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Physical Review A

Time evolution of a charged oscillator with a time-dependent mass and frequency in a time-dependent electromagnetic field

Chi‐Fai Lo

Published Apr 1, 1992• 39 citations
Publisher

Abstract

In this paper we investigate the time evolution of a charged oscillator with a time-dependent mass and frequency in a time-dependent electromagnetic field using the Lie algebraic technique. We obtain an exact form of the time evolution operator which, in turn, enables us to find the exact wave functions and coherent states of the system at any time t. Our analyses indicate that the wave function of the system starts as a coherent state at t=0 and evolves as a squeezed state at a later time.

DOI
10.1103/physreva.45.5262

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In this paper we investigate the time evolution of a charged oscillator with a time-dependent mass and frequency in a time-dependent electromagnetic field using the Lie algebraic technique. We obtain an exact form of the time evolution operator which, in turn, enables us to find the exact wave functions and coherent states of the system at any time t. Our analyses indicate that the wave function of the system starts as a coherent state at t=0 and evolves as a squeezed state at a later time.
The analytic expression of the time evolution wavefunction of the two-dimensional harmonic oscillator with time-dependent mass and frequency in a static magnetic field is obtained using an operator-algebraic method slightly different from the usual Lie algebraic technique. The evolution operator of the one-dimensional harmonic oscillator with time-dependent mass and frequency is established first by forming an operator differential equation with the su (1, 1) Lie algebra, which is deduced from the time-dependent linear unitary transformation for boson operators ( a , a †), and then by comparing this operator equation with the time evolution equation of the one-dimensional oscillator.
In this paper we investigate the time evolution of a general time-dependent harmonic oscillator (TDHO) with variable mass using Feynman path integral approach. We explicitly evaluate the squeezing in the quadrature components of a general quantum TDHO with variable mass. This calculation is further elaborated for three particular cases of variable mass whose propagator can be written in a closed form. We also obtain an exact form of the time-evolution operator, the wave function, and the time-dependent coherent state for the TDHO. Our results clearly indicate that the time-dependent coherent state is equivalent to the squeezed coherent state.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
We study the temporal evolution of a coherent state under the action of a parametric oscillator and a nonlinear Kerr-like medium. We make use of the interaction-picture representation and use an exact time-evolution operator for the time-independent part of the Hamiltonian. We approximate the interaction-picture Hamiltonian in such a way as to make it a member of a Lie algebra. The corresponding time-evolution operator behaves like a squeezing operator due to the temporal dependence of the oscillator’s frequency. We analyze the probability amplitude, the auto-correlation function, and the Husimi distribution function for different Hamiltonian parameters. We find very good agreement between our approximate results and converged numerical calculations.