Quantum force sensing using backaction noise suppression in optomechanical system

Abstract

In this paper, we investigate the quantum force sensing in a single optomechanical system by the use of a coherent quantum noise cancellation scheme. The system consists of a Fabry–Pérot cavity simultaneously coupled to a mechanical oscillator and an ensemble of atom. Accordingly, the dynamics of the systems are studied by utilizing the quantum Langevin equations approach. It is found that the backaction noise can be eliminated from coherent quantum noise cancellation. Remarkably, we examine a force sensor that uses coherent noise cancellation to beat the standard quantum limit. We also investigate the effect of coherent transitions to atomic noise suppresses, and these simultaneous processes can significantly enhance the performance of the quantum force sensor. Our scheme can be generalized to other hybrid optomechanical systems, and these results may have spectacular applications for the realization of the quantum force sensor.

Appendices

Appendix

Effective susceptibilities

In this appendix, we express the effective susceptibilities of the cavity field ($\chi _{a}$), the membrane ($\chi _{m}$), and atomic ensemble ($\chi _{c}$) as:

$$\begin{aligned} \chi _{a}=&(i\omega +\frac{\kappa }{2})^{-1}, \end{aligned}$$

(25a)

$$\begin{aligned} \chi _{m}=&\frac{\omega _{m}}{\omega ^{2}_{m}-\omega ^{2}+i\omega \gamma _{m}}, \end{aligned}$$

(25b)

$$\begin{aligned} \chi _{c}=&\frac{-\omega _{m}}{\omega ^{2}_{m}-\omega ^{2}+i\omega \varGamma +\varGamma ^{2}/4}, \end{aligned}$$

(25c)

and the modified cavity field susceptibility $\chi _{a}^{\prime }$ as:

$$\begin{aligned} \frac{1}{\chi _{a}^{\prime }}=&\frac{1}{\chi _{a}}-\varDelta _{0}\chi _{a}[g^{2}\chi _{m} +G^{2}\chi _{c}-\varDelta _{0}]. \end{aligned}$$

(26a)

The standard quantum limit

The noise spectrum result of a hybrid system with a standard optomechanical setup

$$\begin{aligned} S_\mathrm{stad}^{F}=&\frac{1}{2}\left[ \frac{\kappa }{4g^{2}\gamma _{m}\mid \chi _{m}\mid ^{2}} +\frac{4g^{2}}{\gamma _{m}\kappa }\right] , \end{aligned}$$

(27a)

the backaction noise scales directly proportional to the measurement $g^{2}$ and the shot noise scales inversely proportional to the $\frac{1}{g^{2}}$. In addition, to realize the lower bound of the standard quantum limit of continuous force sensing, we minimize $S_{F}^\mathrm{stad}$ noise spectrum with respect to $g^{2}$ and $T \ne 0$,

$$\begin{aligned} S^\mathrm{stad}_{F}=\frac{1}{\gamma _{m}\mid \chi _{m}\mid }. \end{aligned}$$

(28a)

In view of the frequency and damping rate of the membrane resonator leads to the membrane susceptibility which is efficiently amplified in the force noise spectral density can be significantly reduced which improves the force sensitivity in CQNC condition.