Stochastic and parametric resonance in quantum dots

We study the dynamics of electrons of a quantum dot interacting with a homogeneous time-dependent magnetic field together with the corresponding induced electric field. By assuming a parabolic dot confining potential, we analyze the cases of harmonically oscillating and time-random magnetic fields. In both cases, we solve the semiclassical equations of motion and also construct the quantum-mechanical propagator corresponding to the associated time-dependent Schrödinger equation. In the former case, we find that, depending on the strength of the dot potential and on the parameters characterizing the external magnetic field (amplitude and frequency), the electrons develop a series of dynamical instabilities alternating with regions of bounded motion. In the unstable cases, the electrons absorb energy steadily from the driving electric field (parametric resonance). In the case of the random magnetic field, we consider a Markoff stochastic process with a short autocorrelation time. In this example, it is shown that the electron dynamics is always unstable and the electrons’ mean energy grows exponentially in time with a characteristic time scale determined by the Fourier transform of the field correlation function, and corresponds to stochastic resonance. We analyze the collective behavior of a system of noninteracting dots and derive an expression of direct physical relevance such as energy and magnetization. It is shown that all our conclusions are not affected by the inclusion of the electron-electron coulomb interaction (Kohn theorem).