Discrete time crystals in the absence of manifest symmetries or disorder in open quantum systems

We establish a link between metastability and a discrete time-crystalline phase in a periodically driven open quantum system. The mechanism we highlight requires neither the system to display any microscopic symmetry nor the presence of disorder, but relies instead on the emergence of a metastable regime. We investigate this in detail in an open quantum spin system, which is a canonical model for the exploration of collective phenomena in strongly interacting dissipative Rydberg gases. Here, a semi-classical approach reveals the emergence of a robust discrete time-crystalline phase in the thermodynamic limit in which metastability, dissipation, and inter-particle interactions play a crucial role. We perform large-scale numerical simulations in order to investigate the dependence on the range of interactions, from all-to-all to short ranged, and the scaling with system size of the lifetime of the time crystal.

In this work we establish a link between metastability in open quantum systems [51,52] and DTCs. This provides a simple and generic mechanism (see Fig. 1) for the emergence of a DTC under periodic driving, which does not hinge upon the presence of either disorder or of any manifest symmetry of the generator of the time evolution. To illustrate this mechanism, The phase space of the system is divided into two basins of attraction (bright and dark areas). They are associated with the two stationary states, labeled by ρ ss 1 (blue dot) and ρ ss 2 (red dot), respectively. The combination of an appropriate transformation (L (R) ) and dissipative dynamics (L (0) ) during a time interval T maps one of the stationary state into the other one. The repetition of this composed transformation makes the state oscillate with a doubled period 2T. (b): Time evolution of a typical observable of the system in the DTC phase. Here, in particular, we show the expectation value of S x (t) as a function of time t (units T −1 ) for a dissipative Rydberg gas (see text for details). The period of the oscillations is twice the one of the driving (shaded areas correspond to the two different basins of attraction). Here, Ω (0) we employ an example taken from the physics of dissipative Rydberg gases [53][54][55][56][57][58][59][60][61][62][63][64]. This system displays a stationarystate phase transition in sufficiently large dimensions [62,65,66]. The concomitant closing of the spectral gap leads to metastability. We discuss in detail a protocol for achieving a DTC and investigate its stability as well as its lifetime.
DTCs from metastability -We consider a general Markovian open quantum systems with N degrees of freedom (e.g., spins) whose dynamics is governed by the quantum master equation (QME) ∂ t ρ = L (0) [ρ] [67], with Here, H (0) is the system Hamiltonian while D describes Markovian dissipation. We denote the eigenvalues of the "superoperator" L (0) by {λ k , k = 1, 2, ...} and order them by decreasing real part, i.e. Re(λ k ) ≥ Re(λ k+1 ). The (complete) positivity and trace-preserving properties of L (0) guarantee that λ 1 = 0. Its associated right eigenmatrix ρ ss represents the stationary state of the dynamics, i.e. L (0) [ρ ss ] = 0 [67], which we assume to be unique at any finite size N < ∞. In the following, we focus on systems with vanishing gap for N → ∞, displaying metastable behavior [51,52]. Specifically, we require that, for some choice of the parameters of L (0) , λ 2 be real and λ 2 → 0 while lim inf N →∞ |Re(λ 3 )| > 0. This leads to a separation of timescales for large N. Defining in fact τ m = 1/|Re(λ m )|, one can distinguish three different regimes: For t τ 3 there is a transient dynamics strongly depending on the initial state. For t τ 2 the system instead approaches stationarity and its state converges to ρ ss . Under our assumptions, one can find a third time-frame, τ 3 t τ 2 , which defines a so-called metastable regime: Here, the dynamics can be effectively reduced to a space spanned by the eigenspaces of λ 1 and λ 2 . Denoting by R 2 (L 2 ) the right (left) eigenmatrix of L (0) corresponding to λ 2 this means that [51,52] with The dynamics in the r.h.s. takes place in this reduced space and it can be described in terms of classical jumps between the two extreme metastable states (eMSs) [51,52] with c max 2 (c min 2 ) the maximum (minimum) eigenvalue of L 2 . In the thermodynamic limit (N → ∞) the gap closes (λ 2 → 0), determining a phase transition between two phases characterized by the properties of the two eMSs. At the transition point, the system becomes bistable and the two phases coexist on equal terms. Individual quantum trajectories will asymptotically approach either one or the other eMS, identifying the corresponding basins of attraction. Importantly, on timescales t τ 2 (and N large enough), the system tends to behave as if it were in a bistable regime [51,52]. The eMSs can therefore be approximately regarded as two effective stationary states.
As sketched in Fig. 1(a), this phenomenology can be exploited to engineer a DTC. The key step consists of identifying a second dynamics, generated e.g. by a Lindbladian L (R) , which mapsρ 1 to the basin of attraction ofρ 2 and vice versa in a given time t R . Fixing a period T > t R such that T − t R τ 3 , one can define the dynamics via the prescription with m ∈ N. For simplicity, we assume that the system starts from one of the two eMSs (say,ρ 1 ). For t ≥ 0 the dynamics is clearly T-periodic, but the state of the system will instead evolve with doubled period 2T, which is the hallmark of a DTC. The underlying mechanism can be understood in a pictorial way from Fig. 1(a): By assumption, applying L (R) toρ 1 for t R will bring the system into the basin of attraction ofρ 2 . The subsequent action of L (0) for a time τ 3 will bring the system to its metastable regime and therefore close toρ 2 after the first driving period T. The second application of L (R) will then displace the state into the basin of attraction ofρ 1 and the second instance of L (0) will bring it back (close to)ρ 1 , closing the cycle at time 2T. The repetition of these four steps will then reproduce the same physics, leading indeed to a 2T-periodic dynamics and to the emergence of DTC order. Dissipative Rydberg model -To test the general mechanism outlined above, we study here a model taken from the physics of dissipative Rydberg gases [54,55,68]. This model displays bistable behavior at the mean-field level [57,60,69], associated with the presence of an underlying first-order phase transition [62,66]. In finite dimensions, bistable-like behavior as observed in experiments [58] can be interpreted in terms of the coexistence region of the two phases across the transition, with the boundaries being the corresponding spinodal lines [61]. Interestingly, despite the absence of a manifest Z 2 symmetry, an emergent symmetry is present in the stationary limit of the dynamics and the first-order line terminates at an Isinglike critical point [61,70]. We remark that the exploitation of an emergent, rather than manifest, symmetry differs from earlier works which assumed the latter as a necessary requirement for the appearance of a DTC [15, 17-19, 23, 29, 30, 46].
For the connection between the model we use here and the physics of Rydberg atoms we refer the reader, e.g., to Refs. [54,57,61,69,71]. Here we limit ourselves to introduce it as a 1 2 -spin system governed by a QME where k and p are site indices, σ x/y/z k denote the Pauli matrices acting on the k−th spin and n k = (I k + σ z k )/2 is the k−th site number operator. The last term of Eq. (4) is the only one connecting spins at different sites and describes a two-body time-independent interaction with coupling V k p . The remaining parameters Ω ≡ Ω x , Ω y , ∆ are T-periodic according to (3): with Ω (0) = Ω (0) x , 0, ∆ (0) and Ω (R) = Ω (R) x , Ω (R) y , ∆ (R) two sets of constants. Dissipation is produced here by a time-independent term with σ ± k = (σ x k − iσ y k )/2, describing independent spin decay at rate Γ.
We now briefly recall the features of the mean-field stationary phase diagram under the Rydberg dynamics L (0) (see e.g. [57,61,69]). This allows to build a basic intuition on the phase structure and the emergence of a metastable regime. The uniform mean-field equations of motion -reproduced further below -are defined in terms of the expectation values S µ = σ µ k , µ = x, y, z, and n = n k where the site index k is dropped assuming translational invariance. In an extended region of parameter space, a slice of which (at ∆ (0) fixed) is enclosed by the gray contour in Fig. 2(a), these equations feature two stable asymptotic solutions M ss 1 = (S x 1 , S y 1 , S z 1 ) and M ss 2 = (S x 2 , S y 2 , S z 2 ). Outside this region, the stationary values are unique. In any given slice of parameter space, the bistable mean-field region (if present) is delimited by two spinodal lines coalescing with zero net angle into a critical point. Defining the mean-field interaction coupling V = 2N −1 k p V k p , the critical point can be identified (at given ∆ and Γ) by the relations A first-order line, passing through the critical point, is present within the bistable region where M ss 1 and M ss 2 can be related via an emergent Z 2 symmetry [61]. Within the mean-field approximation, the equations of motion corresponding to a generic set of constants Ω = Ω x , Ω y , ∆ , can be straightforwardly generalized to the periodically-driven case simply by updating the parameters in time according to the rules defined in (5). Implementation of the DTC protocol -The remaining step is to define the rotational dynamics L (R) consistently with our requirements, i.e. such that it connects the two basins of attractions of M ss 1 and M ss 2 . Instead of attempting to map out the latter two, which looks unfeasible, we take a more physical approach: Since the stationary solutions are defined in terms of two vectors M ss 1 and M ss 2 , we look for a global rotation U exchanging their respective directions and look for a regime where this is sufficient to map a stationary state into the basin of attraction of the other. By defining the versors m ss i = M ss i / M ss i , U can be described as a rotation by π around their bisecant. In the spin representation, is defined such that D = m ss 1 + m ss 2 and d = D/|D|. We then choose Ω (R) in such a way that the non-interacting part of the Hamiltonian H (R) would perform precisely the rotation U in a time t R , namely Clearly, this does not guarantee that each stationary state is mapped in the other's basin of attraction; however, the effectiveness of this choice can be verified a posteriori and it works for a wide range of parameter values. Notice that t R can be freely tuned to be small so that interactions and dissipation have negligible effects. We remark that demanding each stationary state to be mapped by exp[L (R) t R ] into the basin of attraction of the other is a much looser requirement than demanding the exact mapping between the two stationary state solutions, i.e. M ss 1 → M ss 2 and vice versa. Hence, imperfections in the rotation procedure will not be relevant as long as its end points [squares in Fig. 1(a)] are in the correct basin of attraction. Indeed, the subsequent evolution, for times t τ 3 , guarantees that the state is driven again close to the desired stationary point. This clearly adds to the robustness (or rigidity) of the DTC phase in the proposed mechanism.
In Fig. 1(b) we show the typical 2T-periodic evolution of an observable in the DTC phase of the mean-field equations (7). The stationary phase diagram in the Ω (0) x −V plane, obtained via numerical solution of the same equations, is instead displayed in Fig. 2(a). The colored area corresponds to the bistable region of the mean-field model, where a DTC can be constructed via our procedure. As an order parameter we consider a normalized Fourier component dτ S x (τ) e − 2π j K T τ with the waiting time t w long enough to avoid the transient part of the dynamics, K an integer 1 and j ∈ Z. With our specific choice of L (R) , DTC order is indeed displayed over a finite region of the parameter space. We also studied the robustness of the DTC phase against fluctuations of the parameters of the rotation, for instance Ω (R) x,y (ε) = Ω (R) x,y +εΩ (R) x,y with |ε| < 1. The DTC remains stable over a reasonably wide range of ∼ 0.1.
Finite-size systems -We now turn to the case of finite-size systems to explore how the DTC phase emerges as the number of spins N is increased. First, we focus our attention on a fullyconnected model with V k p = V 0 /N ∀k, p, which is expected to match the mean-field predictions in the thermodynamic limit. With this choice of the interactions, the model becomes permutationally symmetric [72,73] and one can study its dynamics in the totally-permutationally-symmetric subspace [74]. In Fig. 2(b) we display the inverse gap τ 2 = 1/|Re(λ 2 )| in the same range of parameters used in panel (a) for a system of N = 28 spins. The dark zone shows a closing of the gap of L (0) which nicely fits with the mean-field bistable region. Within the same region, τ 2 increases with N, whereas outside it seems to converge to a size-independent value. In the same range of parameters, |Re(λ 3 )| does not strongly depend on N, leading to the emergence of a metastable regime for large enough N. In Fig. 2(c) we show the stroboscopic dynamics of S x (t) (collecting data points only every period T) generated by the fully-connected model, where we set the parameters of L (R) to the mean-field ones. Here, a DTC phase emerges only at short times and eventually dies out exponentially fast. The typical lifetime of the oscillations τ DTC , however, increases with the system size, consistently with the expectation that the model should reproduce the mean-field results in the thermodynamic limit.
Finally, we briefly inspect what happens in a system with short-range interactions, focusing on the nearest-neighbor case (V k p = V 0 δ p,k±1 ). The latter can be efficiently investigated by employing a time-evolving block-decimation algorithm [75][76][77]. As in the previous case, the stroboscopic dynamics of a typical observable displays an oscillatory behavior with period 2T and amplitude decaying exponentially in time (not shown). However, in this case, the lifetime of oscillations does not diverge as N → ∞ but saturates instead to a finite value (see  Fig. 3). This fact is rooted in the underlying phase transition of the model, which features a lower critical dimension of 2 [60,61], so that the gap is not expected to close in one dimension. This is compatible with the behavior shown in the inset of Fig. 3. Nonetheless, in Fig. 3 we see how the lifetime of the DTC is connected to the gap. The smaller the latter becomes, the longer the DTC structure survives. From this one may conjecture that metastable open quantum systems with a closing gap develop, under appropriate periodic driving, DTC phases also in low dimensions.
Conclusions -We have discussed a general mechanism for engineering a DTC in driven open quantum systems featuring a metastable regime. The latter does not require disorder or the presence of any explicit symmetry in the system, although the phase transition associated to the closing of the gap may display an emergent one. We have shown the emergence of a DTC order in a specific case taken from the physics of dissipative Rydberg gases. This, in turn, means that Rydberg systems may represent in the future an interesting platform for the study of DTC phases.