Chaos and dynamical complexity in the quantum to classical transition

Abstract

We study the largest Lyapunov exponents λ and dynamical complexity for an open quantum driven double-well oscillator, mapping its dependence on coupling to the environment Γ as well as effective Planck’s constant β2. We show that in general λ increases with effective Hilbert space size (as β decreases, or the system becomes larger and closer to the classical limit). However, if the classical limit is regular, there is always a quantum system with λ greater than the classical λ, with several examples where the quantum system is chaotic even though the classical system is regular. While the quantum chaotic attractors are generally of the same family as the classical attractors, we also find quantum attractors with no classical counterpart. Contrary to the standard wisdom, the correspondence limit can thus be the most difficult to achieve for certain classically chaotic systems. These phenomena occur in experimentally accessible regimes.

Multiparameter estimation along quantum trajectories with sequential Monte Carlo methods

“…The properties of the quantum trajectories generated by the Duffing oscillator have been studied extensively in terms of the appearance of chaotic behavior from quantum systems in the classical limit [42][43][44][45][46][47][48][49][50][51][52], but it is also a model used for a number of other practical systems where quantum effects in classical nonlinear systems are of interest. For example, it has been used to describe the motion of a levitated particle in an electromagnetic trap [53,54], and is the basis for the analysis of the properties of vibrating beam accelerometers [55][56][57].…”

Observing quantum chaos with noisy measurements and highly mixed states

“…The system that we consider is a standard example from classical chaos: the Duffing oscillator. This system has been studied for pure states and efficient measurements, and it has been shown to make a transition from nonchaotic to chaotic motion as the action is increased relative to [5,6,8,[11][12][13][14]17,18]. To achieve this, one must change the mass of the oscillator, the potential, and any driving forces in such a way that the dynamics remain the same up to a scaling of the coordinates and time, while the area of the phase space increases with respect to .…”

“…In the former case, the evolution approaches that of the probability density in phase space for the equivalent classical system as the action is increased [4,7,15]. Continuous observation turns this probability density into individual trajectories that follow the nonlinear classical motion with the requisite Lyapunov exponents [12,15,17,18].…”

“…There has been a great deal of interest in purely quantum systems, displaying unitary evolution, and nonunitary open quantum systems. This paper is concerned with open quantum systems whose classical counterparts are chaotic and make a transition to chaotic behavior as their size (more precisely their action) is increased so as to be large compared to [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. This transition is enabled by their interaction with the environment or when they are subjected to continuous observation.…”

Quantum chaos in a Bose-Hubbard dimer with modulated tunneling

“…We note that an alternative solution is possible for continuously monitored quantum systems [2,10,11], where individual stochastic quantum trajectories, representing the dynamics conditioned on measurement, can be used to derive effective quantum Lyapunov exponents. In this work, we study exclusively the isolated dynamics of the two-site lattice, where such an approach based on continuous measurement readout is not available.…”