Possible many-body localization in a long-lived finite-temperature ultracold quasi-neutral molecular plasma
PPossible many-body localization in a long-lived finite-temperature ultracoldquasi-neutral molecular plasma
John Sous
1, 2 and Edward Grant
1, 3, ∗ Department of Physics & Astronomy, University of British Columbia, Vancouver, BC V6T 1Z3, Canada Stewart Blusson Quantum Matter Institute, University of British Columbia,Vancouver, British Columbia, V6T 1Z4, Canada Department of Chemistry, University of British Columbia, Vancouver, BC V6T 1Z3, Canada
We argue that the quenched ultracold plasma presents an experimental platform for studyingquantum many-body physics of disordered systems in the long-time and finite energy-density limits.We consider an experiment that quenches a plasma of nitric oxide to an ultracold system of Rydbergmolecules, ions and electrons that exhibits a long-lived state of arrested relaxation. The qualitativefeatures of this state fail to conform with classical models. Here, we develop a microscopic quantumdescription for the arrested phase based on an effective many-body spin Hamiltonian that includesboth dipole-dipole and van der Waals interactions. This effective model appears to offer a way toenvision the essential quantum disordered non-equilibrium physics of this system.
Introduction. — Quantum mechanics serves well todescribe the discrete low-energy dynamics of isolatedmicroscopic many-body systems [1]. The macroscopicworld conforms with the laws of Newtonian mechanics[2]. Quantum statistical mechanics [3] bridges theserealms by treating the quantum mechanical properties ofan ensemble of particles statistically and characterizingthe properties of the system in terms of state properties(temperature, chemical potential, etc.), in an approachthat implies a complex phase space of trajectories withergodic dynamics [4]. However, this is not always thecase, and the macroscopic description of quantum many-body systems that fail to behave as expected statisticallyremains today as a key unsolved problem [5, 6].Ergodicity, when present in an isolated quantum many-body system, emerges as the system thermalizes in aunitary evolution that spreads information among allthe subspaces of the system. The subspaces act asthermal reservoirs for each other. Most known many-body systems thermalize in this fashion, obeying theEigenstate Thermalization Hypothesis (ETH) [4, 6–10]which holds that the eigenstates of corresponding many-body Hamiltonians are thermal.Exceptions include fine-tuned integrable systems [11],and the class of so-called many-body localized (MBL)systems [6, 12], which have attracted intense interest inrecent years. Such systems do not thermalize at finiteenergy densities and are therefore non-ergodic. Disorderin a landscape of interactions preserves memory of theinitial local conditions for infinitely long times. MBLphases cannot be understood in terms of conventionalquantum statistical mechanics [13, 14].MBL has been observed in deliberately engineeredexperimental systems with ultracold atoms in one andtwo-dimensional optical lattices [15–20]. In such cases,tuning of the lattice parameters allows investigation of ∗ Author to whom correspondence should be addressed. Electronicmail: [email protected] the phase diagram of the system as a function of disorderstrength. However, such ultracold systems suffer fromdecoherence, confining localization to short timescalesand low energy densities.It is important to determine experimentally whetherconditions exist under which MBL can persist for longtimes at finite temperatures, and to understand if such arobust macroscopic quantum many-body state can occurnaturally in an interacting quantum system withoutdeliberate tuning of experimental parameters. Such arealization could pave the way to exotic quantum effects,such as entangled macroscopic objects and localization-protected quantum order [21, 22], which could havesocietal and technological implications [23].Motivated by these questions, we have exploredthe quenched ultracold molecular plasma as an arenain which to study quantum many-body effects inthe long-time and finite energy-density limits [24,25]. The ultracold plasma system offers complexity,as encountered in quantum materials, but evolvesfrom state-selected initial conditions that allow for adescription in terms of a specific set of atomic andmolecular degrees of freedom.Experimental work has recently established laboratoryconditions under which a high-density molecularultracold plasma evolves from a cold Rydberg gas of nitricoxide, adiabatically sequesters energy in a reservoir ofglobal mass transport, and relaxes to form a spatiallycorrelated, strongly coupled plasma [25, 26]. Thissystem naturally evolves to form an arrested phasethat has a long lifetime with respect to recombinationand neutral dissociation, and a very slow rate of freeexpansion. These volumes exhibit state propertiesthat are independent of initial quantum state anddensity, parameters which critically affect the timescaleof relaxation, suggesting a robust process of self-assemblythat reaches an arrested state, far from conventionalthermal equilibrium.Departure from classical models suggests localizationin the disposition of energy [25]. In an effort to explain a r X i v : . [ phy s i c s . p l a s m - ph ] F e b this state of arrested relaxation, we have developed aquantum mechanical description of the system in termsof power law interacting spin model, which allows for thepossibility of slow dynamics or MBL Experiment. — Double-resonant pulsed-laser excitationof nitric oxide entrained in a supersonic molecular beamforms a characteristic Gaussian ellipsoid volume of state-selected Rydberg gas that propagates in z with a well-defined velocity, longitudinal temperature ( T || = 500 mK) transverse temperature ( T ⊥ < mK) and preciselyknown initial density in a range from ρ = 10 to cm − (See Figure 1 and References [27, 28]).Rydberg molecules in the leading edge of the nearest-neighbour distance distribution interact to produce NO + ions and free electrons [29]. Electron-Rydberg collisionstrigger an ionization avalanche on a time scale fromnanoseconds to microseconds depending on initial densityand principal quantum number, n .Inelastic collisions heat electrons and the systemproceeds to a quasi-equilibrium of ions, electrons andhigh-Rydberg molecules of nitric oxide. This relaxationand the transient state it produces entirely parallelsthe many observations of ultracold plasma evolution inatomic systems under the conditions of a magneto-opticaltrap (MOT) [30].We see this avalanche unfold directly in sequencesof density-classified selective field ionization spectrameasured as a function of delay after initial formation ofthe Rydberg gas [25]. For a moderate ρ = 3 × cm − ,the ramp-field signal of the selected Rydberg state, n gives way on a 100 ns timescale to form the selective fieldionization (SFI) spectrum of a system in which electronsbind very weakly to single ions in a narrow distributionof high Rydberg states or in a quasi-free state held bythe plasma space charge [28].The peak density of the plasma decays for asmuch as 10 µ s until it reaches a value of ∼ × cm − , independent of the initially selected n and ρ . Thereafter the number of charged particlesremains constant for at least a millisecond. Onthis hydrodynamic timescale, the plasma bifurcates,disposing substantial energy in the relative velocity ofplasma volumes separating in ± x , the cross-beam axis oflaser propagation [26].The avalanche to plasma proceeds at a rate predictedwith accuracy by semi-classical coupled rate equations[25, 28]. This picture also calls for the rapid collisionalrelaxation of Rydberg molecules, accompanied by anincrease in electron temperature to 60 K or more.Bifurcation accounts for a loss of electron energy. But,the volumes that remain cease to evolve, quenchinginstead to form an arrested phase that expands slowly, ata rate reflecting an initial electron temperature no higherthan a few degrees Kelvin. These volumes show no sign ofloss owing to the fast dissociative recombination of NO + ions with electrons predicted classically for low T e [31],or predissociation of NO Rydbergs, well-known to occurwith relaxation in n [32].
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0 15 30 mm mm mm mm mm mm mm mm R e c o il v e l o c i t y ( m s - ) ω pulse energy ( µ J) F i e l d ( V c m - ) Time (µs) a) b)d) e) ρ ( c m - ) N + = N + = c) FIG. 1. a) Double-resonant selection of the initial quantumstate of the n f (2) Rydberg gas. b) Laser-crosseddifferentially pumped supersonic molecular beam. c) Selectivefield ionization spectrum after 500 ns evolution, showing thesignal of weakly bound electrons combined with a residualpopulation of f (2) Rydberg molecules. After 10 µ s, thispopulation sharpens to signal only high- n Rydbergs andplasma electrons. d) Integrated electron signal as a functionof evolution time from 0 to 160 µ s. Note the onset of thearrest phase before 10 µ s. Timescale compressed by a factorof two after 80 µ s. e) x, y -integrated images recorded aftera flight time of 400 µ s with n = 40 for initial Rydberg gaspeak densities varying from × to × cm − . All ofthese images exhibit the same peak density, × cm − . Thus, from the experiment, we learn that 5 µ s afteravalanche begins, Rydberg relaxation ceases. We detectno sign of ion acceleration by hot electrons and thesurviving number of ions and electrons remains constantfor the entire remaining observation period, extendingto as long as 1 ms. With the vast phase spaceavailable to energized electrons and neutral nitrogen andoxygen atom fragments, this persistent localization ofenergy in the electrostatic separation of cold ions andelectrons represents a very significant departure froma thermalized phase. Current experimental evidencethus points strongly to energy localization and absenceof thermalization within the accessible time of theexperiment. Molecular physics of the arrested phase. — Directmeasurements of its electron binding energy togetherwith its observed expansion rate establish experimentallythat the bifurcated plasma contains only high-Rydbergmolecules ( n > ) and NO + ions in combinationwith cold electrons (initial T e < K) bound by thespace charge. As noted above, semi-classical modelsmixing these species in any proportion predict thermalrelaxation, electron heating, expansion and dissipationon a rapid timescale with very evident consequencescompletely unobserved in the experiment. Instead,beyond an evolution time of 10 µ s or less, we find thatthe plasma settles in a state of arrested relaxation of N O + N O + e - e - r ij d i d j FIG. 2. Schematic representation of NO + core ions, pairedwith extravalent electrons to form interacting dipoles d i and d j , separated by r ij = r i − r j . canonical density and low internal energy manifested bya slow free expansion.To describe this apparent state of suppressedrelaxation, we proceed now to develop a formalrepresentation of the predominant interactions in thisarrested phase. Under the evidently cold, quasi-neutral conditions of the relaxed plasma, ions pair withextravalent electrons to form dipoles which interact asrepresented schematically in Figure 2.Assuming an intermolecular spacing that exceeds thedimensions of individual ion-electron separations, we candescribe the Coulomb interactions represented in Figure2 in terms of a simple Hamiltonian: H = X i (cid:18) P i m + h i (cid:19) + X i,j V ij (1)where h i describes the local relationship of each electronwith its proximal NO + core. This local representationextends to account for the interactions of a boundextravalent electron with vibrational, rotational andelectronic degrees of freedom of the core, as described,for example, by Multichannel Quantum Defect Theory[33]. Each ion-electron pair has momentum, P i and V ij ≡ V ( r i − r j ) describes the potential energy of theinteracting multipoles, represented in Figure 2 to lowestorder as induced dipoles with an interaction definedby V dd ij = [ d i · d j − d i · r ij )( d j · r ij )] / r ij , where forsimplicity we average over the anisotropy of the dipole-dipole interaction.The plasma also very likely includes ion-electron pairsof positive total energy. This implies the existenceof local Hamiltonians of much greater complexity thatdefine quasi-Rydberg bound states with dipole andhigher-order moments formed by the interaction of anextravalent electron with more than one ion.Representing the eigenstates of h i by | e i i , we can writea reduced Hamiltonian for the pairwise dipole-dipoleinteractions [34, 35] in the arrested phase: H dd = X i P i m + X i,j V dd ij (2)where we evaluate V dd ij in the | e i i basis. Note that such a Hamiltonian usually refers to the casewhere a narrow bandwidth laser prepares a Rydberg gasin which a particular set of dipole-dipole interactionsgive rise to a small, specific set of coupled states[36–38]. By contrast, the molecular ultracold plasmaforms spontaneously by processes of avalanche andquench to populate a great many different states thatevolve spatially without the requirement of light-mattercoherence or reference to a dipole blockade of any kind.This system relaxes to a quenched regime of ultracoldtemperature, from which it expands radially at arate of a few meters per second. Dipolar energyinteractions proceed on a much faster timescale [39–42]. Cross sections for close-coupled collisions areminuscule by comparison [43]. We can thus assume thatthe coupled states defined by dipole-dipole interactionsevolve adiabatically with the motion of ion centres.This separation of timescales enables us to write aneffective Hamiltonian describing pairwise interactionsthat slowly evolve in an instantaneous frame of slowlymoving ions and Rydberg molecules: H eff = P P i,j V ddij ,where P represents a projector onto the low-energydegrees of freedom owing to dipole-dipole coupling. Effective many-body Hamiltonian. — Consideringpairwise dipolar interactions between ion-electron pairs,we choose a set of basis states (cid:12)(cid:12) e (cid:11) , (cid:12)(cid:12) e (cid:11) , ... (cid:12)(cid:12) e L (cid:11) thatspans the low-energy regime. The superscript with lower(higher) integer label refers to the state with larger(smaller) electron binding energy.Quenching gives rise to a vast distribution of rareresonant pair-wise interactions, creating a randompotential landscape. Dipole-dipole interactions in thisdense manifold of basis states cause excitation exchange.In the disorder potential, these processes are dominatedby low energy-excitations involving L states in number,where we expect L to be small (from 2 to 4). The mostprobable interactions select L -level systems composed of different basis states from dipole to dipole.In a limit of dipole-dipole coupling, we can representpairwise excitations by spins with energies, (cid:15) i , andexchange interactions governed by an XY modelHamiltonian [28, 44] that describes these interactions interms of their effective spin dynamics: H eff = X i (cid:15) i ˆ S zi + X i,j J ij ( ˆ S + i ˆ S − j + h.c. ) (3)where ˆ S in each case denotes a spin- L operator definedas ˆ S γ = ¯ h ˆ σ γ / , for which σ γ is the corresponding spin- L Pauli matrix that spans the space of the L active levelsand γ = x, y or z . h.c. refers to Hermitian conjugate.This Hamiltonian reflects both the diagonal and off-diagonal disorder created by the variation in L -levelsystem from dipole to dipole. The first term in H eff describes the diagonal disorder arising from randomcontributions to the on-site energy of any particulardipole owing to its random local environment. Inspin language, P i (cid:15) i ˆ S zi represents a Gaussian-distributedrandom local field of width W . The representative SFIspectrum in Figure 1 directly gauges a W of ∼ GHzfor the quenched ultracold plasma.In the second term, J ij = t ij /r ij determines the off-diagonal disordered amplitudes of the spin flip-flops.To visualize the associated disorder, recognize thatthe second term varies as t ij ∝ | d i || d j | , where everyinteraction selects a different d i and d j . Over the presentrange of W , a simple pair of dipoles formed by s and p Rydberg states of the same n couple with a t ij of 75GHz µ m [45]. Note that t ij falls exponentially with thedifference in principal quantum numbers, ∆ n ij [46]. Induced Ising interactions.
In the limit | J ij | <
ACKNOWLEDGMENTS
This work was supported by the US Air Force Officeof Scientific Research (Grant No. FA9550-17-1-0343),together with the Natural Sciences and Engineeringresearch Council of Canada (NSERC) and the StewartBlusson Quantum Matter Institute (SBQMI). JS gratefully acknowledges support from the Harvard-Smithsonian Institute for Theoretical Atomic, Molecularand Optical Physics (ITAMP). We have benefited fromhelpful interactions with Rahul Nandkishore, ShivajiSondhi and Alexander Burin. We also appreciatediscussions with Joshua Cantin and Roman Krems. [1] J. J. Sakurai,
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1, 2 and Edward Grant
1, 3, ∗ Department of Physics & Astronomy, University of British Columbia, Vancouver, BC V6T 1Z3, Canada Stewart Blusson Quantum Matter Institute, University of British Columbia,Vancouver, British Columbia, V6T 1Z4, Canada Department of Chemistry, University of British Columbia, Vancouver, BC V6T 1Z3, Canada
I. DOUBLE-RESONANT PRODUCTION OF ASTATE SELECTED MOLECULAR RYDBERGGAS
Laser pulses, ω and ω , cross a molecular beam todefine a Gaussian ellipsoidal volume in which a sequenceof resonant electronic transitions transfer populationfrom the X Π / ground state of nitric oxide to anintermediate state, A Σ + with angular momentumneglecting spin, N = 0 , and then to a specified levelin the mixed n f (2) Rydberg series to create a state-selected Rydberg gas of nitric oxide, in which quantities(0) and (2) refer to rotational quantum numbers of theNO + 1 Σ + cation core.The intensity of ω determines the density of Rydbergmolecules formed by saturated absorption of ω . For agiven ω intensity, the peak Rydberg gas density varieswith ω − ω delay according to the well-known decayrate of the A Σ + state. Choosing I ω and ∆ t ω − ω , weprecisely control the initial peak density of the Rydberggas ellipsoid over a two-decade range from ρ = 10 to cm − [S1].In the core of this ellipsoid, Rydberg molecules,propagating in the molecular beam have a locallongitudinal temperature of T || = 500 mK and atransverse temperature, T ⊥ < mK. These moleculesinteract at a density-determined rate to form NO + ionsand free electrons. Initially created electrons collide withRydberg molecules to trigger electron-impact avalancheon a time-scale that varies with density from nanosecondsto microseconds (see below). II. SELECTIVE FIELD IONIZATIONSPECTROSCOPY OF ELECTRON BINDINGENERGY
Selective field ionization (SFI) produces an electronsignal waveform that varies with the amplitude ofa linearly rising electrostatic field. Electrons in aRydberg state with principal quantum number, n , ionizediabatically when the field amplitude reaches the electronbinding energy threshold, / n [S2]. ∗ Author to whom correspondence should be addressed. Electronicmail: [email protected]
For low density Rydberg gases, SFI has served as anexacting probe of the coupling of electron orbital angularmomentum coupling with core rotation. Studies of nitricoxide in particular have shown that nf (2) Rydberg statesof NO traverse the Stark manifold to form NO + inrotational states N + = 2 and 0 [S3].Experiments described in the main text operate in adiabatic regime, employing a slew rate of 0.7 V cm − ns − . Under these conditions, SFI features that appearwhen the field rises to an amplitude of F V cm − measureelectrons bound by energy E b in cm − , according to E b =4 √ F .Quasi-free electrons, weakly bound in the attractivepotential of more than one cation, ionize at a low fieldthat varies with the number of excess ions in the plasma.The SFI spectrum presented in the text as Figure1(c) and shown here as Figure S1 maps the electronbinding energy as a function of the initial Rydberg gasdensity for a molecular nitric oxide ultracold plasmaafter 500 ns of evolution. At a glance, the spectrumat higher density ( cm − ) shows direct evidence ofeither electrons bound to an increasing space charge or abroader distribution of high- n Rydberg states.This contracts to a narrower distribution of veryweakly bound electrons in plasmas of lower density( cm − ). Here we observe the spectrum of aresidue of molecules with the originally selected principalquantum number of the Rydberg gas, shifted slightly todeeper apparent binding energy by evident l -mixing orslight relaxation in n .We have used SFI measurements like these tocharacterize the avalanche and evolution dynamics of agreat many Rydberg gases of varying density and initialprincipal quantum number. Relaxation times vary, butall of these spectra evolve to form the same final spectrumof weakly bound electrons with traces of residual Rydberggas for systems of low initial density. III. COUPLED RATE-EQUATIONSIMULATIONS OF THE ELECTRON-IMPACTAVALANCHE TO ULTRACOLD PLASMA IN AMOLECULAR RYDBERG GAS
The semi-classical mechanics embodied in a systemof coupled rate equations serves well to describe theavalanche of a molecular Rydberg gas to ultracoldplasma. In this picture, Rydberg molecule densities, a r X i v : . [ phy s i c s . p l a s m - ph ] F e b F i e l d ( V c m - ) ρ ( c m - ) N + = N + = FIG. S1. Selective field ionization spectrum spectrum asa function of initial Rydberg gas density, ρ , after 500 nsof evolution, showing the signal of weakly bound electronscombined with a residual population of f (2) Rydbergmolecules, (initial principal quantum number, n = 49 , inthe f Rydberg series converging to NO + ion rotational state, N + = 2 ). After 10 µ s, this population sharpens to signalhigh- n Rydbergs and plasma electrons, with a residue of theinitial Rydberg population, shifted slightly to deeper bindingenergy by l -mixing and perhaps some small relaxation in n .The prominent feature that appears at the lowest values ofthe ramp field gauges the potential energy of electrons inhigh Rydberg states bound to single NO + ions, combinedwith electrons bound to the space charge of more than oneion. Notice the binding effect of a slightly greater excesspositive charge at the highest initial Rydberg gas densities.The red feature extends approximately to the binding energyof n = 80 or 500 GHz. labeled ρ i , evolve over a ladder of principal quantumnumbers, n i , according to: − dρ i dt = X j k ij ρ e ρ i − X j k ji ρ e ρ j + k i,ion ρ e ρ i − k i,tbr ρ e + k i,P D ρ i (S1)The free-electron density changes as: dρ e dt = X i k ion ρ e − X i k itbr ρ e − k DR ρ e (S2)A variational reaction rate formalism determines T e -dependant rate coefficients, k ij , for electron impacttransitions from Rydberg state i to j , k iion , forcollisional ionization from state i and k itbr , for three-body recombination to state i [S4, S5]. Unimolecularrate constants, k i,P D , describe the principal quantumnumber dependant rate of Rydberg predissociation [S6–S8], averaged over azimuthal quantum number, l [S9]. k DR accounts for direct dissociative recombination [S10] The relaxation of molecules in the manifold of Rydbergstates determines the temperature of electrons releasedby avalanche. Conservation of total energy per unitvolume requires: E tot = 32 k B T e ( t ) ρ e ( t ) − R X i ρ i ( t ) n i + 32 k B T ρ
DRe − R X i ρ P Di n i (S3)where R is the Rydberg constant for NO, and ρ DRe and ρ P Di represent the number of electrons and Rydbergmolecules of level i lost to dissociative recombination andpredissociation, respectively [S11, S12].To realistically represent the density distributionproduced by crossed-beam laser excitation of thecylindrical distribution of NO ground-state moleculesin the molecular beam, we use a concentric systemof 100 shells of defined density spanning a Gaussianellipsoid to 5 σ in three dimensions. Avalanche proceedsas determined by the initial Rydberg molecule densityof each shell. Each shell conserves the combined densityof stationary molecules, ions and neutral fragmentationproducts. Electrons satisfy local quasi-neutrality, but areotherwise assumed mobile, and thermally equilibratedover the entire volume [S13]. A. The semi-classical evolution of an n = 80 Rydberg gas
Figure S2 shows the global evolution of particledensities and electron temperature calculated for an n = 80 Rydberg gas at an initial density of × cm − [S13], representing one limit of the SFIspectrum obtained as above for an ultracold plasma inits arrest state after an evolution time of 10 µ s. Bythis point, the real system begins a phase of unchangingcomposition and very slow expansion that lasts at leasta millisecond – as long a period as we can observe it.The the semi-classical simulation result shown inFigure S2 tells us that the SFI spectrum shown in FigureS1 cannot possibly signal a conventional gas of long livedhigh-Rydberg molecules. Instead, a proven semi-classicalrate model configured for the density distribution of theexperiment, predicts the decay of such a high-Rydberggas to plasma on the timescale of a microsecond or less.In the model, predissociation consumes residualRydbergs in all n -levels within a few microsecondsand the formation of neutral atomic products quicklyslows. This must occur conventionally because therising electron temperature stabilizes the classical plasmastate by suppressing three-body recombination. The realarrested state, however, shows no sign of an electrontemperature higher than a few degrees K. T e m pe r a t u r e ( K ) Time (µs)
0 5 10 15 20
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* T e m pe r a t u r e ( K ) Time (µs)
0 5 10 15 20
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) T e m pe r a t u r e ( K ) Time (µs)
0 5 10 15 20
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* T e m pe r a t u r e ( K ) Time (µs)
0 5 10 15 20
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) FIG. S2. (lower) Numbers of ions and electrons, Rydbergmolecules and neutral dissociation products N( S) and O( P)as a function of time during the avalanche of an n = 80 Rydberg gas of NO to form an ultracold plasma, as predictedby a shell-model coupled rate equation simulation. Here werepresent the initial density distribution of the Rydberg gasby a 5 σ Gaussian ellipsoid with principal axis dimensions, σ x = 1 . mm, σ y = 0 . mm, σ z = 0 . mm and peak densityof × cm − , as measured for a typical experimentalplasma entering the arrest state after am evolution of 10 µ s.The simulation proceeds in 100 concentric shells enclosing setnumbers of kinetically coupled particles, linked by a commonelectron temperature that evolves to conserve energy globally.(upper) Global electron temperature as a function of time. B. The semi-classical evolution of a fully ionizedultracold plasma with T e (0) = 5 K Let us instead test instead the kinetic stability of aconventional ultracold plasma composed entirely of ionsand electrons. Again, we assume initial conditions thatfit with the observed properties of the arrest state: NO + and electrons present at a density of × cm − in an ellipsoid with Gaussian dimensions, σ x = 1 . mm, σ y = 0 . mm, σ z = 0 . mm, representedby simulations evolving in 100 shells, with electrontemperature equilibration [S13]. In keeping with thevery slow rate of plasma expansion observed in theexperiment, we set the initial electron temperature to5 K.Figure S3 shows how this classical arrest state evolvesin time. The formation and rapid decay of NO Rydberg T e m pe r a t u r e ( K ) Time (µs)
10 15 20 25 30
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* T e m pe r a t u r e ( K ) Time (µs)
0 5 10 15 20
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) T e m pe r a t u r e ( K ) Time (µs)
10 15 20 25 30
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* N u m be r o f pa r t i c l e s Time (µs) x 10 N u m be r o f pa r t i c l e s Time (µs)x 10 N ( S) + O ( P)NO + + e - NO* T e m pe r a t u r e ( K ) Time (µs)
0 5 10 15 20
Time (µs) E l e c t r on t e m pe r a t u r e ( K ) FIG. S3. (lower) Numbers of ions and electrons, Rydbergmolecules and neutral dissociation products N( S) and O( P)as a function of time during the evolution of an ultracoldplasma of NO + ions and electrons, as predicted by a shell-model coupled rate equation simulation. Here we representthe initial density distribution of the plasma by a 5 σ Gaussianellipsoid with principal axis dimensions, σ x = 1 . mm, σ y =0 . mm, σ z = 0 . mm, peak density of × cm − and initial electron temperature, T e (0) = 5 K, as measuredfor a typical experimental plasma entering the arrest stateafter am evolution of 10 µ s. The simulation proceeds in 100concentric shells enclosing set numbers of kinetically coupledparticles, linked by a common electron temperature thatevolves to conserve energy globally. (upper) Global electrontemperature as a function of time. molecules signifies an immediate process of three-bodyrecombination, which decreases the charged particledensity of the plasma, Predissociation reduces the steady-state density of Rydberg molecules to a value of nearlyzero, but three-body recombination persists as shown bythe rising density of neutral atom fragments. Eventually,this process slows as the electron temperature rises.Could this hot-electron ultracold plasma represent theend state of arrested relaxation? Absolutely not. Asdetailed in the next section, a plasma with an electrontemperature of 60 K would expand to a volume largerthan our experimental chamber in less than 100 µ s. IV. AMBIPOLAR EXPANSION IN A PLASMAWITH AN ELLIPSOIDAL DENSITYDISTRIBUTION
The self-similar expansion of a spherical Gaussianplasma is well-described by an analytic solution ofthe Vlasov equations for electrons and ions with self-consistent electric fields. For a distribution of width σ ,in the limit of T e (cid:29) T i , this solution reduces to [S14]: e ∇ φ = k B T e ρ − ∇ ρ = − k B T e rσ (S4)In essence, the thermal pressure of the electron gasproduces an electrostatic force that radially acceleratesthe ion density distribution according to the gradientin the electrostatic potential. In approximate terms,the expanding electrons transfer kinetic energy to theions, accelerating the distribution to an average ballisticvelocity, k B T e ≈ m i (cid:10) v i (cid:11) (S5)The velocity varies linearly with radial distance, ∂ t r = γr , where γ falls with time as the distribution expands,and the electron temperature cools according to ∂ t T e = − γT e .To model the ellipsoidal plasma, we represent itscharge distribution by a set of concentric shells. Inthis shell model, the density difference from each shell j to shell j + 1 establishes a potential gradient thatdetermines the local electrostatic force in each principalaxis direction, k [S15]: em i ∇ φ k,j ( t ) = ∂u k,j ( t ) ∂t = k B T e ( t ) m i ρ j ( t ) ρ j ( t ) − ρ j +1 ( t ) r k,j ( t ) − r k,j +1 ( t ) (S6)where ρ j ( t ) represents the density of ions in shell j .The radial coordinates of each shell evolve accordingto its instantaneous velocity along each axis, u k,j ( t ) . ∂r k,j ( t ) ∂t = u k,j ( t ) = γ k,j ( t ) r k,j ( t ) (S7)which in turn determines shell volume and thus itsdensity, ρ j ( t ) . The electron temperature supplies thethermal energy that drives this ambipolar expansion.Ions accelerate and T e falls according to: k B ∂T e ( t ) ∂t = − m i P j N j X k,j N j u k,j ( t ) ∂u k,j ( t ) ∂t (S8)Figure S4 compares the ambipolar expansion of anellipsoidal plasma, simulated for an initial volume withthe starting dimensions described above and an initialelectron temperature of 60 K, compared with the timeevolution of the Gaussian width measured in z byexperiment. Note that the choice of a large initial σ ( t ) ( µ m ) Time ( µ s) FIG. S4. Hydrodynamic expansion of a Gaussian ellipsoidwith the dimensions measured at 10 µ s for the typical arrestedplasma described above, modeled by a 100-shell simulation,assuming an electron temperature that rises to 60 K, withcurves, reading from the bottom on the left, for σ y ( t ) , σ z ( t ) and σ x ( t ) . The lower curve with data shows the measuredexpansion of a typical molecular NO ultracold plasma with aVlasov fit for T e = 3 K. volume intrinsically slows the simulated expansion. Yet,nevertheless, the electron heating that arises inevitablyfrom three-body recombination in a classical ultracoldplasma demands a rate of expansion that is completelyunsupported by experimental observation. V. EFFECTIVE MANY-BODY HAMILTONIAN
Experimental observations tell us that the molecularultracold plasma of nitric oxide evolves to a stateof arrested relaxation in which extravalent electronsoccupy a narrow distribution of weakly bound states.This distribution of states supports a vast distributionof pair-wise interactions, creating a random potentiallandscape. Resonant dipole-dipole interactions in thisdense manifold of basis states cause excitation exchange.In the disorder potential, these processes are dominatedby low energy-excitations involving L states in number,where we expect L to be small (from 2 to 4). The mostprobable interactions select L -level systems composed ofdifferent basis states from dipole to dipole. Thus, thestates (cid:12)(cid:12) e (cid:11) , (cid:12)(cid:12) e (cid:11) ... (cid:12)(cid:12) e L (cid:11) vary from one dipole to the nextand from time to time.Representing excitations by spins, we can write an XYmodel [S16] that describes these interactions in terms oftheir effective spin dynamics H eff = X i (cid:15) i ˆ S zi + X i,j J ij ( ˆ S + i ˆ S − j + h.c. ) (S9)where ˆ S in each case denotes a spin- L operator definedas ˆ S γ = ¯ h ˆ σ γ / , for which σ γ is the corresponding spin- L Pauli matrix that spans the space of the L active levelsand γ = x, y or z . h.c. refers to Hermitian conjugate.Let us now consider specific examples of thisconstruction. A. L = 2 case Figure S5 diagrams a case that is uniquely definedfor every pair of interacting dipoles. In the limitof isolated pairs, this two-level interaction is exactlyresonant. Conditions described below randomly displacethese energy level positions.For each particular dipole i , described by states (cid:12)(cid:12) e i (cid:11) and (cid:12)(cid:12) e i (cid:11) , let us define a projection operator for thehigher-energy state (which we will call spin-up) ˆ σ e i = (cid:12)(cid:12) e i (cid:11) (cid:10) e i (cid:12)(cid:12) = (1 + ˆ σ zi ) / and the lower-energy state (spin-down) ˆ σ e i = (cid:12)(cid:12) e i (cid:11) (cid:10) e i (cid:12)(cid:12) = (1 − ˆ σ zi ) / . Thus, we canrepresent the two levels of a dipole i , with an energyspacing (cid:15) i , by a one-body operator (cid:15) i ˆ S zi = (¯ h(cid:15) i / σ zi .This defines an energy ± ¯ h(cid:15) i / depending on which state (cid:12)(cid:12) e i (cid:11) or (cid:12)(cid:12) e i (cid:11) is occupied, respectively, i.e. (cid:12)(cid:12) e i (cid:11) ≡ |↑ i i and (cid:12)(cid:12) e i (cid:11) ≡ |↓ i i . • • • • S i+ S j- i j i j | e i > | e j >| e i > | e j > N = 2 | e j >| e j > | e i >| e i > FIG. S5. Schematic diagram representing two Rydbergmolecules, i and j , dipole coupled in the two-levelapproximation. In every case, the disorder in the environmentof each molecule perturbs the exact energy level positions of | e i i and | e j i . The onsite energy is given by (cid:15) i = E i + D i [S17], where E i is the energy separation between the two states (cid:12)(cid:12) e i (cid:11) and (cid:12)(cid:12) e i (cid:11) evaluated for the local Hamiltonian h i . h i varieswith the random potential landscape from one dipole tothe next and thus is responsible for the diagonal disorderin the onsite term. D i = P j = i (cid:10) e i , e j (cid:12)(cid:12) V ddi,j (cid:12)(cid:12) e i , e j (cid:11) − (cid:10) e i , e j (cid:12)(cid:12) V ddi,j (cid:12)(cid:12) e i , e j (cid:11) represents the shift in a dipole’senergy due to dipole-dipole interactions [S17]. This termis identically zero for parity-conserving states [S18].Lowering and raising operators, ˆ σ − i = (cid:12)(cid:12) e i (cid:11) (cid:10) e i (cid:12)(cid:12) andits Hermitian conjugate ˆ σ + i = (cid:12)(cid:12) e i (cid:11) (cid:10) e i (cid:12)(cid:12) , define a resonantspin flip-flop between dipoles i and j : J ij ( ˆ S + i ˆ S − j + h.c. ) =(¯ hJ ij / σ + i ˆ σ − j + h.c. ) with amplitude J ij = t ij /r ij ; t ij = (cid:10) e i , e j (cid:12)(cid:12) V ddi,j (cid:12)(cid:12) e i , e j (cid:11) . This refers to the dipole-dipolemediated transfer of excitation [S17] represented by, forexample, ˆ S + i ˆ S − j |↓ i i |↑ j i = |↑ i i |↓ j i i.e. (cid:12)(cid:12) e i (cid:11) (cid:12)(cid:12) e j (cid:11) ˆ S + i ˆ S − j −−−−→ (cid:12)(cid:12) e i (cid:11) (cid:12)(cid:12) e j (cid:11) . We can expect this class of matrix element tobe non-zero for many of the local eigenstates of h i and h j , as the dipole-dipole operator couples states of differentparity, limited only by a few selection rules [S18].Additionally, we note that dipole-dipole interactionslead to a two-body Ising term of the form ˆ S zi ˆ S zj .This term originates from dipole-dipole induced shiftsof pairs of dipoles [S17] and has an amplitude (cid:10) e i , e j (cid:12)(cid:12) V ddi,j (cid:12)(cid:12) e i , e j (cid:11) + (cid:10) e i , e j (cid:12)(cid:12) V ddi,j (cid:12)(cid:12) e i , e j (cid:11) . This term isalso identically zero for parity conserving states [S18].Since, the arrested phase includes no external parity-breaking fields and neglecting local field fluctuations, weassume D i = 0 → (cid:15) i = E i and no dipole-dipole inducedIsing interaction. B. L > cases We can easily imagine systematic coupling schemesthat involve three or four L -level interactions. Excitationtransfer still governs the dynamics via terms like J ij ( ˆ S + i ˆ S − j + h.c. ) , where the ˆ S operators live in theactive L -dimensional subspaces. Figures S6 and S7schematically detail examples of these interactions. •• S i+ S j- i j i j | e i > | e j >| e i > | e j > N = 3 •• | e i > | e j >| e j >| e j >| e j > | e i >| e i >| e i > FIG. S6. Schematic diagram representing two Rydbergmolecules, i and j , dipole coupled in the limits of L = 3 . In thevery high state density of the quenched ultracold plasma, thedisplacement of (cid:12)(cid:12) e i (cid:11) and (cid:12)(cid:12) e j (cid:11) will will lessen the significanceof L = 3 interactions compared with the case of L = 4 . Figure S6 represents an interaction of overwhelmingimportance in studies of Rydberg quantum optics.Typically, a narrow bandwidth laser excites a resonantpair state, such as P / + 23 P / ↔ s + 24 s in Cs[S19]. Excitation transfer in this L = 3 case operates forexample as: ˆ S + i ˆ S − j | S i = − i | S j = 1 i = | S i = 0 i | S j = 0 i , (S10) i.e. (cid:12)(cid:12) e i (cid:11) (cid:12)(cid:12) e j (cid:11) ˆ S + i ˆ S − j −−−−→ (cid:12)(cid:12) e i (cid:11) (cid:12)(cid:12) e j (cid:11) For a gas of Rydberg molecules occupying a densemanifold of disordered states, the case of L = 3 becomesan operationally indistinguishable special case of themore general L = 4 interaction, which maps onto a spinof / .Here, we represent the interaction as an excitationtransfer that operates as: ˆ S + i ˆ S − j | S i = − / i | S j = 3 / i = | S i = − / i | S j = 1 / i , (S11) S i+ S j- i j i j | e i > | e j >| e i > | e j > N = 4 •• | e i > | e j > •• | e i > | e j >| e j >| e j >| e j >| e j > | e i >| e i >| e i >| e i > FIG. S7. Schematic diagram representing two Rydbergmolecules, i and j , dipole coupled in the limits of L = 4 .The high state density and strong disorder in the quenchedultracold plasma gives this case of L = 4 greater significancethan the restrictive limit of L = 3 i.e. (cid:12)(cid:12) e i (cid:11) (cid:12)(cid:12) e j (cid:11) ˆ S + i ˆ S − j −−−−→ (cid:12)(cid:12) e i (cid:11) (cid:12)(cid:12) e j (cid:11) .We can extend such sequences to higher L , but low-energy resonant dipole-dipole excitation exchange in thedense manifold of basis states will most prominentlyinvolve a small number of L -levels per dipole. VI. INDUCED VAN DER WAALSINTERACTIONS
For | J ij | << W , sequences of interactions add Isingterms that describe a van der Waals shifts of pairs ofdipoles [S20]. Consider, for example, three mutuallynearest-neighbour spins i , j and k in the L = 2 case. Athird-order process couples spins i and j via spin k in thefollowing fashion: |↓ i , ↑ j , ↑ k i ˆ S + i ˆ S − j −−−−→ |↑ i , ↓ j , ↑ k i ˆ S + j ˆ S − k −−−−→|↑ i , ↑ j , ↓ k i ˆ S + k ˆ S − i −−−−→ |↓ i , ↑ j , ↑ k i ; defining a self interactionthat changes the pairwise energies of i , j . U ij is inherently random owing to the randomness in J ij . It is also important to note that this limit gives riseto additional perturbative processes that renormalize thelocal onsite fields by van der Waals terms and slightlyaffect the pairwise flip-flop amplitudes [S20–S22]. Wesimply absorb these effects in the definitions of (cid:15) i and J ij .Taken together with Eq (S9) this result yields ageneral spin model with dipole-dipole and van der Waalsinteractions: H eff = X i (cid:15) i ˆ S zi + X i,j J ij ( ˆ S + i ˆ S − j + h.c. )+ X i,j U ij ˆ S zi ˆ S zj (S12)where U ij = D ij /r ij and D ij = t ij e J/W .The appearance of this third term underlines themany-body nature of Eq (S9). Even in this extreme limit,its dynamics are non-trivial, clearly involving more thanspin flip-flops with emergent correlations between spins. Non-resonant spin-spin interactions —
Theappearance of the term, P i,j U ij ˆ S zi ˆ S zj , underlines the many-body nature of this model. One obtains thisterm by treating J ij as a perturbation in Eq (S9) [S20].For the L = 2 case, this occurs at the third order,while for all other L , this term appears at the secondorder [S20]. Thus, such a term arises generally in the | J ij | (cid:28) W limit in three dimensions.The van der Waals interactions occur with anamplitude, U ij ≈ J ij e J/W , where e J estimates J ij atthe average distance separating spins. We do not expectthese interactions to depend strongly on the off-diagonaldisorder, as they arise from the off-resonant part of P i,j J ij ( ˆ S + i ˆ S − j + h.c. ) , which presumably does not causereal transitions [S20]. Thus, we can rationalize the use of e J here as an average weighting term. We leave the taskof studying the effect of off-diagonal disorder to futurework. Non-resonant onsite interactions —
It is alsoimportant to note that this limit gives rise to additionalperturbative processes that renormalize the local onsitefields P i (cid:15) i ˆ S zi by van der Waals terms [S20].Similar considerations from a completely differentatomistic perspective verify that this term isapproximately P l = i hC ij /r ij where h is the Planckconstant and C ij denotes the C coefficients for the vander Waals interaction between the off-resonant dipoles i and j [S21, S22].The induced onsite terms will also vary randomlyowing to the randomness in the potential landscape. Wesimply absorb such terms in the definition of (cid:15) i . VII. RESONANCE COUNTING AND THENUMBER OF DIPOLES IN THE QUENCHEDULTRACOLD PLASMA
Ref [S20] considers the problem of delocalization viaresonance counting arguments in the model of Eq S12for the general case of α < β , under conditions for which d > d c . Here α refers to the power law that regulates J ij and β refers to U ij . d and d c stand for dimensionalityand critical dimensionality. This work concludes thatdelocalization occurs at arbitrary disorder given sufficientsystem size.For local disorder, W , and average spin flip-flopamplitude, e J , the resonant pair criterion defines, N c ,a critical number of dipoles above which the systemdelocalizes. Here, we compare this theoretical estimatewith an accurate experimental measure of the numberof dipoles present in the arrest state of the quenchedultracold plasma.Controlled conditions of supersonic expansion preciselydefine the cylindrical density distribution of nitric oxidein the molecular beam [S1]. Co-propagating laser beams,Gaussian ω and ω , cross orthogonally in the x, y planeto define a Gaussian ellipsoidal excitation volume.When ω saturates the second step of doubleresonance, the intensity of ω controls the peak densityof the Rydberg gas volume up to a maximum of × cm − , obtained upon saturation of the first step.Density varies from shot to shot, and we have developedan accurate means of classifying and binning individualSIF traces according to initial Rydberg gas peak density,as displayed in Figure S1. Coupled rate simulationsdescribing the kinetics of the avalanche of Rydberg gasto plasma confirm these estimates of peak density. TABLE S1. Distribution of ions in an idealized Gaussianellipsoid shell model of a quenched ultracold plasma of NO asit enters the arrest state with a peak density of × cm − , σ x = 1 . mm, σ y = 0 . mm and σ z = 0 . mm. At thispoint, the quasi-neutral plasma contains a total of . × NO + ions (NO Rydberg molecules). Its average density is . × cm − and the mean distance between ions is 3.32 µ m.Shell Density Volume Particle Fraction a ws Num cm − cm Number × µ m . × . × − . × .
04 1 .
812 3 . × . × − . × .
23 1 .
833 3 . × . × − . × .
75 1 .
864 3 . × . × − . × .
87 1 .
935 2 . × . × − . × .
52 2 .
086 2 . × . × − . × .
40 2 .
267 1 . × . × − . × .
46 2 .
498 1 . × . × − . × .
81 2 .
789 7 . × . × − . × .
28 3 . . × . × − . × .
56 3 . . × . × − . × .
85 4 . . × . × − . × .
94 7 . . × . × − . × .
27 14 . . × . × − . × .
12 38 . . × . × − . × .
00 176 . Two methods of plasma tomography determine theevolution of plasma size and relative density distributionas a function of time. In the SFI apparatus,a perpendicular imaging grid that translates in themolecular beam propagation direction, z , yields anelectron signal waveform that gauges the changingplasma density and width as a function of evolution time.This waveform, followed to a point of evident arrest atabout 5 µ s, and well beyond, as illustrated by Figure 1in the main text, establish a case for arrested relaxation.Images projected in the x, y plane together withwaveforms in z , recorded after nearly 0.5 ms of flight,detail a slow ballistic expansion in Cartesian coordinatesthat we extrapolate back to an evolution time of 10 µ s to determine the absolute density distribution of thearrested ultracold plasma, described by the shell modelpresented in Table S1. This representation neglectsthe redistribution of charge density associated with theinitial stages of bifurcation. The total number of ionsrepresented by this distribution remains constant foras long as we can measure it in our long flight-pathinstrument, at least a half millisecond.The ion density averaged over shells determines h| r ij |i .This average distance between dipoles, combined with aa rough upper-limiting estimate of the average dipole- dipole matrix element, h t ij i , based on values computedfor a ∆ n = 0 Föster resonant interaction in Li [S22],yields an upper-limiting estimate of e J .However, interaction with charged particles in theplasma environment perturbs the electronic structureof individual Rydberg molecules. This diminishes theprobability of finding resonant target states, decreasingthe real value of e J , and giving rise to a rarityand randomness of resonant dipole-dipole interactionsdistributed over a huge state space defined by themeasured distribution of electron binding energies, W .As noted in Figure S1, a simple measure of the widthof the plasma feature in the delayed SFI spectrumdetermines W . Table S2 summarizes this and otherparameters of the arrest state derived from experiment,including the e J for Li under our conditions as an upperlimit.For short range interactions in a one-dimensional spinchain, perturbative arguments applied to disorderedinteracting spin models, such as the one above, predictmany-body localization [S23]. However, in higherdimensions especially, long-range resonant interactionsplay an important role in defining the conditions underwhich localization can occur. It is generally accepted thatinteractions governed by a coupling amplitude, J ij thatdecreases with distance as /r βij delocalizes any systemat finite temperature for which the dimension, d exceeds β/ .However, building on ideas introduced by Anderson[S24] and Levitov [S25], Burin [S20] offers a means bywhich to test a dimensionally constrained system forconditions that favor the onset of delocalization. He usesa perturbation approach that defines limits over whichlocalization can occur in a system as modeled above inwhich delocalization proceeds by the Ising interaction ofextended resonant pairs.In this picture, a system that violates the dimensionconstraint delocalizes for an arbitrary size of disorderwhenever the number of dipoles exceeds a criticalnumber, N c , which is determined by the disorder width, W and the average coupling strength, ˜ J . Coupling termsin the Hamiltonian defined by Eq S12 scale in r accordingto α = 3 , β = 6 and d = 3 . This sets a critical numberof dipoles defined by the quantity, N c = ( W/ e J ) [S20].For the arrest state defined by the density distributiondescribed by the elliptical shell model in Table S1, themeasured W taken with our upper limiting estimate for e J , yields N c = 3 . × .Considering this value of N c in relation to the averagedensity of the system at arrest defines R ∗ , an effectivedistance between resonant dipoles at which point thisoccurs [S20]. For the conditions described in TableS1, this coupling would occur in a system large enoughto contain . × dipoles a distance of 4 mm ormore. At this distance, our upper-limiting dipole-dipolematrix element would predict a characteristic irreversibletransition time, τ ∗ on the order of one second [S26]. TABLE S2. Resonance counting parameters in the arreststate of the quenched ultracold plasma. The disorder width W , taken directly from the width of the plasma feature in theSIF spectrum, combined with e J – derived from a rough upper-limiting estimate of the average dipole-dipole matrix element, h t ij i , based on values computed for ∆ n = 0 interactions inalkali metals [S22], together with the mean distance betweenNO + ions in the shell model ellipsoid – determines N c . acritical number of dipoles required for delocalization. R ∗ describes the length scale for delocalization and τ ∗ denotesthe delocalization time, given a sufficient number of dipolesat the average density of the experiment. Note that theultracold plasma quenched experimentally contains an orderof magnitude fewer than N c dipoles. W h t ij i h| r ij |i e J N c R ∗ τ ∗ GHz GHz( µ m) µ m GHz µ m s500 75 3.3 2.0 . × We note that the quenched ultracold plasma formedexperimentally relaxes to a volume that contains an orderof magnitude fewer dipoles than N c , as determined forthis case by the model of Ref [S20].As we attempt to convey above, the experiment yieldsplasmas of well defined density distribution and totalnumber of dipoles. However, the precise nature ofthe associated quantum states and their dipole-dipoleinteraction is much less well known. This limits thecertainty with which we can determine N c . What’s moreperturbation theory in a locator expansion formulationmay not accurately define the limiting conditions forMBL in higher dimensions [S27].Rare thermal regions (Griffiths regions) are thoughtto destabilize MBL systems of higher dimension [S28–S32], creating a glassy state, characterized by a slow evolution to a delocalized phase. However, other resultscontradict this notion, and support the possibility oflocalization in all dimensions [S33]. An added feature inthe self-assembly of the molecular ultracold plasma maypreclude destabilization by rare thermal regions: Shouldthe quenched plasma develop a Griffiths region as a sitefor delocalization to occur, the predissociation of relaxingNO molecules would promptly deplete that region to avoid of no consequence.In any event, the quenched plasma seems consistentlyable to find the conditions necessary for arrestedrelaxation. A great many different avalanche startingconditions, defined by varying initial Rydberg gasdensity and initial principal quantum number, all evolveto retain comparable internal energy and yield anarrest state with much the same density distributionas that described by the shell model detailed in Table S1. ACKNOWLEDGMENTS
This work was supported by the US Air Force Officeof Scientific Research (Grant No. FA9550-12-1-0239),together with the Natural Sciences and Engineeringresearch Council of Canada (NSERC), the CanadaFoundation for Innovation (CFI), the British ColumbiaKnowledge Development Fund (BCKDF) and theStewart Blusson Quantum Matter Institute (SBQMI).JS gratefully acknowledges support from the Harvard-Smithsonian Institute for Theoretical Atomic, Molecularand Optical Physics (ITAMP). We have benefited fromhelpful interactions with Rahul Nandkishore, ShivajiSondhi and Alexander Burin. We also gratefullyacknowledge discussions with Rafael Hanel, JamesKeller, Rodrigo Vargas, Arthur Christianen, MarkusSchulz-Weiling, Hossein Sadeghi and Luke Melo. [S1] M. Schulz-Weiling, H. Sadeghi, J. Hung, and E. R.Grant, J Phys B , 193001 (2016).[S2] T. F. Gallagher, Rydberg Atoms (Cambridge UniversityPress, 2005).[S3] R. Patel, N. Jones, and H. Fielding, Phys Rev A ,043413 (2007).[S4] P. Mansbach and J. Keck, Phys. Rev. , 275 (1969).[S5] T. Pohl, D. Vrinceanu, and H. R. Sadeghpour, Phys.Rev. Lett. , 223201 (2008).[S6] M. Bixon and J. Jortner, Journal of Modern Optics ,373 (1996).[S7] E. Murgu, J. D. D. Martin, and T. F. Gallagher, J.Chem. Phys. , 7032 (2001).[S8] F. Remacle and M. Vrakking, J Phys Chem A , 9507(1998).[S9] W. A. Chupka, J Chem Phys , 4520 (1993).[S10] I. F. Schneider, I. Rabadán, L. Carata, L. Andersen,A. Suzor-Weiner, and J. Tennyson, J Phys B , 4849(2000).[S11] N. Saquet, J. P. Morrison, M. Schulz-Weiling,H. Sadeghi, J. Yiu, C. J. Rennick, and E. R. Grant, J Phys B , 184015 (2011).[S12] N. Saquet, J. P. Morrison, and E. Grant, J Phys B ,175302 (2012).[S13] R. Haenel, M. Schulz-Weiling, J. Sous, H. Sadeghi,M. Aghigh, L. Melo, J. Keller, and E. Grant, PhysRev A , 023613 (2017).[S14] D. S. Dorozhkina and V. E. Semenov, Exact solutionsfor matter-enhanced neutrino oscillations , 2691(1998).[S15] H. Sadeghi and E. R. Grant, Phys Rev A , 052701(2012).[S16] S. Sachdev, Quantum phase transitions (Wiley OnlineLibrary, 2007).[S17] V. M. Agranovich,
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