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International Journal of Quantum Chemistry

On the dynamics of a linear and a nonlinear quantum oscillator with randomly changing harmonic frequency

Pranab Sarkar, S. P. Bhattacharyya

Published Jan 1, 1997• 1 citation
Publisher

Abstract

Numerical experiments with a nonlinear (λχ4) oscillator which has its harmonic frequency changing randomly with time reveal certain interesting features of its dynamics of quantum evolution. When λ = 0, the level populations are seen to oscillate. But, as the nonlinear coupling is switched on (λ > 0), a threshold is reached at λ = λc when the evolution is seen to be characterized by an abrupt transition dominantly to the highest available state of the unperturbed (initial) oscillator. It is shown that this transition probability is maximized at a particular value of λ. The time threshold for this transition decreases with increasing nonlinear coupling strength. The numerically obtained structures of the underlying quantum-phase spaces of the linear and nonlinear random oscillators are examined. Possible use of these results in a problem of chemical origin is explored. © 1997 John Wiley & Sons, Inc.

DOI
10.1002/(sici)1097-461x(1997)62:3<265::aid-qua4>3.0.co;2-u

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We consider a system of linear oscillators, or quantum states, described by random matrix theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values, and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Owing to universal features of random matrix theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.
The temporal evolution of the state vector relative to a harmonic oscillator with time-dependent frequency is examined. The Schrödinger equation is solved by choosing the instantaneous eigenstates of the Hamiltonian as the basis, thus getting an infinite set of coupled linear differential equations. This formulation is particularly suitable for studying the cases in which the Hamiltonian undergoes a very slow or a sudden variation from an initial constant value into a final one. A rigorous proof of adiabatic invariance to all orders in the slowness parameter is given for the transition probabilities. An application to the evolution of an initially coherent state is made.
Harmonic oscillators with multiple abrupt jumps in their frequencies have been investigated by several authors during the last decades. We investigate the dynamics of a quantum harmonic oscillator with initial frequency ω0, which undergoes a sudden jump to a frequency ω1 and, after a certain time interval, suddenly returns to its initial frequency. Using the Lewis−Riesenfeld method of dynamical invariants, we present expressions for the mean energy value, the mean number of excitations, and the transition probabilities, considering the initial state different from the fundamental. We show that the mean energy of the oscillator, after the jumps, is equal or greater than the one before the jumps, even when ω1<ω0. We also show that, for particular values of the time interval between the jumps, the oscillator returns to the same initial state.
Abstract The quantum dynamics of the evolution of a system of a few coupled nonlinear oscillators is studied. The spectral entropy and the quantal information entropy, as well as their correlation functions, are computed and used to identify distinctive phases of the evolution. The two entropy functions lead to similar conclusions. It appears that adiabatic switching of the nonlinear interaction does not affect the nature of the short time dynamics but affects the dynamics over intermediate time scales. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
Abstract In this work, we give a quantitative answer to the question: how sudden or how adiabatic is a frequency change in a quantum harmonic oscillator (HO)? We do that by studying the time evolution of a HO which is initially in its fundamental state and whose time-dependent frequency is controlled by a parameter (denoted by ϵ ) that can continuously tune from a totally slow process to a completely abrupt one. We extend a solution based on algebraic methods introduced recently in the literature that is very suited for numerical implementations, from the basis that diagonalizes the initial Hamiltonian to the one that diagonalizes the instantaneous Hamiltonian. Our results are in agreement with the adiabatic theorem and the comparison of the descriptions using the different bases together with the proper interpretation of this theorem allows us to clarify a common inaccuracy present in the literature. More importantly, we obtain a simple expression that relates squeezing to the transition rate and the initial and final frequencies, from which we calculate the adiabatic limit of the transition. Analysis of these results reveals a significant difference in squeezing production between enhancing or diminishing the frequency of a HO in a non-sudden way.
We investigate the entanglement dynamics of two harmonic oscillators with a weak time-dependent coupling. We find strong connections between the dynamical behaviors of the entanglement and the stability of the normal modes. An abrupt transition of the entanglement dynamics from periodic and bounded to monotonically increasing occurs, when one of the normal modes becomes dynamically unstable as the modulation amplitude crosses the critical value. In particular, when the modulation frequency is roughly twice the oscillator frequency, an explicit critical value for the modulation strength is derived to specify the transition. Moreover, it is found that one can significantly change the dynamical behavior of the entanglement by adding an extremely weak time modulation with specific frequency to a constant coupling.