Gyroscope with two-dimensional optomechanical mirror

We propose an application of two-dimensional optomechanical oscillator as a gyroscope by detecting the Coriolis force which is modulated at the natural frequency of the optomechanical oscillator. Dependence of gyroscope's sensitivity on shot noise, back-action noise, thermal noise, and input laser power is studied. At optimal input laser power, the gyroscope's sensitivity can be improved by increasing the mass or by decreasing the temperature and decay rate of the mechanical oscillator. When the mechanical oscillator's thermal occupation number, nth, is zero, sensitivity improves with decrease in frequency of the mechanical oscillator. For n th ≫ 1 , the sensitivity is independent of the mechanical oscillator's frequency.


Introduction
Detection of absolute rotation [1,2] has significant importance in fundamental physics for testing gravitation theories [3,4] and in practical applications for improving navigation systems [5,6]. The Sagnac effect [7][8][9][10][11] and detection of rotation induced pseudo forces [12,13] are the two most commonly used methods for absolute rotation detection. Pseudo forces that arise due to rotation are centrifugal force, Coriolis force, and force due to angular acceleration. The angular acceleration force disappears in a system rotating with constant angular velocity. Estimating centrifugal force requires additional procedures to locate the axis of rotation [14]. Hence, Coriolis force [15] measurement is more suitable for absolute rotation detection. In this work, we propose an optomechanical gyroscope by detecting the Coriolis force which is modulated at the natural frequency of the optomechanical oscillator.
Optomechanical cavities [16,17] are very sensitive to any external forces acting on them, and hence they were extensively studied for gravitational wave detection [18,19], weak force detection [20][21][22][23][24][25], and displacement sensors [20,26,27]. In general, the freely oscillating mirror of the optomechanical cavity (OMC) is subject to the radiation pressure force [16,17,28] of the intra-cavity field. This radiation pressure force changes the length of the cavity and thus the optical response of the cavity itself. In this work, we consider a two dimensional optomechanical mirror [29], which is free to oscillate along x-axis and y-axis. Such an oscillator can be realized by connecting the one-dimensional oscillator, which oscillates along the x-axis, to an electrical circuit. The electrical circuit oscillates the mirror along the direction perpendicular to x-axis. The two dimensional oscillator is placed on a rotating table and driven by a laser field along the x-axis. When the table rotates, oscillator is driven by radiation pressure force and Coriolis force. The frequency of Corilis force acting on the mirror is equal to the frequency of optomechanical mirrorʼs oscillation along the y-axis. Figure 1 shows the schematics of the two-dimensional optomechanical gyroscope. The two-dimensional optomechanical mirror has mass m and angular frequency w m along x-axis and y-axis. The OMC, which is placed Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Optomechanical gyroscope
in arm 2 of the interferometer, is driven by an optical fieldÊ. Arm-2 is parallel to the x-axis of the rotating table. When the table is not rotating, the equation of motion of the optomechanical mirror oscillating along the x-axis is given as where gĉ , mb , x andF are the decay rate of the mechanical oscillator, cavity field annihilation operator, optomechanical mirror annihilation operator, and thermal noise operator, respectively.
with ÿ as Planck constant, l as the length of OMC and w o is the eigen frequency of the cavity. The OMC is mounted on a stage which is driven with velocityȳ along the y-axis. When the table starts to rotate with angular velocity q , then fictitious forces begin to act on the optomechanical mirror. In presence of the fictitious forces, equation (1) is given as [15] w g c c c q q q = --- In equation (2), q term represents Coriolis force, q 2 term represents centrifugal force and q term represents angular acceleration force. We set q = 0 by assuming that the table is rotating with uniform angular velocity. We are interested in detecting the rotation rates that are much smaller than 1 Hz. Hence the q contribution to the centrifugal force, which has q 2 dependence, is negligible in comparison with the Coriolis force term which has q dependence. With these simplifications, the final simplified form of equation (2) can be written as The cavity field equation of motion is given as We linearize equations (4) and ( in presence of the Coriolis force. Moreover, the noise term in the Coriolis force, which is given by d qṁ 2 y , contributes to the second order and can be neglected in this work. The OMC is mounted on a stage which is driven co-sinusoidally so that the instantaneous velocity of the mechanical oscillator along y-axis isȳ. When the table rotates with angular velocity q , the Coriolis force acts on the mechanical oscillator along the x-axis and is equal to qm y 2 . The phase shift, corresponding to the change in the optomechanical cavity length due to the action of Coriolis force, is measured at the detectors D1 and D2 to estimate the angular velocity of the table.
By setting , we can write the solutions ofb andc as The mean field at the output of the OMC, represented byc o , can be obtained by using input-output formalism [30,31] as The solution for the equation of motion of d c can be obtained in the Fourier frequency space by using the transform function  . Then the fluctuation in the cavity field output, denoted by d ŵ ( ), can be obtained by using input-output formalism as  where

Signal evaluation
We add a p 2 phase to the field coming out of the OMC so that the difference in the photo-detector reading is given as whereˆˆÎ I c , , o 2 1 1 andĉ o are the intensity at detector D2, intensity at detector D1, output field from FM and output field from OMC, respectively. We linearize the equation (8)  Assuming an ideal 50:50 beam splitter, fields entering into arm-2 and arm-1 are given as in equation (9). Hence, in Fourier frequency domain, equation (9) is given as Using equation (6), the mean value of equation (10) is given as Equation (11) represents the signal from the gyroscope. The velocity of the optomechanical mirror along the y-axis of the rotating table, shown byȳ in equation (11), is treated as a general function of time. We fix the optomechanical mirror to a co-sinusoidal oscillator such that the optomechanical mirror is being oscillated with an angular frequency of w m along the y-axis of the Substituting equations (11) and (12) in (13), we can write . We assumed that w g  m m in writing S o . The presence of w t m term in equation (14) indicates that the signal is due to the Coriolis force which is modulated at angular frequency w m .

Results and discussion
In an experiment, the angular velocity of absolute rotation can be estimated by measuring the amplitude of the signal (equation (14)) which is oscillating with angular frequency w m . The rotation detection sensitivity can be estimated by dividing the noise at w m with the amplitude S o . Hence, the sensitivity q w q ȯ ( )Ṅ On the RHS of equation (18), inside the second square root, the first term shows the shot noise contribution, while the second term represents the back-action noise [32] contribution and the third term gives the optomechanical mirror's noise contribution. The mirror noise contribution does not depend on the input laser intensity I l . The q s is directly proportional to the square root of I l for the back-action noise contribution and inversely proportional to the square root of I l for the shot noise contribution. Figure 2 shows the dependence of rotation detection sensitivity as a function of input intensity. The green line, which shows the shot noise contribution, indicates that the sensitivity is being improved with increase of I l . The Blue line, which shows the back-action noise, indicates that the sensitivity is being decreased with the increase of I l . So for a change in I l value: if the shot noise increases then the back-action noise decreases and vice versa. As a result, there exists an optimum intensity I o at which both the shot noise and radiation pressure noise contribute equally to the sensitivity. At I l =I o , we get the best sensitivity which corresponds to the lowest point in the Purple curve. Minimizing equation (18) (18), at I l =I o , the rotation detection sensitivity is given as  (20) is in the same order of magnitude as the sensitivity achieved in the squeezed-vacuum gyroscope [15]. In the squeezed-vacuum gyroscope, squeezed vacuum field is injected in to two optomechanical cavities to suppress the shot noise and back-action noise and thus improving the sensitivity of the gyroscope. In the present work, the frequency of the Coriolis force is modulated such that it is on resonance with the optomechanical mirror's natural frequency and hence the mechanical oscillator more responsive to the Coriolis force. As a consequence, without using squeezed vacuum field, the gyroscope in the present work is able to attain the similar sensitivity as that in the squeezed-vacuum gyroscope. It is important to note that the contributions from shot noise, back-action noise and mirror noise are almost equal in equation (20), hence the sensitivity given in equation (20) can not be improved significantly by suppressing shot noise and back-action noise, at optimal input laser intensity, by using the squeezed light technique described in [15]. Equation (20) implies that the sensitivity can be improved by decreasing w m . Decreasing w m reduces the restoring force and hence the optomechanical mirror becomes more sensitive to the Coriolis force. The sensitivity can also be improved by increasing the mass of the oscillator. Increasing m leads to increase in I o according to equation (19), but this can be compensated by decreasing g m such that the product g m m remains constant in equation (19). So increasing m and decreasing g m such that the product g m m remains constant can enhance the gyroscopes sensitivity without changing I . The thermal occupation number, represented by n th , of the mechanical oscillator is assumed to be zero in equation (20). Given the recent progress in cooling [33,34] the mechanical oscillator to its ground state [35], the result shown in equation (20) is practically relevant. However, in many practical cases we have ¹ n 0 th and hence it is necessary to consider the temperature dependence. Taking T as the temperature and K b as the Boltzmann constant, temperature dependence of the rotation detection sensitivity is given as Hence for q ṅ 1, s th is independent of w m and the sensitivity can not be improved indefinitely, unlike equation  Using these parameters the gyroscope's sensitivity is estimated as 7×10 −15 rad s -1 -/ Hz 1 2 at 0 K temperature and -1.5 10 9 rad s -1 -/ Hz 1 2 at 300 K temperature. Other prominent method for rotation detection uses Sagnac effect [37]. Improving the performance of fiber optic Sagnac-gyros requires increasing the length of the optical fiber through which light travels. Such a design leads to increased scattering and absorption of light by the optical fiber. Sagnac based ring laser gyros, which detect rotation by measuring frequency difference at the output, are limited by laser line-width. Moreover, the sensitivity of the Sagnac effect based gyroscopes is limited by the area [38] of the closed loop. The method described in this work do not depend on the dimensions of the optical path. Hence this scheme can be realized in miniature structures more easily when compared with Sagnac gyroscopes. The two dimensional optomechanical mirror can be realized by coupling a one-dimensional oscillator to an electrical circuit or by designing two dimesional oscillators as described in [39,40].

Conclusion
We propose an application of two-dimensional optomechanical oscillator as a gyroscope to detect absolute rotation. We showed that the two-dimensional optomechanical gyroscope's sensitivity is better than onedimensional optomechanical gyroscope [15] by a factor of g w m m . Using the parameters from [36], the sensitivity of the gyroscope is estimated as 7×10 −15 rad s -1 -/ Hz 1 2 at 0 K temperature. Temperature dependence of the gyroscope's sensitivity is studied and at 300 K temperature, the sensitivity is approximately equal to´-1.5 10 9 rad s -1 -/ Hz 1 2 . Given the recent progress in the field of optomechanics, the work described in this manuscript can be used to design very sensitive gyroscopes.