The Sagnac effect [Sagnac 1913b; Sagnac (1913a)] was intended by Sagnac itself to disprove the validity of Einstein Special Relativity (SR) [Pascoli (2017)] and, in particular, to prove the existence of the luminiferous aether. More than 100 years later, Sagnac gyroscopes have reached sensitivity levels high enough to test some General Relativity (GR) effect (Di Virgilio et al., 2020; Di Virgilio et al., 2021; Di Virgilio et al., 2022), like De Sitter and Lense–Thirring ones, and to set upper bounds to the validity of alternative theories of gravity [Capozziello et al. (2021)].

A Sagnac Ring Laser Gyroscope (RLG) is a ring optical cavity that includes an active laser medium volume. This is intrinsically a quantum mechanical system where two laser emissions, travelling in opposite directions, are emitted with different frequencies as they experience two optical cavities of effective different length due to the rotation of the ring. The RLG, then, is a quantum device where the measured quantity is the difference in frequency of the two counter–propagating optical waves. The latter is the so called Sagnac frequency, proportional to the rotation rate of the RLG. If the rotation is described in the GR frame it is possible to evidence the role of GR in the measured frequency. The Sagnac effect can be easily understood in term of wave-optic while a detailed description of the system would require a quantum optical approach that takes into account the presence of the gain medium and all the possible mechanisms, like back–scattering, that may couple the two beams. A complete quantum mechanical model may also enlighten the use of RLGs for investigating the interplay between rotation, and in general non–inertial reference frames, and quantum optical effects like entanglement and superposition.

The role of RLGs in Fundamental Physics Research has been established since the early days of these devices (Scully et al., 1981; Stedman 1997), while only recently the interplay between quantum effects in the optical domain and rotation has gained some interest (Restuccia et al., 2019; Toroš et al., 2020; Toroš et al., 2022; Kish and Ralph 2022; Cromb et al., 2023).

Both types of effect, GR and quantum, are tiny and require high sensitivity device.

We found, recently, that the sensitivity of GINGERINO [Di Virgilio et al. (2023)], an active RLG installed at the INFN Gran Sasso National Laboratory, shows an experimental upper limiting noise close to 2 × 10⁻¹⁵ rad/sec for ∼ 2 × 1 0 5 s of integration time. This value is surprisingly almost one order of magnitude below the value calculated following the quantum model of RLG, developed under the so called independent beams concept (Dorschner et al., 1980; Cresser et al., 1982b,Cresser et al., 1982a; Cresser 1982).

GINGERINO is a prototype RLG in view of the on-going construction of the GINGER (Gyroscopes IN General Relativity) experiment: an array of independently oriented RLGs. Based on the sensitivity observed for GINGERINO, GINGER should reach a sensitivity of 1 part in 10¹¹ of the Earth rotation rate, two orders of magnitude better than the one required to measure GR effects.

In this paper we recap the so far published result in the use of RLGs in GR and beyond tests. Before presenting the last experimental results in term of GINGERINO sensitivity we briefly discuss the physical mechanisms that may give rise to theoretically unexplored correlation causing the above mentioned inconsistency. Looking at GINGERINO data, we also discuss the impact of data analysis on noise estimation.

Measuring the Earth rotation with high accuracy would pave the way to testing some gravitational effects predicted by GR. Prerequisite to this goal is having at disposal a rotation sensor that reaches a sensitivity better than 10^–9 times the Earth rotation rate. As mentioned, a RLG can play this role and, actually, GINGERINO has reached this highly demanding sensitivity.

Despite its extraordinary predictive power, GR suffers from some difficulties in describing Nature at the ultraviolet (UV) and far-infrared (IR) scales. Concerning the UV (high-energy) regime, where the quantum nature of matter cannot be ignored, there is currently no known coherent and self-consistent reformulation of GR as a standard Quantum Field Theory (QFT) [Goroff and Sagnotti (1986)]. On the other hand, the formation and dynamics of cosmic structures and the evolution of the observable Universe require the addition of dark matter and dark energy, whose fundamental nature is currently unknown (Peebles and Ratra 2003; Frieman et al., 2008).

In order to overcome GR shortcomings, several alternative theories have been proposed (Stelle 1977; Capozziello and De Laurentis 2011; Clifton et al., 2012), but their ever-increasing number makes it necessary to develop very precise experiments aimed at constraining their free parameters and discarding non-viable models. This is where a RLG array, as planned for the GINGER experiment [Altucci et al. (2023)], fits in. In fact, some features of gravitational theories, including violation of Lorentz symmetry, may be detected as non-reciprocal effects experienced by counter-propagating light beams inside a ring optical cavity. In the following, we will illustrate how this is possible from a theoretical point of view.

To this end, we adopt the geometrical optics approximation on curved spacetimes (Misner et al., 1973; Santana et al., 2017), valid in both GR and generic metric-affine theories [Capozziello and De Laurentis (2011)] (i.e. theories in which gravitation is described by a Lorentzian metric and/or by a linear connection defined on a spacetime manifold). In particular, we will assume that a Sagnac interferometer consists of an appropriate set of mirrors that force two light rays to circulate on two null geodesics whose spatial projection is the same closed path S travelled in opposite directions, as seen from an observer at rest on S . We also assume, for simplicity, that the spacetime is endowed with a stationary symmetric Lorentzian metric that, in the coordinates adopted, can be written in the form ds ² = g _μν dx ^μ dx ^ν , where the components g _μν are independent of the time coordinate. The difference between the roundtrip (proper) time of the counter-propagating beams inside the optical cavity, as measured from the observer mentioned before, is given by Kajari et al. (2009): Δ τ = − 2 c g 00 q r ∫ S g 0 i g 00 d s i (1) where q ⁱ are the spatial coordinates of the point (on S ) in which the observer is located.

Restricting to the case of metric theories, adopting a proper reference system, i.e. one in which the observer is at rest in the origin of coordinates, metric components assume the following form: g 00 = 1 − 2 c 2 a i x i + O | x | 2 (2) g 0 j = 1 c ϵ j k l x k Ω l + O | x | 2 (3) g i j = η i j + O | x | 2 (4) where x ^(μ) are the coordinates in the proper reference system and η _(μν) the components of the Minkowski tensor. The worldline of the observer is described by the set of equations x ⁽ⁱ⁾ = 0. In Eq. (2), a ⁽ⁱ⁾ are the spatial components of the 4-acceleration of the observer, while in Eq. (3) Ω⁽ⁱ⁾ are the spatial components of the angular velocity 4-vector of the orthonormal tetrad carried by the observer relative to Fermi-Walker transported tetrads, i.e. relative to ideal test gyroscopes carried by the observer along its world line. Terms quadratic in the distance from the world line are proportional to the components of the Riemann tensor. Physical effects we are interested in are encoded in the spatial part of the angular velocity 4-vector, and, as we will see in a moment, they can be derived when confronting the local angular rotation rate measured by the RLG and those measured relative to distant inertial observes. We remark on the fact that, given the construction of the proper reference frame [as reported for example in Misner et al. (1973)], Eqs 2, 3 are valid for a general metric theory of gravity provided that the details of the theory (e.g. parameters of the PPN formalism) are encoded in the relations between metric components and the matter sources via Ω ⃗ , for example.

For a sufficiently small planar Sagnac interferometer, Eq. 1 can be rewritten in the following form: Δ τ = 4 c 2 Ω ⃗ ⋅ A ⃗ (5) where A ⃗ is the area vector associated to the cavity.

The quantity given by Eq. 5 is the Sagnac time delay. However, the quantity measured by an actual RLG is the Sagnac frequency, that is the frequency difference between the counter-propagating beams, given by: ω s = 8 π Ω ⃗ ⋅ A ⃗ λ P (6) where P is the perimeter of the cavity and λ is the (mean) wavelength of the beams. Eq. 6 can be derived from Eq. 5 taking into account resonance condition for the cavity.

Sagnac frequency is proportional to Ω ⃗ , the angular velocity of the rest frame of the observer (so also of the interferometer) relative to local inertial frames. This vector provides information on both the motion of the observer in spacetime and the structure of the spacetime itself.

The components of Ω ⃗ can be read off from time-space components of the metric once they have explicitly written it in appropriate coordinates, as Eq. 3 suggests. Adopting a linear perturbation approach, it can be shown that for a Sagnac interferometer at rest on Earth’s surface Ω ⃗ is given by: Ω ⃗ = Ω ⃗ ⊕ + Ω ⃗ I + Ω ⃗ loc + Ω ⃗ T h + Ω ⃗ d S + Ω ⃗ L T (7)

Here Ω ⃗ ⊕ is the Earth angular velocity (with contributions from tides and polar motion) and Ω ⃗ loc provides information on local deformations. These terms are derived from geophysics and geodesy. They are usually provided by the International Earth Rotation System (IERS) through VLBI (Very Long Baseline Interferometry) technique, so they represent rotation rates relative to distant stars (asymptotic inertial frame).

The term Ω ⃗ I is the contribution coming from spurious rotation of the device (relative to the laboratory frame) due to external perturbation.

The last three terms in Eq. 7 derive from the relativistic nature of gravitational theories and they all represent the rotation rates of ideal test gyroscopes carried by the observer (at rest in the laboratory frame) relative to the frame of distant stars. Ω ⃗ T h is the Thomas precession angular velocity and it affects gyroscopes in non-linear accelerated motion. It can be derived from SR alone and it essentially encodes the non-group nature of Lorentz boosts in different directions. Ω ⃗ d S is the De Sitter precession angular velocity and it is due to the coupling between the velocity of a gyroscope and the static part of the gravitational field due to the presence of a central mass. Ω ⃗ L T is the Lense-Thirring precession term and is a manifestation of the dragging of (local) inertial frames due to spinning masses.

De Sitter and Lense-Thirring effects are known as gravitoelectric and gravitomagnetic terms, respectively, as they can be derived adopting a linear perturbation approach to Einstein field equations and solving equations of motion for a test gyroscope in analogy with those for a spinning electric charge in electromagnetic field (Thorne 1988; Ciufolini and Wheeler 1995; Mashhoon 2007). In more detail, restricting to the GR case and provided a suitable choice of coordinates and reasonable properties of the matter sources, metric components can be related to the gravitoelectric Φ and gravitomagnetic potentials A ⃗ . The first is the Newtonian potential in the weak field limit, while the latter has components proportional to the time-space elements of the metric g _0i . Given Φ and A ⃗ , in the same way as in electromagnetism, the so called gravitoelectromagnetic fields E ⃗ and B ⃗ can be defined. In terms of E ⃗ and B ⃗ , Einstein equations are rewritten in a form very similar to Maxwell equation, so precession effects for a test gyroscope in the gravitational field are derived in exact analogy with precession terms for a spinning charge moving in the electromagnetic field. In particular, Ω ⃗ L T is the analogous of the Larmor precession angular velocity, so it is proportional to B ⃗ , while Ω ⃗ D S is due to the (gravito) magnetic field induced in the rest frame of the spinning charge moving in a static external (gravito) electric field, so it is proportional to v ⃗ × E ⃗ Thorne (1988).

Sagnac time delay has been calculated in the context of various alternative theories of gravity Capozziello et al. (2021). For instance, retaining only the lowest order corrections to GR, a generic metric theory of gravity in PPN formalism predicts the following Sagnac time delay Bosi et al. (2011): Δ τ P P N = 4 A c 2 Ω ⊕ cos θ + α − 1 + γ G N M c 2 R ⊕ sin ⁡ θ ⁡ sin ⁡ α − C 4 G N I ⊕ c 2 R ⊕ 3 2 ⁡ cos ⁡ θ ⁡ cos ⁡ α + sin ⁡ θ ⁡ sin ⁡ α (8) where γ and C = − 1 2 ( 4 + 4 γ + α 1 ) are parameters to be constrained. In Eq. 8, θ is the colatitude and α denotes the angle between the radial direction and the normal vector of the cavity u ̂ n , defined as u ̂ n = cos ⁡ α u ̂ r + sin ⁡ α u ̂ θ . In deriving Eq. 8 we have assumed the Earth as an isolated spherical spinning body of radius R _⊕ and mass M _⊕ and we have discarded the contribution coming from Ω ⃗ I and Ω ⃗ loc . The first term is the kinematic term given by Earth rotation, the second is the total contribution of Thomas and De Sitter precessions, the latter is the Lense-Thirring one.

GR is recovered for γ = 1, C = −4. In this case, it can be shown that each relativistic contribution to 8 is roughly 1 part in 10^(9÷10) of the Earth’s rotation term (Ω_⊕ ≈ 7.29 × 10⁻⁵  rad/s).

Gravitomagnetic effects generated by isolated spinning Earth have also been derived in the context of the low energy limit of a specific Horava–Lifshitz gravity model Radicella et al. (2014), f(R) scalar-tensor gravity (Capozziello et al., 2015; Capozziello et al., 2021) and Standard Model Extension Moseley et al. (2019). Moreover, some authors have derived the gravitoelectric contribution generated by the Sun in the case of light scalars coupled conformally and disformally to matter Benisty et al. (2023). It is interesting to note that Horava-Lifshitz gravity and the Standard Model Extension are theories with broken local Lorentz invariance. For both theories, non-reciprocal effects entering the Sagnac frequency arise from Lorentz violating terms in the Lagrangian. So, a measure of Ω ⃗ could provide insight into the local symmetries of the spacetime.

It has been demonstrated that the expected performances of GINGER Altucci et al. (2023) will allow to test all theories listed above at a sensitivity level comparable, if not better, to present satellite tests. Moreover, the measurements are not averaged and do not require a map of Earth’s gravity, since the RLG is located at a fixed latitude.

According to studies conducted on GINGERINO, GINGER will be able to measure the De Sitter and Lense-Thirring effects with a precision of 1 part in 10⁴ and in 10³ of their general relativistic values, respectively Capozziello et al. (2021). Details on how the above precision levels translate into constraints on the free parameters of the theories listed above can be found in Moseley et al., 2019; Capozziello et al., 2021; Benisty et al., 2023.

The sensitivity of any measurement based on the detection of photons is limited by an intrinsic limit: the shot–noise level (SNL). This limit comes from the quantum nature of light and is often referred as standard quantum limit (SQL). The inherent fluctuations in the number of photons that stochastically are converted in photo-electrons set an intrisic limit to the precision of the measurement. This limit depends on the effective description of the measurement in a quantum model. While rarely SQL becomes an effective limit to the precision one can achieve in an optical apparatus, it is well known that it limits interferometric measurements Caves (1981)) and methods exist to overcome it Aasi et al. (2013).

Generally the limit to the sensitivity, is assumed to be the one obtained for a coherent beam optical probe, i.e. with all the involved light beams in a coherent state of the quantum field. If there are more than one beam involved in the measurement, they are assumed mutually independent so that the noise is the squared sum of the SNL relative to each of the beam.

As it is now clear, a RLG relies on the interference of two beams of slightly different frequency. In the hypothesis of two independent beams, the shot noise has been calculated by Cresser et al. (1982b). There, the two beams phase noise Langevin equations are uncoupled and driven by two commuting and delta correlated Langevin forces coupled independently one to each mode. This amounts to have two independent phase (Wiener) diffusion processed inside the laser medium (Stedman, 1997, Section 4).

At the same is widely accepted that the classical amplitude of the two beams are governed by coupled equations (Wilkinson, 1987, Section 4). This happens because in the RLG dynamics there are more than one physical mechanisms that, already at the classical level, may couple the two counter–propagating modes. Two of them, passive back–scattering from cavity mirrors and scattering mediated by the gain medium, have been recently studied by A. Mecozzi from a quantum perspective Mecozzi (2023) in the frame of a full quantum model where both cavity field and laser emitters, in terms of atomic transitions, are represented by quantum operators while the optical cavity is modeled in the standard input–output formalism that gives reason of the cavity linewidth. This approach leads to a modification of the noise behavior of the Sagnac signal and, in particular, modifies the Allan variance for the frequency difference. The latter deviates from the one calculated by the uncoupled modes model in function of the coupling parameter between the two modes. This model is not easily applicable to our system where the laser medium is an isotopic mixture of He-Ne whose effect is to favorite decoupling of the two counter-propagating emission. In this case to calculate the weight of the coupling requires a precise model of the gain profile and of the laser threshold values for the two isotopes.

GINGERINO is a RLG with a square optical cavity (see Figure 1). Its sensitivity limit, in the frame of the independent beam model, is expected to be 18 prad/(s Hz^1/2) [(Schreiber and Wells, 2013, Table 2)] (GINGERINO parameters p = 14.4 m, A = 12.96 m², Q = 1.8 × 10¹² (corresponding to a cavity decay time of ≈600 μs), a power ouput p = 40 nW, ω _l = 2π × 4.74 ⋅ 10¹⁵ Hz). It operates rigidly attached to the ground through a granite basement directly installed on the bed rock of one of the LNGS tunnel.

Four vacuum chambers are located at the corner of a square. Each chamber hosts a super-mirrors (R = 99.999%) so to define a square high finesse optical cavity. The corner boxes are connected by pipes, vacuum tight. The whole tank is filled with an isotopic mixture of Helium Neon gas. At the midpoint middle of one of the side is the pyrex capillary provided with external electrodes, used to excite the laser medium by radio frequency. The laser emits around 632.8 nm. Mirrors are equipped with piezoelectric actuators to control the geometry, but in general the RLG can run uncontrolled. Outside one of the mirror the transmitted light beams interferes at a cube beam-splitter, the interference contains the beat–note. At another corner two photodiodes monitor single beam intensities and are used, then, for analysing the back-scattering signal that appears as a modulation of the single beam intensity. This modulation is at the Sagnac frequency, implicitly confirming that this type of coupling impacts on the Sagnac signal itself. The Sagnac frequency is, then, reconstructed using the beat note signal and the mono–beams, in order to correct the typical systematic of the laser: backscatter and null–shift.

The interference signal is acquired in a differential scheme. Two photodiodes measure the two interference signal outing the two ports of the mixing beam–splitter. This configuration allows to have two distinct Sagnac measurements that, if summed, give a SNR 2 higher than a single one, if used in a differential configuration permit detailed noise floor analysis.

Since 2020 (Di Virgilio et al., 2020; Di Virgilio et al., 2021) we had experimental evidence that the ultimate sensitivity of the GINGERINO prototype is not fully consistent with the shot–noise calculated by the above mentioned independent beams model.

The experimental noise limit of GINGERINO, a square RLG operating underground at the INFN National Gran Sasso Laboratory has been measured by a dual signal scheme. The measured noise is below what has been thought so far as the SQL for a an active RLG. The theoretical expectation come from the stringent assumption of two independently generated and counter–propagating laser beams inside the ring cavity. Renouncing to this hypothesis implies including in the fully quantum system equations all possible physical mechanisms that may couple the two beams. The noise behavior of GINGERINO is a conclusive experimental proof that its noise floor is not consistent with the independent beam models. The slope of its frequency dependent Allan variance, in some frequency ranges, is not permitted by delta correlated noise sources. These two results require a complete quantum model where all the possible couplings among the two optical modes are taken into account for understanding the nature of the evident correlation. Eventually, the sensitivity reached by GINGERINO is a jumping–off point for the use of RLG for fundamental physics research including the study of optical quantum effect in non-inertial frames.