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

We carry out a quantum analysis of a dc superconducting quantum interference device (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.

Figure 1

Layout of the dc SQUID displacement detector. The dimensions of the SQUID have been enlarged relative to the transmission line to show the key characteristics employed in the analysis.

Figure 2

(Color online) Bistable region (shaded) of the cavity-oscillator system for negative spring-softening Duffing nonlinearity. The drive current and detuning are expressed in units of the bistability onset critical values $I_{\text{bi}}$ and $\left|\Delta {\omega }_{\text{bi}}\right|$. The labeled straight-line traces correspond to detection $\left(d\right)$ and cooling $\left(c\right)$ current drive-detuning parameter examples considered in Secs. 5, 6. The arrows give the direction in which the drive current is varied in order to enter the bistable region on the small amplitude branch.

Figure 3

(Color online) Detector noise versus signal response at $\Delta \omega =0$ for harmonic $\left(K_d=0\right)$ transmission line, Duffing $\left(K_d<0\right)$ transmission line $\left(d_1\right)$, and Caves’ bound (Ref. 42) (black-dashed line). Noise for the nonlinear transmission line is also evaluated for blue detunings: $\Delta \omega =+0.2\left|\Delta {\omega }_{\text{bi}}\right|$ $\left(d_2\right)$, and $+0.4\left|\Delta {\omega }_{\text{bi}}\right|$ $\left(d_3\right)$. The labeled curves correspond to the traces in Fig. 2. The dashed colored lines give Caves’ bound (Ref. 42) for the corresponding detuning values.

Figure 4

(Color online) Cartoon indicating the “radiation pressure” force exerted on the mechanical oscillator by the transmission line mode during one cycle of mechanical motion: (a) the harmonic transmission line mode approximation; (b) approaching the onset of bistability. The work done on the oscillator is proportional to the area swept out during each cycle, considerably exaggerated here for clarity. Positive mechanical damping (red detuning) results on the positive slope side of the curves. Negative mechanical damping (blue detuning) results on the negative slope side. A spring-softening nonlinearity can result in improved cooling for red detuning and improved signal-to-noise amplification for blue detuning.

Figure 5

(Color online) Mechanical oscillator damping renormalization factor $R_{\gamma }$ for detunings both above and below the bistable detunings $\Delta {\omega }_{\text{bi}}$. The amplification region corresponds to negative back-action damping, i.e., $R_{\gamma }<1$.

Figure 6

(Color online) (a) Detector noise effective back-action occupation number versus current drive when red detuned at $\Delta \omega =-\sqrt{{\omega }_m^2+{\gamma }_{pT}^2}$, $\left|\Delta \omega \right|<\left|\Delta {\omega }_{\text{bi}}\right|$, with a Duffing nonlinearity (solid line), without a Duffing nonlinearity (dashed line), and without both Duffing and ponderomotive nonlinearities (dotted line). (b) Oscillator coupling renormalization factor $R_{\gamma }$ for the corresponding back-action occupation number curves. These plots are obtained for the straight-line trace labeled $c_1$ in Fig. 2.

Figure 7

Transmission line resonator response curve for $Q_T=300$ restricted to the small amplitude solution branch. The example drive currents from bottom to top: $I/I_{\text{bi}}=0.8$, 0.95, 1.15, and 1.3. The jump between small (red-detuned) and large (blue-detuned) amplitude solutions is indicated by the dotted lines.

Figure 8

(Color online) (a) Detector noise effective occupation number versus current drive when red detuned at $\Delta \omega =1.3\Delta {\omega }_{\text{bi}}$, corresponding to straight-line trace $c_1^{\prime }$ in Fig. 2. The Duffing nonlinear transmission line resonator occupation number (solid line) rapidly decreases as the resonator is driven toward the upper bistable boundary, assuming the resonator can be maintained on the small amplitude metastable stable solution branch. In contrast, a harmonic transmission line resonator (dashed line) or a cavity with neither Duffing nor ponderomotive nonlinearities in its mean-field microwave mode equations (dotted line) shows no such decrease in the occupation number. (b) Mechanical oscillator damping renormalization factor for the same fixed detuning and drive current range. The dashed vertical lines indicate the boundaries of the bistable region for the given transmission line resonator parameters.

Figure 9

(Color online) (a) Net mechanical occupation number at $\Delta \omega =-\sqrt{{\omega }_m^2+{\gamma }_{pT}^2}$ for a harmonic transmission line resonator. External bath temperatures: $T=1$ (solid line), 10 (dashed line), 50 (dotted line), and 100 (dotted-dashed line) mK. (b) Dependence of the net mechanical oscillator occupation number on current drive for a Duffing transmission line with detuning $\Delta \omega =1.3\Delta {\omega }_{\text{bi}}$. The bistable region boundaries are indicated by the dashed vertical lines.

Figure 10

(Color online) (a) Detector noise effective occupation number versus current drive when red detuned at $\Delta \omega =-\sqrt{{\omega }_m^2+{\gamma }_{pT}^2}$, $\left|\Delta \omega \right|<\left|\Delta {\omega }_{\text{bi}}\right|$, for a Duffing nonlinear (solid line) harmonic (dashed line) transmission line and with the effects of frequency pulling due to both ponderomotive and Duffing nonlinearities neglected (dotted line). The vertical dashed lines give the bistable region boundaries for the Duffing and harmonic transmission line resonators. This plot is obtained for the straight-line trace labeled $c_2$ in Fig. 2b. Oscillator coupling renormalization factor $R_{\gamma }$ for optimal harmonic detuning. (c) Detector occupation number detuned at twice the harmonic optimum, $\Delta \omega =2.2\Delta {\omega }_{\text{bi}}$, and locked to the lower stable amplitude solution. This plot is obtained for the straight-line trace labeled $c_2^{\prime }$ in Fig. 2. (d) Corresponding back-action damping rate when driven to the upper bistable boundary.

Figure 11

(Color online) (a) Net mechanical occupation number at $\Delta \omega =-\sqrt{{\omega }_m^2+{\gamma }_{pT}^2}$ for a harmonic transmission line resonator. External bath temperatures: $T=1$ (solid line), 10 (dashed line), 50 (dotted line), and 100 (dotted-dashed line) mK. (b) Dependence of the net mechanical oscillator occupation number on current drive for a Duffing transmission line with detuning $\Delta \omega =2.2\Delta {\omega }_{\text{bi}}$.