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Journal of Optics B Quantum and Semiclassical Optics

Quantum coupled oscillators versus forced oscillator

Antonio S. M. de Castro, V. V. Dodonov

Published Jul 9, 2001• 6 citations
Publisher

Abstract

We compare the time evolutions of two quantum systems: a quantum harmonic oscillator driven by a classical force and two coupled quantum harmonic oscillators. We find the conditions under which the first system can be considered as a `mean-field approximation' to the second one. This happens in the weak coupling limit provided that the `master' oscillator was initially in a highly excited quantum state with small fluctuations of the canonical variables (such as the coherent state or displaced thermal state), while the `subordinate' oscillator must not be `too excited'. The essential difference between the resonance and nonresonance cases is discussed. In particular, the effect of the resonance-quantum-state exchange is demonstrated.

DOI
10.1088/1464-4266/3/4/306

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We compare the time evolutions of two quantum systems: a quantum harmonic oscillator driven by a classical force and two coupled quantum harmonic oscillators. We find the conditions under which the first system can be considered as a mean-field approximation' to the second one. This happens in the weak coupling limit provided that the master' oscillator was initially in a highly excited quantum state with small fluctuations of the canonical variables (such as the coherent state or displaced thermal state), while the subordinate' oscillator must not be too excited'. The essential difference between the resonance and nonresonance cases is discussed. In particular, the effect of the resonance-quantum-state exchange is demonstrated.
We consider a forced harmonic oscillator in one-dimension. Using coherent states, we show that the treatment of the system is simplified, that the relationship between the classical and quantum solutions becomes transparent, and that the evolution operator of the system can be calculated easily as the free evolution operator of the harmonic oscillator followed by a displacement operator that depends on the classical solution. In addition, we consider the system in the rotating-wave-approximation (RWA), an application of the Averaging Theorem. We determine the relationship between the exact solution and the one in the RWA, test the accuracy of the RWA, and explain why the RWA gives accurate results in the realm of cavity quantum electrodynamics. Finally, we apply the results to a charged particle interacting with an electromagnetic field.
We investigate the exact dynamics of the damped quantum harmonic oscillator under the (un)correlated initial conditions. The master equation is generalized to the cases of the arbitrary factorized state and/or Gaussian state. We show that the variances of the factorized Gaussian state do not sensitively depend on the initial oscillator-bath correlation, which however can remarkably affect the mean values even at high temperature. We also illustrate that the correlations among the factorized states still give rise to the initial dips during the purity evolutions, which can be smoothed out by increasing the amount of correlation to some extent. We finally study the effects of repeated measurements on the time evolution of the damped oscillator analytically, which are compared with the weak coupling results to indicate that they give rather different transient behaviors even for an intermediate coupling.
Recent experiments on quantum behavior in microfabricated solid-state systems suggest tantalizing connections to quantum optics. Several of these experiments address the prototypical problem of cavity quantum electrodynamics: a two-level system coupled to a quantum harmonic oscillator. Such devices may allow the exploration of parameter regimes outside the near-resonance and weak-coupling assumptions of the ubiquitous rotating-wave approximation (RWA), necessitating other theoretical approaches. One such approach is an adiabatic approximation in the limit that the oscillator frequency is much larger than the characteristic frequency of the two-level system. A derivation of the approximation is presented, together with a discussion of its applicability in a system consisting of a Cooper-pair box coupled to a nanomechanical resonator. Within this approximation the time evolution of the two-level-system occupation probability is calculated using both thermal- and coherent-state initial conditions for the oscillator, focusing particularly on collapse and revival phenomena. For thermal-state initial conditions parameter regimes are found in which collapse and revival regions may be clearly distinguished, unlike the erratic evolution of the thermal-state RWA model. Coherent-state initial conditions lead to complex behavior, which exhibits sensitive dependence on the coupling strength and the initial amplitude of the oscillator state. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the weak-coupling limit, which provides insight into the differences between the thermal- and coherent-state models. Finally, potential experimental observations in solid-state systems, particularly the Cooper-pair box---nanomechanical resonator system, are discussed and found to be promising.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the PPP representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the PPP representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.
This paper is concerned with the dynamics of coupled quantum oscillators in the presence of a classical harmonic driving force. By means of a theorem which describes the evolution of Gaussian distributions of coherent states, the solutions of the equations of motion are presented in a very compact form. Specific applications deal with forced oscillation and parametric amplification. The theoretical models include systems of damping oscillators with a continuous frequency distribution. The approach to thermal equilibrium in the absence of a driving force is discussed as a special case.