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Harmonic oscillators with multiple abrupt jumps in their frequencies havebeen investigated by several authors during the last decades. We investigatethe dynamics of a quantum harmonic oscillator with initial frequency$\omega_0$, that undergoes a sudden jump to a frequency $\omega_1$ and, after acertain time interval, suddenly returns to its initial frequency. Using theLewis-Riesenfeld method of dynamical invariants, we present expressions for themean energy value, the mean number of excitations, and the transitionprobabilities, considering the initial state different from the fundamental. Weshow that the mean energy of the oscillator, after the jumps, is equal orgreater than the one before the jumps, even when $\omega_1<\omega_0$. We alsoshow that, for particular values of the time interval between the jumps, theoscillator returns to the same initial state.

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