Force measurement in squeezed dissipative optomechanics in the presence of laser phase noise

Wen-ju Gu; Wen-ju Gu; Yue-yuan Wang; Zhen Yi; Wen-Xing Yang; Wen-Xing Yang; Li-hui Sun

doi:10.1364/OE.389854

1. Introduction

Cavity optomechanical system is of growing interest for its significant potential in ultra-sensitive measurements approaching or beating the standard quantum limit (SQL) [1–5], explorations of fundamental physics [6–9], and applications in quantum information processing [10–13]. In general, there exist two optomechanical interaction mechanisms. First, dispersive coupling in which mechanical motion changes the resonance frequency of the cavity has been extensively investigated, including mechanical cooling [14–16], optomechanically induced transparency [17,18], ponderomotive squeezing [19,20], and optomechanical steering [21–26], etc. Besides, dissipative coupling in which the mechanical motion modifies the cavity decay rate has recently broadened the scope for some novel phenomena, such as strong optomechanical cooling beyond the sideband-resolved regime [27–29] and generating a stable optical spring in gravitational-wave (GW) detectors [30]. Dissipative coupling has been observed in a wide variety of devices [31–34]. Particularly in the subject of quantum measurement, much attention has been attracted to overcome the SQL in dispersive optomechanics, and there are various demonstrations such as exploitation of quantum correlations [35] and quantum backaction-evading measurements [36,37]. Quantum speed meter is a quantum non demolition (QND) meter which allows to overcome SQL, and enable the force detection via output optical field containing information about difference $\sim \left [x(t)-x(t-\tau )\right ]\simeq \tau \dot {x}(t)$ (where $x$ is the position, $\tau$ is a deley time). A variety of conceptual topologies based on dispersive coupling have been proposed, where the subtraction is realized by interaction of a mechanical degree of freedom with two coupled optical modes and it increases complexity of experimental realization [38–40]. Pure dissipative coupling provides possibility to realize subtraction naturally, enabling a simple way of measurement and suitable for a sensitive detection of a classical force [41,42]. Realization of pure dissipative coupling on the basis of Michelson-Sagnac interferometer is suggested in [27] and observed in [28].

The precision of cavity optomechanical sensors is generally influenced by thermal noise, shot noise (SN), and quantum backaction noise (QBA). Squeezed optomechanics, which can be realized by intracavity optical squeezing with a nonlinear medium [43–45] or injecting squeezed light, allows to reduce noise floor if SN is dominant. Thus, squeezed optomechanics can improve the force sensitivity and broaden the range of the detection frequencies [46]. For instance, squeezed light is first recognized as a quantum sensing resource to improve the sensitivity of large-scale GW interferometers beyond the SN limit [47] and experimentally demonstrated thereafter [48–51]. Recent advances in nano- or micromechanical oscillator have also allowed transduction sensitivity below the SN level [52,53]. For the dissipative optomechanical speed meter, the QBA can be avoided through an appropriate choice of homodyne phase angle [41]. To further reduce the SN through enhancement of optomechanical interaction, a strong laser is usually employed in practice, which gives rise to another source of noise: laser phase noise. Laser phase noise is inevitable and related to the pump power, and has been a subject of considerable interest, including cooling and coherent evolution of optomechanical systems [54,55], parametric instability [56], quantum memory [57], and optomechancial entanglement [58,59]. In optomechanically-based sensing schemes, an important question is how to reduce the effect of laser phase noise on the measurement sensitivity if the floor of SN and QBA has been lowered. Fortunately, besides lowering the SN floor, squeezed light is also able to contribute to measurement sensitivity at a lower optical probe power [60], which is beneficial for these systems where the use of high optical powers is limited by technical reasons or increasing optical powers gives rise to an additional noise [61]. Taken together, in dissipative optomechanically-based detection schemes given the avoidance of QBA, squeezed injection could exponentially reduce the SN and laser phase noise through decreasing pump power.

Here we study the squeezing-enhanced force detection sensitivity based on a direct quantum measurement of speed of a free mass in a dissipatively coupled optomechanical system, with the presence of technical noises including laser phase noise and optical losses. With the use of optical squeezed field and homodyne detection of the generalized quadrature of transmitted field, SN can be suppressed owning to the reduction of imprecision noise on the squeezed quadrature, and QBA induced by antisqueezing on the conjugate quadrature can be evaded through an appropriate choice of homodyne phase angle. Mechanical thermal noise can be reduced through improving mechanical Q factor and cooling the oscillator. However, laser phase noise is proportional to pump power according to the linearized treatment, similar to the QBA. The minimum sensitivity is obtained by balancing reduced SN and phase noise, and finally improved by squeezed light, in which pump power is shown to be exponentially lowered. The sub-SQL detection is achievable with the current experimental parameters. In addition, optical losses are also an important limiting factor, because high sensitivity is delicate and easily destroyed by optical losses which lead to decoherence [62], and effect of squeezing can be saturated by losses. Therefore, high outcoupling and optical detection efficiencies are necessary in practice.

The paper is organized as follows. In Sec. 2 we introduce the squeezed dissipative optomechanical speed meter in the presence of laser phase noise. In Sec. 3 the measurement sensitivity with the injection of squeezed light and outcoupling and detection losses is calculated, sensitivity improved by the squeezed light without consideration of outcoupling and detection losses is first analyzed, and then the limits set by these losses are also discussed. In Sec. 4 some discussions are presented and at last the conclusion is drawn.

2. Squeezed dissipative optomechanics in the presence of laser phase noise

We consider a dissipative optomechanical system consisted of a mechanical system of mass $m$ coupled to a cavity field $\hat {c}$ with resonance frequency $\omega _c$ [41,63], as shown in Fig. 1. When Michelson-Sagnac interferometer (MSI) operates at a point where the transmissivity of the effective mirror is close to zero, the transmissivity will then depend sensitively on the membrane displacement $x$. Combining with a perfect mirror $M_1$ will form an effective Fabry-Pérot interferometer (FPI) whose linewidth depends on $x$ dominantly via transmissivity [27]. The dissipation rate of cavity mode can be written in the form $\kappa (\hat {x})=\kappa _{R}(1+\eta {\hat {x}})$, where $\kappa _R$ is the dissipation rate from right-hand side of the cavity for ${\hat {x}}=0$, and $\eta$ characterizes the dissipative coupling strength in unit of $\kappa _R$ that is controlled by position of membrane and reflectivity of the central BS. In addition, the cavity field is pumped by a bright squeezed light from right-hand side, with carrier frequency $\omega _l$, coherent amplitude $\epsilon _l$, and a broadband squeezed-vacuum resource $\hat {c}_{\textrm {in,R}}$. In the free-mass regime, i.e., detection frequency $\omega$ much larger than mechanical frequency $\omega _m$, the Hamiltonian of system in the rotating frame of frequency $\omega _l$ is given by

(1)$$\begin{aligned}\hat{H}&=\hbar(\omega_c-\omega_l)\hat{c}^\dagger\hat{c}+\frac{\hat{p}^2}{2m}-f_{\textrm{ex}}{\hat{x}}\\ &+ i\hbar\sqrt{\kappa({\hat{x}})} \left[\hat{c}^\dagger\left(\epsilon_l+\hat{c}_{\textrm{in,R}}e^{{-}i\theta_s(t)}\right)e^{{-}i\theta_l(t)}-\hat{c}\left(\epsilon_l+\hat{c}^\dagger_{\textrm{in,R}}e^{i\theta_s(t)}\right)e^{i\theta_l(t)}\right], \end{aligned}$$

where $\hat {p}$ is the momentum of free mass, and $f_{\textrm {ex}}$ is an external force with zero mean value acting on the free mass. The square root of dissipation rate approximately equals $\sqrt {\kappa ({\hat {x}})}\simeq \sqrt {\kappa _R}(1+\frac {\eta }{2}{\hat {x}})$, given that the parameter $\eta$ is about $4.182\times 10^8~\textrm {m}^{-1}$ in the experimental setup [28]. Here, a bright squeezed light source, which is experimentally achievable at a wavelength of $1064$ nm [64], would be compatible with dissipative optomechanical setup [28]. However, we take into consideration of the phase noise of laser field $\theta _l(t)$ where $\dot \theta _l(t)$ is the frequency noise and phase noise of squeezed light $\theta _s(t)$ relative to laser field. Since in practice, phase noise and mechanical thermal noise would impose limitations to ground-state cooling, coherent evolution of optomechanical systems, as well as the measurement sensitivity. But, squeezer phase noise $\theta _s(t)$ is negligible because in current experiment $\theta _s(t)$ is on the scale of $\textrm {mrad}$ when the pump phase is locked to the seed light [65]. In the following we focus on the laser phase noise $\theta _l(t)$ that is proportional to laser power in the linearized treatment. Phase noise was treated as white noise in theory at first [66,67], which would produce damaging effects on cooling and quantum features. However, the noise actually has a bandpass filter form, and following a more rigorous derivation with experimental parameters, the phase noise must be considered as a color noise with finite bandwidth [54,56]. At a certain frequency $\omega$, the phase noise $\theta _l[\omega ]$ is related to the frequency noise $\dot {\theta }_l[\omega ]$ by $\dot {\theta }_l[\omega ]=-i\omega \theta _l[\omega ]$, so the phase noise spectrum is expressed in the form of frequency noise spectrum $\bar {S}_{\theta _l}[\omega ]=\bar {S}_{\dot {\theta }_l}[\omega ]/\omega ^2$ (here correlation corresponds to classical ensemble averages), where $\bar {S}_{\dot {\theta }_l}[\omega ]=\frac {2\gamma \gamma _c^2}{\gamma _c^2+\omega ^2}$, with the laser linewidth $\gamma$, the cutoff frequency $\gamma _c$ in the laser spectrum. In general noise at high frequencies $\vert {\omega }\vert >\gamma _c$ is suppressed [55].

Fig. 1. Dissipative optomechanics consisted of an effective Fabry-Pérot cavity with a fixed mirror ($\textrm {M}_1$) and an input mirror formed by Michelson-Sagnac interferometer (MSI). Bright squeezed light is used to improve the measurement and the output light is monitored by an homodyne detection apparatus.

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Taking into account of laser phase noise and on cavity resonance $\omega _c=\omega _l$, we introduce an unitary operator $\hat {U}(t)=\exp [-i\theta _l(t)\hat {c}^\dagger \hat {c}]$, and achieve the transformed Hamiltonian $\hat {H}_{\textrm {trans}}=\hat {U}^\dagger (t)\hat {H}\hat {U}(t)+i\hbar [\partial _t\hat {U}^\dagger (t)]\hat {U}(t)$ that is

(2)$$\hat{H}_{\textrm{trans}}={-}\hbar\dot{\theta}_l(t)\hat{c}^\dagger\hat{c}+\frac{\hat{p}^2}{2m}+i\hbar\sqrt{\kappa_R}\left(1+\frac{\eta}{2}{\hat{x}}\right) \left[\hat{c}^\dagger\left(\epsilon_l+\hat{c}_{\textrm{in,R}}\right)-\hat{c}\left(\epsilon_l+\hat{c}^\dagger_{\textrm{in,R}}\right)\right]-f_{\textrm{ex}}{\hat{x}}.$$

Then the qauntum Langevin equation (QLE) of the system is given by

(3)$$\begin{aligned}\dot{{\hat{x}}}&=\frac{\hat{p}}{m},\\ \dot{\hat{p}}&={-}i\hbar\sqrt{\kappa_R}\frac{\eta}{2}\left[\epsilon_l(\hat{c}^\dagger-\hat{c})+(\hat{c}^\dagger\hat{c}_{\textrm{in,R}}-\hat{c}^\dagger_{\textrm{in,R}}\hat{c})\right]-\gamma_m\hat{p}+\hat{\xi}+f_{\textrm{ex}},\\ \dot{\hat{c}}&=\sqrt{\kappa_R}\left(1+\frac{\eta}{2}{\hat{x}}\right)(\epsilon_l+\hat{c}_{\textrm{in,R}})-\frac{\kappa_R}{2}\eta{\hat{x}}\hat{c}-\frac{\kappa_0}{2}\hat{c}+\sqrt{\kappa_L}\hat{c}_{\textrm{in,L}}+i\dot{\theta}_l(t)\hat{c}, \end{aligned}$$

where $\gamma _m$ is the mechanical damping rate, $\hat {\xi }$ is the mechanical thermal noise, $\kappa _L$ indicates the optical losses through the end mirror $M_1$ and absorption inside the interferometer, $\hat {c}_{\textrm {in,L}}$ is the corresponding vacuum noise, and $\kappa _0=\kappa _L+\kappa _R$ is the total optical losses rate. We define the cavity outcoupling $\eta _c=\kappa _R/\kappa _0$ to indicate degraded escape efficiency, and high outcoupling is necessary in practice.

Then we follow the linearization procedure by writing each operator as the sum of steady-state value and fluctuation operator ${\hat {x}}=x_s+\delta {\hat {x}}$, $\hat {p}=p_s+\delta \hat {p}$, $\hat {c}=c_s+\delta \hat {c}$, where $x_s$, $p_s$, and $c_s$ are the steady-state values, and $\delta {\hat {x}}$, $\delta \hat {p}$, and $\delta \hat {c}$ are the small fluctuation operators. Under the assumption of the steady-state displacement of the free mass $x_s=0$, the steady-state values of momentum and amplitude of cavity field are $p_s=0$ and $c_s=\frac {\sqrt {\kappa _R}}{\kappa _0/2}\epsilon _l$. Here we introduce the amplitude and phase quadratures of cavity field as $\delta {\hat {x}}_c=(\delta \hat {c}+\delta \hat {c}^\dagger )/\sqrt {2}$ and $\delta \hat {p}_c=(\delta \hat {c}-\delta \hat {c}^\dagger )/i\sqrt {2}$, and quadratures of cavity input noise as ${\hat {x}}_{\textrm {in,j}}=(\hat {c}_{\textrm {in,j}}+\hat {c}_{\textrm {in,j}}^\dagger )/\sqrt {2}$ and $\hat {p}_{\textrm {in,j}}=(\hat {c}_{\textrm {in,j}}-\hat {c}_{\textrm {in,j}}^\dagger )/i\sqrt {2}$, $j=L,R$. Then after linearization the fluctuation quadrature operators yield the QLE

(4)$$\begin{aligned}\delta\dot{{\hat{x}}}&=\frac{\delta\hat{p}}{m},\\ \delta\dot{\hat{p}}&=-\hbar\sqrt{\frac{\kappa_R}{2}}\eta(\epsilon_l\delta\hat{p}_c-c_s\hat{p}_{\textrm{in,R}})-\gamma_m\delta\hat{p}+\hat{\xi}+f_{\textrm{ex}},\\ \delta\dot{{\hat{x}}}_c&=-\frac{\kappa_0}{2}\delta{\hat{x}}_c-\frac{\sqrt{2}}{4}\kappa_R\eta c_s\delta{\hat{x}}+\sqrt{\kappa_R}{\hat{x}}_{\textrm{in,R}}+\sqrt{\kappa_L}{\hat{x}}_{\textrm{in,L}},\\ \delta{\dot{\hat{p}}}_c&=-\frac{\kappa_0}{2}\delta\hat{p}_c+\sqrt{\kappa_R}\hat{p}_{\textrm{in,R}}+\sqrt{\kappa_L}\hat{p}_{\textrm{in,L}}+\sqrt{2}c_s\dot{\theta}_l. \end{aligned}$$

The detection of output light from right-hand side of cavity is determined by the standard input-output relation

(5)$$\begin{aligned}\delta{\hat{x}}_{c}^{\textrm{out,R}}&={-}{\hat{x}}_{\textrm{in,R}}+\sqrt{\kappa_{R}}\delta{\hat{x}}_{c}+\sqrt{\kappa_{R/2}}\eta c_{s}\delta{\hat{x}},\\ \delta{\hat{p}}_{c}^{\textrm{out,R}}&={-}\hat{p}_{\textrm{in,R}}+\sqrt{\kappa_{R}}\delta{\hat{p}}_{c}. \end{aligned}$$

In the frequency domain via employing Fourier transformation $\hat {f}[\omega ]=\frac {1}{\sqrt {2\pi }}\int _{-\infty }^\infty \hat {f}(t)e^{i\omega t}dt$, we have the steady-state output amplitude quadrature

(6)$$\delta{\hat{x}}_c^{\textrm{out,R}}[\omega]=\left(\frac{\kappa_R}{\kappa_0-i\omega}-1\right){\hat{x}}_{\textrm{in,R}}[\omega] +\frac{\sqrt{\kappa_R\kappa_L}}{\kappa_0/2-i\omega}{\hat{x}}_{\textrm{in,L}}[\omega] +\sqrt{\kappa_R/2}\eta c_s\frac{(\kappa_0-\kappa_R)/2-i\omega}{\kappa_0/2-i\omega}\delta{\hat{x}}[\omega].$$

In practice to obtain as much of the cavity light as possible, the reflectivities of two cavity mirrors differ significantly, forming a strongly asymmetric optical resonator. For example, the cavity outcoupling efficiency $\eta _c=\kappa _R/\kappa _0$ reaches $0.95$ in [68]. In dissipative setup the power reflectivity of $M_1$ is $r_{M_1}^2=0.9997$, and optical losses inside the interferometer are $t^2_{\textrm {loss}}=5\times 10^{-3}$ [28]. Thus the cavity outcoupling efficiency $\eta _c=\kappa _R/\kappa _0$ can even approach unity. Under this condition, $\kappa _0\simeq \kappa _R$, and output field approximately equals to

(7)$$\delta{\hat{x}}_c^{\textrm{out,R}}[\omega]\simeq\frac{\kappa_0/2+i\omega}{\kappa_0/2-i\omega}{\hat{x}}_{\textrm{in,R}}[\omega]+\frac{\kappa_0\sqrt{1-\eta_c}}{\kappa_0/2-i\omega}{\hat{x}}_{\textrm{in,L}}[\omega] +\sqrt{\kappa_0/2}\eta c_s\frac{-i\omega\delta{\hat{x}}[\omega]}{\kappa_0/2-i\omega},$$

where information on speed of the probe mass $-i\omega \delta {\hat {x}}$ if $\kappa _0\gg \omega$ (not displacement) is provided naturally based on dissipative coupling [41]. Finally via solving QLE in frequency domain in the limit of $\kappa _R\gg \kappa _L$, the steady-state solutions for quadratures of transmitted field are

(8)$$\begin{aligned}\delta\hat{p}_{c}^{\textrm{out,R}}[\omega]&=\frac{\kappa_{0}/2+i\omega}{\kappa_{0}/2-i\omega}\left(\hat{p}_{\textrm{in,R}}[\omega]+\frac{\kappa_{0}\sqrt{1-\eta_{c}}}{\kappa_{0}/2+i\omega} \hat{p}_{\textrm{in,L}}[\omega]+\frac{\sqrt{2\kappa_{0}}c_{s}}{\kappa_{0}/2+i\omega}\omega\theta_{l}[\omega]\right),\\ \delta{\hat{x}}_{c}^{\textrm{out,R}}[\omega]&=\frac{\kappa_{0}/2+i\omega}{\kappa_{0}/2-i\omega}{\bigg [}\hat{x}_{\textrm{in,R}}[\omega]+\frac{\kappa_{0}\sqrt{1-\eta_{c}}}{\kappa_{0}/2+i\omega}{\hat{x}}_{\textrm{in,L}}[\omega] +Q[\omega]u_m[\omega]\hat{p}_{\textrm{in,R}}[\omega]\\ &\quad+\frac{\sqrt{2Q[\omega]}}{F_{\textrm{SQL}}[\omega]}u[\omega]u_{m}[\omega]\left(\hat{\xi}[\omega]+f_{\textrm{ex}}[\omega]\right)+Q[\omega]\frac{\sqrt{\kappa_{0}/2}c_{s}}{i\omega-\gamma_{m}}\omega\theta_{l}[\omega]{\bigg ]}, \end{aligned}$$

with the parameters

(9)$$\begin{aligned}Q[\omega]&=P\frac{\kappa_0^2/4}{\kappa_0^2/4+\omega^2},\hspace{5pt} P=\frac{2\hbar\eta^2c_s^2}{m\kappa_0}, \hspace{5pt}u_m[\omega]=\frac{i\omega}{i\omega-\gamma_m},\\ u[\omega]&=i\frac{\kappa_0/2-i\omega}{\sqrt{\kappa_0^2/4+\omega^2}}\frac{\vert{\omega}\vert}{\omega}, \hspace{5pt} F_{\textrm{SQL}}[\omega]=\sqrt{2\hbar m\omega^2}. \end{aligned}$$

Here $P$ is dimensionless and related to the pump power by $\mathcal {P}=\frac {m\kappa _0^2\omega _l}{8\eta ^2}P$, $F_{\textrm {SQL}}[\omega ]$ is the SQL for a free mass sensitivity to a weak classical force.

3. Measurement sensitivity improved by squeezed light

The output field is detected by the balanced homodyne measurement with a local oscillator (LO), which separates from the same laser source as pump laser but without going through the optomechanical system. Also taken into consideration of phase noise of laser source, the detected field operator is

(10)$$\hat{i}=\alpha_\textrm{LO}e^{{-}i\theta_l(t)}e^{i\phi}\hat{c}^\dagger_\textrm{out,R}+\alpha_\textrm{LO}e^{i\theta_l(t)}e^{{-}i\phi}\hat{c}_\textrm{out,R},$$

where $\alpha _{\textrm {LO}}$ is the amplitude of LO, $\theta _{l}(t)$ is the same phase noise as laser field, and $\phi$ is the homodyne angle. After the rotation of phase noise in transformation, the detected field becomes $\hat {i}=\alpha _{LO}\left (\delta {\hat {x}}_{c}^{\textrm {out,R}}\cos \phi +\delta \hat {p}_{c}^{\textrm {out,R}}\sin \phi \right )$. However, due to the inefficient detection of optical field, the output light is usually modelled by a beam-splitter mixing to uncorrelated vacuum noise. If the total fractional detection efficiency is $\eta _{\textrm {det}}$, the effect of such inefficiency changes the quadratures of output field to

(11)$$\begin{aligned}\delta{\hat{x}}_{c}^{\textrm{out,R}}&\rightarrow\sqrt{\eta_{\textrm{det}}}\delta{\hat{x}}_{c}^{\textrm{out,R}}+\sqrt{1-\eta_{\textrm{det}}}{\hat{x}}_{c}^{\textrm{v}},\\ \delta\hat{p}_{c}^{\textrm{out,R}}&\rightarrow\sqrt{\eta_{\textrm{det}}}\delta\hat{p}_{c}^{\textrm{out,R}}+\sqrt{1-\eta_{\textrm{det}}}\hat{p}_{c}^{\textrm{v}},\end{aligned}$$

where ${\hat {x}}_{c}^{\textrm {v}}$ and $\hat {p}_{c}^{\textrm {v}}$ are the introduced vacuum noise. In total, the output quadrature measured by a balanced homodyne receiver is

(12)$$\begin{aligned}\delta\hat{z}_{c}^{\textrm{out,R}}[\omega]&=\sqrt{\eta_{\textrm{det}}}\left(\delta{\hat{x}}_{c}^{\textrm{out,R}}[\omega]\cos\phi+\delta\hat{p}_{c}^{\textrm{out,R}}[\omega]\sin\phi\right)\\ &\quad+\sqrt{1-\eta_{\textrm{det}}}\left({\hat{x}}_{c}^\textrm{v}[\omega]\cos\phi+\hat{p}_{c}^\textrm{v}[\omega]\sin\phi\right). \end{aligned}$$

Therefore, the measured force from the detection, which is obtained by expressing the quadrature in unit of force, is

(13)$$\hat{f}_\textrm{meas}[\omega]=\hat{f}_{\textrm{add}}[\omega]+\hat{f}_{\textrm{ex}}[\omega],$$

where the force noise added by the measurement process $\hat {f}_{\textrm {add}}[\omega ]$ is

(14)$$\begin{aligned}\hat{f}_{\textrm{add}}[\omega]&=\frac{F_{\textrm{SQL}}[\omega]}{\sqrt{2Q[\omega]}u[\omega]}\Bigg[\frac{{\hat{x}}_{\textrm{in}}[\omega]}{u_{m}[\omega]}+\left(\tan\phi+Q[\omega]\right)\hat{p}_{\textrm{in}}[\omega]\\ +&\quad\left(\frac{\sqrt{2\kappa_{0}}}{\kappa_{0}/2+i\omega}\tan\phi+\frac{\sqrt{\kappa_{0}/2}Q[\omega]}{i\omega-\gamma_{m}}\right)\frac{-i\omega\theta_{l}[\omega]c_{s}}{u_{m}[\omega]}\\ +&\quad\frac{\kappa_{0}\sqrt{1-\eta_{c}}}{(\kappa_{0}/2+i\omega)u_{m}[\omega]}\left({\hat{x}}_{\textrm{in,L}}[\omega]+\tan\phi\hat{p}_{\textrm{in,L}}[\omega]\right)\\ +&\quad\sqrt{\frac{1-\eta_{\textrm{det}}}{\eta_{\textrm{det}}}}\frac{1}{u_{m}[\omega]}\left({\hat{x}}_{c}^{\textrm{v}}[\omega]+\tan\phi\hat{p}_{c}^\textrm{v}[\omega]\right)\Bigg]+F_{\textrm{SQL}}[\omega]\tilde{\hat{\xi}}[\omega]. \end{aligned}$$

Thus the limit in measurement sensitivity of weak force $f_{\textrm {ex}}[\omega ]$ is determined by the added force noise $\hat {f}_{\textrm {add}}[\omega ]$, which contains the noise of amplitude quadrature (SN), phase quadrature induced by the optomechanical interaction (QBA), laser phase noise, outcoupling and detection inefficiencies, and mechanical thermal noise. Thermal noise $\tilde {\hat {\xi }}[\omega ]=\hat {\xi }[\omega ]/F_{\textrm {SQL}}[\omega ]$ is scaled by SQL, and fulfills the correlation function

(15)$$\langle\tilde{\hat{\xi}}[\omega]\tilde{\hat{\xi}}[\omega^\prime]\rangle=\frac{k_BT}{\hbar\omega^2}\gamma_m\delta(\omega+\omega^\prime),$$

with the Boltzmann constant $k_B$ and the temperature of environment $T$.

The symmetrized spectrum density of $\hat {f}_{\textrm {add}}[\omega ]$ is defined as

(16)$$\bar{S}_{\textrm{add}}[\omega]\delta(\omega+\omega^\prime)=\frac{1}{2}\langle\{\hat{f}_{\textrm{add}}[\omega],\hat{f}_{\textrm{add}}[\omega^\prime]\}\rangle.$$

To obtain $\bar {S}_{\textrm {add}}[\omega ]$, the correlations of input noise should be clarified, where the amplitude-squeezed field injected from the right-hand side fulfills

(17)$$\begin{aligned} \langle{\hat{x}}_{\textrm{in,R}}[\omega]{\hat{x}}_{\textrm{in,R}}[\omega^\prime]\rangle &=\frac{1}{2}e^{{-}2r}\delta(\omega+\omega^\prime),\hspace{5pt} \langle\hat{p}_{\textrm{in,R}}[\omega]\hat{p}_{\textrm{in,R}}[\omega^\prime]\rangle=\frac{1}{2}e^{2r}\delta(\omega+\omega^\prime),\\ \langle{\hat{x}}_{\textrm{in,R}}[\omega]\hat{p}_{\textrm{in,R}}[\omega^\prime]\rangle &=-\langle\hat{p}_{\textrm{in,R}}[\omega]{\hat{x}}_{\textrm{in,R}}[\omega^\prime]\rangle=\frac{i}{2}\delta(\omega+\omega^\prime), \end{aligned}$$

with the squeezing parameter $r$, and the vacuum noises due to the outcoupling and detection inefficiencies fulfill

(18)$$\begin{aligned}\langle{\hat{x}}_{\textrm{in,L}}[\omega]{\hat{x}}_{\textrm{in,L}}[\omega^\prime]\rangle &=\frac{1}{2}\delta(\omega+\omega^\prime), \hspace{5pt} \langle\hat{p}_{\textrm{in,L}}[\omega]\hat{p}_{\textrm{in,L}}[\omega^\prime]\rangle=\frac{1}{2}\delta(\omega+\omega^\prime),\\ \langle{\hat{x}}_{\textrm{in,L}}[\omega]\hat{p}_{\textrm{in,L}}[\omega^\prime]\rangle &={-}\langle\hat{p}_{\textrm{in,L}}[\omega]{\hat{x}}_{\textrm{in,L}}[\omega^\prime]\rangle=\frac{i}{2}\delta(\omega+\omega^\prime),\\ \langle{\hat{x}}_{c}^{\textrm{v}}[\omega]{\hat{x}}_{c}^{\textrm{v}}[\omega^\prime]\rangle &=\frac{1}{2}\delta(\omega+\omega^\prime), \hspace{5pt} \langle\hat{p}_{c}^{\textrm{v}}[\omega]\hat{p}_c^{\textrm{v}}[\omega^\prime]\rangle=\frac{1}{2}\delta(\omega+\omega^\prime),\\ \langle{\hat{x}}_c^{\textrm{v}}[\omega]\hat{p}_c^{\textrm{v}}[\omega^\prime]\rangle &=-\langle\hat{p}_c^{\textrm{v}}[\omega]{\hat{x}}_c^{\textrm{v}}[\omega^\prime]\rangle=\frac{i}{2}\delta(\omega+\omega^\prime). \end{aligned}$$

Then the symmetrized noise spectrum density $\bar {S}_{\textrm {add}}[\omega ]$ in units of $F_{\textrm {SQL}}[\omega ]^2$ becomes

(19)$$\begin{aligned}\frac{\bar{S}_{\textrm{add}}[\omega]}{F_{\textrm{SQL}}[\omega]^{2}}&=\frac{1}{4Q[\omega]}{\bigg \{}e^{{-}2r}+\left[\left(\tan\phi+Q[\omega]\right)e^{r}-\frac{\gamma_m}{\omega}e^{{-}r}\right]^{2} +\frac{1+\tan^{2}\phi}{\vert{u_m[\omega]}\vert^{2}}\left[\frac{\kappa_{0}^{2}(1-\eta_c)}{\kappa_{0}^{2}/4+\omega^{2}}+\frac{1-\eta_\textrm{det}}{\eta_\textrm{det}}\right]\\ &+\vert{\frac{2\tan\phi}{\kappa_{0}/2+i\omega}+\frac{Q[\omega]}{i\omega-\gamma_m}}\vert^{2}\frac{\kappa_0c_{s}^{2}\omega^{2}}{\vert{u_{m}[\omega]}\vert^{2}}\bar{S}_{\theta_{l}}[\omega]{\bigg \}} +\frac{k_{BT}}{\hbar\omega^2}\gamma_{m}. \end{aligned}$$

3.1 Measurement sensitivity without outcoupling and detection losses

Optical losses would set limits on sensitivity, and increasing efficiency is necessary to study the ultimate limits. Thus, we first consider an ideal case of perfect outcoupling and detection efficiencies, i.e., $\eta _{c}=1$ and $\eta _{\textrm {det}}=1$, in which the sensitivity is constrained by squeezing-suppressed SN, antisqueezing-enlarged QBA, laser phase noise, and thermal noise. To surpass SQL, which means that noise spectrum $\bar {S}_{\textrm {add}}[\omega ]$ should be less than $F_{\textrm {SQL}}[\omega ]^{2}$, we should first eliminate the antisqueezing-enhanced QBA because SQL occurs at the tradeoff between SN and QBA. Thus, the optimal homodyne angle has to fulfill

(20)$$\tan\phi_{\textrm{opt}}={-}Q[\omega_{0}]+\frac{\gamma_{m}}{\omega_{0}}e^{{-}2r}$$

at the detection frequency $\omega _{0}$. Usually for the ultrahigh Q mechanical oscillator, such as $\gamma _{m}/\kappa _{0}\sim 10^{-6}-10^{-7}$ in [28], the optimal homodyne angle is approximately equal to

(21)$$\tan\phi_{\textrm{opt}}\approx{-}Q[\omega_{0}].$$

The noise spectrum at the detection frequency $\omega _{0}$ after evading QBA becomes

(22)$$\frac{\bar{S}_{\textrm{add}}[\omega_{0}]}{F_{\textrm{SQL}}[\omega_{0}]^{2}}=\frac{1}{4Q[\omega_{0}]}\left(e^{{-}2r}+Q[\omega_{0}]^{2}\kappa_0c_s^{2}\bar{S}_{\theta_{l}}[\omega_{0}]\right)+\frac{k_{BT}}{\hbar\omega_{0}^{2}}\gamma_{m}.$$

Therefore, the squeezed light cannot keep improving the sensitivity by increasing squeezing degree or pumping power, which is limited by the thermal noise and laser phase noise. Thermal noise is only controllable by increasing mechanical Q factor and optical cooling techniques, and here we focus on the tradeoff between SN and phase noise that is balanced by pump power $P$. At the detection frequency $\omega _{0}=\kappa _{0}/2$ (without considering thermal noise and setting $\bar {S}_{\theta _{l}}=\bar {S}_{\theta _{l}}[\omega _{0}]$ hereafter),

(23)$$\frac{\bar{S}_{\textrm{add}}[\omega_{0}]}{F_{\textrm{SQL}}[\omega_{0}]^{2}}=\frac{e^{{-}2r}}{2P}+\frac{m\kappa_{0}^2\bar{S}_{\theta_{l}}}{16\hbar\eta^2}P^2.$$

To beat SQL, phase noise and the optimal pump power should fulfill

(24)$$\bar{S}_{\theta_{l}}\leq\frac{256\hbar\eta^2}{27m\kappa_{0}}e^{4r},\hspace{5pt} P_{\textrm{opt}}=\sqrt[3]{\frac{4\hbar\eta^2}{m\kappa_{0}^{2}\bar{S}_{\theta_{l}}}e^{{-}2r}},$$

where the demands on phase noise and optimal pump power are exponentially reduced at the same time. With the choice of experimental parameters [28] $m=80~\textrm {ng}$ and $\kappa _{0}=1~\textrm {MHz}$, it indicates that to obtain sub-SQL detection phase noise should fulfill $\bar {S}_{\theta _l}\leq 1.7\times 10^{-20}~\textrm {Hz}^{-1}$ without squeezing. However, the phase noise $\bar {S}_{\theta _{l}}$ in optomechancial cooling experiment is on the scale of $10^{-16}~\textrm {Hz}^{-1}$ [69]. To further reduce the phase noise, the laser should pass through an additional filter cavity with linewidth $\kappa _{f}<10~\textrm {kHz}$ substantially lower than detection frequency, where the phase noise is passively filtered by a factor of $(2\omega /\kappa _{f})^{2}$ when the cavity is locked on resonance with the laser. With the use of squeezed light, the sub-SQL sensitivity is achievable due to the balance between squashed SN and reduced phase noise at a lower pump power. In Fig. 2, we present the spectrum density $\bar {S}_{\textrm {add}}[\omega _{0}]$ in unit of $F_{\textrm {SQL}}[\omega _{0}]^2$ as a function of pump power $P$ with different squeezing parameters. Sub-SQL detection is unreachable without squeezing for the given parameters, but large squeezing could help to realize it. It also clearly indicates that the increasing of squeezing contributes to detection sensitivity and decrease of optimal pump power.

Fig. 2. Noise spectrum $\bar {S}_{\textrm {add}}[\omega ]$ in unit of $F_{\textrm {SQL}}[\omega ]^2$ at the detection frequency $\omega _{0}=\kappa _{0}/2$ versus the pump power $P$ with different squeezing parameters. Here $m=80~\textrm {ng}$, $\kappa _{0}=1~\textrm {MHz}$, $\eta =4.182\times 10^8~\textrm {m}^{-1}$, and $\bar {S}_{\theta _{l}}=10^{-16}~\textrm {Hz}^{-1}$. Sub-SQL sensitivity is indicated by the colored region.

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Moreover, variational readout, in which the optical readout quadrature is changed as a function of frequency ($\phi _{\textrm {opt}}[\omega ]$), has demonstrated in broadband displacement detections [35,70]. Here in variational readout the optimal phase angle $\phi _{\textrm {opt}}[\omega ]=-\arctan Q[\omega ]$ changes with detection frequencies, and the noise spectrum becomes (also ignoring the effect of thermal noise)

(25)$$\frac{\bar{S}_{\textrm{add}}[\omega]}{F_{\textrm{SQL}}[\omega]^2}=\frac{1}{4Q[\omega]}\left(e^{{-}2r}+Q[\omega]^2\kappa_{0}c_{s}^{2}\bar{S}_{\theta_{l}}[\omega]\right).$$

Also to beat SQL, limits on phase noise and optimal pump power are both frequency dependent and are given by

(26)$$\bar{S}_{\theta_{l}}[\omega]\leq\frac{512}{27}\frac{\hbar\eta^2}{m\kappa_{0}^{2}}\frac{\kappa_{0}^{2}/4}{\kappa_{0}^{2}/4+\omega^2}e^{4r}, \hspace{5pt} P_{\textrm{opt}}[\omega]=\sqrt[3]{\frac{\hbar\eta^2}{m\kappa_{0}^{2}\bar{S}_{\theta_{l}}[\omega]}\left(\frac{\kappa_{0}^{2}/4+\omega^2}{\kappa_{0}^{2}/4}\right)^2 e^{{-}2r}}.$$

At high frequencies ($\omega \gg \gamma _{c}$), the phase noise spectrum rolls off as $\bar {S}_{\theta _{l}}[\omega ]\simeq \frac {2\gamma \gamma _{c}}{\omega ^{4}}\propto 1/\omega ^{4}$. Therefore, the phase noise at high frequencies is smaller than that of lower frequencies, and compared with noise spectrum $\bar {S}_{\theta _{l}}$ at $\omega _{0}=\kappa _{0}/2$, $\bar {S}_{\theta _{l}}[\omega ]\simeq \bar {S}_{\theta _{l}}\omega _{0}^{4}/\omega ^4$. Therefore, at optimal pump power the noise spectrum approximately equals to

(27)$$\frac{\bar{S}_{\textrm{add}}[\omega]}{F_{\textrm{SQL}}[\omega]^2}\simeq\frac{3}{8}\sqrt[3]{\frac{m\bar{S}_{\theta_{l}}}{4\hbar\eta^2}} \frac{\kappa_{0}^{4/3}}{\omega^{2/3}}e^{{-}4r/3},$$

which clearly indicates the improvement of sensitivity with the increasing detection frequency and squeezing. However, the optimal pump power also increases with the increasing squeezing given by in Eq. (26). In Fig. 3 we present the numerical and analytical solutions for variational measurement which are consistent at high frequencies, and sub-SQL detection is also achievable with increasing detection frequency due to the roll-off phase noise in regions (II) and (III). But in the high-frequency region (III), the requirement on high-power pump laser is demanding in experiment.

Fig. 3. Noise spectrum $\bar {S}_{\textrm {add}}[\omega ]$ (variational readout) in unit of $F_{\textrm {SQL}}{[\omega ]^{2}}$ (left axis) and the corresponding optimal pump power $P_{\textrm {opt}}[\omega ]$ (right axis) versus the detection frequency $\omega$. Solid and dashed lines indicate the numerical and analytical results, respectively. The squeezing parameter $r=2$, and all the other parameters are the same as in Fig. 2.

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3.2 Measurement limits set by outcoupling and detection losses

The sensitivity discussed above is obtained by measuring some mixed quadratures other than normal phase, which trades off some of the mechanical signal and makes it delicate and easily destroyed by optical losses. Therefore in practice, outcoupling and detection losses should be lowered, high outcoupling and optical detection efficiencies are necessary in this way. Explicitly, after evading QBA the noise spectrum becomes (also ignoring thermal noise and in ultrahigh mechanical Q limit)

(28)$${\frac{\bar{S}_{\textrm{add}}[\omega]}{F_{\textrm{SQL}}{[\omega]^{2}}}}={\frac{1}{4Q[\omega]}}\left\{{e^{{-}2r}}+Q{[\omega]^{2}}\kappa_{0}{c_{s}^{2}}\bar{S}_{\theta_{l}}[\omega]+\left(1+Q{[\omega]^{2}}\right)\left[\frac{{\kappa_{0}^{2}}(1-\eta_{c})}{\kappa_{0}^{2}/4+\omega^{2}}+\frac{1-\eta_{\textrm{det}}}{\eta_{\textrm{det}}}\right]\right\}.$$

With the efficiencies $\eta _{c}=0.95$ and $\eta _{\textrm {det}}=0.77$ in experiment [35], losses noise may dominate over squeezed SN, and improvement by squeezing is saturated by losses, similar to the limits set on the detected ponderomotive squeezing [19]. For example, the minimum limit set by losses is ${\frac {1}{2}}\frac {1-\eta _{\textrm {det}}}{\eta _{\textrm {det}}}$ that is about $0.15$ at $Q[\omega ]=1$, which is larger than that of squeezed SN at $r>0.255$ ($2.2$ dB squeezing). Then the sensitivity is determined by the balance between the phase noise and losses. In Fig. 4, we show that the sensitivity tends to saturation with the increase of squeezing, which is set is by losses. The optimal sensitivity cannot beat SQL even at high frequency $\omega =10\omega _{0}$ in Fig. 4(a). Thus the increase of detection efficiency is also essential to sensitivity, and the overall detection efficiency $0.85$ has been reached in [46]. We numerically show that the SQL can be surpassed with the high squeezing as shown in Fig. 4(b) when the detection efficiency reaches $\eta _{\textrm {det}}=0.95$.

Fig. 4. Noise spectrum $\bar {S}_{\textrm {add}}[\omega ]$ in unit of $F_{\textrm {SQL}}[\omega ]^{2}$ at the detection frequency $\omega =10\omega _{0}$ versus the pump power $P$ with different squeezing parameters. The outcoupling and detection efficiencies are $\eta _{c}=0.95$ and $\eta _{\textrm {det}}=0.77$ (a), and $\eta _{c}=0.95$ and $\eta _{\textrm {det}}=0.95$ (b). All the other parameters are the same as in Fig. 2.

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3.3 Measurement sensitivity under ideal conditions

At last, we study the ultimate sensitivity limits based on ideal conditions, i.e., without technical and thermal noises. The noise spectrum becomes a simplified form

(29)$${\bar{S}_{\textrm{add}}[\omega]}={\frac{F_{\textrm{SQL}}[\omega]^{2}}{4Q[\omega]}}\left\{e^{{-}2r}+\left[Q[\omega]+\tan\phi\right]^{2} {e}^{2r}\right\},$$

which is constrained by the squeezed SN and antisqueezed QBA. Similarly, the antisqueezed QBA is eliminated by tuning the homodyne angle to fulfill $\tan {\phi _{\textrm {opt}}}=-Q[\omega ]$, which is

(30)$$\tan\phi_{\textrm{opt}}[\omega]=\frac{-P\kappa_{0}^{2}/4}{\kappa_{0}^{2}/4+\omega^{2}},$$

and the optimal sensitivity

(31)$$\frac{\bar{S}_{\textrm{add}}[\omega]}{F_{\textrm{SQL}}[\omega]^{2}}=\frac{\kappa_{0}^{2}/4+\omega^{2}}{P\kappa_{0}^{2}}e^{{-}2r}$$

is squeezing-improved. In Fig. 5 we present the sensitivities for $\phi =-26.5^{o}$ readout quadrature (corresponding to optimal homodybe angle at detection frequency $\omega _{0}=\kappa _{0}/2$) and varied (variational readout) quadrature. It is obvious that variational readout offers useful opportunities to advance broadband detection.

Fig. 5. Noise spectrum $\bar {S}_{\textrm {add}}[\omega ]$ in unit of $\hat {F}_{\textrm {SQL}}[\omega ]^{2}$ for at $\phi =-26.5^{o}$ quadrature (corresponding to optimal homodyne angle at frequency $\omega _{0}=\kappa _{0}/2$) and varied quadrature versus the detection frequency $\omega$. The pump power is $P=1$, and the squeezing parameter $r=2$.

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4. Discussions

Here the system operates in the free-mass regime, where detection frequency is much larger than mechanical frequency. Because for a mechanical oscillator with effective mass $m$, resonance frequency $\omega _{m}$, and damping rate $\gamma _{m}$, the mechanical susceptibility is $\chi _{m}[\omega ]={m^{-1}}({\omega _{m}}^{2}-{\omega ^{2}}-i{\gamma _{m}}{\omega })^{{-1}}$. When the detection frequency $\omega$ is much larger than mechanical frequency $\omega _{m}$, i.e., in the limit of $\omega \gg \omega _{m}$, mechanical susceptibility approximately equals to $\chi _{m}[\omega ]\simeq {m^{-1}}(-\omega ^{2}-i\gamma _{m}\omega )^{-1}$ by ignoring mechanical frequency, which is the form of free-mass limit. Therefore, on the characteristic timescale of detection, the mirror can be well thought of as free mass. Currently, the dissipative optomechanical setup is far from sideband-resolved regime [28], where a fundamental frequency of oscillation is $\omega _{m}=136~\textrm {kHz}$ and cavity bandwidth $\kappa _{0}/2\pi$ is tunable between $0.7$ and $1.5~\textrm {MHz}$. Here we consider the detection frequency $\omega$ is on the scale of $\kappa _{0}$, where mechanical frequency $\omega _{m}$ become negligible and system operates sufficiently into the free-mass regime.

In addition, we consider the optical mode is resonantly pumped in dissipative optomechanics. Resonant field can enhance dynamic backaction and thus reduce the SN, and importantly, recent experiment demonstrates cooling of mechanical motion on cavity resonance [28]. Moreover, herein the sensitive detection is based on the idea of speed meter, which is a QND meter to beat SQL. Assuming off-resonantly pumped with a detuning $\Delta$, ideal outcoupling and detection efficiencies and without phase noise, the optical operator after linearization fulfills

(32)$$\delta\dot{\hat{c}}={-}\left(\kappa_{0}/2+i\Delta\right)\delta\hat{c}-\left(\frac{\kappa_{0}}{2}-i\Delta\right)\frac{\eta c_{s}}{2}\delta{\hat{x}}+\sqrt{\kappa_{0}}\hat{c}_{\textrm{in}}.$$

The output field is achievable with the input-output relation $\delta \hat {c}_{\textrm {out}}=-\hat {c}_{\textrm {in}}+\sqrt {\kappa _{0}}\delta \hat {c}+\frac {\sqrt {\kappa _{0}}}{2}\eta c_{s}\delta {\hat {x}}$, and in the frequency domain is

(33)$$\delta\hat{c}_{\textrm{out}}[\omega]=\frac{\kappa_{0}/2-i(\Delta-\omega)}{\kappa_{0}/2+i(\Delta-\omega)}\hat{c}_{\textrm{in}}[\omega]+\frac{\sqrt{\kappa_{0}}\eta c_{s}}{2}\frac{i(2\Delta-\omega)\delta{\hat{x}}[\omega]}{\kappa_{0}/2+i(\Delta-\omega)}.$$

Obviously in the case of $\Delta =0$, the optomechanical interaction can provide the pure speed information $-i\omega \delta {\hat {x}}[\omega ]$.

The external squeezing technique to enhance the detection precision, has been recently demonstrated in the LIGO and cavity optomechanics setup [52], where SN is squeezed without influencing the signal enhancement. Alternatively, intracavity squeezed optomechanics where squeezing is created directly inside the cavity by a nonlinear medium has also been demonstrated in [46]. The squeezing occurs inside optical cavity linewidth, in which noise is squeezed while signal is deamplified. Moreover, by now it seems difficult to extend the implement directly to intracavity-squeezed dissipative optomechanics for the large nonlinear crystal and different fundamental wavelengths. In contrast, intensive researches have been carried out to improve the squeezed source, and a $15$ dB at $1064$ nm squeezed state based on periodically poled PPKTP is detected [64]. Furthermore, bright squeezed light at a wavelength of $1064$ nm has been used to allow a $20\%$ improvement in magnetic field sensitivity [53] and quantum enhanced feedback cooling of a mechanical oscillator [71]. Bright squeezed source should be also compatible to the dissipative optomechanical setup that also operates at the wavelength of $1064$ nm [28], and thus externally-injected squeezing seems more practical on experiments.

5. Conclusions

To conclude, we have investigated how to significantly improve force measurement sensitivity in a dissipatively coupled optomechanical speed meter with the use of squeezed light in presence of laser phase noise and optical losses. Squeezed light could produce a reduced SN on the squeezed quadrature and an increased QBA due to the antisqueezed conjugate quadrature. Through appropriately choosing the phase angle in homodyne detection, QBA is cancellable, and the optimal sensitivity is dominated by the balance between SN and laser phase noise. For SN, squeezed quadrature can exponentially reduce the noise floor, and the balance needs the reduction of laser phase noise, leading to the exponentially reduction of optimal laser power accordingly. Thus, sub-SQL detection is achievable at a lower pump power. However, the improvement by squeezing is limited by optical losses, and thus high outcoupling and detection efficiencies are necessary to beat SQL in practice.

Funding

National Natural Science Foundation of China (11504031, 11774054, 61505014); Basic Research Program of Jiangsu Province (BK20161410); Yangtze Funds for Youth Teams of Science and Technology Innovation (2015cqt03).

Disclosures

The authors declare no conflicts of interest.

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Force measurement in squeezed dissipative optomechanics in the presence of laser phase noise

Abstract

1. Introduction

2. Squeezed dissipative optomechanics in the presence of laser phase noise

3. Measurement sensitivity improved by squeezed light

3.1 Measurement sensitivity without outcoupling and detection losses

3.2 Measurement limits set by outcoupling and detection losses

3.3 Measurement sensitivity under ideal conditions

4. Discussions

5. Conclusions

Funding

Disclosures

References

Cited By

Figures (5)

Equations (33)

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