We study the dynamics of a nonlinear oscillator near the critical point where period-two vibrations are first excited with the increasing amplitude of parametric driving. Above the threshold, quantum fluctuations induce transitions between the period
-two states over the quasienergy barrier. We find the effective quantum activation energies for such transitions and their scaling with the difference of the driving amplitude from its critical value. We also find the scaling of the fluctuation correlation time with the quantum noise parameters in the critical region near the threshold.
We investigate a resonantly modulated harmonic mode, dispersively coupled to a nonequilibrium few-level quantum system. We focus on the regime where the relaxation rate of the system greatly exceeds that of the mode, and develop a quantum adiabatic a
pproach for analyzing the dynamics. Semiclassically, the dispersive coupling leads to a mutual tuning of the mode and system into and out of resonance with their modulating fields, leading to multistability. In the important case where the system has two energy levels and is excited near resonance, the compound system can have up to three metastable states. Nonadiabatic quantum fluctuations associated with spontaneous transitions in the few-level system lead to switching between the metastable states. We provide parameter estimates for currently available systems.
We show how frequency fluctuations of a vibrational mode can be separated from other sources of phase noise. The method is based on the analysis of the time dependence of the complex amplitude of forced vibrations. The moments of the complex amplitud
e sensitively depend on the frequency noise statistics and its power spectrum. The analysis applies to classical and to quantum vibrations.
We study resonant response of an underdamped nanomechanical resonator with fluctuating frequency. The fluctuations are due to diffusion of molecules or microparticles along the resonator. They lead to broadening and change of shape of the oscillator
spectrum. The spectrum is found for the diffusion confined to a small part of the resonator and where it occurs along the whole nanobeam. The analysis is based on extending to the continuous limit, and appropriately modifying, the method of interfering partial spectra. We establish the conditions of applicability of the fluctuation-dissipation relations between the susceptibility and the power spectrum. We also find where the effect of frequency fluctuations can be described by a convolution of the spectra without these fluctuations and with them as the only source of the spectral broadening.
We investigate the relaxation of a superconducting qubit for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The qubit relaxation rates are different in diff
erent states. They can display strong dependence on the qubit frequency and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven oscillator changes the effective temperature of the qubit.
Dynamics of a system that performs a large fluctuation to a given state is essentially deterministic: the distribution of fluctuational paths peaks sharply at a certain optimal path along which the system is most likely to move. For the general case
of a system driven by colored Gaussian noise, we provide a formulation of the variational problem for optimal paths. We also consider the prehistory problem, which makes it possible to analyze the shape of the distribution of fluctuational paths that arrive at a given state. We obtain, and solve in the limiting case, a set of linear equations for the characteristic width of this distribution.
