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arXiv:2105.10640 · Chaos An Interdisciplinary Journal of Nonlinear Science

Revival of oscillation and symmetry breaking in coupled quantum oscillators

Biswabibek Bandyopadhyay, Tanmoy Banerjee

Published Jun 1, 2021• 18 citations
arXivHTMLPDF

Abstract

Restoration of oscillations from an oscillation suppressed state in coupled oscillators is an important topic of research and has been studied widely in recent years. However, the same in the quantum regime has not been explored yet. Recent works established that under certain coupling conditions, coupled quantum oscillators are susceptible to suppression of oscillations, such as amplitude death and oscillation death. In this paper, for the first time, we demonstrate that quantum oscillation suppression states can be revoked and rhythmogenesis can be established in coupled quantum oscillators by controlling a feedback parameter in the coupling path. However, in sharp contrast to the classical system, we show that in the deep quantum regime, the feedback parameter fails to revive oscillations, and rather results in a transition from a quantum amplitude death state to the recently discovered quantum oscillation death state. We use the formalism of an open quantum system and a phase space representation of quantum mechanics to establish our results. Therefore, our study establishes that the revival scheme proposed for classical systems does not always result in restoration of oscillations in quantum systems, but in the deep quantum regime, it may give counterintuitive behaviors that are of a pure quantum mechanical origin.

DOI
10.1063/5.0055091

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Restoration of oscillations from an oscillation suppressed state in coupled oscillators is an important topic of research and has been studied widely in recent years. However, the same in the quantum regime has not been explored yet. Recent works established that under certain coupling conditions, coupled quantum oscillators are susceptible to suppression of oscillations, such as amplitude death and oscillation death. In this paper, for the first time, we demonstrate that quantum oscillation suppression states can be revoked and rhythmogenesis can be established in coupled quantum oscillators by controlling a feedback parameter in the coupling path. However, in sharp contrast to the classical system, we show that in the deep quantum regime, the feedback parameter fails to revive oscillations, and rather results in a transition from a quantum amplitude death state to the recently discovered quantum oscillation death state. We use the formalism of an open quantum system and a phase space representation of quantum mechanics to establish our results. Therefore, our study establishes that the revival scheme proposed for classical systems does not always result in restoration of oscillations in quantum systems, but in the deep quantum regime, it may give counterintuitive behaviors that are of a pure quantum mechanical origin.
We study the quantum manifestations of homogeneous and inhomogeneous oscillation suppression states in coupled identical quantum oscillators. We consider quantum van der Pol oscillators coupled via weighted mean-field diffusive coupling and, using the formalism of open quantum systems, we show that, depending on the coupling and the density of mean-field, two types of quantum amplitude death occurs, namely, squeezed and nonsqueezed quantum amplitude death. Surprisingly, we find that the inhomogeneous oscillation suppression state (or the oscillation death state) does not occur in the quantum oscillators in the classical limit. However, in the deep quantum regime we discover an oscillation death-like state which is manifested in the phase space through the symmetry-breaking bifurcation of the Wigner function. Our results also hint toward the possibility of the transition from quantum amplitude death to oscillation death state through the "quantum" Turing-type bifurcation. We believe that the observation of quantum oscillation death state will deepen our knowledge of symmetry-breaking dynamics in the quantum domain.
Quantum aging transition has recently been reported, whereby the oscillatory dynamics of a network of globally coupled quantum self-sustained oscillators is degraded sharply when the fraction of inactive nodes exceeds a critical value. Here we explore the reverse process: the revival of rhythmic dynamics from an aged state in a quantum oscillator network by introducing a controlled incoherent single-boson pumping channel. Using a self-consistent quantum master equation in the Lindblad form for globally coupled active--inactive quantum Stuart--Landau oscillators, we show that increasing the pumping strength progressively suppresses the aging transition and eventually eliminates it, restoring rhythmicity even when all oscillators are inactive. Remarkably, complete compensation of the coupling-induced dissipative loss is not required; a substantially weaker pumping strength is sufficient to eliminate the aging transition. We show that the revival is governed by a redistribution of the Fock-state population of the inactive oscillators toward higher-occupation states. Our results establish controlled incoherent pumping as an effective mechanism for restoring dynamical robustness in quantum oscillator networks.
An important transition from a homogeneous steady state to an inhomogeneous steady state via the Turing bifurcation in coupled oscillators was reported recently [Phys. Rev. Lett. 111, 024103 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.024103]. However, the same in the quantum domain is yet to be observed. In this paper, we discover the quantum analog of the Turing bifurcation in coupled quantum oscillators. We show that a homogeneous steady state is transformed into an inhomogeneous steady state through this bifurcation in coupled quantum van der Pol oscillators. We demonstrate our results by a direct simulation of the quantum master equation in the Lindblad form. We further support our observations through an analytical treatment of the noisy classical model. Our study explores the paradigmatic Turing bifurcation at the quantum-classical interface and opens up the door toward its broader understanding.
We consider models of open quantum systems in which the system couples to the reservoir at two spatial locations, effecting a feedback loop. In particular, we consider two different models for the interaction of a harmonic oscillator with a feedback reservoir: one with the standard quantum-optical approximations, and one without. The former approach yields the surprising result that damping of the oscillator can be partially or fully suppressed in certain parameter regimes, but in the latter case the situation is more complicated.
We consider models of open quantum systems in which the system couples to the reservoir at two spatial locations, effecting a feedback loop. In particular, we consider two different models for the interaction of a harmonic oscillator with a feedback reservoir: one with the standard quantum-optical approximations, and one without. The former approach yields the surprising result that damping of the oscillator can be partially or fully suppressed in certain parameter regimes, but in the latter case the situation is more complicated.
We study parametrically driven quantum oscillators and show that, even for weak coupling between the oscillators, they can exhibit various many-body states with broken time-translation symmetry. In the quantum-coherent regime, the symmetry breaking occurs via a nonequilibrium quantum phase transition. For dissipative oscillators, the main effect of the weak coupling is to make the switching rate of an oscillator between its period-2 states dependent on the states of other oscillators. This allows mapping the oscillators onto a system of coupled spins. For identical oscillators, the stationary state can be mapped on that of the Ising model with an effective temperature $\ensuremath{\propto}\ensuremath{\hbar}$, for low temperature. If the oscillators are different and are away from the bifurcation parameter values where the period-2 states emerge, the stationary state corresponds to having a microscopic current in the spin system. Close to the bifurcation point the coupling cannot be considered weak and the system maps onto coupled overdamped Brownian particles performing quantum diffusion in a potential landscape.