Inertial navigation^[1,2] systems have many applications such as guiding missiles, satellites, and so on. At a fundamental level, an inertial navigation system contains an accelerometer, which is used to measure instantaneous velocity, and a gyroscope which plays an important role as a direction sensor. Different kinds of gyroscopes like mechanical gyroscopes [e.g., gyroscopes based on microelectromachanical systems (MEMSs)^[3,4] utilizing the Coriolis force], optical gyroscopes (e.g., ring-laser gyroscopes and fiber-optic gyroscopes, which utilize the Sagnac effect,^[5]) and nuclear magnetic resonance gyroscopes^[6,7] or atom interferometer gyroscopes^[8,9] which utilize the quantum effect, have been studied extensively.

On the other hand, because of the recent advances in nano-fabrication techniques for designing high-quality optical and mechanical resonators, optomechanics has emerged as a rapidly developing field for exploring the coupling between optical fields and mechanical oscillations. One of the most promising applications of optomechanics is for designing highly sensitive optical detection of small forces,^[10–13] displacements^[14–17] or masses^[18–20] by manipulating the mechanical motion of a macroscopic object in the quantum regime.^[21–28] Hence, optomechanical systems are used to detect the angular velocity of absolute rotation,^[29] by optically monitoring the Coriolis force acting on the two-dimensional (2D) mechanical resonator.

Gyroscopes based on the Coriolis force detection were discussed in Refs. [3] and [29]. The analysis provided in Ref. [29] is based on classical mechanics and the effects of quantum noise, such as optomechanical back-action noise, are neglected. Recently, we presented theoretical analysis for the application of the optomechanical cavities for absolute rotation detection^[30–32] by considering the effects of shot noise, back-action noise, and thermal noise.^[33] In this work, we propose an optomechanical gyroscope model, where one mode of the 2D mechanical resonator is coupled to an optical cavity. Our model is similar to that in Refs. [31] and [32] except that here we do not use an external drive on the mechanical resonator. A strong external mechanical drive used in previous optomechanical gyroscopes^[29,31,32] and thus the driven mode of the 2D mechanical resonator is treated classically by ignoring the effects of the related thermal noise and Coriolis force. In the present work, without using such an external drive, we treat all the three-modes of the system quantum mechanically and provide an alternative method for measuring angular velocity.

The gyroscope is made of a three-mode optomechanical system, shown in Fig. 1(a), consisting of two parts: a 2D mechanical oscillator which does simple harmonic motion along the x axis and y axis, and a single-mode cavity field which is used to monitor the 2D mechanical oscillator’s displacement along the x axis via optomechanical coupling of the radiation pressure force. The gyroscope is placed on a platform rotating with angular velocity Ω. Ω is along the z axis and goes through the original point, e.g., the center of the platform.

Fig. 1.

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Fig. 1. (color online) (a) Schematics of the gyroscope containing the three-mode optomechanical system. A 2D mechanical oscillator (without mechanical drive) is placed on a platform rotating with angular velocity Ω (Ω is parallel to the z axis). The two mechanical modes are coupled by the Coriolis effect due to the absolute rotation. The mechanical motion in the x direction is coupled to an optical cavity by radiation pressure. (b) Schematics of the balanced homodyne measurement. The output field from the cavity is mixed with a much stronger classical reference laser at the input of a 50:50 beam splitter. The output from the beam splitter is fed to two photo-detectors D1 and D2. The difference in the photodetector readings gives the phase of the output field from the cavity.

In the coordinate system fixed to the rotating platform, the Hamiltonian of the 2D mechanical oscillator is given as 1 where x and y represent the displacements of the mechanical oscillator along the x axis and y axis, respectively. m and ω_x (ω_y) are the mass and the frequency of the mechanical oscillator along the x axis (y axis), respectively. The equilibrium position of the mechanical oscillator is placed at the original point. The terms depending on Ω in Eq. (1) result from the Coriolis force acting, in the x–y plane, on the 2D mechanical oscillator.

Noteworthily, we have omitted the centrifugal force in Eq. (1) by assuming it is much smaller than the Coriolis force, that is . Actually, even though the centrifugal force is not negligible compared with the Coriolis force, it will also not bring a significant effect on our result of angular velocity detection. As we will show later, e.g., in Eqs. (31)–(39), the angular velocity is deduced by detecting the optical output spectrum. Though the Coriolis force will change the output spectrum in all the frequency range, we only focus on the spectrum around ω = ω_y for a realistic detection. In contrast, the centrifugal force will only change the zero-frequency component of the output spectrum (for a constant angular velocity) and thus will not affect the detection result of angular velocity. For the sake of simplicity, we do not take the centrifugal force into account in this work.

The canonical momenta conjugate to x and y are given as^[34,35] 2 where and represent the mechanical momenta of the mechanical oscillator along the x axis and y axis, respectively. Hence, the corresponding commutation relations are [x,p_x] = iℏ and [y,p_y] = iℏ with ℏ the reduced Planck constant.

The model under consideration is a three-mode optomechanical system involving two mechanical modes and one cavity field mode. After taking the cavity field into account, the Hamiltonian for the whole system is given as 3 where a is the annihilation operator for the cavity field, and g = ω_c/L is the single-photon radiation pressure coupling strength with L the length of the cavity and ω_c the eigen-frequency of the cavity. The cavity is coupled to an external driving field whose frequency and the driving strength are given as ω_d and , respectively.

After making a rotating frame at drive frequency ω_d, the Heisenberg–Langevin equations of motion are given as 4 5 6 7 8 with κ being the decay rate of the cavity field and γ_x(γ_y) being the decay rate of the mechanical oscillation along the x axis (y axis). Δ₀ = ω_c − ω_d is the detuning between the cavity field mode and the drive field. ξ_x,y and δa_in are the noise operators with zero mean values, and their non-zero correlations^[36,37] are given as 9 10 where k_B is the Boltzmann constant and T is the bath temperature. We assumed that k_BT ≫ ℏω_x,y in Eq. (10).

Note that the present optomechanical gyroscope model is different from the ones used in previous works^[29,31,32] in the following way. In the previous works,^[29,31,32] a classical external drive is applied on the mechanical resonator along the y axis. In the presence of such a strong classical external drive, the mechanical motion along the y axis is treated classically (e.g., the classical motion with sinusoidal velocity^[29,32] or constant velocity^[31] along the y direction) by ignoring the effects of the related thermal noise and Coriolis force. Thus, the corresponding Heisenberg–Langevin equations of motion contain only a single cavity mode and a single mechanical mode in the x axis. In the present work, due to the absence of such a classical external drive in our model, the Heisenberg–Langevin equations of motion for the complete system includes the motion of all the three modes (one cavity mode and two mechanical modes) as given in Eqs. (4)–(8).

Equations (4)–(8) can be linearized by writing all the operators as (A = a,x,y,p_x,p_y), where and δA are the classical mean value and quantum fluctuation of A. Thus the steady state solutions for the mean value equations of motion of Eqs. (4)–(8) can be obtained by setting as 11 12 13 where the effective detuning is 14 As can be seen from Eq. (13), the mean photon number, which is given by , inside the cavity is independent of Ω. Hence we cannot estimate Ω by monitoring the mean intensity of the cavity output field. Thus, we shall focus on the equations of motion of the fluctuation operators in order to measure Ω. When Ω ≠ 0, the Coriolis force coupling between the two mechanical modes modifies the spectrum of the fluctuations in the output optical field via radiation-pressure optomechanical coupling. Thus the absolute rotation of the platform can be monitored from the spectrum of the fluctuations in the output optical field.

The linearized equations of motion for the fluctuation operators in Eqs. (4)–(8) are 15 16 17 18 19 where the effective coupling strength has been assumed to be real without loss of generality. Note that in writing Eqs. (15)–(19), the higher-order terms like δxδa are neglected. This is reasonable since we can make by changing the intensity of the driving laser, .

By defining the Fourier transform of operator A as with , we obtain the solution to the linearized equations (15)–(19) in the frequency domain as 20 with 21 22 23 24 Here we defined Making use of the input–output relationship^[38] , we can obtain the fluctuation and steady-state mean value of the output cavity field as 25 26 with and .

The phase of the output field, represented by a_out, from the cavity is measured using a homodyne setup as shown in Fig. 1(b). The output field a_out is mixed with a classical reference laser field,^[39] whose amplitude is α_R, at a 50:50 beam splitter as shown in Fig. 1(b). The output from the beam splitter is fed onto photo-detectors D₁ and D₂. The readings in D₁ and D₂ are given as and , respectively, with and . The difference in the photo detector readings is given as 27 where α_R has been taken to be real.

We rewrite I as a sum of its mean value and fluctuation by writing . So we obtain 28 which is independent of Ω. The fluctuation δI in I is given as 29 The spectrum of δI is given as 30 By using Eqs. (9), (10), (20), (25), and (29), we can obtain the spectrum 31 with 32 33 34 In Eq. (31), S_add includes both the optical shot noise and the optomechanical back-action noise, S_xx gives the thermal noise in the mechanical oscillator’s motion along the x axis, and S_yy gives the effect of the Coriolis force on the mechanical oscillator’s motion along the x axis due to the thermal vibrations along the y axis. Note that χ(ω) is dependent on Ω, which means S_xx and S_yy (as well as S_add) include the effect of the Coriolis forces.

In principle, the fact that all the three parts (S_add, S_xx, and S_yy) in the spectrum S_II depend on Ω, allows one to measure Ω by monitoring the optical output spectrum. Here, we will just focus on the following simple case 35 When equation (35) is satisfied, χ(ω) is independent of Ω and is given as 36 Thus, both S_add(ω) and S_xx(ω) are independent of Ω, and S_yy(ω) can be rewritten as 37 with the transmission coefficient 38

The fact that S_yy is proportional to Ω² provides us with a method to estimate the angular velocity as , from the homodyne measurement results of the optical output spectrum. The other two parts in the spectrum S_II(ω), i.e., S_add and S_xx, determine the noise in the optical output spectrum. Hence the minimum measurable angular velocity Ω_noise is given as 39 In principle, the above minimum measurable angular velocity will be obtained when the measurement time t_m → ∞. Note that the signal spectrum S_yy(ω) as well as the transmission coefficient σ(ω) is a Lorentz peak with full width at half maximum being of the order of γ_y. In a realistic detection, when the measurement time is long enough such that t_m ≫ 1/γ_y, one can obtain approximately the minimum measurable angular velocity Ω_noise.

In what follows, we will only consider two limiting cases: S_add(ω) ≪ S_xx(ω) and S_add(ω) ≫ S_xx(ω).

When 40 the minimum detectable angular velocity 41 is independent of T, Δ, and ω_x.

According to Eq. (41), Ω_noise(ω) reaches its minimum value when ω = ω_y. For simulation, we consider the parameters:^[40]m = 6.9 × 10⁻⁴ kg, ω_y/2π = 84.8 Hz, γ_x/2π = γ_y/2π = 1.9 × 10⁻³ Hz, κ/2π = 4 × 10⁶ Hz, and g/2π = 1.46 ×10¹⁷ Hz/m. Temperature is as T = 300 K and the effective optomechanical coupling strength is G/2π = 2.19 × 10²² Hz/m (with the corresponding ). The minimum detectable angular velocity is shown in Fig. 2(a) with the optimal value Ω_noise(ω_y)/2π = 9.5 × 10⁻⁴ Hz. The corresponding transmission coefficient σ(ω_y) is optimized when Δ = 0 as shown in Fig. 2(b).

Fig. 2.

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Fig. 2. (color online) (a) The minimum detectable angular velocity in Eq. (41). (b) The corresponding transmission coefficient σ(ωy) versus Δ for different values of ωx. (c) Simulation of the condition given in Eq. (35) with ω = ωy for different ωx. (d) The ratio of Sadd(ωy) in Eq. (32) to Sxx(ωy) in Eq. (33) as a function of ωx and Ω with fixed Δ = 0 and T = 300 K. The other parameters are:[40]m = 6.9 × 10−4 kg, ωy/2π = 84.8 Hz, γx/2π = γy/2π = 1.9 × 10−3 Hz, κ/2π = 4 × 106 Hz, and g/2π = 1.46 × 1017 Hz/m. The effective optomechanical coupling strength is taken as G/2π = 2.19 × 1022 Hz/m (with the corresponding ). For these parameters, the optimal value of Ωnoise(ωy)/2π = 9.5 × 10−4 Hz, also see the vertical line in panel (c).

Note that the above analysis for estimating the minimum detectable angular velocity as well as the optimal transmission coefficient is valid only when equations (35) and (40) are satisfied. As shown in Fig. 2(c), the condition in Eq. (35) is not fulfilled for the resonance case of ω_x = ω_y when Ω is larger than the optimal minimum detectable angular velocity Ω_noise. For the non-resonance case of ω_x = 2ω_y or ω_x = 0.5ω_y, the condition in Eq. (35) is satisfied well when Ω/2π is less than about 0.1 Hz (∼10⁻³ω_y). Since the condition of Eq. (40) is always fulfilled for the above parameters [see Fig. 2(d)], we can know that for the typical simulation parameters as given in the caption of Fig. 2 and the non-resonance case of ω_x ≃ 2ω_y, the optimal minimum detectable angular velocity of our optomechanical gyroscope is Ω_noise(ω_y)/2π = 9.5 × 10⁻⁴ Hz with the corresponding working range of the angular velocity being Ω_noise/2π ≤ Ω/2π ≲ 0.1 Hz.

Now we consider the second limiting case: when 42 In this case the minimum detectable angular velocity reduces to 43 which is dependent on T, Δ, and ω_x.

In Fig. 3(a), we plot the minimum detectable angular velocity in Eq. (43) for the parameters:^[41]m = 1.9 × 10⁻⁷ kg, ω_y/2π = 8 × 10⁵ Hz, γ_x/2π = γ_y/2π = 80 Hz, κ/2π = 1 × 10⁶ Hz, and g/2π = 1.17 × 10¹⁷ Hz/m. Temperature is T = 300 K and the effective optomechanical coupling strength is G/2π = 3.51 × 10²⁰ Hz/m (with the corresponding ). The optimal minimum detectable angular velocity appears at ω = ω_y [as similar to the case in Fig. 2(a)] with Ω_noise/2π = 92.6 Hz, 40.0 Hz, and 336.4 Hz for ω_x = 0.5ω_y, ω_y, and 2ω_y, respectively. The corresponding transmission coefficient σ(ω_y) is shown in Fig. 3(b) with its optimal value appearing at Δ = 0.

Fig. 3.

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Fig. 3. (color online) (a) The minimum detectable angular velocity in Eq. (43) for different values of ωx. (b) The corresponding transmission coefficient σ(ωy) versus Δ for different ωx. (c) Simulation of the condition given in Eq. (35) with ω = ωy for different ωx. (d) The ratio of Sadd(ωy) in Eq. (32) to Sxx(ωy) in Eq. (33) as a function of ωx and Ω. Here Δ = 0 in panels (a) and (d). The other parameters are:[41]m = 1.9 × 10−7 kg, ωy/2π = 8 × 105 Hz, γx/2π = γy/2π = 80 Hz, κ/2π = 1 × 106 Hz, T = 300 K, and g/2π = 1.17 × 1017 Hz/m. The effective optomechanical coupling strength is taken as G/2π = 3.51 × 1020 Hz/m (with the corresponding ).

The above analysis of the minimum detectable angular velocity as well as the optimal transmission coefficient is only valid when equations (35) and (42) are satisfied. It is shown in Fig. 3(c) that the condition in Eq. (35) is not fulfilled for the resonance case of ω_x = ω_y when Ω is larger than the optimal Ω_noise. However, this condition is satisfied well for the non-resonance case of ω_x = 2ω_y or ω_x = 0.5ω_y when Ω/2π is less than 800 Hz (∼10⁻³ω_y). It is shown in Fig. 3(d) that the condition of Eq. (42) is always fulfilled for the above considered parameters. Thus we know for the typical parameters as given in the caption of Fig. 3, and under the non-resonance case ω_x ≃ 0.5ω_y, the optomechanical gyroscope works well when Ω_noise(ω_y)/2π ≥ Ω/2π ≲ 800 Hz and the optimal minimum detectable angular velocity Ω_noise(ω_y)/2π = 92.6 Hz.

In conclusion, we studied the application of a three-mode optomechanical system as a gyroscope for detecting absolute rotation. Our model of an optomechanical gyroscope, where a cavity field mode is coupled to one of the two mechanical modes of a 2D resonator, is similar to the previous ones^[29–31] except that here we do not use any additional external mechanical drive on the mechanical resonator. When our optomechanical system is placed on a rotating platform, the mechanical oscillation modes of the 2D mechanical oscillator are coupled due to the Coriolis force. Using the Heisenberg–Lengivin equations of motion, the dynamics of all the three modes (one cavity field mode and two mechanical oscillator modes) are studied. The angular velocity of the absolute rotation is estimated by monitoring the spectrum of the output field from the cavity via homodyne measurement. The noises in the output spectrum, which determine the minimum detectable angular velocity, include the thermal noise of the mechanical oscillator (S_xx), and the additional noise (S_add) from the optical shot noise and optomechanical back-action. We further analyzed the cases when S_xx ≫ S_add and S_xx ≪ S_add, and estimated the optimal minimum detectable angular velocity. The corresponding working ranges of the angular velocity in these two cases are also discussed.

We would like to remark that our analysis is based on the condition in Eq. (35), which requires that the angular velocity should not be too strong. That is, the noises in the optical output spectrum are required to keep approximately unchanged when the system rotates. This limits the maximum value of the angular velocity to be detected in our model. Actually, similar conditions also exist in the previous optomechanical gyroscopes.^[29,31,32] In future, we will further investigate the application of an optomechanical gyroscope in the case of large angular velocity which does not satisfy the condition in Eq. (35).