Quantum oscillator with fluctuating time-dependent frequency

The dynamics of the harmonic oscillator, with a fluctuating, time-dependent frequency, is studied in the quantum framework. For periodic fluctuations of period tau, it is found that the mean energy diverges exponentially in time, provided tau falls into certain "bands," centered around "quasiresonant" values, for which the rate of increase is maximum. In agreement with the correspondence principle, this rate is shown to coincide exactly with the one obtained for the classic oscillator, in the same conditions [Loris Ferrari, Phys. Rev. B 56, 593 (1997)]. The notion of adiabaticity has been revisited in the case of exponentially diverging mean energy. It is found that the Heisenberg form of the Hamiltonian operator can be expressed as a new effective Hamiltonian, which is still harmonic, but has exponentially diverging coefficients. It is argued that the same results hold in general (that is, without any quasiresonance condition) if the frequency fluctuation in time is disordered. A brief discussion is presented about nonharmonic interaction's leading to dissipative effects. Applications to real physical systems are outlined, with special reference to the electrons in a uniform and constant magnetic field, to which a small fluctuating component is added.