key: cord-1045448-xwz5zrex authors: Ray, Sayak; Vardi, Amichay; Cohen, Doron title: Quantum signatures in quench from chaos to superradiance date: 2022-01-19 journal: Phys Rev Lett DOI: 10.1103/physrevlett.128.130604 sha: 2fead64ad0cbccf5fd3198f4ff602165d506bfff doc_id: 1045448 cord_uid: xwz5zrex The driven-dissipative Dicke model features normal, superradiant, and lasing steady-states that may be regular or chaotic. We report quantum signatures of chaos in a quench protocol from the lasing states. Within the framework of a classical mean-field perspective, once quenched, the system relaxes either to the normal or to the superradiant state. Quench-from-chaos, unlike quench from a regular lasing state, exhibits erratic dependence on control parameters. In the quantum domain this sensitivity implies an effect that is similar to universal conductance fluctuations. The essence of chaos is often presented as a butterfly effect: a small variation in a control parameter h leads to a drastically different outcome, with seemingly erratic deterministic dependence. For example, a particle is launched into a chaotic cavity and is either transmitted (Q=1) or reflected (Q=0). The classical dependence Q(h) looks uncorrelated on a scale that is larger than some exponentially small δh c . Alternatively, one may consider a coin tossing experiment that involves a dissipative quench to the binary final outcome due to the proverbial coin-ground interaction. In the present work, we consider a quench from chaos (QFC) to bistability for atoms in a lasing cavity. The control parameter h is a pre-quench preparation time t prep , and the post-quench outcome is either a normal state (NS) [Q=0] or a supperradiant (SR) state [Q =0]. The observable Q is the number of photons in the cavity, namely, Q = n(t m ) where t m is the time-to-measurement, i.e. the duration of the quench. Within the framework of a classical (Mean Field) perspective, for an appropriate tuning of the atom-field interaction, the dependence of Q on h is erratic, as illustrated in Fig.1 . We seek for the signature of this dependence in the quantum regime. The simplest quantum version of QFC is a semiclassical phase-space picture. The wavepacket spreads over the chaotic sea, and therefore the erratic dependence of Prob(Q=1) on h is smeared away: in the classical meanfield context this probability is either 0% or 100%, while in the semiclassical truncated Wigner approximation perspective it equals a number p that reflects the relative volume of the basin leading to the SR state. However, interference between semiclassical trajectories should result in irregular dependence on h in the exact quantum many-body dynamics, see Fig.1 . Fluctuations due to QFC are analogous to universal conductance fluctuations (UCF) [1, 2] and chaos-assisted tunneling (CAT) [3] . In the UCF context Q is the transmission (conductance) through a chaotic cavity, and h is the magnetic field, while in the CAT context Q is the tunneling rate, and h is the scaled Planck constant. In all those cases (QFC, UCF, CAT) the systematic nonsemiclassical fluctuations in the output signal constitute In the classical (mean-field) limit the outcome of the measurement (blue line) is binary and erratically depends on the parameter that controls the preparation protocol (in our demonstration it is the preparation time tprep). In the semiclassical (truncated Wigner) approximation, this erratic dependence is smoothed away (black line). The measured Q reflects the relative volume of the basin that leads to the Q=1 attractor. In the proper quantum treatment the outcome (red line) manifests fluctuations that arise from interference of trajectories. However, any mesoscopic system eventually relaxes, such that for t=∞ the expectation value Q reflects a thermal equilibrium that does not depend on the initial preparation. quantum signature of chaos. However, in QFC we have the extra complication due to dissipation, and one wonders whether any memory of chaos survives after the quench. The availability of both regular and chaotic lasing steady states in the driven-dissipative Dicke model [4] [5] [6] [7] [8] [9] [10] [11] [12] offers an opportunity to directly contrast the QFC with a quench from a quasi-periodic regular orbit and show how the h dependence of the quench outcome indicates whether the prepared state was regular or chaotic. Outline.-We first review the regime diagram of the dissipative Dicke model, highlighting NS, SR, as well as regular and chaotic lasing regions. Relaxation towards the NS-SR bistability is then considered as a measurement protocol. In the full QFC scheme, we choose the pre-quench preparation time (t prep ) as a control parameter. This QFC scenario is contrasted with the quench from dynamically regular motion. In particular we aim to clarify the significance of the quench duration (t m ). The Dicke model.-The model describes N two level atoms (exitation energy E) that interact with a single cavity mode (frequency Ω) [13, 14] . The Hamiltonian involves, respectively, the bosonic field operatorâ, Steady state phase diagram. The vertical axis is theg/g ratio that reflects coherent pumping. In panel (a) the horizontal axis is the normalized incoherent collective pumping. We assume Ω=E=1 and g=2, while κ=2 and γc=0.5. The label NS * indicates a stable all-atom-excited state. The labels LC and Chaos indicate a regular limit cycle and a chaotic lasing state, respectively. With vanishing dissipation, bistability appears forg/g ≤ 0.5, and the energy landscape has 3 attractors (NS and two SR fixed points), while with finite dissipation this range is shifted. Panel (b) shows the dependence of the bistability region on κ, for g=2, while fc=γc=0. The symbols are based on numerical analysis, while the lines are based on stability analysis (see SM). and the Pauli matricesσ i , with the common subscripts i = x, y, z, ±. The couplings g andg denote the strength of the co-rotating and counter-rotating terms of atomphoton interaction. Namely, We define the mode oocupation operatorn =â †â , and the collective excitation operatorsŜ i = (1/2) rσ r i , (i=x, y, z) that generate a spin algebra with angular momentum ≤N/2. It is well known [14] [15] [16] [17] that the ground state of the Dicke model undergoes a quantum phase transition from a normal state (NS) with n = 0 to a pair of superradiant (SR) states with n = 0. Moreover, depending on (g,g), the model exhibits an excited state quantum phase transition [18, 19] . Dissipative dynamics.-Several loss and incoherent processes are associated with the Dicke system [5] [6] [7] [8] [9] [10] [11] [12] . The corresponding dissipative dynamics can be studied within the framework of a Lindblad master equation, where The incoherent dynamics in Eq.(2) arises from the cavity-photon We start with all the atoms in the ground state, while n ∼ 0. In the left panelsg/g = 0.75, and the relaxation is towards SR. In the right panelsg/g = 0.48, and the relaxation is towards NS-SR bistability. The other parameters are g=2, and κ=2, and γc=0.5 and fc=0.04. In the quantum simulation we have N =16 atoms (meaning =8), and use N b =80 truncation for the bosonic mode. The semiclassical results of (a,b) and the quantum results of (c,d), are compared in (e,f). The waiting time up to the measurement is t = tm = 20. Solid black line is the semiclassical distribution, while dashed red line is the quantum distribution. The classical SR fixed points are marked by horizontal dashed lines in (a-d) and by arrowheads in (e-f). Note the n=0 peak at (f). loss L[â] with a rate κ, and from local incoherent decay and pumping transitions L[σ r − ] and L[σ r + ] with rates γ ↓ and γ ↑ , respectively. Apart from the local incoherent processes, there are also incoherent collective processes L[Ŝ − ] and L[Ŝ + ], with rates γ c ↓ and γ c ↑ , respectively. Below, we focus on collective incoherent transitions, and neglect local incoherent processes. The collective decay/pumping for the Dicke model is justified when the atoms are concentrated in a spatial region much smaller than the wavelength of the coupled cavity modes [5] . The total spin then becomes a constant of motion. Per our preparation we focus on the = N/2 multiplet. The reduced Hamiltonian can be written in terms of the S i operators. For large N the classical approximation is ob- 4 . The prepared state. The system is prepared in a non-dissipative chaotic state with g=1 andg=0.48. This is done by launching a coherent state with sz=sx=1/ √ 8, and sy=0, while n≈0, followed by a long waiting time 50 < tprep < 1000. In the quantum simulation we have N =16 atoms (meaning =8), and use N b =80 truncation for the bosonic mode. Panel (a) is the quantum Husimi distribution of the prepared state in the [Re(a) − Im(a)] plane at t = tprep = 50. On top we display the corresponding cloud of classical points. The latter are color-coded based on the post-quench outcome: blue for those that belong to the NS basin, and red/magenta for those of the SR basins. Panel (b) displays the associated sz=0 Poincare section (the sy, a > 0 branch) projected on the (n − sx) plane, with added blue/red/magenta circles that indicate the attractors. For the quench we assumed g=2, but kept the sameg/g, with dissipation parameters κ=2 and γc=0.5, and with incoherent pumping fc=0.04. tained by treating them as classical coordinates. We define scaled variables s :=Ŝ − /N , and s x,y,z :=Ŝ x,y,z /N , such that s 2 x +s 2 y +s 2 z = 1/4. We also scale the bosonic coordinates as a :=â/ √ N . Consequently, the classical equations of motion arė where the net incoherent pumping is f c = γ c ↑ − γ c ↓ , while the total incoherent rate of transition is γ c = γ c ↑ + γ c ↓ . In Fig. 2 we present phase-diagrams obtained by stability analysis and numerical long-time propagation of Eq.(3). The phase-diagram includes NS, SR, as well as regular and chaotic lasing phases. Moreover, there is a bistable NS-SR phase that we are going to utilize for the measurement protocol. The NS-SR Bistability.-An energy landscape E(n, s z ) for the cavity can be obtained by minimizing H D for a given (n, s z ) under the constraint s 2 x +s 2 y +s 2 z = 1/4, see SM. For small g this landscape exhibits a stable NS minimum at n=0 and s z = − 1/2 that becomes an attractor for κ > 0. For (g+g) > √ ΩE, the NS becomes an energetic saddle point rather than a local minimum, but ifg/g < 1−[ √ ΩE/g] it maintains dynamical stability and remains an attractor. The transition of the NS to a saddle point is accompanied by the appearance of a pair of broken symmetry n =0 SR minima. These two SR states remain attractors provided κ is not too large. For quantitative details, including a (κ,g/g) regime diagram, see SM and Fig.2b . Relaxation towards bistability.-In Fig.3 we inspect the distribution P (n) of the cavity mode's occupation. In the quantum simulation we start with all the atoms in the ground state, while n ∼ 0. In the semiclassical simulation we prepare an initial cloud centred near the south pole of the Bloch sphere s z ∼ −1/2, with photon number n ∼ 0, and let the cloud relax. We compare the outcome of relaxation towards a SR steady state, to the relaxation in the bistable NS+SR phase. In the latter case P (n) exhibits two distinct peaks, that exhibit broadening in the quantum simulation. The quantum SR/NS peak ratio is tilted towards the NS with respect to the classical one due to the quantum spilling from the metastable SR state. It is important to realize that this broadening and peak-ratio tilting are not a signature of true quantum interference: similar broadening would have been captured semiclassically, if Langevin noise terms were included [20] . By contrast, the quantum-interference signature we seek can not be captured by means of stochastic semiclassical simulations. Quench from chaos (QFC).-Having gathered all the necessary ingredients, we turn to discuss the full scenario, including a preparation stage and a quench stage. The purpose of the measurement is to detect chaos in the preparation stage. The quench is to a bistable phase in order to amplify small fluctuations in the prepared state. The preparation of the chaotic state is demonstrated in Fig.4 . Panel (a) demonstrates qualitatively the rather good correspondence that we have between the quantum distribution and the semiclassical cloud. The points are color-coded according to which basin they belong: upon quench the blue points will reach the NS fixed point, while the red/magenta points will reach the two SR fixedpoints. The phase-space location of the basins is better resolved in the Poincare section of panel (b). The quench is an abrupt change in the model parameters. Specifically we force the system to relax towards bistability by setting the parameters (g,g, κ, γ c , f c ) to the values specified for Fig.3b . This is followed by a wait time t m , during which the system evolves under the dissipative dynamics with the new parameters. At the end of the waiting time, a measurement of Q =n(t m ) is preformed. Zero quench time (t m =0) formally means that there is no quench process, and accordingly the observable is Q =n(0) =n. For sufficiently large t m , disregarding the quantum/noisy broadening effect, the measured quantity is a sum of a projector on the NS basin, and a projector on the SR basin, weighted by n NS = 0 and n SR = 0: . We clearly see that chaos is reflected in the outcome of the QFC scenario, in accordance with the discussion of Fig.1 . In contrast, the flutuation due to quench from a regular state, are non-erratic and merely reflect the spectral context of the quai-regular dynamics. Memory loss.-In a mesoscopic device the information is eventually blurred due to noisy hopping between the fixed points. The outcome of the measurement is presented in Fig.6a for several choices of t m . We observe memory loss gradually with increasing t m . For short t m the systematic variation of Q as a function of t prep is apparent. Furthermore, due to our choice of observable, the outcome is partially correlated with the t m =0 measurement of n . This is demonstrated in Fig.6b . We would like to provide a semiclassical procedure for the analysis of this correlation. In the semiclassical simulation, the ergodized cloud does not show any fluctuations, and therefore, the postquench dynamics does not depend on the preparation time. However, we can mimic the quantum fluctuations by giving each "point" of the semiclassical cloud a weight w j ∝ (1 + Cn j ), where the proportionality constant is determined such that w j = 1. Using the semiclassical equations of motion we can determine the mapping n j → n j (t m ). Then we can calculate For each t prep the parameter C is adjusted such that n(0) sc = n qm . Then we can predict the outcome for finite t m . The result of this phenomenological theory is incorporated in Fig. 6b . The departure of the symbols from the calculated lines (e.g. blue as opposed to red symbols) is the signature that fluctuations over the Q of Eq.(4) do not reflect trivially fluctuations of n. On the other hand, the memory loss due to noisy hopping between the fixed-points is reflected by the "flattening" of the outcome (e.g. green symbols). Discussion.-A realistic measurement, unlike an idealized projective measurement, involves a dissipative quench process. In a macroscopic reality a tossed-coin, or a ferromagnetic pointer, will always point "up" or "down" at the end of the quench. For a non-violent quench, a relatively large t m is required in order to reach the attractor, allowing differentiation between initially similar states. Thermal and quantum fluctuations can be ignored. But in a mesoscopic context, the time of the quench (t m ) should be optimized in order to keep the information about the measured (pre-quench) state (it should be "large" but not too large). Our emphasis was on QFC, looking for the quantum signature of chaos, and clarifying the physical significance of t m . Per our construction the "large" t m measurement was strongly correlated with the t m =0 measurement, but clearly this is not a general feature. In general the "basins" of Q are not correlated with a simple observable of the system. Either way, we have demonstrated the manifestation of irregular quantum fluctuations in the outcome, providing signature for chaos in the "measured" state. These fluctuations resemble CAT and UCF. They are completely diminished in the semiclassical picture, and come instead of the classical exponential sensitivity that one would expect if reality were not quantum-mechanical. But unlike UCF and CAT, they are endangered by memory loss due to relaxation. 0. Given n and the constrain s 2 x +s 2 y +s 2 z = 1/4, and assuming thatg < g, the minimum is obtained at ϕ=0, and we find E(n) = minimum H(n, ϕ; s x , s y , s z ) = Ωn − E 2 2 + g 2 + n (S- 15) We see that the NS fixed-point (n=0) is no longer the minimum if g + > √ ΩE. This is a necessary condition for bistability. The reason for having bistability is that the NS, while being a saddle in the energy landscape, is still a dynamically stable fixed-point, that becomes an attractor for finite κ. We plot in Fig.2b the boundaries that we have found for the regions where the NS and the SR fixed points are stable. The two regions overlap. In the "SR+NS" overlap region we have bistability, as demonstrated in the simulation of Fig.S1 . For vanishing dissipation (κ → 0) and g=2 the bistability region is 0