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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Transition to deterministic chaos in a periodically driven quantum system and breaking of the time-reversal symmetry

Uwe Schwengelbeck, Farhad H. M. Faisal

Published May 1, 1997• 8 citations
Publisher

Abstract

The phenomenon of transition from regular to chaotic dynamics in a periodically driven quantum system is demonstrated. The associated quantum Lyapunov numbers are determined analytically. A numerical experiment is made to test the nature of time evolution predicted by the theory. In contrast to the evolution in the regular domain, the passage to deterministic quantum chaos is found to break the time-reversal symmetry of the quantum dynamics, whenever the latter cannot be followed with infinite precision.

DOI
10.1103/physreve.55.6260

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The phenomenon of transition from regular to chaotic dynamics in a periodically driven quantum system is demonstrated. The associated quantum Lyapunov numbers are determined analytically. A numerical experiment is made to test the nature of time evolution predicted by the theory. In contrast to the evolution in the regular domain, the passage to deterministic quantum chaos is found to break the time-reversal symmetry of the quantum dynamics, whenever the latter cannot be followed with infinite precision.
We study the time evolution of driven quantum systems using analytic, algebraic, and numerical methods. First, we obtain analytic solutions for driven free and oscillator systems by shifting the coordinate and phase of the undriven wave function. We also factorize the quantum evolution operator using the generators of the Lie algebra comprising the Hamiltonian. We obtain coupled ODE's for the time evolution of the Lie algebra parameters. These parameters allow us to find physical properties of oscillator dynamics. In particular we find phase-space trajectories and transition probabilities. We then search for chaotic behavior in the Lie algebra parameters as a signature for dynamical chaos in the quantum system. We plot the trajectories, transition probabilities, and Lyapunov exponents for a wide range of the following physical parameters: strength and duration of the driving force, frequency difference, and anharmonicity of the oscillator. We identify conditions for the appearance of chaos in the system.
We propose a new diagnostic for quantum chaos. We show that the time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behavior.
In the context of quantum chaos, both theory and numerical analysis predict large fluctuations of the tunneling transition probabilities when irregular dynamics is present at the classical level. Here we consider the nondissipative quantum evolution of cold atoms trapped in a time-dependent modulated periodic potential generated by two laser beams. We give some precise guidelines for the observation of chaos-assisted tunneling between invariant phase space structures paired by time-reversal symmetry.
Dynamical chaos in a periodically driven, dissipative soft impact oscillator is investigated in the quantum regime using the complex-number quantum Langevin equation (c-number QLE). The averaged system dynamics are analyzed through a comprehensive suite of time-series diagnostics, including bifurcation diagrams, Lyapunov exponents, Fourier spectra, and the 0-1 test. Systematic variation of the wall position reveals a rich sequence of dynamical transitions and grazing bifurcations, progressing from periodic to multiperiodic motion and culminating in chaotic behavior. These results demonstrate the persistence of impact-induced chaos under quantum dissipation and elucidate how environmental fluctuations influence non-linear dynamics in open quantum systems.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.