Quantum Noise Interference and Backaction Cooling in Cavity Nanomechanics

We present a theoretical analysis of a novel cavity electromechanical system where a mechanical resonator directly modulates the damping rate $\kappa$ of a driven electromagnetic cavity. We show that via a destructive interference of quantum noise, the driven cavity can effectively act like a zero-temperature bath irrespective of the ratio $\kappa /{\omega }_M$, where ${\omega }_M$ is the mechanical frequency. This scheme thus allows one to cool the mechanical resonator to its ground state without requiring the cavity to be in the so-called good cavity limit $\kappa \ll {\omega }_M$. The system described here could be implemented directly using setups similar to those used in recent experiments in cavity electromechanics.

Figure 1

(a) Backaction force noise spectral density for $\Delta =\kappa /2$, for different ratios of the dispersive to dissipative optomechanical couplings. The solid blue curve corresponds to $\tilde{A}/\tilde{B}=10$, the long-dashed green curve to $\tilde{A}/\tilde{B}=0.75$, and the short-dashed red curve to $\tilde{A}/\tilde{B}=0$. Each curve has been scaled by its maximum value. (b) Number of oscillator quanta $n_{\mathrm{osc}}$ versus drive detuning $\Delta$, obtained by solving the full equations of motion for a purely dissipative optomechanical coupling (i.e., $\tilde{A}=0$). The dotted blue curve is the Bose-Einstein factor $n_{\mathrm{eff}}$ corresponding to the backaction effective temperature $T_{\mathrm{eff}}$ when $\kappa ={\omega }_M$ [i.e., $n_{\mathrm{eff}}=S_{FF}(-{\omega }_M)/[S_{FF}({\omega }_M)-S_{FF}(-{\omega }_M)]$]. The remaining curves correspond to $\tilde{B}\overline{a}=0.2$, $n_{\mathrm{eq}}=50$, $\gamma ={10}^{-6}{\omega }_M$, and $\kappa /{\omega }_M=0.2$ (green solid curve), $\kappa /{\omega }_M=1.0$ (purple long-dashed curve), $\kappa /{\omega }_M=5.0$ (red short-dashed curve).

Figure 2

(a) Number of mechanical quanta $n_{\mathrm{osc}}$ versus coupling strength $\tilde{B}|\overline{a}|$, for $\tilde{A}=0$, $n_{\mathrm{eq}}=50$, $\kappa /{\omega }_M=1$ and for an optimal detuning $\Delta ={\omega }_M/2$; the mechanical damping $\gamma$ is as marked. Dashed curves are results from the linear-response, quantum noise approach, whereas solid curves are obtained from the full solutions of the equations of motion. (b) Schematic of a cavity (modeled as an $LC$ resonator) coupled to a transmission line (impedance $Z_0$) via an $x$-dependent capacitance $C_1(x)$.