Shattered Time: Can a Dissipative Time Crystal Survive Many-Body Correlations?

We investigate the emergence of a time crystal in a driven-dissipative many-body spin array. In this system the interplay between incoherent spin pumping and collective emission stabilizes a synchronized non-equilibrium steady state which in the thermodynamic limit features a self-generated time-periodic pattern imposed by collective elastic interactions. In contrast to prior realizations where the time symmetry is already broken by an external drive, here it is only spontaneously broken by the elastic exchange interactions and manifest in the two-time correlation spectrum. Employing a combination of exact numerical calculations and a second-order cumulant expansion, we investigate the impact of many-body correlations on the time crystal formation and establish a connection between the regime where it is stable and a slow growth rate of the mutual information, signalling that the time crystal studied here is an emergent semi-classical out-of-equilibrium state of matter. We also confirm the rigidity of the time crystal to single-particle dephasing. Finally, we discuss an experimental implementation using long-lived dipoles in an optical cavity.

Introduction -Experimental progress in the control and preparation of quantum cold gases [1] has opened a new era in which non-equilibrium phenomena have a central role. In particular, time crystals (TCs) [2][3][4][5][6][7][8][9][10][11][12][13][14][15] -phases of quantum matter which spontaneously break time translational invariance and which can only exist in out-of-equilibrium systems [16] -have recently attracted significant attention. A system hosting a crystalline time phase should be many-body and exhibit an order parameter, φ( r, t), whose unequal time correlation function approaches, in the thermodynamic limit, a non-trivial periodic, oscillating function of time [16]: φ( r, t)φ( r , 0) → f (t), at sufficiently large distances | r − r |. Such behavior must be robust to imperfections of the system parameters or external perturbations.
In this work, we propose a paradigm shift in the concept of time crystals by considering an incoherently driven array of long lived dipoles in a cavity which are subject to collective dissipative decay (superradiance) and elastic long range interactions (see Fig. 1). Our scheme is similar in spirit to recently proposed quantum TCs with dissipation [14,15,28,29] and connected to the emergence of a time-periodic steady state in the thermodynamic limit of an open quantum system. However, the key difference is that by applying incoherent pumping in- 1. An ensemble of N spin 1/2 particles pumped at rate W , experiencing collective emission at rate ∝ f Γ and collective spin-exchange interactions (orange), ∝ gΓ, form the basis of the superradiant TC. The elastic interactions imprint collective spin oscillations at frequency ω spontaneously breaking the time translation symmetry (manifested as persistent oscillations in the unequal time spin-spin correlation function in the thermodynamic limit).
stead of a coherent, periodic drive we do not impose an external frequency which explicitly breaks the time translational symmetry ab initio; rather, the latter is only spontaneously broken by the subtle interplay between collective interactions and driving processes. Moreover, the incoherent drive allows for the population of a much larger Hilbert space compared to the fully symmetric Dicke manifold.
Within our framework, collective emission prevents unwanted heating and fulfills the role of disorder-induced arXiv:1805.03343v2 [quant-ph] 15 May 2018 localization in the Floquet TC (see Fig. 1). The balance of pumping and dissipation leads to the stabilization of a non-equilibrium synchronized steady-state [30][31][32] and allows for the formation of a TC that is robust to imperfections or environmental disturbances in the presence of finite but moderate elastic interactions. While the TC exists only when elastic interactions are present, we also find that if they are too strong they can destroy the periodic order. The TC thus only exists within a finite window of interaction strengths with a width which we show grows as the square-root of the particle number. The melting of the TC can be understood from the population of low-lying eigenvectors of the Liouvillian operator that have a finite negative real component: we find that these eigenvalues can be linked to the growth of mutual information in the transient dynamics.
Model -We consider an ensemble of N spin-1/2 particles, whose evolution is described by a master equation for the density matrixρ, whereŜ ± ≡ N i=1σ ± i andσ a i are the Pauli matrices (a = x, y, z) acting on spin i = 1, ..., N . The first term in H describes a collective spin-exchange interaction, whilst the second describes a static disordered magnetic field along theẑ-direction. For simplicity, but without loss of generality, we assume the δ i 's are distributed according to a Lorentzian of width ∆ and zero mean. The dissipator L[ρ] = L W [ρ] + L f [ρ] encodes two channels: local, incoherent pumping described by The parameter Γ sets the scale of the spin-spin interactions, while g and f are dimensionless parameters characterizing the relative strength of their corresponding elastic and dissipative part respectively. The incoherent nature of the pumping preserves the U (1) phase symmetry of the dynamics (which can be seen by the invariance of L[ρ] under the transformation σ + j →σ + j e iφ ) and thus by itself does not break the time translation symmetry. In fact, in the steady state the condition σ + i (t → ∞) = 0, is always satisfied. Moreover, the incoherent pumping allows coupling between states with different total S, with S(S + 1) the eigenvalues of the Ŝ · Ŝ operator, and thus the dynamics is not restricted to the collective S = N/2 manifold.
Mean-field analysis -We start with a simple mean-field analysis which illustrates how synchronization emerges in a dissipative setting. It assumes that the many-body density matrix can be written as a tensor product of singlespin density matrices and thus neglects spin-spin correlations. This is equivalent to a description of the system in In the mean-field treatment the effect of the elastic and dissipative interactions generates a self-adjusting effective complex magnetic field identically experienced by each spin in the ensemble due to interactions with the other spins. The corresponding non-linear Bloch equations are presented in the SM [33]. Here, we focus on the dynamics of the azimuthal phases, with δφ ij = φ j − φ i . From direct inspection, we see that in Eq. (3) the term proportional to f can be identified with a similar term in the Kuramoto model [34], the iconic model used to describe the emergence of phase synchronization in classical non-linear oscillators. For synchronization to occur, the coupling strength per oscillator, here proportional to f Γs i > 0 , must be positive and large enough to compensate for the dephasing generated by the different single particle frequencies. The term proportional to g, arising from the real part of the effective magnetic field, also present in the Kuramoto-Sakaguchi model [35], is responsible for imprinting a collective spin rotation on the non-equilibrium steady state. The effective field not only induces a net collective precession but also favors spin alignment and self-rephasing against the depolarization induced by the inhomogeneous field as theoretically and experimentally demonstrated in prior work [36][37][38][39][40][41][42]. Therefore, both f and g are responsible for the rigidity of the time crystal. Diagnosis of the TC regime at the mean-field level proceeds by assuming the existence of a synchronized solution of the mean-field equations, namely setting δ i to zero and determining self-consistently the associated frequency which governs the collective oscillatory dynamics. Later we will restore the detunings and show robustness of the TC to those imperfections. We define the normalized collective order parameter S + (t) ≡ N j=1 R j e iφj , and assume the following scaling form S + (t)/N = Ze iωMFt , looking for a solution of the equations of motion which is stationary in the frame corotating at the angular frequency ω MF . This mean-field solution can feature collective oscillations in the order parameter S + (t) breaking the U (1)-symmetry of the microscopic dynamics (Eq. (1)). This is not the case in the exact solution that must preserve the U (1)-symmetry, Ŝ + (t → ∞) = 0.
The system synchronizes when Z acquires a positive real value, which is self-consistently determined from the system's parameters [30,32]. For W < f Γ the meanfield equations do not admit a synchronized solution for Z, as single particle emission dominates the dynamics and the system depolarizes completely. The same occurs if W > f N Γ, since in this regime pumping dominates, heating the system into a trivial incoherent state  2. (a) Real part of C(τ ) as a function of the characteristic time η ≡ f N Γτ at optimal pumping and g/f = 1/2 for a system of N = 10 (blue), N = 50 (red), and N = 100 (orange) spins along with the finite size scaling prediction in the thermodynamic limit (purple). (b) Extracted ratio of the absolute value of C(τ ) angular frequency |ω| over its spectral width B (see SM [33]) vs system size N and interaction coupling g/f . We also show a frequency contour corresponding to ω opt MF /f Γ ∼ 5 (purple) and a contour of mutual information growth corresponding to f Γ/I (η = 0.03) ∼ 80 (yellow). (c) Growth rate of two-particle mutual information at short characteristic times (here we set η = 0.03) starting from a maximally coherent array σ x i (0) = 1.
fully polarized along theẑ-direction; however, for N > 8 there always exists a finite window of pumping parameters in which the system enters a synchronized phase, with non-vanishing order parameter featuring collective, synchronous, oscillatory dynamics at the angular frequency ω MF . There exists an optimal pumping rate, W opt = f N Γ/2 (assuming N 1), for which the magnitude of the order parameter reaches a maximum value, Z 1/ √ 8. While Z is found to be independent of g, ω MF is proportional to g and approaches ω opt MF = gN Γ/2 at optimal pumping. The rigidity of this frequency will be discussed below.
Quantum model -We now extend our understanding of the formation of the TC beyond the mean-field approximation. Specifically, we study the order parameter for the TC given in terms of the two-time correlation function, Our analysis is based on an efficient exact numerical solution of the master equation [Eq.
(1)] that uses the spin permutation symmetry to drastically improve the exponential scaling of the Liouville space from 4 N to O(N 3 ) [43,44]. C(τ ) is computed via the linear quantum regression theorem [45], . The square root of the equal time correlator, C(0) ≡ Z Q , corresponds to the quantum analog of Z. In close agreement with the mean-field solution, Z Q is found to be nonzero within a window of pumping W where the system synchronizes, and reaches a maximum value, Z opt Q ≈ 1/ √ 8, almost independently of g at W opt , when the system is maximally synchronized. On the other hand C(τ > 0) is highly dependent on g. At optimal pumping, large N and finite but moderate interactions, 1/N g/f √ N , the order parameter C(τ ) oscillates at the mean-field angular frequency ω opt MF . The oscillations slowly decay but appear to become persistent in the thermodynamic limit, thus signaling the emergence of a time crystal. In other words, in units of the TC periodicity, the decay time goes to infinity in the thermodynamic limit. Figure 2  Large many-body correlations can lead to melting of the time crystal [8,12]. For the purposes of quantifying the window of stability of the time crystal, we use the absolute value of the ratio between C(τ ) oscillation angular frequency, ω, and corresponding decay rate, B (or bandwidth) which is portrayed in Fig. 2(b). In our open system we can understand this behavior directly from the quantum regression theorem. While the lowest energy eigenstates of the Liouvillian with pure imaginary eigenvalue are the ones that determine the oscillatory character in the late-time synchronized state, excited eigenstates can contribute to the dynamics of the unequal time correlator through the term e L τ . For moderate interactions, low-lying Liouvillian eigenvectors will have eigenvalues with nonzero imaginary part and small, negative real part. These eigenvalues are dominant and determine the oscillation frequency. As interaction strength increases, however, so does the magnitude of the real part, which results in damping of the TC [33]. Such mechanism is also responsible for the growth of manybody correlations during the transient dynamics and thus directly manifest in the mutual information defined as is the entanglement entropy computed from the reduced density matrixρ α of the subsystem α = A, B, AB (AB is the joint subsystem).
To establish a more formal connection between the TC stability and the growth of many-body correlations, we have computed the derivative of the mutual information for the case when A and B are single spin subsystems, starting from a maximally coherent initial state (all spins pointing along thex-direction). We find that in the large N limit and short characteristic times η 1 (respect to the oscillation period), with η ≡ f N Γτ , the growth rate of the mutual information approaches dI AB /dτ ∼ f Γ 2 1 N + 4g 2 N f 2 η and thus remains irrelevant for g/f √ N . This parameter regime is consistent with the range of g values where we observe that the time crystal forms. Outside this region, I AB grows rapidly with increasing g/f (see Fig. 2c). We explicitly indicate the contours g/f ∝ 1/N and g/f ∝ √ N set by f Γ/ω opt MF and the onset of fast mutual information growth, respectively, in Fig. 2(b). From these considerations we can conclude that the superradiant crystal only exists in the parameter regime where many-body correlations are subdominant and thus it can be regarded as an emergent semi-classical non-equilibrium state of matter.
Robustness to disorder -We now investigate the impact of inhomogeneous dephasing, δ i = 0. For this, we resort to a second-order cumulant expansion [46], since in the absence of permutation symmetry exact numerical calculations are constrained to small systems, N 15. The complete set of equations of motion and equations for the two-time correlations are presented in the SM [33]. We find excellent agreement between the cumulant expansion and exact numerical solution of C(τ ) in the region where a stable TC is expected for the homogeneous (δ i = 0) case, as well as for small system sizes in the presence of disorder [33].
In Fig. 3(a) we show the robustness of the averaged equal time correlator Z Q = C(0) evaluated at optimal pumping W opt to weak disorder ∆/(f Γ) within the relevant window of interactions g/f . One observes that finite elastic interactions protect the synchronized state against disorder, preserve phase coherence and favor spin alignment. While similar phase locking effects in the transient dynamics have been experimentally reported in cold atom experiments [36][37][38][39][40][41][42], the interesting feature observed here is that the phase locking is achieved in the steady state of a driven dissipative system. Panels (b) and (c) of  0), of the averaged two-time correlation function determined from the cumulant expansion. The observed rigidity of the frequency also agrees with the simpler mean-field predictions [33] which allow us to derive an analytic expression for the protection in the weak disorder limit: where we observe the 1/N suppression gained from the collective nature of the elastic and dissipative interactions.
Experimental Realization and Outlook-The superradiant crystal can be directly realized using an array of incoherently pumped atomic dipoles tightly trapped by a deep optical lattice that is supported by an optical cavity. The cavity couples two relevant internal states of the atoms, and operates in the bad cavity limit where the bare atomic linewidth γ is significantly smaller than the cavity linewidth κ. In this regime the cavity photons do not participate actively in the dynamics but instead mediate collective dissipative decay (superradiant emission) [30,31,47], with f Γ ∝ κ/(4δ 2 c + κ 2 ), and elastic exchange interactions, with gΓ ∝ δ c /(4δ 2 c + κ 2 ), which can be independently controlled by varying the cavity detunning δ c from the atomic transition. The signature of the TC can then be directly observed in the spectrum of the light leaked from the cavity [41]. A similar implementation can be realized by replacing the cavity photons by phonons in an ion crystal [48] Having demonstrated the rigidity of the TC to dephasing, now we discuss its rigidity to variations in the system's parameters. For the proposed implementation, ω opt MF ∝ N δ c /(4δ 2 c + κ 2 ). From this expression, one can see ω opt MF is not highly sensitive to variations in the cavity linewidth, κ, but on the contrary it is linearly sensitive to variations on δ c and N . Systematics in the cavity detuning, nevertheless, can be currently controlled at the subhertz level by locking the cavity to a state-of-the-art clock laser [49]. Fluctuations in N can be also suppressed by operating the system in a three dimensional optical lattice in the band or Mott insulator regimes [50] and spectroscopically selecting a fixed region of the atomic array [51].
In summary, we have proposed and investigated the emergence of a TC in a many-body driven dissipative quantum system. By investigating its stability to quantum correlations we showed that it only exists in the parameter regime where many-body correlations are subdominant and thus can be regarded as an emergent semiclassical non-equilibrium state of matter. However, it is important to emphasize that this system is fundamentally distinct from the prototypical laser. This can be seen from the fact that lasing is possible even in the absence of coupling between the atomic dipoles. The superradiant TC, on the contrary, is a genuine many-body phenomenon that happens due to the intrinsic competition and cooperation of collective interactions. It can produce spectrally pure light without stimulated emission and might find direct applications in "quantum-interaction enhanced" sensing.