantum vector N. There is no quantum number associated withN. The total angular momentum of the system - combination of electron spin,electronic orbital angular momentum and nuclear rotation - is always designated J.Different classifications for different modes of coupling for J were first introducedby Hund. Figure A.1 shows Hund’s cases (a), (b) and (d), which play a role in ourexperiment.Hund’s case (a) assumes that the interaction of nuclear rotation with theelectronic motion Ω is weak. Thus, J, constant in magnitude and direction, isa resultant formed by the nutation of Ω and N about J. A different way of under-standing Hund’s case (a), is that the precession of L and S about the internuclearaxis is much faster than above mentioned nutation.In Hund’s case (b), the electronic spin S is only very weakly coupled to themolecular axis. This is the case when Λ = 0 (absence of internal magnetic field)or even if Λ 6= 0 in the case of particularly light molecules. Thus, Ω is not defined.103Figure A.1: Hund’s cases: Coupling of rotation and electronic motion -J is the total angular momentum, N is the angular momentum ofthe nuclear rotation. K is the total angular momentum apart fromspin. L is the total electronic orbital angular momentum and Λ itsprojection on the molecular axis. S is the total electronic spin and Σit’s projection on the molecular axis. In Hund’s case (d), N becomesR, a good quantum number. (Figure (c) does not show the couplingof S and K to J.) Credit: J.P. MorrisonNow, Λ (if nonzero) and N together form the resultant K.K = Λ,Λ+1,Λ+2, ... (A.4)is the total angular momentum apart from spin. K and S together form resultantJ.J = (K+S),(K+S−1),(K+S−2), ..., |K−S| (A.5)Hund’s case (d) refers to the case where the coupling between L and thenuclear axis is very weak while that between L and the nuclear rotation is strong.This is usually the case for molecular high Rydberg states. Here, the angularmomentum of nuclear rotation is called R (rather than N) and has magnitude√R(R+1)}/2pi . Possible values for R are 0, 1, 2, ... . Vector addition of R104and L yield K, which can have values:K = (R+L),(R+L−1),(R+L−2), ..., |R−L| (A.6)The angular moments K and S together form total angular momentum J. In gen-eral (except for K<S), each level with a given K consist of 2S+1 subcomponents.Nitric oxide has a total of fifteen electrons and forms an electronic groundstate in the (1σ)2(2σ)2(3σ)2(4σ)2(5σ)2(1pi)4(2pi)1 configuration, as shown inFigure A.2. NO has eleven valence electrons but only the pi∗2px electron is unpaired.Thus, total orbital angular momentum projection isΛ= 1 and total spin is S= 1/2.The NO ground state is (mostly) Hund’s case (a). Spin-orbit interaction yields amultiplet Ω= Λ+Σ= 3/2,1/2. The notation for the resulting states are:2Π3/2 and 2Π1/2Figure A.3 shows the NO level diagram and our excitation pathway. As typicalfor Hund’s case (a), spin-orbit splitting between Ω = 3/2 and Ω = 1/2 is largecompared to the rotational spacing. As mentioned previously, levels with Λ 6= 0are double degenerate. Interactions between nuclear rotation and L causes thisdegeneracy to lift and is called Λ-type doubling. As a result, each J value splitsinto two components. Designations e and f label the rotationless parity as positiveand negative, respectively.The first excited state in NO has the configuration (1σ)2(2σ)2(3σ)2(4σ)2(5σ)2(1pi)4(6σ)1 and corresponds to a 2Σ electronic state. Since Λ = 0, thisstate is Hund’s case (b). For S=1/2 the multiplicity is 2. Similar to above case,molecular rotation induces a small internal magnetic field which splits the spindegeneracy. The spectroscopic labels F1(K) and F2(K) refer to components withJ=K+1/2 and J=K-1/2, respectively.105Figure A.2: NO molecular orbital diagram - Only the open shell electronlevels are displayed. The excitation pathway for ω1 photons isindicated.For any electric dipole transition, the following selection rules apply:∆J = 0,±1, with the restriction J = 0⇒ J = 0Parity: only pos neg∆S = 0The first two rules are rigorous. A very cold (<1 Kelvin) spectrum of the NO Xto A-state transitions consists of four lines:pQ11(1/2) : J“ = 1/2,K“ = 1,F1 f ⇒ J‘ = 1/2,K‘ = 0,F1eqR11(1/2) : J“ = 1/2,K“ = 1,F1e⇒ J‘ = 3/2,K‘ = 1,F1eqQ21(1/2) : J“ = 1/2,K“ = 1,F1e⇒ J‘ = 1/2,K‘ = 1,F2 frR21(1/2) : J“ = 1/2,K“ = 1,F1 f ⇒ J‘ = 3/2,K‘ = 2,F2 fTransitions are designated P, Q, R for changes in J of +1, 0 , -1; p, q, r for changes106in K. The subscripts 11 (or 21) label transitions to F1 (or F2) from F1.In our experiment, we tune laser ω1 to the pQ11(1/2) transition (∼226nm)to populate the rovibronic ground-state of the NO A-state. Subsequent photonabsorption of ω2 light populates the high-Rydberg manifold between principalquantum numbers 35 to 80. The ionic core of such Rydberg molecules has con-figuration (1σ)2(2σ)2(3σ)2(4σ)2(5σ)2(1pi)4 and corresponds to 1Σ.107Figure A.3: NO level diagram for X, A and Ry-states and transitions A←X,Ry←A. - Labeling information for energy levels and transitions arefound in the text.108Appendix BSupplementary Materials forChapter 2In this appendix, I show enlarged version of histograms related to all SFI densitycalibration experiment that I have collected. These plots are the same shown indifferent frames of Figure 2.10.109ω1 energy2 μJΔtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsFigure B.1: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 2 µJ, for different delaysof the second laser.110Δtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsω1 energy4 μJFigure B.2: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 4 µJ, for different delaysof the second laser.111Δtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsω1 energy6 μJFigure B.3: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 6 µJ, for different delaysof the second laser.112Δtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsω1 energy8 μJFigure B.4: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 8 µJ, for different delaysof the second laser.113Δtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsω1 energy10 μJFigure B.5: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 10 µJ, for different delaysof the second laser.114Δtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsω1 energy12 μJFigure B.6: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 12 µJ, for different delaysof the second laser.115Δtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsω1 energy15 μJFigure B.7: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 15 µJ, for different delaysof the second laser.116Δtω2 = 0 ns Δtω2 = 50 nsΔtω2 = 100 ns Δtω2 = 150 nsΔtω2 = 250 nsω1 energy20 μJFigure B.8: Histograms of SFI density calibration experiment with 1000traces. PQN=44, first laser pulse energy of 20 µJ, for different delaysof the second laser.117Appendix CMATLAB Codes and Functions1 % ******************************2 % DA.m3 % ******************************45 % DA class stands for data analysis done for NO experiments6 % at Prof. Grant's laboratory. This class call essential7 % functions that loads data from a selected directory.8 % There are some functions that facilitate9 % plotting and some other tasks.10 classdef DA11 methods (Static)12 function d=readspec(path)13 % readspec take path as input. The path should14 % indicate the location of the folder that15 % contains pfi, w2, or w1 spectrum1617 % find files that end with .wf18 f=dir(fullfile(path,'*.wf'));1920 % initiate a data structure21 d=struct;1182223 % loop over all .wf files24 for i=1:length(f)25 % print the file name in the command window26 fprintf('%s\n',f(i).name);2728 % read dataset from a set of files29 % the inputs are the main path to the30 % folder and the name of each file31 % with .wf in the folder path32 % the outputs are 2D data,33 %specifications of the waveform,34 % time array, and the scanning35 % variable (delay, wavelength,...)36 [d(i).data,d(i).wf,d(i).t,d(i).wl]=...37 DAIO.readdat(path,f(i).name);3839 % add a new field called name which40 % is the same as the file41 % name except the date at the42 % beginning and .wf at the end43 d(i).name=f(i).name(13:end-3);44 end45 end46 function d=readPS(path, lines)47 % readPS take path and lines as input and48 % returns pulse shape data. Path is the49 % folder location of the pulse shapes and50 % lines is the number of comment lines on51 % top of the data. The code will eliminate52 %the top part of the file as specified by53 % number of lines and reads the rest as data5455 % find files that end with .wf56 f=dir(fullfile(path,'*.wf'));5758 % initiate a data structure11959 d=struct;60 for i=1:length(f)61 % readwl is the core function that reads a62 % pulse shape file.63 % It take the original path and the name of64 % each pulseshape in the folder and the65 % number of lines of comments and66 % processes the data67 [d(i).t,d(i).v,d(i).wf]=...68 DAIO.readwl(path,f(i).name, lines);6970 % make sure that there is no offset to the71 % voltage72 d(i).v=d(i).v-d(i).v(1);7374 % add a new field name from the file name75 d(i).name=f(i).name;7677 % EF (electric field) used to be voltage78 % divided by distance. Right now EF is79 % redundant as it is the same as80 % v. Let's keep it that way for now.81 d(i).EF=d(i).v;8283 % fit a smooth function to the shape of the84 % electric field without worrying much85 % about its functional form86 d(i).cf=fit(d(i).t(:),sort(d(i).EF(:)),...87 fittype('smoothingspline'));88 end89 end90 function x=fitns(wl,g)91 % this function take a set of wavelength at92 % maxiumum absorbtion and an initial guess of93 % g for the n0 = quantum number and94 % ∆ = quantum defect and attempts to fit95 % IP, ∆, and quantum number12096 % wavelengths must be consequtive. It can be97 % ascending or descending. Wavelength is of98 % the doubled frequency and vacuum99 % corrected100101 % no matter what or wl is, make is ascending102 wl=sort(wl);103104 % compute wavenumber from double frequency105 % wavelength in vacuum106 wavenum=1e7./wl;107 % input is in nm. output is in cm^-1108109 % Rydberg constant cm^-1110 Ryd=109737.316;111112 % an objective function that returns the error113 % of fit114 function o=obj(x)115 % quantum defect116 ∆=x(3);117118 % build a range of numbers that when added119 % to n0 = g(1)120 % will give the range of quantum numbers.121 n=length(wl)-1:-1:0;122123 % convert the range of numbers to range124 % of quantum numbers125 n=n+fix(x(2));126127 % compute difference of experimental128 % wavenumbers and the129 % computed one130 o=wavenum'-(x(1)-Ryd./(n- ∆).^2);131 end132121133 % minimize error of the objective function134 % to find the best fit135 % IP n0 ∆136 % x is 3 by 1 array: IP, n0, and ∆137 x=lsqnonlin(@(x)obj(x),[30544.2333070219,g(1),g(2)]);138139 end140 function x=fitnsqdfixed(wl,g)141 % this function take a set of wavelength at142 % maximum absorption and an initial guess of143 % g for the n0 = quantum number and144 % ∆ = quantum defect and attempts to fit IP, ∆,145 % and quantum number146 % wavelengths must be consecutive. It can be147 % ascending or descending. Wavelength is of148 % the doubled frequency and vacuum149 % corrected150151 % no matter what or wl is, make is ascending152 wl=sort(wl);153154 % compute wavenumber from double frequency155 % wavelength in vacuum156 wavenum=1e7./wl;157 % input is in nm. output is in cm^-1158159 % Rydberg constant cm^-1160 Ryd=109737.316;161162 % quantum defect163 ∆=g(2);164165166 % an objective function that returns the error of167 % fit168 function o=obj(x)169122170 % build a range of numbers that when added to171 % n0 = g(1)172 % will give the range of quantum numbers.173 n=length(wl)-1:-1:0;174175 % convert the range of numbers to range of176 %quantum numbers177 n=n+fix(x(2));178179 % compute difference of experimental180 % wavenumbers and the computed one181 o=wavenum'-(x(1)-Ryd./(n- ∆).^2);182 end183184 % minimize error of the objective function185 % to find the best fit186 % IP n0 ∆187 % x is 3 by 1 array: IP, n0, and ∆188 x=lsqnonlin(@(x)obj(x),[30544.2333070219,g(1)]);189190 end191 function wl = ntowl(ns,x)192 % compute the wavelength of the double193 % frequency light in vacuume194 % (make sure multiply by 2 if you're working195 % with the fundamental wavelength out of ND6000196 wl = 1e7./(x(1)-cons.Ryd/100./(ns-x(2)).^2);197 end198 end199 end1 % ******************************2 % DAIO.m3 % ******************************41235 classdef DAIO6 % DAIO stands for data analysis input output7 % this class contains core functions that are called by8 % DA class.9 % It is seldom necessary to call DAIO directly from10 % your script1112 methods (Static=true)1314 function [dat,wf,t,wl] = readdat(path, fname)15 % this function read the .dat file containing16 % the 2D intensity plot1718 % read the .wf files to get the parameters of19 % the waveform20 % filename should be imported as something .wf21 $ (handled by DA class)22 wf = DAIO.readwf(path, fname);2324 % the 3rd parameter from .wf file is the # of25 $ points in a single trace, same as the26 % length of the time array27 num_points = wf(3);2829 % now keep the same file name but swap the30 % extension. The new filename will look for31 % a .dat file with the same name as .wf32 fname = regexprep(fname, '.wf', '.dat');3334 % the full file name is path + name35 file = [path, fname];3637 % open a file as read-only and return an ID to38 % the file handle39 fid = fopen(file);4041 % read from the file with the ID = fid. Inf12442 % specifies the number of bytes to read, in43 % this case everything. 'uint16'44 % specifies the format of the binary data45 % which is unsigned 16 bit integer46 % (the number of points on the y axes of the47 % scope starting from zero. 'b' stands for48 % bigendian. Labview uses IEEE standard to49 % record numbers as binary, we tell matlab how50 % to read that number as an integer51 rawdat = fread(fid, inf, 'uint16', 'b');5253 % close the connection to the file54 % (a good practice to always do that)55 fclose(fid);5657 % rawdata will be a long array of integers.58 % The processdat function in this class will59 % reshape and scale the data to60 % give voltages in mV. I have eliminated the61 % need to offset62 % data in the core function.63 dat=DAIO.processdat(rawdat,num_points,wf);6465 % build time array based on the number of points,66 % division per point and the offset67 t=DAIO.buildt(wf,num_points);6869 % this function will read the wl file associated70 % with other files that have the same name but71 % ends in .wl72 wl=DAIO.wlfileuni(path,fname);7374 % make sure all the connection to files are closed75 fclose('all');76 end77 function t=buildt(wf,num_points)78 t0=wf(6); % initial scope time12579 tinc=wf(5); % time increment80 tf=tinc*num_points+t0; % final time8182 % generate an array with specified start,83 % end and number of points84 t=1e6*linspace(t0,tf,num_points);85 end86 function dat=processdat(rawdat,num_points,wf)87 % reshape the rawdata based on the number of ...point in each88 % single trace89 dat = reshape(rawdat, num_points, ...max(size(rawdat))/...90 num_points);91 %just reshape it9293 % convert integers to double94 dat = double(dat'); %convert to double9596 % convert raw values to mV97 dat=-dat*wf(8); %convert to mV98 end99100 function wf=readwf(path,fname)101 % this function will read the waveform file ...that containst some102 % information about the103104 % build full path to the file105 fname = [path fname];106107 % open file as read-only108 fID=fopen(fname);109110 % read one unit of binary data in the format ...specified as 'b' =111 % bigendian. The unit will depend on the ...126format. int16 wil read112 % 2 bytes, while double will read 8 bytes113 wf(1)=double(fread(fID,1,'int16=>int16','b'));114 wf(2)=fread(fID,1,'int16=>int16','b');115 wf(3)=fread(fID,1,'int32=>int32','b');116 wf(4)=fread(fID,1,'int32=>int32','b');117 wf(5)=fread(fID,1,'double=>double','b');118 wf(6)=fread(fID,1,'double=>double','b');119 wf(7)=fread(fID,1,'int32=>int32','b');120 wf(8)=fread(fID,1,'double=>double','b');121 wf(9)=fread(fID,1,'double=>double','b');122 wf(10)=fread(fID,1,'int32=>int32','b');123124 % close connection to all files125 fclose('all');126 end127 function [t,d,wf]=readwl(path,fname,lines)128 fname2=regexprep(fname,'.wf','.wl');129 d=importdata([path fname2],'\t',lines);130 wf=DAIO.readwf(path,fname)';131 d=d.data;132 t=(d(:,1)'+wf(6))*1e6;133 d=d(:,2)'*wf(8);134 fclose('all');135 end136 function [t,d,wf]=readwlraw(path,fname,lines)137 fname2=regexprep(fname,'.wf','.wl');138 d=importdata([path fname2],'\t',lines);139 wf=DAIO.readwf(path,fname)';140 d=d.data;141 t=(d(:,1)'+wf(6))*1e6;142 d=d(:,2)';143 fclose('all');144 end145 function wl=wlfileuni(path,fname)146 % this function does the same thing as147 % readwl. However, it will not require you to ...127specify the148 % number of lines of commnets149150 % prepare the name of the file with the same ...name as the .dat151 % file but with .wl extension152 fname2=regexprep(fname,'.dat','.wl');153 % fname2=[fname(1:end-3) '.wl'];154155 % open the file with read-only permission156 fid=fopen([path fname2]);157 % [path fname2]158 % get the first line159 tline = fgetl(fid);160161 % while the first line or the next one are not162 % #START continue reading lines163 while ¬strcmp(tline,'#START')164 tline=fgetl(fid);165 if tline==-1166 break;167 end168 end169170 q=1;171172 % the curser to the fid must be right after ...#START, where173 % comments end and data begins174 while ¬feof(fid)175 % read a new line176 tline=fgetl(fid);177 % split the line by \t = tab between the ...numbers178 tline=strsplit(tline,'\t');179 % convert the array of text cells to an ...array of doubles128180 wl(q,:)=str2double(tline);181182 % advance the index183 q=q+1;184 end185 % close connection to all files186 fclose(fid);187 fclose('all');188 %%% wombat189 % ,.--""""--.._190 % ." .' `-.191 % ; ; ;192 % ' ; )193 % / ' . ;194 % / ; `. `;195 % ,.' : . : )196 % ;|\' : `./|) \ ;/197 % ;| \" -,- "-./ |; ).;198 % /\/ \/ );199 % : \ ;200 % : _ _ ; )201 % `. \;\ /;/ ; /202 % ! : : ,/ ;203 % (`. : _ : ,/"" ;204 % \\\`"^" ` : ;205 % ( )206 % akg ////207 end208 end209210 end1 % ******************************2 % cons.m3 % ******************************1294 classdef cons5 properties (Constant=true)6 Avo=6.02214129e23;7 R=8.314462145468951;8 F=96485.3365;9 el=1.60217657e-19;10 epsilon=8.854187817e-12;11 epsilon2d=7.323564369075211e-18;12 kB=1.3806488e-23;13 mi=4.981733643307871e-26; % mass of NO in kg14 me=9.10938291e-31;15 mp=1.67262178e-27;16 h=6.62606957e-34;17 hbar=cons.h/(2*pi);18 G=6.67384e-11;19 c=299792458; %m/s20 a0=5.2917721092e-5;21 kBau=0.012374764324710;22 Ryd=10973731.6;23 Rydhc=2.179872000000000e-18;%2.181381270723222e-18;24 RydhcAU=1.955173475509261e+03;25 RydkB=1.579968251682269e+05;26 NOrot=1.67195; %cm-127 NOprot=1.9971945;28 NOIP=30522.45;29 NOIPN2=30522.45+11.9;30 end31 methods (Static)32 function y=NN(r,den)33 y=4*pi*den*r.^2.*exp(-4*pi*den*r.^3/3);34 end35 function [pf, eden, rden]=penningfraction(n,den)36 Rn0=n.^2*cons.a0;37 % radius of Rydb. by bohr model using ...semi-classical method38 Rmax=1.8*(Rn0*2);39 % Robicheaux paper, within this distance, 90% ...130penning ionize40 pf=1-exp(-4*pi*den*Rmax.^3/3);41 % proportion between 0 and Rmax42 eden=pf/2*den;43 % the den of electron produce is half the ...proportion44 % (1e- per partner)45 rden=(1-pf)*den;46 % this is remaining density of rydbergs47 end48 function [r,rp]=randonsphere49 x=(rand-.5)*2;50 y=(rand-.5)*2*sqrt(1-x.^2);51 z=sqrt(1-x.^2-y.^2);52 if rand>.553 z=-z;54 end55 r=[x;y;z];56 r=r(randperm(3));5758 x=(rand-.5)*2;59 y=(rand-.5)*2*sqrt(1-x.^2);60 z=sqrt(1-x.^2-y.^2);61 if rand>.562 z=-z;63 end64 rp=[x;y;z];65 rp=rp(randperm(3));66 rp=rp-r*sum(rp.*r);67 rp=rp/sqrt(sum(rp.^2));68 end69 function [r,rp]=randoncircle70 % create a vector pointing randomly71 x=(rand-.5)*2;72 y=sqrt(1-x.^2);73 if rand>.574 y=-y;13175 end76 r=[x;y];77 r=r(randperm(2));78 % do it one more time79 x=(rand-.5)*2;80 y=sqrt(1-x.^2);81 if rand>.582 y=-y;83 end84 rp=[x;y];85 rp=rp(randperm(2));8687 % subtract the common part of the two vector ...to create normal88 % ones that are prependicular89 rp=rp-r*sum(rp.*r);90 % normalize the new prependicular vector91 rp=rp/sqrt(sum(rp.^2));92 end93 function GHz=cmtoGHz(cm)94 GHz=cons.c*cm*100;95 end96 function mm=GHztomm(GHz)97 mm=cons.c/(GHz*1e9)*1000;98 end99 function cm=nmtocm(nm)100 cm=1e7./nm;101 end102 function y=lambdanm(n1,n2)103 y=1e9./(10968800*abs(1./n1.^2-1./n2.^2));104 end105 function y=yukawa(r,l)106 y=cons.el.^2./(4*pi*cons.epsilon*r).*exp(-r/l);107 end108 function y=aws(den)109 y=(3./(4*pi*den)).^(1/3);110 end132111 function y=a2d(den)112 y=(1./(4*pi*den)).^(1/2);113 end114 function y=awsden(aws)115 y=1./(4*pi*aws.^3/3);116 end117 function y=debye(Te,ne)118 y=sqrt(cons.epsilon*cons.kB.*Te./(ne*cons.el^2));119 end120 function y=debnum(Te,ne)121 y=4*pi*ne*cons.debye(Te,ne)^3/3;122 end123 function y=yukawaTene(r,Te,ne)124 y=cons.yukawa(r,cons.debye(Te,ne));125 end126 function y=ncritical(T)127 y=round(sqrt(cons.Rydhc/cons.kB./T));128 end129 function y=scaledT(Ti,Te,ne)130 y=cons.kB*4*pi*cons.epsilon*Ti*cons.debye(Te,ne)/cons.el^2;131 end132 function y=scaledn(ni,Te,ne)133 y=4*pi*ni*cons.debye(Te,ne).^3/3;134 end135 function y=g(den,Te)136 y=cons.el^2./(4*pi*cons.epsilon*cons.kB*Te*cons.aws(den));137 end138 function y=we(den)139 y=sqrt(den*cons.el^2/(cons.me*cons.epsilon));140 end141 function y=wpi(den)142 y=sqrt(den*cons.el^2/(cons.mi*cons.epsilon));143 end144 function y=av(r,N)145 y=sqrt(sum(r.^2.*N,2)./sum(N,2)/3);146 end147 function y=scaledtoEn(scEn,den)133148 y=scEn*cons.me*cons.aws(den).^2.*cons.we(den).^2/cons.kB;149 end150 function y=scaledtot(sct,den)151 y=sct/cons.we(den);152 end153 function y=encm(n)154 y=cons.Ryd/100/n^2;155 end156 function y=dErot(J1,J2,B)157 y=B*(J1*(J1+1)-J2*(J2+1));158 end159 function y=EF(n,J,a)160 y=((cons.encm(n)-cons.dErot(2,J,2))/a)^2;161 end162 function n=boundton(En,den)163 n=(-cons.scaledtoEn(En,den)*cons.kB/cons.Rydhc).^(-1/2);164 end165 function str=gettimedate()166 str=datestr(now);167 str=strrep(str,':','_');168 str=strrep(str,'-','_');169 str=strrep(str,'.','_');170 str=strrep(str,' ','_');171 end172 function cc=fitgauss(t,data,tlim,tag)173 [a,¬]=size(data);174 ind=tlim(1):tlim(2);175 t=t(ind)';t=t(:);176 data=data(:,ind);177 ft=fittype('gauss1');178 cc=zeros(a,3);179180 parfor i=1:a181 y=data(i,:);y=y(:);182 fo(i)=fitoptions(ft);183 fo(i).Lower=[0 min(t) 0];184 ub=max(y)-min(y)+eps;134185 fo(i).Upper=[ub*1.2 max(t) max(t)-min(t)];186 if strcmp(tag,'ramp')187 [¬,b]=findpeaks(y,'minpeakdistance',100,...188 'minpeakheight',mean(y)/2,'sortstr','descend');189 if length(b)≥1190 b=b(1);191 fo(i).StartPoint=[y(b) t(b) 1];192 else193 fo(i).StartPoint=[mean(y)/2 ...mean(t) 1];194 end195 elseif strcmp(tag,'fixed')196 fo(i).StartPoint=[mean(y)/2 mean(t) ...max(t)-min(t)];197 end198 cf=fit(t,y,ft,fo(i));199 cc(i,:)=coeffvalues(cf);200 end201 end202 function EF=ntoEF(n,tp)203 % n=(5.14E+9./(16*EF)).^(1/4) -> ...EF=5.14E+9./(16*n.^4)204 if strcmp(tp,'a')205 EF=5.14E+9./(16*n.^4);206 elseif strcmp(tp,'d');207 EF=5.14E+9./(9*n.^4);208 end209 end210 function y=hgauss(x,cc)211 a=cc(1);212 b=cc(2);213 c=cc(3);214 y=a*exp(-(x-b).^2/c^2);215 end216 function tau = rydperiod(n)217 %\tau^2 = {4\pi^2\mu \over kZe^2}a^3218 tau = ...135sqrt(16*pi^3*cons.epsilon*cons.me/(cons.el^2)...219 *(cons.a0*1e-6*n^2)^3);220 end221 function tau = rydperiodscaled(n,den)222 %\tau^2 = {4\pi^2\mu \over kZe^2}a^3223 tau = cons.rydperiod(n)*cons.we(den);224 end225 end226 end1 % ******************************2 % SFI_data_analysis.m3 % *******************************45 %% 1) INITIALIZATION SECTION6 originaldate=''; % ***UPDATE THIS***78 list_of_PQN=[55]; % list of all principal quantum numbers ...that have been collected and I want to analyze9 list_of_grid_positions=[110]; % list of all grid positions ...that have been collected and I want to analyze10 list_of_w1_powers=[4]; % list of all w1 powers that have ...been collected and I want to analyze in uJ11 list_of_ramp_delays=[0]; % list of all ramp_delays that ...have been collected and I want to analyze12 list_of_w1_w2_delays=[0]; % list of all w1_w2_delays that ...have been collected and I want to analyze. the 10000 ...is the second 0 delay after retaken.1314 number_of_traces=3000; %the first X number of traces to ...analyze.1516 for rommel=list_of_PQN17 for markus=list_of_grid_positions18 for luke=list_of_w1_powers13619 for rafael=list_of_ramp_delays20 for mahyad=list_of_w1_w2_delays2122 PQN=rommel; %23 grid_position=markus; % in [mm]24 w1_power=luke; % in [uJ]25 ramp_delay=rafael; % in [ns]26 w1_w2_delay=mahyad; % in [ns]272829 path=strcat(['C:\Users\...\Data\2018\01jan\02\'])303132 traces=DA.readspec([path 'pfi-5kohm-3000V\',...33 'PQN' num2str(PQN),'\',...34 'grid' num2str(grid_position),'mm\',...35 'w1 Power ' num2str(w1_power),'uJ\',...36 'RampDelay' num2str(ramp_delay) ...'n\w1w2Delay' num2str(w1_w2_delay) 'n\']);373839 ps=DA.readPS([path 'ps-5kohm-3000V\'], 19);404142 filename=strrep(traces.name,'_',' '); % replace ...underscores (!)4344 % 2) Modified data: Only keeping the first 1000 ...traces. It returns an error if too few traces are ...collected45 if length(traces.wl)<number_of_traces46 error('Not Enough Traces in Collected Data')47 end4849 modified_data=zeros(number_of_traces,length(traces.t));50 for p=1:number_of_traces51 for pp=1:length(traces.t)13752 modified_data(p,pp)=traces.data(p,pp);53 end54 end555657 clf;58 clearvars eliminate_first p pp;59 %removing the variables which are not going to60 % be used anymore. To keep the WorkSpace as less61 %crowded as possible, and to make the memory free.6263 % 3) Background subtraction, finding the maximum data ...value, and the total number of valid traces.6465 modified_data=modified_data - ...mean(modified_data(:,1:50),2) * ...ones(1,length(traces.t));6667 datamax=max(modified_data(:));6869 numbertraces=size(modified_data,1);7071 % 4) Calculating sum of signal on each trace, from a ...start point to an end point.7273 trace_start=1; % ADJUST AS NEEDED74 trace_end=4950; % ADJUST AS NEEDED75 total=zeros(numbertraces,1); % preallocation7677 for k=1:numbertraces78 index(k)=k;79 for kk=trace_start:trace_end80 if modified_data(k,kk)<081 modified_data(k,kk)=0; % eliminating ...negative array values82 end83 total(k)=total(k) + modified_data(k,kk); ...138%calculating sum of the signal in the k'th trace.84 end85 end8687 clearvars trace_start trace_end k kk; %removing the ...variables which are not going to be used anymore.88 % 5) Sorting traces based on the total signal in each ...trace89 [sortedtotal,order]=sort(total,1,'descend'); % Gives ...two matrices: one has the sorted total signal ...(sortedtotal). The other one (order) has the same ...size as "total" and describes the arrangement of ...the elements of "total" into "sortedtotal" along ...the sorted dimension (which is 1 here)90 sortedtraces=modified_data(order,:); %Then here we use ..."order" indexes to sort the traces.91 % % 6) Plot unsorted data92 %imagesc(traces.t,index,modified_data);93 %axis([0 1.5 1 numbertraces]);94 %s1='--unsorted';95 %s=[filename s1];96 %title(s);97 %xlabel('Arrival time (\musec)','fontsize',12);98 %ylabel('Trace number (random)','fontsize',12);99 %set(gca,'YTickLabel',num2str(get(gca,'YTick').'))100 %%pause(2);101102 clearvars s s1;103 % % 7) Plot sorted data104 %imagesc(traces.t,sortedtotal,sortedtraces);105 %axis([0 1.5 min(sortedtotal) max(sortedtotal)]);106 s2='--sorted';107 s=[filename s2];108 %title(s);109 %xlabel('Arrival time (\musec)','fontsize',12);110 %ylabel('Integrated signal (relative)','fontsize',12);111 %set(gca,'YTickLabel',num2str(get(gca,'YTick').'))139112 %set(gca,'YDir','normal');113 %%pause(2);114115 clearvars s2;116 % 8) Calculate the Electric Field117118 dist=(155.5-grid_position)*1e-1; %distance between G1 ...and G2 dist=(155.5-grid_position)*1e-1 where ...grid_position is in [mm]119120 t=traces.t;121 shift=0.042; % 42 ns for earliest electron arrival 0.042122 EF = feval(ps.cf,t-shift)/dist; % ***PICK CORRECT PS ...TO ANALYZE***123124 clearvars m offset shift t;125126 % 10) Calculating N^+=0 and N^+=2 fild tresholds (Why ...do I get them in wrong order?)127128 R=109737.30; % Rydberg constant in (cm^-1)129 B=1.98; % Rotational constant for excited NO.130 BE_0_N=R/(PQN^2); % Binding Energy of N+=0131 BE_2_N=(R/(PQN^2))-(B*2*(2+1)); %Binding Energy of N+=2132 F_0_N=round((BE_0_N/4.12)^2); % Field treshold for N+=0133 F_2_N=round((BE_2_N/4.12)^2); % Field treshold for N+=2134135 clearvars n R B BE_0_N BE_2_N;136 % 10) Plot raw data as pcolor image137138 %Electro counts on different regions of the SFI ...spectrums (Plasma, Rydberg1, and Rydberg2 peaks)139 plasma_start=0;140 plasma_start_index=100; %this is the first index when ...plotting the "pcolor" below.141142 plasma_end=F_2_N-25; %the constant needs to be changed ...140for each PQN.143 plasma_end_index=find(EF>plasma_end-0.1 & ...EF<plasma_end+0.1,1);144145 N_zero_start=plasma_end;146 N_zero_start_index=plasma_end_index+1;147148 N_zero_end=F_0_N-25; %the constant needs to be changed ...for each PQN.149 N_zero_end_index=find(EF>N_zero_end-0.1 & ...EF<N_zero_end+0.1,1);150151 N_two_start=N_zero_end;152 N_two_start_index=N_zero_end_index+1;153154 N_two_end=F_0_N+70; %the constant needs to be changed ...for each PQN. Stop right before the ions peak.155 N_two_end_index=find(EF>N_two_end-0.1 & ...EF<N_two_end+0.1,1);156157 total_plasma = sum(sum(sortedtraces(:, ...plasma_start_index:plasma_end_index)));158 total_rydberg_zero = sum(sum(sortedtraces(:, ...N_zero_start_index:N_zero_end_index)));159 total_rydberg_two = sum(sum(sortedtraces(:, ...N_two_start_index:N_two_end_index)));160161 clf;162163 %Y(1,:)=[numbertraces:-1:1];164 %pcolor(EF(175:end),Y,sortedtraces(:,175:end)); %not ...sure if this is working properly. should the ...plasma not be at the very left of the plot?165 %axis([0 350 0 numbertraces]); % Ramp field appearance ...thresholds166167 figure('Visible','off')141168 pcolor(EF(100:end), sortedtotal, sortedtraces(:,100:end));169 axis([0 350 min(sortedtotal) max(sortedtotal)]); % ...Ramp field appearance thresholds (I do not ...understand this comment!?)170 colormap('jet');171 shading interp;172 title(s);173 xlabel('Ramp Field (V/cm)','fontsize',12);174 ylabel('Integrated signal (relative)','fontsize',12);175176 vline(F_2_N,'w'); % ADJUST for each n value177 vline(F_0_N, 'w'); % ADJUST for each n value I ...should read about N^+=0 and 2.!!!178179 vline(plasma_start,'r*'); % ADJUST for each n value180 vline(plasma_end,'r*'); % ADJUST for each n value181 vline(N_zero_start,'yo'); % ADJUST for each n value182 vline(N_zero_end,'yo'); % ADJUST for each n value183 vline(N_two_start,'gx'); % ADJUST for each n value184 vline(N_two_end,'gx'); % ADJUST for each n value185186 text(220,0.9*max(sortedtotal),strcat(['Total Plasma ...electrons = ' num2str(round(total_plasma,-3)) ...'.']),'color','magenta','FontSize',13);187 text(220,0.85*max(sortedtotal),strcat(['Total N^+=0 ...electrons = ' ...num2str(round(total_rydberg_zero,-3)) ...'.']),'color','magenta','FontSize',13);188 text(220,0.80*max(sortedtotal),strcat(['Total N^+=2 ...electrons = ' num2str(round(total_rydberg_two,-3)) ...'.']),'color','magenta','FontSize',13);189190 text(F_2_N+1,min(sortedtotal),'N^+=0','color', ...'magenta','FontSize',12);191 text(F_0_N+1,min(sortedtotal),'N^+=2','color', ...'magenta','FontSize',12);192 set(gca,'YDir','normal');142193194 clearvars Y;195 aaaaa_last_completed_Section=10;196197 saveas( gcf, filename, 'png' )198 end199 end200 end201 end202 end143

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