Many-body physics with ultracold plasmas: Quenched randomness and localization

The exploration of large-scale many-body phenomena in quantum materials has produced many important experimental discoveries, including novel states of entanglement, topology and quantum order as found for example in quantum spin ices, topological insulators and semimetals, complex magnets, and high-$T_c$ superconductors. Yet, the sheer scale of solid-state systems and the difficulty of exercising exacting control of their quantum mechanical degrees of freedom limit the pace of rational progress in advancing the properties of these and other materials. With extraordinary effort to counteract natural processes of dissipation, precisely engineered ultracold quantum simulators could point the way to exotic new materials. Here, we look instead to the quantum mechanical character of the arrested state formed by a quenched ultracold molecular plasma. This novel class of system arises spontaneously, without a deliberate engineering of interactions, and evolves naturally from state-specified initial conditions, to a long-lived final state of canonical density, in a process that conflicts with classical notions of plasma dissipation and neutral dissociation. We take information from experimental observations to develop a conceptual argument that attempts to explain this state of arrested relaxation in terms of a minimal phenomenological model of randomly interacting dipoles of random energies. This model of the plasma forms a starting point to describe its observed absence of relaxation in terms of many-body localization (MBL). The large number of accessible Rydberg and excitonic states gives rise to an unconventional web of many-body interactions that vastly exceeds the complexity of MBL in a conventional few-level scheme. This experimental platform thus opens an avenue for the coupling of dipoles in disordered environments that will demand the development of new theoretical tools.


Introduction
When a transform-limited laser pulse excites an ensemble of isolated small molecules to a superposition state, the system evolves periodically with a well-defined pattern of quantum beats [1]. But, a wavepacket similarly created in a large molecule dissipates arXiv:1808.07479v1 [cond-mat.quant-gas] 22 Aug 2018 Many-body physics with ultracold plasmas: Quenched randomness and localization 2 exponentially to a state of maximum entropy [2]. The Eigenstate Thermalization Hypothesis (ETH) explains this difference by arguing that the unitary dynamics of a superposition formed in a dense manifold of states always yield thermal expectation values as a time average [3][4][5][6][7][8]. Experiments confirm this idea, finding ergodic dynamics in quantum systems as small as three transmon qubits [9]. Statistical intramolecular energy redistribution forms the bedrock of more than 50 years of success in unimolecular reaction rate theory [10].
But, just as methods of coherent control can overcome ergodicity and direct particular pathways for the chemical transformation of molecules, theory has described conditions under which certain many-body systems lack intrinsic decoherence, preserve spatial order and localize energy in highly excited states [11][12][13][14][15][16]. In these systems, transport paths for energy flow destructively interfere, causing a quenched system of quantum states to get stuck in a corner of its phase space. The eigenstates of a manybody localized (MBL) phase retain a memory of their initial conditions for arbitrarily long times. This possibility to form enduring quantum superpositions embedded in interacting many-body ensembles suggests a potential as a means of preserving quantum information [17,18].
Experimental quantum systems that fail to thermalize thus command a great deal of interest. A small number of highly engineered examples investigating the dynamics of ultracold atoms have confirmed the principle of MBL [19][20][21][22][23][24][25][26][27]. These results invite an intriguing question: In systems with stronger interactions and stronger disorder, could a state of many-body localization arise naturally?
MBL acts in a profound way to constrain transport processes in a quantum system. In the MBL phase, any local operator, time-evolved with the Hamiltonian and averaged over time, Ô , becomes a local integral of motion (LIOM) [28][29][30][31][32]. Following any perturbation in a state of MBL, local integrals of motion associated withÔ survive for long times. The LIOM refer to conserved quantities, for instance, a local particle or energy density. As a system quenches, emergent LIOM determine the propagation of these conserved quantities in the MBL phase. This naturally raises the question, in the progress of a quench, can locally emergent conservation laws act to guide the propagation of a spatially evolving quantum system to form a global many-body localized state?
In an experimental test of this question, we have studied the dynamics of a stateselected Rydberg gas of nitric oxide as it evolves to form a plasma [33][34][35][36]. On a microsecond timescale, this plasma bifurcates, irreversibly disposing electron energy to a reservoir of mass transport [37]. This evidently quenches the system to form an ultracold, quasi-neutral plasma in a state of arrested relaxation, far from thermal equilibrium [38]. Classical models for plasma rate processes that describe the initial stages of Rydberg gas avalanche and plasma dissipation fail to account for the long lifetime and low electron temperature of this system [39,40]. This anomalous stability signifies a cessation of classical energy transport, which invites the theoretical question: Does the arrested relaxation of this system signify self-assembly to a state of many-body localization? Here, we detail a conceptual model [41] that we have developed in an effort Many-body physics with ultracold plasmas: Quenched randomness and localization 3 to explain these contradictory dynamics.

Experimental evidence for a state of arrested relaxation
The molecular ultracold plasma forms a state of arrested relaxation in a sequence of steps, that begins with the electron impact avalanche of a prolate Rydberg gas ellipsoid. A fraction of Rydberg molecules form pairs spaced within a critical distance, r P . These interact, releasing Penning electrons. Starting in the core, electron-Rydberg collisions drive an electron-impact ionization avalanche. Some fraction of Rydberg molecules relax, releasing energy that heats electrons. The electron gas expands, exhausting energy by radially accelerating NO + ions. The moving NO + ions efficiently exchange charge with stationary NO * Rydberg molecules. This redistributes momentum, and causes velocitymatched ions, cold electrons and Rydberg molecules to accumulate in the wings of the ellipsoid.
We see these dynamics directly in three-dimensional images of plasma bifurcation. Here, the acceleration of opposing plasma volumes disposes kinetic energy initially deposited in the electron gas. This sequence of events quenches and compresses these volumes to form ultracold, correlated distributions of NO + ions, electrons and Rydberg molecules.
We use selective field ionization (SFI) spectroscopy to resolve the evolution of the electron binding energy as a function of time and initial Rydberg gas density, ρ 0 . The dynamics of this electron-impact avalanche fit well with coupled-rate-equation simulations that consider ρ 0 and n 0 , the selected principal quantum number of the Rydberg gas. Figure 1 shows a typical example. SFI traces at lower initial density first exhibit the field ionization spectrum of a Rydberg gas in the initially selected 44f (2) state. The spectrum then evolves owing to the effects of electron-Rydberg collisions. Over a period of 300 ns, the spectrum directly shows evidence of electron-Rydberg collisional l-mixing, which shifts the signal produced at lower density by residual n 0 Rydberg molecules to a higher appearance potential. On the same timescale, SFI traces show that the avalanche at higher density consumes all the 44f (2) Rydberg molecules, creating a narrow electron binding-energy distribution of width W , which peaks within 500 GHz of the zero-field ionization threshold, and tails to reflect a high-Rydberg population of intermediate n.
After 300 ns, electron collisional redistribution of the Rydberg population over n apparently stops. The distribution of molecules with near-threshold binding energy rises while the residual population of 44f (2) molecules decays, and we see little or no change in the high-density distribution of intermediate-n Rydberg molecules, and n = 44 features evidently evolve to a state of very low binding energy in a single step without populating Rydberg states at intermediate energy.
As displayed in Figure 1, the total amplitude of the detected electron signal falls with time from 0 to 5 µs. This drop accords with rate-theory models for charge neutralization by NO + + e − dissociative recombination and NO * Rydberg Many-body physics with ultracold plasmas: Quenched randomness and localization    (2) Rydberg gases of nitric oxide, obtained with fields that rise linearly at 0.6 V ns −1 , starting 0, 300 and 450 ns after excitation. The figure for each ramp-field delay displays a stack of 4,000 traces ordered from top to bottom by decreasing initial Rydberg gas density. The characteristic pair of features observed for Rydberg gases of lower density correspond to the field ionization of the single selected 44f (2) state to form ions with rotational quantum numbers of N + = 0 and 2 respectively. Note in particular how effects of electron-Rydberg collisions evident in shaping the distribution of Rydberg binding energies at high densities, and shifting the l-mixed 44f (2) signal to higher appearance potential between 0 and 300 ns, cease to matter in the evolution to 400 ns, while at lower density, n = 44 features evolve to a state of very low binding energy without populating Rydberg states at intermediate energy. bottom: Integrated electron signal as a function of ultracold plasma flight times from 0 to 400 µs. See reference [38] predissociation. After 5 µs, neutral dissociation stops, and the ultracold plasma begins a phase of arrested relaxation that lasts for a time as long as we can make measurements. While the rate processes that define this sequence of events depend on initial conditions, the system evolves to the same final state density and binding energy regardless of ρ 0 and n 0 . Experiment thus affords a direct view of a robust self-assembly in which a Rydberg gas undergoes a spontaneous avalanche, bifurcation and quench to form a well-defined ensemble in which ultracold, correlated NO + ions and electrons occupy a binding energy distribution that extends 500 GHz below the ionization energy of nitric oxide. Densitycalibrated, time-resolved SFI spectra directly map the propagation of the electronimpact avalanche as it consumes the initially selected Rydberg gas.
Images of this evolution projected in the x, y plane perpendicular to the propagation axis, capture a cross-beam bifurcation that quenches the plasma to form separating volumes that expand very slowly. This slow rate of expansion tells us that free electrons, if present, must have an exceedingly low temperature. But, for these conditions, a classical nitric oxide plasma would be highly unstable: Coupled rateequation simulations predict the rapid evolution of such a system to an inevitable steady state of high electron temperature with rapid expansion and dissipation to neutral nitrogen and oxygen atoms.
The long lifetime, low binding energy and slow rate of expansion might suggest a system quenched to a gas of very high-n Rydberg molecules. But again, for the conditions of these observations, such a Rydberg gas would be highly unstable with respect to classical collisional rate processes. Coupled rate-equation simulations predict a rapid evolution essentially to the same steady state of high electron temperature and dissipation to neutral nitrogen and oxygen atoms.
Thus, the experiment captures definitive images of a plasma that has been quenched to a long-lived state that should be both thermodynamically unstable and kinetically unstable with respect to well-defined thermochemical rate processes. We proceed now to develop a theoretical model for this state of arrested relaxation.

Single-particle basis states
From time-resolved SFI spectra, we find that the ellipsoidal nitric oxide Rydberg gas avalanches to a plasma, bifurcates, quenches, and compresses to form the same ultracold, correlated distributions of NO + ions, electrons and Rydberg molecules, regardless of its initial principal quantum number or density.
In an effort to describe the many-body state of this system, let us start by adopting a simple basis consists of the high-Rydberg states of NO that gives way for very low binding energy to a further set of excitonic states of electrons bound to ions in a dielectric medium containing some small excess fraction of NO + ions. Let us define this basis by the eigenfunctions of a Hamiltonian, H d : obtained within the narrow binding-energy interval, W . Within H d , a local Hamiltonian, h i , defines the eigenstates of each extravalent electron in the field determined by its NO + core in a dielectric medium defined by the surrounding ions and electrons. For each dipole, i, h i is exceedingly complex. Even for the limit of a completely isolated NO + and Rydberg electron, this Hamiltonian must account for electron orbital interactions with the rotational, vibrational and electronic degrees of freedom of the core ion, including in particular coupling to predissociative channels of irreversible decay to neutral products, N( 4 S) and O( 3 P). Extending h i to allow for interactions in the transport phase and excitonic binding adds considerable complexity. But, we can expect the eigenstates of h i to exhibit continuity as a function of binding energy extending from a regime of isolated, central-field Rydberg molecules to one of Rydberg-like excitons.
Many-body physics with ultracold plasmas: Quenched randomness and localization Together, these form a basis of dipoles, |e i , with momentum P i Figure 2 schematically diagrams this basis. Here we show the interval W defined by SFI spectra and indicate a domain of lower energy, where more deeply bound Rydberg states predissociate on a microsecond timescale, and one of higher energy where free electrons drive substantial rates of plasma expansion. Experimental observations of very long plasma lifetimes and very slow rates of plasma expansion exclude states in these domains from the basis we need to describe this system.
Bifurcation quenches the plasma to occupy levels in the interval, W , which for an average binding energy of 250 GHz, form a manifold with more than two states per MHz. Over a period of 10 µs, the plasma evolves to an ellipsoidal arrested state with a spatially averaged density of ∼ 1.4 × 10 10 cm −3 , at which point a distance of 3.3 µm separates the average pair of Rydberg molecules. The plasma quench distributes molecules randomly over states. Rydberg-Rydberg interactions determine the properties of these states and the dynamics of energy transport with which the system continues to evolve. We now consider these interactions and develop a model for the long-time dynamics.

Intermolecular interactions
Rydberg and Rydberg-like molecules represented by the states in our zeroth-order basis undergo interactions governed in large part by the electrostatic potentials formed by NO + ion -electron pairs. In a limit for which the intermolecular distance exceeds the dimensions of the individual Rydberg charge distributions, we can represent this by a multipole potential, V ij = V (r i − r j ), which, to lowest order reduces to the anisotropic dipolar interaction defined by Many-body physics with ultracold plasmas: Quenched randomness and localization 7 Note that any pair of neighbouring dipoles i and j each occupy random states of arbitrary energy, total angular momentum and parity. Their dipole-dipole interaction encompasses hundreds if not thousands of paired, dipole-allowed transitions. The strength of this interaction varies with the distance between NO + ions and the relative orientation of their internuclear axes. Little of what follows depends upon the exact determination of matrix elements for particular interactions. Where necessary for illustration, we shall average over solid angle in the molecular frame and estimate coupling strengths for partners separated by the average distance for a given density, as determined by the Wigner-Seitz radius, a ws = (3/4πρ) 1/3 . Generally speaking, the amplitudes of individual dipole-dipole interactions increase with the average effective principal quantum number, approximately as [(n i + n j )/2] 2 , and fall off quadratically with the difference in principal quantum number [42,43]. A small subset of resonant transitions dominate the interaction of any particular pair of Rydberg/excitons.

Dipole-dipole Hamiltonian
Taking H d together with this interaction potential, the Hamiltonian up to pairwise dipole-dipole interactions is We can reduce this Hamiltonian to describe the pairwise interactions evaluated in the |e i basis [44,45]: Hamiltonians such as (3) and (4) usually refer to the case where a high-resolution laser prepares a Rydberg gas in which some dipole-dipole interaction gives rise to a particular set of coupled states [46][47][48]. In the present case, avalanche and quench spontaneously form a molecular ultracold plasma in a process that populates a great many different states. These states evolve spatially and interact without reference to a light-matter coherence or a dipole blockade of any kind 3.2.2. Projection to a regime of lower-order resonance The experimental system relaxes in a few microseconds to a quenched state of ultracold temperature, from which it expands at no more than a few tens of meters per second. Dipole-dipole coupling proceeds on an instantaneous timescale [49][50][51][52]. During the first 100 ns of the avalanche, under conditions well-described by classical coupled rate-equation kinetics, the SFI spectrum shows obvious effects of electron-Rydberg collisional l-mixing [38]. Long-range NO + ion -NO Rydberg collisions act to redistribute ion radial momentum and accelerate electron cooling. Thereafter, as the system approaches a state of arrested relaxation, all signs of electron-Rydberg interaction disappear. In particular, SFI spectra show no sign of electron-Rydberg n-changing collisions. Heavy-particle, Rydberg-Rydberg inelastic collisions occur with many orders of magnitude lower frequency. We thus assume that Many-body physics with ultracold plasmas: Quenched randomness and localization 8 dipole-dipole exchange dominates energy transport during this latter phase of relaxation, and that these instantaneous interactions evolve adiabatically with the motion of ion centres.
These well separated timescales allow us to write an effective Hamiltonian that describes pairwise interactions evolving in a frame of slow-moving ions and Rydberg molecules: H eff = P i,j V dd ij , in which P represents a projector onto a Hilbert subspace characterized by low-order resonances. In the plasma, no laser field acts to specify the particular states that couple in the interaction between any pair of dipoles, i and j. Solving the problem in full therefore requires a diagonalization of the complete dipolar matrix, encompassing all the states in a dense manifold that extends from n ∼ 80 up to the weakest bound electronic state. The subset of active levels varies widely from dipole to dipole in the plasma. The use of P simplifies this task by focusing attention on the comparably sparse set of most probable dipolar transitions in the arrest phase. Note that the model keeps all low-order transitions, even those that involve interacting molecules with drastically different quantum numbers.

3.2.3.
Dipole-dipole coupling during the energy transport phase Let us consider an individual pair of dipoles in arbitrary states interacting under delocalized conditions. The dipole operator couples these molecules in an appropriate subspace spanned by the full set of accessible product states of the Rydberg single-particle wavefunctions. This gives rise to a process of state mixing that we can represent as energy transport by a sequence of virtual one-photon exchanges. Thus, a molecule of low principal quantum number can interact with one of high principal quantum number to exchange energy by instantaneous resonant interactions involving a total of L states, |e 1 , |e 2 ... |e L . Likewise, two molecules that initially occupy states of similar principal quantum number couple in a comparable sequence, that transports one molecule to high n and the other to low n (where predissociation occurs with greater likelihood). The experimental SIF spectrum in Figure 1 may well show direct evidence for such a process, in which for lower initial densities, two molecules with n = 44 interact to transport one to a longlived state with n > 80 and the other to one to a dissociative level in the region of hydrogenic n = 34.

Dipole-dipole coupling in the arrested phase
The avalanche of the Rydberg gas to plasma, its hydrodynamic disposal of electron energy, its quench, correlation and transport to a regime of weak electron binding presages localization. Interactions that occur in the transport phase alter the energy level structure of every molecule. These perturbations that arise from disorder act to decrease or increase the degree of resonance in any particular coupling sequence. To occur with substantial effect, dipole-dipole interactions at any point in this process must have coupling amplitudes that exceed the energy deficit of transitions paired on i and j.
The projector acts to limit the number of sequences in the scheme of dipole-dipole coupling permitted in the arrested phase. The most extreme case, L = 2, considers only Many-body physics with ultracold plasmas: Quenched randomness and localization 9 single-step transitions involving randomly selected basis states, |e 1 i , |e 2 i , e 1 j and e 2 j . The case of L = 3 couples three levels per molecule, in sequences of as many as two transitions. For L = 4, the active space increases to four basis states and includes the possibility of three transitions per molecule. Each such calculation selects states, 1, 2...L, randomly on each molecule, without bias. It seems reasonable to expect conditions of resonance to limit the effectiveness of sequences with increasing order, but this remains to be determined. Rare higher-order resonances could prove important.
For now, we describe the arrested phase as an ensemble of random interacting dipoles in which pairwise interactions couple finite subsets of L-level basis states |e 1 , |e 2 , ... e L that span the full space of all dipolar levels. The superscript with lower (higher) integer label refers to the state with larger (smaller) electron binding energy. Because these interactions occur between molecules quenched to random quantum states, the active levels, as they determine L in each case, vary from one pair to the other. Thus, our model imagines an effective dynamic quasi-resonant description with random dipolar matrix elements.
The Rydberg-Rydberg coupling driven by V dd ij causes interacting states of nearly equal energy to repel. The magnitude of the repulsion varies with the average intermolecular distance, as determined by sample density, and by the principal quantum number dependent average dipole-dipole coupling matrix element,τ . In the regime of energy transport, this level repulsion acts to produce a detectable void in the spectrum of levels populated in the dense system of states very near the ionization threshold. The width of this void, as it varies with density, provides an experimental gauge ofτ , and from this, the average dipole-dipole coupling amplitude,J at any given density.
Note that the projection we adopt here does not neglect Rydberg or excitonic dipoles of any particular binding energy. On the contrary, P spans the full set of basis states, operating to pair all allowed low-order dipolar resonances. Thus, dipoles in the arrested state of the ultracold molecular plasma dynamically couple via random dipolar interactions involving transitions in a L-level coupling schemes, where the L levels change from dipole pair to dipole pair and from time to time in a random distribution from one pair of interacting dipoles to another.

Effective many-body spin model
3.3.1. Spin Hamiltonian The quenched system supports a vast distribution of rare, resonant pair-wise interactions that occur in a random landscape of Rydberg/exciton energies. Dipole-dipole coupling in this dense system of basis states causes excitation exchange. Near-resonant interactions predominate. We assume these to involve L states on each molecule, where L is a small number in the range from 2 to 4. These interactions form L-level systems drawing from different basis states from dipole to dipole: The states |e 1 , |e 2 ... e L vary from one dipole to another and from time to time.
To describe this ensemble, we adopt the representation of a many-body Fock space of interacting dipoles [53], representing pairwise excitations by spins with energies, i , Many-body physics with ultracold plasmas: Quenched randomness and localization 10 and exchange interactions governed by an XY model Hamiltonian that describes these interactions in terms of their spin dynamics: whereŜ γ in each case denotes a spin operator acting in the space of L active states, and γ = x, y or z. h.c. refers to Hermitian conjugate. This Hamiltonian reflects the diagonal and off-diagonal disorder created by the variation in the level system from dipole to dipole. In what follows, we shall seth = 1.

On-site local spin term
3.3.3. Nature of disorder in i Quenching results in a strongly disordered landscape of Rydberg/exciton energies that varies randomly from dipole to dipole. The timeresolved SFI spectrum affords an electron binding-energy distribution that describes the range over which these level separations vary, and this directly reflects the disorder in these on-site energies. In spin language, our Hamiltonian term i iŜ z i represents a Gaussian-distributed random local field that spans a width, W , which experimentally defines the range of m S i . The representative SFI spectrum in Figure 1 directly gauges W ∼ 500 GHz by the range of electron binding energies observed in the quenched ultracold plasma. We see then that the first term in H eff describes the diagonal disorder arising from random contributions to the on-site energy of any particular dipole owing to its random quantum state and the random environment of neighbouring dipoles.

Resonance and spin-exchange
In the coupled-state subspace of a pair of interacting dipoles, the dipole-dipole matrix elements describe a resonant transfer of excitation from i to j with weight J ij [54][55][56]. We represent this excitation exchange by lowering and raising operators in the L subspace:σ − i and its Hermitian conjugate (h.c.) Many-body physics with ultracold plasmas: Quenched randomness and localization 11 σ + i . An excitation transfer between dipoles i and j therefore corresponds to a spin flip- . We emphasize here that this term provides for the collisionless propagation of excitations in the plasma by resonant and thus evidently rare spin flip-flop interactions. The interplay between the dimensionality and rarity of interactions owing to disorder eventually determines the terminal state of (de)localization of the plasma.
We thus can represent the two levels of a dipole i with spacing i , as in Figure 3, in terms of a onebody operator iŜ j . This refers to the dipole-dipole mediated transfer of excitation [54] represented by, for We can expect this class of matrix element to be non-zero for many of the local eigenstates of h i and h j , as the dipole-dipole operator couples states of opposite parity, limited only by selection rules governing the conservation of energy and total angular momentum [57]. L = 3 case Dipole-dipole coupling schemes link sequences of transitions in cases of L = 3 and 4. Excitation transfer still governs the dynamics via terms like Many-body physics with ultracold plasmas: Quenched randomness and localization 12 , where theŜ operators live in the active L-dimensional subspaces. For example, a molecule initially in the random state |e 1 i and one in the state e 3 j can interact via transitions involving a common final state, |e 2 i = e 2 j , defining the sequence for L = 3 diagrammed in Figure 4. This type of three-state interaction figures prominently in studies of Rydberg quantum optics, in which a narrow bandwidth laser drives a resonant pair state. Consider for example, the case of 58d 3/2 + 58d 3/2 ↔ 60p 1/2 + 56f 3/2 in 87 Rb. In zero field, at large separations these two-atom states of Rb exhibit a quasi-degeneracy, with an energy difference less than 7 MHz. Using dipole traps to control the distance between pairs of 58d 3/2 Rydberg atoms, Gaëtan et al [58] have showed that a separation of 4 µm breaks the degeneracy, forming a set of three eigenstates, mixed in the p, d, f basis and separated by 50 MHz. Excitation transfer operates in this case as: More generally, for L = 3, a system randomly interacting dipoles couples to yield a mix of configurations, L = 4 case For a disordered Rydberg gas occupying a dense manifold of states, the interaction described above as L = 3 represents a special case of the more general L = 4 coupling scheme diagrammed for particular levels of molecules i and j Figure 5. Here, the interaction of i and j give rise to eigenstates arising from the pair configurations, The particular interaction diagrammed above, which links a transition from |e 2 i to |e 1 i with the transition from e 3 j to e 4 j , describes coupling in a subspace of L = 4 with characteristics of the S = 1/2 flip-flop interaction diagrammed in Figure 3: Figure 5. Schematic diagram representing a sequence in the coupling of two Rydberg molecules, i and j, in a case of four active levels per molecule. The random population distribution over a very large number of Rydberg/exciton levels in the quenched ultracold plasma adds to the likelihood of L = 4 and higher-order interactions. Note that the state density increases rapidly with the elevation of population to higher principal quantum numbers. This, accompanied by the growing rate of predissociation in the population of levels descending to n, gives the L = 4 pathway a particular importance as the system relaxes to occupy a band of low binding energies during the transport phase that precedes arrest with an excitation transfer: i.e. e 2 i , e 3 j . These partial coupling schemes link sets of flip-flop transitions on interacting dipoles. A fuller treatment would apply the complete S = 1 and 3/2 Pauli matrices to the cases of L = 3 and 4, respectively, and extend similar sequences to higher L. But, to a first approximation, we might assume that low-order resonant dipole-dipole excitation exchange in the dense manifold of basis states most prominently involves pairs of levels in these three low-L cases.

Randomness in
ij determines the off-diagonal disordered amplitudes of the spin flip-flops. To visualize the associated disorder, recognize that the second term varies as t ij ∝ |d i ||d j |, where every interaction selects a different d i and d j . The C 3 of 3230 MHz µm 3 measured by Gaëtan et al. [58] for the 58d 3/2 + 58d 3/2 ↔ 60p 1/2 + 56f 3/2 interaction in 87 Rb scales as n 2 to yield a dipoledipole coupling energy of 2 GHz at n = 146 for a density 0.1 µm −3 . This coupling width, representative of ∆n = 2 interactions at this binding energy, rises at high n to span a large set of | i − j | energy deficits. Yet, this J ij is clearly much smaller than W ≈ 500 GHz, the range of i , which is determined by the maximum separation between |e 1 i and e L i , as measured by the width of the binding energy distribution in the arrested state.

Induced Ising interactions
For this generic situation of |J ij | << W , levels interact in sequence to add Ising terms that cause van der Waals shifts of pairs of Many-body physics with ultracold plasmas: Quenched randomness and localization 14 dipoles [59]. Figure 6 diagrams an exemplary pairwise interaction of three nearestneighbour spins i, j and k for the case of L = 2. Spins i and j couple via spin k in a third-order process that proceeds as follows: resulting in a self interaction that shifts the energies of i, j. These processes occur with an amplitude, U ij ≈ J 2 ij J/W 2 , where J estimates J ij , for an average value of t ij at an average distance separating spins. e, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have ten it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve dipole-dipole coupling of a transition between two levels on one molecule with an gy conserving inverse on the other. We can extend such sequences to higher L, but energy resonant dipole-dipole excitation exchange in the dense manifold of basis s will most prominently involve a small number of L-levels per dipole.
. Induced Ising interactions For |J ij | << W most relevant to the experimental tion, sequences of interactions add Ising terms that describe a van der Waals shifts airs of dipoles ?. Consider, for example, three mutually nearest-neighbour spins and k in the L = 2 case. A third-order process couples spins i and j via spin k Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dipole coupled under conditions of L = 4. The high state density and strong disorder in the quenched ultracold plasma gives this case of L = 4 greater significance than the restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve the dipole-dipole coupling of a transition between two levels on one molecule with an energy conserving inverse on the other. We can extend such sequences to higher L, but low-energy resonant dipole-dipole excitation exchange in the dense manifold of basis states will most prominently involve a small number of L-levels per dipole.  nse, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have ritten it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve e dipole-dipole coupling of a transition between two levels on one molecule with an ergy conserving inverse on the other. We can extend such sequences to higher L, but w-energy resonant dipole-dipole excitation exchange in the dense manifold of basis ates will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the experimental tuation, sequences of interactions add Ising terms that describe a van der Waals shifts pairs of dipoles ?. Consider, for example, three mutually nearest-neighbour spins j and k in the L = 2 case. A third-order process couples spins i and j via spin k Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dipole coupled under conditions of L = 4. The high state density and strong disorder in the quenched ultracold plasma gives this case of L = 4 greater significance than the restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve the dipole-dipole coupling of a transition between two levels on one molecule with an energy conserving inverse on the other. We can extend such sequences to higher L, but low-energy resonant dipole-dipole excitation exchange in the dense manifold of basis states will most prominently involve a small number of L-levels per dipole.  sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you ha written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case invol the dipole-dipole coupling of a transition between two levels on one molecule with energy conserving inverse on the other. We can extend such sequences to higher L, b low-energy resonant dipole-dipole excitation exchange in the dense manifold of ba states will most prominently involve a small number of L-levels per dipole.  sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve the dipole-dipole coupling of a transition between two levels on one molecule with an energy conserving inverse on the other. We can extend such sequences to higher L, but low-energy resonant dipole-dipole excitation exchange in the dense manifold of basis states will most prominently involve a small number of L-levels per dipole.   Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dipole coupled under conditions of L = 4. The high state density and strong disorder in the quenched ultracold plasma gives this case of L = 4 greater significance than the restrictive limit of L = 3 e, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have ten it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve dipole-dipole coupling of a transition between two levels on one molecule with an gy conserving inverse on the other. We can extend such sequences to higher L, but energy resonant dipole-dipole excitation exchange in the dense manifold of basis s will most prominently involve a small number of L-levels per dipole.
. Induced Ising interactions For |J ij | << W most relevant to the experimental tion, sequences of interactions add Ising terms that describe a van der Waals shifts airs of dipoles ?. Consider, for example, three mutually nearest-neighbour spins and k in the L = 2 case. A third-order process couples spins i and j via spin k  Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dipole coupled under conditions of L = 4. The high state density and strong disorder in the quenched ultracold plasma gives this case of L = 4 greater significance than the restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve the dipole-dipole coupling of a transition between two levels on one molecule with an energy conserving inverse on the other. We can extend such sequences to higher L, but low-energy resonant dipole-dipole excitation exchange in the dense manifold of basis states will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the experimental situation, sequences of interactions add Ising terms that describe a van der Waals shifts of pairs of dipoles ?. Consider, for example, three mutually nearest-neighbour spins i, j and k in the L = 2 case. A third-order process couples spins i and j via spin k  Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dipole coupled under conditions of L = 4. The high state density and strong disorder in the quenched ultracold plasma gives this case of L = 4 greater significance than the restrictive limit of L = 3 nse, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have ritten it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve e dipole-dipole coupling of a transition between two levels on one molecule with an ergy conserving inverse on the other. We can extend such sequences to higher L, but w-energy resonant dipole-dipole excitation exchange in the dense manifold of basis ates will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the experimental tuation, sequences of interactions add Ising terms that describe a van der Waals shifts pairs of dipoles ?. Consider, for example, three mutually nearest-neighbour spins j and k in the L = 2 case. A third-order process couples spins i and j via spin k  Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dipole coupled under conditions of L = 4. The high state density and strong disorder in the quenched ultracold plasma gives this case of L = 4 greater significance than the restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve the dipole-dipole coupling of a transition between two levels on one molecule with an energy conserving inverse on the other. We can extend such sequences to higher L, but low-energy resonant dipole-dipole excitation exchange in the dense manifold of basis states will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the experimental situation, sequences of interactions add Ising terms that describe a van der Waals shifts of pairs of dipoles ?. Consider, for example, three mutually nearest-neighbour spins i, j and k in the L = 2 case. A third-order process couples spins i and j via spin k  Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dip coupled under conditions of L = 4. The high state density and strong disorder the quenched ultracold plasma gives this case of L = 4 greater significance than t restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you ha written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case invol the dipole-dipole coupling of a transition between two levels on one molecule with energy conserving inverse on the other. We can extend such sequences to higher L, b low-energy resonant dipole-dipole excitation exchange in the dense manifold of ba states will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the experiment situation, sequences of interactions add Ising terms that describe a van der Waals shif of pairs of dipoles ?. Consider, for example, three mutually nearest-neighbour spi i, j and k in the L = 2 case. A third-order process couples spins i and j via spin plasma, the displacement of e 2 i and e 2 j will will lessen the significance of L = 3 interactions compared with the case of L = 4. Figure 5. Schematic diagram representing two Rydberg molecules, i and j, dipole coupled under conditions of L = 4. The high state density and strong disorder in the quenched ultracold plasma gives this case of L = 4 greater significance than the restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you have written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case involve the dipole-dipole coupling of a transition between two levels on one molecule with an energy conserving inverse on the other. We can extend such sequences to higher L, but low-energy resonant dipole-dipole excitation exchange in the dense manifold of basis states will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the experimental situation, sequences of interactions add Ising terms that describe a van der Waals shifts of pairs of dipoles ?. Consider, for example, three mutually nearest-neighbour spins i, j and k in the L = 2 case. A third-order process couples spins i and j via spin k sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case in the dipole-dipole coupling of a transition between two levels on one molecule wi energy conserving inverse on the other. We can extend such sequences to higher L low-energy resonant dipole-dipole excitation exchange in the dense manifold of states will most prominently involve a small number of L-levels per dipole.  sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies a written it to di↵erentiate these situations of L = 2, 3 and 4, which in each c the dipole-dipole coupling of a transition between two levels on one molec energy conserving inverse on the other. We can extend such sequences to hi low-energy resonant dipole-dipole excitation exchange in the dense manifo states will most prominently involve a small number of L-levels per dipole.  sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as y written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case the dipole-dipole coupling of a transition between two levels on one molecule energy conserving inverse on the other. We can extend such sequences to highe low-energy resonant dipole-dipole excitation exchange in the dense manifold states will most prominently involve a small number of L-levels per dipole.  sense, as in Figure ?? Let's be sure that your Pauli matrix algebra appli written it to di↵erentiate these situations of L = 2, 3 and 4, which in eac the dipole-dipole coupling of a transition between two levels on one mo energy conserving inverse on the other. We can extend such sequences to low-energy resonant dipole-dipole excitation exchange in the dense ma states will most prominently involve a small number of L-levels per dipo sense, as in Figure ?? Let's be sure written it to di↵erentiate these situa the dipole-dipole coupling of a tran energy conserving inverse on the oth low-energy resonant dipole-dipole e states will most prominently involve sense, as in Figure ?? Let's be sure that y written it to di↵erentiate these situations the dipole-dipole coupling of a transition energy conserving inverse on the other. W low-energy resonant dipole-dipole excitat states will most prominently involve a sm   Figure 5. Schematic diagram representing two Rydberg molecules, i and j, coupled under conditions of L = 4. The high state density and strong disor the quenched ultracold plasma gives this case of L = 4 greater significance th restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as you written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case in the dipole-dipole coupling of a transition between two levels on one molecule wi energy conserving inverse on the other. We can extend such sequences to higher L low-energy resonant dipole-dipole excitation exchange in the dense manifold of states will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the experim situation, sequences of interactions add Ising terms that describe a van der Waals of pairs of dipoles ?. Consider, for example, three mutually nearest-neighbour i, j and k in the L = 2 case. A third-order process couples spins i and j via s  Figure 5. Schematic diagram representing two Rydberg molecules, i coupled under conditions of L = 4. The high state density and stron the quenched ultracold plasma gives this case of L = 4 greater significa restrictive limit of L = 3 sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies a written it to di↵erentiate these situations of L = 2, 3 and 4, which in each c the dipole-dipole coupling of a transition between two levels on one molec energy conserving inverse on the other. We can extend such sequences to hi low-energy resonant dipole-dipole excitation exchange in the dense manifo states will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the ex situation, sequences of interactions add Ising terms that describe a van der W of pairs of dipoles ?. Consider, for example, three mutually nearest-neigh i, j and k in the L = 2 case. A third-order process couples spins i and j  Figure  sense, as in Figure ?? Let's be sure that your Pauli matrix algebra applies as y written it to di↵erentiate these situations of L = 2, 3 and 4, which in each case the dipole-dipole coupling of a transition between two levels on one molecule energy conserving inverse on the other. We can extend such sequences to highe low-energy resonant dipole-dipole excitation exchange in the dense manifold states will most prominently involve a small number of L-levels per dipole.

Induced Ising interactions
For |J ij | << W most relevant to the expe situation, sequences of interactions add Ising terms that describe a van der Waa of pairs of dipoles ?. Consider, for example, three mutually nearest-neighbo i, j and k in the L = 2 case. A third-order process couples spins i and j vi  sense, as in Figure ?? Let's be sure that your Pauli matrix algebra appli written it to di↵erentiate these situations of L = 2, 3 and 4, which in eac the dipole-dipole coupling of a transition between two levels on one mo energy conserving inverse on the other. We can extend such sequences to low-energy resonant dipole-dipole excitation exchange in the dense ma states will most prominently involve a small number of L-levels per dipo

Induced Ising interactions
For |J ij | << W most relevant to the situation, sequences of interactions add Ising terms that describe a van de of pairs of dipoles ?. Consider, for example, three mutually nearest-ne i, j and k in the L = 2 case. A third-order process couples spins i an sense, as in Figure ?? Let's be sure that y written it to di↵erentiate these situations the dipole-dipole coupling of a transition energy conserving inverse on the other. W low-energy resonant dipole-dipole excitat states will most prominently involve a sm  Taken together with Eq (5) this result yields a general spin model with dipole-dipole and van der Waals interactions: where U ij = D ij /r 6 ij and D ij = t 2 ij J/W 2 . With the addition of this term, the Hamiltonian develops many-body characteristics, which give rise to non-trivial dynamics involving emergent correlations between spins coupled by flip-flops. For L = 2, this term appears at the third order in perturbation theory, but at second order for all other L [59]. Thus, Ising interactions develop naturally in the |J ij | W limit in three dimensions. These interactions arise from the off-resonant part of i,j J ij (Ŝ + iŜ − j + h.c.), which cannot cause real transitions [59]. This rationalizes the use of an average dipole-dipole coupling, J, as a weighting term in U ij . These interactions have been shown to arise from an atomic physics picture of level-matching dipolar interaction [60] and conform with the van der Waals interactions of atomic and molecular physics.

Non-resonant on-site interactions
We note as well that that this limit includes further perturbative processes that renormalize on-site fields by van der Waals shifts and affect the pairwise flip-flop amplitudes [59][60][61]. We simply absorb these effects in the definitions of i and J ij . In a large-defect limit of Rydberg interactions, we can approximate the Ising term by l =i C ij 6 /r 6 ij in which C ij 6 represents the C 6 coefficient describing the van der Waals interaction between off-resonant dipoles [60,61].

Solution of the many-body Hamiltonian
Even with its simplifying approximations, the complexity of the effective Hamiltonian places an exact solution of Eq (8) beyond reach for a plasma of 10 8 molecules. The number of basis set wavefunctions needed for a full computation for all possible initial spin state configurations scales as L N plasma or L 10 8 . Nonetheless we can gauge some likely properties of such a solution by analytical arguments.

Locator expansion approach and resonance counting arguments
To proceed this this respect, we start with an assumed state of complete localization and then ask whether delocalizing perturbations can destabilize this phase. Here, we cast the problem in the form of the Hamiltonian, Eq (5), and expand about the infinite disorder limit where a localized phase must exist. We ignore the disorder in J ij , and focus on dipole-dipole interactions in the L = 2 limit. Thus, with reference to Eq (5), we develop a locator expansion in dipole-dipole perturbations, , about the localized state defined by H 0 = i iŜ z i . We consider transport by dipole-dipole interactions along a chain of n dipoles from a position, π(0) = a to π(n) = b. The resolvent [62]: propagates the wavefunction to point b, given its properties at a. Here, π:a→b,|π|=n sums over paths on the web of dipoles from a to b, and Σ {π(0),... ,π(s−1)} π(s) denotes the selfenergy, having incorporated the contributions of different delocalizing paths to π(s) and removed the points π(0), . . . , π(s − 1). This procedure involves no double counting of paths. The self-energies converge to 0 in the infinite disorder limit as propagation of excitations completely breaks down. For more details, see reference [62].
The expansion defined by the resolvent accounts for all possible pathways for the propagation of excitations. Delocalizing interactions, which in effect increase this number to infinity, cause G(b, a, ω) to diverge. The complexity of the present problem prevents a complete computation of the resolvent. As an alternative, we can gauge the divergence of G(b, a, ω) by counting the number of resonances defined by a first-order perturbative condition in which the dipole-dipole coupling between two distant sites i and j, J ij = t ij /r 3 ij , substantially exceeds the energy deficit of the coupled transition Many-body physics with ultracold plasmas: Quenched randomness and localization 16 . A number of resonances as a function of system size that diverges in the thermodynamic limit indicates a breakdown of the locator expansion. Such divergence would suggest an infinite number of pathways for excitations to propagate in the system, and thus delocalization on some finite timescale. In the development that follows, we consider the resonance structure that develops for the specific case of dipole-dipole interactions in a three-dimensional system.

Single-particle problem
In the limit of a single excitation, the Ising interaction has zero amplitude, and the problem reduces to the single-particle dipolar XY Hamiltonian. According to locatorexpansion arguments, dipolar interactions give rise to extended states with subdiffusive dynamics in three dimensions [63,64]. Recent results show that while dipolar interactions in three dimensions can lead to extended states, such systems exhibit non-ergodic behviour [65]. Moreover, algebraic localization (as opposed to exponential Anderson localization) arises in systems with off-diagonal disorder in the long-range spin flip-flop interactions in one dimension [66]. Thus, in a model like Eq. (8), with a mixture of diagonal and off-diagonal disorder, we might expect to find some form of localization in the single-excitation limit.

Many-body problem
For a system of many coupled excitations, Ising interactions form off-diagonal matrix elements in the resonantly coupled pair states [67], which allow energy to propagate from one pair to the other. The progression of these interactions can ultimately lead to delocalization.
In the development that follows, we utilize the treatments of references [59,68] to explore the particular consequences of dipole-dipole coupling in the three-dimensional system described by Eq (8). We represent typical values of t ij and D ij by t and D respectively, and refer again to an average dipole-dipole coupling term, J, which constrains this picture to a limit without off-diagonal disorder. To start, we simply explore the effect of the xy spin flip-flops, and then consider the additional effects of z Ising interactions on these couplings, as sketched below.

Spin flip-flop interactions
Delocalization driven by spin flip-flop interactions proceeds via matrix elements coupling pairs of dipoles that exceed the energy deficit associated with the corresponding transition: t ij /|r 3 ij | ≥ δ ij = | i − j |, where i and j span the measured interval of energy, W . We consider a spherical shell surrounding each dipole that extends from a radial distance R xy to 2R xy . On average, each central dipole sees a number of resonant spins for R xy < |r ij | < 2R xy defined by: Many-body physics with ultracold plasmas: Quenched randomness and localization 17 or, where ρ is the density of spins (dipoles) in the plasma. Note, because the number of particles in a shell increases as R 3 xy while the resonance width decreases as R −3 xy , dipoledipole coupling in three dimensions forms a critical case in which, for a given density, ρ, the number of resonant dipoles, N xy does not change with radial distance.

Ising interactions
These resonant pairs, mixed by flip-flop interactions, define a new degree of freedom, termed a pseudospin, with an average energy splitting, J(R xy ) = t/R 3 xy . Ising interactions couple pairs of pseudospins to an average degree determined by U (R z ) = D/R 6 z , where R z describes the distance between pseudospins and D again represents an average value of the van der Waals matrix element defined by Eq (8).
The quantity (R xy ) = ρN xy (R xy ) defines the density of pseudospins of radial size, R xy . For such a pseudospin, we can determine a number of Ising resonances present within a shell from distance R z to distance 2R z by, If N z (R xy , R z ) diverges as R z → ∞, the corresponding system delocalizes owing to a diverging number of energy transfer pathways. To apply Eq (12) in a test for this, we must know the typical pseudospin size. As pointed out by Yao and coworkers [68], the effective size of a central pseudospin, R xy , ought to grow with R z . Enforcing an extended-pairs condition for this case yields R xy ∼ (t/D) 1/3 R 2 z [68], which determines a number of Ising resonances that grows with distance as: This quantity diverges like the volume as we take R z → ∞.
4.3.3. Critical system size As described above, for a three-dimensional system of fixed density spin flip-flop interactions mix dipoles to create a number of pseudospins that is critical in the system size. In other words, we find a number of pseudospins that is invariant with coupling distance. Ising interactions couple these pseudospins, mixing and energetically displacing states in the pseudospin basis. The number of Ising resonances grows as the third power of the interaction distance R. For conditions under which strength of these interactions exceeds the coupling energy of the pseudospins, they create a web of dynamic coupling that leads to delocalization [59]. The average energy splitting of resonant pairs coupled by spin flip-flop interactions is J(R) = t/R 3 where t = J/ρ. The density of these resonant pairs with any size R is simply, Many-body physics with ultracold plasmas: Quenched randomness and localization 18 Defining an average Ising interaction D = U /ρ 6/3 = U /ρ 2 , we compute the Ising interacting energy density between pseudospins as Generally speaking dipole-dipole coupling is stronger in first order than second order. But, for the system of pseudospin pairs, we see above that the quantity U (R), which gauges the delocalizing effect of Ising interactions, decreases with R more slowly than J(R), the energy that couples dipoles to form the average pseudospin.
So, in a system of finite size, there must exist a range R c , that encompasses a critical number of dipoles, N c , above which the coupling energy of the dipole pairs falls below the strength of the Ising interactions, allowing the system to delocalize [59].
To determine N c we compare U (R), the average Ising interaction energy coupling two extended pseudospins to J(R), the energy on average that couples the dipoles to form pseudospins. The value, R c , at which the two energies match determines a maximum system size for which delocalization overcomes MBL for a given strength of disorder W . Thus, to find R c , we solve U (R c ) = J(R c ) to obtain: and the resulting condition is where we used the definition D ij = t 2 ij J/W 2 to estimate U = J 3 /W 2 in the last step.

4.3.
4. An estimate of N c for the experimental system Let us compare this theoretical estimate with an experimental determination of the number of dipoles present in the quenched state of the ultracold plasma. With precise control of the laser-crossed molecular beam geometry, we can fix the distribution of nitric oxide density in the volume illuminated to form an initial Rydberg gas [69]. Generally speaking, ω 2 saturates the second step of double resonance, so a varied intensity of ω 1 determines the peak density of the Rydberg gas volume up to a maximum of 6 × 10 12 cm −3 (6 µm −3 ). This Rydberg gas density fluctuates, but we have managed to calibrate a measure of the sum of counts in each SFI trace to classify the corresponding initial density of every shot. Coupled rate simulations that describe the kinetics of the avalanche of Rydberg gas to plasma conform well with the observed evolution in electron binding energy as a function of Rydberg gas density estimated in this way [38,70]. The information combined from two means of detection measures the evolution of this density distribution as a function of time. An imaging grid in our short flightpath apparatus, mounted perpendicular to the axis of the molecular beam, translates Table 1. Radial distribution of ion density in a Gaussian ellipsoid shell model for a quenched ultracold plasma of NO as it enters the arrest state with a peak density of 4 × 10 10 cm −3 (0.04 µm −3 ), σ x = 1.0 mm, σ y = 0.55 mm and σ z = 0.70 mm. Here, the quasi-neutral plasma contains a combined total of 1.9 × 10 8 NO + ions and NO Rydberg molecules. The average density of this plasma is 1.4 × 10 10 cm −3 and the mean distance between NO + ions is 3.32 µm. in z to extract an electron signal waveform that traces the width of the plasma in the propagation direction as a function of its time of flight from 1 to 180 µs. We also collect images that detail the two-dimensional projection in the x, y plane, and determine the width in z after nearly 500 µs of flight. We extrapolate the slow ballistic expansion of the plasma charge distribution back to shorter evolution time of 10 µs to estimate the density distribution in a typical plasma ellipsoid, as represented by the shell model presented in Table 1. Here we neglect the initial stages of bifurcation, which acts to redistribute the charge density in an ideal Gaussian ellipsoid. A plasma with the total number of ions represented in Table 1 retains that integrated mass for a time as long as our long flight-path instrument can measure it, at least 500 µs. An average of the ion density over shells determines, |r ij | , the average distance between dipoles. C sp 3 values computed for ∆n = 0 Föster resonant interactions in Li at n ≈ 95 [60] provide a rough estimate of the average high-n dipole-dipole matrix element, t ij . Combining these quantities yields an upper-limiting estimate of J. We can expect the perturbation of individual molecules by the background of charged-particles in the plasma. This will diminish the probability of finding resonant target states and decrease the value of J for symmetric exchange. Thus, we can expect a Many-body physics with ultracold plasmas: Quenched randomness and localization 20 quenched system in which dipole-dipole interactions occur with rarity and randomness, sampling the huge state space defined by the measured distribution of electron binding energies, W .

Shell
As noted above in the caption to Figure 1, the SFI spectrum, which displays a distribution of electron binding energies in the arrested state of the plasma, provides a direct measure of W . Table 2 summarizes this information together with other properties of the quenched plasma derived from experiment, including J, as estimated for Li under our conditions as an upper limit. Table 2. Resonance counting parameters under conditions of arrested relaxation in the quenched ultracold plasma. We take the disorder width W directly from the width of the plasma feature in the SIF spectrum. J is derived as a rough upper-limiting estimate, based on the average dipole-dipole matrix element, t ij computed for ∆n = 0 interactions in alkali metals [60], together with the mean distance between NO + ions in the shell model ellipsoid. These parameters determine N c , a critical number of dipoles required for delocalization. R c describes the length scale for delocalization and τ c denotes the delocalization time, given a sufficient number of dipoles at the average density of the experiment. Note that the ultracold plasma quenched experimentally contains an order of magnitude fewer than N c dipoles. 3 µm GHz µm s 500 75 3.3 2.0 3.6 × 10 9 4000 0.85 In the locator expansion picture, a system with arbitrary disorder delocalizes whenever the number of dipoles exceeds a critical number, N c , determined by the disorder width, W and the average coupling strength, J. The coupling terms in the Hamiltonian defined by Eq (8) scale with R in a three-dimensional system to set a critical number of dipoles defined by a quantity, N c = (W/ J) 4 derived above. Thus, a quenched state described by the elliptical shell model, with a density distribution as in Table 1, a measured W = 500 GHz, and an upper limiting estimated J = 2 GHz, requires N c = 3.6 × 10 9 interacting dipoles to delocalize. This number of dipoles, deemed necessary for resonance delocalization, exceeds the experimental number of excited molecules in the quenched ultracold plasma by more than an order of magnitude.
Returning to Eq (17), we can figure for a system with a peak density of dipoles, ρ = 4 × 10 10 cm −3 that a threshold a critical number of dipoles, N c = 3.6 × 10 9 must attain a critical interaction distance or minimum system size, R c ≈ 4 mm, to delocalize. For such a range, an upper-limiting dipole-dipole matrix element, t ij = 75 GHz(µm) 3 , calls for an irreversible transition time, τ c , of about 1 second [67]. Note that the quenched ultracold plasma volume formed experimentally contains an order of magnitude fewer dipoles than N c , as determined by the model of reference [59].
While the experimental plasma detailed above forms with a well-defined total number of dipoles, we cannot describe its constituent quantum states or their dipole-Many-body physics with ultracold plasmas: Quenched randomness and localization 21 dipole interaction with precision. This limits the accuracy with which we can specify a value of N c . Moreover, it remains to be seen whether resonance-counting arguments can serve to define the limiting conditions for MBL in higher dimensions [71].

Beyond locator-expansion approaches
The analysis above follows from a simple perturbative argument in which a diverging number of resonances calls for a breakdown of the locator expansion, signalling delocalization. However, Nandkishore and Sondhi have pointed out [71] that a breakdown of the locator expansion need not imply an absence of localization. In their work, they studied a long-range Coulomb model for which they argue that non-perturbative short-range interactions can arise in a regime for which locator expansion arguments can be correctly applied, concluding that many-body localization may well extend to long-range interacting systems of higher dimension [72].
We have presented a conservative model based on resonance counting arguments that argues for disorder-driven localization in absence of off-diagonal disorder (in J ij ). The disorder in J ij ought to lead to unconventional resonance structure, that will very likely prevent delocalization, as it becomes very difficult for excitations to propagate amongst a chain of dipoles involving very different quantum states and very different transition amplitudes. Moreover, extensions studying non-perturbative effects in the spirit of Nandkishore and Sondhi, while not easily generalizable to dipolar systems, seem to suggest that if localization survives Coulomb interactions, it should in principle survive shorter-range dipolar interactions.

Rare thermal regions
Our analysis completely ignores the potential effects of so-called Griffiths region. These rare regions are thought to appear spontaneously as a product of delocalizing fluctuations in the localized phase, forming thermal reservoirs that ultimately destabilize MBL systems of higher dimension [73][74][75][76][77][78]. These regions lead to a delocalization scenario in which the thermal region grows, driving the MBL phase to form a glass, that evolves at best with slow dynamics. An important element of the self-assembly of the molecular ultracold plasma may diminish the effect of rare thermal regions as a destabilizing factor. As the quenched ultracold plasma approaches a state of arrested relaxation, SFI spectra show that electron-Rydberg collisions cease to drive energy transport, defying classical models of plasma kinetics. If a localized plasma happens to develop a Griffiths region, thermal relaxation can be expected to produce a local volume of electron gas, heated by inelastic collisions with Rydberg molecules. These collisions would rapidly transport the Rydberg population to a domain of low n, where strong intramolecular Rydberg-valence coupling causes rapid predissociation [79,80]. Such a loss of Rydberg NO to form neutral N( 4 S) and O( 3 P) would drive a dissipation of the delocalized region to form a void of no consequence.

Conclusions
Experiment paints a very clear picture of arrested relaxation in an ultracold neutral plasma of NO + ions and electrons. Formed by electron-impact avalanche in a stateselected Rydberg gas of supersonically cooled nitric oxide, this plasma spontaneously bifurcates, quenching the energy distributions of ions and electrons to form compressed volumes of canonical density and ultracold temperature. Experimental images detail the time evolution of the electron binding energy and three-dimensional density distribution. These metrics directly show evidence for an initial redistribution of Rydberg population driven by collision with energetic electrons. Thereafter, all sign of free electrons disappears, and the relaxation of the system ceases. These dynamics fail to conform with conventional models of plasma coupled-rate processes and NO predissociation.
The present work proceeds from this well-defined base of experimental observations to formulate a conceptual rationale for localization as it might occur in an ensemble of strongly interacting random dipoles. We refer to the selective field ionization (SFI) spectrum of electron binding energies to gauge the ensemble disorder in the energies associated with dipole-dipole resonant couplings, and construct a scheme to describe the resulting web of interactions in terms a projection to some low number, L, of random coupled transitions. The vast array of accessible pathways, and the randomness of the system of states populated in the quenched plasma far exceeds the complexity typically encountered in few-level quantum optics systems. Nevertheless, we take the simplest possible approach of an XY Hamiltonian for interacting dipolar spins with emergent Ising interactions. Applying the idea of a locator expansion, we consider the conditions required for delocalization in an L = 2 limit. We find N c , a critical number of dipoles required to sustain delocalization, that exceeds the number of dipoles in the plasma by more than an order of magnitude. We thus argue that the plasma ought to exhibit localization on a long observational timescale at least.
We represent this quenched ultracold plasma as a class of system in which localization arises naturally, in which locally emergent conservation laws act in concert with a quench to guide the propagation of the system toward a global many-body localized state. This evolution must draw upon a subtle interplay between dissociation and resonant but rare dipole-dipole interactions giving rise to novel many-body phenomena. Tracking the state of the molecular ensemble, as it proceeds from the semi-classical conditions of a spectroscopically defined Rydberg gas, provides a unique opportunity to observe stages in the development of localization. While the construction of a picture of the full many-body Fock space of the random Rydberg/excitonic dipoles will require new theoretical tools, strong evidence of dipole-dipole interactions remains, providing a logical basis for the projection to a simple leading order of resonance and a minimal effective model for localization. Further work is required to wrestle with the theoretical complexity of the quenched plasma. Effects such as Griffiths regions and dissociation may prove to have singular importance in this molecular system. The emergence of discerning experimental tools promise both greater fundamental