Many-body dynamics of holes in a driven, dissipative spin chain of Rydberg superatoms

Strong dipole-dipole interactions between atoms in high-lying Rydberg states can suppress multiple Rydberg excitations within a micron-sized trapping volume and yield sizable Rydberg level shifts at larger distances. Ensembles of atoms in optical microtraps then form Rydberg superatoms with collectively enhanced transition rates to the singly excited state. These superatoms can represent mesoscopic, strongly-interacting spins. We study a regular array of such effective spins driven by a laser field tuned to compensate the interaction-induced level shifts between neighboring superatoms. During the initial transient, a few excited superatoms seed a cascade of resonantly facilitated excitation of large clusters of superatoms. Due to spontaneous decay, the system then relaxes to the steady state having nearly universal Rydberg excitation density $\rho_{\mathrm{R}} = 2/3$. This state is characterized by highly-nontrivial equilibrium dynamics of quasi-particles -- excitation holes in the lattice of Rydberg excited superatoms. We derive an effective many-body model that accounts for hole mobility as well as continuous creation and annihilation of holes upon collisions with each other. We find that holes exhibit a nearly incompressible liquid phase with highly sub-Poissonian number statistics and finite-range density-density correlations.


Introduction
Strongly interacting many-body systems subject to external driving and coupled to (possibly tailored) reservoirs offer a new route to create and stabilize interesting states of matter. As a simple example, a quantum state can be made immune to particle losses if it is the stationary state of an open system coupled to a particle reservoir. Furthermore, the competition between coherent driving and dissipation can lead to exotic steady states [1,2] and phase transitions in open many-body systems [3][4][5][6][7][8].
Rydberg atoms [9] are well suited to study the interplay between strong interaction and coupling to coherent laser fields and dissipative environments. They are thus prime candidates to investigate many-body physics of driven, dissipative spin models. A prominent and well studied consequence of the strong, long-range interaction between atoms in Rydberg states is the so-called blockade phenomenon, whereby a Rydberg excited atom suppresses further excitations within a certain blockade distance [10]. Rydberg blockade in a dilute gas or in a regular array of single atoms leads to short-range spatial ordering of excitations, as was studied theoretically [11][12][13][14] and demonstrated experimentally [15,16]. In the so-called anti-blockade regime, successive excitation of atoms at a certain distance from each other can be resonantly enhanced [17][18][19][20][21][22][23], which led to the lively debate on the possibility of attaining bistable steady states [22][23][24][25][26].
When many atoms are confined within the blockade distance from each other, they form an effective twolevel system-Rydberg superatom-that can accommodate at most one Rydberg excitation [10,13,27]. The coupling of a superatom to the laser radiation is collectively enhanced, while the steady-state probability of a single Rydberg excitation can approach unity. This permits the level of control of single collective spins represented by superatoms far exceeding that for individual atoms. Moreover, being composed of many atoms, superatoms are relatively insensitive to atom number fluctuations and losses. Regular arrays of spins represented by superatoms can then be prepared with less experimental effort, which should be contrasted with the sophisticated dynamical preparation techniques used to realize defect-free arrays of individual Rydberg atoms [28,29]. This, together with the strong, long-range interactions between the Rydberg excitations, makes superatoms ideal building blocks for realizing dissipative many-body spin models and analyzing their dynamics.
Single Rydberg superatoms have been observed in several experiments [16,[30][31][32][33][34]. A two-dimensional square lattice of superatoms with nearest-neighbor excitation blockade was studied in [35], demonstrating the possibility of phase transition to an anti-ferromagnetic steady state with spontaneously broken lattice symmetry. In the complementary interaction regime of the Rydberg anti-blockade, little is known about the many-body dynamics and the steady state of a lattice of superatoms. Here we study a one-dimensional lattice of Rydberg superatoms (see figure 1(a)), in which an already excited superatom facilitates resonant excitation of its neighbor, but the presence of two excited neighbors suppresses the excitation. This systems exhibits interesting excitation dynamics and a highly non-trivial steady state characterized by an almost universal density 2 3 R r = of Rydberg excitations with strongly suppressed number fluctuations. We show that this behavior can be explained in terms of mobile excitation holes on the background of Rydberg excited lattice. The holes behave as a nearly incompressible liquid of hard rods with characteristic two-particle correlations (see figure 1(b)). We derive and verify an effective many-body model for holes. Varying the parameters of the effective model, we find a crossover between a liquid of holes with density-density correlations peaked at the distance of two lattice periods, a 2 , and the onset of crystalline order with period a 3 . In both cases the density of holes is 1 3 h r = with highly suppressed number fluctuations.
The paper is organized as follows. In section 2, we formulate the model for a regular array of superatoms and derive the formalism for the efficient treatment of the system. In section 3 we present the results of numerical simulations of the dynamics of the chain of superatoms (driven spin chain) and introduce the effective hole model that leads to an intuitive physical picture for the equilibrium phase of the system. Section 4 summarizes our results. Technical derivations are deferred to the appendices A-C. , with a B being the Rydberg blockade distance. Atoms in the ground state (open dots) are excited to the Rydberg state (red filled dots) by a uniform laser field with Rabi frequency Ω and detuning Δ. We tune Δ to compensate the interaction-induced level shift of Rydberg states of neighboring superatoms leading to resonantly facilitated excitation at distance r a fac = . (b) In the steady state of a continuously driven lattice of superatoms, having nearly universal density 2 3 R r = of Rydberg excitations (red filled circles), the typical two-particle correlation function g d 2 ( ) ( ) for the excitation holes (blue filled circles) corresponds to a liquid of hard rods of length a 2 .

Chain of Rydberg superatoms
In this section, and in appendices A and B, starting from the fully quantum many-body master equation, we derive rate equations for the chain of laser-driven and mutually interacting Rydberg superatoms. These rate equations will then be used in section 3 for numerical simulations of the dynamics and steady state of the manybody system.

The microscopic model
We consider an ensemble of cold atoms trapped in a regular array of microtraps [32,33,[36][37][38] or a longwavelength optical lattice with the period a of several microns. Each lattice site j contains on average N atoms, see figure 1(a). A laser field of carrier frequency ω drives the atoms on the transition from the ground state gñ | to the excited Rydberg state eñ | with the Rabi frequency Ω and detuning eg w w D = -. In the frame rotating with frequency ω, the coherent excitation dynamics of the atoms is described by the Hamiltonian | is the transition (m n ¹ ) or projection (m n = ) operator for the kth atom, and | is the interaction potential between pairs of atoms at positions r r , k k¢   excited to the Rydberg state eñ | . The usual van der Waals interaction corresponds to p=6, while the static dipole-dipole interaction to p=3 [9].
Atoms excited to the Rydberg state eñ | spontaneously decay to the ground state with the rate s G , and are dephased with a typically much larger rate d G due to atomic collisions and motion in the inhomogeneous trapping potential, collisions of the Rydberg electron with the ground state atoms [39][40][41], or intermediate state decay if g e ñ  ñ | | is a two-photon transition [15,42]. The dissipative dynamics is described by the master equation for the density matrix r of the system,

Rate equations for Rydberg superatoms
In equation (1) we can split the sum over all the atoms into two parts: the sum over the lattice sites j, and the sum over the atoms k j in each lattice site. We assume that all N atoms within each lattice site are confined within a small spatial interval r a d  , such that the interatomic interaction energy C r p p d ( ) exceeds all the relevant energy scales pertaining to the atoms, namely, the laser Rabi frequency Ω and detuning Δ, as well as the atomic spontaneous decay s G and dephasing d G rates and the resulting Rydberg state excitation linewidth [13]. This allows us to neglect all the multi-atom states containing more than one Rydberg excitation per lattice site [10,13]. If on the spatial scale r 1 m  d m of such a Rydberg superatom the laser field can be assumed uniform, it would couple the collective ground state G gg g N  is its center of mass coordinate.
Within each Rydberg superatom, the dephasing couples incoherently the symmetric single excitation state In turn, the reverse coupling of the non-symmetric states to the symmetric state has the rate N d G . All single excitation states decay back to the ground state Gñ | with rate s G , see figure 2(a). In appendix A we derive a simple rate equation model describing the excitation dynamics of a superatom in the limit of strong dephasing d  G W.
Starting from the density matrix equations for a single Rydberg superatom, we adiabatically eliminate all coherences and the population of the symmetric excited state, which scales as N 1 . The superatom then reduces to an effective two-level system, see figure 2(b), and its dynamics is governed by the rate equations for the populations of the ground GG j r and excited EE j r states, where the excitation and de-excitation rates are given by is the effective detuning of superatom j which includes the Rydberg level shift due to the interaction with all the other superatoms in the Rydberg state. In figure 2(c), and in more detail in appendix A, we compare the dynamics of a single superatom as obtained from the exact solution of the complete set of the density matrix equations and the solution of the rate equations. We observe that the rate equation model approximates well the relaxation timescale of a superatom and the steady-state population of the excited state, EE ex ex de r = G G + G . Remarkably, unlike for a single two-level atom, the excitation probability of the superatom under continuous (near-)resonant driving and in the presence of strong dephasing  g W can approach unity, 1

Facilitated excitation of superatoms
The laser irradiates continuously and uniformly the 1D chain of superatoms. We set the detuning Δ of the laser to be equal to the interaction strength V a C a V between neighboring superatoms, V N D = . We neglect the interaction between the next to nearest neighbors. The conditions for validity of our treatment of the chain of superatoms and the parameters for suitable experimental systems are discussed in appendix B. In this so-called facilitation regime [18], an already excited superatom shifts the frequency of its nearest neighbor into resonance with the laser. The excitation G  and de-excitation G  rates for the facilitated superatom, having one and only one, excited neighbor, are On the other hand, a superatom surrounded by non-excited neighbors is non-resonant with the laser and has a much lower excitation rate But once excited, it will play the role of a seed for a rapid growth of a cluster of excited superatoms. Naively, such clusters will grow until they collide and nearly all the superatoms in a lattice will be excited. However, spontaneous decay of superatoms with rate s G will produce excitation holes-ground state superatoms surrounded by excited superatoms. A hole cannot be resonantly excited as its Rydberg state is shifted by the interaction with two excited neighbors by V 2 N , leading to the effective detuning V 2 eff N D = D -= -D. Hence, the rate to refill the hole turns out to be the same highly suppressed seed rate seed G .
In the facilitation regime that we consider, the typical hierarchy of the relevant rates is ,

The many-body dynamics
Upon turning on the excitation laser, the chain of superatoms under the facilitation conditions described above will exhibit transient dynamics on the timescale of t s 1 Gand then settle in a steady state. The excitation transients will be described later in this section. First we discuss the steady state characterized by the average Rydberg excitation density of 2 3 R r  and highly non-trivial equilibrium dynamics of the excitation holes. is spatially uniform. We observe that, to a good approximation, holes behave as hard rods of length a 2 , with the average density of rods being 1/3.

Effective model for holes
To understand the results of the numerical simulations for the hole density and correlation function, we have derived an effective model for the equilibrium dynamics of the holes. The derivation, details of which are given in appendix C, is based on adiabatic elimination of short-lived configurations involving two or more neighboring superatoms in the ground state. Such configurations appear when an excited superatom next to a hole decays to the ground state with rate s G  ( ) , but the lifetime of these configurations is very short, t 1 G  -, due to the excitation facilitation with the fast rate s G G    ( ) . Hence, in the effective model, holes cannot be located on the neighboring sites, which gives the physically intuitive picture as to why they behave as hard rods of length a 2 . There are three fundamental processes affecting the many-body dynamics of the holes on the lattice, as illustrated in the inset of figure 3. We describe these processes in terms of the Lindblad jump operators acting in the corresponding subspace for the holes.
where j s  ( ) is the hole creation/annihilation operator for site j of the lattice, and n h j j j s s º + -ˆ( ) ( ) ( ) is the number operator. The last two terms on the right-hand side ensure that a hole cannot be created next to an existing one.
(ii) When two holes are separated by one excited superatom, one of the holes can be annihilated. This process is initiated by spontaneous decay of the middle superatom which triggers fast excitation of two superatoms from one or both sides of the three-site region. The hole annihilation is described by with the total annihilation probability given by a s G = G . The remaining hole can then occupy either the middle site with half of the total probability, or one of the side sites, each with quarter of the total probability.
(iii) Holes can hop from site to site. This process is mediated by de-excitation of the superatom next to a hole with rate G  followed by facilitated excitation of one of the ground state superatoms. The hole transport is described by where 2 t G = G  is the transport rate and the last term ensured that the hole cannot hop to a site next to an existing hole. In figure 4 we compare the spatial correlation function g d 2 ( ) ( ) for ground state superatoms obtained from the numerical simulations of the full superatom model and the effective hole model. We observe very good agreement between the full and effective models, including g 1 0

Liquid-crystal crossover for lattice holes
Although in the present setup the hole hopping rate cannot be made arbitrary small, t ca ,  G G , it is instructive to analyze how varying t G would affect the many-body steady state.
Consider first the hypothetical case of no hole transport, 0 t G  . We can also neglect for now the probability of refilling the hole, due to smallness of the corresponding rate seed G . Then the only stable configuration corresponds to holes on every third site of the lattice, since neither the hole creation nor annihilation processes of equations (9) and (10)  with short correlation length a 1  x , see figure 4. Since holes cannot come closer than two lattice sites, they start to behave as mobile hard rods of length a 2 . We finally note that when the interaction strength V N , and thereby the laser detuning V N D = , are not sufficiently larger than the excitation linewidth, the seed rate seed G is not negligible and holes can refill, leading to 1 3 h r < , as can be seen in figure 3.

Transient excitation dynamics of the system
As promised above, we now discuss the dynamics of the system of superatoms initially in the ground state upon switching on the excitation laser. In figure 6  , since holes can be refilled with appreciable seed rate seed G .  figure 6, we put initially one seed excitation in order to start the facilitated excitation dynamics at a well-defined time, rather than at some uncertain (within 1 seed G ) time when the seed excitation appears with small rate seed G . As seen in that figure, after switching on the laser, the density of Rydberg excitations can peak at a large value and then relax to the lower steady-state density. At first sight, this observation is surprising, since such a behavior is reminiscent to partially coherent dynamics of a quantum system, such as, e.g., damped Rabi oscillations, while here our system is completely governed by rate equations and no coherences are involved. We now outline a macroscopic model that will explain the nature of the peak and the associated peak time t 0 . We consider three basic states of the system: the ground state (g), the fully excited state (e) and the final steady state (s). The corresponding probabilities are denoted by p g , p e , and p s . Initially the system is in the ground state, p 1 g = , and to start the dynamics we need a seed excitation. The probability of the seed p seed is governed by the equation p t p t seed g seed ¶ = G ( ) . For short times, we can assume p 1 g  and obtain linear growth of the seed probability, p t p t 0 seed seed seed . For longer times, this approximation breaks down, but once p g is depleted, the role of the seed becomes unimportant, as will become clear shortly.
The equations of motion for the probabilities of the three basic states are Each seed excitation in the lattice triggers fast growth of facilitated excitations in both lattice directions. The ground state is then being depleted with the rate 2G  and its population is transferred to the fully excited state. In turn, the Rydberg excitations in the fully excited state decay with rate s G . Since the steady state corresponds to configurations with, on average, every third site non-excited, the decay rate of p e to p s is 3 s G , which we have also verified via numerical simulations of the full system with all the superatoms initially excited. We also include in the above equations the process of refilling the holes with the corresponding rate seed G . Then the solution of these equations approximates remarkably well the mean density of Rydberg excitations t p t p t R e 2 3 s r = + ( ) ( ) ( ), even in the regime of sizable seed G , as can be seen in the main panel of figure 6.
The peak in the excitation density R r is reached when the ground state probability p g is depleted and the majority of superatoms are transferred to the excited state, p 1 e  . Integration of p t g ¶ suggests the peak time The scaling of the peak time with the interaction strength V w N is verified in the inset of figure 6. Note finally that our macroscopic model neglects cluster collisions, also producing holes, and thereby slightly overestimates the excitation density R r obtained from the simulations of the complete microscopic model.

Conclusions and outlook
To summarize, we have studied the excitation dynamics and steady state of a lattice of Rydberg superatoms driven by a laser in the facilitation regime. We have shown that the steady state of the system has nearly universal Rydberg excitation density of 2 3 R r = . More interestingly, it corresponds to an equilibrium dynamics of mobile quasi-particles-excitation holes. We have derived an effective hole model which involves hole creation and pair annihilation of holes separated by two lattice sites. We have found that the number fluctuations of the holes are characterized by the Mandel Q parameter Q 0.81 - , and their spatial correlations decay on a distance of a 1  x comparable to the lattice constant a. That negative values of Q, which signify pronounced sub-Poissonian number distribution, have their origin in the hard rod constraint, has been pointed out already in [11] considering a Gibbs ensemble of Rydberg excitations. In our system, however, the pair annihilation of holes leads to a state with much stronger suppressed fluctuations. In the Gibbs state of [11], we would have to choose a rod length of about d = 0.611 to obtain the average hole density 1 3 h r = , but this would then lead to Q 0.63 - which is significantly larger than what we obtain from our simulations.
Our model corresponds to an experimentally amenable regime with a large hole transport rate t ca , periodicity. Such a regime can in principle be achieved, but at the expense of more complicated experimental setup involving additional lasers coupling the ground state of superatoms to a different long lived Rydberg or metastable state which would make the holes immobile.
Another interesting direction of research is to consider different lattice geometries, e.g., a two-dimensional square or triangular lattice. The latter might simulate dissipative frustrated spin models. Finally, the rate equation approach, amenable to large scale numerical simulations, is applicable in the regime of strong dephasing. Coherence effects might lead to interesting dynamics and yield long-range correlations and entanglement. However, fully quantum many-body simulations are limited to small system sizes. r r r r =  ( ) which obeys the equation of motion where the excitation and de-excitation rates are given by This is illustrated in figure 2(b).
In figure A1(a) we compare the excitation dynamics of a resonantly driven superatom as obtained from the solution of the rate equations and the exact solution of the master equation for the density operator using the Monte Carlo wavefunction approach in the truncated single excitation Hilbert space. We observe that the rate equations provide accurate description of the dynamics of the system when N  g W, while they cannot account for the (damped) Rabi oscillations between the ground and the excited states when N g W  .
Nevertheless, the rate equations model approximates well the relaxation timescale and the steady-state population of the superatom.
In figure A1(b) we show the spectrum of eigenvalues k l of the matrix Λ in equation (A2). The eigenvalue 0 0 l = corresponds to the steady state of the system, while the (negative) real parts of the other eigenvalues, Re k l [ ]for k 1, 2, 3, 4 = , characterize the relaxation rates of the superatom towards the steady state. In that figure, we also show the total relaxation rate of a superatom towards the steady state as given by the rate equations model, tot ex de G = G + G , which compares favorably with the exact relaxation rate for a broad range of parameters.

Appendix B. Excitation facilitation conditions and experimental considerations
With the interatomic potential V r C r p p = ( ) (assuming repulsive interaction C 0 p > ) and the single-atom excitation linewidth w, the Rydberg blockade distance is defined as a C w p p B 1 º ( ) . Our model assumes that each superatom can accommodate at most one Rydberg excitation. We therefore require the spatial extent of a superatom to be small compared to blockade distance, r a B d  .
Clearly, the superatom excitation probability is maximal at resonant driving and is suppressed when the laser is detuned by more than its excitation linewidth w w N 2 SA  . We consider a one-dimensional chain of superatoms driven by spatially uniform laser with detuning Δ. We set the laser detuning to be equal to the interaction strength between neighboring superatoms separated by the lattice constant a, V a D = ( ). Then an already excited superatom will facilitate the excitation of the nearest neighbors by shifting the Rydberg energy level into resonance with the laser field. On the other hand, we require that w SA D > in order to suppress excitation of superatoms that have either two non-excited neighbors or two excited neighbors. Since w w SA > , the facilitation distance r C p p fac 1 = D ( ) , and thereby the lattice constant a r fac = , are smaller than the blockade distance a B .
The interaction potential V(r) is a convex monotonic function. Our approach is valid when the interactioninduced Rydberg level shifts of all the atoms of the facilitated superatom are within the superatom linewidth w SA . Linearizing the interaction potential around the facilitation distance r fac , we obtain the following condition on the spatial extent r d of the superatom . This can be rewritten as a condition on the power-law scaling p of the interaction potential,