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arXiv:2201.06340 · Physical Review A

Out-of-time-order correlator in the quantum Rabi model

Aleksandrina V. Kirkova, Diego Porras, Peter A. Ivanov

Published Mar 25, 2022• 20 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.105.032444

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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.
Abstract Quantum chaos is an intriguing topic and has attracted a great deal of interests in quantum mechanics and black hole physics. Recently, the exponential growth of out‐of‐time‐ordered correlator (OTOC) has been proposed to diagnose quantum chaos and verify the correspondence principle. Here, good correspondence is found between the linear entanglement entropy and the semiclassical phase space structures in the anisotropic quantum Rabi model. The Loschmidt echo in the chaotic sea decays more faster than that in the stable island. However, the OTOCs grow exponentially at early times for the initial states centered both in the chaotic and stable regions. The exponential growth of the OTOC is attributed to quantum collapse that provides a novel mechanism of yielding exponential growth of the OTOC in quantum systems. Moreover, the quantum collapse effect is more obvious for the initial states centered in the chaotic one. The results show that in the anisotropic quantum Rabi model, the linear entanglement entropy, and Loschmidt echo are more effective than OTOC for diagnosing quantum chaotic signals.
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.
Abstract As one of the famous effects in the quantum Rabi model (QRM), Rabi oscillation may lead to the occurrence of quantum dynamical behaviors without classical dynamic counterparts, such as quantum collapse and revival effects. In this paper, we focus on studying the long-time quantum signatures of chaos in the large atom-light frequency ratios of the Rabi model. It is shown that the saturated values of the entanglement entropy for initial states located in chaotic sea are higher than that in the regular regions, and the Husimi Q function are more dispersed in phase space. Moreover, we observed that the long-time average entanglement entropy and spin variance correspond well with the semiclassical phase space. Our results imply that the correspondence principle is not invalidated by quantum collapse and revival effects in the large atom-light frequency ratios Rabi model.
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-1/21/21/2 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/N1/N1/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.
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.
We study the quantum critical behavior of the Dicke Hamiltonian with finite number of atoms and explore the signature of quantum chaos using measures like the ground-state fidelity and the Loschmidt echo and the time-averaged Loschmidt echo. We show that these quantities clearly point to the classically chaotic nature of the system in the superradiant (SR) phase. While the ground-state fidelity shows aperiodic oscillations as a function of the coupling strength, the echo shows aperiodic oscillations in time and decays rapidly when the system is in the SR phase. We clearly demonstrate how the time-averaged value of the echo already incorporates the information about the ground-state fidelity and stays much less than unity, indicating the classically chaotic nature of the model in the SR phase.