Crystalline structures and frustration in a two-component Rydberg gas

We study the static behavior of a gas of atoms held in a one-dimensional lattice where two distinct electronically high-lying Rydberg states are simultaneously excited by laser light. We focus on a situation where interactions of van-der-Waals type take place only among atoms that are in the same Rydberg state. We analytically investigate at first the so-called classical limit of vanishing laser driving strength. We show that the system exhibits a surprisingly complex ground state structure with a sequence of compatible to incompatible transitions. The incompatibility between the species leads to mutual frustration, a feature which pertains also in the quantum regime. We perform an analytical and numerical investigation of these features and present an approximative description of the system in terms of a Rokhsar-Kivelson Hamiltonian which permits the analytical understanding of the frustration effects even beyond the classical limit.

Introduction.-Atoms in highly excited states -socalled Rydberg atoms -interact via power-law potentials, which in conjunction with an external laser drive give rise to intricate many-body phenomena. Recent experiments with Rydberg gases have revealed that the dynamical behavior of these systems is of surprising variety. Examples are the emergence of bistable behavior [1,2], the observation of correlated aggregation of excitations [3] and of coherent excitation transport [4][5][6]. Strong interactions also become manifest in the structure of the ground state of Rydberg ensembles. Several theoretical works have predicted and studied the formation and the melting of crystalline phases [7][8][9][10][11][12][13][14][15]. But only recently it was shown that these crystalline states can be actually accessed experimentally [16].
Currently there is a surge of interest in atomic systems in which multiple Rydberg states are excited. One of the main motivations is the presence of an exchange interaction between Rydberg states of different atoms that results in coherent transport dynamics [4][5][6], the nonlocal propagation of light [17] as well as a non-trivial collapse and revival dynamics [18,19]. Interesting physics emerges also in the absence of exchange interactions. The reason is that depending on the Rydberg states the interaction between two atoms can vary by several orders of magnitude [20]. This mechanism is exploited for example in the all-optical transistors for light pulses [21,22].
In this work we explore the many-body ground states of an atomic lattice gas in which two Rydberg statesor species -are simultaneously excited. We focus on a situation where interactions are present only between atoms of the same species. This is reminiscent of the Potts model [23], for which however only few studies exist that consider interactions that extend beyond nearest neighbors [24][25][26][27]. We show that the ground state of the two-species Rydberg lattice gas features a surprisingly complex structure with a series of commensurate to incommensurate transitions. In the commensurate case the two species can occupy the lattice such that they can both minimize their interaction energy independently. In the incommensurate case this is not possible and it leads to mutual frustration. We provide analytical expressions for the regions of stability of the (in)commensurate phases in the classical regime, i.e. in the limit of vanishing laser driving strength. The quantum regime is studied numerically as well as analytically with the help of an approximate Hamiltonian of Rokhsar-Kivelson form. These considerations show how frustration persists also in the quantum regime. Description of the system.-We consider a onedimensional gas of atoms trapped in a lattice of spacing a with a single atom per site. The relevant internal level structure of the atoms is shown in Fig. 1a. The ground state |g is coupled to two Rydberg nS-states (|s and |w ) via two laser fields with Rabi frequencies Ω α and detuning ∆ α (α = s, w). The fundamental interaction between two such atoms is to leading order given by the dipole-dipole potential. However, for Rydberg atoms in nS-states, as considered here, this interaction results in a (second order) van-der-Waals energy shift [29]. For two atoms in the pair state |αα ≡ |α |α and separated by the distance R it reads V α (R) = C (α) 6 R −6 . The vander-Waals coefficients C (α) 6 scale with with the eleventh power of the principal quantum number n [29][30][31]. This generates strong interactions among Rydberg states and moreover permits to achieve a scenario in which the interaction between two atoms in the pair-state |ss is much larger than the interaction between two atoms in |ww . Henceforth we focus on such a case, which is in practice achieved by choosing the principal quantum number of state |s (the strong species) to be larger than that of state |w (the weak species). Moreover, such large difference in the principle quantum numbers results in a strongly suppressed interspecies interactions [32] as experimentally shown in Ref. [20]. The Hamiltonian of the system can then be written (within the rotating-wave approximation) as a sum of Hamiltonians of the strong and where n (α) = |α α| and σ x(α) = |α g| + |g α|, and V α = C (α) 6 /a 6 . Note, that H w and H s do not commute due to the fact that both Rydberg species are excited from a common ground state. This is the origin of the commensurate to incommensurate transitions as well as the frustration effects which we will discuss in the following.
Commensurate to incommensurate transition.-We start by studying the case Ω α = ∆ α = 0 in the thermodynamic limit, i.e. an infinite lattice, where Eq. (1) reduces to the Hamiltonian of a classical two-species Pottsmodel with (convex) 1/r 6 interactions and inequivalent couplings (V s = V w ). We are interested in finding the microscopic state which minimizes the interaction energy for given filling fractions of excitations ρ s and ρ w . For a single species and a general convex potential this problem was studied in [28,[33][34][35]. Given the filling fraction the convexity of the potential forces the system into the most homogeneous configuration achievable accounting for the lattice constraint, as shown by Hubbard in Ref. [28].
When two species are present it is in general not possible to minimize at the same time the interaction energy of both. An example is given in Fig. 1(b) for ρ s = 1/3 and ρ w = 1/2. Finding the actual ground state is simplified by our assumption V s V w . Here, the configuration of the s-species can be considered as "frozen" and not constrained by that of the w-species [36]. Atoms of the s-species will then arrange following the single species prescription of Ref. [28] which will in general lead to a deformed lattice formed by the remaining empty sites on which the minimum-energy arrangement for atoms of the w-species needs to be found.
For the example shown in Fig. 1(b) the w-atom sitting in the doubly occupied site has to be moved to an empty site in a way that minimizes the increment in interaction energy. As shown in Fig. 1(c) the strong species leaves a distorted lattice with a density of empty sites given by 1 − ρ s = 2/3 and lattice spacings a and 2a. The minimum interaction energy is then obtained by arranging the w-atoms according to the single species prescription of Ref. [28] considering that the filling fraction of the w species on the distorted lattice is ρ w / (1 − ρ s ) = 3/4. This finally leads to the state depicted in Fig. 1(c). The mathematical proof of this method we provide in a separate publication, see Ref. [37]. Some examples of minimum interaction-energy configurations are shown in Fig.  1(d). From these considerations one finds that two cases, which depend on the filling fractions ρ α , need to be distinguished: (i) The commensurate case in which the two species assume their minimum-energy dispositions independently without interfering with each other. (ii) The incommensurate case in which the strong species prevents the atoms of w-species from assuming their minimum energy configuration.
Understanding whether two filling fractions are commensurate is generally a hard task which requires the study of the full arrangement of excitations. In the following analytical analysis we will consider filling fractions of the form ρ α = 1/q α , where both q α are positive integers. Clearly, the minimum energy configuration is achieved when the α-atoms are arranged uniformly with distance q α . The filling fractions are commensurate only if they share a common divisor (see [37]). In the incommensurate case the distribution of the w-atoms will be distorted and can be represented by repeated strings of repeated intervals of lengths ...q w (q w − 1)(q w + 1)q w ..., as shown for the example in Fig. 1(c).
Let us now turn to the discussion of the stability of the (in)commensurate phases. So far we have assumed that the filling fractions ρ α of the individual species are externally imposed. However, in practice the filling fraction is controlled by the parameters ∆ α in Hamiltonian (1) which act as chemical potentials [8,14]. The question is then which state, or filling fraction, is actually stabilized in a given region of the ∆ s − ∆ w manifold. As the distribution of the s-species is effectively independent from that of the w-atoms, we can study its stability with the single species method [14,34,35]. The stability of a given filling fraction ρ w on the other hand is less simple to analyze, as it depends on the configuration of the s-atoms. We can obtain an analytical result for the transition to an incommensurate state in the thermodynamic limit starting from a commensurate arrangement of atoms for which ν = q w /q s is integer. The region of stability is delimited by [37] On the one hand a transition to an incommensurate state takes place when ∆ w = ∆ (+) w , the value for which the introduction of one more excitation is energetically favorable when keeping ∆ s fixed. On the other hand there is a transition at ∆ w = ∆ (−) w at which the w species loses an excitation.
Quantum fluctuations and frustration energy. -We now address the question as to how quantum fluctuations introduced by a laser of finite driving strength affect the commensurate to incommensurate transitions and in particular the emergence of frustrated states. To this end we consider a non-zero value of Ω s . This changes the state of the s-species from a classical one to a superposition of configurations with different excitation number. We are interested in how this impacts the classical arrangements of the w-species, i.e. at Ω w = 0. To quantify this we introduce the frustration energy which measures by how much the strong species prevents the w-species from reaching its minimum energy configuration if it were alone. Here E 1sp (∆ w ) is the energy of the classical configuration of the w-species in the absence of s-atoms.
In the classical limit, Ω α = 0, the frustration energy is zero when the detunings ∆ α are chosen such that they stabilize filling fractions ρ α which are commensurate. E fr becomes in general larger with increasing Ω s as increasing density fluctuations in the s-species force the w-atoms to assume configurations with increased energy. In order to study this behaviour in more detail we diagonalize Hamiltonian (2) for a chain of length L = 8 with periodic boundary conditions. This approach has the typical drawbacks of a small scale numerical study: the presence of finite size effects (though minimized by the periodic boundaries), and the fact that one is limited to filling fractions of the form p L for p ≤ L. We will see, however, that the results provide an intuitive understanding of the physics at work. Note furthermore, that the considered relatively small system size in fact comes close to what is currently realizable experimentally in the context of Rydberg atoms [16,38].
For our numerical study we focus on a regime where the strong species in the classical limit forms a crystal with filling fraction ρ cl s = 1/2. With increasing Ω s this crystal melts as is seen in Fig. 2(a) where we show a density plot of the transverse magnetization . The magnetization displays the typical lobe-structure and the formation of a (longitudinally) paramagnetic state (m s = −1) at large Ω s . Here it is evident that the state of the strong species is formed by a superposition of states with different number of excitations.
Let us now have a look at the frustration energy when the weak species is added. The corresponding data is shown in Fig. 2(b). At Ω s = 0, i.e. the classical limit, E fr is zero for commensurate densities and finite for incommensurate ones. With increasing Ω s , the w-species becomes more and more frustrated which is reflected in an increase of E fr . Interestingly, this increase is step-wise and each step is accompanied by a change of the filling fraction ρ w of w-atoms, whose value is provided in Fig.  2(b) for each plateau of E fr . Furthermore, we show the k |G as function of ξw in the absence of s-atoms. Here ρw saturates to its unfrustrated value 1/2. (c) In the presence of s-atoms the w-atoms can only assume a maximum filling fraction ρs = 1/3 and form a frustrated state with exponentially many configurations. The inset shows two example configurations with filling ρw = 1/3. Here red ellipsoids (blue rectangles) sketch the dimers (trimers) of the RK construction.
behavior of the weak filling fraction in the entire Ω s − ∆ s plane in Fig. 2(c,d). Note, that contrary to ρ s [shown as contours in Fig. 2(a)], ρ w exhibits a lobe like structure which is not symmetric. This can be understood as follows: For a given finite Ω s the filling fraction ρ s decreases with decreasing ∆ s . The resulting smaller number of s-atoms permits the accommodation of a larger number of w-atoms. This effect is stronger the larger Ω s leading to an increasing asymmetry of the lobes in Figs. 2(c,d). Moreover, for any ∆ s , ρ w decreases stepwise with increasing Ω s and eventually vanishes. The decrease in ρ w can be qualitatively explained as follows: As Ω s increases the ground state begins to contain basis states with s-atom numbers that are larger than in the classical case. This forces the w-atoms to assume a lower filling fraction. The fraction of admixed basis states with increased s-atom number becomes larger the larger ∆ s [see contours in Fig. 2(a)] which results in the asymmetry of the lobes in Figs. 2(c,d).
We expect the qualitative features, namely the increase of frustration with Ω s to hold also in the thermodynamic limit. To study this a semi-analytical treatment of quantum fluctuations based on a perturbative expansion of the ground state around the classical limit could in principle be performed. This task is, however, quite involved due to the fact that the filling fraction of the weak species is in fact a function of Ω s . Nevertheless, one can still gain analytical insight into the quantum regime by considering the description of the system in terms of an approximate Hamiltonian, as is shown in the following.
Rokhsar Kivelson approximation.-In the following we conduct an analytical study of the ground state by means of a Rokhsar-Kivelson (RK) Hamiltonian H RK , generalizing the approach used in Refs. [11,39,40] to charac-terize the statics of a Rydberg gas. The central idea is to represent Hamiltonian (1) as a sum of local positive semi-definite projective Hamiltonians h k . In this case a state |G can be found which is annihilated by each of them, i.e. h k |G = 0, and thus represents the ground state of H RK . This approach is an approximation. It works when the system parameters (Ω α , ∆ α , V α are located on the so-called RK manifold (see further below) and under the omission of the long-range tails of the interaction. Due to the latter H RK does not capture correctly the ground state in the classical limit. However, it can be used to study the effect of quantum fluctuations on crystalline states and, as we will see, also shows the emergence of frustrated states for non-zero Ω α .
We consider a situation where in the classical limit the densities of the weak and strong species are given by ρ w = 1/2 and ρ s = 1/3, which can then be thought of as dimers and trimers, respectively [40]. In this case where the parameters ξ α = f α V α /Ω α with f s = 1/(2 × 3 6 ) and f w = 1/2 6 parameterize the RK manifold [see Fig. 3(a)]. Large (small) values of ξ α result in a ground state with high (low) density of the respective species. The explicit form of the RK manifold is where g s = 10 and g w = 6. The local Hamiltonians are h where the projection operators Π k . The ground state |G of H RK can be represented explicitly as matrix product state (see e.g. [40]).
By construction H RK is frustration free, i.e. the frustration energy (2) is always zero. In fact this is an artifact stemming from the omission of long-range tails. Nevertheless, the interspecies frustration still manifests in the behaviour of the filling fractions, Fig. 3. Here we fix the parameter ξ s such that the strong species (trimers) fills up the lattice at density ρ s = 1/3. We now increase ξ w which increases the density of the weak species. We see that ρ w saturates at 1/3 and not at 1/2 as it would in the absence of the strong species. The weak species therefore does not crystallize but is in a superposition of exponentially many states with density 1/3, two of which are shown in the inset Fig. 3. This is a signature of frustration.
Summary and outlook.-We have investigated the statics of a two component Rydberg gas at zero temperature, finding that its ground state is of frustrated nature. In the classical regime we identified regions in parameter space where the two Rydberg species form commensurate and incommensurate arrangements. In the quantum case we introduced the frustration energy to quantify frustration. We found that this quantity shows a staircase pattern as the system visits different phases characterized by different filling fractions. Finally, we performed an approximate analytical study of the quantum regime by means of an RK Hamiltonian, which further corroborated the existence of frustrated states.