Response of parametrically-driven nonlinear coupled oscillators with application to micro- and nanomechanical resonator arrays

The response of a coupled array of nonlinear oscillators to parametric excitation is calculated in the weak nonlinear limit using secular perturbation theory. Exact results for small arrays of oscillators are used to guide the analysis of the numerical integration of the model equations of motion for large arrays. The results provide a qualitative explanation for a recent experiment [Buks and Roukes, cond-mat/0008211, to appear in J. MEMS (2002)] involving a parametrically-excited micromechanical resonator array. Future experiments are suggested that could provide quantitative tests of the theoretical predictions.Comment: 27 pages (in preprint format), 8 figure

Dephasing due to Intermode Coupling in Superconducting Stripline Resonators

The nonlinearity exhibited by the kinetic inductance of a superconducting stripline couples stripline resonator modes together in a manner suitable for quantum non-demolition measurement of the number of photons in a given resonator mode. Quantum non-demolition measurement is accomplished by coherently driving another resonator mode, referred to as the detector mode, and measuring its response. We show that the sensitivity of such a detection scheme is directly related to the dephasing rate induced by such an intermode coupling. We show that high sensitivity is expected when the detector mode is driven into the nonlinear regime and operated close to a point where critical slowing down occurs

Quantum Analysis of a Linear dc SQUID Mechanical Displacement Detector

We provide a quantum analysis of a dc SQUID mechanical displacement detector within the subcritical Josephson current regime. A segment of the SQUID loop forms the mechanical resonator and motion of the latter is transduced inductively through changes in the flux threading the loop. Expressions are derived for the detector signal response and noise, which are used to evaluate the position and force detection sensitivity. We also investigate cooling of the mechanical resonator due to detector back reaction

Nonlinear resonance in a three-terminal carbon nanotube resonator

The RF-response of a three-terminal carbon nanotube resonator coupled to RF-transmission lines is studied by means of perturbation theory and direct numerical integration. We find three distinct oscillatory regimes, including one regime capable of exhibiting very large hysteresis loops in the frequency response. Considering a purely capacitive transduction, we derive a set of algebraic equations which can be used to find the output power (S-parameters) for a device connected to transmission lines with characteristic impedance $Z_0$.Comment: 16 pages, 8 figure

Quantum analysis of a nonlinear microwave cavity-embedded dc SQUID displacement detector

We carry out a quantum analysis of a dc SQUID mechanical displacement detector, comprising a SQUID with mechanically compliant loop segment, which is embedded in a microwave transmission line resonator. The SQUID is approximated as a nonlinear, current dependent inductance, inducing an external flux tunable, nonlinear Duffing self-interaction term in the microwave resonator mode equation. Motion of the compliant SQUID loop segment is transduced inductively through changes in the external flux threading SQUID loop, giving a ponderomotive, radiation pressure type coupling between the microwave and mechanical resonator modes. Expressions are derived for the detector signal response and noise, and it is found that a soft-spring Duffing self-interaction enables a closer approach to the displacement detection standard quantum limit, as well as cooling closer to the ground state

Phonon number quantum jumps in an optomechanical system

We describe an optomechanical system in which the mean phonon number of a single mechanical mode conditionally displaces the amplitude of the optical field. Using homodyne detection of the output field we establish the conditions under which phonon number quantum jumps can be inferred from the measurement record: both the cavity damping rate and the measurement rate of the phonon number must be much greater than the thermalization rate of the mechanical mode. We present simulations of the conditional dynamics of the measured system using the stochastic master equation. In the good-measurement limit, the conditional evolution of the mean phonon number shows quantum jumps as phonons enter and exit the mechanical resonator via the bath.Comment: 13 pages, 4 figures. minor revisions since first versio