Robust quantum correlations in out-of-equilibrium matter-light systems

High precision macroscopic quantum control in interacting light-matter systems remains a significant goal toward novel information processing and ultra-precise metrology. We show that the out-of-equilibrium behavior of a paradigmatic light-matter system (Dicke model) reveals two successive stages of enhanced quantum correlations beyond the traditional schemes of near-adiabatic and sudden quenches. The first stage features magnification of matter-only and light-only entanglement and squeezing due to effective non-linear self-interactions. The second stage results from a highly entangled light-matter state, with enhanced superradiance and signatures of chaotic and highly quantum states. We show that these new effects scale up consistently with matter system size, and are reliable even in dissipative environments.

Many-body quantum dynamics are at the core of many natural and technological phenomena, from understanding of superconductivity or magnetism, to applications in quantum information processing as in adiabatic quantum computing [1]. Critical phenomena, defect formation, symmetry breaking, finite-size scaling are all aspects that emerge from the collective properties of the system [2]. Spin networks, many-body systems composed of the simplest quantum unit, are an obvious starting point to understand those phenomena, as they enclose much of their complex behavior in a highly controllable and tractable way. However, if the system under investigation includes a radiation subsystem, new opportunities arise for monitoring and characterizing the resulting collective phenomena [3,4]. By devising driving protocols of the light-matter interaction, high precision macroscopic control then becomes a possibility, regardless of whether the focus is on matter subsystem, the light, or the composite manipulation of both. This is particularly true for the Dicke model (DM) [5], which is the subject of the present work.
The DM describes a radiation-matter system which, despite its simplicity, exhibits a wide arrange of complex collective phenomena, many of them specifically associated with the existence of a quantum phase transition (QPT) [6]. Experimental realizations of the DM have been presented in different settings, from proposed realizations in circuit quantum electrodynamics [7], to the recent very successful demonstrations of DM superradiance in various cold atom experiments [8]. While current knowledge of the light and matter properties in the equilibrium ground state are relatively well known [9][10][11][12][13], its fully quantum out-of-equilibrium critical behavior is just starting to be understood [14,15].
In the DM, both the matter and field are known to act as mediators of an effective non-linear self-interaction with each other [10,13]. These non-linear interactions produce interesting phenomena in both atomic and optical systems [16,17]. Among the most relevant effects, there is the strong collapse and revival of squeezing [18,19], which in many matter states can be related to atom-atom entanglement [20]. Applications of such effects are widespread, including high precision quantum metrology [21], and quantum information technologies [22]. As the non-linear interactions are only effective, it is of essential relevance to understand how the eventual matter-field correlations could affect the generation of the desirable quantum squeezing in each subsystem. We will theoretically address this problem in the particular setting of the finite size DM, when it is ramped across the critical threshold starting from its initial equilibrium noninteracting state.
Some previous works have already examined the dynamical emergence of quantum effects on each subsystem of the DM. However they focused on the semi-classical limit and a static coupling after a sudden quench [23][24][25][26], or on small dynamic oscillations around a phase space region [14]. Our work goes beyond this by giving a fully quantum analysis of the process of continuously turning on the interaction, in order to assess the emergent effects of the non-linear self interaction in each subsystem. We have carried this out across all dynamical regimes: from the very slow adiabatic regime, where the equilibrium ground state results apply, to the sudden quench regime, which is the initial preparation scheme for most previous non-equilibrium results. In between these limits, we have found an unexplored yet very rich intermediate set of annealing velocities featuring remarkable amplification of critical quantum effects as compared to the near-equilibrium results. This magnification of critical properties is followed by a novel chaotic dynamical phase, characterized by giant light-matter entanglement [27], negative Wigner quasi-distributions, and signatures of quantum chaos. This paper is structured as follows. Section 1 presents the model, dynamical setting, and methods of analysis. Section 2 explains the general profile of the dynamical evolution under the perspective of matter and light quantum properties, namely squeezing and the order parameter (OP). Section 3 gains further insight from the point of view of phase space representations of the subsystems. Section 4 establishes the consistency of our results, even under dissipative effects, and for different system sizes, by means of a power-law relation. Section 5 concludes with some final general remarks.

Theoretical framework
The DM describes a totally symmetric interaction of N matter qubits with a radiation mode. Its Hamiltonian is [5], represent collective operators of the qubits, and symbolâ † (â) is the creation (annihilation) operator of the radiation field. Coefficients and ω are single qubit and single mode excitation energies respectively. From now on, we set resonant energies, = ω = 1. The thermodynamic limit (N → ∞) phase boundary of the interaction parameter is λ c = √ ω/2 = 0.5, which is slightly different at finite values of N [11]. The total system S, represented by a density matrixρ S , evolves under a unitary part generated byĤ, and a dissipative part caused by radiation losses of the field cavity. The time evolution is then expressed by a master equation [28], where κ is the damping rate of the cavity, andn is the thermal mean photon number. At initial time t = 0 the system starts at where e −β =n/ (n + 1). The initial state is the equilibrium state at λ(0) = 0. The interaction parameter will change with an annealing velocity (AV) υ by a linear ramping: λ(t) = υt. The interval of interest is λ ∈ [0, 2], well across the QPT. This work is focused on subsystem properties. Any subsystem A will be described by a density matrix ρ A , defined as the trace with respect to the other degrees of freedom:ρ A (t) = tr S−A {ρ S }.
We explore the dynamical development of the quantum properties in subsystems of the matter-light system, caused by this ramped QPT crossing, without resorting to any common simplifications like mean-field, rotating wave or semi-classical approximations. Our main results lie at AV values outside the well-known adiabatic and sudden quench regimes, where a novel magnification of subsystem quantum properties arises. Exact numerical solutions have been obtained by integrating Eq. 2. When κ = 0, evolution lies on the pure Hilbert space and it is generated by the DM Hamiltonian. Otherwise, evolution lies in the space of density matrices and its generator is the total Liouvillian superoperator of Eq. 2 [28]. Thus, the dimension of the numerical evolution vector space is squared as soon as the unitary condition is broken. As the dimension of the field Hilbert space is infinite, the Fock basis {|n } is truncated up to a number where numerical results converge. We have taken advantage of every simplifying condition, such as parity conservation when κ = 0.
The state of any set of M qubits is the same (all qubits are equivalent) and lies on the maximal Dicke manifoldĴ 2 = (J + 1)J, with J = M/2. The first quantum property of matter that we address is entanglement, which for a 2 qubit subsystem is measured by the Wootters concurrence c W [29]. The second one is spin squeezing, which is highly related to multipartite entanglement and usually expressed in terms of a squeezing parameter ξ q [30]. Under unitary evolution (κ = 0), when parity of the total system is even and conserved, c W and ξ q are explicitly related whenever the concurrence is different from zero [31], The factor N − 1 in the concurrence is a direct manifestation of the monogamy of entanglement [32], since each qubit is equally entangled to any other one and hence the finite amount of possible entanglement is evenly distributed. Any qubit state can be visualized by the Agarwal-Wigner function (AWF), which is a Bloch sphere representation of the qubit's density matrix [33], and Y l,m are the spherical harmonics. Terms T l,m are the expected values of the multipole operator,T where j = N/2, and is the Wigner 3j symbol.
In addition to the matter subsystems, the DM has the presence of a radiation mode, which is usually much more experimentally accessible, thanks to the radiation leaked by the cavity and properties disclosed by tomographic techniques [3]. The general state of the mode can be condensed in the form of its Wigner quasi-probability distribution [34,35], whereD (α) = e αâ † −α * â is the displacement operator, and α ∈ C. There is also an analogous squeezing parameter in the field mode, though it cannot be directly related to a form of entanglement. It is expressed in terms of the variance and covariance of field quadratures [36], The displacement operator in Eq. 7 is related to the quadratures by √ 2α = x +ιp.

Out-of-equilibrium enhancement of quantum correlations
A general picture of the time evolution of the matter quantum properties is revealed by means of spin squeezing (related to matter entanglement) and the expected value of its OP in Figs. 1a-b. There all the relevant AV are considered, from the adiabatic limit (bottom) to the sudden quench (top). If the adiabatic condition is fulfilled, properties of the system are those known from the equilibrium QPT [9,37]. In the sudden quench condition, no changes are appreciable because the system stays as it started. In between those limits, the actual dynamical regime occurs. Near the thermodynamic limit equilibrium phase transition λ = 0.5, adiabatic evolution exhibits a maximum value of entanglement followed by its decay and then the growth of the OP. The anti-correlation between squeezing and OP is present in the dynamical regime as well, so that Figs. 1a and 1b are negative images of each other. As the range of high AV is entered, the position of the maximum point of squeezing and the onset point of the OP are pushed toward higher values of λ. Any value of υ beyond the adiabatic limit leads to the dynamical regime, provided that the interaction parameter is ramped up to a high enough value. In other words, the sudden quench condition is only a consequence of the upper bound of λ.
More important than the position of maximum concurrence in the dynamical regime, is its value. It reaches up to three times the corresponding value of the adiabatic evolution (see Fig. 1a). This is a remarkable improvement for the collective generation of such a distinctive quantum property, and is due entirely to the system being in the non-equilibrium dynamical regime. This magnified entanglement scales very well with the number of qubits N (see inset of Fig. 1a), keeping the spin squeezing parameter almost identical, and gets modestly better as the system size grows. The sudden death point of concurrence is also virtually size-independent. The scaling properties of the dynamical regime will be discussed in section 4. Matter OP also shows very good scaling collapse in its dynamical evolution (inset of Fig. 1a).// With respect to the maximum matter OP in Fig. 1b, it is bounded. Therefore independent of the regime, it cannot be increased. The kind of enhancement revealed by the OP is a magnification of superradiance, which is the traditional way to describe the QPT of DM [38]. The OP indicates the scaled number of excitations within the subsystem, which is bounded because of finiteness of N . In the dynamical regime, excitations develop more quickly (see blue curve in inset of Fig. 1b). Seen by the eventual leakage of excitations through the cavity, it manifests itself as a sharp burst of photons which are suddenly released. Bursts like these are excellent indicators of the DM QPT in cold atom experimental realizations [8].
Qubit quantum properties are very important for applications in quantum computation and quantum information. However, the finite character of the matter subsystem conceals much of the complexity of the evolution. The field subsystem, with its own relevance in highly controlled quantum optics, does not have this restriction. Field dynamical profiles analogous to that of Fig. 1 are shown in Figs. 2a-b. Similarities with Figs. 1a-b are noticeable, though with relevant differences. A large dynamical magnification of the field quadrature squeezing is also present, but it dies off later than its matter counterpart. Thus the development of the field OP is delayed as compared to that of the qubits. Superradiance, as seen by value of the field OP, is now not only sharper, but it can be as much as twice the value attainable with adiabatic ramping at a given λ. This greatly enhances the intensity of superradiant bursts. Radiation intensity enhancements of this kind have also been predicted in the DM submitted to sudden quenches and a lossy cavity [35]. Dynamically enhanced and suppressed superradiance alternate after the first burst (see blue curve in inset of Fig. 2b), a behavior totally absent in the equilibrium QPT. A wavy plateau in figure 1b coincides with that alternating stage. The dynamical phase marked by these oscillations is characterized by giant light-matter entanglement, with emergent quantum properties, beyond the critical ones [27]. b , as defined in Eq. 8, whenever it is greater than zero (squeezed radiation). (b) Scaled OP of light â †â /N . Major tendencies of Fig. 1 are replicated here. However, the unbounded nature of the field OP allows an appreciation of the field intensity's oscillation (associated with supperradiance) around the equilibrium values, with intervals of significant dynamical enhancement of superradiant bursts. The oscillatory behavior is related to a combined light-matter quantum chaotic stage.
Magnification of squeezing and the other effects discussed so far are caused by a prolonged realization of effective non-linear interactions within both subsystems, after a preparation stage where the sudden quench approximation holds. Then, the non-linear processes are realized as one axis squeezing with a transverse field for matter [19,30], and Kerr-like interactions for radiation [17]. During these separate squeezing processes, both radiation and matter act as effective interaction mediators for each other. This extends to the non-equilibrium case static correspondences with matter-only [10], or radiation-only systems [13]. Separate squeezing, which is related to internal quantum entanglement, suddenly dies as soon as combined matter-light entanglement emerges, because of monogamy and the breaking of the effective non-linear interaction condition.
As the field has more information capacity, its squeezing can survive longer. By contrast, in the adiabatic regime, entanglement within and between subsystems compete against each other, because they occur simultaneously at the critical point.

Matter-light quasi-probability behavior
Complete state representations of the subsystems state reveal more details of the processes involved in the dynamical regime. Figure 3 show several snapshots of Wigner and AWF quasi-distributions at different instants. The enormous difference in squeezing is now clear graphically, as well as a difference in the direction of squeezing between the dynamical regime and the adiabatic one. At any regime, death of squeezing is associated with a splitting in half of the distributions, caused by the spontaneous symmetry breaking of the QPT [27]. This splitting is symmetrical along the x and −x direction in both subsystems. The system's density of excitations, indicated by the OP, increases as the distributions become displaced away from the initial state. Features of the complex light-matter entangled stage are clarified by Fig. 3. The dynamical splitting of the distribution leads to negative scars and complex patterns for both subsystems, though the phenomenon is far more conspicuous in radiation. Donut shapes with a negative centered AWF such as that in Fig. 3v, have been experimentally obtained in 3000 atoms with just a single photon [4]. In addition, round-tailed interference patterns such as that in Fig. 3V have been obtained for light in Kerr-like media following a Fokker-Planck equation [18], which confirms the presence of non-linear effective interactions. These results mean that the field is able to exhibit chaotic behavior itself, regardless of its entanglement to the matter subsystem. Despite being a single radiation mode, the field acts as a reservoir that dissipates the quantum correlations present in the squeezed states of each subsystem. If the qubits were not coupled to the field, squeezing could have revivals after its sudden death [19]. The field Wigner function is not only full of negatives regions (a marker of non-classical light), but it also contains abundant so-called sub-Planckian structures which have been related to quantum chaos [39]. The DM is non-integrable, and its ordered phase has been connected to chaotic features [6,40]. As spin squeezing is very sensitive to chaos [26], it is no surprise that its sudden death becomes irreversible once the light-matter entangled stage arises.

Scaling and robustness of non-equilibrium quantum correlations
So far, unitary evolution have been assumed in this work. Figures 4a-e address how the presence of losses in the cavity affect the dynamics. All the main results found in κ = 0 evolution survive very well if decoherence towards the environment is two order of magnitudes lower than the main energy scale. Even if dissipation is at values of just an order of magnitude below, spin squeezing effects remain highly robust, with increasing noise resistance with system size. Field squeezing surprisingly survives to dissipation regimes comparable to the Hamiltonian dynamics itself. On the other hand, detailed features of the chaotic stage are far more sensitive to decoherence, such as OP oscillations, negative regions, and Sub-Planck structures. Such sensitivity has been proposed as a tool for measuring very weak forces [41]. In our analysis, we have found that small finite values ofn (such as those typical at the ultra-low temperatures of most experimental realizations) do not change dramatically the conclusions; they just slightly intensify the process of decoherence. Previous works on non-equilibrium DM under different settings have also confirmed that some major phenomena present here survive losses. Enhanced superradiance can be detected [35], and squeezing can still have time to develop [18,24]. We note that, despite perturbing the quantum state of the system, a lossy cavity compensates by providing the experimental possibility to monitor the dynamic evolution of the field in time [3].
Our results exhibit substantial size invariance, as the insets of Figs. 1 and 2 show. Even the small system sizes in lossy cavity results in Figs. 4b-e are highly size indepen- dent, which justifies our claim that the conclusions drawn there can be taken as general trends. Figures 4f-g show that the onset point of the dynamical regime occurs at points λ d following a power law, (λ d − λ c ) ∝ υ 2/3 . This kind of scaling relation is typical in QPT critical properties [2]. This scaling laws extends our previous results on adiabatic scaling theory [15] to the fast regime. Even though we have used the same kind of scaling variables as in the near adiabatic case, many results in the dynamic regime collapse very well without any re-scaling of λ.
The origin of the scaling exponent can be understood in a rather simple way, using concepts from the Kibble-Zurek mechanism [1]. The scaled time is the instant where the healing time of the system (as measured by the inverse mean energy gap in the spectrum δ −1 ) has the same order of magnitude, i.e., when tδ ≈ 1. As δ ∝ (λ − λ c ) 1/2 in the ordered phase [10], and t ∝ (λ − λ c )υ −1 , we get the exponent 2/3 present in all lines of Figs. 4f-g. The fact that equilibrium critical exponents have been used reveals that, even though the emergence of the dynamical regime is well inside the λ > 0.5 range, its mechanism is still related to the QPT, and so many critical phenomena get dynamically enhanced. On the other hand, the fact that the evolutions are very similar in the chaotic region, despite the different sizes, has to do with the relatively equal structure of the interaction well inside the superradiant phase, provided it has not bounced against the finite limits of the matter subsystem.

Conclusions
We investigated the details of the microscopic properties of both matter and light subsystems of the DM when macroscopically driven into the superradiant phase from an initial non-interacting equilibrium state. We have found that, in between the traditional driving schemes (either near-adiabatic or sudden quench), a previously unnoticed intermediate annealing regime emerges, and that this regime can be seen as advantageous for many quantum control processes. The onset of this intermediate phase is marked by induced non-linear self-interactions in both matter and field subsystems, developing squeezed states in each of them that are related to internal entanglement. This squeezing process is much stronger than what can be achieved in near-adiabatic evolutions, since the internal entanglement does not have to compete against the matter-field entanglement that arises in later stages.
We have also found that the dynamical squeezing mechanism is succeeded by the development of a chaotic and entangled radiation-matter dynamical phase. Clear signatures of quantum chaos such as fragmented Wigner functions, have been identified in this phase. This stage has its own dynamically enhanced properties such as superradiance, phase order, and sensibility to weak forces. We have established the robustness of all the quantum enhancing processes under dispersive conditions and its invariance to system size. We have shown that this last invariance condenses into a power-law relation associated with critical exponents. We hope this understanding will prove important for developing schemes to generate squeezed and other entangled states, as have been proposed in the contexts of quantum metrology and quantum information processing.