A table-top high-sensitivity gyroscope based on slow light and cavity enhanced photon drag

The sensitive gyroscope measuring the rotation of an object or the frame can be used to test fundamental physical effects^1–3 and also promise important applications such as directional positioning,^4,5 inertial navigation systems,^6–9 and sensing length-of-day variation of the Earth.^10,11 Currently, various methods are developed to measure slow rotation by using microelectromechanical gyroscopes,^12–16 surface acoustic wave gyroscopes,^17–20 ring laser gyroscopes (RLGs),^21–25 fiber optic gyroscopes (FOGs),^26–28 atomic-optical hybrid gyroscopes,^29,30 and nanophotonic optical gyroscopes.³¹ Among them, optical gyroscopes based on the Sagnac effect are the most reliable and show the highest sensitivity thus far. Its principle involves two counter-propagating waves in the same closed loop, generating a beat frequency reflecting the rotation rate.³² 

A Sagnac-effect-based gyroscope can reach a high sensitivity but typically requires a large size. The ROMY RLG with six 12 m arms displays a sensitivity of $∼80prad/s/Hz$⁠.^21,33 Thus, it can analyze small ground disturbances caused by earthquakes and oceans. The GINGERINO with four 3.6 m long sides shows a sensitivity up to the level of $prad/s/Hz$⁠.³⁴ The ring gyroscope G using a 16-m-circumference ring cavity achieves $12prad/s/Hz$ and a resolution of several milliseconds in detecting disturbances on the Earth’s surface after 120 days of continuous measurements.³⁵ These RLGs utilize mirrors with ultra-low optical loss to construct high finesse optical cavities and use two counter-propagating light beams to reduce bias drift errors.^36,37 Despite great success in rotation measurement, these RLGs are large in size thus far, costly, and fixed on a basis. In contrast, FOGs shrink the volume by circling a long optical fiber to a many-turn coil and, therefore, are portable, but the obtained sensitivity is relatively lower. The state-of-the-art FOG can only achieve a sensitivity of $9.7×10−8rad/s/Hz$⁠,³⁸ several orders lower than the RLG counterparts. Because of the long optical path, the application of FOGs is constrained by the asymmetric excitation in the middle of the fiber circuit caused by temporal and local perturbation, resulting in the non-negligible dynamic errors.^39,40 Such asymmetric perturbation is a common challenge in the Sagnac-effect-based gyroscope with a big ring. Precise measurement of daily variations of the Earth’s rotation rate (Ω_e = 72.92 μrad/s) requires a sensitivity approaching the level of tens $prad/sHz$⁠, which has only been achieved by the large-size RLG.^2,21,34,35 Therefore, it is highly desirable to develop a portable gyroscope with such sensitivity but remains a challenge.⁴¹ 

Propagation of linearly polarized light in a spinning medium results in the change of polarization plane, known as the photon drag effect, because the rotation of the medium causes circular birefringence.⁴² The slow-light medium can significantly enhance the photon drag and narrow the linewidth of a cavity by several orders.^43,44 However, it is thus far yet to explore the interplay of slow light and the photon drag effect for a sensitive gyroscope.

In this work, we theoretically propose a photon-drag-based gyroscope by embedding a slow-light medium in a 20 cm long Fabry–Pérot (FP) cavity. By measuring the transmitted light power of the cavity, we can, in principle, achieve the remarkable gyro sensitivity. Our proposal provides a table-top device for measuring daily variations in the Earth’s rotation rate and testing the effects of Lorentz violation and general relativity on Earth.

The paper is organized as follows: The system and model are described and developed in Sec. II. Then, the results are presented in Sec. III. The feasibility of implementing our system is discussed in Sec. IV. In the end, this paper is concluded in Sec. V.

The system of our gyroscope is depicted in Fig. 1(a). An Er³⁺-doped glass rod is inserted into the FP cavity by two highly reflective mirrors, M₁ and M₂. By detecting the field component orthogonal in polarization to the input field, this system can work as a gyroscope. We assume that the two mirrors have the same reflectivity R and cause a decay rate κ_e to the cavity each. We also assume that the cavity has an intrinsic loss with rate κ_i. The overall decay rate of the cavity is then given by κ = 2κ_e + κ_i. The two ends of the rod are coated with anti-reflective film. A horizontally polarized (HP) probe laser beam, denoted as $âinH$⁠, is input to the cavity via the M₁ mirror. $âout$ denotes the transmitted field. When the frame or the glass rod spins at a rate Ω, the photon drag causes a difference Δn between the refractive indices n_l and n_r “seen” by the left-hand circularly polarized (LCP) and the right-hand circularly polarized (RCP) cavity eigenmodes, respectively. Because of the unidirectional mechanical angular momentum, this difference Δn leads to a nonreciprocal phase shift and lifts the resonance frequencies of the LCP and RCP eigenmodes to non-degenerate. As a result, the LCP and RCP components of the input field accumulate a non-zero relative phase after transmitting through the cavity. A vertically polarized (VP) component $âoV$ appears in the transmitted field. It is then separated by a polarization beam splitter (PBS). By detecting its power with a photodetector, we can determine the spinning rate Ω. In addition, quantum noise (⁠$êin,1$ and $êin,2$⁠) also enters the photodetector and causes errors.

We consider the slowly rotating frame, e.g., Earth’s rotation. Then, the photon drag is very weak. To amplify the signal, another control laser beam $b̂inH$ is used to increase the group velocity index of the probe field to n_g ≫ 1 in the glass rod via the coherent population oscillation (CPO) process of the Er³⁺ ions. The enhancement resulting from slow light is twofold: enhancing the photon drag by several orders⁴³ and significantly narrowing the cavity linewidth.⁴⁴ The combination of both allows us to make a table-top gyroscope with sensitivity exceeding the large RLGs. Here, we neglected the influence of the photon drag due to the spin of the mirrors and the PBS because it is much smaller than the slow-light enhanced photon drag in the Er³⁺-doped glass rod.

Figure 1(b) reflects the group refractive index n_g as a function of the power of the control laser field for various mirror reflectivity. The peak values of n_g are very close for different reflectivities, but the required control laser power, P_s,in, increases as the reflectivity increases. When R = 0.99, the peak n_g ≈ 3.7 × 10⁶ is attained at P_s,in = 440 mW. When the reflectivity is improved to R = 0.999, which is much harder, the peaked n_g is about 3.7 × 10⁶, and the optimal power P_s,in decreases to $∼274mW$⁠, corresponding to 2.1 × 10¹⁸. Thus, it is experimentally convenient to choose the reflectivity between 0.99 and 0.999. Figure 1(c) shows the dependence of n_g on the input photon flux of the control light for various frequency differences δ. The peak n_g first increases rapidly and then decreases after reaching the maximum value as the control laser power increases. The peak value also decreases as the modulation frequency δ increases. From Figs. 1(b) and 1(c), the group refractive index can reach the level of 10⁶.

The gyro sensitivity relies on the input photon flux (N_in), the group refractive index (n_g), the duty ratio (η), and the decay rate (κ). In order to obtain optimal sensitivity, the optimal operation is discussed next.

The performance of our gyro crucially relies on the power of the input probe light. Figure 2 shows the photon flux of the output VP component and the gyroscope sensitivity. According to the photon drag effect, the deflection phase caused by a rotating medium is proportional to the group refractive index n_g.⁵³ Thus, the giant group refractive index induced in the Er³⁺-doped glass rod can essentially enhance the polarization rotation of light due to the photon drag,⁴³ greatly amplifying the output power to a detectable level. Figure 2(a) shows the output photon flux as a function of the probe power and the rotation rate. Taking the experimentally accessible values for parameters R = 0.998, L = 10 cm, L₀ = 20 cm, and η = 0.5, the maximum output photon flux of the VP component can reach $PoutV/ℏωp≈1011photons/s(≈13nW)$ when Ω/Ω_e ≈ 10⁻⁹. In this case, the total transmitted photon flux is approximately P_out/ℏω_p ≈ 5.5 × 10¹⁶ photons/s (≈7 mW). This level of photon flux can be well detected experimentally. The peak output photon flux is attainable for Ω/Ω_e < 1.5 × 10⁻⁹. The required input power is less than P_p,in/ℏω_p < 4 × 10¹⁷ photons/s, corresponding to a probe field power $<52mW$⁠. The performance of our gyroscope is determined by the duty ratio η but not the cavity length L₀. Throughout the following investigation, we fix L₀ = 20 cm.

According to Eq. (12), it can be seen that the gyro sensitivity is proportional to κ and inversely proportional to η, $ng2$⁠, and $Nin$⁠. Therefore, the sensitivity tends to be improved as the input power increases, as shown in Fig. 2(b), but reaches the optimal value of $ΔΩmin=26frad/s/Hz$ when the input photon flux P_p,in/ℏω_p ≈ 2 × 10¹⁷ photons/s (≈26 mW), and then reduces. The reason is that n_g first increases rapidly and then decreases after reaching the maximum value as the input power increases; see Figs. 1(b) and 1(c). It implies that to obtain the best sensitivity, the gyro needs to operate at the optimal input power.

According to Fig. 1(b), we are interested in R = 0.99–0.999, which allows us to achieve the maximum n_g with low control laser power. Figure 3(a) shows the gyro sensitivity vs the mirror reflectivity R and the probe power P_p,in. The gyro is most sensitive when R ≈ 0.998 and P_p,in/ℏω_p ≈ 2 × 10¹⁷ photons/s; see Fig. 3. Figure 3(b) clearly shows the obtained optimal sensitivity of $ΔΩmin≈26frad/s/Hz$ at R ≈ 0.998. A higher reflectivity of the cavity mirrors not only leads to a lower cavity loss⁵⁴ but also significantly affects n_g, as shown in Fig. 1(b). As a trade-off, the mirror reflectivity is optimal at R ≈ 0.998. Note that the slow-light enhances n_g and, thus, the photon drag effect by a factor of 10⁶⁴³ and significantly narrows the intracavity linewidth.⁴⁴ Thanks to this double improvement, the gyro sensitivity can reach nine orders of magnitude higher than Earth’s rotation rate, enabling precise detection of tiny variations of the Earth’s rotation and its wobbling.³⁵ 

Figure 4 presents the dependence of the gyro sensitivity on the duty ratio η. For a given η, the gyro sensitivity first improves as the probe photon flux increases. After reaching an optimal point, its value increases. As predicted by Eq. (12), the sensitivity ΔΩ is inversely proportional to the duty ratio η. An example for P_p,in/ℏω_p = 2 × 10¹⁷ photons/s is presented in Fig. 4(b). Obviously, a larger duty ratio yield is preferable for achieving high performance. However, considering practical experimental losses and operational, we chose a spinning medium length of L = 10 cm. On this basis, the gyro sensitivity can reach $26frad/s/Hz$⁠. This level of sensitivity is accurate enough to detect small variations in the Earth’s rotation rate and the effects of Lorentz violation and general relativity in gravity research.

Figure 5 illustrates the relationship between gyro sensitivity and cavity length L₀ with a fixed duty ratio (η = 0.5). By extending both the cavity length and medium length proportionally, sensitivity is further enhanced, reaching $4.3frad/s/Hz$ at a cavity length of L₀ = 120 cm. This indicates that greater cavities and medium lengths are advantageous for achieving high sensitivity. However, the medium length is subject to technical limitations. Moreover, longer cavity lengths may affect stability in practical applications. There are still significant challenges in experimental implementation. Therefore, we chose a cavity length of L₀ = 20 cm and a medium length of L = 10 cm to achieve a high-sensitivity gyroscope.

Note that we measure the VP component of the transmitted field. Because the accumulated relative phase of the LCP and RCP fields is very small, this VP component is only a fraction of the total transmitted field. Most of the transmitted photons are HP. Moreover, some intracavity photons are dissipated into the environment and the cavity does not store many photons.

In this work, we use a FP cavity embedded with a spinning medium to construct a sensitive gyro. The medium is an Er³⁺-doped glass rod, and slow light can be obtained with a modulated laser beam. The length of the Er³⁺-doped glass rod can be set to L = 10 cm, for example. The refractive index of the glass rod is about n₀ ≈ 1.48. According to Ref. 46, the relaxation time of the excited state is T₁ = 10.5 ms, and the corresponding unsaturated absorption coefficient is α₀ = 0.16 cm⁻¹. We set the cavity mirror reflectivity of the FP cavity R = 0.998. The cavity decay rate is calculated to be κ ≈ 5 MHz. The input photon flux P_p,in/ℏω_p ≈ 2 × 10¹⁷ photons/s (≈26 mW). This laser field at wavelength $∼1536nm$ is sinusoidally modulated by the function generator. We fix P_p/P_s = 0.08, which is experimentally accessible.⁴⁹ We can obtain the output photon flux of the VP component is $PoutV/ℏωp≈1011photons/s(≈13nW)$ and total transmitted photon flux is approximately P_out/ℏω_p ≈ 5.5 × 10¹⁶ photons/s (≈7 mW). On this basis, the resulting gyro sensitivity can reach $26frad/s/Hz$⁠. Moreover, this method is also applicable to a ruby. Using an argon ion laser functioning at 514.5 nm, the population is pumped from the ground state to the wide absorption band of ⁴F₂. Then, slow light based on spectral holes can be observed.⁴⁷ Actually, the slow-light enhanced photon drag effect using a ruby has been experimentally observed.⁴³ Therefore, our theoretically proposed scheme can be implemented with an Er³⁺-doped glass rod or a ruby. To measure the modulated sideband light, the output VP component can be measured via the homodyne detection. In particular, the output VP light is first interfered with by a local light field to generate an electric signal oscillating at frequency δ. Then, this oscillating signal is filtered by an electronic circuit and measured.

Vibration of the Earth’s rotation rate is currently measured mainly by using a large ring-laser gyroscope or fiber optic gyroscopes based on the Sagnac effect. However, these gyroscopes are large, costly, and subject to errors induced by asymmetric perturbation and common-mode noises. Here, we theoretically propose to implement a table-top gyroscope with unprecedented high sensitivity up to $26frad/s/Hz$ by exploring the nonreciprocity of photon drag in a rotating medium. The proposed gyroscope is small in volume and, thus, cost-effective. The two circularly polarized components of the laser beam co-propagate in the system, therefore eliminating the effects of the asymmetric perturbation and common-mode noise. This work opens up the door for a portable gyroscope measuring the fluctuation of Earth’s motion.

Moreover, in the measurement, the photon dragging effect is proportional to the group refractive index of the rotating part. The group refractive index of the glass rod is enhanced by slow-light to an order of 10⁶, far larger than other components of the system. Thus, we only need to consider the photon drag effect of the glass rod and can neglect the influence of other components. There are a few technical challenges in the implementation of our gyroscope. Inserting a solid slow-light medium in a high-quality cavity can be difficult. Nevertheless, slow-light in an Er³⁺-doped glass rod and a ruby has been experimentally demonstrated.^43,47 Slow-light with atomic ensembles in a high-quality cavity has also been realized.⁵⁵ Second, the proposed gyroscope needs a stable laser, the cavity, and the ambient temperature. Despite these challenges, the concept of exploring the photon drag in a solid slow-light medium embedded in a stable, high-quality cavity for a gyroscope is experimentally feasible.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 92365107 and 12305020), the National Key R&D Program of China (Grant No. 2019YFA0308700), the Program for Innovative Talents and Teams in Jiangsu (Grant No. JSSCTD202138), the China Postdoctoral Science Foundation (Grant No. 2023M731613), and the Jiangsu Funding Program for Excellent Postdoctoral Talent (Grant No. 2023ZB708). We thank the High Performance Computing Center of Nanjing University for allowing the numerical calculations on its blade cluster system.

The authors have no conflicts to disclose.

Min She: Conceptualization (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Jiangshan Tang: Conceptualization (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). Keyu Xia: Conceptualization (lead); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Based on the CPO in a two-level system,^47,48 we can derive the density matrix equation for the Er³⁺-doped glass rod. Er³⁺ can be excited by a laser beam with wavelength λ ≈ 1536 nm from the ground state ⁴I_15/2 to the substable state ⁴I_13/2 as a two-level system.^45,46 Due to the long lifetime of metastable level ⁴I_13/2 of Er³⁺, the system exhibits a capacity for rapid nonlinear response within a narrow spectrum. When the control and probe fields with a small frequency difference are input simultaneously, the medium exhibits the phenomenon of saturable absorption. In this situation, the number of ground state particles undergoes periodic oscillations, resulting in spectral hole burning in the absorption spectrum. At the burning point, the refractive index of light rapidly changes. According to the Kramers–Kronig relationship, the group velocity also undergoes drastic changes.