1 Introduction

High precision rotation rate measurement for navigation purposes is done by very sensitive gyroscopes. There are different types of gyroscopes [1], each of which uses different physical principles such as mechanical, MEMSFootnote 1 [2], optical [3,4,5] and cold atom [6,7,8] types. Among the existing physical methods, using the Coriolis force, which is formed as a virtual force in non-inertial reference frames, is a suitable option for measuring rotation rate. This force is often used in MEMS gyroscopes and has good results in various applications from civilian to military samples [9].

Using this virtual force along with very sensitive physical phenomena, will help us in very accurate measurement of rotation rate. One of these sensitive phenomena is the interaction of radiation with the mechanical oscillator in the quantum optomechanical cavity. Optomechanical cavities are devices in which there is a bilateral interaction between electromagnetic radiation and mechanical vibrations of a moving part inside them. This interaction is called optomechanical coupling. Optomechanical coupling leads to the formation of various phenomena, such as mechanical cooling [10], optical spring effect [11], multi-stability of optomechanically induced transparency [12], etc. The mutual interaction of light and mechanical oscillator, leads to increase the sensitivity of the optomechanical cavity in the measurement of physical parameters such as force, mass, displacement, rotation rate, acceleration, magnetic intensity, etc. [13].

So far, several works have been done in the field of quantum optomechanical gyroscope, all of which have been investigated under two optical effects. One case is related to the use of Sagnac interference effect [14], which is obtained by combining an optomechanical cavity with a ring laser [15]. In this method, the optical path length will change due to rotation, and then extra phase shift will Because of the freedom of moving mirrors of the optomechanical cavity in the direction perpendicular to the cavity axis, and since the cavity is located on the rotating frame, the Coriolis force is formed along the length of the cavity. The amount of this force depends on the rotation rate of the reference frame. This force affects the interaction of radiation with the mirror [16,17,18].

In this paper, the rotation rate of cavity is calculated from the Coriolis force. The optomechanical cavity considered in this research consists of a simple Fabry-Perot cavity with two moving mirrors, all of which are located on a rotating table. Cavity mirrors are free to move in the x direction. Also, in order to apply the effect of Coriolis force on the vibrations of the mirrors, with the help of an electric circuit, the entire cavity is made to vibrate sinusoidally in the y direction with an angular velocity equal to the angular velocity of the natural oscillations of the mirrors. As a result of the motion created in the y direction and the rotating motion of the table, the virtual Coriolis force is applied to the mirrors in the x direction. This virtual force, together with the radiation pressure force inside the cavity, which is applied to the mirrors, lead to the change of the optical path length of the radiation inside the cavity, and as a result, the optical response of the cavity will change. By studying the variation of optical response, the minimum and sensitivity of the cavity rotation rate are calculated.

In this paper, we will first introduce our gyroscope arrangement in detail and then will present the theory of the gyroscope in the quantum regime. After that, we will obtain the minimum measurable rotation rate and its sensitivity. Finally, we will simulate the output of our gyroscope by the Allan deviation curve.

2 Introducing the gyroscope

The general arrangement of our optomechanical gyroscope is shown in Fig. 1. A fixed mirror at the end of the first arm and an optomechanical cavity with two moving mirrors are placed at the end of the second arm of the Michelson interferometer. This interferometer is located on a rotating table with angular velocity \(\dot{\theta }\).

Fig. 1
figure 1

Optomechanical gyroscope arrangement. a A rotating table with angular velocity \(\dot{\theta }\) on which the Michelson interferometer is located. b A fixed mirror is placed at the end of the first arm of the interferometer and an optomechanical cavity with two moving mirrors is placed at the end of the second arm. Two oscillating motion with velocities of \(\dot{\bar{y}}_1=\dot{\bar{y}}_0\sin (\omega _mt)\) and \(\dot{\bar{y}}_2=\dot{\bar{y}}_0\cos (\omega _mt)\) in the y direction are applied to the cavity mirrors. As a result of the rotation of the table, the Coriolis forces of magnitude \(2m\dot{\bar{y}}_k\dot{\theta }\) are applied to the mirrors in the x direction where \(k=1, 2\). These virtual forces, in addition to the existing radiation pressure force, lead to the phase shift of the radiation inside the cavity by changing its length. This phase shift depends on \(\dot{\theta }\) and its effect is recorded in the measurement by detectors \(D_1\) and \(D_2\)

Full size image

According to Fig. 1, the optomechanical cavity mirrors have freedom to move in the x direction, and also two micro-piezoelectric devices apply the sinusoidal motions in the y direction to the mirrors with the velocities of \(\dot{\bar{y}}_1=\dot{\bar{y}}_0\sin (\omega _mt)\) and \(\dot{\bar{y}}_2=\dot{\bar{y}}_0\cos (\omega _mt)\). The working process of this arrangement is as follows: first, the laser beam inside the Michelson interferometer is divided into two equal beams by a 50:50 beam splitter. One of the beams enters the first arm of the interferometer, which, totally reflected at the end of arm. The second beam is transferred into the optomechanical cavity. The mirror on the left side of the cavity has a partial transmission coefficient, while the right mirror has total reflection. The two return beams are combined and decomposed again in the beam splitter and finally directed to the balanced Homodyne detector [19]. The electrical signals from two detectors of the Homodyne system are subtracted. We calculate the rotation rate and its sensitivity by using the average value and the fluctuations of the final signal.

3 Theoretical modeling of optomechanical gyroscope

To start dealing with the theoretical issues of each optomechanical cavity, we must first obtain its Hamiltonian, then solve the Heisenberg–Langevin equations of motion. To obtain the optomechanical cavity Hamiltonian introduced in Fig. 1, first the Hamiltonian of the mirrors is obtained. The velocity transformation law in the inertial reference frame (laboratory) and the non-inertial reference frame (rotating table), is [20]:

$$\begin{aligned} \begin{aligned} \vec{v}=\frac{d\vec{r}}{dt}=\frac{d^*\vec{r}}{dt}+\dot{\theta }\hat{k}\times \vec{r} =\left( \frac{d^*x^*}{dt}-\dot{\theta }y^*\right) \hat{i}^*+\left( \frac{d^*y^*}{dt}+\dot{\theta }x^*\right) \hat{j}^* \end{aligned} \end{aligned}$$
(1)

the asterisked parameters are related to the non-inertial reference frame. From now on, for convenience, we avoid writing asterisks on parameters. In the following, for the Lagrangian of two mirrors in the inertial reference frame, we have:

$$\begin{aligned} \begin{aligned} L&=T-V=\sum _{i=1}^{2}\left[ \frac{1}{2}m_i\left( \frac{dx_i}{dt}-\dot{\theta }(y_i-y_{i0})\right) ^2\right. \\&\quad \left. +\frac{1}{2}m_i\left( \frac{dy_i}{dt}+\dot{\theta }(x_i-x_{i0})\right) ^2\right. \\&\quad \left. -\frac{1}{2}m_i\omega _m^2(x_i-x_{i0})^2-\frac{1}{2}m_i\omega _m^2(y_i-y_{i0})^2\right] \end{aligned} \end{aligned}$$
(2)

where \(\omega _m\) is the angular frequency of vibration of mirrors. With the help of definition for the momentum, i.e. \(P_{q_i}=\frac{\partial L}{\partial \dot{q}_i}\), the mirror’s Hamiltonian in the inertial reference frame is [20]:

$$\begin{aligned} \begin{aligned} H_m&=\sum _{k}^{} \dot{q}_k\frac{\partial L}{\partial \dot{q}_k}-L=\sum _{i=1}^{2}\left[ \frac{P_{x_i}^2}{2m_i} +\frac{P_{y_i}^2}{2m_i}\right. \\&\quad \left. +\frac{1}{2}m_i\omega _m^2(x_i-x_{i0})^2\right. \\&\quad \left. +\frac{1}{2}m_i\omega _m^2(y_i-y_{i0})^2+\left( P_{x_i}(y_i-y_{i0})\dot{\theta }-P_{y_i}(x_i-x_{i0})\dot{\theta }\right) \right] \end{aligned} \end{aligned}$$
(3)

In the following, by considering two movable mirrors as quantum harmonic oscillators, one can write the creation (and hence the annihilation) operator of the mirrors as follows:

$$\begin{aligned} \hat{b}_{x_i}= & {} \frac{\hat{x}_i-x_{i_0}}{\chi _{q_i}}+\frac{i \hat{P}_{x_i}}{\chi _{p_i}} \end{aligned}$$
(4)
$$\begin{aligned} \hat{b}_{y_i}= & {} \frac{\hat{y}_i-y_{i_0}}{\chi _{q_i}}+\frac{i \hat{P}_{y_i}}{\chi _{p_i}} \end{aligned}$$
(5)

where, \(\chi _{p_i}=\sqrt{2\hbar m_i\omega _m}\) and \(\chi _{q_i}=\sqrt{2\hbar /(m_i\omega _m)}\) represent a coefficient of zero-point fluctuations of momentum and position of the moving mirrors, respectively [19]. The Hamiltonian of mirrors is written as follows:

$$\begin{aligned} \begin{aligned} \hat{H}_m&=\sum _{i=1}^{2}\left[ \hbar \omega _m(\hat{b}_{x_i}^\dagger \hat{b}_{x_i}+\frac{1}{2})+\hbar \omega _m(\hat{b}_{y_i}^\dagger \hat{b}_{y_i} +\frac{1}{2})\right. \\&\quad \left. +\frac{\chi _{p_i}\chi _{q_i}}{2i}(\hat{b}_{x_i}\hat{b}_{y_i}^\dagger -\hat{b}_{x_i}^\dagger \hat{b}_{y_i}) \dot{\hat{\theta }}\right] \end{aligned} \end{aligned}$$
(6)

3.1 Calculation of rotation rate and sensitivity

In the previous step, we obtained the quantum Hamiltonian of two moving mirrors, then for the total Hamiltonian, of course in a frame of reference rotating with an angular frequency equal to the laser frequency by using the unitary transformation,Footnote 2 we have [19]:

$$\begin{aligned} \begin{aligned} \hat{H}&=\hat{H}_{cav}+\hat{H}_m+\hat{H}_{int}+\hat{H}_{drive}\\&=-\hbar \Delta \hat{c}^\dagger \hat{c}+\sum _{i=1}^{2}\left( \hbar \omega _m \hat{b}_{x_i}^\dagger \hat{b}_{x_i}+\hbar \omega _m \hat{b}_{y_i}^\dagger \hat{b}_{y_i}\right. \\&\quad \left. +\frac{\chi _{p_i}\chi _{q_i}}{2i}(\hat{b}_{x_i}\hat{b}_{y_i}^\dagger -\hat{b}_{x_i}^\dagger \hat{b}_{y_i})\dot{\hat{\theta }}\right. \\&\quad \left. +(-1)^i\frac{\chi _o \chi _{q_i}}{2}\hat{c}^\dagger \hat{c}(\hat{b}_{x_i}+\hat{b}_{x_i}^\dagger )\right) +i\hbar \sqrt{\zeta _e} \hat{E}(\hat{c}^\dagger -\hat{c}) \end{aligned} \end{aligned}$$
(7)

where \(\hat{H}_{cav}\), \(\hat{H}_{int}\), \(\hat{H}_{drive}\), \(\hat{c}\), \(\hat{c}^\dagger\) and \(\omega _d\) are the intracavity field Hamiltonian, the mirror-intracavity field interaction Hamiltonian, the Laser drive Hamiltonian, the annihilation and creation operators of intracavity field and the laser frequency respectively. In Eq. 7, we have omitted the constant terms within the Hamiltonian of mechanical oscillators and optical cavity, because they will be removed in the Heisenberg’s equations of motion and in fact have no effect on the dynamics of the system. Also, \(\chi _o=-\hbar \omega _c/L\), \(\omega _c\) is the resonant angular frequency of the radiation inside the cavity, L is the length of the cavity with two mirrors in the stationary state, and \(\omega _d\) is the angular frequency of the driving laser. In Eq. 7, the quantity \(\Delta =\omega _d-\omega _c\) is the detuning of the angular frequency of the laser relative to the resonance angular frequency inside the cavity.

The operators \(\hat{b}_{x_i}\) are defined in the inertial reference frame of the laboratory, which include non-inertial operators. Since it is easier to work with the operators of the rotating reference frame attached to the rotating table, we separate the mentioned operators according to their parameters:

$$\begin{aligned} \begin{aligned} \hat{b}_{x_i}=\frac{\hat{x}_i-x_{i_0}}{\chi _{q_i}} +\frac{im_i\dot{\hat{x}}_i}{\chi _{p_i}}-\frac{im_i(\hat{y}_i-\hat{y}_{i0})\dot{\hat{\theta }}}{\chi _{p_i}} =\hat{b}_i-\frac{im_i(\hat{y}_i-\hat{y}_{i0})\dot{\hat{\theta }}}{\chi _{p_i}} \end{aligned} \end{aligned}$$
(8)
$$\begin{aligned} \begin{aligned} \hat{b}_{y_i}&=\frac{\hat{y}_i-y_{i_0}}{\chi _{q_i}}+\frac{im_i\dot{\hat{y}}_i}{\chi _{p_i}} +\frac{im_i(\hat{x}_i-\hat{x}_{i0})\dot{\hat{\theta }}}{\chi _{p_i}} \end{aligned} \end{aligned}$$
(9)

where \(\hat{b}_i=(\hat{x}_i-x_{i_0})/\chi _{q_i}+im_i\dot{\hat{x}}_i/\chi _{p_i}\). Finally, since we are dealing with an open quantum system, according to the Langevin equation of motion, we must add the contribution of the dissipative and input terms to the Heisenberg equation of motion [21]. The input radiation term for the equation of motion related to \(\dot{\hat{c}}\) has been previously considered by the contribution of \(H_{drive}\) in the total Hamiltonian, according to Eq. 7, and we only need to add the contribution due to dissipation to it. On the other hand, for the two related equations to \(\dot{\hat{b}}_k\) \((k=1, 2)\), we must write both the dissipation and input contributions, hence we have:

$$\begin{aligned} \begin{aligned} \dot{\hat{b}}_k&=(-i\omega _m-\frac{\gamma _{m_k}}{2})\hat{b}_k -(-1)^ki\frac{\chi _o \chi _{q_k}}{2\hbar }\hat{c}^\dagger \hat{c}\\&\quad +\frac{im_k}{\chi _{p_k}}(2\dot{\hat{y}}_k \dot{\hat{\theta }}+(\hat{y}_k-\hat{y}_{k0}) \ddot{\hat{\theta }}+(\hat{x}_k-\hat{x}_{k0}) \dot{\hat{\theta }}^2)+\sqrt{\gamma _{m_k}}\hat{F_k} \end{aligned} \end{aligned}$$
(10)
$$\begin{aligned} \begin{aligned} \dot{\hat{c}}=\left[ i\left( \Delta +\frac{\chi _o \chi _{q_1}}{2\hbar }(\hat{b}_1+\hat{b}_1^\dagger )-\frac{\chi _o\chi _{q_2}}{2\hbar }(\hat{b}_2 +\hat{b}_2^\dagger )\right) -\frac{\zeta }{2}\right] \hat{c}+\sqrt{\zeta _e} \hat{E} \end{aligned} \end{aligned}$$
(11)

where \(\gamma _{m_k}\) is the mechanical loss rate of mirrors 1 and 2, \(\zeta =\zeta _i+\zeta _e\) is the sum of the internal and external radiation loss rates and where \(\zeta _i\) is the loss rate caused by the internal events of the cavity and \(\zeta _e\) is the loss rate caused by the left mirror of the cavity witch has partial reflection. Also, \(\hat{F_k}\) is the input noise operator to the mechanical oscillators 1, 2, caused by their heat exchange with the environment. In Eq. 10, the terms containing \(\dot{\hat{\theta }}\), \(\dot{\hat{\theta }}^2\) and \(\ddot{\hat{\theta }}\) are caused by the Coriolis force, the centrifugal force, and the angular acceleration force respectively. If we assume that we are dealing with a constant rotation rate and on the other hand, if its value is much smaller than one, the effect of angular acceleration and centrifugal forces can be ignored. In addition, we investigate the classical effects of the Coriolis force on this quantum system, so instead of using the operator \(\dot{\hat{\theta }}\), we use its classical mean value, i.e. \(\dot{\theta }\). With these explanations, for the equation of motion 10, we have:

$$\begin{aligned} \begin{aligned} \dot{\hat{b}}_k&=\left(-i\omega _m-\frac{\gamma _{m_k}}{2}\right)\hat{b}_k-(-1)^ki\frac{\chi _o \chi _{q_k}}{2\hbar } \hat{c}^\dagger \hat{c}\\& \quad +\,\frac{im_k}{\chi _{p_k}}(2\dot{\hat{y}}_k \dot{\theta })+\sqrt{\gamma _{m_k}}\hat{F_k} \end{aligned} \end{aligned}$$
(12)

Now, to calculate the output signal and noise of the cavity that reach the detectors, we must solve the equations of motion 11 and 12. For this purpose, due to the small fluctuations of the quantum operators inside the cavity compared to their classical average values [22], we consider the operators as \(\hat{c}=\bar{c}+\hat{\delta }_c, \hat{b}_k=\bar{b}+\hat{\delta }_{b_k}, \dot{\hat{y}}_1=\dot{\hat{y}}_2=\dot{\bar{y}}+\dot{\hat{\delta }}_y, \hat{E}=\bar{E}+\hat{\delta }_E, \hat{F}=\hat{\delta }_F\), in which the parameters \(\bar{O}\) are the classical mean values of the operators and \(\hat{\delta }_O\) are the first-order fluctuation operators. Also, according to the general model of optomechanical interaction [23], we consider the internal and external loss rates in the cavity as the following linear approximations:

$$\begin{aligned} \zeta _{e,i}=\bar{\zeta }_{e,i}+\zeta _{om}^{e,i}\left[ \frac{\chi _{q_1}}{2}(\hat{\delta }_{b_1} +\hat{\delta }_{b_1}^\dagger )+\frac{\chi _{q_2}}{2}(\hat{\delta }_{b_2}+\hat{\delta }_{b_2}^\dagger )\right] \end{aligned}$$
(13)

where \(\bar{\zeta }_{e,i}\) are the average values of the external and internal loss rates. Also \(\zeta _{om}^{e,i}\) are the external and internal dissipation couplings of the cavity. By putting the mentioned linear approximations and Eq. 13 in 11 and 12, two sets of steady state and fluctuation equations are obtained. Each set of these equations includes three parts. On the other hand, for the Coriolis force in the equations of motion, since the term \(2\dot{\hat{\delta }}_y\dot{\theta }\) is of the second-order fluctuation type, we ignore it and only keep its first-order fluctuation term i.e. \(2\dot{\bar{y}}\dot{\theta }\). Also, we replace the Taylor expansion of the root of the Eq. 13 by considering only its linear variations in Eq. 11; therefore, we have:

$$\begin{aligned} \sqrt{\zeta _e}\sim \sqrt{\bar{\zeta }_e}\left[ 1+\frac{\zeta _{om}^e}{2\bar{\zeta }_e}\left( \frac{\chi _{q_1}}{2}(\hat{\delta }_{b_1} +\hat{\delta }_{b_1}^\dagger )+\frac{\chi _{q_2}}{2}(\hat{\delta }_{b_2}+\hat{\delta }_{b_2}^\dagger )\right) \right] \end{aligned}$$
(14)

The laser frequency can be adjusted in such a way that the condition \(\Delta =-(\chi _o \chi _{q_1}/2\hbar )(\bar{b}_1+\bar{b}_1^*)+(\chi _o \chi _{q_2}/2\hbar )(\bar{b}_2+\bar{b}_2^*)\) is established, according to the steady state equations, for the values of \(\bar{c}, \bar{b}_1, \bar{b}_2\) we have:

$$\begin{aligned} \bar{c}= & {} \frac{2\sqrt{\bar{\zeta }_e}\bar{E}}{\bar{\zeta }} \end{aligned}$$
(15)
$$\begin{aligned} \bar{b}_k= & {} \frac{(-1)^{k+1}\frac{i\chi _o \chi _{q_k}}{2\hbar } |\bar{c}|^2}{i\omega _m+\frac{\gamma _{m_k}}{2}} \end{aligned}$$
(16)

and cavity’s input–output relationship is equal to [21, 24]:

$$\begin{aligned} \hat{c}_o= & {} \hat{E}-\sqrt{\zeta }_e \hat{c} \end{aligned}$$
(17)
$$\begin{aligned} \hat{c}_1^o= & {} -\hat{E}_1 \end{aligned}$$
(18)

Therefore, by Eq. 17, the average output of the optomechanical cavity becomes \(\bar{c}_o=\bar{E}-\sqrt{\bar{\zeta }_e}\bar{c}=\bar{E}(1-\frac{2\bar{\zeta }_e}{\bar{\zeta }})\). Also, since at the end of the first arm of the Michelson interferometer, there is only one fixed mirror with nearly \(100\%\) reflection coefficient, hence \(\bar{c}_1=-\bar{E}_1\). On the other hand, with the help of the introduced linear approximation, we have the following output operators:

$$\begin{aligned} \begin{aligned} \hat{\delta }=\hat{\delta }_E-\sqrt{\bar{\zeta }_e}\left[ \hat{\delta }_c+\frac{\zeta _{om}^e}{2\bar{\zeta }_e} \bar{c}\left( \frac{\chi _{q_1}}{2}(\hat{\delta }_{b_1}+\hat{\delta }_{b_1}^\dagger ) +\frac{\chi _{q_2}}{2}(\hat{\delta }_{b_2}+\hat{\delta }_{b_2}^\dagger )\right) \right] \end{aligned} \end{aligned}$$
(19)
$$\begin{aligned} \hat{\delta }_1=-\hat{\delta }_{E_1} \end{aligned}$$
(20)

In the following, the fluctuations equations are obtained by applying the linear approximation in Eqs. 11 and 12:

$$\begin{aligned} \begin{aligned} \dot{\hat{\delta }}_c&=i\Delta \hat{\delta }_c+i\frac{\chi _o}{\chi _{p_1}}\bar{b}_1\hat{\delta }_c -i\frac{\chi _o}{\chi _{p_2}}\bar{b}_2\hat{\delta }_c+i\frac{\chi _o}{\chi _{p_1}}\bar{c}\hat{\delta }_{b_1} -i\frac{\chi _o}{\chi _{p_2}}\bar{c}\hat{\delta }_{b_2}\\&\quad +i\frac{\chi _o}{\chi _{p_1}}\bar{b}_1^*\hat{\delta }_c-i\frac{\chi _o}{\chi _{p_2}}\bar{b}_2^*\hat{\delta }_c +i\frac{\chi _o}{\chi _{p_1}}\bar{c}\,\hat{\delta }_{b_1}^\dagger -i\frac{\chi _o}{\chi _{p_2}}\bar{c}\, \hat{\delta }_{b_2}^\dagger -\frac{\bar{\zeta }}{2}\hat{\delta }_c\\&\quad -\left( \frac{\zeta _{om}^i+\zeta _{om}^e}{2}\right) \bar{c}\left( \frac{\chi _{q_1}}{2}(\hat{\delta }_{b_1}+\hat{\delta }_{b_1}^\dagger ) +\frac{\chi _{q_2}}{2}(\hat{\delta }_{b_2}+\hat{\delta }_{b_2}^\dagger )\right) \\&\quad +\sqrt{\bar{\zeta }_e}\left[ \hat{\delta }_E+\frac{\zeta ^e_{om}}{2\bar{\zeta }_e}\bar{E}\left( \frac{\chi _{q_1}}{2}(\hat{\delta }_{b_1} +\hat{\delta }_{b_1}^\dagger )+\frac{\chi _{q_2}}{2}(\hat{\delta }_{b_2}+\hat{\delta }_{b_2}^\dagger )\right) \right] \end{aligned} \end{aligned}$$
(21)
$$\begin{aligned} \begin{aligned} \dot{\hat{\delta }}_{b_k}&=\left( -i\omega _m-\frac{\gamma _{m_k}}{2}\right) \hat{\delta }_{b_k} -(-1)^ki\frac{\chi _o}{\chi _{p_k}}\bar{c}^* \hat{\delta }_c\\ &\quad-(-1)^ki\frac{\chi _o}{\chi _{p_k}}\bar{c} \hat{\delta }_c^\dagger +\sqrt{\gamma _{m_k}}\hat{\delta }_{F_k} +\frac{2im\dot{\bar{y}}_k\dot{\theta }}{\chi _{p_k}} \end{aligned} \end{aligned}$$
(22)

since these equations are complex, by calculating their complex conjugation, two other equations for \(\dot{\hat{\delta }}_c^\dagger\) and \(\dot{\hat{\delta }}_{b_k}^\dagger\) will be obtained. On the other hand, because in Eqs. 21 and 22, and their complex conjugations, we have only first-order time differential equations, for the simplicity of the solution, we transfer the equations to the frequency space with the help of the Fourier transform, i.e. \(\mathcal {F}(f(t))=\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{\infty } f(t) e^{i \omega t} \,dt\). Hence, we have the following matrix equation:

$$\begin{aligned} A \begin{bmatrix} \hat{\delta }_c(\omega )\\ \hat{\delta }_c^\dagger (-\omega )\\ \hat{\delta }_{b_1}(\omega )\\ \hat{\delta }_{b_1}^\dagger (-\omega )\\ \hat{\delta }_{b_2}(\omega )\\ \hat{\delta }_{b_2}^\dagger (-\omega ) \end{bmatrix} = \begin{bmatrix} \sqrt{\bar{\zeta }_e}\hat{\delta }_E(\omega )\\ \sqrt{\bar{\zeta }_e}\hat{\delta }_E^\dagger (-\omega )\\ \sqrt{\bar{\gamma }_{m_1}}\hat{\delta }_{F_1}(\omega )+\mathcal {F}\left( \frac{2im_1\dot{\bar{y}}_1\dot{\theta }}{\chi _{p_1}}\right) \\ \sqrt{\bar{\gamma }_{m_1}}\hat{\delta }_{F_1}^\dagger (-\omega )-\mathcal {F}\left( \frac{2im_1\dot{\bar{y}}_1\dot{\theta }}{\chi _{p_1}}\right) \\ \sqrt{\bar{\gamma }_{m_2}}\hat{\delta }_{F_2}(\omega )+\mathcal {F}\left( \frac{2im_2\dot{\bar{y}}_2\dot{\theta }}{\chi _{p_2}}\right) \\ \sqrt{\bar{\gamma }_{m_2}}\hat{\delta }_{F_2}^\dagger (-\omega )-\mathcal {F}\left( \frac{2im_2\dot{\bar{y}}_2\dot{\theta }}{\chi _{p_2}}\right) \\ \end{bmatrix} \end{aligned}$$
(23)

where the symbol \(\mathcal {F}\) is the Fourier transform and A is the matrix of coefficients of our equations in frequency space. To solve this matrix equation, we must first find the inverse of A, so we define \(M=A^{-1}=\{m_{ij}\}\). This matrix is a link between the input fluctuations operators to the cavity and the fluctuations operators of the cavity and two mirrors. Next, to find the noise, we need the operators \(\hat{\delta }_1\) and \(\hat{\delta }\) which are the fluctuations of the output operators \(\hat{c}_1^o\) (from the fixed mirror at the end of the first arm of the Michelson interferometer) and \(\hat{c}^o\) (from the optomechanical cavity), respectively. According to Eqs. 19 and 20 and using the matrix equation 23, after some calculations, we can get output fluctuation operators and their complex conjugations, i.e. \(\hat{\delta }_1\), \(\hat{\delta }^{\dagger }_1\), \(\hat{\delta }\), \(\hat{\delta }^{\dagger }\).

Now, in order to find the signal and quantum fluctuations of the output of the optomechanical cavity, we remove the classical noises of the laser source by the balanced Homodyne arrangement, and to calculate the quadrature of the amplitude of the output field from the cavity, we add the \(\pi /2\) phase to its output, just before the beam-splitter [25], hence for the signal difference obtained from photo-detectors, we have:

$$\begin{aligned} \hat{I}_2-\hat{I}_1=\hat{c}^{o\dagger }\hat{c}_1^o+\hat{c}_1^{o\dagger }\hat{c}^o \end{aligned}$$
(24)

where \(\hat{I}_1\), \(\hat{I}_2\), \(\hat{c}^o\) and \(\hat{c}_1^o\) are the intensity in detector \(D_1\), intensity in detector \(D_2\), the output field operator from the optomechanical cavity and the output field operator due to reflection from the fixed mirror in arm 1 of the Michelson interferometer respectively. In the following, we use the first-order linear approximation of \(\hat{c}^o=\bar{c}^o+\hat{\delta }\), \(\hat{c}_1^o=\bar{c}_1^o+\hat{\delta }_1\), where \(\hat{\delta }, \hat{\delta }_1\) are the same first-order fluctuations of the two output operators \(\hat{c}^o, \hat{c}_1^o\). Now, with this linear approximation, the Eq. 24 can be written as follows by removing the second-order fluctuations:

$$\begin{aligned} \hat{I}_2-\hat{I}_1\sim \bar{c}^{o*}\bar{c}_1^o+\bar{c}_1^{o*}\bar{c}^o+\hat{\delta }^\dagger \bar{c}_1^o +\hat{\delta }_1^\dagger \bar{c}^o+\bar{c}^{o*}\hat{\delta }_1+\bar{c}_1^{o*}\hat{\delta } \end{aligned}$$
(25)

Since a 50:50 beam splitter is used in the center of the Michelson interferometer in the arrangement shown in Fig. 1, the input field operators to each Michelson arm are \(\hat{E}=(i\hat{E}_d+\hat{E}_v)/\sqrt{2}\), \(\hat{E}_1=(\hat{E}_d+i\hat{E}_v)/\sqrt{2}\), where \(\hat{E}_d\) is the laser driver field operator and \(\hat{E}_v\) is the vacuum field operator. Therefore, the average values and fluctuations of these two operators are equal to \(\bar{E}=i\bar{E}_d/\sqrt{2}\), \(\bar{E}_1=\bar{E}_d/\sqrt{2}\). On the other hand, according to the same argument, since \(\bar{c}_1^o=-\bar{E}_d/\sqrt{2}\), \(\bar{c}^o=i\bar{E}_d/\sqrt{2}(1-\bar{\zeta }_e/(\bar{\zeta }/2))\), as a result, the first two terms of the Eq. 25 cancel each other and only the last four terms remain. Finally, for the stable signal measured by the Homodyne detector, we have:

$$\begin{aligned} S=\langle \bar{c}_1^o \hat{\delta }^\dagger +\bar{c}^o \hat{\delta }_1^\dagger +\bar{c}^{o*} \hat{\delta }_1+\bar{c}_1^{o*} \hat{\delta } \rangle \end{aligned}$$
(26)

For the Eq. 26 which contains fluctuation terms, the averages of the \(\langle \hat{\delta }_1 \rangle\), \(\langle \hat{\delta }_1^\dagger \rangle\), \(\langle \hat{\delta }_E \rangle\), \(\langle \hat{\delta }_E^\dagger \rangle\), \(\langle \hat{\delta }_{F_i} \rangle\), \(\langle \hat{\delta }_{F_i}^\dagger \rangle\) are zero, and the only term that contains a non-zero average value is due to the term \(\mathcal {F}(2im_k\dot{\bar{y}}_k\dot{\theta }/\chi _{p_k})\) that appears in the \(\hat{\delta }, \hat{\delta }^\dagger\) operators. Therefore, by simplifying, we finally have:

$$\begin{aligned} \begin{aligned} S&=\frac{-2i\bar{I}_d\sqrt{\bar{\zeta }_e}}{D\Gamma _0\bar{\zeta }^3} \left[ \alpha _1\mathcal {F}\left( \frac{2im_1\dot{\bar{y}} \dot{\theta }}{\chi _{p1}}\right) -\alpha _2\mathcal {F}\left( \frac{2im_2\dot{\bar{y}}\dot{\theta }}{\chi _{p2}}\right) \right] \end{aligned} \end{aligned}$$
(27)
$$\begin{aligned} \begin{aligned} \alpha _1&=(\Gamma _1-\Gamma _3)\chi _{q_1}\left[ \Gamma _0\Gamma _2\Gamma _4\bar{\zeta }^2(4ig_{om} \bar{\zeta }_e+\bar{\zeta }\zeta _{om}^e)\right. \\& \quad\left. -\,8g_{om}^2\bar{I}_d(\Gamma _2-\Gamma _4)\bar{\zeta }_e^2(\zeta _{om}^e+\zeta _{om}^i)\chi _{q_2}^2\right] \end{aligned} \end{aligned}$$
(28)
$$\begin{aligned} \begin{aligned} \alpha _2&=(\Gamma _2-\Gamma _4)\chi _{q_2}\left[ \Gamma _0\Gamma _1\Gamma _3 \bar{\zeta }^2(4ig_{om}\bar{\zeta }_e-\bar{\zeta }\zeta _{om}^e)\right. \\& \quad\left. +\,8g_{om}^2\bar{I}_d(\Gamma _1-\Gamma _3)\bar{\zeta }_e^2(\zeta _{om}^e+\zeta _{om}^i)\chi _{q_1}^2\right] \end{aligned} \end{aligned}$$
(29)

where \(D=\frac{4\sqrt{\bar{\zeta }_e}}{\Gamma _0}Det(A)\), \(\Gamma _0=-i \omega +\bar{\zeta }/2\), \(\Gamma _1=-i(\omega -\omega _m)+\gamma _{m_1}/2\), \(\Gamma _2=-i(\omega -\omega _m)+\gamma _{m_2}/2\), \(\Gamma _3=-i(\omega +\omega _m)+\gamma _{m_1}/2\), \(\Gamma _4=-i(\omega +\omega _m)+\gamma _{m_2}/2\), \(\bar{I}_d=\bar{E}_d^2/2\) and \(g_{om}=\chi _o/\hbar\). On the other hand, considering that \(\dot{\bar{y}}_1=\dot{\bar{y}}_0\sin \omega _m t\) and \(\dot{\bar{y}}_2=\dot{\bar{y}}_0\cos \omega _m t\), with the Fourier transform of \(\mathcal {F}(\frac{2im_k\dot{\bar{y}}_k\dot{\theta }}{\chi _{p_k}})\) in Eq. 27, it can be written:

$$\begin{aligned} S& {}= \frac{4\bar{I}_d\sqrt{\bar{\zeta }_e}\,\dot{\bar{y}}_0\,\dot{\theta }}{D\Gamma _0\bar{\zeta }^3} \sqrt{\frac{\pi }{2}}\left[ \left( \frac{m_1\alpha _1}{\chi _{p1}}-i\frac{m_2\alpha _2}{\chi _{p2}}\right) \delta (\omega -\omega _m)\right. \nonumber \\{} & {}\quad \left. +\left( \frac{m_1\alpha _1}{\chi _{p1}}+i\frac{m_2\alpha _2}{\chi _{p2}}\right) \delta (\omega +\omega _m)\right] \end{aligned}$$
(30)

Now, with the help of Fourier transform, the signal obtained in the frequency space can be transferred to the time space. For simplicity, we consider the moving mirrors to be completely similar; that is, \(m_1=m_2=m\), \(\gamma _{m_1}=\gamma _{m_2}=\gamma _m\), hence \(\chi _{q_1}=\chi _{q_2}=\chi _q\) and \(\chi _{p_1}=\chi _{p_2}=\chi _p\) and if we consider the over-coupled regime, i.e. for \(\bar{\zeta }\simeq \bar{\zeta }_e\gg \bar{\zeta }_i\) and the dispersive regime, i.e. for \(g_{om}\gg \zeta ^e_{om}\simeq \zeta ^i_{om}\simeq 0\), for our optomechanical cavity, the signal in the time domain will be as follows:

$$\begin{aligned} \begin{aligned} S&=\frac{1}{\sqrt{2\pi }}\int _{-\infty }^{\infty }e^{-i\omega t}S(\omega )\, d\omega \\&=\frac{16\sqrt{2}\,g_{om}\bar{I}_d\,m\,\dot{\bar{y}}_0\,\dot{\theta }\,\chi _q\,\omega _m}{\gamma _m\,\chi _p\,\sqrt{(\bar{\zeta }_e^2/4+\omega _m^2)(\gamma _m^2/4+4\omega _m^2)}}\cos (\omega _mt-\Theta ) \end{aligned} \end{aligned}$$
(31)

where we have,

$$\begin{aligned} \Theta =\arctan \left( \frac{\bar{\zeta }_e(\gamma _m-4\omega _m)-2\omega _m(4\omega _m+\gamma _m)}{-\bar{\zeta }_e(4\omega _m+\gamma _m)-2\omega _m(\gamma _m-4\omega _m)}\right) \end{aligned}$$
(32)

According to Eq. 31, the signal is modulated with the frequency \(\omega _m\), caused by the Coriolis force applied to the optomechanical cavity mirrors. In the limit of approximately stable oscillation for mirrors, i.e., \(\omega _m \gg \gamma _m\), the signal amplitude is approximated as follows:

$$\begin{aligned} S_0=\frac{16\sqrt{2}\,\bar{I}_d\,\dot{\bar{y}}_0\,\dot{\theta }\,g_{om}}{\gamma _m\,\omega _m\,\sqrt{\bar{\zeta }_e^2+4\omega _m^2}} \end{aligned}$$
(33)

Based on the amplitude of the signal obtained in our cavity, this value is \(\sqrt{2}\) times the output signal of the cavity, in which only one moving mirror is considered [17].

Next, to find the noise at the Homodyne measurement method, we must obtain the variance of the Eq. 25. The square root of the variance gives the noise in the signal. For this purpose, we have:

$$\begin{aligned} \begin{aligned} V \delta (\omega +\omega ^\prime )&=\langle \Delta \hat{I}^\dagger (-\omega ) \Delta \hat{I}(\omega ^\prime ) \rangle \\&=\langle (\hat{I}_2-\hat{I}_1)^\dagger (-\omega ) (\hat{I}_2-\hat{I}_1)(\omega ^\prime ) \rangle \\&-\langle (\hat{I}_2-\hat{I}_1)^\dagger (-\omega ) \rangle \langle (\hat{I}_2-\hat{I}_1) (\omega ^\prime )\rangle \end{aligned} \end{aligned}$$
(34)

The second term in Eq. 34 is the product of two complex conjugate signals at two separate frequencies \((-\omega )\) and \((\omega ^\prime )\). The obtained signal in the frequency domain (Eq. 30) includes two Dirac delta functions, with arguments \((\omega -\omega _m)\) and \((\omega +\omega _m)\). Now considering it as a function \(S(\omega )=f(\omega )[\delta (\omega -\omega _m)+\delta (\omega +\omega _m)]\) and since it is easy to prove \(\langle (I_2-I_1)^\dagger (-\omega ) \rangle \equiv S(\omega )\), we have:

$$\begin{aligned} \begin{aligned} V \delta (\omega +\omega ^\prime )&=\langle (\hat{I}_2-\hat{I}_1)^\dagger (-\omega ) (\hat{I}_2-\hat{I}_1)(\omega ^\prime ) \rangle \\&\quad -f(\omega ) f(\omega ^\prime )\left[ \delta (\omega -\omega _m)+\delta (\omega +\omega _m)\right] \\&\quad \times \left[ \delta (\omega ^\prime -\omega _m)+\delta (\omega ^\prime +\omega _m)\right] \end{aligned} \end{aligned}$$
(35)

Now, to calculate the first term of Eq. 35, we consider the following correlation relations [19]:

$$\begin{aligned}{} & {} \langle \hat{a}_{in}(\omega ^\prime )\hat{a}_{in}^\dagger (-\omega ) \rangle =\delta (\omega +\omega ^\prime ) \end{aligned}$$
(36)
$$\begin{aligned}{} & {} \langle \hat{a}_{in}^\dagger (-\omega )\hat{a}_{in}(\omega ^\prime ) \rangle =0 \end{aligned}$$
(37)
$$\begin{aligned}{} & {} \langle \hat{b}_{in}(\omega ^\prime )\hat{b}_{in}^\dagger (-\omega ) \rangle =(n_{th}+1)\delta (\omega +\omega ^\prime ) \end{aligned}$$
(38)
$$\begin{aligned}{} & {} \langle \hat{b}_{in}^\dagger (-\omega )\hat{b}_{in}(\omega ^\prime ) \rangle =n_{th}\delta (\omega +\omega ^\prime ) \end{aligned}$$
(39)

where \(n_{th}=1/(e^{\hbar \omega /(k_B T)}-1)\) is the mean phonon occupation number, \(\hat{a}_{in}\) is the input radiation operator, \(\hat{b}_{in}\) is the input thermal operator, and on the other hand, we assume that there is no correlation between the laser field and the vacuum field, as well as between the fields of the two arms of the Michelson interferometer. If we consider the two dispersive and over-coupled regimes mentioned later, after a little calculation we have:

$$\begin{aligned} \begin{aligned}&\langle (\hat{I}_2-\hat{I}_1)^\dagger (-\omega ) (\hat{I}_2-\hat{I}_1)(\omega ^\prime ) \rangle \\&\quad =\left\{ |\bar{c}_1^o|^2 \left[ (1-\bar{\zeta }_em_{11}(\omega )) (1-\bar{\zeta }_em_{22}(\omega ^\prime ))+\bar{\zeta }_e^2 m_{21}(\omega ) m_{12}(\omega ^\prime ) \right. \right. \\&\left. \left. \qquad + \gamma _m \bar{\zeta }_e \left( m_{13}(\omega ) m_{24}(\omega ^\prime ) +m_{23}(\omega ) m_{14}(\omega ^\prime ) +m_{15}(\omega ) m_{26}(\omega ^\prime ) \right. \right. \right. \\&\left. \left. \left. \qquad + m_{25}(\omega )m_{16}(\omega ^\prime )\right) (n_{th}+1) \right. \right. \\&\left. \left. \qquad + \gamma _m \bar{\zeta }_e \left( m_{14}(\omega ) m_{23}(\omega ^\prime ) +m_{24}(\omega ) m_{13}(\omega ^\prime ) +m_{16}(\omega ) m_{25}(\omega ^\prime ) \right. \right. \right. \\&\left. \left. \left. \qquad + m_{26}(\omega )m_{15}(\omega ^\prime )\right) (n_{th})\right] \right. \\&\left. \qquad + (\bar{c}_1^{o*})^2 \left[ -\bar{\zeta }_e m_{12}(\omega ^\prime )(1-\bar{\zeta }_em_{11}(\omega ))\right. \right. \\&\left. \left. \qquad + \gamma _m \bar{\zeta }_e(m_{13}(\omega ) m_{14}(\omega ^\prime )+m_{15}(\omega )m_{16}(\omega ^\prime ))(n_{th}+1)\right. \right. \\&\left. \qquad + \gamma _m \bar{\zeta }_e(m_{14}(\omega ) m_{13}(\omega ^\prime )+m_{16}(\omega )m_{15}(\omega ^\prime ))n_{th}\right] +|\bar{c}^o|^2\\&\left. \qquad + (\bar{c}_1^o)^2 \left[ -\bar{\zeta }_e m_{21}(\omega )(1-\bar{\zeta }_em_{22}(\omega ^\prime ))\right. \right. \\&\left. \left. \qquad + \gamma _m \bar{\zeta }_e(m_{23}(\omega ) m_{24}(\omega ^\prime )+m_{25}(\omega )m_{26}(\omega ^\prime ))(n_{th}+1)\right. \right. \\&\left. \left. \qquad + \gamma _m \bar{\zeta }_e(m_{24}(\omega ) m_{23}(\omega ^\prime )+m_{26}(\omega )m_{25}(\omega ^\prime ))n_{th}\right] \right\} \delta (\omega +\omega ^\prime ) \end{aligned} \end{aligned}$$
(40)

where \(m_{ij}\)’s are the elements of matrix M. We ignore the second term of Eq. 35, because it depends to \(\dot{\theta }^2\). Also, with two selected regimes mentioned above, it can be written according to Eqs. 15, 17 and 18, \(\bar{c}=i \sqrt{2/\bar{\zeta }_e} \bar{E}_d\), \(\bar{c}_1^o=-\bar{E}_d/\sqrt{2}\), \(\bar{c}^o=-i\bar{E}_d/\sqrt{2}\). Now, by simplifying the Eq. 40 by these quantities and considering \(\omega =\omega _m\) (because the signal oscillates with frequency \(\omega _m\)), finally, we have:

$$\begin{aligned} N=\sqrt{V(\omega =\omega _m)}=\sqrt{2\bar{I}_d+a \bar{I}_d^3+b \bar{I}_d^2} \end{aligned}$$
(41)

in which we have:

$$\begin{aligned}{} & {} a=\frac{1024 g_{om}^4 \chi _q^4}{\gamma _m^2(\bar{\zeta }_e^2+4\omega _m^2)^2} \end{aligned}$$
(42)
$$\begin{aligned}{} & {} b=\frac{64 g_{om}^2 \chi _q^2(1+2n_{th})}{\gamma _m(\bar{\zeta }_e^2+4\omega _m^2)} \end{aligned}$$
(43)

Now, by finding the signal and noise of this gyroscope, its rotation rate sensitivity can be obtained according to the following definition [17]:

$$\begin{aligned} \dot{\theta }_s=\frac{N \dot{\theta }}{S}=\frac{\gamma _m\,\omega _m \sqrt{\bar{\zeta }_e^2+4\omega _m^2}}{16\sqrt{2}\,\dot{\bar{y}}_0\,g_{om}}\sqrt{\frac{2}{\bar{I}_d}+a\bar{I}_d+b} \end{aligned}$$
(44)

In Eq. 44, the terms inside the square root from left to right are due to shot noise, radiation pressure noise, and thermal noise, respectively.

In the following, to check the compatibility of the experimental data mentioned in Table 1 with the general model of optomechanical interaction [23], we used the fluctuation-dissipation theorem to find the displacement noise of the mirror in an optomechanical cavity with one moving mirror introduced in [26], that we used its experimental data in our cavity.

Table 1 Experimental parameters of an optomechanical cavity with one movable mirror [26]
Full size table

This noise was investigated in two situations, i.e., with the presence of the optical spring effect and its absence. The results are presented in Fig. 2.

Fig. 2
figure 2

Mirror displacement noise in an optomechanical cavity with only one moving mirror as a function of its vibration frequency based on the general model of optomechanical interaction [23] and the experimental values mentioned in Table 1. The dashed curve corresponds to the absence of the optical spring effect, and the solid line curve corresponds to the presence of the optical spring effect

Full size image

As it is clear from Fig. 2 and Fig. 1 in [26], using the general model of optomechanical interaction, along with the selected cavity parameters, gives the same results for the displacement noise of the mirror. In the \(\Delta _0=\chi _o \chi _q/(2\hbar )(\bar{b}+\bar{b}^*)\) or \(\Delta _{eff}=\Delta _0-\chi _o \chi _q/(2\hbar )(\bar{b}+\bar{b}^*)=0\) state, there is no optical spring effect and the noise diagram in Fig. 2 is obtained at the frequency \(\nu _m=84.8\, \textrm{Hz}\) as in Fig. 1a diagram in [26]. On the other hand, in the \(\Delta _0 \equiv -0.5 \times \chi _o \chi _q/(2\hbar )(\bar{b}+\bar{b}^*)\) or \(\Delta _{eff}=\Delta _0-\chi _o \chi _q/(2\hbar )(\bar{b}+\bar{b}^*)=-1.5 \times \chi _o \chi _q/(2\hbar )(\bar{b}+\bar{b}^*)\) state, the effect of the optical spring is formed and leads to the transfer of the noise diagram in Fig. 2 to the frequency \(\nu _m=85.5\,\textrm{Hz}\), as this issue can be seen in the noise diagram of Fig. 1b in [26]. In fact, the effect of the optical spring, which is caused by the mutual interaction of the radiation and the mirror, leads to the displacement of the resonant frequency of the moving mirror. Therefore, we can use this both the general optomechanical cavity model and these experimental parameters in our optomechanical cavity and obtain our final results based on them.

According to the Eqs. 42–44, increasing the mass causes a decrease in \(\dot{\theta }_s\) and increases the sensitivity, so a cavity should be considered with the mass of its mirrors as large as possible and of course not so much that we go out of the range of quantum effects. This problem is also a reason for the use of experimental cavity parameters that mentioned in Table 1, where \(m=69 \times 10^{-5}\,\textrm{kg}=0.69\,\textrm{gr}\). The sensitivity diagram of our gyroscope was obtained according to these parameters in Fig. 3.

Fig. 3
figure 3

The sensitivity curve of the quantum optomechanical gyroscope with two moving mirrors at \(T=0K\) with respect to the input laser intensity. The dashed, thin, dotted, and thick curves are related to the sensitivity caused by shot noise, radiation pressure noise, thermal noise, and the total sensitivity of the rotation rate of the gyroscope, respectively. The considered parameters are the same as mentioned in Table 1

Full size image

In Fig. 3, it is clear that the sensitivity caused by shot noise, increases with the laser intensity, but almost at the intensities \(\bar{I}_d > 10^{10}\,{\rm Hz}\), intensity increase becomes almost ineffective. The sensitivity that is caused by radiation pressure noise shows a decreasing trend with laser intensity. Also, the sensitivity caused by thermal noise has no dependence on laser intensity, and practically this noise does not contribute to the change in the rotation rate sensitivity. The amount of laser intensity where the two quantities of shot noise and radiation pressure noise are equal is called the Standard Quantum Limit (SQL), and at this optimal intensity, the sensitivity reaches its lowest value. The value of this quantity is shown in Fig. 3 at \(\bar{I}_d=1.62 \times 10^{9}\, \textrm{Hz}\). This quantity for the case where two mirrors are similar can be obtained by Eq. 44, therefore we have:

$$\begin{aligned} \bar{I}_{opt}=\sqrt{\frac{2}{a(\omega _m)}}=\frac{\gamma _m\left( \frac{\bar{\zeta }_e^2}{4}+\omega _m^2\right) mL^2 \omega _m}{8\sqrt{2}\,\hbar \,\omega _c^2} \end{aligned}$$
(45)

Now, by replacing the optimal intensity and the values of \(a(\omega _m)\) and \(b(\omega _m)\) in Eq. 44, one can obtain the lowest sensitivity of the rotation rate of the gyroscope. Therefore, we have:

$$\begin{aligned} \dot{\theta }_{s, min}=\frac{1}{\dot{\bar{y}}_0}\sqrt{\frac{\hbar \gamma _m \omega _m}{4m}\left[ \sqrt{2}+1+2n_{th}\right] } \end{aligned}$$
(46)

According to the above equation, the reduction of \(\omega _m\) leads to the reduction of \(\dot{\theta }_{s,min}\) and actually increases the sensitivity of the gyroscope. This is due to the reduction of the performance of the spring force on the mechanical oscillators of the optomechanical cavity, and as a result, the sensitivity of the oscillators to the Coriolis force increases compared to the spring force. On the other hand, increasing the mass of the moving mirrors in the optomechanical cavity, according to the Eq. 46, increases the sensitivity, but in Eq. 45, it increases the optimal intensity. To prevent the change of the optimal intensity, it is enough to reduce the loss rate of the mirrors \((\gamma _m)\) with respect to Eq. 45 so that the product of \(m \times \gamma _m\) remains constant. On the other hand, fortunately, reducing \(\gamma _m\) according to Eq. 46 itself leads to an increase in sensitivity. Therefore, it is possible to increase the sensitivity by keeping the optimal intensity constant.

The previously presented cases are fully valid for temperatures \(T \approx 0K\), because at this temperature, \(n_{th} \approx 0\). Reaching temperatures close to zero kelvin is possible with the advances made today in the field of cooling. On the other hand, for high temperatures where \(\hbar \omega _m \ll k_B T\), \(n_{th} \approx k_B T/(\hbar \omega _m) \gg 1\). In this situation, together with the unresolved side-band regime(\(\bar{\zeta }_e \gg \omega _m\)) [26], the sensitivity equation is equal to:

$$\begin{aligned} \dot{\theta }_s \approx \frac{1}{\dot{\bar{y}}_0}\sqrt{\frac{\gamma _m k_B T}{4m}} \end{aligned}$$
(47)

In this case, the sensitivity does not depend on \(\omega _m\) and it is no longer possible to increase the sensitivity value by decreasing \(\omega _m\). This problem is shown by drawing the curves of \(\dot{\theta }_s\) in terms of \(\omega _m\) at three temperatures \(1\,\textrm{mK}\), \(10\,\textrm{K}\), \(300\,\textrm{K}\).

As shown in Fig. 4, for \(\omega _m \ge 10^{14}\, \mathrm{rad/s}\), the sensitivity of the gyroscope is the same at all temperatures. As the frequency of oscillations of moving mirrors decreases, the lower the temperature, the higher the sensitivity (its value in the curves decreases), so that at temperature \(T=1\,\textrm{mK}\), the curve of sensitivity shows a significant decreasing slope. At high temperatures, the curves have almost zero slope, and in fact, the change of \(\omega _m\) will no longer have a significant effect on the change of sensitivity. Of course, at high temperatures, it is possible to increase the sensitivity by increasing the mass of the mirrors (m) to extent that the quantum effects are preserved, reducing their loss rate \((\gamma _m)\) and increasing the inserted velocity amplitude of the cavity in the y direction (\(\dot{\bar{y}}_0\)). According to Fig. 4, at temperature \(T=300\,\textrm{K}\), the sensitivity value of the gyroscope is \(\dot{\theta }_s<7.5 \times 10^{-10}\,(\mathrm{rad/s})/\sqrt{\textrm{Hz}}\), which compared to the gyroscope presented in the [17], we improved the sensitivity well.

Fig. 4
figure 4

Gyroscope sensitivity curves with respect to the frequency of moving mirrors oscillations at three temperatures \(T=1\,\textrm{mK}\) (dashed curve), \(T=10\,\textrm{K}\) (thin curve) and \(T=300\,\textrm{K}\) (thick curve). The considered parameters are the same as mentioned in Table 1

Full size image

According to Table 2, the values obtained for the sensitivity in rotation rate at three temperatures \(0\,\textrm{K}\), \(1\,\textrm{mK}\), and \(300\,\textrm{K}\) show that our cavity compared to the cavity of paper [17] with the same parameters mentioned in Table 1, due to considering the motion for both mirrors, by more than \(56 \%\) improved at \(0\,\textrm{K}\) and equal to \(50 \%\) at \(300\,\textrm{K}\).

Table 2 Calculated rotation rate sensitivity (\(\dot{\theta }_s\)) in three temperatures according to the numerical values presented in [26](mentioned in Table 1)
Full size table

On the other hand, the effect of temperature on the sensitivity of rotation rate according to experimental parameters mentioned in Table 1, in the temperature range of \(T=1\,\textrm{mK}\) and \(T=300\,\textrm{K}\) is shown in Fig. 5. As it is known, with the decrease in temperature, the sensitivity change is almost linear, but for temperatures lower than \(10\,\textrm{K}\), the changes are formed with a very steep slope. This problem indicates that decreasing in temperature has a very favorable effect on increasing the sensitivity of the optomechanical gyroscope due to the reduction of the effect of thermal fluctuations.

Fig. 5
figure 5

Variation of rotation rate sensitivity with temperature from \(T=1\,\textrm{mK}\) to \(T=300\,\textrm{K}\). The considered parameters are the same as mentioned in Table 1

Full size image

4 Gyroscope output simulation

In this section, we want to simulate the performance of a gyroscope made on the basis described in the previous section. The minimum rotation rates, obtained for the different temperatures, are considered as mean values and to simulate the noise effects, a random fluctuation around them is considered. We can use the usual algorithm for estimating the performance of the gyroscope called WLS.Footnote 3 This algorithm is a method to determine the noise parameters that take place in the operation of gyroscopes [27,28,29]. In this method, after measuring the data, a logarithmic diagram is drawn under the name of Alan’s deviation, whose slope in 5 regions represents the different noises in it. The most important noises in this diagram are ARWFootnote 4 and BS,Footnote 5 which respectively show the amount of white noise in the gyroscope output and its measurement accuracy.

For this purpose, we consider a specific rotation rate for the rotating table, which we denote by \(\dot{\theta }_{table}\). On the other hand, for the hypothetical measurement of the rotation rate of the table by the gyroscope, we consider a fluctuation equal to the minimum rotation rate that can be measured by the gyroscope. The minimum rotation rate(\(\dot{\theta }_{min}\)) is equal with the sensitivity in Eq. 44 when \(N=S\), hence, we can use of parameters of Table 3 for \(\dot{\theta }_{min}\). As a result, with the help of these two parameters (\(\dot{\theta }_{table}\), \(\dot{\theta }_{min}\)) and the use of the random function in the Mathematica software, we generate ten thousand random rotation rates measured by our gyroscope hypothetically. Then by the WLS algorithm, we draw the Allan deviation graph of our gyroscope and finally for the three temperatures, by curve fitting on the raw data, we calculate two parameters ARW and BS.

For \(T=300\,\textrm{K}\), the minimum rotation rate is \(\dot{\theta }_{min}=7.55 \times 10^{-10}\,\mathrm{rad/s}\) and the rotation rate of the table is considered to be \(\dot{\theta }_{table}=1 \times 10^{-4}\,\mathrm{rad/s}\). Now to generate hypothetical gyroscope measurements we have:

$$\begin{aligned} \dot{\theta }_i=\dot{\theta }_{table}+RandomInteger[\{-1,1\}] \times \dot{\theta }_{min} \end{aligned}$$
(48)

where \(\dot{\theta }_i\) is the hypothetical ith measured rotation rate of our gyroscope. Also we consider the temperatures \(T=1\,\textrm{mK}\) and \(T=0K\) where the minimum measurable rotation rates of our gyroscope obtained \(\dot{\theta }_{min}=1.38 \times 10^{-12}\,\mathrm{rad/s}\) and \(\dot{\theta }_{min}=3.06 \times 10^{-15}\,\mathrm{rad/s}\) respectively, hence with the help of equation 48, again the hypothetical measurements of the gyroscope are generated and the Allan deviation diagram according to the WLS algorithm is drawn.

Figures 6, 7 and 8 show the Allan deviation curves and their fitting at the three mentioned temperatures. These curves are very similar to the real data curves of built gyroscopes, like the curves in [27]. By fitting the raw cluster data with Allan deviation noise model defined in [28], we calculate ARW and BS quantities. These quantities are shown at three temperatures in Table 3. As shown in curve Fig. 6, \(ARW=1.87 \times 10^{-10} (\mathrm{rad/s})/ \sqrt{\textrm{Hz}}=6.43 \times 10^{-7} \deg /\sqrt{\textrm{hr}}\) in \(T=300K\). The amplitude of the bias stability noise equals \(BS=2.13 \times 10^{-11}\, \mathrm{rad/s}=4.39 \times 10^{-6}\, \mathrm{deg/hr}\). The amount of bias stability noise at \(T=300\,\textrm{K}\) compared to other gyroscopes is two orders of magnitude better than the strategic gyroscopes [1].

Fig. 6
figure 6

Allan deviation of cluster data and curve fitting at \(T=300\,\textrm{K}\). By curve fitting, \(ARW=1.87 \times 10^{-10}\,(\mathrm{rad/s})/\sqrt{\textrm{Hz}}\) and \(BS=2.13 \times 10^{-11}\,\mathrm{rad/s}\)

Full size image
Fig. 7
figure 7

Allan deviation of cluster data and curve fitting at \(T=1\,\textrm{mK}\). By curve fitting, \(ARW=3.58 \times 10^{-13} (\mathrm{rad/s})/\sqrt{\textrm{Hz}}\) and \(BS=2.1 \times 10^{-14}\,\mathrm{rad/s}\)

Full size image
Fig. 8
figure 8

Allan deviation of cluster data and curve fitting at \(T=0K\). By curve fitting, \(ARW=8.05 \times 10^{-16} (\mathrm{rad/s})/\sqrt{\textrm{Hz}}\) and \(BS=7.24 \times 10^{-17}\,\mathrm{rad/s}\)

Full size image
Table 3 Calculated ARW and BS quantities of our gyroscope in three temperatures
Full size table

According to curve Fig. 7, in \(T=1\,\textrm{mK}\), the amount of white noise equal to \(ARW=3.58 \times 10^{-13} (\mathrm{rad/s})/\sqrt{\textrm{Hz}}=1.23 \times 10^{-9}\,\textrm{deg}/\sqrt{hr}\) and bias stability equal to \(BS=2.1 \times 10^{-14}\, \mathrm{rad/s}=4.33 \times 10^{-9} \, \mathrm{deg/hr}\) was obtained. As we expected, by reducing the temperature of the gyroscope and subsequently reducing the thermal noises, these two noises in the gyroscope output were significantly reduced.

According to curve Fig. 8, in \(T=0K\), the amount of white noise \(ARW=8.05 \times 10^{-16}\,(\mathrm{rad/s})/\sqrt{\textrm{Hz}} = 2.77 \times 10^{-12}\,\textrm{deg}/\sqrt{\textrm{hr}}\) and bias stability \(BS=7.24 \times 10^{-17}\,\mathrm{rad/s}=1.49 \times 10^{-11} \,\mathrm{deg/hr}\) was obtained. As expected at this stage, with the complete disappearance of the thermal noise, these two noises have a very significant reduction in the output of the gyroscope.

In this section, we simulated the performance of gyroscope, but it is important to note that we only considered the accuracy of the gyroscope fluctuations as a measurement error, and this is an ideal state, while the presence of optical and electronic devices such as lasers in the gyroscope will be involved in the error caused by its measurement. In this section, the effect of all other disturbances has been ignored to check the performance of the gyroscope for the presence of inherent optomechanical cavity noises, and in an ideal situation to obtain the best values for two critical parameters in the evaluation of the gyroscope, namely ARW, and BS.

5 Conclusion

In this paper, we introduced a gyroscope using optomechanical cavity with two moving mirrors. Our calculations were done based on three regimes named unresolved side-band, over-coupled, and dispersive, and by using appropriate experimental parameters in [26]. In this work, by considering two moving mirrors instead of only one moving mirror, although the radiation pressure and thermal noises are increased, but due to the greater growth of the signal, the minimum sensitivity of the rotation rate of our gyroscope improved by more than \(56\%\) at \(T=0\,\textrm{K}\) and \(50\%\) at \(T=300\,\textrm{K}\). With this gyroscope, we were able to achieve a minimum sensitivity of \(7.55 \times 10^{-10} (\mathrm{rad/s})/\sqrt{\textrm{Hz}}\) at ambient temperature \((T=300\,\textrm{K})\). If we lower the temperature of the gyroscope to \(T=1\,\textrm{mK}\) with cooling technologies, the minimum sensitivity will be \(1.38 \times 10^{-12}\,(\mathrm{rad/s})/\sqrt{\textrm{Hz}}\). Also, at the ideal temperature \(T=0 K\), a sensitivity value of \(3.06 \times 10^{-15}\,(\mathrm{rad/s})/\sqrt{\textrm{Hz}}\) obtained.

We also showed that by reducing the temperature of the optomechanical cavity, due to the reduction of the effect of thermal fluctuations, the sensitivity of the gyroscope increases almost linearly (the numerical value of \(\dot{\theta }_s\) decreases) and at temperatures lower than \(T=10\,\textrm{K}\), the trend of increasing the sensitivity in terms of temperature is very steep. This problem shows the strong effect of cooling at temperatures lower than \(T=10\,\textrm{K}\) on the performance of our gyroscope.

In the last part, by using the WLS algorithm and based on the minimum rotation rates obtained at three temperatures \(T=0\,\textrm{K},\, 1\,\textrm{mK},\, 300K\) and assuming the rotation rate of \(1 \times 10^{-4}\,\mathrm{rad/s}\) for the table, we drew the Allan deviation diagrams. In this section, we assumed that by ignoring all kinds of disturbances in a real gyroscope, such as the presence of optical, electronic devices, etc., only by considering the inherent fluctuations of our optomechanical gyroscope, i.e. shot noise, radiation pressure noise, and thermal noise, find the most optimal value for the two parameters ARW and BS, which are important parameters in the evaluation of gyroscopes. Despite the data availability [1], the obtained results showed the excellent performance of this gyroscope, especially in terms of thermal noise reduction.