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Photon waiting time statistics: a keyhole into dissipative quantum chaos

Igor Yusipov, Olga Vershinina, S. Denisov, M. V. Ivanchenko

Published May 11, 2019• 0 citations
OpenAlex

Abstract

Quantum systems, when interacting with their environments, can exhibit complex non-equilibrium states that are tempting to be interpreted as quantum versions of chaotic attractors. Here we propose an approach to open cavity dynamics based on the unraveling of the corresponding master equation into an ensemble of quantum trajectories. By using the concept of ``quantum Lyapunov exponents'' [I. I. Yusipov et al., arxiv: 1806.09295], we demonstrate that `chaotic' and `regular' regimes of the intra-cavity dynamics can be identified. The chaotic regimes are marked by the emergence of power-law intermediate asymptotics in the distribution of photon waiting times. The photon counting statistics can be retrieved by monitoring photon emission in experiment. Therefore, chaotic regimes can be identified without additional measurements (and thus disturbance) of the intra-cavity dynamics.

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Quantum systems, when interacting with their environments, can exhibit complex non-equilibrium states that are tempting to be interpreted as quantum versions of chaotic attractors. Here we propose an approach to open cavity dynamics based on the unraveling of the corresponding master equation into an ensemble of quantum trajectories. By using the concept of ``quantum Lyapunov exponents'' [I. I. Yusipov et al., arxiv: 1806.09295], we demonstrate that chaotic' and regular' regimes of the intra-cavity dynamics can be identified. The chaotic regimes are marked by the emergence of power-law intermediate asymptotics in the distribution of photon waiting times. The photon counting statistics can be retrieved by monitoring photon emission in experiment. Therefore, chaotic regimes can be identified without additional measurements (and thus disturbance) of the intra-cavity dynamics.
Open quantum systems can exhibit complex states, for which classification and quantification are still not well resolved. The Kerr-nonlinear cavity, periodically modulated in time by coherent pumping of the intracavity photonic mode, is one of the examples. Unraveling the corresponding Markovian master equation into an ensemble of quantum trajectories and employing the recently proposed calculation of quantum Lyapunov exponents [I. I. Yusipov et al., Chaos 29, 063130 (2019)], we identify "chaotic" and "regular" regimes there. In particular, we show that chaotic regimes manifest an intermediate power-law asymptotics in the distribution of photon waiting times. This distribution can be retrieved by monitoring photon emission with a single-photon detector so that chaotic and regular states can be discriminated without disturbing the intracavity dynamics.
Open quantum systems can exhibit complex states, for which classification and quantification are still not well resolved. The Kerr-nonlinear cavity, periodically modulated in time by coherent pumping of the intracavity photonic mode, is one of the examples. Unraveling the corresponding Markovian master equation into an ensemble of quantum trajectories and employing the recently proposed calculation of quantum Lyapunov exponents [I. I. Yusipov et al., Chaos 29, 063130 (2019)], we identify “chaotic” and “regular” regimes there. In particular, we show that chaotic regimes manifest an intermediate power-law asymptotics in the distribution of photon waiting times. This distribution can\nbe retrieved by monitoring photon emission with a single-photon detector so that chaotic and regular states can be discriminated without disturbing the intracavity dynamics.
Quantum systems, when interacting with their environments, may exhibit nonequilibrium states that are tempting to be interpreted as quantum analogs of chaotic attractors. However, different from the Hamiltonian case, the toolbox for quantifying dissipative quantum chaos remains limited. In particular, quantum generalizations of Lyapunov exponents, the main quantifiers of classical chaos, are established only within the framework of continuous measurements. We propose an alternative generalization based on the unraveling of quantum master equation into an ensemble of "quantum trajectories," by using the so-called Monte Carlo wave-function method. We illustrate the idea with a periodically modulated open quantum dimer and demonstrate that the transition to quantum chaos matches the period-doubling route to chaos in the corresponding mean-field system.
The agenda of dissipative quantum chaos is to create a toolbox that would allow us to categorize open quantum systems into "chaotic" and "regular" ones. Two approaches to this categorization have been proposed recently. One of them is based on the spectral properties of generators of open quantum evolution. The other one utilizes the concept of Lyapunov exponents to analyze quantum trajectories obtained by unraveling this evolution. By using two quantum models, we relate the two approaches and try to understand whether there is an agreement between the corresponding categorizations. Our answer is affirmative.
When applied to dynamical systems, both classical and quantum, time periodic modulations can produce complex non-equilibrium states which are often termed "chaotic." Being well understood within the unitary Hamiltonian framework, this phenomenon is less explored in open quantum systems. Here, we consider quantum chaotic states emerging in a leaky cavity when the intracavity photonic mode is coherently pumped with the pumping intensity varying periodically in time. We show that a single spin when placed inside the cavity and coupled to the mode can moderate transitions between regular and chaotic regimes-that are identified by using quantum Lyapunov exponents or features of photon emission statistics-and thus can be used to control the degree of chaos.
Quantum chaos, a phenomenon that began to be studied in the last century, still does not have a rigorous understanding. By virtue of the correspondence principle, the properties of the system that lead to chaotic dynamics at the classical level must also be present in the underlying quantum system. In the classical case, the exponential divergence of nearby trajectories in time is described in terms of the Lyapunov exponent. However, in the quantum case, a similar description of chaos is, strictly speaking, impossible due to the absence of trajectories. There are different approaches to remedy this situation, but the universal criterion of quantum chaos is absent. We propose a quantum chaos definition in a manner similar to the classical one using decoherent histories as a quantum analog of trajectories. For this purpose, we consider the model of an open quantum kicked top interacting with the environment, which is a bosonic bath, and illustrate this idea. Here, the environment plays the role of a trajectory-recording device. For the kicked-top model at the classical level, depending on the kick strength, crossover occurs between the integrable and chaotic regimes. We show that for such a model, the production of entropy of decoherent histories is radically different in integrable and chaotic regimes. Thus, the entropy of an ensemble of quantum trajectories can be used as a signature of quantum chaos.