Exact solution of a time-dependent quantum harmonic oscillator with two frequency jumps via the Lewis-Riesenfeld dynamical invariant method

TL;DR

This paper derives exact solutions for a quantum harmonic oscillator experiencing two sudden frequency jumps, using the Lewis-Riesenfeld invariant method, revealing energy and state transition behaviors.

Contribution

It provides explicit formulas for energy, excitation number, and transition probabilities for a quantum oscillator with two frequency jumps, extending previous studies to non-fundamental initial states.

Findings

Mean energy after jumps is greater or equal to before

Oscillator can return to initial state for specific time intervals

Results apply to initial states beyond the fundamental

Abstract

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 $\omega_0$, that undergoes a sudden jump to a frequency $\omega_1$ 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 $\omega_1<\omega_0$. We also show that, for particular values of the time interval between the jumps, the oscillator returns to the same initial state.