Pulse excitation of a harmonic oscillator: dependence on the parameters of an exciting force

We study the dependence of the excitation energy of a harmonic oscillator with damping under the action of a pulse of a periodic force over the entire duration of pulse on its duration, carrier frequency, and shape of the envelope. Pulses with exponential and Gaussian envelope shapes are considered. Simple analytical expressions describing this process are obtained. It is established that in the case of an exponential pulse, the dependence of the excitation energy on the pulse duration can have an inflection point, while in the case of the Gaussian pulse, extrema can occur at a sufficiently large detuning of the carrier frequency of the pulse from the eigenfrequency of the oscillator. It is shown that when a quantum oscillator is excited, the average number of quanta and the amplitude of its coordinate averaged from the operator are determined by the same dependences using the parameters of the exciting pulse as in the case of a classical oscillator.