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arXiv:2011.00118 · Physical review. E

Chaos in the quantum Duffing oscillator in the semiclassical regime under parametrized dissipation

Andrew Maris, Bibek Pokharel, Sharan Ganjam Seshachallam, Moses Misplon, Arjendu K. Pattanayak

Published Aug 6, 2021• 4 citations
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Abstract

We study the quantum dissipative Duffing oscillator across a range of system sizes and environmental couplings under varying semiclassical approximations. Using spatial complexity metrics based on Kullback-Leibler distances between phase-space attractors and temporal complexity metrics based on the Lyapunov exponent, we isolate the effect of the environment on quantum-classical differences. Moreover, we quantify the system sizes where quantum dynamics cannot be simulated using semiclassical or noise-added classical approximations. We find that a remarkable parametrically invariant meta-attractor emerges at a specific length scale and noise-added classical models deviate strongly from quantum dynamics below this scale. We also generalize the previous surprising result that classically regular orbits can have the greatest quantum-classical differences in the semiclassical regime and show that the dynamical growth of quantum-classical differences is not determined by the degree of classical chaos.

DOI
10.1103/physreve.104.024206

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We study the quantum dissipative Duffing oscillator across a range of system sizes and environmental couplings under varying semiclassical approximations. Using spatial complexity metrics based on Kullback-Leibler distances between phase-space attractors and temporal complexity metrics based on the Lyapunov exponent, we isolate the effect of the environment on quantum-classical differences. Moreover, we quantify the system sizes where quantum dynamics cannot be simulated using semiclassical or noise-added classical approximations. We find that a remarkable parametrically invariant meta-attractor emerges at a specific length scale and noise-added classical models deviate strongly from quantum dynamics below this scale. We also generalize the previous surprising result that classically regular orbits can have the greatest quantum-classical differences in the semiclassical regime and show that the dynamical growth of quantum-classical differences is not determined by the degree of classical chaos.
The effect of a large environment on a finite-size quantum mechanical system is two-fold: It brings dissipation, but also fluctuations of thermal and quantum origin. While dissipation tends to stabilize the dynamics, we question if and how environmental quantum fluctuations can generate chaos in an otherwise classically non-chaotic system. We work out a paradigmatic model of quantum optics: the dissipative Dicke model, where a large spin interacts with a dissipative harmonic mode. We dial in the classical/quantum correspondence by working in the semiclassical regime at large but finite spin. We demonstrate that, starting from a classically regular phase space in the superradiant regime, quantum noise can generate a strange attractor with fractal dimension and a positive Lyapunov exponent. We unveil the deep connection with shear-induced chaos that was recently developed in the mathematical community.
In classical mechanics, driven systems with dissipation often exhibit complex, fractal dynamics known as strange attractors. This paper addresses the fundamental question of how such structures manifest in the quantum realm. We investigate the quantum Duffing oscillator, a paradigmatic chaotic system, using the Caldirola-Kanai framework, where dissipation is integrated directly into a time-dependent Hamiltonian. By employing the Husimi distribution to represent the quantum state in phase space, we present the first visualization of a quantum strange attractor within this model. Our simulations demonstrate how an initially simple Gaussian wave packet is stretched, folded, and sculpted by the interplay of chaotic dynamics and energy loss, causing it to localize onto a structure that beautifully mirrors the classical attractor. This quantum "photograph" is inherently smoothed, blurring the infinitely fine fractal details of its classical counterpart as a direct consequence of the uncertainty principle. We supplement this analysis by examining the out-of-time-ordered correlator, which shows that stronger dissipation clarifies the exponential growth associated with the classical Lyapunov exponent, thereby confirming the model's semiclassical behavior. This work offers a compelling geometric perspective on open chaotic quantum systems and sheds new light on the quantum-classical transition.
We investigate the quantum dissipative dynamics near the stable states (attractors) of a driven Duffing oscillator. A refined perturbation theory that can treat two perturbative parameters with different orders is developed to calculate the quantum properties of the Duffing oscillator near the attractors. We obtain the perturbative analytical results that go beyond the standard linearization approach for the renormalized level spacings, the orbital displacements, and the effective temperature near the classical attractor. Furthermore, we demonstrate that strong damping induces additional slight renormalization of level spacings and the Bose distribution together with dephasing. Our work provides new insights into the quantum dynamics of the driven Duffing oscillator and offers a theoretical framework that can be applied to related quantum systems near their stable states.
We study the dynamical complexity of an open quantum driven double-well oscillator, mapping its dependence on effective Planck's constant ℏeff≡β\hbar_{eff}\equivβℏeff​≡β and coupling to the environment, ΓΓΓ. We study this using stochastic Schrodinger equations, semiclassical equations, and the classical limit equation. We show that (i) the dynamical complexity initially increases with effective Hilbert space size (as βββ decreases) such that the most quantum systems are the least dynamically complex. (ii) If the classical limit is chaotic, that is the most dynamically complex (iii) if the classical limit is regular, there is always a quantum system more dynamically complex than the classical system. There are several parameter regimes where the quantum system is chaotic even though the classical limit is not. While some of the quantum chaotic attractors are of the same family as the classical limiting attractors, we also find a quantum attractor with no classical counterpart. These phenomena occur in experimentally accessible regimes.
Quantum trajectories given by the quantum state diffusion theory are used to study the dynamics of a pair of qubits interacting with the quantized Duffing oscillator. The latter is a nonlinear dissipative and driven quantum system that in the semiclassical limit displays typical bifurcations and chaotic behavior of the classical model. It is shown that the oscillator part of the qubits-oscillator system in the semiclassical limit is well described by the classical model despite the oscillator-qubit interaction. On the other hand, exact quantum dynamics of the qubits pair is completely different from the classical model as long as there is non-negligible qubit-oscillator interaction. Dynamics of the interqubits entanglement and the qubits pair von Neumann entropy is studied in the deeply quantum and various degrees of semiclassical regimes and for different values of the oscillator's bifurcation parameter. It is concluded that the decoherence of the qubits pair by the dissipative nonlinear oscillator is more effective when the oscillator is in the more classical regime, and if the semiclassical oscillator dynamics is chaotic rather than regular.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.