Two Dimensional Isotropic Harmonic Oscillator on a Time-dependent Sphere

TL;DR

This paper studies a 2D isotropic harmonic oscillator on a dynamic spherical surface, revealing how background fluctuations induce quantum transitions similar to stimulated emission and absorption, applicable to various quantum systems.

Contribution

It introduces a model for a quantum oscillator on a fluctuating spherical background and derives transition probabilities due to background fluctuations, extending understanding of quantum interactions with dynamic geometries.

Findings

Transition probabilities depend on energy and background frequency relations.

Oscillator interacts with background as a quantum field at matching frequencies.

Results are applicable to arbitrary quantum systems on fluctuating symmetric backgrounds.

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.