Classical counterparts of quantum attractors in generic dissipative systems

Chaos: An Interdisciplinary Journal of Nonlinear Science

Vershinina

Денисов

et al. 2019

Quantum systems, when interacting with their environments, may exhibit non-equilibrium 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.It is one of the pillar concepts of Chaos theory that complex deterministic dynamics is rooted in the local instability which forces two initially close trajectories to diverge. This divergence is conventionally quantified with Lyapunov exponents (LEs), a powerful tool to quantify dynamical chaos. The history of attempts to generalize LEs to quantum dynamics is nearly as old as the history of Quantum Chaos. Most of this history is about the Hamiltonian limit, where the spectral theory of Quantum Chaos [1] was established first. The corresponding generalizations range from early ideas to use quasi-probability functions and define quantum LEs in terms of a "distance" between them [2-4] to very recent advances based on out-of-time correlation functions [5][6][7]. When a quantum system is open and its dynamics is modeled with a quantum master equation [8], the evolution of the system's density operator can be unraveled into an ensemble of evolving trajectories, each one described by a wave function [8]. Dynamics of these wave functions is essentially stochastic; therefore, LEs could be introduced in a more intuitive way than in the Hamiltonian limit. But will so-defined exponents make sense? Here we define a particular type of quantum LEs and give a positive answer to this question. Since quantum trajectories [9] are not just a formal trick but a part of reality, e.g., in optical [10] and microwave [11] cavity systems, we believe that our results will be of interest to the theoreticians (and, hopefully, to the experimentalists) dealing with these systems.

Section: Introductionmentioning

confidence: 99%

Quantum Lyapunov exponents beyond continuous measurements

Chaos: An Interdisciplinary Journal of Nonlinear Science

Vershinina

Денисов

et al. 2019

“…We consider a system of N indistinguishable interacting bosons, that hop between the sites of a periodically rocked dimer. This model is a popular theoretical testbed [39][40][41], recently implemented in experiments [42][43][44], known to exhibit regular and chaotic regimes [24][25][26]30]. Its unitary dynamics is governed by the Hamiltonian…”

Section: Model and Methodsmentioning

confidence: 99%

“…To date, several quantum counterparts of dissipative bifurcations have been described: pitchfork, saddle-node, and period doubling 27–29,31–33 . This is commonly done in the Markovian framework, and the dynamics of a model system is described with Lindblad equation 21,23,34,35 .…”

Section: Introductionmentioning

confidence: 99%

“…These systems interact with their environments (or are subjected to actions from outside), and therefore their dynamics is essentially dissipative [17,18]. Although such systems evolve to a unique asymptotic state under broad conditions [19,20], the dynamics has proved to be no less complex then the unitary one, framed by a set of eigenstates, yielding the states structurally and dynamically similar to classical chaotic attractors [21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quantum Neimark-Sacker bifurcation

Ivanchenko

2019

Sci Rep

Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on “quantumtorus” and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.

“…Known as Floquet states, they are distinctly different from the quantum states exhibited by the same system in the stationary limit [7,8]. The interplay between decoherence and periodic modulation creates novel quantum attractor states, which understanding is still far from complete [10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Chaotic spin-photonic quantum states in an open periodically modulated cavity

Chaos: An Interdisciplinary Journal of Nonlinear Science

Денисов

Ivanchenko

2021

When applied to dynamical systems, both classical and quantum, time periodic modulations can produce complex nonequilibrium 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 state emerging in a leaky cavity, when an intracavity photonic mode is coherently pumped with the 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 -and thus can be used to control the degree of chaos. In an experiment, these transitions can be detected by analyzing photon emission statistics.A passage connecting Chaos Theory 1 and many-body quantum physics is provided by the mean-field ideology 2-4 and different semiclassical approximations 5,6 . They declare that, when the number N of quantum degrees of freedoms is systematically increased, the exponentially complex evolution of a quantum model can be approximated with a fixed size system of classical non-linear differential equations. These equations model the dynamics of the expectation values of relevant observables and the model becomes exact in the thermodynamic limit N → ∞. A degree of chaos in the original quantum system can be quantified by calculating standard classical quantifiers (usually maximal Lyapunov exponents 6,7 ) for the corresponding classical system. In the case of an open quantum system, the mean-field approach can be realized on the level of density matrices 8 . Alternatively, the adjoint form of a Markovian master equation, governing the evolution of the system density matrix 4 , can be employed [9][10][11] . What if we are dealing with an open model and do not want (or simply do not have a possibility) to go into the (semi)classical limit or resort to a mean-field description? When the evolution of the system is modeled with a master equations of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form 4 , a recently proposed idea of quantum Lyapunov exponents 12 provides a possibility to quantify the degree of chaos in a straightforward manner. Here we implement this idea and demonstrate how chaotic regimes of a system with N 1 states (a photonic mode in an open cavity) can be controlled by coupling it to a single spin.