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

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.