Absolute rotation detection by Coriolis force measurement using optomechanics

In this article, we present an application of the optomechanical cavities for absolute rotation detection. Two optomechanical cavities, one in each arm, are placed in a Michelson interferometer. The interferometer is placed on a rotating table and is moved with a uniform velocity of y ¯ ˙ with respect to the rotating table. The Coriolis force acting on the interferometer changes the length of the optomechanical cavity in one arm, while the length of the optomechanical cavity in the other arm is not changed. The phase shift corresponding to the change in the optomechanical cavity length is measured at the interferometer output to estimate the angular velocity of the absolute rotation. An analytic expression for the minimum detectable rotation rate corresponding to the standard quantum limit of measurable Coriolis force in the interferometer is derived. Squeezing technique is discussed to improve the rotation detection sensitivity by a factor of γ m / ω m at 0 K temperature, where γ m and ω m are the damping rate and angular frequency of the mechanical oscillator. The temperature dependence of the rotation detection sensitivity is studied.


Introduction
Detection of absolute rotation has significant importance in fundamental physics for testing gravitation theories [1] and in practical applications for improving navigation systems. The Sagnac effect [2] is one of the well known phenomena used for the detection of rotation. In the Sagnac effect, absolute rotation introduces a time delay [2][3][4] between two counter propagating laser beams placed on a rotating table. The time delay is manifested in the form of phase difference in a fiber optic laser gyroscope [5,6], and as frequency beat signal in an active laser gyroscope [7]. The Sagnac effect is also extended to atomic interferometry [8,9] for improved rotation detection. A review of rotation detection schemes can be found in [10]. Correlated spontaneous emission theory is developed in [11] to improve the rotation detection by reducing phase noise. The Sagnac effect is generalized to linear motion in [12]. Another prominent method for detecting rotations is by measuring the effects of rotation induced pseudo forces like centrifugal force and Coriolis force. Hence, a very sensitive force detector can potentially be used as a rotation detector. In our previous work [13], Coriolis force induced displacements are measured using enhanced longitudinal Fizeau drag effect [14] to detect the rotation. In this paper, we propose a rotation detector based on the effects of the Coriolis force on the optomechanical cavity placed on a rotating table. We also discuss the application of squeezed light for improving the rotation detection sensitivity. Temperature dependence of the rotation detection sensitivity is also discussed.
Optomechanical cavities are based on the principles of radiation pressure force, but these systems are very sensitive to any external forces acting on them. Hence they were extensively studied for gravitational wave detection [15][16][17][18]. In general, an optomechanical cavity consists of a Fabry-Perot cavity with one of its two cavity mirrors freely oscillating. When such a cavity is driven by an external laser field, the freely oscillating mirror of the cavity will be displaced by the radiation pressure force [19][20][21] of the intra cavity field. This changes the length of the cavity and thus the optical response of the cavity itself. Optomechanical cavities are found to be extremely useful as displacement sensors [22][23][24][25] and as weak force sensors [26][27][28][29][30][31][32]. Recently, optomechanical systems have become a focus for a broad range of research activities [20,21] and several interesting phenomena like phase conjugation [33], squeezing [34], super-radiance [35], laser cooling [36][37][38], optomechanically induced transparency [39][40][41][42][43], etc, have been predicted in optomechanics.
In the first part of this paper, we derive the Hamiltonian for an optomechanical cavity placed on a rotating table [44]. Then we derive the equations of motion for a mechanical oscillator in the presence of non-inertial forces arising due to rotation. Then we derive analytic expression for the minimum detectable rotation rate by neglecting the thermal effects. In the second part, we show that the main contribution to the output noise comes from the vacuum fluctuations entering through the empty port of a beam splitter. Then we show that squeezing the input vacuum fluctuations can improve the rotation detection sensitivity by a factor of g w m m , where g m and w m are the damping and angular frequency of the mechanical oscillator, at 0 K temperature. In the last part we discuss the effects of temperature in our system.

Optomechanical oscillator in rotating frame
Consider a perfectly reflecting optomechanical mirror which is moving with velocity (˙˙) x y , on a rotating table. (x, y) represent the instantaneous position co-ordinates of the mechanical oscillator in a reference frame corotating with the rotating table. The velocity of the optomechanical mirror in a non-rotating fixed lab frame is given as where v  is velocity of the mechanical mirror observed from the lab frame,î andĵ are the unit vectors in the frame of reference co-rotating with rotating table. The Lagrangian of the optomechanical mirror in the lab frame is given as where m and w m are the mass and angular frequency of the optomechanical mirror. The co-ordinates ( ) a a , x y represent the equilibrium position in the frame of reference rotating with the rotating table of the mechanical mirror when the table is not rotating. The angular velocity of the rotating table is given by q. The Lagrangian conjugate variables corresponding to the generalized co-ordinates x and y are given as . 3 x y Using equations (2) and (3), the classical Hamiltonian for the optomechanical mirror [44] in the lab frame is given as In the lab frame of reference, the corresponding quantum mechanical Hamiltonian for the mirror oscillator is given as y y are the annihilation (creation) operators for the simple harmonic motion along x-axis and y-axis, respectively. The non-zero commutation relations are given as 1, , 1 .
x x y y

Rotation detection
Two optomechanical cavities, cavity-1 and cavity-2, are placed along the arm-1 and arm-2 of a Michelson interferometer as shown in figure 1(b). The interferometer is placed on the rotating table in such a way that arm-1 coincides with the y-axis, and the arm-2 is parallel to the x-axis, of the rotating table (shown in figure 1(a)). A laser source with amplitudeÊ and with angular frequency w d drives both the optomechanical cavities. For simplicity, we assume that the input and output fields for the cavity-1 and cavity-2 are along y-axis and x-axis, respectively. Using equation (5) and the optomechanical Hamiltonian description in [45], the Hamiltonian for the interferometer placed on the rotating table, in the lab frame of reference, is given as where c w = - l o o , l is the length of the optomechanical cavities when the optomechanical mirrors are in equilibrium position. The annihilation(creation) operator for cavity filed in cavity-a is a a ( ) † c c . a b x and b yα are the annihilation operators for the optomechanical mirror oscillation along the x-axis and y-axis, respectively, in cavity-a. The equations of motion for the cavity fields after making rotating wave approximation at input laser frequency w d are given as where z and â E are the decay rate and input for the cavity field oscillator in cavity-a, respectively. The equations of motion for theb x2 andb y1 are given as In the case of absolute rotation detection, the detector, the laser source, and the entire experiment is on the rotating table. Hence, it is more logical to transform the equations of motion (equations (7)-(9)) from the lab frame of reference to the frame of reference co-rotating with the rotating table. This can be achieved by writing the annihilation and creation operators as following: 2 . Substituting equations (11)(12)(13)(14) into equations (8-10)) the equations of motion for the mechanical oscillator and cavity field in the frame of reference co-rotating with rotating table are given as where z and g m are the phenomenological damping rate for the cavity field oscillator and mirror oscillator, respectively. â E and â s are the inputs for the cavity field oscillator and mirror oscillator. The second term on the RHS of the equations (15) and (16) indicates the presence of non-inertial force due to the rotation of the table. In equations (15) and (16), the q 2 term represents the centrifugal force, q term represents the Coriolis force and q term represents the force due to the angular acceleration of the rotating table. We set q = 0, by assuming that the table is rotating with constant angular velocity. We are interested in detecting the small rotation rates, hence the contribution from the centrifugal force term, which has q 2 dependence, is negligible in comparison with the Coriolis force term which has q dependence. The mechanical oscillator in cavity-1 oscillates only along the yaxis of the rotating table, hence =ẋ 0 1 in equation (15). The mechanical oscillator in cavity-2 oscillates only along the x-axis of the rotating table. Hence =˙ȳ y 2 , whereȳ is the velocity with which the interferometer is moving along the y-axis of the rotating table, in equation (16). We want to investigate the effect of classical Coriolis force, hence we replace q in equations (15) and (16) with its classical mean value q . Using the above arguments, the equations (15) and (16) can be simplified as , , , b c E s represents the fluctuation in mirror oscillator, cavity field oscillator, cavity field oscillator input and mirror oscillator input for the cavity-a, respectively. a a a ā¯¯b c E s , , , represents the steady state classical values for mechanical oscillator, cavity field oscillator, cavity field oscillator input and mechanical oscillator input for the cavity-a, respectively. We take = ā s 0. The steady state classical values for the cavity field can be obtained by solving equations ((7), (18), (19)) with their time derivatives set to zero. in equations (7) and (8). The fluctuations in the cavity field oscillator and mechanical oscillator are solved using the Fourier transform function The solution to the fluctuations, in frequency domain, for the cavity-a is given as   A π/2 degree phase is added to the output from cavity-2 before reaching the beam splitter. Hence, the intensity of the optical field reaching both the detectors is equal when the table is not rotating. The difference in the photo detector readings is given as The inputs for the cavity field oscillators are taken as the coherent laser field with amplitudeÊ and the vacuum field entering through the empty port of the beam splitter. For a lossless 50:50 beam splitter, the amplitude of fields entering the cavity-a is given as whereÊ v is the amplitude of the vacuum field. Linearizing equation (30) gives the steady state classical values for the cavity field oscillator's inputs as =Ē E 2 1 and =Ē E i 2. 2 Assuming that the input laser fieldÊ is real, and using equation (26), the equation (28) is given as  I  E  2 2 is the input laser intensity. The input power, represented by P, corresponding to the input intensity is given as o We assume that z D  and set z z D + » 2 2 2 in equation (31). From equation (30), the amplitude of fluctuations in the input fields are given as Assuming that the input vacuum and laser fields are uncorrelated at all times, the only non-zero correlations terms we have are d w d w . We also neglect the thermal noise by Using these correlation functions and equation (22), the quantum noise spectrum can be calculated from the variance of d d - where w z G = i .

Results and discussion
The quantum noise spectrum at the interferometer output is given by V . The v o in equation (35) gives the quantum noise at the signal. In the RHS of equation (35), the first term represents shot noise due to the discrete nature of photons in both the arms, the second term represents the back action noise due to the radiation pressure fluctuations [48,49] in both the cavities, the third term represents the mirror noise originating from the mechanical oscillator input in both the cavities. With thermal noise neglected, the significant contribution to the noise comes from the shot noise and from the radiation pressure noise. The shot noise can not be manipulated by tuning the optomechanical cavity parameters. On the other hand, the radiation pressure noise can approximately be set to the shot noise level when the following condition is satisfied. Signal in equation (31) requiresĪ to be large, but equation (36) requires theĪ to be small. Hence, there exists an optimum intensity [50], say I opt , at which we obtain the best signal to noise ratio. I opt is the maximum valueĪ can take with out violating the condition in equation (36). Hence, we can write Note that, for z D  , the contribution from the shot noise and the radiation pressure noise in equation (38) are equal [51]. By equating the signal from equation (31) with noise in equation (38), the minimum detectable angular velocity is given as The shot noise and the radiation pressure noise in equation (43) can be adjusted by the squeeze parameters f and r. Minimizing equation (43) with respect to f gives the optimum phase squeeze parameter f o , which is given as The q s represents the minimum detectable angular velocity when the vacuum field entering through the empty port of the beam splitter is squeezed. The first term on the RHS of equation (47) comes from the shot noise, the second term comes from the radiation pressure noise and the third term comes from the mirror noise. The shot noise and the radiation pressure noise terms can be set to zero [58] for r 1  , but the mirror noise can not be reduced by the squeezing. Hence the mirror noise sets the lower limit on q s value. Since we are working with optomechanical mirrors with high quality factor, g w m m is much smaller than one. For sufficiently large r, » e 0 r 2 and the rotation detection sensitivity is improved by a factor of g w m m as given in equation (47). It is important to note that squeezing can enhance sensitivity only if the mirror noise is less the shot noise and radiation pressure noise. In practical situations, like at room temperature, noise from the mechanical oscillator can be much larger than the shot noise and the radiation pressure noise. In such cases, squeezing the input vacuum field can not enhance the rotation detection sensitivity. The temperature dependence of minimum measurable angular velocity is discussed in the next section.

Temperature dependence
Equations (39) and (47) estimate the minimum measurable angular velocity at 0 K temperature. In practical situations, especially when w w m d  , the thermal noise can not be neglected as it can be much larger than the shot noise and the radiation pressure noise. The thermal noise can be estimated by using the correlation When the squeezed vacuum is sent through the empty port of the beam splitter, the thermal dependence of q s is given as

 
where q sT is the minimum detectable angular velocity at temperature T when the vacuum entering through the empty port of the beam splitter is squeezed. The effect of temperature with squeezed vacuum is discussed below in case (i) and case (ii).
At r=0, the mechanical oscillator noise in equation (51) is smaller than the shot noise and the radiation pressure noise and hence the total noise is determined by the shot noise and radiation pressure noise. By squeezing the input vacuum such that f f = o and r 1  , the shot noise and the radiation pressure can be approximately set to zero. Then the total noise decreases to the level of the mirror oscillator noise, and hence the rotation detection sensitivity can be improved beyond equation (39) by using squeezed vacuum. At r=0, the mechanical oscillator noise in equation (51) is larger than the radiation pressure noise and shot noise. Hence the total noise is determined by the mechanical oscillator noise.
Since squeezing the input vacuum can not reduce the mechanical oscillator noise, the rotation detection sensitivity can not be improved. At temperature T, the squeezing technique described in this work can improve the rotation detection sensitivity only if w g , by substituting equation (37) in to equation (51), the minimum detectable angular velocity is given as (50) and (52), we can conclude that the rotation detection sensitivity can not be improved by using the squeezed vacuum technique when the mirror oscillator noise is much larger than the shot noise and radiation pressure noise. However, the rotation detection sensitivity can be improved by increasing the mass of the mechanical oscillator. Increasing the mass of the mechanical oscillator also increases the optimum intensity required to achieve the best sensitivity. According to equation (37), the increase in I opt value due to an increase in m can be compensated by using a cavity with large w z o value. Note that for the optomechanical cavity parameters used in this simulation, w g  kT 2 m m  at room temperature. Hence, the mechanical oscillator noise at room temperature is much larger than the shot noise and radiation pressure noise. In this case, the squeezed vacuum technique can not enhance the sensitivity at room temperature.

Conclusion
Application of optomechanical cavities for rotation detection via measurement of Coriolis force induced effects is discussed in this paper. Analytic expressions to estimate the optimum power and minimum measurable rotation rate are derived. The implementation of squeezed vacuum technique for improving the rotation detection sensitivity discussed. We showed that at zero Kelvin temperature, using the squeezing technique improves the rotation detection sensitivity by a factor of g w m m . Temperature dependence of the minimum measurable angular velocity is studied. We showed that for w g  k T 2  , the noise from the mechanical oscillator is larger than the shot noise and the radiation pressure noise, hence squeezing the input vacuum do not improve the sensitivity in this case.