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Chinese Physics Letters

Dynamics of Coupled Quantum–Classical Oscillators

He Wei-Zhong, Xu Liu-Su, Zou Feng-wu

Published Jul 30, 2004• 1 citation
Publisher

Abstract

The dynamics of systems consisting of coupled quantum–classical oscillators is numerically investigated. It is shown that, under certain conditions, the quantum oscillator exhibits chaos. When the mass of the classical oscillator increases, the chaos will be suppressed; if the energy of the system and/or the coupling strength between the two oscillators increases, chaotic behaviour of the system appears. This result will be helpful to understand the probability of the emergence of quantum chaos and may be applied to explain the spectra of complex atoms qualitatively.

DOI
10.1088/0256-307x/21/8/006

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The dynamics of systems consisting of coupled quantum–classical oscillators is numerically investigated. It is shown that, under certain conditions, the quantum oscillator exhibits chaos. When the mass of the classical oscillator increases, the chaos will be suppressed; if the energy of the system and/or the coupling strength between the two oscillators increases, chaotic behaviour of the system appears. This result will be helpful to understand the probability of the emergence of quantum chaos and may be applied to explain the spectra of complex atoms qualitatively.
Abstract We investigate the chaotic dynamics of a coupled three-level atom–molecular Bose–Einstein condensate system composed by one molecular mode and two atomic modes. With the increase of atom–molecular coupling strength, we reveal the emergence of chaotic oscillations of the relative population difference between two atomic modes, which can be proven by the broad windows with a huge number of frequencies in spectral density and the chaotic trajectories in phase space diagrams. The different effects of initial states on atomic population oscillations are revealed, where for more particles in the initial state of the molecular model, chaos appears in the larger parameter region of system dynamics. Furthermore, we find that strong intermolecular interaction strength can suppress chaos resulting from strong atom–molecule coupling. This is due to the difficult transformation between atomic and molecular modes, as well as the relatively independent dynamic evolution of atoms and molecules.
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.
Quantum domain behaviour of classically chaotic systems is studied using the quantum theory of motion in teh sense of clasical interpretationof quantum mechanics as developed by de Broglie and Bohm. Dynamics of quantum Henon- Heiles oscillatorm barbanis oxcillatior and CTW oscillator are ananysed with the help of quantum Lypunov exponent and Kolmogorov-Sinai entropy defined in terms of the distance between two initially close Bohmian trajectories. Standard diagnostics of quantum chaos like auticorrelation function and the assocated power spectrum, nearest neighbour spacing distribution phase space volume, spectral rigidity, etc. support these results. Quantum theory of motion provides an alternative route for understanding quantum chaos. Nonlinear dynamics of integrable systems in quantum domain is also properly taken care of within this framework.
A Kerr-nonlinear parametric oscillator (KPO) can generate a quantum superposition of two oscillating states, known as a Schr\"odinger cat state, via quantum adiabatic evolution and can be used as a qubit for gate-based quantum computing and quantum annealing. In this work, we investigate complex dynamics, i.e., chaos, in two coupled nondissipative KPOs at a few-photon level. After showing that a classical model for this system is nonintegrable and consequently exhibits chaotic behavior, we provide quantum counterparts for the classical results, which are quantum versions of the Poincar\'e surface of section and its lower-dimensional version defined with time integrals of the Wigner and Husimi functions and also the initial and long-term behavior of out-of-time-ordered correlators. We conclude that some of them can be regarded as quantum signatures of chaos, together with energy-level spacing statistics (conventional signature). Thus, the system of coupled KPOs is expected to offer not only an alternative approach to quantum computing but also a promising platform for the study on quantum chaos.
A semiquantum system, composed of a quantum part coupled to a classical part, can exhibit dynamical chaos in the motion of the quantum state vector. However, there has been disagreement as to whether this mathematical chaos is physically real or merely an artifact of the semiquantum approximation. It is shown, for a model of a quantum spin coupled to an approximately classical nonlinear oscillator, that the semiquantum chaos disappears rapidly as the mass of the oscillator is increased to make it more classical. The time interval during which the semiquantum approximation remains accurate increases with the mass, but is not closely related to the Lyapunov time.
In this thesis, a comparison between quantum and classical mechanics was made for two systems whose classical behaviour can be chaotic.The general correspondence between classical mechanics and quantum mechanics in the classical regime has always been assumed but never proved.Recent research in the field of quantum chaos has brought this assumption into question.For some systems, such as the kicked rotor in the classically chaotic regime, quantum mechanics does not display the classical behaviour of unbounded momentum.It has been suggested that quantum mechanics will fail to produce classical chaos for a broad range of simple systems.A comparison of quantum and classical mechanics was made using numerical simulations, without recourse to taking limits such as 7i going to zero.The two systems studied were a driven pendulum and a pair of pendula joined by a spring.The quantum and classical expectation values for a number of observables were computed, as were the probability distributions in phase space.For periodic motion of the driven pendulum it was found that the theories agreed well for times up to 150 driving periods.Similar agreement was found for initial conditions whose classical behaviour was chaotic, but only for times up to a few tens of driving periods.For longer times, it was found that the quantum system displayed a boundedness in position and/or momentum that was absent in the classical system.A possible explanation was given in terms of the quasienergy expansion.In the coupled-pendula system no such disagreements between the theories were found.I would like to thank Dr. Ballentine for his continued support during this project and his help in editing and revising this thesis.I would like to thank the members of my committee, Drs.Bechhoefer, Boa1 and Loss for their many suggestions.For many helpful discussions, I would like to thank Dominic Mimnagh and Jim Zibin.