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S c i e n c e A d v a n c e s | R e s e ar c h A r t i c l e

PHYSICS
Experimental observation of Earth’s rotation with quantum entanglement
Raffaele Silvestri1,2,3, Haocun Yu1,3*, Teodor Strömberg1,2,3, Christopher Hilweg1,3, Robert W. Peterson1,3, Philip Walther1,3,4*
Precision interferometry with quantum states has emerged as an essential tool for experimentally answering fundamental questions in physics. Optical quantum interferometers are of particular interest because of mature methods for generating and manipulating quantum states of light. Their increased sensitivity promises to enable tests of quantum phenomena, such as entanglement, in regimes where tiny gravitational effects come into play. However, this requires long and decoherence-free processing of quantum entanglement, which, for large interferometricareas,remainsunexploredterritory.Here,wepresentatable-t opexperimentusingmaximallypath-entangled quantum states of light in a large-scale interferometer sensitive enough to measure the rotation rate of Earth. The achieved sensitivity of 5 μrad s−1 constitutes the highest rotation resolution ever reached with optical quantum interferometers. Further improvements to our methodology will enable measurements of general-r elativistic effects on entangled photons, allowing the exploration of the interplay between quantum mechanics and general relativity, along with tests for fundamental physics.

Copyright © 2024 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution License 4.0 (CC BY).

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INTRODUCTION

For more than a century, interferometers have been important instru-

ments to experimentally test fundamental physical questions. They

disproved the luminiferous ether, helped establish special relativity

(1, 2), and enabled the measurement of tiny ripples in space-time it-

self known as gravitational waves (3). With recent advances in tech-

nology, interferometers can nowadays also operate using various

different quantum systems including electrons (4, 5), neutrons (6),

atoms (7–11), superfluids (12, 13), and Bose-Einstein condensates (14–

16). Quantum interferometers are of interest for two main reasons:

First, the exploitation of quantum entanglement allows for super-

resolving phase measurements that go beyond the standard quantum

limit (17, 18). Second, the enhanced sensitivity of quantum inter-

ferometers opens up opportunities for precision measurements that

can explore new frontiers in physics. These include setting constraints

on dark-energy models (19), testing quantum phenomena in non-

inertial reference frames (20–22), and investigating the interplay be-

tween quantum mechanics and general relativity (23–28).

Optical systems are particularly well suited for realizing quantum

interferometers, owing to mature techniques available for generating

a variety of quantum states, ranging from squeezed vacuum (29–32)

to maximally path-e ntangled photons (33–35). The N00N states be-

long

to

the

latter

category,

represented

by

√1 N

(∣

N

⟩a∣

0⟩b

+

∣

0⟩a∣

N

⟩b) ,

wherein N photons exist in a superposition of N photons in mode a

with zero particles in mode b, and vice versa (17). These states be-

have similar to those of a single photon with N times the energy,

enabling precision in phase measurements at the Heisenberg limit

1University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), Vienna, Austria. 2University of Vienna, Faculty of Physics and Vienna Doctoral School in Physics, Boltzmanngasse 5, A-1090 Vienna, Austria. 3University of Vienna, Faculty of Physics and Research Network Quantum Aspects of Space Time (TURIS), Boltzmanngasse 5, A-1 090 Vienna, Austria. 4Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, A-1 090 Vienna, Austria. *Corresponding author. Email: haocun.yu@univie.ac.at (H.Y.); philip.walther@ univie.ac.at (P.W.)

√ that scales as 1/N and thus goes beyond the1 ∕ N scaling of the standard quantum limit (18). Another advantage of photonic systems is that fiber-optical interferometers offer a clear pathway for expanding the interferometric area while maintaining a low level of quantum decoherence.
In this work, we report the design and operation of a large-scale quantum-optical fiber interferometer exploiting N00N states that reaches a sensitivity in the range of μrad s−1, sensitive enough to measure the rotation of Earth. We inject two-p hoton N00N states into a 715-m 2 Sagnac interferometer, using quantum interference to demonstrate super- resolution while extracting Earth’s rotation rate. This goes beyond previous laboratory demonstrations of measurements probing Sagnac interferometers with quantum states of light, which involved fiber interferometers with at most hundred-m eter-length fibers (20–22, 32, 36) and were only used to measure synthetic and controllable signals. We are able to confirm an acquired Sagnac phase from Earth’s rotation with an enhancement factor of two because of the two-p hoton entangled state. To the best of our knowledge, this is the largest and most sensitive quantum-optical Sagnac interferometer in the world, surpassing previous state-of-the-art rotation sensors using two-p article entanglement by three orders of magnitude (see also Results) (22).
We chose the detection of Earth’s rotation as a benchmark for our large-scale fiber interferometer, as its minute rate, fixed direction, and the absence of ways to manipulate its behavior make it particularly challenging to observe. On the other hand, the ubiquitous presence of acoustic-and seismic vibrations and thermal fluctuations transduce directly into phase noise in optical fiber (37) and drive the motion of the large apparatus. To solve these problems, we build our rotatable fiber interferometer with an optical switch to turn Earth’s rotation signal on and off, allowing us to fully characterize the angle- dependent Sagnac phase (Fig. 1).
According to the Sagnac effect (38, 39), the flying times of photons traveling in opposite directions around a rotating encircled path are different, inducing a measurable phase difference

ϕs

=

8πΩE AcosΘ λc

(1)

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Here, ΩE is the rotation angular frequency of Earth. A is the interferometer’s effective area of 715 m2 (for the calibration of the ap-
paratus, see details in Materials and Methods). Θ is the angle between the area vector of the fiber loop and the angular velocity vector of
Earth, and λ is the photon wavelength of 1546 nm. In a Sagnac interferometer with a rotationally induced phase shift
ϕs, a two-mode coherent state ∣α⟩a∣α⟩b evolves to ∣α⟩a∣eiϕsα⟩b, where a and b are the two propagation directions in the interferometer and
α is the complex amplitude of the state. After interfering on the beam splitter, the normalized intensity in the output arm α of the interferometer, previously used as the input, is Pa = [1 + cos (ϕs)]/2. We contrast this with multiphoton interference, whic√h occurs when we inject the entangled state (∣ N⟩a∣ 0⟩b + ∣ 0⟩a∣ N⟩b) ∕ 2 into the in-
terferometer, where the N photons are in a superposition of being in either of the two modes. After propagating through th√e interferometer, the state evolves to (∣ N⟩a∣ 0⟩b + eiNϕs ∣ 0⟩a∣ N⟩b) ∕ 2 . At the in-
terferometer output, the probability of finding exactly N photons in
the output arm oscillates at N times the frequency: PN,a ∝ [1 + cos (Nϕs)] (34). For a single-photon input state, corresponding to N = 1,

the detection probability coincides with the normalized intensity of coherent light. However, for N ≥ 2, an enhancement of the phase shift by a factor of N is observed.
RESULTS Experimental implementation The two-photon path-e ntangled state is realized by exploiting the polarization correlation of photon pairs emitted by a type-I I spontaneous parametric down-conversion (SPDC) source (40). The photon pairs, centered at 1546 nm, are created in the product state ∣1⟩H∣1⟩V, where H and V denote horizontal and vertical polarization, respectively. A half-wave plate (HWP) oriented at 22.5° (with respect to the horizontal axis) transforms this product state into the polarizat√ion- entangled two-photon N00N state (∣ 2⟩H ∣ 0⟩V − ∣ 0⟩H ∣ 2⟩V ) ∕ 2 , where the cross terms cancel out because of the indistinguishability of the photons. Subsequently, this state is converted into a path- entangled state at a polarizing beam splitter (PBS), which separates the H and V photons into clockwise and counterclockwise

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Fig. 1. Earth’s rotation measured using entangled photons. (A) A rotatable 715-m 2 Sagnac fiber interferometer is built in a laboratory located in Vienna, Austria. (B) Simplified schematic of the experimental setup. Orthogonally polarized photon pairs are converted to path-entangled N00N states in the Sagnac interferometer via a half-wave plate (HWP) followed by a polarizing beam splitter (PBS). The frame angle Θ is defined as the angle between Earth’s angular velocity vector <20><>⃗ E and the fiber loop area vector <20>A⃗<41>  . The signal is extracted by observing the phase shift of quantum interference fringes induced by Earth’s rotation, using a set of quarter-wave plates (QWP) and a HWP, in combination with single-p hoton coincidence counting (&). (C) An optical switch (OS) is used to toggle Earth’s rotation signal on and off independent of the frame angle Θ. This is achieved by controlling the propagation direction (clockwise or counterclockwise) of photons in one half of the fiber spool.

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propagating modes. After passing through the 2-km fiber loop, the clockwise-traveling photons pick up a Sagnac phase shift ϕs induced by Earth’s rotation relative to the counterclockwise-traveling ones.
The same PBS then converts the state back into the polarization-
entangled state

1 √

(∣ 2⟩H ∣ 0⟩V − ei2ϕs ∣ 0⟩H ∣ 2⟩V )

2

(2)

Interference takes place again at the 22.5° HWP, leading to the output state

1 √

sinϕs(∣ 2⟩H ∣ 0⟩V + ∣ 0⟩H ∣ 2⟩V ) − icosϕs∣ 1⟩H ∣ 1⟩V

2

A set of wave plates is used to control the detection probabilities by introducing a bias phase ϕ0. This artificially adds a relative phase between the H and V polarization components, allowing us to scan
the full interference fringe and also project the measurements onto
any polarization basis, turning the state in Eq. 2 into

1 √

[2H 0V − ei(2ϕ0+2ϕs)0H 2V ]

2

(3)

To perform a projective measurement onto the ∣1⟩H∣1⟩V component of the state, we analyze the twofold coincidence probability PHV by collecting photons in both output ports of the PBS before the
detectors

PHV

=

1 [1 2

+

cos(2ϕ0

+

2ϕs)]

(4)

This gives an enhancement factor of two in the observed Sagnac phase, as well as in the bias phase.
The central component of the Sagnac interferometer consists of 2-k m fibers wound around a 1.4-m square aluminum frame (yellow; Fig. 1B). Because the detectable Sagnac phase shift induced by Earth’s rotation depends on the direction of the area vector A<><41>⃗ , the frame is designed to be rotatable in both pitch and yaw dimensions. This allows for a series of measurements to be taken at different values of Θ.
To more distinctly manifest the rotation signal, an optical switch is incorporated to toggle the effective area of the interferometer. The optical fiber is divided into two equal 1-k m fiber segments (orange and blue), which are connected by the four-port optical switch. As shown in Fig. 1C, flipping the optical switch reverses the direction of light propagation in one of the fiber loops. When the optical switch is in the “OFF” state, the Sagnac phase shift is canceled out because of the opposite directions of light propagation in the two fiber segments, resulting in two area vectors with opposite signs and a zero effective area. By comparing the measurements in the optical switch “ON” and “OFF” states, it can be confirmed that the observed phase shifts are exclusively caused by Earth’s rotation.
From Eq. 1, the Sagnac phase is maximized when the interferometer is oriented in a way that Earth’s rotation vector perpendicularly intersects the plane of the interferometer area. This orientation is determined from a calibration procedure with classical light in the interferometer (see details in Materials and Methods). Figure 2 shows the data for the Sagnac phase shifts induced by Earth’s rotation

at Θ = 2.5°. The data points are acquired for one-and two-p hoton N00N states propagating through the interferometer. For the two- photon entangled states, 11 different data points were taken while continuously switching between the two operating modes: with and without Earth’s rotation signal (switch on and off, respectively). When alternating operation between the two modes at a frequency of 0.1 Hz, Earth’s rotation signal is resolved by comparing the interference fringes of the two modes. To further confirm that the phase shift is solely due to Earth’s rotation, additional data are acquired at various frame angles, thereby enabling curve fitting and precise phase difference extraction, as depicted in Fig. 3.

Earth’s rotation-induced phase extraction
Figure 2 shows quantum interference fringes of the Sagnac interferometer at Θ = 2.5°. In the central figure, the red and orange marks represent normalized two-photon coincidence counts between the H and V photons measured with the optical switch on and off, respectively. These data were generated from 11 sets of 30-m in contiguous integration periods. Each dataset was taken with a specific value of ϕ0, ranging from −π/8 to 2π + π/8 to cover a full interference fringe. The blue and green marks are heralded single-p hoton measurements, with 11 (7 shown) 15-m in contiguous integration periods, ranging from −π/4 to 2π + π/4, serving as the reference measurement. The uncertainties for each data point are represented by ±1 SDs, which were calculated from Monte Carlo simulations using 105 samples of Poisson-distributed photon coincidence counts (see details in Materials and Methods for a comprehensive error analysis).
For the two-photon measurements, the 11 data points obtained in each switch mode are fit to an interference fringe model. Earth’s rotation signal is extracted by calculating the phase shift between the two curves (red and orange). On the basis of Eq. 4, the data are fit with

Nswitch oﬀ(ϕ) = N0 1 + cos(2ϕ)

(5)

Nswitch on(ϕ) = N0[1 + cos(2ϕ + ϕ(s2))]

(6)

where N0 is the amplitude of the photon interference, 𝒱 is the interference visibility, and ϕ(s2) is Earth’s rotation-induced phase shift to
be measured. The extracted phase difference between two interference fringes is ϕ(s2) = 5.5(5) mrad . A similar fitting and phase ex-
traction procedure is used for single-photon reference measurements
(blue and green), resulting in ϕs = 2.8(2) mrad. In the two-p hoton measurement, the phase shift is enhanced by a factor of two because
of the presence of two-photon path entanglement.
Sagnac phase shift measurements at five additional frame angles
Θ are presented in Fig. 3. This plot explicitly shows two things: First, the Sagnac phase shift induced by Earth’s rotation is proportional to
cos(Θ) as expected from Eq. 1. Second, the two-p hoton measurements consistently exhibit a doubled phase compared to the single-
photon measurements for all the different frame angles. For each
value of Θ, the Sagnac phase shift is extracted by comparing the interference fringes with the optical switch on and off, following a pro-
cedure identical to that used for Θ = 2.5°. The data for the five additional angles were acquired with a shorter integration time
compared to Fig. 2, resulting in correspondingly larger statistical

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Fig. 2. Quantum interference measurement revealing the Sagnac phase shift induced by Earth’s rotation. (Middle) Normalized quantum interference fringes of single-p hoton and two-photon entangled state measurements. The red and orange (blue and green) marks show the normalized two-p hoton (one-p hoton) coincidence counts with the Earth rotation signal switched on and off, respectively. The corresponding curves are least-s quares fits to the data using a model of the experiment (see the Supplementary Materials). The doubled fringe frequency of the two-p hoton curves reveals the super-resolution due to quantum entanglement. (Left and right) Sagnac phase shifts induced by Earth’s rotation at Θ = 2.5°, zooming in around ϕ = π, π/2, and 0 for single-photon measurement (left) and around ϕ = π, 3π/4, and π/2 for two-p hoton measurement (right). The widths of the vertical lines indicate the size of uncertainties due to uncorrelated photon counting noise. Because the same phase bias ϕ0 has been applied to both one-photon and two-p hoton measurements, the doubled Sagnac phase shift does not manifest in the plots. 1, one-photon state; 2, two-p hoton N00N state; M, maximum; m, minimum; q, quadrature.

uncertainties. The red and blue curves are the least-squares fits using Eq. 1. From these fits, the maximum Sagnac phase shift induced by Earth’s rotation in the two-p hoton N00N state is 5.5(4) mrad, which corresponds to an Earth’s rotation rate of ΩE = 7.1(5) × 10−5 rad s−1, compared with 2.8(1) mrad or ΩE = 7.2(3) × 10−5 rad s−1 in the one- photon measurement. Both agree with the internationally accepted value 7.3 × 10−5 rad s−1 (41). The experimentally determined enhancement factor due to two-photon quantum entanglement is 1.96(15).
The achieved phase resolution in our experiment is primarily hindered by scale factor instability, with the most detrimental contributions coming from mechanical vibrations of the frame due to its extensive surface area, thermal fluctuations, and acoustic noise. Scaling to larger interferometric areas will be possible by incorporating design lessons from cutting-edge fiber-optic gyroscopes (FOGs). FOGs using classical light have reached phase resolutions of less than a nanoradian, translating to rotation rates below 0.1 nrad s−1, with a stable signal over more than a month (42, 43). Combining proposals for next- generation FOGs, such as the Giant-F OG with an area of 15,000 m2 (44), with state-of-the-art single-photon sources (45), we anticipate that a phase resolution of about 20 prad s−1 could be reached with quantum states of light, which is within two orders of magnitude of the general-relativistic rotation rate correction term ΩGR = 10−9 ΩE because of the frame-dragging and the geodetic effect (see Fig. 4) (46, 47).
DISCUSSION We have demonstrated the largest and most precise quantum-optical Sagnac interferometer to date, exhibiting sufficient sensitivity to

measure Earth’s rotation rate. Our work advances the state of the art of entanglement-b ased rotation sensors by three orders of magnitude and introduces a signal-s witching technique that can be used to modulate the effective area of the interferometer. This enables a self- referenced measurement of the fixed-rate rotation signal, without the need for building an additional small-a rea interferometer (48), thereby going beyond previous work focusing on sensing an induced motion relative to the surrounding laboratory.
When comparing the use of two-photon entangled states to single-p hoton states, we observe an improvement by a factor of two in the measured phase value due to super-r esolution. Our approach using a polarization encoding is readily scalable to N00N states with higher photon numbers (34), with the main limitations being the large amount of transmission loss of the experimental setup and the generation of the multiphoton states.
Our methods pave the way for other technically challenging proposals, such as dynamically generating entanglement from the underlying space-time (48), directly probing gravitationally induced phase shifts (49, 50), rotational and gravitational decoherence in the quantum interference of photons (51), testing fundamental symmetries in quantum field theory (52), investigating local Lorentz invariance violation (53), detecting exotic low-m ass fields from high-energy astrophysical events (54), and dark matter searches such as axion- photon coupling (52). It is predicted that two photons can transition to axions in the presence of an external magnetic field. In an optical Sagnac interferometer, where two orthogonal polarizations counterpropagate, the component parallel to the magnetic field would then be retarded with respect to the other, leading to a nonreciprocal

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Fig. 3. Sagnac phase shifts induced by Earth’s rotation measured at six interferometer frame angles. Θ’s range from −67.5° to +25°, evenly spaced by 22.5°. (Top) Each data point is obtained with the same measurement sequence and extraction method as Fig. 2. At each angle Θ, the Sagnac phase shift measured using two-p hoton entangled N00N states (red triangle marks) is found to be doubled compared with single-p hoton states (blue circle marks). The blue and red curves are the least-s quares
fits to Eq. 1 of the one-photon and two-p hoton N00N state measurements, respectively. (Bottom) Representation of different angles between the area vector of the in-
terferometer (blue dashed line) and Earth’s rotation angular velocity vector (red arrow). The Sagnac phase shift induced by Earth’s rotation can be increased and decreased
as the frame rotation angle is varied.

observable phase shift or the loss of two photons. Our interferometer constitutes an excellent testbed for this phenomenon, allowing investigations both with classical light and entangled photons.
In conclusion, the successful observation of Earth’s rotation using entangled states of light, a century after the first local observation of Earth’s rotation-induced fringe shift with a Sagnac interferometer (2), constitutes a milestone toward the goal of probing the interface between quantum mechanics and general relativity. The zero-area switching technique that we have introduced, which allows the rotation signal to be referenced to an effectively nonrotating frame, is a key technical advancement over previous works (20). This is manifested in the greatly improved sensitivity over previous entanglement-based sensors, which, in turn, shows the promise of our approach for measuring general-relativistic non-inertial effects on quantum states.
MATERIALS AND METHODS Characteristics of the experimental setup A detailed experimental setup is provided in fig. S1. A periodically poled KTiOPO4 crystal produces orthogonally polarized photon pairs centered at 1545.76 nm in a type-II SPDC process. The crystal is pumped by continuous wave (CW) Ti:Sapphire laser (Coherent Mira HP) emitting at 772.88 nm (40). The photon source pump power is set to 145 mW, leading to a detected photon pair coincidence rate of approximately 400 kHz. The photons in each generated pair are combined on a PBS and are overlapped temporally using a

delay line in one of the input ports of the PBS. The total loss of the entire experimental setup is 90% (10 dB). The Sagnac loop introduces 5 dB of losses, out of which 1 dB is the optical switch insertion loss, 1 dB from the 2-km polarization-m aintaining (PM) fiber (≈0.5 dB/ km), and 3 dB from fiber connections, while the input and output of the optical setup contribute the remaining 5 dB. In detection, photons from the output paths are coupled into single-mode fibers connected to superconducting nanowire single-photon detectors, housed in a 1 K cryostat, with a detection efficiency of roughly 95% and a dark-count rate around 300 Hz. Amplified detection signals are counted using a time tagging module with a timing resolution of 156.25 ps, and two-photon coincidence events are extracted using a coincidence window of 3.75 ns. In the two-photon N00N state measurements, when both photons are propagating through the interferometer, the detected photon pair rate is around 4 kHz, consistent with the expected exponential fragility to losses of a two-p hoton N00N state 1 − η2i ≈ 99 % , where ηi = 0.1 is the total transmission efficiency of the interferometer. In the one-p hoton measurements, one photon of the pair is used as a trigger while the other propagates through the interferometer. The total heralded single-p hoton rate in the two detection ports is around 20 kHz, which is compatible with the overall losses 1 − ηtηi ≈ 95%, where ηt = 0.5 is the transmission coefficient of the trigger photon fiber path.
Interferometer calibration
In the laboratory, the axis normal to the fiber spool plane when vertically oriented with respect to the horizon is pointed north. The

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Fig. 4. Rotation rate resolutions and enclosed areas of existing and predicted quantum optical Sagnac interferometers. The plot is divided into three sensitivity re-
gimes: sensitivity below Earth rotation ΩE (white zone), sensitivity above ΩE but below general relativistic effects ΩGR (blue zone), and sensitivity above ΩGR (orange zone). Diamond markers represent existing interferometric platforms, while star markers are proposed platforms but yet to be realized. Solid markers represent performed experi-
ments with quantum states of light, while empty markers represent experiments yet to be performed. Bertocchi et al. (36): Lf = 550 m and P = 0.63 m; Restuccia et al. (20): Lf = 100 m and P = 2.85 m; Fink et al. (22): Lf = 270.5 m and P = 0.49 m; this work: Lf = 2 km and P = 5.6 m; Lefèvre (42): Lf = 3 km and P = 0.63 m; Mead and Mosor (43): Lf = 8 km and P = 2.15 m; de Toldi et al. (44): Lf = 15 km and P = 12.57 m; our proposed experiment: Lf = 47.5 km and P = 6 km, where Lf is the fiber length and P is the perimeter. The photon pair generation rate is 1 GHz for the (42, 43) and 10 GHz for the (44) and our proposed experiment, with integration times on the order of a month (for more details see
the Supplementary Materials).

rotational degree of freedom of the fiber loop frame introduces the
opportunity for experimental calibration of the interferometer by
estimating its scale factor S, while assuming Earth’s rotation rate
(ΩE) as a known quantity, with ϕs = SΩE. A set of phase measurements are performed with a CW light source at telecom wavelength at six different angular positions Θk of the fiber loop frame spaced by 22.5° (see Fig. 5), allowing us to find the frame angle that maximizes the Sagnac phase (Θ = 0°). H-p olarized light is injected into the interferometer, which is converted into diagonal polarization by a
HWP before entering the Sagnac interferometer. Because of Earth’s
rotation, the H and V components acquire a relative Sagnac phase
ϕs, which is encoded in the polarization state ellipticity angle χ (55), such that ϕS = 2χ. A compact free-s pace polarimeter is used to fully characterize the polarization state after the wave plates, which com-
pensate first for the polarization rotation in the output fiber circula-
tor path (see fig. S2). As in the measurements with quantum light,
the optical switch is driven by a 0.1-Hz square wave. The recorded
time trace of χ is partitioned into two sets by demodulating it using the driving signal. For each frame angle Θk, the differential average between the two traces δχk = χkon − χkoﬀ is used to calculate the Sagnac phase ϕkS and its associated uncertainty σk. As part of a Monte Carlo simulation resampling the phase values ϕkS using their uncer-
tainties, the data are fit to the model function ϕE(Θ) = SΩE cos (Θ + Θ0), where ΩE = 7.29 × 10−5 rad s−1 is the known value of Earth’s rotation rate and S and Θ0 are free parameters. The Monte Carlo simulation estimating these parameters additionally samples the frame angles Θk from uniform probability distributions [Θk − 1, Θk + 1].
The extracted fit parameters are the scale factor S = 38.8(1) and the

angular offset Θ0 = 0.03(33)°. The CW measurements are compared with the photon measurements in Fig. 3.
Noise mitigation The interferometer frame is fixed on an air-floated optical table to dampen the transduction of ambient seismic vibrations into the frame. The fiber spools are covered with layers of insulation material (Thinsulate) to mitigate temperature-and air current-induced spatial gradients and time-v arying fluctuations of the fiber length and refractive index. This passive isolation increases the scale factor stability in time by stabilizing the enclosed interferometric area.
More crucially, the optical switching method is also a fundamental and powerful tool in our experimental implementation. By zeroing the interferometer’s effective area, we are able to obtain a reference measurement, allowing for the distinction and elimination of spurious signals arising from various technical and background noise sources, which include laser intensity fluctuations, imperfections in the input photon state, nonideal polarization rotations during light propagation out of the fiber loop, and variations in mechanical stresses applied to the frame structure across its angular orientations. Furthermore, the modulation of the signal at a specific frequency helps mitigate slow frequency drifts in the measured phase via postprocessing, thereby increasing its long-term stability over acquisition times spanning hours.
Phase estimation and uncertainty analysis The phase shifts and associated uncertainties presented in Fig. 3 are estimated using a Monte Carlo simulation accounting for photon counting noise and uncertainties in the phase offset. In each round of the

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Fig. 5. Comparison of photon measurements and CW measurements as calibration. Data of one-p hoton state (blue) and two-photon N00N state data (red) are inher-
ited from Fig. 3. Green square marks are obtained using a polarimeter with CW light at six different Θ’s ranging from −90° to +22.5°, evenly spaced by 22.5°. The green solid curve is the least-s quares fit of CW measurements with fitting function as ϕE(Θ) = SΩE cos (Θ + Θ0). This measurement allows us to find the frame angle that maximizes the Sagnac phase (Θ = 0°). The green dashed curve is plotted as 2SΩE cos (Θ + Θ0) to compare with the two-photon N00N state measurements.

simulation, the tion with mean

photon and SD

coofuNnkstsanarde√saNmksp , lreedspuescintigvealyP,owishseorne

distribu-

Ns
k

is

the

number of recorded photon counts for the offset phase ϕk0 and switch

state s ∈ {on, off}. In addition, phase-offset noise, correlated be-

tween the on and off states, is sampled from a distribution derived

from the waveplate motor repeatability and added to the offsets ϕk0 .

For each sampled dataset, a least-s quares fit is performed, using the

amplitude N0, fringe visibility 𝒱, and phase shift ϕ as free parame-

ters. Last, the values and uncertainties of these parameters are estimated using the mean and SD, respectively, taken over 105 repetitions

of the simulation.

Supplementary Materials
This PDF file includes: Supplementary Text Figs. S1 and S2 Tables S1 to S3 References
REFERENCES AND NOTES
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Acknowledgments: We thank P. Schiansky, I. Agresti, and L.A. Rozema for help with the photon source and the photon detectors. We also thank Q. Zhuang, A. Brady, Z. Yin, H. Cao, and T. Mieling for discussions. Funding: R.S. acknowledges support from Uni:Docs fellowship program, hosted by the University of Vienna. H.Y. acknowledges funding from the European Union HORIZON TMA MSCA Postdoctoral Fellowships -European Fellowships under grant agreement no. 101064373 (MAGIQUE). C.H. and R.W.P. acknowledge support from the ESQ Discovery program (Erwin Schrödinger Center for Quantum Science and Technology), hosted by the Austrian Academy of Sciences (ÖAW). P.W. acknowledges support from the research

network TURIS and funding from the European Union (ERC, GRAVITES, No 101071779). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. This research was funded in whole or in part by the Austrian Science Fund (FWF) [10.55776/COE1] and the European Union – NextGenerationEU, by the Austrian Science Fund (FWF) [10.55776/F71 and 10.55776/FG5]. For open access purposes, the author has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. Author contributions: R.S., H.Y., and R.W.P. implemented the experiment and performed data analysis with leading input from R.S. T.S. and C.H. provided help with the theoretical ideas and experimental implementation. T.S. assisted with the data analysis. R.W.P. conceived the experiment. R.W.P., C.H., and P.W. supervised the project. All authors contributed to the preparation of the manuscript, with leading input from H.Y. and T.S. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data pertaining to this work are publicly available on Zenodo at https://doi.org/10.5281/zenodo.10811006.
Submitted 12 January 2024 Accepted 10 May 2024 Published 14 June 2024 10.1126/sciadv.ado0215

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