Two Dimensional Isotropic Harmonic Oscillator on a Time-dependent Sphere

Abstract

In this paper, we investigate a two dimensional isotropic harmonic oscillator on a time-dependent spherical background. The effect of the background can be represented as a minimally coupled field to the oscillator's Hamiltonian. For a fluctuating background, transition probabilities per unit time are obtained. Transitions are possible if the energy eigenvalues of the oscillator $E_i$ and frequencies of the fluctuating background $\omega_n$ satisfy the following two simple relations: $E_{j}\simeq E_{i}-\hbar\omega_{n}$ (stimulated emission) or $E_{j}\simeq E_{i}+\hbar\omega_{n}$ (absorption). This indicates that a background fluctuating at a frequency of $\omega_n$ interacts with the oscillator as a quantum field of the same frequency. We believe this result is also applicable for an arbitrary quantum system defined on a fluctuating maximally symmetric background.