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arXiv:1808.02038 · Physical Review A

Scrambling in the Dicke model

Yahya Alavirad, Ali Lavasani

Published Apr 1, 2019• 43 citations
arXivHTMLPDF

Abstract

The scrambling rate ${\ensuremath{\lambda}}_{L}$ associated with the exponential growth of out-of-time-ordered correlators can be used to characterize quantum chaos. Here we use a particular Majorana fermion representation of spin-$1/2$ systems to study quantum chaos in the Dicke model. We take the system to be in thermal equilibrium and compute ${\ensuremath{\lambda}}_{L}$ throughout the phase diagram to leading order in $1/N$. We find that the chaotic behavior is strongest close to the critical point. At high temperatures ${\ensuremath{\lambda}}_{L}$ is nonzero over an extended region that includes both the normal and superradiant phases. At low temperatures ${\ensuremath{\lambda}}_{L}$ is nonzero in (a) close vicinity of the critical point and (b) a region within the superradiant phase. In the process we also derive an effective theory for the superradiant phase at finite temperatures. Our formalism does not rely on the assumption of total spin conservation.

DOI
10.1103/physreva.99.043602

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The scrambling rate \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ associated with the exponential growth of out-of-time-ordered correlators can be used to characterize quantum chaos. Here we use a particular Majorana fermion representation of spin-1/21/21/2 systems to study quantum chaos in the Dicke model. We take the system to be in thermal equilibrium and compute \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ throughout the phase diagram to leading order in 1/N1/N1/N. We find that the chaotic behavior is strongest close to the critical point. At high temperatures \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ is nonzero over an extended region that includes both the normal and superradiant phases. At low temperatures \ensuremathλL{\ensuremath{\lambda}}_{L}\ensuremathλL​ is nonzero in (a) close vicinity of the critical point and (b) a region within the superradiant phase. In the process we also derive an effective theory for the superradiant phase at finite temperatures. Our formalism does not rely on the assumption of total spin conservation.
Abstract We discuss the onset of quantum chaos and thermalization in the two-mode Dicke model, which describes the dipolar interaction between an ensemble of spins and two bosonic modes. The two-mode Dicke model exhibits normal to superradiant quantum phase transition with spontaneous breaking either of a discrete or continuous symmetry. We study the behaviour of the fidelity out-of-time-order correlator (FOTOC) derived from the Loschmidt echo signal in the quantum phases of the model. We show that the exponential growth of the FOTOC in the beginning of the time evolution cannot be related to a classical unstable point in the general case. Furthermore, we find that the collective spin observable in the two-mode Dicke model quickly saturates to its long-time average value, and shows very good agreement between its diagonal ensemble average and microcanonical average even for a small number of spins. We show that the temporal fluctuations of the expectation value of the collective spin observable around its average are small and decrease with the effective system size, which leads to thermalization of the spin system.
Recently, the out-of-time-ordered correlator (OTOC) has gained much attention as an indicator of quantum chaos. In the semi-classical limit, its exponential growth rate resembles the classical Lyapunov exponent. The quantum–classical correspondence has been supported for the one-body chaotic systems as well as realistic systems with interactions, as in the Dicke model, a model of multi-two-level atoms and cavity field interactions. To this end, we calculate the OTOC for different variations of the Dicke model in an open quantum system setting. The connection between the superradiant phase transition of the Dicke model and the OTOC is studied. Further, we establish a relation between the OTOC and the second-order coherence function. This becomes important for the experimental studies of the OTOC and quantum chaos in the models of quantum optics.
Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalization in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analyzing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behavior. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic-driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behavior in quantum systems, beyond the initial exponential growth in OTOCs.
Early studies on the Dicke model linked superradiance to quantum chaos, whereas later anisotropic Dicke and Rabi studies reached apparently conflicting conclusions. We resolve this tension by distinguishing global chaos from local instability. Along the isotropic Dicke path, a sweep at fixed low excitation energy crosses both the superradiant boundary and a nearby chaotic window, explaining the reported correlation. However, the anisotropic Dicke model can be chaotic in the normal phase and regular in the superradiant phase, making superradiance neither necessary nor sufficient for global chaos. A fidelity out-of-time-order correlator (FOTOC), in contrast, probes the local stability of the stationary point selected by the initial wave packet. The superradiant transition destabilizes the origin, whereas a change in anisotropy can restore its stability and create displaced saddles. Wave packets centered at these saddles exhibit exponential FOTOC growth even on a classically regular Dicke energy shell with Poisson statistics. This local sensitivity also enables quantitative spectroscopy: combined with the topology of stationary points and local density of states relations, FOTOC dynamics reconstructs the absolute amplitudes of spectral singularities in both models without fitting the density of states.
Fast scrambling, quantified by the exponential initial growth of out-of-time-ordered correlators (OTOCs), is the ability to efficiently spread quantum correlations among the degrees of freedom of interacting systems and constitutes a characteristic signature of local unstable dynamics. As such, it may equally manifest both in systems displaying chaos or in integrable systems around criticality. Here we go beyond these extreme regimes with an exhaustive study of the interplay between local criticality and chaos right at the intricate phase-space region where the integrability-chaos transition first appears. We address systems with a well-defined classical (mean-field) limit, as coupled large spins and Bose-Hubbard chains, thus allowing for semiclassical analysis. Our aim is to investigate the dependence of the exponential growth of the OTOCs, defining the quantum Lyapunov exponent λ_{q} on quantities derived from the classical system with mixed phase space, specifically the local stability exponent of a fixed point λ_{loc} as well as the maximal Lyapunov exponent λ_{L} of the chaotic region around it. By extensive numerical simulations covering a wide range of parameters we give support to a conjectured linear dependence 2λ_{q}=aλ_{L}+bλ_{loc}, providing a simple route to characterize scrambling at the border between chaos and integrability.
We investigate signatures of chaos and equilibration in the quantum Rabi model, which exhibits a quantum phase transition when the ratio of the atomic level-splitting to bosonic frequency grows to infinity. We show that the out-of-time-order correlator derived from the Loschmidt echo signal quickly saturates in the normal phase and undergoes exponential growth in the superradiant phase, which is associated with the onset of quantum chaos. Furthermore, we show that the effective time-averaged dimension of the quantum Rabi system can be large compared with the spin system size, which leads to suppression of the temporal fluctuations and equilibration of the spin system.