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.

Figure 1

Scheme for the displacement detector showing the pump-probe line $p$, transmission line resonator $T$, and dc SQUID with mechanically compliant loop segment having effective mass $m$ and fundamental frequency ${\omega }_m$. Note that the scale of the dc SQUID is exaggerated relative to that of the stripline for clarity.

Figure 2

Displacement detector noise spectral density (solid line) and lower bound (dashed line) versus drive current amplitude. The noise densities are evaluated at $\omega ={\omega }_p+R_{\omega }{\omega }_m$, corresponding to phase preserving detection.

Figure 3

Force detector noise spectral density versus drive current amplitude for detuning $\Delta \omega =0$ (solid line), $\Delta \omega =-5{\omega }_m$ (dashed line), and $\Delta \omega =-10{\omega }_m$ (dotted line). The noise densities are evaluated at $\omega ={\omega }_p+R_{\omega }{\omega }_m$, corresponding to phase preserving detection.

Figure 4

Net effective average occupation number of the mechanical oscillator versus drive current. The solid curve is for external bath temperature $T=100\mathrm{mK}$ $[n\left(R_{\omega }\omega \right)=523]$, the dashed curve is for $T=10\mathrm{mK}$ $[n\left(R_{\omega }\omega \right)=52]$, and the dotted curve is for $T=1\mathrm{mK}$ $[n\left(R_{\omega }\omega \right)=4.8]$.