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Journal of the Korean Physical Society

Classical and Quantum Oscillator Model of an Atom in a Classical Field with Time-Dependent Damping

Kyu-Hwang Yeon, Seok Seong Kim, Suc-Kyoung Hong

Published Aug 14, 2007• 0 citations
Publisher
DOI
10.3938/jkps.51.599

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A variety of dynamics in this world can be approximately treated by a model: a driven and damped parametric oscillator. An intensive investigate of this time-dependent model from both classical and quantum mechanical points of view provides a dynamical insight to resolve the classical and quantum dynamics in order to understand the more complicated time-dependent evolutions not only happens in the macroscopic classical scale for the synchronized behaviors but also in the coherent microscopic scale for a quantum state control. By using a Lie transformation on a general time-dependent quadratic Hamiltonian, we exactly solve the dynamical behaviors of the driven and damped parametric oscillator to give a universal physical mechanism for the classical parametric control on the quantum state evolution in a dissipation system.
The vacuum-field Rabi splitting of an optical mode in a cavity interacting with a system of N excitations (atomic or electronic) represented by harmonic oscillators is obtained quantum mechanically both without and with coupling to dissipation. The optical lineshapes and the scaling of the Rabi splitting with the number of oscillators and its dependence on the damping parameters are given. Corresponding results are given for a classical treatment of the oscillators, and it is shown that to the leading order in the dissipation, the quantum mechanical results for absorption are the same as the classical ones.
In this paper we use the algebraic and the invariant method to study the time-dependent damped harmonic oscillator from classical and quantum points of view. The solution of the classical equation of motion and the wavefunction solving the time-dependent Schrödinger equation are found explicitly. We show that the original time-dependent quantum-mechanical problem is completely related to the well known time-independent harmonic oscillator. In addition, we elucidate the intimate connection between the damped harmonic oscillator (DHO) and the generalized harmonic oscillator (GHO). More importantly, the evolution of the states of the DHO cannot be cyclic, in contradistinction with the states of the GHO. Explicit expressions for both the dynamical and the geometric angles and phases are deduced in the adiabatic limit. The coherent states describing the invariant-angle variables of the classical DHO are constructed; they allow us to recover the classical evolution and invariant from the quantum evolution.