Homodyne coherent quantum noise cancellation in a hybrid optomechanical force sensor

In this paper we propose an experimentally viable scheme to enhance the sensitivity of force detection in a hybrid optomechanical setup assisted by squeezed vacuum injection, beyond the standard quantum limit (SQL). The scheme is based on a combination of the coherent quantum noise cancellation (CQNC) strategy with a variational homodyne detection of the cavity output spectrum in which the phase of the local oscillator is optimized. In CQNC, realizing a negative-mass oscillator in the system leads to exact cancellation of the backaction noise from the mechanics due to destructive quantum interference. Squeezed vacuum injection enhances this cancellation and allows sub-SQL sensitivity to be reached in a wide frequency band and at much lower input laser powers. We show here that the adoption of variational homodyne readout enables us to enhance this noise cancellation up to $40\mathrm{dB}$ compared to the standard case of detection of the optical output phase quadrature, leading to a remarkable force sensitivity of the order of ${10}^{-19}\mathrm{N}/\sqrt{\mathrm{Hz}}$, about $70\%$ enhancement compared to the standard case. Moreover, we show that at nonzero cavity detuning, the signal response can be amplified at a level three to five times larger than that in the standard case without variational homodyne readout, improving the signal-to-noise ratio. Finally, the variational readout CQNC developed in this paper may be applied to other optomechanical-like platforms such as levitated systems and multimode optomechanical arrays or crystals as well as Josephson-based optomechanical systems.

Figure 1

Schematic description of the system under consideration. The system consists of a Fabry-Pérot cavity, in which a single-mode MO is coupled to the radiation pressure of the cavity field. Furthermore, the cavity contains an ensemble of effective two-level atoms with an effective transition rate ${\omega }_{\sigma }={\omega }_m$ that can be controlled by a classical laser field with Rabi frequency ${\mathrm{\Omega }}_R$. The atomic ensemble behaves effectively as a negative-mass oscillator under Faraday interaction and bosonization process [54]. An external classical force ${\tilde{F}}_{\text{ext}}$ is exerted on the MO acting as a sensor or test mass. The cavity is also driven by a coherent light field with power $P_L$ and frequency ${\omega }_L$. A squeezed vacuum light at resonance with the cavity mode, ${\omega }_{\text{sq}}={\omega }_c$, is also injected into the cavity. The cavity output field enters the homodyne detection setup in order to extract information on the external force and imprinted in the modulated phase of the cavity output field.

Figure 2

Force noise spectral density versus $\omega /{\omega }_m$, with an optimized squeezed injected light, i.e., $\left|M\right|=\sqrt{N(N+1)}$ with $N=10$, and $\varphi ={\varphi }_{\mathrm{opt}}$. The results are plotted for the case of off-resonance cavity driving, with $y=1/2$ (red dashed line for $\theta =0$ and purple dotted line for $\theta ={\theta }_{\mathrm{opt}}$) and $y=1$ (blue dash-dotted line for $\theta =0$ and orange dash–double-dotted line for $\theta ={\theta }_{\mathrm{opt}}$). The black solid line corresponds to the SQL. The other parameters are ${\omega }_m/2\pi =300\mathrm{kHz}, {\gamma }_m/2\pi =30\mathrm{mHz}, g_0/2\pi =300\mathrm{Hz}, {\lambda }_L=780\mathrm{nm}, P_L=24\mu \mathrm{W}$ (i.e., $g/g_0=4.91\times {10}^3$), and $\kappa /2\pi =10\mathrm{MHz}$.

Figure 3

Noise reduction advantage $\delta S_{\text{ICQNC}}$ in decibel scale versus $\omega /{\omega }_m$, with an optimized squeezed injected light, i.e., $\left|M\right|=\sqrt{N(N+1)}$ with $N=10$, and $\varphi ={\varphi }_{\mathrm{opt}}$. Different curves correspond to $y=0$ (black dotted line), $y=0.5$ (blue solid line), and $y=1$ (red dashed line). The other parameter values are the same as those in Fig. 2.

Figure 4

Noise reduction advantage in decibel scale versus $\omega /{\omega }_m$. Here we set the normalized detuning $y=\frac{1}{2}$ and consider different values for the squeezing parameter: $N=0$ (purple dotted line), $N=5$ (blue dashed line), $N=15$ (red dash-dotted line), and $N=25$ (black solid line). The other parameter values are the same as those in Fig. 2.

Figure 5

Force noise spectral density versus ${(g/g_0)}^2$, at the frequency $\omega ={\omega }_m+4{\gamma }_m$. Here we set the normalized detuning $y=\frac{1}{2}$ and consider different values for the squeezing parameter: $N=0$ (green solid line for $\theta ={\theta }_{\mathrm{opt}}$ and brown dotted line for $\theta =0$), $N=5$ (blue dashed line for $\theta ={\theta }_{\mathrm{opt}}$ and orange densely dashed line for $\theta =0$), $N=15$ (red dash–triple-dotted line for $\theta ={\theta }_{\mathrm{opt}}$ and yellow dash-dotted line for $\theta =0$), and $N=25$ (gray dash–double-dotted line for $\theta ={\theta }_{\mathrm{opt}}$ and black densely dash–double-dotted line for $\theta =0$). The other parameter values are the same as those in Fig. 2.

Figure 6

Noise reduction advantage in decibel scale versus ${(g/g_0)}^2$, at the frequency $\omega ={\omega }_m+4{\gamma }_m$. Here we set the normalized detuning $y=\frac{1}{2}$ and consider different values for the squeezing parameter: $N=0$ (purple dotted line), $N=5$ (blue dashed line), $N=15$ (red dash-dotted line), and $N=25$ (black solid line). The other parameter values are the same as those in Fig. 2.

Figure 7

Spectral density of the force noise versus $\omega /{\omega }_m$, in the case of the resonant cavity driving ${\mathrm{\Delta }}_c=0$, with an optimized squeezed injected light with $\varphi ={\varphi }_{\mathrm{opt}}\left(0\right)=0, \left|M\right|=\sqrt{N(N+1)}$, and $N=0.2$. We choose the mismatches between the MO and the effective NMO parameters as $(G-g)/g={10}^{-3}$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=-0.2$ (blue dash-dotted line for $\theta =0$ and red dashed line for $\theta ={\theta }_{\mathrm{opt}}$). The green dotted line and black solid line correspond to the perfect CQNC and SQL, respectively. The other parameter values are the same as those in Fig. 2.

Figure 8

Noise reduction advantage (in decibels), at off-resonance frequency $\omega ={\omega }_m-4{\gamma }_m$ versus the coupling mismatch $G/g$ (horizontal axis) and decay rate mismatch $\mathrm{\Gamma }/{\gamma }_m$ (vertical axis). Here we consider an optimized squeezed injected light with $\varphi ={\varphi }_{\mathrm{opt}}\left(0\right)=0, \left|M\right|=\sqrt{N(N+1)}$, and $N=0.126$. In addition, we assume that the cavity is driven at resonance ${\mathrm{\Delta }}_c=0$. The other parameters are the same as those in Fig. 2.

Figure 9

Noise reduction advantage (in decibels) at off-resonance frequency $\omega ={\omega }_m-4{\gamma }_m$ versus the dissipation rate mismatch $\mathrm{\Gamma }/{\gamma }_m$ (vertical axis) and the squeezing parameter $N$ (horizontal axis) in the case of resonant cavity driving ${\mathrm{\Delta }}_c=0$. The coupling rate mismatch is $(G-g)/g=0.02$ and an optimized squeezed light with $\varphi ={\varphi }_{\mathrm{opt}}\left(0\right)=0$ and $\left|M\right|=\sqrt{N(N+1)}$ is considered. The other parameter values are the same as those in Fig. 2.

Figure 10

Force noise spectrum versus ${(g/g_0)}^2$ at off-resonance frequency $\omega ={\omega }_m+4{\gamma }_m$ in the case of resonant cavity driving ${\mathrm{\Delta }}_c=0$. An optimized squeezed injected light with $\varphi ={\varphi }_{\mathrm{opt}}\left(0\right)=0, \left|M\right|=\sqrt{N(N+1)}$, and $N=0.1245$ is considered. Different curves correspond to $G=g$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$ (green solid line for $\theta ={\theta }_{\mathrm{opt}}$ and brown dotted line for $\theta =0$), $(G-g)/g=0.01$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$ (blue dashed line for $\theta ={\theta }_{\mathrm{opt}}$ and orange densely dashed line for $\theta =0$), $(G-g)/g=-(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$ (red dash–double-dotted line for $\theta ={\theta }_{\mathrm{opt}}$ and yellow densely dash–double-dotted line for $\theta =0$), and $(G-g)/g=0.2$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=-0.1$ (purple dash-dotted line for $\theta ={\theta }_{\mathrm{opt}}$ and black dash–triple-dotted line for $\theta =0$). The other parameter values are the same as those in Fig. 2.

Figure 11

Noise reduction advantage (in decibels) versus ${(g/g_0)}^2$ at off-resonance frequency $\omega ={\omega }_m+4{\gamma }_m$ in the case of resonant cavity driving ${\mathrm{\Delta }}_c=0$. An optimized squeezed injected light with $\varphi ={\varphi }_{\mathrm{opt}}\left(0\right)=0, \left|M\right|=\sqrt{N(N+1)}$, and $N=0.1245$ is considered here. Different curves correspond to $G=g$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$ (green dotted line), $(G-g)/g=0.01$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$ (blue dashed line), $(G-g)/g=0.05$ and $\mathrm{\Gamma }={\gamma }_m$ (red densely dashed line), $(G-g)/g=0.1$ and $\mathrm{\Gamma }={\gamma }_m$ (purple solid line), $(G-g)/g=(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$ (orange dash-dotted line), and $(G-g)/g=0.2$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=-0.1$ (black dash–double-dotted line). The other parameter values are the same as those in Fig. 2.

Figure 12

Spectral density of the force noise $S_F\left(\omega \right)$ versus the normalized frequency $\omega /{\omega }_m$, in the case of the off-resonance cavity driving with ${\mathrm{\Delta }}_c/\kappa =1$. An optimized squeezed injected light with $\left|M\right|=\sqrt{N(N+1)}$ and $N=1$ is considered. Different curves correspond to $(G-g)/g={10}^{-5}$ and $\mathrm{\Gamma }={\gamma }_m$ (blue dashed line for $\theta =0$ and red dotted line for $\theta ={\theta }_{\mathrm{opt}}$); the black solid line corresponds to the SQL. The other parameter values are the same as those in Fig. 2.

Figure 13

Noise reduction advantage (in decibels) versus the coupling rate mismatch (horizontal axis) and the parameter $N$ (vertical axis), at off-resonance frequency $\omega ={\omega }_m-4{\gamma }_m$ and in the case of off-resonant cavity driving with ${\mathrm{\Delta }}_c/\kappa =1$. An optimized squeezed injected light with $\left|M\right|=\sqrt{N(N+1)}$ is considered here. We also assume that the decay rate mismatch is $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.2$. The other parameter values are the same as those in Fig. 2.

Figure 14

Force noise spectrum versus ${(g/g_0)}^2$ at off-resonance frequency $\omega ={\omega }_m-4{\gamma }_m$ and in the case of off-resonant cavity driving ${\mathrm{\Delta }}_c/\kappa =1$. An optimized squeezed vacuum injected light with $\varphi ={\varphi }_{\mathrm{opt}}\left(1\right), \left|M\right|=\sqrt{N(N+1)}$, and $N=20$ is considered. Different curves correspond to $(G-g)/g=0.05$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$ (green solid line for $\theta ={\theta }_{\mathrm{opt}}$ and brown dotted line for $\theta =0$), $(G-g)/g=0.1$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.15$ (blue dashed line for $\theta ={\theta }_{\mathrm{opt}}$ and orange dash-dotted line for $\theta =0$), $(G-g)/g=0.15$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.2$ (purple dash–double-dotted line for $\theta ={\theta }_{\mathrm{opt}}$ and red dash–triple-dotted line for $\theta =0$). The other parameter values are the same as those in Fig. 2.

Figure 15

Noise reduction advantage (in decibels) versus ${(g/g_0)}^2$ at off-resonance frequency $\omega ={\omega }_m-4{\gamma }_m$ and in the case of off-resonant cavity driving with ${\mathrm{\Delta }}_c/\kappa =1$. An optimized squeezed vacuum injected light with $\varphi ={\varphi }_{\mathrm{opt}}\left(1\right), \left|M\right|=\sqrt{N(N+1)}$, and $N=20$ is considered. Different curves correspond to $(G-g)/g=+0.05$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=+0.1$ (purple solid line), $(G-g)/g=-0.05$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=-0.1$ (red dotted line), $(G-g)/g=+0.1$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=+0.2$ (blue dash-dotted line), $(G-g)/g=-0.1$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=-0.2$ (green dashed line), and $(G-g)/g=+0.2$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=+0.3$ (orange dash-double-dotted line). The other parameter values are the same as those in Fig. 2.

Figure 16

Power of signal response versus $\omega /{\omega }_m$ under the (a) perfect and (b) imperfect CQNC conditions. Different curves in (a) correspond to ${\mathrm{\Delta }}_c=\kappa$ (red dashed line for $\theta ={\theta }_{\text{opt}}$ and black dotted line for $\theta =0$) and ${\mathrm{\Delta }}_c=0$ (blue solid line for $\theta ={\theta }_{\text{opt}}$ and green dash-dotted line for $\theta =0$). In (b) we consider $(G-g)/g={10}^{-4}$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.01$ for coupling rate and decay rate mismatches. Different curves in (b) refer to ${\mathrm{\Delta }}_c=0$ (blue solid line for $\theta ={\theta }_{\text{opt}}$ and green dash-dotted line for $\theta =0$) and ${\mathrm{\Delta }}_c=\kappa$ (red dashed line for $\theta ={\theta }_{\text{opt}}$ and black dotted line for $\theta =0$). In both cases we consider an optimized vacuum injected light with $\varphi ={\varphi }_{\text{opt}}\left(y\right), \left|M\right|=\sqrt{N(N+1)}$, and $N=10$. The other parameter values are the same as those in Fig. 2.

Figure 17

Signal amplification advantage versus $\omega /{\omega }_m$, considering the imperfect CQNC conditions with $(G-g)/g={10}^{-4}$ and $(\mathrm{\Gamma }-{\gamma }_m)/{\gamma }_m=0.1$. Different curves correspond to ${\mathrm{\Delta }}_c=0$ (green dashed line) and ${\mathrm{\Delta }}_c=\kappa$ (blue solid line). The other parameter values are the same as those in Fig. 16.

Figure 18

Optical susceptibility ${\chi }_a\left(\omega \right)$ versus the scaled frequency $\omega /\kappa$. Different curves correspond to $\mathrm{Re}\left[{\chi }_a\left(\omega \right)\right]$ (red solid line), $\mathrm{Im}\left[{\chi }_a\left(\omega \right)\right]$ (blue dashed line), and ${\chi }_a\left(\omega \right)\sim 2/\kappa$ (black dash-dotted line).