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The evolution from Rydberg gas to plasma in an atomic beam of Xe : with comparative simulations to a… Hung, Jachin 2015

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The Evolution from Rydberg gas toPlasma in an Atomic Beam of XeWith Comparative Simulations to a Strongly BlockadedRydberg Gas of RbbyJachin HungB.Sc., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)May 2015c© Jachin Hung 2015AbstractWe study a supersonic beam of cold, dense, xenon Rydberg atoms as it evolves to an ultracoldplasma. At early times, while the free electron density is low, d-series Rydbergs atoms undergolong-range `-mixing collisions producing states of high orbital angular momentum. These high-`states drive dipole-dipole interactions where Penning ionization provides a seed of electrons in acloud of Rydberg atoms excited into the 51d state. The electron density increases and reachesthe threshold for avalanche into plasma at 25 µs. After 90 µs the plasma becomes fully formeddeveloping rigidity to a 432 V/cm ionizing field as well as sensitivity to a weak 500 mV/cm field.A shell model was developed to understand the dynamics behind this process.In addition, in collaboration with the Weidemu¨ller group, a model was developed using Pen-ning ionization to seed the spontaneous avalanche of a cloud of strongly blockaded Rydberg atomsin a MOT.iiPrefaceThis thesis was a result of the collaborative effort from the Ultracold plasma group of the UBCGrant Lab. The experiments were done in collaboration with PhD students Hossein Sadeghi andMarkus Schulz-Weiling. My responsibilities included planning and performing the experimentsas well as analysing the data we obtained.I wrote MATLAB simulations describing the experiment built upon previous MATLAB codeimplemented by Hossein. Furthermore in collaboration with the Weidemu¨ller group in Heidelberg,I developed simulations to describe the spontaneous ionization of a strongly blockaded cloud ofRydberg atoms in a MOT.Finally this manuscript was written by myself, helped by the many thoughtful suggestionsprovided by the current members of the Grant lab.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Ultracold Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Rydberg Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 General Background and History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 The Bohr model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Rydberg Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Rydberg Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Angular Momentum Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Physics of Ultracold Neutral Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Penning Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Ion, Electron, and Rydberg Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Vlasov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.1 Zeroth moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 First moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.3 Plasma parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Debye Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20ivTable of Contents4 The Creation and Detection of a Supersonic UCP . . . . . . . . . . . . . . . . . 234.1 Creation of a Supersonic UCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Detection of a Supersonic UCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Formation of a Supersonic UCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.1 Selection of Rydberg atoms in state n0, `0 . . . . . . . . . . . . . . . . . . 285.1.2 The time evolution of Rydberg atoms . . . . . . . . . . . . . . . . . . . . . 305.2 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.1 Kinetics of `-mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.2 Electron avalanche and the evolution to plasma . . . . . . . . . . . . . . . 355.3 Comparison with Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Simulation of a Strongly Blockaded Rydberg Gas into Plasma . . . . . . . . . 396.1 Creation of a Strongly Blockaded Rydberg Gas [20] . . . . . . . . . . . . . . . . . 396.2 Detection of a Strongly Blockaded Rydberg Gas [20] . . . . . . . . . . . . . . . . . 406.3 Observation of Spontaneous Ionization [20] . . . . . . . . . . . . . . . . . . . . . . 406.4 Original Rate-Equation Modelling Avalanche Dynamics [20] . . . . . . . . . . . . 416.5 Updated Rate-Equation Modelling Avalanche Dynamics . . . . . . . . . . . . . . . 436.6 Seeding with Penning Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.7 Attractive 55d states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Conclusion and Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57vList of Tables2.1 Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Rydberg Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11viList of Figures1.1 Types of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Standard Radial Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1 Schematic Diagram of UCP Formation . . . . . . . . . . . . . . . . . . . . . . . . . 154.1 Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Excitation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Ionizing Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Weak Forward Bias Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.1 Selection of Metastable 6p Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Selection of Rydberg States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Typical Waveform of Rydberg Detection . . . . . . . . . . . . . . . . . . . . . . . . 305.4 FI Response of 51d Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 FI Spectrum as a Function of n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.6 Survival Fraction with FI Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.7 Electron Signal with Pulse Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.1 Ionization Dynamics of a Blockaded Rydberg Gas. . . . . . . . . . . . . . . . . . . 416.2 Rydberg Evolution in a MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Updated Rydberg Evolution in a MOT with Temperature Change . . . . . . . . . 466.4 Rydberg Evolution in a MOT with Temperature Change and Expansion . . . . . . 476.5 Simulation with Penning Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.6 Simulation with Weighted Penning Ionization . . . . . . . . . . . . . . . . . . . . . 516.7 Ground State Population Pumping into 55s states at the Full Volume . . . . . . . 526.8 Ground State Population Pumping into 55s states at the Half Volume . . . . . . . 536.9 Ground State Population Pumping into 40s states at the Half Volume . . . . . . . 546.10 Log-log Scale of tcrit vs Atomic Density . . . . . . . . . . . . . . . . . . . . . . . . 546.11 Predicted and Experimental Data of Ground State Atoms for 55d-states . . . . . . 55viiAcknowledgementsI am deeply grateful to Dr. Edward Robert Grant for his guidance and support as I underwentthis journey in the Grant Lab. He has been more than what is expected from a supervisor. Inaddition to his insight, opportunities, and resources for research, he has provided genuine interestin my career and life as well.I am also indebted to Hossein Sadeghi for his always thoughtful insight and for his patiencein teaching me experimental (and theoretical) plasma research. To Markus Schulz-Weiling, whoeffectively maintained his laser and gave us time to use it for our experiments. To Martin andthe Weidemu¨ller group for their collaboration with the Rb experiment.I need to also thank Julian Yiu, Ashton Christy, Najme Tavassoli, Zhiwen Chen, and AllisonBain, for the encouragement and much fun had with you guys.This journey could not have been completed without any of the these people. Thanks!viiiDedicationTo my brother Boaz for the thoughtful discussions regarding whatever and whenever. You’vetaught me much of what I know about life. Thank you.ixChapter 1IntroductionPlasma forms as a distinct state of matter when a collection of ionized and neutral particlesreaches a charge density sufficient to exhibit collective behaviour. The charged particles mayequilibrate or quasi-equilibrate with the neutral atoms or molecules giving a plasma its stateproperties that distinguishes it from the three standard states of matter: solid, liquid, and gas.Its unique properties and great natural abundance characterizes it as a unique state of matter.The plasma state exists in a wide variety of environments and spans a large range of densities,and temperatures. We encounter plasma in our everyday lives in the form of fire, lightning,fluorescent lights, televisions, and in a great variety of other natural and engineered environments.More distant but no less import, plasmas give rise to natural phenomena like the Aurora Borealisin the ionosphere to the genesis of planets in space.Research on plasma provides insight on exotic environments such as the interiors of planetsand stars and has lead to applications in materials fabrication, light sources, and fusion energy.1.1 The Ultracold PlasmaThe Saha equation, seen in equation 1.1, describes the degree of ionization in a gas.ninn= 2.4× 1015T 3/2nie−U/kBT . (1.1)Here ni is the density (cm−3) of ionized gas, nn is the density of the neutral gas (cm−3), T is thetemperature in Kelvin, kB is Boltzmann’s constant, and U is the ionization energy of the gas.Under standard room temperature conditions, this fraction is very low [1]ninn≈ 10−122.Usually this fraction only becomes significant at high temperatures or in the presence of non-thermal sources of energy. As a result, plasmas typically exist in a state of high temperatureequilibrium. Naturally, plasmas form at 150 K in the ionosphere to while thermonuclear fusionignites a plasma with a temperature of 15×106 K in the core of the sun.Figure 1.1 plots a variety of plasmas at different temperatures and densities. A black lineindicates the case where a parameter, Γ, has a value of 1. The coupling parameter, Γ, defined by11.1. The Ultracold Plasmathe ratio of the Coulomb interaction energy to the average kinetic energy [3]Γ =q24pi0rskBT, (1.2)where 0 is the permittivity of free space and rs is the Wigner-Seitz radius indicates the degree ofcoupling. This factor distinguishes systems said to be strongly couple when Γ is greater than one[3]. For typical plasmas in a laboratory or in the solar corona, we find Γ ≈ 10−5 and Γ ≈ 10−7respectively [3]. These plasmas are weakly coupled and their physics can be explained using wellunderstood ideal plasma models.Figure 1.1: A survey of neutral plasmas at different temperatures and densities. Plasma existsin a large range of temperatures and densities. The solid line indicates the case where Γ = 1.Strongly coupled plasmas exhibit spatial-correlations manifested by short-range particle inter-actions which arises when the potential energy becomes greater than the thermal kinetic energy.These systems provide experimental models on fundamental physics such as many body systems,collective modes, instabilities and transport. Recently laboratories have created a new form ofplasma known as an ultracold neutral plasma (UCP) [2] with temperatures as low as 1K. Asseen in figure 1.1, UCP expands the range of plasma conditions to access the regime of strongcoupling.Crossing this boundary between traditional plasmas created in the laboratory and more exotic21.1. The Ultracold Plasmastrongly coupled plasmas, UCPs offer a way to reach the strongly correlated regime in a tabletop setting. Furthermore, owing to their very low temperature, UCPs can be prepared with welldefined initial conditions which provide a suitable platform for the study of creation and evolutiondynamics.The first ultracold plasma was created in 1999 by researchers at the National Institute ofStandards and Technology (NIST) [4]. They achieved this by photoionizing laser-cooled xenon(Xe) in a magneto-optical trap (MOT). Subsequently, UCPs have been created in a MOT usingrubidium, caesium, strontium, and calcium; practically any atom that can be laser cooled.In a MOT, densities of 109 cm−3 and ion temperatures of millikelvin can be easily reached.Naively using these parameters in equation 1.2 would result in a correlation number upwardsof Γ ≈ 6000! Unfortunately, this promise has proven elusive. When the initially randomlydistributed atoms become ionized (as in a plasma), the positively charged ions repel and movetowards a position of lower energy. This phenomena is called disorder-induced heating (DIH) andheats the system up to ≈ 2 K on the order of hundreds of nanoseconds [2].Recently, the Grant lab created an ultracold neutral plasma of molecular nitric oxide (NO) ina supersonic beam [5]. Although the molecular beam creates a plasma from initial temperaturesmuch higher than a MOT, disorder-induced heating is thought to quickly bring the temperaturein the two types of systems to a similar point.The supersonic beam method provides great versatility in creating a UCP. It creates UCPsat higher densities than a MOT and allows molecular systems to be studied as well. This givesthe possibility of intrinsically molecular phenomena such as dissociative recombination and pre-dissociation.To date, no experiment directly compares the UCP created in a supersonic beam with oneformed in a MOT. This thesis is the culmination of work in creating a UCP using Xe, the originalUCP atom, with a supersonic beam. It compares the similarities and differences between aMOT UCP and a supersonic beam UCP. The signatures across the two types of systems are wellmaintained.This thesis ends by describing simulations created to explain the formation of plasma froma cloud of Rydberg atoms in a MOT. These simulations were performed to explain evolutiondynamics observed by the Weidemu¨ller group in Heidelberg.This thesis consists of 7 chapters. Chapter 1 - Introduction: introduces and motivates the workdone in this thesis. Chapter 2 - Rydberg Atoms: provides the general background of Rydbergphysics and detection methods to understand the experiment. Chapter 3: Physics of UltracoldNeutral Plasmas: describes the basic physics in understanding ultracold plasmas. Chapter 4 -The Creation and Detection of a Supersonic UCP: outlines the method used to create an ultracoldneutral plasma. Chapter 5 - Formation of a Supersonic UCP: discusses the evolution dynamicsfrom a cloud of Xe Rydberg states into an ultracold plasma. Chapter 6 - Simulation of a StronglyBlockaded Rydberg Gas into Plasma: describes a simulation to understand the spontaneous31.1. The Ultracold Plasmaionization of a cloud of strongly blockaded Rydberg gas in a MOT. Chapter 7 - Conclusion andFuture Outlook: provides a summary of the work done in this thesis and covers what future workcan be done on UCPs.4Chapter 2Rydberg AtomsHigh lying Rydberg atoms can be formed when recombination takes place between ions and elec-trons. In a plasma where ions and electrons are both present in abundance, Rydberg atoms canbe commonly found. They possess a large orbital radius which causes them to be highly suscep-tible to perturbations and this affects the static charge and kinetic energy of the plasma. Theionization of Rydberg atoms through ionizing collisions, photoionization, autoionization has beenused as a mechanism to create UCPs. Understanding the behaviour of plasmas thus necessitatesthe understanding of Rydbergs as well.This chapter provides some background information on basic Rydberg physics which will helpto explain the formation of ultracold neutral plasmas.2.1 General Background and HistoryRydberg atoms are excited atoms in a state of high principal quantum number (PQN) n. Theseatoms exhibit many exaggerated and unique properties. Consequently they have been the subjectof much interest over many years.The first appearance of Rydberg atoms in literature comes from the study of atomic spec-troscopy. In 1885, the Balmer’s formulaλ =bn2n2 − 4, (2.1)where λ is the wavelength, b = 3645.6A˚, and n is an integer greater than 2, relates the wavelengthsbetween different transitions seen in atomic hydrogen (H) [6]. This quantitative analysis ledto similar classifications being done with different Alkali atoms. When Hartley was studyingmagnesium (Mg) spectra, he realized that Balmer’s formula could be rewritten in the form ofwavenumbers instead of wavelengthν =(14b)(14−1n2), (2.2)where ν is the wavenumber. With this, it soon became evident that this formula reflects thetransitions between the n = 2 state with higher n states.Following this work, Rydberg realized that these Alkali spectrums could be expressed in terms52.1. General Background and Historyof different series: s (sharp), p (principal), and d (diffuse). Each series could then be expressedin terms of a series constantνl = ν∞l −R(n− δl)2, l = s, p, d (2.3)where ν∞l is the series constant, δl is the quantum defect, and R is the universal Rydberg constant.This realization paved the way for constructing energy diagrams in early atomic spectroscopy.However the significance of this work was not truly realized until 1913, when Bohr proposedhis model of the hydrogen atom [6]. Although we now know that the Bohr model of atoms doesnot fully explain the properties of atoms, it highlights the basic properties of Rydberg atoms thatmake them interesting.2.1.1 The Bohr modelThe Bohr model consists of a classical electron orbiting a positive core. Equating the centripetaland the Coulomb forcemv2r=Ze24pi0r2, (2.4)where Z is the ion charge, 0 is the permittivity of free space, Bohr quantized the angularmomentum in integer units of Planck’s constant (h),mvr = nh2pi= n~ . n = 1, 2, 3, . . . (2.5)Furthermore Bohr changed classical theory such that accelerating charges do not continuouslygive off radiation, but instead energy is released in discrete transitions. Accepting this, theexpected orbital radius, r, is found to ber =n2~24pi0Ze2m= a0, (2.6)where a0 is the now famous Bohr radius. The total energy of the system, kinetic plus potentialenergy, can be expressed in these termsE =mv22−Ze24pi0r2= −Z2e4m32pi220n2~2, (2.7)where the negative final energy indicates that the electron is in a bound state. Similarities canbe seen to the Rydberg formula in equation 2.3 if we rewrite the Rydberg constant asRy =Z2e4m32pi220~2. (2.8)The Bohr formalism was constructed for the simplest case of hydrogen. However a Rydberg62.2. Rydberg Wavefunctionatom is simply an atom with an electron in an excited state. Far away from the core, the electronfeels an effective charge of +1 resembling the situation of the electron in hydrogen. As a result,the behaviour of Rydberg atoms follows the trends predicted by the Bohr model well.From equation 2.6, the radius grows as n2 and consequently the geometric cross-section asn4. This leads to very large cross-sections as n increases. These large cross-sections can leadto dipole-dipole interactions and van der Waals interactions. Furthermore from equation 2.7,high n states have loosely bound electrons which makes Rydberg atoms very sensitive to electricfields. These exaggerated properties have made Rydberg atoms a great interest for classical andquantum mechanical study. Currently they have become the subject of many exotic quantummechanical measurements.2.2 Rydberg WavefunctionAlthough the semi-classical Bohr model explains the most distinguishing properties of a Rydbergatom, a more complete quantum mechanical description requires solving the Schro¨dinger equation.In order to the expressions, we use standard atomic units defined in table 2.1Quantity Atomic Unit ValueMass Electron mass 9.1×10−28gCharge Electron charge eEnergy Twice the ionization potential of hydrogen 27.2eVLength Bohr radius 0.529 A˙Electric field Field at first Bohr orbit 5.14×109 V/cmTable 2.1: Commonly used atomic units in Rydberg physics [6].We approach the problem by noting the similarities between a Rydberg atom and a hydrogenatom. The hydrogen atom consists of an electron surrounding a positive point charge. On theother hand, a many-electron Rydberg atom features an electron orbiting a positive core. Thedifference is that the hydrogen has a point sized core while the Rydberg’s core is of finite size. ForRydberg states of high n, where the electron is excited far away from the core, it is only sensitiveto the net charge imbalance. In the most common cases, this is just the same potential as forhydrogen. Therefore we will develop the Rydberg wavefunction by first solving the Schro¨dingerequation for hydrogen.Assuming that the atom is neutral and the electron only feels a point ionic core of Z = +1charge, the Schro¨dinger equation is(−∇22−1r)ψ = Eψ, (2.9)72.2. Rydberg Wavefunctionwhere ∇2 is the Laplacian, r is the distance from the core to the electron, and E is the energy.In spherical coordinates, the Laplace operator becomes∇2 =∂∂r2+2r∂∂r+1r2 sin θ∂∂θ(sin θ∂∂θ)+1r2 sin2 θ∂2∂φ2. (2.10)Following the traditional methods in solving equation 2.9, the solution is assumed separable andan ansatz of the form ψ = Y (θ, φ)R(r) is made. Substituting this into 2.9 and dividing throughby RY/r2, the Schro¨dinger equation becomesr2R[∂2R∂r2+2r∂R∂r+ 2(E +1r)R]+1Y[1sin θ∂∂θ(sin θ∂Y∂θ)+1sin2 θ∂2Y∂φ2], (2.11)where the first term is the radial component and the second term is the angular component. Sincethese terms are independent of each other, they must be equal to a negative constant, λ, of eachother. The angular component can be solved by further substituting another separable ansatz,Y (θ, φ) = Θ(θ)Φ(φ), giving the expression1sin θ∂∂θ(sin θ∂Θ∂θ)Φ +Θsin2 θ∂2Φ∂2φ= −λΘΦ. (2.12)The solutions to this equation are the normalized spherical harmonics, Ylm(θ, φ). Expressed interms of the Legendre polynomial, Pml (x), the spherical harmonics areΘ(θ)Φ(φ) = Ylm(θ, φ) =√(l −m)!(l +m)!2l + 14piPml (cos θ)eimφ. (2.13)Here l is a non-negative integer, m takes on integer values between −l to l, and λ is constrainedto be l(l + 1).Substituting this value for λ into the radial term of equation 2.11, the radial equation can berewritten as∂2R∂r2+2r∂R∂R+[2E +2r−l(l + 1)r2]R = 0. (2.14)This can be further simplified if R(r) is assumed take the standard form of a Coulomb charge,R(r) = U(r)/r. With this substitution the radial equation becomes∂2U∂r2+[2E +2r−l(l + 1)r2]U = 0. (2.15)This can be rewritten in a form that is suggestive of the original Schro¨dinger’s equation(−12∂2∂r2−1r+l(l + 1)2r2)U(r) = EU(r), (2.16)82.2. Rydberg Wavefunctionwhere the electron experiences an attractive Coulomb potential, 1/r, and a repulsive centrifugalpotential, l(l + 1)/r2. This potential can be understood qualitatively by looking at figure 2.1.Far away from the core, the effective potential is dominated by the Coulomb potential. Near thecore, the centrifugal potential plays the larger effect.Figure 2.1: The potential felt by an electron orbiting a positive core. At every point the attractiveCoulomb potential is summed with the repulsive centrifugal potential to give an effective potential.A suggested property of a solution to the radial part of Schro¨dinger’s equation can be solvedby looking at the asymptotic behaviour of r. As r →∞, equation 2.16 becomes∂2∂r2U(r) = −2EU(r). (2.17)Considering only bound states where E < 0, the general solution isU(r) = Ae−√2Er +Be√2Er. (2.18)As r →∞, e√2Er blows up so B = 0. Therefore the solution must be proportional to e−kr. Theexplicit solution can be found using the conventional power series method where the solutions are92.2. Rydberg Wavefunctionthe associated Laguerre polynomialLq(x) ≡ ex(ddx)q(e−xxq), (2.19)where q is the qth derivative [8]. The boundary condition as r →∞ ensures the series terminatesand limits the energy to beE = −12n2. (2.20)Putting together the radial and angular solution, the normalized hydrogen wave functions areψnlm =√(2na0)3 (n− l − 1)!2n[(n+ l)!]3e−r/na(2rna)l [L2l+1n−l−1(2r/na)]Y ml (θ, φ). (2.21)So far this derivation of the wave function has assumed the potential to be a positive pointcharge. This is true for a hydrogen atom, but in the case of the Rydberg atom the ionic corehas a finite size. Far away from the core, the Coulomb potential felt by the orbiting electron isindistinguishable from a point charge. However near the core, the finite size of the core causesthe electron to feel a deeper potential.As in the case of a point charge, this potential is still spherically symmetric and only dependson r. Therefore Schro¨dinger’s equation is still separable. The spherical harmonics remain thesolution to the angular equation while the radial equation replaces the Coulomb, −1/r, potentialwith an effective potential, VRydberg. To solve the radial equation, quantum defect theory [9] mustbe applied. The solution often cannot be solved analytically and numerical methods such as theNumerov method must be used.The inclusion of a finite size potential core consequently shifts the allowed energies. Theallowed energies in equation 2.20 are calculated by replacing the quantum number n with theeffective quantum number n∗n∗ = n− δl, (2.22)where δl is the quantum defect which depends on the angular momentum l. The magnitude ofthe quantum defect can be understood by looking at figure 2.1. Due to the repulsive l(l+ 1)/2r2centrifugal potential, high l states do not sample areas of small r as much. As a result electronswith a large angular momentum feel a potential only slightly shifted from a positive point charge.This is reflected by a small quantum defect for high l states. By a similar argument, low l stateshave higher quantum defects.Having set the groundwork formalizing the Rydberg wavefunction, many other properties canbe derived. This will not be done here, however many important trends are summarized in table2.2 expressed in terms of their n dependence [6].102.3. Rydberg DetectionProperty n dependenceBinding energy n−2Energy between adjacent n states n−3Orbital radius n2Geometric cross section n4Dipole moment < nd|er|nf > n2Polarizability n7Radiative lifetime n3Fine-structure interval n−3Period (τ) 2pin3Table 2.2: General properties of Rydberg atoms [6].2.3 Rydberg DetectionThe use of field ionization techniques remain useful in the study of Rydberg atoms as it is efficientand state selective. Therefore it is important to understand the evolution of a Rydberg atomunder the influence of an external field.Assuming a uniform field, F , is applied in the z direction, the potential felt by the excitedelectron of a hydrogenic Rydberg atom in atomic units isV =−1r+`(`+ 1)2r2+ Fz. (2.23)First assuming that there is no centrifugal force (` = 0), we can solve for the saddle point inequation 2.23 on the z axis. This occurs at z = −1/√F where the electron feels a potential ofV = −2√F .Knowing that the electron is bound by an energy, E, we re-write the classical field ionizingstrength to beF =E24. (2.24)Ignoring stark shifts for a given state, we can further express the field ionization threshold interms of the initial principal quantum number nF =116n4. (2.25)This threshold condition is only valid when there is no centrifugal force (m = 0). For states withhigher |m|, there exists a 1/(x2 + y2) potential that raises the threshold for ionization. Whencompared to the m = 0 state, it has been calculated that a fractional increase of∆FF=|m|√E√2=|m|2n(2.26)112.4. Angular Momentum Mixingoccurs [7]. This classical approach for the calculation of field ionization thresholds has 2 maindefects. First it ignores Stark shifts if we use the |m|/2n expression of equation 2.26. Second,the classical approach in calculating the field ionization threshold ignores the spatial distributionof the wavefunction. As can be seen in a typical Stark map, higher energy blue states require alarger field to ionize than a lower energy red state of the same n. Briefly, blue states have highm, central states have medium m, and red states have low m values.The ionizing field can be further analyzed by solving the Schro¨dinger equation in paraboliccoordinates. This reveals the required ionizing field to beF =E24Z2, (2.27)where Z2 is the effective charge. The colour of the state can be classified using the parabolicquantum numbers n1, n2, where n1, n2 are related to n and m byn = n1 + n2 + |m|+ 1. (2.28)A blue state where n1 − n2 ≈ n experiences Z2 = 1/n. A central state where n1 ≈ n2 ≈ n/2experiences Z2 = 1/2. Finally a red state where n2−n1 ≈ n experiences Z2 = 1. Expressed in thismanner, the lower energy red state is seen to ionize the easiest. A physical picture can be drawnin understanding why red states ionize faster than blue states. For blue states, the electron is ina position of the atom away from the saddle point. Meanwhile, red states are located adjacent tothe saddle point [6].2.4 Angular Momentum MixingA cloud of Rydberg atoms prepared with initial PQN, n0, and orbital angular momentum, `0,can undergo various collisional processes. Of note adiabatic non-ionizing `-changing collisionsbetween Rydberg states can occur. This is illustrated byA(n0, l0) +A′ → A(n0, l1) +A′. (2.29)The transfer of Rydberg atoms into different high-` states produces a cloud of `-mixed atomsfrom an initially pure state. This process is understood as follows: a Rydberg atom, whenapproached by passing electrons of nearby Rydberg atoms, experiences an increase in electricfield [10] which intersects its hydrogenic manifold with nearby n states. As the electron passes,the electric field decreases, but the Rydberg atom may remain in the crossed hydrogenic manifoldchanging its initial angular momentum from `0 into `1.Since the behaviour of `-mixing occurs from the interactions between two nearby Rydbergatoms, the `-mixing cross section strongly depends on the geometric size of the atom and the122.4. Angular Momentum Mixinginitial quantum number. The `-mixing cross section, σl, has been previously been calculated tobe [10]σl ≈ pin5a50. (2.30)With it’s strong n5 dependence, `-mixing is expected to occur faster for higher principal quantumnumbers for the same density of Rydberg atoms.These `-changing effects have been observed in a cloud of 51d 85Rb Rydberg atoms inside aMOT by Raithel [11]. They obtained field ionization spectra at a fixed pulsed field ionizationdelay of 38 µs after Rydberg creation. At an intermediate density of 5.5 × 108 cm−3, the fieldionization spectrum develops secondary features at higher field strengths ascribed to `-mixedRydberg states. Higher `-states requiring a higher field to ionize are described by equation 2.26in chapter 2. The limit at which this secondary peak appeared was observed to be approximately4 times the original threshold ionization field strength.13Chapter 3Physics of Ultracold Neutral PlasmasThe first UCP was created by trapping metastable xenon in a MOT by Rolston [4] at the NIST.They achieved this by photoionizing a small portion of the neutral atomic cloud above threshold.Immediately after photoionization, the charge distribution remains neutral in the cloud of Xe.But due to the extra kinetic energy given to the electrons, these electrons redistribute and escapethe cloud. As each electron escapes, it leaves behind an increasing potential well felt by theremaining electrons.This well grows in depth until it reaches the threshold condition where its potential equalsthe initial kinetic energy of the electrons. This condition depends on the initial kinetic energyof the electrons, density, and the trap size. At this point, the well becomes strong enough totrap any remaining electrons wishing to escape. Since they are unable to escape, the remainingenergetic electrons undergo collisions with other ions and atoms in the cloud. The net effect ofthese collisions build up the electron population until the remaining atoms ionize into plasma.This is illustrated schematically in figure 3.1.The growth in electron population is usually seen as a sigmoid shape. The sharp rise in electronpopulation relates to the time when the critical electron population is reached for avalanche intoa plasma.Although escaping electrons develop a potential well, the remaining gas is still consideredquasi-neutral. Since under ultracold conditions, when < 1% of electrons escape, it creates asufficiently deep well to trap the remaining electrons. The remaining sections in this chapterconsiders the physics of ultracold neutral plasmas from formation to long term dynamics.3.1 Penning IonizationIn addition to the creation of UCP through photoexcitation, clouds of high-lying Rydberg atomstrapped in a MOT have also spontaneously evolved into plasma [12]. The ionizing mechanismsfor the cloud of neutral atoms have been attributed to blackbody photoionization and ionizingcollisions between hot and cold Rydberg atoms. Similar Rydberg systems in molecular beams,lacking hot Rydberg atoms, have also evolved into plasma [5]. Furthermore since the black bodyphotoionization rate is too slow [6], another seeding mechanism, Penning ionization, has beenproposed.Penning ionization can occur between two nearby Rydberg atoms when the deexcitation of143.1. Penning IonizationFigure 3.1: Schematic diagram indicating plasma formation. Top: Ionized electrons escapethe gas cloud leaving behind a trapping potential well. The trap grows in strength until theremaining energetic electrons are unable to escape. Bottom: Electrons in the cloud escape untilthe potential well develops the strength to trap additionally ionized electrons. These trappedelectrons undergo ionizing collisions with the remaining atoms, causing an avalanche ionization.The rise in electron numbers indicates plasma releases sufficient energy to ionize the other. This process is shown symbolically byA∗ + A′∗ → A+ + A′(deactivated) + e−, (3.1)where A is a Rydberg atom with PQN = n0. The deactivated atom goes into state n with adistribution proportional to n5. This distribution reaches a maximum at n = n0/√2 due toenergy conservation [13].For a cloud of dense Rydberg atoms, Penning ionization possibly provides a seed for ionizationin a cloud of initially neutral atoms. In a cloud of statistically mixed `,m Rydberg states, therate for Rydbergs to come together has been calculated using Hamilton’s equations to be [13]T ≈ 20µs×√M(amu)R50(µm)/n2, (3.2)where R0 is the separation between atoms. For Rb in n = 55 with a separation distance of 5 µm,the time to reach strong interaction is approximately 3.4 µs.The probability for 90% ionization between 2 Rydberg atoms as a function of distance andtime has also been calculated using Monte Carlo simulations. In 400 Rydberg periods this occursup to a maximum distance of 1.6× 2n2a0, in 800 Rydberg periods up to 1.8× 2n2a0 and in 1200153.2. Ion, Electron, and Rydberg DynamicsRydberg periods up to 2.1× 2n2a0 where a0 is the Bohr radius [13].3.2 Ion, Electron, and Rydberg DynamicsAs electrons and ions form from Rydberg atoms, an array of interactions between them can occur.These effects must be considered in order to understand the dynamics of plasma formation. Asmentioned in the previous section, the build up of electrons lead to ionizing collisions betweenelectrons and Rydbergs. However a process called three body recombination (TBR) combats thecreation of ions.This involves the creation of a high lying Rydberg atom and an energetic electron throughthe interaction of 2 electrons and an ion. The competing process is illustrated asAn∗ + e−kion−−−⇀↽ −kTBRA+ + e− + e−. (3.3)This effect is particularly important in UCP physics as the energy taken away from the electronin TBR leads to heating of the plasma. The change in temperature is important as it affects therate of interactions between the ions, electrons, and atoms.In addition to TBR and ionizing collisions, n changing collisions can occur. This results whenan electron interacts with a Rydberg in state ni and produces a Rydberg atom in state nf 6= niand an electron. This is illustrated asA∗(ni) + e− k(ni,nf )−−−−−→ A∗(nf ) + e−. (3.4)In 1979, Mansbach and Keck (MK) did seminal work investigating these rates using MonteCarlo simulations [14]. These rates have been recently updated by Pohl, Vrinceanu, and Sadegh-pour in 2008 [15] to include the consequences of small energy transfers. They fitted Monte Carlosimulations to obtain the following rate expressions for n changing collisionsk(ni, nf ) = k0ε5/2i ε3/2fε5/2>e−(εi−ε<)[22(ε> + 0.9)7/3+9/2ε5/2> ∆ε4/3], (3.5)for three body recombinationktbr(ni) =11(R/kBT )1/2k0e−εiε7/3i + 4.38ε1.72i + 1.32εin2iΛ3ρeeεi , (3.6)and for ionizing collisionskion(ni) =ktbr(ni)n2iΛ3ρeeεi, (3.7)where R ≈ 13.6eV is the Rydberg constant, k0 = e4/(kBT√mR), εi(f) = R/n2i(f)kBT , ∆ε =163.3. Vlasov Equation|εf − εi|, ε< = min(εi, εf ), ε> = max(εi, εf ), Λ =√h2/2pimkBT is the thermal de Brogliewavelength, ρe is the electron density, m is the mass, and kB is the Boltzmann constant.3.3 Vlasov EquationIn order to develop a deeper understanding of the plasma, the velocity distribution of the particlesf = f(r,v, t) (3.8)must be considered. Explicitly, f(r,v, t) refers to the number of particles at position r and timet with velocity between vx + dvx, vy + dvy, and vz + dvz. In the simplifying limit where all thevelocities are assumed to be Maxwellian everywhere, fluid dynamics suffices and f is reduced to4 variables f(r, t).In cases where this is not true, kinetic theory must be applied where solutions to f(r,v, t)must satisfy the Boltzmann equation∂f∂t+ v · ∇f +Fm·∂f∂v=(∂f∂t)c. (3.9)Here F is the force on the particles, (∂f/∂t)c is the rate of change of f due to collisions, and∂/∂v is the gradient in velocity space.The physical meaning of the Boltzmann equation becomes clear by applying the chain rulefor the total time derivative on f . Expanding df/dt, we havedfdt=∂f∂t+∂f∂xdxdt+∂f∂ydydt+∂f∂zdzdt+∂f∂vxdvxdt+∂f∂vydvydt+∂f∂vzdvzdt. (3.10)The first term ∂f∂t is just the explicit time dependence while the next 3 terms can be seen as v ·∇fin equation 3.9. The last 3 terms are can be understood using Newton’s lawdvdt=Fm. (3.11)It is clear this can be rewritten as F/m · ∂f/∂v which is just the third term of the Boltzmannequation. Therefore the Boltzmann equation merely says that df/dt stays the same unless thereare collisions.In the simplifying assumption that collisions do not occur (true in many hot plasmas), thecollisional term (∂f/∂t)c = 0 and equation 3.9 simplifies to the well known Vlasov equation∂f∂t+ v · ∇f +qm(E + v ×B) ·∂f∂v= 0. (3.12)The force has been rewritten to be entirely electromagnetic in the absence of collisions.173.3. Vlasov Equation3.3.1 Zeroth momentThe zeroth moment of the Boltzmann equation is solved by integrating equation 3.9 over dv. It isso named because it is multiplied by a factor of v0 whereas the higher nth moments are multipliedby a factor of vn.Doing so, the Boltzmann equation becomes∫∂f∂tdv +∫v · ∇fdv +∫qm(E + v ×B) ·∂f∂vdv =∫ (∂f∂t)cdv. (3.13)The first term is rewritten such that∫∂f∂tdv =∂∂t∫fdv =∂n∂t, (3.14)where n is the density of the cloud at position r and time t.Similarly the second term is rewritten by taking ∇ out of the integrand which is over velocityspace ∫v·∇fdv = ∇ ·∫vfdv = ∇ · (nu). (3.15)Here u is defined as the average velocity. The third term in equation 3.13 can be shown to vanishusing the divergence theorem∫E ·∂f∂vdv =∫∂∂v· (fE)dv =∫S∞fE · dS = 0. (3.16)This must be true since f → 0 at the surface infinitely far away in order for the distribution tobe finite. The v ×B from equation 3.13 is also shown to be 0 using the product rule∫(v ×B) ·∂f∂vdv =∫∂∂v· (fv ×B)dv −∫f∂∂v· (v ×B)dv = 0. (3.17)Finally the collisional term in equation 3.13 is∫ (∂f∂t)cdv =∂∂tcn = 0 (3.18)since collisions cannot change the number of particles over time. Combining these results yieldsthe famous continuity equation∂n∂t+∇ · (nu) = 0. (3.19)183.3. Vlasov Equation3.3.2 First momentThe first moment is calculated by multiplying the Boltzmann equation by mv and integratingover dv. This turns equation 3.9 intom∫v∂f∂tdv +m∫vv · ∇fdv + q∫v(E + v ×B) ·∂f∂vdv =∫mv(∂f∂t)cdv. (3.20)The solution to this equation can be found in any traditional textbook, such as [1], to give thefluid equationmn[∂u∂t+ (u · ∇)u]= qn(E + u×B)−∇ ·P + Pij , (3.21)where P ≡ mnww is the stress tensor and Pij is the change in momentum due to collisions.3.3.3 Plasma parametersIn the context of ultracold neutral plasmas, the plasma is assumed to be of Gaussian phase spacedensity since they are either held in MOTs or spontaneous ionized Rydberg states excited bylasers. For a Gaussian phase space density, an ansatz offi ∝ exp(−∑kr2k2σ2k)exp(−∑kmi(vk − γkrk)22kBTi,k), (3.22)where k = x, y, z can be used as a solution to equation 3.12. Substituting this spherical Gaussianform into equation 3.12, 3.13, and 3.20 yields the following rate equations for the plasma width∂∂tσ2 = 2γσ2, (3.23)and hydrodynamic ion velocity∂∂tγ2 =(kBTe + kBTi)miσ2− γ2. (3.24)As the Vlasov equation only holds for non-collisional plasmas, the full Boltzmann equation mustbe solved for collisional processes. Including three-body recombination, electron impact ioniza-tion, and electron induced transitions, changes the evolving ion velocity to∂∂tγ2 =(kBTe + kBTi)miσ2NiNi +Na− γ2, (3.25)where Na is the total number of atoms [2]. The Ni/(Ni +Na) factor results from the energy andmomentum exchange when repulsive ions recombine to form neutral atoms due to TBR. Finally193.4. Debye Shieldingthe energy conservation condition isEtot =32Ni(kBTe + kBTi +miγ2σ2)−∑nNa(n)Rn2. (3.26)Taking the full time derivative on the energy equation and applying the chain rule allows oneto calculate the evolving electron temperature and ion temperatures. Since mi  me the iontemperature is often assumed negligible when compared to the electron temperature. The elec-trons are also assumed to equilibrate instantaneously when compared to the ion dynamics. Thesetechniques will be applied in the development of the plasma simulations in chapters 5 and 6.3.4 Debye ShieldingA fundamental property of plasma lies in its ability to screen out external electric potentials. Thisbehaviour stems from the plasma’s mobile electrons which move freely in response to an electricfield. For example, if a positively charged plate was inserted inside a plasma, free electronswill immediately move to surround the positive plate (assuming a dielectric barrier prevents theelectrons and ions from recombining). They will move until the potential well supplied by thecharged plate becomes quenched by the electrons.At zero temperatures, the electrons surrounding the plate will occupy an infinitesimally shortlength. A sharp boundary would exist where outside, no field would exist. This is the case ofperfect shielding. However this would not be the case if the temperature was finite. Near theedge of the cloud, the field from the charged plate is weak. Therefore at the edge, electrons withthermal energy kBT can escape the potential well. This leads to incomplete shielding.The thickness of this shielding length can be computed by conventional electrodynamics meth-ods. The calculation assumes that the mass of the ions mi  me = m such that ions do notmove on this time scale while electrons are assumed mobile. Beginning with Poisson’s equation∇2φ =(∂2∂x2+∂2∂y2+∂2∂z2)φ = f(x, y, z), (3.27)where φ is the potential distribution and f(x, y, z) is the charge distribution, we simplify this intoone dimension such that∇2φ =d2φdx2= −4pie(ni − ne) (3.28)and define ni as the ion density and ne as the electron density. Far away from the charged surface,we know that the plasma is unperturbed, thereforeni = n∞,where n∞ is defined as the ion density far away. The electron charge distribution in the presence203.4. Debye Shieldingof a potential φ can be shown to bef(u) = Aexp(−12mu2 + eφkBTe), (3.29)where A is a normalization constant. This can be understood intuitively as it predicts areas ofhigher potential to have more electrons than areas of lower potential. This must be true sinceelectrons move to areas of higher potential.The density of electrons can be found by integrating over all velocitiesn =∫ +∞−∞f(u)du. (3.30)Integrating our electron charge distribution and applying the boundary condition far away, suchthat φ→ 0 and ne = ni = n∞ due to quasi-neutrality, we find the electron density to bene = n∞exp(eφ/kBTe). (3.31)Substituting this into equation 3.28, we write the Poisson equation in the form ofd2φdx2= 4pien∞[exp(eφkBTe]− 1). (3.32)At the edge of the cloud, we assume that eφ/kBTe << 1 so we can expand the electron densityas a Taylor series:d2φdx2= 4pien∞[1 +eφkBTe+12(eφkBTe)2+ . . .− 1]. (3.33)Near the charged plate the kinetic energy and potential are comparable so we cannot expand thepotential as a Taylor series. Fortunately we will see that the potential falls off quickly from thecharged plate and this distance does not contribute much to the thickness of the electron cloud.Keeping only the first non-zero term in equation 3.33, the Poisson equation becomesd2φdx2≈4pin∞e2kBTeφ =1λ2Dφ, (3.34)where the Debye length, λD, has been defined asλD ≡√kBTe4pine2. (3.35)213.4. Debye ShieldingThe solution to the ordinary differential equation in equation 3.34 isφ = Aexp(−|x|/λD) +Bexp(|x|/λD). (3.36)Since the potential cannot grow further away from the charged plate, we know that B = 0. Wealso define define A = φ0 to be the potential at the charged sheet. This brings us to the finalsolution of equation 3.34 to beφ = φ0 exp(−|x|/λD). (3.37)This can be understood as follows, at distances much larger than λD, electrons do not feelthe potential of the charged surface due to electron screening. We check the solution with ourintuition. With an increase in electron density, λD decreases since there would be more electronsin each layer of plasma. Furthermore λD increases with T since electrons have more energy toescape the potential. Also if T → 0, the shielding length becomes infinitely thin, confirming theassumption at the beginning of the section.22Chapter 4The Creation and Detection of aSupersonic UCPThis chapter outlines the experimental realization of an UCP from a supersonic beam of XeRydberg atoms. It describes a cloud of Xe atoms as it leaves the reservoir tank, to its excitationinto the Rydberg manifold, and finally to its detection and evolution to plasma.4.1 Creation of a Supersonic UCPA pulsed beam of xenon is released from a reservoir tank of pure xenon by actuating a solenoidvalve. The valve opens for 200 µs releasing a cloud of xenon from a stagnation pressure of 1 bar.It expands as it travels downstream until the free jet passes through an electroformed skimmer3 centimetres downstream. A hydrodynamic model estimates the skimmed supersonic beam tohave a longitudinal temperature (T|| along the axis of propagation) of 0.4 K and diverges atless than 9 ms−1. The measurement of Rydberg divergence returns a higher temperature of 4K. This disagreement arises from the possibility of cluster formation of Xe dimers. Setting thehydrodynamic model at 4 K fits the experimental divergence measurements well. Under theseconditions, the on axis density of Xe atoms is calculated to be 1014 cm−3.After passing through the skimmer, the beam passes through an orifice in the first field plate(G1) entering into the excitation chamber. This region allows precise control over the electricfield due to the mutual presence of G1, G2, and G3. The excitation and detection apparatus isshown in figure 4.1.The excitation chamber, between G1 and G2, is maintained at an operational pressure of 10−4mbar using a 500 litre per second turbo pump. Here it is excited by ω1, a Continuum ND6000 dyelaser pumped by a Q-switched Powerlite 800 DLS Nd:YAG laser. It selects the state, describedby the JJ coupling limit as 5p5[2P◦3/2]6p 2 [5/2] J = 0, through a two-photon transition fromXe 5p6 S0. This transition appears at a total energy of 78 119.718 cm−1 and is metastable witha lifetime of 40 ns [16].Assuming that significant saturation effects do not occur before 3 ω1 photoionization whendriving the 2 photon transition into the metastable 6p, the expressionNex =12σ2NgsF2dt, (4.1)234.1. Creation of a Supersonic UCPFigure 4.1: Schematic diagram showing the experimental apparatus. A skimmed supersonic beampasses through an orifice in G1 where it is intersected by lasers ω1 and ω2. The gas travels untilG2 where electrons are extracted, by the application of a potential on grid G3, into the MCP.The dotted area indicates a movable grid assembly.where Nex is the number density of 6p states, σ2 is the 2 photon cross section, Ngs is the numberdensity of 5p states, F is the photon flux, and dt is the interaction time allows us to predict thedensity of the 6p state for our beam conditions.The 2 photon cross section was calculated to be 7.9×10−45 cm4s for our laser conditionsfollowing the method outlined in [17]. The laser waist was measured to be 0.1 mm and the Q-switched pumped ω1 releases its energy in ≈ 8 ns. Using these values in equation 4.1 predicts ametastable density of ≈ 8× 108 cm−3.The cloud, now composed of metastable 6p Xenon, is subsequently intersected by ω2, a Lu-monics HyperDYE-350 dye laser pumped by a Q-switched Continuum Surelite Nd:YAG. At alaser power of 90 MW per pulse, it promotes up to half of the 6p states into high lying Rydbergatoms achieving a Rydberg density of 4 ×108cm−3.The second dye laser, which can be tuned over a range from 515 - 508 nm, passes through afrequency doubler and selects for states of defined initial PQN n0 and orbital angular momentum`0. The excitation is illustrated schematically in figure Detection of a Supersonic UCPFigure 4.2: Excitation mechanism of Xe. Two ω1 excitation excites ground state Xe into themetastable 6p state. A third ω1 photon excites the 6p into the continuum. Meanwhile a ω2photon can state select into a Rydberg state n0, `0.4.2 Detection of a Supersonic UCPThe Rydberg atoms travel at the velocity of the beam to pass through a mesh grid, G2, and entersinto the detection region. Similar to the excitation region, the detection region is maintained at10−8 mbar by a second turbo pump. By maintaining G2 at ground and applying a charge onto thethird grid, G3, the cloud of atoms passing G2 feel a field of 130 V cm−1 . The grid produces strayfields nearby the Rydberg orbital radius which causes the cloud to release electrons proportionalto the number of atoms. A highly charged plate between G2 and G3 accelerates and guides theelectrons into the microchannel plate (MCP).The grid and detector assembly moves parallel to the direction of propagation by an externalmotorized control in submicron steps. This varies the time of flight from Rydberg creation todetection at G2. The motorized assembly is illustrated by the dotted area in figure 4.1.The Rydberg cloud can also be detected before it reaches G2 through the application of electricfields. As the cloud travels in the excitation chamber, a transient high voltage negative or positivefield can be applied on G1 or G2 by discharging a Behlke high voltage switch. Combined with aresister, this serves as a resistor-capacitor (RC) circuit which reaches fields of up to 838 V cm−1with a time constant of 3 µs. A representative field is shown in figure 4.3.In a similar manner, weak non-ionizing fields can be applied on G1 or G2 in order to detectfree electrons while having little effects on the Rydberg atoms. A Sony Tektronix AWG2020 isused to create a pulse train of 3 consecutive negative 3 V square waves shown schematically infigure 4.4. Setting G2 at a distance of 56 mm away from G1, produces a field of 500 mV/cm. Afield of this strength only ionizes Rydberg atoms of initial PQN greater than 150.This forward bias field is applied on G1 and accelerates electrons towards the detection as-sembly G2 and the MCP. On the time scale of the experiment, the electrons appear immediately254.2. Detection of a Supersonic UCPFigure 4.3: Typical applied ionizing electric field. A high voltage discharge through a capacitorproduces a strong electric field with a time constant of 3 µs.Figure 4.4: A below threshold forward bias pulse detects electrons while having little effect onRydberg atoms. Setting G2 at a distance of 56 mm away from G1 produces fields of less than500 mV/cm.264.2. Detection of a Supersonic UCPon the oscilloscope trace when the pulse is applied.27Chapter 5Formation of a Supersonic UCPThis chapter presents the observed dynamics as a cloud of supersonic xenon atoms are excited intoRydberg atoms and spontaneously evolve into UCP. It begins with our experimental observationsand analyzes these results in the light of a rate equation shell model including expected Rydbergphysics and UCP behaviour.5.1 Experimental Observations5.1.1 Selection of Rydberg atoms in state n0, `0Figure 5.1: The selection of the metastable 6p state of Xenon. The incoming laser, ω1 is tuneduntil the wavelength selects for the resonant 2 photon transition into the metastable 6p state.The integrated signal received from the 3 ω1 ionization of the metastables are shown.Figure 5.1 indicates the formation of metastable 5p5 6p 2 [5/2] J = 0 states through the 2-photon5p6 S0 transition. The spectrum is produced by resonant three-photon ionization where a 3rd ω1285.1. Experimental Observationsphoton ionizes the 6p metastable state into the continuum. This releases electrons into the MCPwhich integrates the signal over a fixed amount of time.The second incident laser, ω2, tuned over a range of 515 - 508 nm excites the 6p atomsinto Rydberg states of principal quantum numbers from n = 30 to n = 80 with orbital angularmomentum ` = 0 (s) or ` = 2 (d). Figure 5.2 displays the s and d series from PQN 38 - 53obtained by pulsed field ionization 1 µs after laser excitation.Figure 5.2: The selection of excitated Rydberg states. By tuning a counter propagating laser ω2into a beam of metastable 6p Xenon, the s and d Rydberg series are produced. The Rydbergstravel until G2 where the electrons are extracted giving the integrated signal.Omitting the pulsed field, the cloud of Rydberg atoms travels downstream until it comes incontact with G2 and the rest of the detection assembly. The signal from excitation to detectionproduces a characteristic waveform shown in figure 5.3.The small prompt electron signal at ≈ 0 µs results from three-photon ionization and largelydepends on ω1 power. At this time scale, electrons produced in this way appear immediatelyon the oscilloscope. The second larger peak between 40-60 µs marks the time when the cloudof atoms come in contact with G2. This time depends on the distance between the Rydbergexcitation and the G2 detector assembly. This distance divided by the arrival time of the latepeak predicts a beam speed of 385 m/s for our cloud of Xe atoms.295.1. Experimental ObservationsFigure 5.3: Oscilloscope trace showing the electron signal as a function of time at the MCP.The prompt peak at 0 µs indicates the 3 photon ionization or collisional ionization from betweenRydberg atoms. The second and stronger late peak indicates the time when the excited beampasses through G2 and electrons are extracted.5.1.2 The time evolution of Rydberg atomsThe voltage ramp used to obtain the high-Rydberg excitation spectrum in figure 5.2 produceselectrons at a point in time determined by the amplitude of the electrostatic field. Ionization of theselected Rydberg state occurs when the field strength matches the ionizing threshold determinedusing equation 2.25.Figure 5.4 follows the response of a cloud of 51d Rydberg atoms as the pulsed electric field isdelayed over time. The three traces black, dark grey, and grey represent ramp initiation delaysof 0 µs, 20 µs, and 40 µs respectively. The applied electric field is overlayed in blue.For short delays (≈ 0 µs), the ionizing electric field produces a sharp electron signal at atime corresponding with the field required to ionize the 51d state. As the initiation of the fieldis delayed by 20 µs, this sharp peak decreases in intensity. At the same time a weaker, broadersecondary peak develops at a higher field 0.1 µs later. Further delaying the ionization field to 40µs exaggerates the effects of a decreasing sharp peak and a growing secondary peak.The pulsed field ionization spectrum not only depends on the initiation of the field, but alsovaries systematically with initial PQN n0. Figure 5.5 displays the ionization spectra of Rydbergatoms as a function of principal quantum numbers n = 40 − 59 in the nd series. These spectra305.1. Experimental ObservationsFigure 5.4: The evolution of the initially selected 51d state over time. The signal is detectedthrough ionization from a 838 V/cm pulse. An initially sharp peak centered around the associatedionization potential for the 51d state decreases over time while a secondary peak requiring a higherionizing potential appears. The field strength is overlayed in blue.are obtained at the three pulsed-field delays of 0 µs, 20 µs, and 40 µs.As expected Rydberg atoms prepared with higher principal quantum numbers ionize at earliertimes since they require a lower field threshold. The secondary peak also develops earlier forhigher initial principal quantum numbers. In figure 5.5c, atoms with n = 59− 55 have developedsecondary peaks as large as the initial peak at 40 µs. Meanwhile atoms with lower PQN, such asn = 40, have not developed a secondary broader peak. On the contrary, a shoulder at higher fieldsis seen in the field ionization spectrum of lower n states instead of a distinct broad secondarypeak.After pulse field ionization, any remaining volume of Xe atoms propagates along the axis ofthe supersonic beam until it intersects G2. As noted in figure 5.3, in the absence of a pulsed field,the gas emits a large signal as it contacts G2.Applying the pulsed field before the cloud reaches G2 greatly diminishes this signal. Figure5.6 plots the integrated late peak of an initially 42d Rydberg atom as a function of pulse fieldionization delay. The detector assembly beginning at G2 is positioned at a fixed distance of 56mm away from G1. This corresponds with a flight time of 150 µs from the excitation region.In order to avoid saturating the MCP detector, a positive pulse is applied on G1 which createsa reverse bias field with a maximum strength of 432 V cm−1. At short delays, the pulse greatlydiminishes the total area of the signal. As the pulse is delayed, the signal begins a sustained rise315.1. Experimental Observations(a) 0 µs delay.(b) 20 µs delay.(c) 40 µs delay.Figure 5.5: The ionization signal of Rydberg states from n = 40 to n = 59 for a field initiationdelay of 0 µs, 20 µs, and 40 µs. The `-mixed states are seen to develop quicker for higher n thanfor lower n.325.2. Experimental AnalysisFigure 5.6: The extracted electron signal of the 42d Rydberg state as it passes through G2 as afunction of ionization field delay. The excited volume is seen to develop a rigidity to an ionizingelectric field with a rise time of 90 µs. Overlayed is the simulated electron population for peakdensities of 1.2×109cm−3, 8.8×108cm−3, and 6.6×108cm−3 from left to 25 µs until reaching a plateau at 90 µs.The cloud of 42d Rydberg atom’s produces a coincident electron waveform in response to aweak forward bias pulse. The applied pulse, seen in figure 4.4, produces a field amplitude of -500mV cm−1. It is too weak to ionize the selected Rydberg atom but will direct free or loosely boundelectrons towards the MCP. Figure 5.7 displays the results.At short weak-pulsed field delays, the field directs stray electrons into the MCP. These elec-trons likely come from 3 ω1 photoionization of metastable 6p Xe. Similar to figure 5.6, the electronbegins a sustained rise at 25 µs until reaching a plateau at 90 µs. The signal has been rescaledsuch that the signal maximum is at 1.5.2 Experimental AnalysisA cloud of Xenon Rydberg atoms, formed in a supersonic beam through 2 + 1 double-resonantexcitation from the 5p5 6p 2 [5/2] state, is observed as it evolves in time. The pulse fieldionization spectrum reveals a developing secondary broad peak as the pulse initiation is delayed.This second peak develops earlier for higher initial principal quantum numbers. As the pulse isfurther delayed, the surviving fraction of gas reaching the G2 detector begins to increase at 25 µsuntil reaching a maximum at 90 µs. A similar electron response is seen with the application of a335.2. Experimental AnalysisFigure 5.7: The electron signal produced by the application of a small forward bias electric fieldwhich is too weak to affect the excited 42d state. The signal begins to grow at 20 µs until reachinga threshold at 90 µs. Overlayed is the simulated initial Rydberg population for peak densities of1.4×109cm−3, 8.8×108cm−3, and 4.7×108cm−3 from left to right.forward bias non-ionizing field.5.2.1 Kinetics of `-mixingThe pulse field ionization spectrum in figures 5.4 and 5.5 responds consistently with observationsof `-mixing of Rydberg gases in MOT systems [11]. On the axis of the the supersonic beam,we calculate a predicted Rydberg density of 4 × 108 cm−3. Comparing the pulse field ionizationspectrum of our cloud of 51d Xe atoms with Raithel’s field ionization spectrum of 51d 85Rb atomsat a density of 5.5 × 108 cm−3, a similar behaviour in the FI spectrum is observed. This lendsfurther credibility that we are at this density.Specifically figure 5.4 displays a secondary broader peak at higher fields. We follow Raithel’sanalysis and attribute this feature to `-mixing collisions. By performing pulse field ionizationexperiments for multiple n states at different delays, we see from figure 5.5 that the developmentof `-mixed peaks varies systematically according to the selected initial principal quantum number.The rate of `-mixing strongly depends on the initial PQN, n0, as expected from equation 2.30.We see that after 40 µs, high n states such as n0 = 59 have fully developed `-mixed peaks whilestates of lower PQN such as n0 = 49 have no such feature. On the contrary, a shoulder developsat higher fields in the field ionization peak of n0. This shoulder appears at a field strength345.3. Comparison with Simulationscorresponding to the threshold ionization of n = n0 + 1 and results from n-mixing collisions.The lack of features from n = n0 − 1 and n-mixed peaks for higher n0 is due to the quick risingasymmetric electric field and a lack of detection resolution.5.2.2 Electron avalanche and the evolution to plasmaAt sufficiently high densities, attractive dipole-dipole and van der Waals interactions pull Ry-dberg atoms together. Nearby Rydberg atoms can undergo Penning ionization giving rise to adistribution of Rydberg states, ions, and electrons. At our calculated density of 4×108 cm−3, 1%ionization is sufficient to develop a space charge to trap electrons with 60 K.Electrons created after this amount of ionization remain trapped by the potential well andundergo collisional processes such as three-body recombination, n, `-mixing, and impact ioniza-tion. Among other things, this builds up the electron density until a threshold for avalanche intoplasma is reached. In the plasma state, there is an equilibrium between Rydberg atoms, ions,and electrons. The electrons are free to move and shield the ions from an external electric fieldoutside the Debye radius.Analyzing figure 5.6 in this way, we observe the evolution of Rydberg atoms into plasma. Thepulsed 432 V cm−1 field is higher than the ionizing threshold for n0 = 42 of E0 = 103.3 V/cm.It also exceeds the `-mixing limit which requires higher field strengths for ionization [18]. As aresults this field should ionize the Rydberg atoms leaving little of the cloud to be detected atG2. This is demonstrated by the low signal strength between 0 and 25 µs. The sustained growthin the fraction of atoms reaching G2 between 25 and 90 µs indicates the evolution into plasma.During the evolution into plasma, the increasing electron density decreases the Debye radius andimproves the shielding properties of the plasma. As a result, a growing fraction of the cloudsurvives the ionizing pulse and reaches the detector.Similarly figure 5.7 demonstrates a growing electron density from the time of Rydberg creation.As a plasma develops, the population of electrons seeds ionizing processes until an equilibrium isreached between electrons, ions, and atoms. The delayed forward bias pulse is far lower than theionizing threshold for n0 = 42 atoms and has little effect on these atoms. However the forwardbias pulse is able to pick up stray or loosely bound electrons. The growth in electron populationbetween 25 and 90 µs indicates the initiation of avalanche into plasma. The population levelsoff after this time, indicating that the processes have reached quasi-equilibrium. The outputwaveform of the electron signal corresponds accordingly with the development of the Rydbergatom’s ‘rigidity’ to an ionizing field in figure Comparison with SimulationsA shell model using coupled equations was developed to verify the observed plasma formationdynamics. The simulation begins with a cloud of neutral Rydberg atoms with PQN, n0. Following355.3. Comparison with SimulationsRobicheaux’s analysis [13], atoms within a distance of 1.8× 2n2a0 undergo Penning ionization in800 Rydberg periods. From table 2.1 the Rydberg period τ , is calculated by τ = 2pin3. Excitinginto the n0 = 42 state, 800 Rydberg periods occur in ≈ 9 ns. This is assumed to be instantaneousover the time scale of 120 µs in the experiment.Using 30 shells of varying density, each individual shell calculates the fraction of atoms thathave a nearest neighbour within a distance of 1.8 × 2n2a0 in conformation with the Erlangdistribution [19]. These atoms are selected to undergo Penning ionization which produces a seedof electrons and a distribution of Rydberg atoms.This produces the initial conditions for the numbers of electrons and Rydberg atoms. Wherethe initial number of electrons isNe,i = f × ρi ×12, (5.1)where Ne,i is the initial electron population, ρi is the initial density, f is the Penning fraction,and the 1/2 indicates that the number of electrons produced is half the Penning fraction. Thatis, when 2 Rydbergs Penning ionize only 1 electron is released. The starting Rydberg densityafter Penning ionization is thereforeNn,i = (1− f)× ρi , (5.2)while the deactivated states are distributed according to a 5.95x5 distribution from n = 10 ton = n0/√2. The lower limit of n = 10 is an artificial limitation from the simulation where theseRydberg atoms are assumed to deactivate purely from spontaneous emission.The electron temperature in each individual shell is calculated by the difference in binding en-ergy before and after Penning ionization. Although electrons are assumed to be mobile within thecloud, the shells maintain their respective electron densities due to the quasi-neutrality condition.However since electrons are mobile, the temperature they possess should be constant across theshells. The temperature is calculated by weighting the temperature in each shell by the fractionalnumber of electrons. This method calculates not only the initial electron temperature but thetemperature change after each time step as well.The simulation couples the rate equations for Rydberg atoms from n = 10− 89 with the rateequations for electron and temperature. For each individual n state, the rate equationddtNn = −kionNnNe/V + ktbrNeNeNe/V2− Σn’ 6= nk(n, n′)NnNe/V + Σn’ 6= nk(n′, n)NnNe/V, (5.3)takes into account electron-Rydberg ionizing collisions, kionNnNe/V , three-body recombination,ktbrNeNeNe/V 2, and non-ionizing n-changing collisions which can depopulate, Σn’ 6= nk(n, n′)NnNe/V ,or populate Σn’ 6= nk(n′, n)NnNe/V a given n state. The quasi-neutrality condition assumes thenumber of ions equals the number of electrons.365.3. Comparison with SimulationsThe rate equation for the number of electronsddtNe = Σ∀nkionNnNe/V − Σ∀nktbrNeNeNe/V2, (5.4)comes from the same processes as equation 5.3 to conserve the number of particles. The symbolΣ∀n signifies the sum over all n Rydberg levels.The rate equation for the change in temperatureddtT =(Σ∀nRN˙nn2− 1.5kBTN˙e)/(1.5kBNe) , (5.5)assumes that the ion temperature is insignificant compared to the electron temperature andresults from energy conservation. After every time step, the weighted electron temperature iscalculated and used to update the rate constants in the equations.Although the experiment runs over 120 µs, the simulation ignores plasma expansion. Thisis mainly due to the difficulty in using a shell model to simulate expanding shells. However,this assumption is valid since previous simulations indicates that significant expansion does notoccur until the electron avalanche is near completion. This can be understood because a large ionpopulation does not build up until avalanche. Furthermore, the momentum exchange betweenrepulsive electrons forming into neutral Rydberg atoms with ions via TBR limits expansion untilavalanche is almost complete.Simulations are performed for peak initial Rydberg densities of 4.7×108 cm−3 to 1.4×109cm−3. As the density increases, the simulated avalanche times decreases from 60 µs for a densityof 4.7×108 cm−3 to 17 µs for a density of 1.4×109 cm−3. The simulated electron signal over timehas been overlayed with the surviving late peak and the response to a weak forward bias pulsein figures 5.6 and 5.7 respectively. The simulations have been normalized such that they matchthe limits of the experimental data.As the initial Rydberg density is increased from 4.7×108 cm-3 to 1.4×109 cm-3, the time foravalanche into plasma decreases from 60 µs to 18 µs. The avalanche into plasma is representedby a sharp rise in electron population while its asymptopic behaviour indicates a fully formedplasma.The peak density of 8.8×108 cm−3 corresponds best with both the curves in figures 5.6 and5.7. This corresponds with a density of 5.3×108 cm−3 one σ away from the maximum which moreaccurately reflects the average density. This agrees with our calculation of 4 ×108 cm−3 Rydbergatoms from the supersonic beam calculations as well as the expected density from the system’sspectroscopic response to a field ionization pulse [11].The avalanche time of the Rydberg gas into plasma occurs over almost 40 µs. One may expectthat once avalanche begins, electrons quickly build up and ionize Rydberg atoms in a very shorttime. Before the development of the shell model, the simulation predicted the rise time to occur375.3. Comparison with Simulationsover 10 µs. Only when the shell model was introduced did we see the rise time stretch out in thismanner.This can be understood by invoking the quasi-neutrality condition and the necessity of acritical electron density until avalanche. Areas of high densities avalanche first while areas of lowerdensities avalanche later. But by itself the electrons from areas of high densities are not enoughto set the whole gas into avalanche since by the quasi-neutrality condition, the gas maintains anequal amount of electrons and ions in each shell. As the electrons move into a new shell, it onlycommunicates its increased electron temperature while another electron must step in and takeits place. This draws out the avalanche process resulting in a slow avalanche over 40 µs seen infigures 5.6 and 5.7.This simulation of an evolving cloud of Xe Rydberg atoms into plasma corresponds well withthe experiment. The simulated growth in electron signal corresponds well with the respondingelectron signal to an ionizing pulse and the surviving fraction from a delayed ionizing field.Furthermore the density at which the simulated rise times match best corresponds with the ourcalculated beam density.38Chapter 6Simulation of a Strongly BlockadedRydberg Gas into PlasmaExperimental and theoretical work in UCPs have been driven by the potential to reach areas ofstrong coupling. So far the strongly coupled regime has remained out of reach largely due todisorder-induced heating (DIH). Ions created in an initially random distribution quickly redis-tributes due to Coulomb interactions converting potential energy to kinetic energy.Recently the Weidemu¨ller group has observed the spontaneous avalanche ionization of astrongly blockaded Rydberg gas [20]. The strongly blockaded cloud is created using the Rydbergblockade effect. This effect occurs at sufficiently high densities when the interaction between twoRydberg atoms causes a shift in the energy levels of the other. If the shift is sufficiently large andthe excitation laser sufficiently narrow, an exclusion sphere around a Rydberg atom is created.Within this sphere, another Rydberg cannot be excited giving rise to a degree of correlation inthe gas. This effect provides a possibility to mitigate the effects of DIH.This chapter reviews previous work done by the Weidemu¨ller group in creating a plasma froma strongly blockaded Rydberg gas. It also develops a model in collaboration with their group tounderstand formation and long term dynamics of a cloud of Ryberg atoms.6.1 Creation of a Strongly Blockaded Rydberg Gas [20]The Weidemu¨ller group created a strongly blockaded gas of rubidium (Rb) atoms in a MOT viathe intermediate 5p state using a narrow band laser. They loaded up to 1.5×105 87Rb 5S1/2 statesinto an optical dipole trap to reach atomic densities of 2.8 × 1010 cm−3. The atoms were thenheld and allowed to equilibrate at a temperature of T = 230± 30 µK. The first laser, at 780 nm,excites the 5S1/2 state into the 5P3/2 state. This laser is aligned with the second excitation at 480nm which selects for the 55S1/2 Rydberg state [20]. The excitation volume, V , is approximatedby a cylinder of length L = 70 µm and radius R = 28 µm.The blockade effect limits the density of Rydberg excitation because only one atom withinthe radius of an exclusion sphere can be excited. The exclusion radius is defined byRc ≈6√(2C6Γ)/Ω2, (6.1)396.2. Detection of a Strongly Blockaded Rydberg Gas [20]where C6/2pi ≈ 50 GHz µm6, the strength of the van der Waals interaction, and an intermediate-state decay rate Γ/2pi ≈ 6.1 MHz yield Rc ≈ 5 µm.6.2 Detection of a Strongly Blockaded Rydberg Gas [20]The evolution of the system is studied by monitoring the number of ground states, Ng, Rydbergs,Nr, and ions, Ni, separately. The ground state distribution is probed with a light pulse usingresonant absorption imaging between the lower transition. This simultaneously measures the5S1/2 and 5P3/2 states of the system. Their total population forms a quantity labelled as theground state in future simulations below.The Rydberg population is detected by applying an electric field exceeding the field ionizationthreshold of n0 = 55. Ionized atoms are subsequently guided to the MCP. To detect ions alone,a below threshold field guides the ions to the detector. The effect of this field on the n0 = 55Rydberg atoms are negligible.6.3 Observation of Spontaneous Ionization [20]Figure 6.1(a) shows the absorption signal seen as the remaining shadow cast by ground stateatoms following various intervals of 480 nm pumping to the 5S1/2 state. After ≈ 10 µs a holebegins to appear coincident with the focus of the 480 nm laser. Over the next 10-20 µs, this holegrows until the Rb cloud becomes transparent. Figure 6.1(b) estimates the number of groundstate atoms by fitting the absorption images [20]. It shows that the ground state populationdecreases slowly until a critical time tcrit ≈ 12 µs whena rapid depletion occurs.Figure 6.1(c) shows the populations of the Rydbergs, Nr, and ions, Ni, over time is shown.The Rydberg population reaches a steady state Nr ≈ 500 after ≈ 4 µs. At tcrit ≈ 12 µs, Nrdecreases rapidly. Coincidently, the number of ions, Ni, rapidly increases.This can be understood as follows. Ng decreases slowly before 10 µs because the blokadelimits pump into the Rydberg level. During this time, Rydberg atoms undergo ionizing collisionscausing the ion population, Ni to grow. At t ≈ 10 µs, Ni reaches a critical ion density requiredto trap electrons. The cloud of trapped electrons with Rydberg atoms leads to a rapid avalancheionization of the remaining Rydberg atoms causing Nr to drop and Ni to rise. The reduction ofRydberg atoms allow the lasers to freely excite Ng into Rydberg atoms causing the rapid drop inNg during the same time.The application of a small electric field during excitation suppresss ionization. This electricfield removes any accumulating ions indicating that the spontaneous avalanche is related to theaccumulation of charges [20].406.4. Original Rate-Equation Modelling Avalanche Dynamics [20]Figure 6.1: Observed evolution from Rydberg atoms into plasma. a) Absorption spectroscopy ofNg atoms. b) Calculated Ng over time. The dotted line indicates the onset of avalanche. Thesolid line includes the effects of the original coupled rate equation model including the blockademechanism. c) Recorded Rydberg (circle) and ion (triangle) population. The original simulation(solid line) corresponds poorly after avalanche. Image credit: Figure taken from [20].6.4 Original Rate-Equation Modelling Avalanche Dynamics [20]Weidemu¨ller and coworkers published a primitive model describing the avalanche dynamics. Itcouples the ground state, Ng (representing the combind population of 5s and 5p) atoms with the416.4. Original Rate-Equation Modelling Avalanche Dynamics [20]Rydberg state Nr by means of the rate equations:ddtNg = −AφNg +BNr (6.2)ddtNr = AφNg −BNr − γbbiNr − γcolNrNg/V − γavNrNe/V . (6.3)Here Ne is the electron population, V is the excitation volume and γbbi ≈ 150/s is the black-bodyionization rate [21].The model determines the ion population and electron population using particle conservationand defining a critical ion population Ncrit for trappingNi = Ntot −Ng −Nrr (6.4)Ne = Ni −Ncrit if > 0. (6.5)The pumping rate, A, and stimulated emission, B, is constrained by the steady state solutionof the three-level optical Bloch equations to be A/(A + B) = 0.38. The blockade mechanismdetermines the fraction of ground state atoms, φ, available for excitation in terms of the Carnahan-Starling approximation for hard spheres [22]φ = exp(−8η + 9η2 − 3η3(1− η)3), (6.6)where η = piNrR3c/6V is the packing fraction of hard spheres with radius Rc/2. Adjusting thevalue of A yields a best fit with the experimental data for A = (8.7± 0.3)× 10−2 MHz.A simple collision theory expression describes the rate of ionizing collisions between groundstate or intermediate state atoms and Rydberg atoms by:γcol = σcol√16kbTpimRb, (6.7)where σcol = 0.73± 0.21 µm2 fits best with the experimental observations. Note that this is 8-15times larger than the geometric cross section for Rydberg - Rydberg collisions σgeo ≈ pin∗4a20 ≈0.064 µm2.The avalanche rate, γav, is expresed by the product of electron velocity and a geometric crosssection for electron impact ionization of Rydberg atomsγav = σgeo√Ee/me, (6.8)where the electron energy was chosen to be half the binding energy. Finally the number of ionsneeded to trap created electrons was calculated using the electrostatic trapping potential of a426.5. Updated Rate-Equation Modelling Avalanche Dynamicshomogeneous cylinder. WhereNcrit =8EeLpiR20q2[L(−L+√L2 + 4R2) +R2csch−1(2R/L)](6.9)is the number of ions required and 0 is the permittivity of free space.This model replicates the ground state evolution well. Without the Carnahan-Starling ap-proximation, the ground state population decreases faster than observed. The solid line in figure6.1(b) indicates the predicted ground state levels including the Carnahan-Starling approximation.Figure 6.1(c) shows that the model also matches the quasi-steady state population of Rydbergsbefore avalanche. This match up continues until avalanche at ≈ 10 µs. Afterwards the experimentfinds a build up of Rydberg population while the simulated Rydberg population goes to 0. Inthe same figure, the model shows a slow build up of ions until avalanche. At this time thecalculated ion population grows to much higher levels than observed. These discrepancies havebeen attributed by the Weidemu¨ller group to the recombination of electrons and ions to reformRydberg atoms.6.5 Updated Rate-Equation Modelling Avalanche DynamicsA collaboration with the Weidemu¨ller group at the University of Heidelberg formed with theGrant group at UBC in order to better model the long term dynamics of a Rydberg gas intoplasma. This section steps through the major changes to the original assumptions and presentsthe results in steps until reaching a most realistic model at the end.To begin with we remove the phenomenological avalanche rate γav, and add rate equations forthree-body recombination rates, ktbr, electron-Rydberg ionization rates, kion, and non-ionizingn-changing collision rates, k(n, n′), as outlined by Pohl, Vrinceanu, and Sadeghpour [15]. Withthese explicit rate equations, the rate equations expand to include multiple n states instead ofjust the selected n0 = 55.The simulation couples rate equations for each individual nddtNn = AφNg −BNn − γbbiNn − kionNnNe/V− γcolNnNg/V + ktbrNiNeNe/V2− Σn’ 6= nk(n, n′)NnNe/V + Σn’ 6= nk(n′, n)NnNe/V, (6.10)with the ground state populationddtNg = −AφηNg + Σ∀nBNn, (6.11)436.5. Updated Rate-Equation Modelling Avalanche DynamicsFigure 6.2: Rate equation model for ground state (|S1/2〉 and |5P3/2〉) atoms, Rydberg atomsn = 10− 89, and ions including the effects of electron-Rydberg redistribution, impact ionization,and TBR. The Rydberg level n = 10 is not shown since there is an artificial bottleneck at thisPQN due to the simulation limit.and the ion populationddtNi = γbbiNn + γcolNnNg/V+ Σ∀nkionNnNe/V − Σ∀nktbrNiNeNe/V2. (6.12)For cases where n 6= n0, we set A = B = 0. The simulation limits n values between 10 - 89assuming that states below 10 decay purely by spontaneous emission and states higher than 89ionize immediately. In addition, the fraction of available ground states φ is calculated using onlythe number of radiatively coupled Rydbergs in n = n0 = 55.In keeping with the analysis by the Weidemu¨ller group, the number of electrons and its changeover time is calculated byNe =Ni −Ncrit if Ne > 0 (6.13)ddtNe =ddtNi if Ne > 0 . (6.14)The inclusion of these terms changes the simulation from the solid lines in figure 6.1 to figure 6.2.446.5. Updated Rate-Equation Modelling Avalanche DynamicsThe top of figure 6.2 indicates the ground state population while the bottom half plots the ionpopulation in black and each n Rydberg state, Nn, in red. Replacing the phenomenological γavwith the electron-Rydberg ionization rate, kion maintains the avalanche timescales. The groundstate population reaches a quasi-steady state at Ng ≈ 4500 after 3 µs owing to the limitation ofthe blockade effect. At avalanche, the ground state population drops rapidly but reaches steadystate levels of Ng ≈ 2500 which is higher than the experiment. In addition, the number of ionsreaches a maximum of ≈ 1000 instead of the rapid rise to numbers  1500 seen in the previoussimulation. Furthermore the Rydberg population after avalanche does not reach the levels ofrepopulation seen in figure 6.1 of the experiment. This indicates that a mechanism in additionto TBR must be at play.Additionally, as Rydberg states are ionized, the difference in binding energy of a boundRydberg and an ion and electron results in a change of temperature. A more realistic simulationshould allow for temperature evolution since three-body recombination, n-changing collisions andionizing collisions all depend on temperature.This can be tracked by rewriting the energy conservation condition to readddtT = (RcΣ∀n N˙nn2− 1.5kBTN˙e)/(1.5kBNe). (6.15)Additional energy terms take into account the energy gained from ionizing collisions betweenground state and Rydberg atoms collisions and black body ionization. The energy gained fromblack body ionization is assumed to equal the threshold energy for a specific n state. For groundstate-Rydberg collisions, this energy was chosen to be 1.5× the threshold energy. Electronsionized in this way gain a kinetic energy equal to half the binding energy.However this poorly reflects the actual energy gained by the system arising from Rydberg-ground state ionizations (Hornbeck-Molnar ionization) [23]. This process occurs when a molecularion forms in the collision from an excited atom with a ground state atom. This is illustrated asA∗ +B(ground)→ (AB)+ + e−. (6.16)The actual energy released from the formation of a molecular ion equals the bond strength lessany internal energy left in AB+Although the assumption for energy addition is imperfect, the inclusion of these terms allowsthe simulation to explore the effects of a changing temperature in plasma formation.Figure 6.3 displays the results after allowing for temperature change. The time scales foravalanche and the evolving Rydberg population are maintained. Furthermore, the ground statepopulation drops to ≈ 1000 which is similar to what is seen in the experiment [20]. The maineffect of allowing for temperature change is a higher final ion population, Ni ≈ 3500, than before.This is expected as the rise in temperature provides more energy to drive collisional effects.As the plasma forms, Rydbergs ionize forming ions and electrons. The once neutral volume,456.5. Updated Rate-Equation Modelling Avalanche Dynamicsnow charged, experiences the outward force of the expanding electron gas causing the ions toexpand. Assuming a symmetric Gaussian plasma, the width (σ) and hydrodynamic ion velocity(γ) can be derived from the Vlasov equation. The rate equations are well described by Killianand coworker’s [2] to beddtσ2 = 2γσ2, (6.17)andddtγ =kBTmσ2NiΣ∀nNn +Ni− γ2, (6.18)where the changing hydrodynamic velocity in equation 6.18 has been weighted by the fraction ofRydbergs and ions due to momentum exchange from TBR.Figure 6.3: Rate equation model for ground state (|S1/2〉 and |5P3/2〉) atoms, Rydberg atomsn = 10− 89, and ions including the effects of electron-Rydberg redistribution, impact ionization,TBR, and allowing for temperature change. The Rydberg level n = 10 is not shown since thereis an artificial bottleneck at this PQN due to the simulation limit.In addition the total energy must be modified such thatEtot = −RcΣ∀nNnn2+32NikB(Te + Ti) +32Nimiγ2σ2 (6.19)which includes the expanding velocity of the ions.We track the temperature change by taking the time derivative of both sides and setting itequal to the energy gained and lost by through pumping and black body ionization. Equation466.5. Updated Rate-Equation Modelling Avalanche DynamicsFigure 6.4: Rate equation model for ground state (|S1/2〉 > and |5P3/2〉) atoms, Rydberg atomsn = 10− 89, and ions including the effects of electron-Rydberg redistribution, impact ionization,TBR, temperature change, and expansion. All Rydberg levels are plotted.6.20 shows thisddtEtot =ddt(−RcΣ∀nNnn2+32NikB(Te + Ti) +32Nimiγ2σ2)= −RAφNgN20+RBN0N20+ Σ∀nγbbiRNnn2(6.20)Reorganizing the equation 6.20 to isolate the temperature we getddtT =(Σ∀nRN˙nn2−32kBTN˙e −32N˙imγ2σ2 − 3Nimγγ˙σ2 − 3Nimγ2σσ˙− RAφNgN20+RBN0N20+ Σ∀nγbbiRNnn2)/(1.5kBNe) , (6.21)where the ion temperature has been assumed to be negligible when compared to electron tem-perature.Equations 6.17, 6.18, and 6.21 are coupled into the simulations when Ne > 1. The results aredisplayed in figure 6.4. As before the time scale and population for Rydberg atoms are maintained.This final simulation reproduces the time scales and populations of Nn, Ng, and Ni the best. Ofnote, the inclusion of expansion predicts more reasonable long term ion numbers of ≈ 1000. This476.6. Seeding with Penning Ionizationis because energy is used towards expansion which drives the temperature down. With expansion,the ground-state Rydberg collisions also decrease resulting in a final ground state population ofNg ≈ 2500. This is higher than observed in the experiment. On the other hand, the quasi-steadystate Rydberg population, Nn ≈ 500, and the repopulation after avalanche agrees well with theexperiment.The higher number of predicted ions than seen is acceptable since the experiment in figure 6.1does not conserve the number of particles. This may arise from an unaccounted drop in detectionefficiency as the plasma expands. Note as well that the avalanche occurs over a shorter period(≈ 5 µs) in the simulation than in the experiment (≈ 10 µs). The assumption of a the cylindricalcloud of constant density causes this. The xenon model developed in chapter 5 shows that a shellmodel for a Gaussian density distinctly increases the length of the avalanche. This is seen infigure 5.6 and 5.7. The stretching out of the avalanche time in a shell model can be understoodsince areas of higher density avalanche before areas of lower density.Finally, the biased repopulation into the n0 state seen in the experiment [20] is not purelyreflective of three-body recombination. This is understood by comparing figure 6.3 (no expansion)with figure 6.4 (includes expansion). The repopulation of Rydbergs in figure 6.3 does not replicatethe numbers seen in the experiment while figure 6.4 replicates this well. This suggests thatexpansion decreases the ionizing collisions which allows the continuous pumping of the lasers torepopulate n0 causing the observed rise after 20 µs in figure Seeding with Penning IonizationThe Hornbeck-Molnar (ground state-Rydberg) picture for ionization of Rydberg atoms by reactivecollisions with ground state atoms presents a source of problems in the simulation above. Firstthe energy released in forming a molecular ion cannot be properly accounted for. Second thecross section that fits the experimental data is much larger than expected. For example the crosssection at 443 K, for n = 12 collisions with the 5s state of Rubidium measures σcol = 8×10−6µm2[23], but the simulation uses σcol = 7.3× 10−1µm2. This is an order of 105 larger than expected.In addition the fitted cross section is ≈ 15 times larger than the geometric cross section!Penning ionization offers another possible seeding mechanism to create ions in Weidemu¨ller’sMOT experiment. It involves the interaction of 2 adjacent Rydberg atoms to create a deactivatedRydberg, an ion, and an electron. This section investigates the behaviour of blockaded Rydbergatoms when ground state-Rydberg collisions are replaced by Penning ionization.The Penning ionization rate was introduced by following Robicheaux’s 2005 model [13]. Forstatistically random values of ` and m Rydberg atoms in state n reach the strong interactionregion in a timeTpenn ≈ 20µs×√M(amu)R50(µm)/n2, (6.22)where R0 is the distance between the Rydberg atoms. This equation conserves energy released486.6. Seeding with Penning Ionizationin dipole-dipole interactions.The simulation assumes that once the Rydberg atoms reach the strong interaction region,they immediately undergo Penning ionization. This assumption is justified by Robicheaux’scalculations showning that Penning ionization occurs on the time scale of 100’s of Rydberg periodswhich itself is 10-100’s of picoseconds. This is much smaller than the time scale of our experimentwhich occurs over microseconds.The simulation fits a penning rate constant, kpenning, for each individual n by calculating thetime to reach the strong interaction region as a function of density. The density changes theseparation distance according to the Wigner-Seitz radiusR0 =(34pipiR2LΣ∀nNn)1/3, (6.23)where the volume is described by a homogeneous cylinder of radius, R, and length, L. SincePenning ionization is a bimolecular process, the best fit for the Penning ionization rate wasdetermined byT−1penn = kpenningρ, (6.24)where ρ is the density.The Penning rate calculated for n 6= n0 states assumes that Penning ionization occurs withthe populated n0 state and not with other n states. This is reasonable since the n0 populationoverwhelms the other n 6= n0 states. We confirm this by finding that the effects of n−n0 Penningionization in the simulation produces negligible effect when compared to a case allowing onlyn0 − n0 Penning ionization. In addition, Penning ionization creates another atom in a boundstate. Since n0 − n0 collisions drive the dynamics, the simulation only includes the distributionof bound states produced from by n0 − n0 Penning ionization.496.6. Seeding with Penning IonizationFigure 6.5: Rate equation model for ground state (|S1/2〉 and |5P3/2〉) atoms, Rydberg atomsn = 10− 89, and ions including the effects of electron-Rydberg redistribution, impact ionization,TBR, and allowing for temperature change. The simulation replaces the mechanism of Hornbeck-Molnar ionization with Penning ionization.The inclusion of Penning ionization changes the rate equations for Rydberg atoms in state nfrom equation 6.10 toddtNn = AφNg −BNn − γbbiNn − kionNnNe/V + ktbrNiNeNe/V2− Σn’ 6= nk(n, n′)NnNe/V + Σn’ 6= nk(n′, n)NnNe/V− kpenningNnN0 + kpenningN0NnFn, (6.25)where Fn is the fractional distribution of bound states into state n after Penning ionization. Therate equation for Ni in equation 6.12 also changes toddtNi = γbbiNn + Σ∀nkionNnNe/V − Σ∀nktbrNiNeNe/V2+ 0.5× Σ∀nkpenningNnNn. (6.26)The rate equations for the ground state population, width, hydrodynamic velocity, and temper-ature remain the same as before as in equations 6.11, 6.17, 6.18 and 6.21 respectively.Seeding with Penning ionization yields figure 6.5. Here avalnche begins much earlier than seenin Weidemu¨ller’s experiment [20]. The experiment takes longer to avalanche because Weidemu¨ller506.6. Seeding with Penning Ionizationexcites into the 55s state which exhibits a repulsive potential. In contrast, our Penning ionizationrate uses attractive dipole-dipole interactions between Rydberg atoms for its calculation.Since the excitation into repulsive states decreases dipole-dipole interactions and thereforesuppresses the Penning ionization rate, we adjust penning by a weighting factor until we matchexperimental results. Replacing the old Penning ionization rate in equations 6.25 and 6.26 witha weighted Penning ionization rate, such that knew penning = 0.1kpenning, we attain predictionsmatching the experiment in [20]. Figure 6.6 plots these results overlaid with the experimentaldata of the ground state population, Rydbeg population, as well as the ion population.Figure 6.6: Rate equation model for ground state (|S1/2〉 and |5P3/2〉) atoms, Rydberg atoms n =10−89, and ions including the effects of electron-Rydberg redistribution, impact ionization, TBR,and allowing for temperature change. The blue, black, and red traces indicates the population ofground states, ions, and Rydbergs respectively.This weighting factor must reflect from the development of population in higher order `-states. For example, black body radiation can stimulate transitions to adjacent P -states whichhave attractive pair potentials. In addition, we note that while no other Rydberg atom existswithin the blockade radius, this does not hold for ground state atoms. Ground state atomswithin the wavefunction of the Rydberg atom perturbes the Rydberg atom into lower-n, high-`-states while the ground state atom moves into other hyperfine states or simply accelerates.Collisional angular momentum changing and fine structure changing experiments with Rb havebeen previously noted by Gallagher [6].The overall small but non-zero magnitude of the weighting factor required to obtain agreemenwith experiment points to Penning ionization as a reasonable seeding mechanism for ionization516.6. Seeding with Penning Ionizationas opposed to ground state-Rydberg ionizing collisions.Finally we note that many of the interactions discussed above strongly depend on quantumnumber and the wavefunction overlap between Rydberg and ground state atom. As a result,we investigate the dependence of avalanche times on initial ground state densities and performa fit for a density dependent weighting factor for 2 different quantum numbers and interactionvolumes.Using a weighting factor defined by w = 3 × ρ0(µm−3) + 0.03, we attain excellent resultscorresponding to experimental data for 55s Rydberg states at a full (70 µs long) and half (37 µslong) volume. Figures 6.7 and 6.8 plots these results respectively.Figure 6.7: A rate equation model predicting the ground state population pumping into 55s statesat full volume overlaid with the experimental data. A weighting factor of w = 3×ρ0(µm−3)+0.03has been used.Expecting the interaction to strongly depend on quantum number and density, we introduceanother weighting factor, w = 0.08 × ρ0(µm−3) + 0.01, for pumping into the 40s state. Figures6.9 shows the corresponding predictions and data for the half volume. The data for full volumedid not behave as expected so it has not been included.There remains an excellent correspondence for the 55s states for a density range from 0.8 ×1010cm−3 - 16× 1010cm−3 at full and half volumes. While the avalanche times for the 40s statesretain good correspondence across a range of densities, the predicted final ground state densityover predicts the experimental data. This indicates other long time decay factors and detectorefficiency may not be accounted for.526.7. Attractive 55d statesFigure 6.8: A rate equation model predicting the ground state population pumping into 55s statesat half volume overlaid with the experimental data. A weighting factor of w = 3×ρ0(µm−3)+0.03has been used.Using these weighting rules for the 55s and 40s states at full and half volume, we summarizethese results in figure 6.10 plotted on a log-log scale.6.7 Attractive 55d statesWe finish by investigating the avalanche behaviour of a 55d Rydberg gas performed by the Wei-demu¨ller group. The experiment begins by loading atoms into the MOT. Then, released from thetrap, they expand to a given volume determined by the expansion time. This varies the initialnumber of ground state atoms and the interaction volume addressed by the excitation lasers. Apair of probe and coupling laser pulses excites the 5s atoms into the 55d state via 5p for a fixedperiod of 10 µs. After another delay of 10 µs, the detector acquires the image at an exposure of10 µs.We attempt to model this situation by eliminating the distinction between ns and n` statesallowing Penning ionization to proceed at the full Robicheaux rate constant. However we limit thefraction of available Penning partners according to the Erlang distribution. We run the simulationfor 10 µs with pumping (A) and stimulated emission (B) turned on. Then, setting A and B tozero, we run the simulation for 10 µs more, and record the number of ground state atoms.Since Hornbeck-Molnar ionization is turned off, the ground state population remains constant536.7. Attractive 55d statesFigure 6.9: A rate equation model predicting the ground state population pumping into 40s statesat half volume overlaid with the experimental data. A weighting factor of w = 0.08×ρ0(µm−3)+0.01 has been used.Figure 6.10: A log-log plot of avalanche times vs of atomic densities. Dark symbols indicateexperiments performed at the full volume while light symbols indicate experiments at the halfvolume. Triangles indicate excitation into n0 = 40 while circles indicate n0 = 55. The solid linesare the predicted critical times extracted from the model.after Rydberg excitation switches off. Allowing for spontaneous emission at this point producesa negligible difference. Finally we calculate an interaction volume for different densities in accor-546.7. Attractive 55d statesdance with the supplied expansion, loading time, and density measurements.Figure 6.11 overlays the predicted final ground state atom density for different initial densitiesafter 10 µs of laser excitation. Good correspondence is found with the experimental data settingthe critical radius for Penning ionization at the same distance as in the `-mixing model. Howeverwe must increase the pumping rate to 16× 104 s−1 from 8.7× 104 s−1 from pumping into the sstates. Since the model uses an effective population for both 5s and 5p states, it is difficult topredict whether this rate should increase or decrease when selecting 55d as the final state.As the initial ground state density increases, the final ground state density increases at alinear rate. At a density of ≈ 1.5 × 1010 cm−3, Penning ionization occurs dropping the steadystate Rydberg population, which allows for continuous pumping out of the ground state. Thedip indicates this effect. As the density increases further, the final ground state density increaseslinearly occurs as before.Figure 6.11: The fraction of ground state population after 10 µs of excitation at various densitiesand interaction volumes. The dotted black line indicates the simulated final distribution.55Chapter 7Conclusion and Future OutlookA supersonic beam of Rydberg xenon atoms has been shown to evolve into plasma. A cloudof d-series Rydberg atoms develops `-mixed peaks driving Rydberg ionizations into plasma. Itevolves with similar characteristics seen previously in MOT experiments. This provides a linkbetween atomic UCPs in MOTs and the molecular NO plasmas that have been created in asupersonic beam [5]. Although the pulse field ionization and electron density rise times haveexcellent agreement, there remains a lack of correspondence between atomic beam experimentsand molecular beam experiments in terms of time scales. This results from the limited density of6p metastable Xe achieved by two-photon excitation.In the future, supersonic atomic plasmas will be studied using the Even-Lavie valve. Thisvalve generates metastable atoms using an electrical discharge allowing the creation of metastableelements at much higher densities. By excitation to Rydberg states, this provides a simple methodin creating cold atomic plasmas at high densities for a variety of elements. This experiment willfurther bridge the gap between MOT and supersonic plasmas.Rate equation models were also developed to understand the spontaneous ionization of a gasof rubidium Rydberg atoms in a MOT. These simulations were performed in collaboration withthe Weidemu¨ller group in Heidelberg. It was shown that temperature change, expansion, andthree-body recombination have large effects on long-term plasma formation dynamics. Penningionization was also shown to be a possible seeding mechanism for ionization in a gas of repulsiveRydberg atoms. Further investigations must be performed to understand the mechanism drivingionization by excitation into different ` states. Finally the simulation should build in the effectsof correlation by the blockade effect in order to fully explain the observed dynamics.56Bibliography[1] Chen Francis F. Introduction to Plasma Physics. Plenum Press: New York, 1974.[2] Killian TC, Pattard T, Pohl T, Rost JM. Ultracold Neutral Plasma. Physics Reports. 2007;449:77-130.[3] Ichimaru S. Strongly coupled plasmas. Rev. Mod. Phys. 1982; 54: 1017-1059.[4] Killian TC, Kulin S, Bergeson SD, Orozco LA, Orzel C, Rolston SL. Creation of an UltracoldNeutral Plasma. Phys. Rev. Lett. 1999; 83: 4776-4779.[5] Morrison JP, Rennick CJ, Keller JS, Grant ER. Evolution from a Molecular Rydberg Gas toan Ultracold Plasma in a Seeded Supersonic Expansion of NO. Phys. Rev. Lett. 2008; 101:205005.[6] Gallagher TF. Rydberg Atoms. Cambridge University Press: New York, 1994.[7] Cooke WE, Gallagher TF. Dependence of Rydberg-state field-ioinization thresholds on |ml|.PRA. 1978; 17:1226.[8] Griffiths DJ. Introduction to Quantum Mechanics secondd edition. Pearson education: NJ,2005.[9] Seaton MJ. Quantum defect theory. Rep. Prog. Phys. 1983; 46:167-257.[10] Dutta SK, Feldbaum D, Walz-Flannigan A, Guest JR, Raithel G. High-Angular-MomentumStates in Cold Rydberg Gases. PRL. 2001; 85:3993-3996.[11] Walz-Flannigan A, Guest JR, Choi JH, Raithel G. Cold Rydberg-gas Dynamics. PRA. 2004;69:063405.[12] Robinson MP, Tolra BL, Noel MW, Gallagher TF, Pillet P. Spontaneous Evolution of Ryd-berg Atoms into an Ultracold Plasma. PRL. 2000; 85:4466-4469.[13] Robicheaux F. Ionization due to the interaction between two Rydberg atoms. J. Phys. B.2005; 38:S333-S342.[14] Mansbach P, Keck J. Monte Carlo Trajectory Calculations of Atomic Excitations and Ion-ization by Thermal Electrons. Phys. Rev. 1969; 181:275.57Bibliography[15] Pohl T, Vrinceanu D, Sadeghpour HR. Rydberg Atom Formation in Ultracold Plasmas:Small Energy Transfer with Large Consequences. PRL. 2008; 100:223201.[16] Allen L, Jones DGC, Schofield DG. Radiative Lifetimes and Collisional Cross Sections forXe I and II. JOSA. 1969; 59:842-847.[17] Bamford DJ, Jusinski LE, Bischel WK. Absolute two-photon absorption and three-photonionization cross sections for atomic oxygen. PRA. 1986; 34:185-198.[18] Damburg RJ, Kolosov VV. A Hydrogen atom in a uniform electric field III. J. Phys. B. 1979;12:2637-2643.[19] Comparat D, Pillet P. Dipole blockade in a cold Rydberg atomic sample. JOSA B. 2010;27:A208-A232.[20] Robert-de-Saint-Vincent M, Hofmann CS, Schempp H, Gu¨nter G, Whitlock S, Weidemu¨llerM. Spontaneous Avalanche of a Strongly Blockaded Rydberg Gas. PRL. 2013; 110:045004.[21] Beterov II, Tretyakov DB, Ryatsev II, Entin VM, Ekers A, Bezuglov NN. Ionization ofRydberg atoms by blackbody radiation. New Journal of Physics. 2009; 11:013052.[22] Talbot J, Schaaf P, Tarjus G. Random sequential addition of hard spheres. Molecular Physics.1991; 72:1397-1406.[23] Kumar A, Saha BC, Weatherford CA, Verma SK. A systematic study of Hornbeck Molnarionization involving Rydberg alkali atoms. Theochem. 1999; 487:1-9.58


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