Optomechanical noise suppression with the optimal squeezing process

Chang-Wei Wang; Wei Niu; Yang Zhang; Jiong Cheng; Wen-Zhao Zhang

doi:10.1364/OE.477710

1. Introduction

Ultrasensitive detection with enhanced quantum effects has a wide range of applications in the field of precision detection, such as gravitational wave detection [1–4], weak field detection [5–7] and tiny mass detection [8,9]. In these applications, the optomechanical system has attracted extensive attention due to its particular type of interactions. With the help of homodyne detection, the optomechanical system exhibits extremely high detection accuracy near the frequency range of the mechanical oscillator [10,11]. The existence of the uncertainty relation, on the other hand, has set a universal boundary for such kind of detection system. To be specific, there is a restrictive relationship between the backaction noise and shot noise, which is induced by the linearized coupling coefficients $G$. This is the so-called standard quantum limit (SQL) [12].

A lot of schemes have been proposed to achieve or even break the SQL, such as nonlinearity-induced optomechanical force sensing [13], structured environmental optimized ultrasensitive detection [10] and noise cancelled weak signal detection [5,14,15]. Among these schemes, squeezing as a very important quantum resource, has unparalleled advantages in noise suppression [16–21], which can suppress the amplitude of noise at the source [14,15,22–24]. Therefore, noise reduction schemes based on squeezing have been widely studied, i.e., quantum phase sensing by injecting two single-mode squeezed states [25], Kerr phase estimation with two-mode squeezed vacuum states [23] and quantum estimation based on interferometer with squeezed light [26–30]. However, as a quantum resource, squeezing is also very difficult to obtain and maintain [31,32]. In particular, high dimensional squeezed states are difficult to prepare experimentally, optically measured intensity squeezing of $20.3\%$ in pillar-shaped semiconductor microcavities in the strong coupling regime [33] and $15 \text {dB}$ squeezed states of light with a nonmonolithic optical parametric amplifier cavity [34,35] for instance. Thus, maximizing the effect of squeezing in the detection process to reduce noise is a essential task. In this paper, we focus on the optomechanical sensor, as a great application prospect [36,37], to explore the mechanism of optical squeezing-induced noise suppression in weak signal detection, and finally propose an optimal noise reduction scheme based on squeezing. In addition, we also explore the restrictive relationship between optical dissipation and mechanical dissipation.

The paper is organized as follows. In Sec. 2, we introduce the model and Hamiltonian to explore the impact of squeezing in detection. Then, the dynamics of the system is derived to analyze the additional noise spectrums and signal amplification spectrums with three representative detection schemes. In Sec. 3, we explored the limits of the capabilities of squeezing in detection optimization and so as to uncover the sources of different noises. An example of weak angular momentum detection and the restrictive relationship between the two dissipation channels induced by the uncertainty relation are shown in Sec. 4. A summary is given in Sec. 5.

2. Model and Hamiltonian

In our model, as shown in Fig. 1, the angular velocity detector consists of two parts, a hybrid optomechanical system for receiving the angular velocity signal and a homodyne detection system for detecting the output signal. The core device for detection is composed of a microdisk optomechanical system [38–40] and a squeezing medium. In this scheme, the optical squeezing is introduced to reduce the detection noise. However, to exclude the influence of other factors, and to discuss the role of optical squeezing more directly, the Sagnac effect arising from rotation should be eliminated. To this end, we can choose the direction of the angular momentum of the signal to be measured parallel to the symmetry axis ($z$-axis in Fig. 1) of the microdisk. As illustrated in Fig. 1, the hybrid optomechanical system and the corresponding optical auxiliaries are fixed on a rotatable table [41–43] in the $x-z$ plane. The direction of the angular momentum coincides with the axis of symmetry of the whispering-gallery-mode (WGM) optomechanical cavity, i.e. the $z$-axis. Therefore, no Sagnac effect causes the frequency difference between the clockwise and counterclockwise light. We assume that the displacement response in the radial direction of the microdisk to the force is linear. By studying the radiation pressure effect of the optical mode, we can derive the displacement of the mechanical oscillator induced by the centrifugal force (see Appendix A and B for details, we also compare the Sagnac effect with the centrifugal force effect, and the former can be basically neglected in the case of a small microdisk radius). The corresponding Hamiltonian of the system reads

(1)$$\begin{aligned}\hat{H} &=\sum_{j=L,R} \left[\omega_j \hat{a}_j^{\dagger}\hat{a}_j-g_j \hat{a}_j^{\dagger}\hat{a}_j\hat{q}+\xi_j(\hat{a}_j^{\dagger}\hat{a}_j^{\dagger}+h.c.)+\varepsilon_{j}(e^{{-}i\omega_{dj}t}\hat{a}_{j}^{\dagger}+h.c.)\right]\\ &\quad+\frac{\omega_{m}}{2}(\hat{p}^2+\hat{q}^2), \end{aligned}$$

where $\hat {p}$ and $\hat {q}$ are momentum and position operators of the mechanical mode of the microdisk. $\hat {a}_L$ and $\hat {a}_R$ indicate the counterclockwise and clockwise optical modes in the microdisk cavity with frequencies $\omega _{L}$ and $\omega _{R}$, respectively, which are marked with red and blue arrows in Fig. 1. $g_j$ denotes the strength of the single-photon coupling between the mechanical mode and the optical mode transmitted with $j$-direction. $\xi _j$ denotes the squeezing factor of the optical modes, which can be achieved by introducing a squeezing medium inside or outside the cavity [16,44,45]. Both optical modes are driven by the external continuous-wave laser with intensity $\varepsilon _j \equiv 2\sqrt {P_j\kappa _{ex}/\hbar \omega _{dj}}$ and frequency $\omega _{dj}$ [46]. The creation of such a drive in our model can be achieved by proximity coupling of the tapered fibre or waveguide to the whispering gallery cavity [47–52], where the rotation axis of the cavity is perpendicular to the fibre or waveguide. In the frame rotating at the drive frequency and under strong driving conditions, the dynamics of the system can be linearized around it’s steady-state values, i.e., $\hat {a}_j\rightarrow {\langle \hat {a}_j\rangle +\delta \hat {a}_j}$, $\hat {p}\rightarrow {\langle \hat {p}\rangle +\delta \hat {p}}$ and $\hat {q}\rightarrow {\langle \hat {q}\rangle +\delta \hat {q}}$, where $\langle \hat {a}_j\rangle$, $\langle \hat {p}\rangle$ and $\langle \hat {q}\rangle$ denote the average values of the optical mode, mechanical momentum and coordinate, respectively. The quantum Langevin equations characterising the dynamics of the system can be expressed as

(2)$$\begin{aligned} \delta \dot{\hat{a}}_{j}&={-}(i\Delta_j+\frac{\kappa_j}{2})\delta{\hat{a}}_{j}+iG_j\delta \hat{q}-2i\xi_{j}\delta{\hat{a}}_{j}^{\dagger}+\sqrt{\kappa_{j}}\hat{a}_{jin},\\ \delta \ddot{\hat{q}}/\omega_m&={-}\omega_m\delta\hat{q}+\sum_{j=L,R}(G_j^{*}\delta{\hat{a}}_{j}+h.c.)-\gamma \delta{\dot{\hat{q}}}+ F_{in}, \end{aligned}$$

where $\Delta _j=\omega _j-\omega _{dj}-g_j\langle \hat {q}\rangle$ denotes the effective detuning of the mean-field modification under strong driving conditions. $G_j=g_j\langle \hat {a}_{j} \rangle$ is the linearized enhanced optomechanical coupling (In our model, we eliminate the Sagnac effect so that the frequencies of the left- and right-row optical modes remain unchanged, so it is natural to assume that the single-photon coupling coefficients are the same, i.e., $g_j=g$). $\hat {a}_{jin}$ is the input noise operator of the optical modes, satisfying the commutation relation $\left [\hat {a}_{jin},\hat {a}_{kin}^{\dagger }\right ]=\delta (j,k)$. $F_{in}$ represents the external input of the mechanical mode, which can be divided into two parts. The noise input from the environment $\sqrt {\gamma }P_{th}$ and the signal input from the tiny angular frequency $\omega _r$. The environmental noise satisfies the thermal correlation $\langle P_{th}^{\dagger }(\omega )P_{th}(\omega ') \rangle =\delta (\omega,\omega ')n_{th}$, where $n_{th}=[\exp (\hbar \omega _m/k_B T)-1]^{-1}$ denotes the thermal phonon number. A small change in rotation with angular velocity $\omega _r$ results in a displacement shift of the mechanical oscillator and thus be detected by the system, and the corresponding effective energy can be formally written as $f(\omega _r)\hat {q}$, which is similar to the form of the environmental input noise under the Born-Markov approximation. Here, $f(\omega _r)=\omega _r^2 \omega _c/(8g \omega _m)$ denotes the response function of the mechanical mode to the rotation, which can be derived from the rotational force by a semi-classical method (see Appendix A for details). $\omega _c$ is the resonant frequency of the uncoupled optical cavity. To analyze the spectral information of the system in the frequency domain and then detect it, we perform a Fourier transform on Eqs. (2), and the transformed equations are as follows

(3)$$\begin{aligned} \delta\hat{a}_j{(\omega)}&=\chi_j\left[iG_j\delta\hat{q}(\omega)-2i\xi_j\delta\hat{a}_j^{\dagger}(\omega)+\sqrt{\kappa_j}\hat{a}_{jin}(\omega)\right],\\ \delta\hat{q}(\omega)&=\chi_m \left\{ \sum_{j=L,R}\left[G_j^{*}\delta\hat{a}_j(\omega)+h.c.\right]+F_{in}(\omega)\right\}, \end{aligned}$$

where $\chi _j$ and $\chi _m$ denote the susceptibilities of the optical and mechanical modes, respectively. The corresponding expressions are as follows

(4)$$\begin{aligned}\chi_j&=\frac{1}{i(\Delta_j-\omega)+\kappa_j/2}=[\chi_j^{\dagger}(-\omega)]^{*},\\ \chi_m&=\frac{1}{-\omega^{2}/\omega_m+\omega_m-i\gamma\omega/\omega_m}. \end{aligned}$$

Fig. 1. Model diagram of the angular velocity detection system (AVDS). It consists of two parts. One is a hybrid optomechanical system for receiving the angular velocity signal, and the other is an external homodyne detection device. The laser light is divided into two beams by a beam splitter (BS). One of the beams is injected into the detection device as the driving light and the other serves as a bright local oscillator (LO) for homodyne detection. Electro-optic modulator (EOM) is used to adjust the phase of the light. Optical superposition device (OSD) is used to superimpose the output light of the left and right rows of the optomechanical system. The polarisation beam splitter (PBS) transmits horizontal light and reflects vertical light. Two $\lambda /4$ waveplates rotate the polarisation of the output signal from horizontal to vertical. The subgraph shows the geometric relationship between the rotation and centrifugal force.

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Using the standard input-output relation $\hat {O}_{out}=\sqrt {\kappa }\hat {O}-\hat {O}_{in}$ and combined with the model of our scheme, the output operator of the microdisk can be described as $\hat {a}_{out}=x \hat {a}_{Lout}+y\hat {a}_{Rout}$ and the corresponding superposition coefficients satisfy $|x|^2+|y|^2=1$, in which the optical superposition device (OSD) can be implemented with a quantum beam splitter (BS) and $x$ and $y$ then correspond to the transmission and reflection coefficients of the BS. Under the strong-LO limit, the output operator of the balanced homodyne detection in Fig. 1 can be expressed as [53,54]

(5)$$\begin{aligned} \hat{M}_{out}&=\eta[\sin{\theta}\hat{X}_{out}(\omega)+\cos{\theta}\hat{Y}_{out}(\omega)]\\ &=i\eta[\hat{a}_{out}^{\dagger}(-\omega)e^{{-}i\theta}-\hat{a}_{out}(\omega)e^{i\theta}]\\ &=A{(\omega)}\hat{a}_{Lin}+A^{\dagger}{(\omega)}\hat{a}_{Lin}^{\dagger}+B{(\omega)}\hat{a}_{Rin}+B^{\dagger}{(\omega)}\hat{a}_{Rin}^{\dagger}+C{(\omega)}F_{in}, \end{aligned}$$

where $\hat {X}_{out}(\omega )=[(\hat {a}_{out}^{\dagger }(-\omega )+\hat {a}_{out}(\omega )]/\sqrt {2}$ and $\hat {Y}_{out}(\omega )=[(\hat {a}_{out}(\omega )-\hat {a}_{out}^{\dagger }(-\omega )]/i\sqrt {2}$ denotes the position and momentum operators of the output field, respectively. $\eta \in [0,1]$ denotes the output efficiency of the signal due to the imperfect experiment. The output spectrum of the signal is reduced by a total ratio $\eta ^2$. In the following discussion we choose $\eta =1$. $\theta$ represents the phase difference between the two output ports in the homodyne detection. This phase can be controlled by an electro-optic modulator (EOM). With the given $x$ and $\theta$, we can calculate the expression of $\hat {M}_{out}$ (later we will give the exact expression of $A(\omega )$, $B(\omega )$ and $C(\omega )$ in Eq. (9)). According to the above equation, all the information of the signal is in $F_{in}$, while the rest terms of the equation are the output system noise that we do not need. The additional noise operator can therefore be defined as

(6)$$\hat{F}_{add}=A{(\omega)}\hat{a}_{Lin}+A^{\dagger}{(\omega)}\hat{a}_{Lin}^{\dagger}+B{(\omega)}\hat{a}_{Rin}+B^{\dagger}{(\omega)}\hat{a}_{Rin}^{\dagger}.$$

The corresponding signal-to-noise ratio (SNR) operator can be defined as

(7)$$SNR_O=\frac{C{(\omega)}F_{in}}{\hat{F}_{add}+C{(\omega)}\sqrt{\gamma}P_{th}}.$$

Combining the above operators in the frequency domain with the standard spectral definition [54,55], we can obtain the corresponding output spectrum

(8)$$S_{OO}(\omega)=\frac{1}{2}\left[\int d\omega' \langle \hat{O}(\omega) \hat{O}(\omega') \rangle+\int d\omega' \langle \hat{O}(-\omega) \hat{O}(\omega') \rangle\right],$$

where the operator $\hat {O}\in \{\hat {M}_{out},\hat {F}_{add},SNR_O \}$. In the $\hat {a}_{out}$ of Eq. (5), $x$ and $y$ determine the composition of the output light, which is the weight of the left and right mode, respectively. The other parameter, $\theta$, determines the physical quantity to be detected. When $\theta =0$ and $\pi$, $\hat {M}_{out}$ is the momentum operator and the position operator, respectively. A numerical simulation was performed to investigate the effect of these two parameters on the SNR, as shown in Fig. 2. In the simulation, we have only selected the SNR at the optimal detection frequency window, i.e. the maximum value of the SNR spectrum. Since the global phase in the superposition coefficients has no effect, the figure is divided into two groups of $x$ and $y$ with the same and different signs, i.e., $SNR_m$ and $SNR_p$. As shown in Fig. 2, for the same sign case, $x = 1/\sqrt {2}$ is optimal; for the different sign cases, $x = 0$ or $-1$ is optimal, and overall the same sign is better than the different sign. $\theta$ in the two cases are optimal at the multiple of $\pi$ (other cases of different $\xi$ are shown in Fig. 9 in the Appendix, which is consistent with our conclusion here).

Fig. 2. SNR as a function of superposition coefficient $x$ and homodyne detection phase $\theta$. In our calculation, as the global phase of the superposition coefficient $x$ and $y$ is meaningless, the problem can be simplified as the same (p) or different (m) signs of $x$ and $y$. We choose $\omega _r=10^{-2}$Hz, $\xi /\omega _m=0.3$. Other values of $\xi$ are shown in Fig. 9 in the Appendix. Parameters are $\Delta _L/\omega _m=1$, $\Delta _R/\omega _m=1.005$, $G_L=G_R=5.6\times 10^{-3} \omega _m$, $\kappa _L=\kappa _R=10^{-2} \omega _m$, $\gamma /\omega _m=10^{-5}$, $n_{th}=0$.

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To address $\hat {M}_{out}$ specifically, the values of $x$ and $y$ are needed. In the following discussion, we select only three representative parameters, i.e., $x=y=1/\sqrt {2}$, $x=-y=1/\sqrt {2}$, and $x=1$, which are marked with subscripts ‘p’, ‘m’, and ‘s’, respectively. For simplicity, we take $\theta =0$, $G_j=G$ and $\kappa _j=\kappa$. The phase can be adjusted by EOM, and the identical linearized coupling coefficient can be achieved by adjusting the same laser drive intensity in both directions (e.g., using a $50:50$ beam splitter). Thus, the coefficients in the above equation can be derived as

(9)$$\begin{aligned} A^{p}(\omega)&=\frac{\sum_{n=0}^4A^+_n \xi^n}{\sum_{n=0}^4 D_n \xi^n},~A^{m}(\omega)=\frac{\sum_{n=0}^4A^{-}_n \xi^n}{\sum_{n=0}^4 D_n \xi^n},\\ B^{p}(\omega)&=\frac{\sum_{n=0}^4B^+_n \xi^n}{\sum_{n=0}^4 D_n \xi^n},~B^{m}(\omega)=\frac{\sum_{n=0}^4B^{-}_n \xi^n}{\sum_{n=0}^4 D_n \xi^n},\\ C^{p}(\omega)&=\frac{C_0^+{+}C_2^+\xi^2}{\sum_{n=0}^4 D_n \xi^n},~C^{m}(\omega)=\frac{C_0^{-}+C_2^{-}\xi^2}{\sum_{n=0}^4 D_n \xi^n}, \end{aligned}$$

For the ‘s’ case, we just need to take $\chi _R=0$. In the ‘p’ case, to obtain the corresponding expressions of the parameters, i.e., $O^{s}(\omega )=O^{p} (\omega,\chi _R=0),\{ O=A,B,C\}$. The specific parameters are expressed as follows,

(10)$$\begin{aligned} A^{{\pm}}_0&=\frac{i}{\sqrt{2}}D_0+\kappa\chi_L[i-2G^2\chi_m(\chi_L^{\dagger}\pm\chi_R^{\dagger})],\\ A^{{\pm}}_1&=\frac{i}{\sqrt{2}}D_1+2\kappa\chi_L\chi_L^{\dagger}\mp4iG^2\kappa(\chi_L\chi_L^{\dagger}\chi_R\mp\chi_R\chi_R^{\dagger}\chi_L),\\ A^{{\pm}}_2&=\frac{i}{\sqrt{2}}D_2-i4\kappa\chi_L\chi_R\chi_R^{\dagger}+16G^2\kappa\chi_m\chi_L\chi_L^{\dagger}\chi_R\chi_R^{\dagger},\\ A_3^{{\pm}}&=\frac{i}{\sqrt{2}}D_3-8\kappa\chi_L\chi_L^{\dagger}\chi_R\chi_R^{\dagger},\\ A_4^{{\pm}}&=\frac{i}{\sqrt{2}}D_4,\\ B^{{\pm}}_0&=\frac{i}{\sqrt{2}}D_0+\kappa\chi_R[i-2G^2\chi_m(\chi_R^{\dagger}\pm\chi_L^{\dagger})],\\ B^{{\pm}}_1&=\frac{i}{\sqrt{2}}D_1+2\kappa\chi_R\chi_R^{\dagger}\mp4iG^2\kappa(\chi_R\chi_R^{\dagger}\chi_L\mp\chi_L\chi_L^{\dagger}\chi_R),\\ B^{{\pm}}_2&=\frac{i}{\sqrt{2}}D_2-i4\kappa\chi_R\chi_L\chi_L^{\dagger}+16G^2\kappa\chi_m\chi_R\chi_R^{\dagger}\chi_L\chi_L^{\dagger},\\ B_3^{{\pm}}&=\frac{i}{\sqrt{2}}D_3-8\kappa\chi_R\chi_R^{\dagger}\chi_L\chi_L^{\dagger},\\ B_4^{{\pm}}&=\frac{i}{\sqrt{2}}D_4,\\ C^{{\pm}}_0&={-}G\sqrt{\kappa}\chi_m\left[(\chi_L+\chi_L^{\dagger})\pm(\chi_R+\chi_R^{\dagger})\right],\\ C^{{\pm}}_2&=4G\sqrt{\kappa}\chi_m\left[\chi_R\chi_R^{\dagger}(\chi_L+\chi_L^{\dagger})\pm\chi_L\chi_L^{\dagger}(\chi_R+\chi_R^{\dagger})\right],\\ D_0&={-}\sqrt{2}+i\sqrt{2}\chi_m(\chi_L-\chi_L^{\dagger}+\chi_R-\chi_R^{\dagger}),\\ D_1&={-}4\sqrt{2}G^2\chi_m(\chi_L\chi_L^{\dagger}+\chi_R\chi_R^{\dagger}),\\ D_2&=4\sqrt{2}\left\{(\chi_L\chi_L^{\dagger}+\chi_R\chi_R^{\dagger})-iG^2\chi_m\left[\chi_L\chi_L^{\dagger}(\chi_R-\chi_R^{\dagger})+\chi_R\chi_R^{\dagger}(\chi_L-\chi_L^{\dagger})\right]\right\},\\ D_3&=32\sqrt{2}G^2\chi_m\chi_L\chi_L^{\dagger}\chi_R\chi_R^{\dagger},\\ D_4&={-}16\sqrt{2}\chi_L\chi_L^{\dagger}\chi_R\chi_R^{\dagger}, \end{aligned}$$

in which $\chi _j^{\dagger }=\chi _j^{*}(-\omega )$. This comes from the Fourier transform $\left (\int \hat {O}(t) e^{-i2\pi \omega t}\right )^{\dagger }=\int \hat {O}^{\dagger }(t) e^{+i2\pi \omega t}$. From the above formulas, it is derived that the denominators are the same for both the ‘p’ and ‘m’ cases, that is, these two cases are mainly affected by the change in the numerator, which is reflected in the organization of parameters shown in Eqs. (10).

3. Detection capability of our system

In this section, we will discuss the measured signal and the suppressed noise by analyzing the characteristics of the output field. According to the standard spectral definition [5] and the noise correlation function under the Born-Markov approximation, where the photon input noise satisfies the $\delta$-correlation, and the phonon input noise satisfies the thermal correlation, the spectral density of the additional noise and the signal amplification can be obtained by taking Eq. (6),

(11)$$\begin{aligned} S_{add}^k{(\omega)}&=\frac{1}{2}\left[\left|\frac{A^k(\omega)}{C^k(\omega)}\right|^2+\left|\frac{B^k(\omega)}{C^k(\omega)}\right|^2\right],\\ A_{p}^k(\omega)&=|C^k(\omega)|^2, \end{aligned}$$

where $k=\{ p,m,s\}$. The minimum additional noise $\text {Min}[S_{add}^{k}(\omega )]$, as a function of the coupling and squeezing coefficients, is shown in Fig. 3, where (a), (b), and (c) correspond to the ‘p’, ‘m’ and ‘s’ schemes, respectively. As seen in the figure, the lowest additional noise can be achieved by selecting the appropriate $G$ and $\xi$. Besides, comparing the three schemes, the ‘m’ scheme has the highest noise, and the ‘p’ scheme has the lowest noise, namely the ‘p’ scheme is more beneficial for noise reduction overall ,which is also consistent with our conclusion in Fig. 2. To better illustrate the role of squeezing in the detection, the optimization factor is defined to characterize the suppression effect of squeezing on the system noise. For the three different schemes mentioned above, the optimization factor can be expressed as

(12)$$f_{k}=\frac{S_{SQL}^{k}}{\text{Min}[S_{add}^{k}(\omega)]}.$$

where $S_{SQL}^{k}$ denotes the SQL corresponding to the $k$ scheme and is a fixed value in the case of scheme determination, and the breakthrough of SQL denotes the optimization of the squeezing in our model. $\text {Min}[S_{add}^{k}(\omega )]$ denotes the minimum value of the noise spectrum with $k$ scheme. The definition of $f_k$ allows us to compare the optimization ability of squeezing in different protocols, and the optimization effect of squeezing is significantly improved by increasing this factor.

Fig. 3. $\text {Min}[S_{add}^{k}(\omega )]$ as a function of $G$ and $\xi$ for different detection scheme is displayed in (a), (b) and (c) respectively. (d) Noise spectrum as a function of $G$. The lowest point represents the SQL due to the competition between two $G$ related noises when $\xi =0$. (e) $\text {Min}[S_{add}^{k}(\omega )]$ as function of $\xi$. (f) Optimization factor $f_k$ as a function of $\xi$ with different detection scheme. Other parameters are $\Delta _L/\omega _m=1$, $\Delta _R/\omega _m=1.005$, $G_L=G_R=5.6\times 10^{-3} \omega _m$, $\kappa _L=\kappa _R=0.01 \omega _m$, $\gamma /\omega _m=10^{-5}$.

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The SQL for different protocols, comparison of the evolution of $\text {Min}[S_{add}^{k}(\omega )]$ and ‘optimization factor’ $f_k$ with the squeezing strength for different protocols are illustrated in Fig. 3(d), (e) and (f), respectively. As shown in Fig. 3(d), different protocols correspond to different SQLs, and the ‘p’ protocol corresponds to the lowest SQL, which means the ‘p’ protocol has the greatest potential for detection. This advantage is also shown in Fig. 3(e), where the ‘p’ protocol has the lowest noise at $\xi /\omega _m=0.3$ with corresponding $G/\omega _m=5.6\times 10^{-3}$ obtained from Fig. 3(a).

As shown in Fig. 3(f), $f_k=1$ denotes that the noise value meets the SQL. $f_k>1$ indicates that the SQL has been breached due to the squeezing effect. The optimal value of the ‘p’ protocol is $683$ times better than the SQL, while for the ‘m’ protocol, the enhancement of squeezing will amplify the noise instead. For $\xi /\omega _m<0.25$ with ‘p’ and ‘s’, the increase of squeezing strength has a positive impact on the detection, that is, the noise is getting closer to SQL. When $0.25<\xi /\omega _m<0.37$, the noise suppression is determined not only by the squeezing strength, but also by a combined effect of several factors. If $0.37<\xi /\omega _m<0.5$, clearly, the squeezing strength is no longer useful for reducing noise. This phenomenon can be reduced simplified as a matching problem between two directions of measurement and squeezing. According to Eq. (5), the homodyne detection in the $\theta =0$ condition results in a momentum-type measurement of the output operator. In the dynamics of the input and output of the squeezed optical field, the value of the squeezing coefficient not only affects the noise level but also changes the ‘direction’ of the squeezing. When $f_k$ reaches its peak, which is $\xi /\omega _m=0.3$ for the ‘p’ scheme, the squeezing ‘direction’ matches the measurement ‘direction’, and the noise can be maximally reduced by optical squeezing. Conversely, when $f_k$ reaches its valley value, the squeezing ‘direction’ is orthogonal to the measurement ‘direction’, and optical squeezing will amplify the noise instead. Especially for the ‘m’ scheme, increasing $\xi$ will obviously increase the noise. Thus, in summary, it is not better to increase the squeezing factor, because it is optimized only when $f_k$ reaches maximum. For the ‘p’ and ‘s’ schemes, the optimum value is at $\xi /\omega _m=0.3$ with our chosen parameters. Therefore, overall, the noise reduction is the most significant for the ‘p’ scheme due to the optimal optimization factor $f_p$ and the lowest SQL. Therefore, we will only focus on the ‘p’ scheme in the next discussion.

4. Weak angular momentum detection

It is important to evaluate the performance of the system by the noise spectrum and signal amplification spectrum. These two spectrums of the ‘p’ scheme with different squeezing coefficients are shown in Fig. 4. It is obvious that no matter what value of $\xi$ is, the $S_{add}$ and $A_p$ of the detection system are perfectly matched, i.e., both optimal values reach at the sideband frequency $\omega = \omega _m$ (this is because due to the fact that squeezing does not change the effective frequency). In addition, the noise is minimized and the amplification factor is maximized at $\xi /\omega _m=0.3$, which agrees with our conclusion in the previous section.

Fig. 4. Three different squeezing factors under the case of noise suppression and signal amplification. The other parameters are the same as that in Fig. 3. The schematic on the right indicates the relationship between the two types of noise and squeezing during dynamics.

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With this matching relationship between $S_{add}$ and $A_p$ as shown in Eqs. (11), it is easy to find that the denominator of $S_{add}$ is exactly $A_p$. If this matching relationship is maintained for any value of $\xi$, the suppression of noise and enhancement of the signal by the squeezing factor can only be controlled by the coefficient $C(\omega )$. Note that the expression of $C(\omega )$ only includes the $\xi ^2$ term, which means both the noise reduction and the signal amplification in our scheme are based on the suppression of shot noise. This is because the shot noise is the noise in the optical phase readout that induces an imprecision in the $\hat {q}$ measurement and with the squeezing existent in the optical mode. Therefore, it will be counted twice together with the noise effect on $\hat {q}$ and the output of the signal, i.e. the terms contain $\xi ^2$. Correspondingly, the definition suggests that only terms contain $\xi ^1$ are included in the backaction noise. In ultra-sensitive detection systems using mechanical oscillators as a probe, the input of the signal and input of the mechanical mode environment are formally the same, which means they cannot be distinguished by the system during the transformation of the quantum state from the mechanical mode to the output spectrum of the optical mode by optomechanical coupling. Since the optomechanical coupling results in the backaction noise, it is more advantageous to break the standard quantum limit by eliminating the shot noise which is separable from the input signal. In Fig. 4, this advantage is also well demonstrated since the system noise can be reduced when a large signal amplification is maintained.

In addition, we noticed that, $\kappa$ is related to the shot noise and $\gamma$ is related to the backaction noise, so there is some constraint relationship between them. It is also obvious that a competitive relationship between these two noises is shown in the Fig. 5, i.e., the dark region. This is similar to the explanation of SQL, except that the two noises do not depend on each other, which means it is possible to reduce both of them at the same time.

Fig. 5. Logarithmic values of minimal additional noise as a function of cavity dissipation $\kappa$ and mechanical damping $\gamma$. (a) is a 3D plot, and (b) is a 2D plot. The different coloured scales to the right of the coordinates correspond to the $\gamma$-values of the different coloured curves. Other parameters are the same as those in Fig. 3.

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To investigate the relationship between the additional noise and the two dissipations more specifically, the variation of $\text {Min}[S_{add}(\omega )]$ with $\kappa$ at different $\gamma$ is illustrated in Fig. 5. The color of the curve corresponds to the value of $\gamma$ taken on the right side of the coordinate, and the value of $\gamma /\omega _m$ arranged from left to right in order of equal scale variation of $\{1\times 10^{-5},2\times 10^{-5},3\times 10^{-5}\cdots \cdots \}$. It is obvious that all curves have an optimal value of $\kappa$, i.e., a valley, which is consistent with our conclusion in Fig. 5, and the related studies are reported in our previous work (in Fig. 6(d) of Ref. [5]). In addition, it is interesting that the optimal $\kappa$ takes exactly the linear relationship of $\kappa _{op}=N \gamma$ with $N=6.2\times 10^2$, which is also marked in the figure with the ‘$\kappa -\gamma$ optimal relationship curve’. With given parameters, the optimal $\kappa$ can be measured with a small amount of data to obtain the scaling $N$ and thus respond to systems with variable $\gamma$ or $\kappa$. Although we cannot give the specific expression of $N$ analytically, the dependence between $N$ and some noise-related parameters ($G$ and $n_{th}$) can be obtained by a simple analysis. It is known that, $\gamma$ is the parameter describing the noise of the mechanical mode. According to Eq. (3), we have $\delta \hat {q}(\omega ) \propto \chi _mG^* \delta \hat {a}_j(\omega )$ and $\delta \hat {a}_j{(\omega )} \propto \chi _j G\delta \hat {q}(\omega )$. Thus, the intrinsic fluctuations of the mechanical oscillator introduced into the optical mode through the optomechanical coupling are proportional to $\chi _j\chi _mG ^2\propto G^2/\gamma$, which is essentially a part of backaction noise. The total noise registered at the detector reads [12]:

(13)$$S_{xx}^{total}(\omega)= S_{xx}^{th}(\omega)+ S_{xx}^{imp}(\omega)+ S_{FF}(\omega)|\chi_{xx}(\omega)|^2,$$

where $S_{xx}^{th}(\omega )$ denotes the thermal noise spectral density, $S_{xx}^{imp}(\omega )=\frac {\kappa }{16 n_{cav}G^2}(1+4\frac {\omega ^2}{\kappa ^2})$ denotes the quantum noise-limited imprecision noise spectral density (shot noise), $S_{FF}(\omega )$ ($\propto G^2/\gamma$ in our model) denotes the spectral density of backaction noise. Under the condition $\omega \gg \kappa$, we have $S_{xx}^{imp}(\omega ) \propto 1/(G^2 \kappa )$. In general, the product of imprecision noise and backaction noise densities fulfills a fundamental inequality, for example, a variant of the Heisenberg uncertainty relation $S_{xx}^{imp}(\omega )\cdot S_{FF}(\omega )\geq \hbar ^2/4$ (see Ref. [12]). The equality “$=$" is realized, i.e. the cavity displacement detector is as good as that allowed by quantum mechanics. Thus, the minimum value can only be taken when the two noises satisfy this equality condition, and we get $\kappa /\gamma \propto 1/G^4$. The gradient of $\kappa -\gamma$ curve is directly proportional to $G$, which is consistent with the results of our numerical simulation shown in Fig. 6(a)-(c). For the thermal noise $n_{th}$, it is essentially the white noise. As seen in Eq. (5), $F_{in}$ will increase the overall noise or decrease the SNR, and the corresponding numerical simulations are shown in Fig. 6(d)-(f). The increase of thermal noise has no effect on $\kappa -\gamma$ curve. It is also noted that the selection of the three protocols of ’p’, ’m’ and ’s’ does not affect the relationship between $\kappa -\gamma$ and $G$ (this can be confirmed by numerical simulation results in Fig. 10). For the ‘Minimum additional noise curve’ in the Fig. 5(b), it exhibits a complex nonlinear relationship with $\kappa$, and specific calculations using Eq. (11) is required.

Fig. 6. Logarithmic values of minimal additional noise as a function of cavity dissipation $\kappa$ and mechanical damping $\gamma$. The linearized coupling coefficients $G/\omega _m$ for (a), (b) and (c) are $0.01$, $0.005$ and $0.001$, respectively. The thermal noise are $n_{th}=0$. The thermal noise $n_{th}$ for (c), (d) and (e) are $0.1$, $1$ and $10$, respectively. The linearized coupling coefficient is $G/\omega _m=0.005$. The black-dashed line indicates the lowest minimal additional noise. Other parameters are the same as those in Fig. 3.

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The detection capability of our model with the ‘p’ protocol is further verified with results shown in Fig. 7. As shown in Fig. 7(a), it is obvious that the output signal increases monotonously with the angular frequency $\omega _r$. When the input signal is $10^{-2}Hz$ and $10^{3}Hz$, the maximal output signal $S_{out}/\omega _m$ is $1.27\times 10^6$ and $1.27\times 10^{16}$, respectively, and the corresponding SNR is shown in Fig. 7(b). The SNR depends on the strength of the detection signal because the response of the system to the angular frequency is nonlinear ($f(\omega _r)\propto \omega _r^2$, see Appendix A for details), so a stronger signal will bring a higher SNR. In addition, we also discuss the influence of the environment on our detection scheme. The minimum noise $\text {Min}[S_{add}^{k}(\omega )]$ varies with dissipation and temperature as shown in Fig. 7(c). The area enclosed by the white dotted line in the figure represents the parameter range that can surpass the SQL. Thus, to exceed the SQL, in addition to introducing squeezing, it is necessary to ensure a sufficiently low temperature and a sufficiently small mechanical dissipation rate. Figure 7(d) shows the influence of temperature on the SNR. The increase of temperature will make the SNR decrease, with a nearly linear trend. As we have concluded in Fig. 6, thermal noise is a overall noise. When the average thermal phonon approaches $10^3$, the SNR is less than $1000$. Therefore, it is necessary to improve the quality factor of the mechanical oscillator and reduce the environmental thermal noise if we want to maintain a high SNR in detection.

Fig. 7. (a) Output spectrum as a function of different input signals $\omega _r$ (the selected frequencies of the curves from bottom to top are marked in the right side of the figure). (b) The SNR as a function of the input signal $\omega _r$(Hz). (c) The minimum noise $\text {Min}[S_{add}^{k}(\omega )]$ varies with temperature $n_{th}$ and dissipation $\gamma$. The white dotted line represents the SQL. (d) The influence of temperature on SNR for different dissipation $\gamma$ with $\omega _r=10^{-2}$Hz. The resonant frequency of the uncoupled optical cavity is $\omega _c=10^{14}\text {Hz}$. The mechanical eigenfrequency is $\omega _m=10^6\text {Hz}$. The single-photon optomechanical coupling strength is $g=10^3\text {Hz}$. Other parameters are the same with Fig. 3.

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5. Discussion and Conclusion

When the detection scheme is determined, for additional noise reduction, boosting squeezing does not enhance the suppression of noise directly. The squeezing efficiency of the steady state is strongly dependent on the squeezing strength and other parameters ($G$, $x$ and $\theta$), which means that simply increasing the squeezing parameter may increase the detection noise of the system instead. It is only when the optimization factor $f_k$ reaches its maximum that squeezing can play its maximum role, and then the noise is optimally suppressed.

In the model we discussed, the dissipation of the system we choose is $\gamma /\omega _m =10^{-5}$ and $\kappa /\omega _m=10^{-2}$. While the experimental reported parameters in whispering-gallery-mode optical microresonantor [49] are $\gamma /\omega _m =9 \times 10^{-5}$ and $\kappa /\omega _m=8 \times 10^{-2}$ with $\omega _m/2 \pi =4.2\times 10^{7}$Hz, which is close to our requirements. Recently, high-quality factors of cavities ($Q_c$) and resonators ($Q_m$) have been reported in crystalline and whispering gallery optomechanical systems, where $Q_c$ and $Q_m$ can achieve $10^{10}$ and $10^7$ [56,57], respectively. Combined with the conclusion displayed in Fig. 5 that a smaller $\kappa$ and $\gamma$ leads to a lower additional noise, an ultra-high ‘optimization-factor’ can be achieved and shown in Fig. 3 in a high quality factor optomechanical system. Moreover, the experimentally reported optical squeezing reaches $15$dB [34,35], which correspond exactly to the data at $\kappa /\omega _m=0.06$ in the Fig. 5(b) (about $15.23$dB), while the corresponding minimum additional noise is below $10^{-5}$. Furthermore it is also clear that, for a fixed squeezing factor $\xi$, a lower optical dissipation corresponds to a lower additional noise, thanks to the cumulative effect of the large decoherence time $1/\kappa$ on the squeezing in the dynamics. This implies that our scheme has a great potential for guiding experiments. As it should be, different detection schemes have different optimization results. In this paper, only the linear scheme based on homodyne detection is discussed, and the more complex or other detection schemes will be discussed in our subsequent work.

Appendix A Effect of rotation

According to the Ref. [58], after ignoring the Casimir force due to the cancellation of the divergent parts of the vacuum pressure from both sides of the low frequency mechanical oscillator, the Hamiltonian that exhibits the nonlinear nature of the coupling between the field and the moving mirror is expressed as,

(14)$$H_I=\hbar \hat{a}^{\dagger} \hat{a} \frac{n \pi c}{L+\hat{x}'},$$

where $\hat {x}'=\hat {x}+\Delta L$ denotes the total change in cavity length. $\Delta L$ denotes the change in cavity length due to rotation. Here we only care about the interaction between the optical mode and the mechanical mode, so we can naturally ignore the moment of inertia ($\hat {I}\omega _r^2/2$) of the system itself, which does not interact with the optical cavity. $\hat {x}=x_{ZPF} \hat {q}$, where $x_{ZPF}=\sqrt {\hbar /(2m \omega _m)}$ is the zero-point fluctuation amplitude of the mechanical oscillator, and $\hat {q}$ is the position operator with resonance frequency $\omega _m$. According to Eq. (14), the cavity resonance frequency is modulated by the mechanical amplitude. A linearized form of the Hamiltonian is presented when the displacement $\hat {x}'$ is much small compared with $L$.

(15)$$\begin{aligned} H_I&=\hbar \hat{a}^{\dagger} \hat{a} \frac{n \pi c}{L}\left \{1-\frac{\hat{x}+\Delta L}{L}+O(\frac{\hat{x}'}{L})^2 \right \}\\ &\approx \hbar \omega_c \hat{a}^{\dagger} \hat{a}-\hbar g \hat{a}^{\dagger} \hat{a} (\hat{b}+\hat{b}^{\dagger})-\hbar \hat{a}^{\dagger} \hat{a} \omega_c \frac{\Delta L}{L}, \end{aligned}$$

where $\omega _c \equiv \frac {n \pi c}{L}$ denotes the resonant frequency of the uncoupled optical cavity. The second term indicates the commonly used optomechanical coupling term and $g=\frac {\omega _c x_{ZPF}}{L}$ is the single-photon optomechanical coupling strength. The third term indicates the modified energy due to rotation. In addition, a specific expression of the radiation pressure is obtained using the relationship between the energy and the radiation pressure force i.e. $F_{OM}=-d H/ d \hat {q}=\hbar g \hat {a}^{\dagger } \hat {a}/x_{ZPF}$. According to our model shown in Fig. 1 and combined with the expression of classical centrifugal force $m \omega _r^2 r$, an equation is obtained as,

(16)$$F = d m \omega_r^2 r = \frac{m\omega_r^2 R}{2\pi} \cos \alpha d \alpha,$$

where $m$ denotes the mass of the mechanical oscillator with $d m=\frac {m}{2\pi }d\alpha$, and $r=R \cos \alpha$ ($R=L/2\pi$) denotes the radius of rotation at $\alpha$. Since it is assumed that the equilibrium displacement of the mechanical oscillator changes linearly with the force, the change of radius in the rotation can be written as $dr = \frac {\hat {x}}{F_{OM}}\times F_{\perp }$, where $F_{\perp }=F \cos \alpha$. The cavity length after change is $\int (R+dr)d\alpha$ and the change of cavity length caused by the rotation is

(17)$$\begin{aligned}\Delta L&=4\iint_{0}^{\pi/2} (R+dr)d\theta-L\\ &= \frac{\hat{q}}{F_{OM}} \frac{2 m\omega_r^2 R}{\pi} \iint_{0}^{\pi/2} \cos^2\alpha d^2 \alpha\\ &= \frac{1}{8}m \omega_r^2 L x_{ZPF}^2 \frac{\hat{q}}{\hbar g \hat{a}^{\dagger} \hat{a} }, \end{aligned}$$

Bringing the above equation into Eq. (15), we have

(18)$$\begin{aligned}\hbar\hat{a}^{\dagger} \hat{a} \omega_c \frac{\Delta L}{L}&=\hbar\hat{a}^{\dagger} \hat{a} \omega_c \frac{1}{L} \times \frac{1}{8}m \omega_r^2 L x_{ZPF}^2 \frac{\hat{q}}{\hbar g \hat{a}^{\dagger} \hat{a} }\\ &= \hbar \frac{\omega_r^2 \omega_c}{8g \omega_m}\hat{q}\\ &= \hbar f(\omega_r)\hat{q}. \end{aligned}$$

Using the expression $g=\frac {\omega _c x_{ZPF}}{L}$, we can further obtain $f(\omega _r)=\frac {\omega _r^2 L}{8 \omega _m x_{ZPF}}$.

Appendix B Comparison between the centrifugal effect and the Sagnac effect

If the Sagnac effect not eliminated with the plane of rotation designed in $x-z$, the phase difference between the left and right output field of the two ports is,

(19)$$\Delta \phi =\frac{8 \pi^2 R^2 \omega_r}{\lambda c},$$

the corresponding angular frequency difference is expressed as,

(20)$$\Delta \omega = \frac{2 R \omega_r}{c} \omega_c.$$

In fact, this change in the eigenfrequencies of the light in the left and right rows does not essentially change the length of the cavity. According to the expression of $f(\omega _r)=\frac {\omega _r^2 L}{8 \omega _m x_{ZPF}}$, it is quantity-independent of the frequency of the optical cavity, that is, the rotation effect and Sagnac effect are independent, so that we can safely discuss and compare them separately. To compare the proportions of these two effects in the Hamiltonian, the factor $R_{o}=f(\omega _r)/\Delta \omega$ is defined. Combined with the expression of $f(\omega _r)$ and $\Delta \omega$, this can be expressed as a function of the system parameters and the input signal.

(21)$$\begin{aligned} R_{o}&=f(\omega_r)/\Delta \omega\\ &= \frac{\omega_r c}{16 R g \omega_m}. \end{aligned}$$

Figure 8(a) shows the modification of the Hamiltonian induced by the centrifugal force and the Sagnac effect. When $R=10^{-3}\text {m}$ and $\omega _r>0.06$Hz, the effect of the centrifugal force is greater than that of the Sagnac effect. The radius of the microdisk is usually in the order of $\mu \textrm{m}$. As we can see from Eq. (21), if a microdisk of $\mu \textrm{m}$ is used to detect the signal, $R_O\approx 10^{-3}$, then the Sagnac effect can be ignored. A 3D plot of $R_O$ as a function of radius and rotation signal is shown in Fig. 8(b). Obviously, even if the Sagnac effect is not eliminated by a specific experimental design scheme, it can still be eliminated by choosing appropriate system parameters, e.g. reducing the radius of the microdisk.

Fig. 8. Comparison between the centrifugal effect and the Sagnac effect. (a) Modification of the Hamiltonian by the two effects as a function of the rotation frequency. The radius of the microdisk is $R=10^{-3}\textrm{m}$. (b) $\lg (R_O)$ as a function of the rotation frequency and radius of the microdisk. Other parameters are $\omega _c=10^{14}\textrm{Hz}$, $\omega _m=10^6\textrm{Hz}$ and single-photon optomechanical coupling strength $g=10^3\textrm{Hz}$.

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Fig. 9. $\textrm{SNR}_p$ as a function of superposition coefficient $x$ and homodyne detection phase $\theta$ under different $\xi$. Other parameters are the same as those in Fig. 2. It is observed that although $\xi$ changes the function relation of SNR, the optimal value still appears at $x=1/\sqrt {2}$.

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Fig. 10. Logarithmic values of minimal additional noise of ’m’ (above) and ’s’ (below) schemes as a function of cavity dissipation $\kappa$ and mechanical damping $\gamma$. The linearized coupling coefficients $G/\omega _m$ are $0.01$, $0.005$ and $0.001$, respectively. The thermal noise is $n_{th}=0$.

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Funding

National Natural Science Foundation of China (11704026, 11704042, 11704205, 11847128); Natural Science Foundation of Zhejiang Province (LY22A040005).

Acknowledgments

We thank Qiu Hui-Hui, Rui-Jie Xiao and Leng Xuan for instructive discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Optomechanical noise suppression with the optimal squeezing process

Abstract

1. Introduction

2. Model and Hamiltonian

3. Detection capability of our system

4. Weak angular momentum detection

5. Discussion and Conclusion

Appendix A Effect of rotation

Appendix B Comparison between the centrifugal effect and the Sagnac effect

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (21)

Optics Express