# Experimental Observation of Earth’s Rotation with Quantum Entanglement

Raffaele Silvestri  University of Vienna, Faculty of Physics & Vienna Doctoral
School in Physics, Boltzmanngasse 5, A-1090 Vienna, Austria  Haocun Yu
<EMAIL_ADDRESS>University of Vienna, Faculty of Physics & Research
Network Quantum Aspects of Space Time (TURIS), Boltzmanngasse 5, 1090 Vienna,
Austria  Teodor Strömberg  University of Vienna, Faculty of Physics & Vienna
Doctoral School in Physics, Boltzmanngasse 5, A-1090 Vienna, Austria
Christopher Hilweg  University of Vienna, Faculty of Physics & Research
Network Quantum Aspects of Space Time (TURIS), Boltzmanngasse 5, 1090 Vienna,
Austria  Robert W. Peterson  University of Vienna, Faculty of Physics &
Research Network Quantum Aspects of Space Time (TURIS), Boltzmanngasse 5, 1090
Vienna, Austria  Philip Walther<EMAIL_ADDRESS>University of
Vienna, Faculty of Physics & Research Network Quantum Aspects of Space Time
(TURIS), Boltzmanngasse 5, 1090 Vienna, Austria

###### Abstract

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 due to mature methods for
generating and manipulating quantum states of light. The increased sensitivity
offered by these states promises to enable quantum phenomena, such as
entanglement, to be tested in unprecedented regimes where tiny effects due to
gravity come into play. However, this requires long and decoherence-free
processing of quantum entanglement, which has not yet been explored for large
interferometric areas. Here we present a table-top experiment using maximally
path-entangled quantum states of light in an interferometer with an area of
$715\text{\,}\mathrm{m}^{2}$, sensitive enough to measure the rotation rate of
Earth. A rotatable setup and an active area switching technique allow us to
control the coupling of Earth’s rotation to an entangled pair of single
photons. The achieved sensitivity of
$5\text{\,}\mathrm{\SIUnitSymbolMicro}~{}\mathrm{rad}~{}\mathrm{s}^{-1}$
constitutes the highest rotation resolution ever achieved with optical quantum
interferometers, surpassing previous work by three orders of magnitude [1].
Our result demonstrates the feasibility of extending the utilization of
maximally entangled quantum states to large-scale interferometers. Further
improvements to our methodology will enable measurements of general-
relativistic effects on entangled photons [2] opening the way to further
enhance the precision of fundamental measurements to explore the interplay
between quantum mechanics and general relativity along with searches for new
physics [3].

##  I Introduction

For more than a century, interferometers have been important instruments to
experimentally test fundamental physical questions. They disproved the
luminiferous ether, helped to establish special relativity [4, 5] and enabled
the measurement of tiny ripples in space-time itself known as gravitational
waves [6]. With recent advances in technology interferometers can nowadays
also operate using various different quantum systems like electrons [7, 8],
neutrons [9], atoms [10, 11, 12, 13, 14], superfluids [15, 16], and Bose-
Einstein condensates [17, 18, 19]. 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
[20, 21]. Second, the enhanced sensitivity of quantum interferometers opens up
opportunities for precision measurements that can explore new frontiers in
physics. These include setting constraints on dark energy models [22], testing
quantum phenomena in non-inertial reference frames [23, 24, 1], and
investigating the interplay between quantum mechanics and general relativity
[25, 26, 27, 28, 29].

Optical systems are particularly well-suited for realizing quantum
interferometers, thanks to mature techniques available for generating a
variety of quantum states, ranging from squeezed vacuum [30, 31, 32, 33] to
maximally path-entangled photons [34, 35]. The latter are referred to as N00N
states, represented by
$\frac{1}{\sqrt{N}}(\Ket{N}_{a}\Ket{0}_{b}-\Ket{0}_{a}\Ket{N}_{b})$, wherein
$N$ photons exist in a superposition of $N$ photons in mode $a$ with zero
particles in mode $b$, and vice versa [20]. Remarkably, these states behave
like those of a single photon with $N$ times the energy, enabling a phase
sensitivity at the Heisenberg limit that scales as $1/N$, and thus goes beyond
the $1/\sqrt{N}$ scaling of the standard quantum limit [21]. 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.

Figure 1: Earth’s rotation measured using entangled photons. a, A rotatable
$715\text{\,}{\mathrm{m}}^{2}$ 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 $\Theta$ is
defined as the angle between Earth’s angular velocity vector
$\vec{\Omega}_{E}$ and the fiber loop area vector $\vec{A}$. 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 photon coincidence counting (&). c, An optical switch
(OS) is used to toggle Earth’s rotation signal on and off independent of the
frame angle $\Theta$. This is achieved by controlling the propagation
direction (clockwise or counterclockwise) of photons in one half of the fiber
spool.

In this work, we report an experiment measuring the rotation of the Earth
harnessing quantum entanglement in a large-scale optical fiber interferometer.
We inject two-photon N00N states into a $715\text{\,}{\mathrm{m}}^{2}$ 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 that involved centimeter-scale fiber interferometers with at
most hundred-meter-length fibers [23, 24, 1, 33, 36], and which were only used
to measure synthetic signals. We are able to confirm an acquired Sagnac phase
from Earth’s rotation with an enhancement factor of two due to the two-photon
entangled state, achieving a rotation sensitivity of a few
$\text{\,}\mathrm{\SIUnitSymbolMicro rad}\text{\,}{\mathrm{s}}^{-1}$. To the
best of our knowledge, this is the largest quantum-optical Sagnac
interferometer in the world, surpassing previous state-of-the-art rotation
sensors employing two-particle entanglement. This measurement represents a
significant milestone in the development of larger-scale quantum
interferometers.

##  II Quantum optical Sagnac interferometer

Figure 2:  Quantum interference measurement revealing the Sagnac phase shift
induced by Earth’s rotation. Center, normalized quantum interference fringes
of single-photon and two-photon entangled state measurements. The red and
orange (blue and green) marks show the normalised two- (one-)photon
coincidence counts with the Earth rotation signal switched on and off,
respectively. The corresponding curves are least-squares fits to the data
using a model of the experiment (see Appendix B). The doubled fringe frequency
of the two-photon curves reveals the super-resolution due to quantum
entanglement. Sides, Sagnac phase shifts induced by Earth’s rotation at
$\Theta=2.5^{\circ}$, zooming in around $\phi=\pi,\pi/2,0$ for single photon
measurement (left), and around $\phi=\pi,3\pi/4,\pi/2$ for two-photon
measurement (right). The widths of the vertical lines indicate the size of
uncertainties due to uncorrelated photon counting noise. Because the same
phase bias $\phi_{0}$ has been applied to both one-photon and two-photon
measurements, the doubled Sagnac phase shift does not manifest in the plots.
1: one-photon sate; 2: two-photon N00N state; M: maximum; m: minimum; q:
quadrature.

A considerable barrier to detecting Earth’s rotation in large-scale fiber
interferometers is its minute rate, fixed direction, and the inability to
manipulate its behavior. 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], the flying times of photons traveling in
opposite directions around a rotating encircled path are different, inducing a
measurable phase difference:

$\displaystyle\phi_{s}=\frac{8\pi\Omega_{E}A\cos{\Theta}}{\lambda c}.$ (1)

Here, $\Omega_{E}$ is the rotation angular frequency of the Earth; $A$ is the
interferometer’s effective area of $715\text{\,}{\mathrm{m}}^{2}$ (for the
calibration of the apparatus see details in Appendix A); $\Theta$ is the angle
between the area vector of the fiber loop and the angular velocity vector of
the Earth; $\lambda$ is the photon wavelength of $1546\text{\,}\mathrm{nm}$.

In a Sagnac interferometer with an induced phase shift $\phi_{s}$, a coherent
or a single-photon state, represented as
$(\Ket{1}_{a}\Ket{0}_{b}-\Ket{0}_{a}\Ket{1}_{b})/\sqrt{2}$, evolves to
$(\Ket{1}_{a}\Ket{0}_{b}-e^{i\phi_{s}}\Ket{0}_{a}\Ket{1}_{b})/\sqrt{2}$. After
interference, a projective measurement at the output with two modes $a$ and
$b$ gives rise to the probabilities of detecting a single photon
$P_{1,a}=(1+\cos(\phi_{s}))/2$ and $P_{1,b}=(1-\cos(\phi_{s}))/2$ in each of
the two output modes. Multi-photon interference occurs when we inject the
entangled state $(\Ket{N}_{a}\ket{0}_{b}-\Ket{0}_{a}\Ket{N}_{b})/\sqrt{2}$
into the interferometer, where the $N$ photons are in a superposition of being
in either of the two modes. After transmission through the interferometer, the
state evolves to
$(\Ket{N}_{a}\ket{0}_{b}-e^{iN\phi_{s}}\Ket{0}_{a}\Ket{N}_{b})/\sqrt{2}$. This
results in the probability of finding photons in two modes oscillating at $N$
times the frequency $P_{N,a}=(1+\cos(N\phi_{s}))/2$ and
$P_{N,b}=(1-\cos(N\phi_{s}))/2$, enhancing the observed phase shift by a
factor of $N$.

Figure 3: Sagnac phase shifts induced by Earth’s rotation measured at six
interferometer frame angles. $\Theta$’s range from $-67.5^{\circ}$ to
$+25^{\circ}$, evenly spaced by $22.5^{\circ}$. Top, each data point is
obtained with the same measurement sequence and extraction method as Fig. 2.
At each angle $\Theta$, the Sagnac phase shift measured using two-photon
entangled N00N states (red triangle marks) is found to be doubled compared
with single-photon states (blue circle marks). The blue and red curves are the
least-squares fits to equation (1) of the one-photon and two-photon N00N state
measurements, respectively. Bottom, representation of different angles between
the area vector of the interferometer (blue 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.

##  III Experimental implementation

The two-photon path-entangled state is realized by exploiting the polarization
correlation of photon pairs emitted by a type-II spontaneous parametric down-
conversion (SPDC) source [39]. The photon pairs, centered at
$1546\text{\,}\mathrm{nm}$, are created in the product state
$\Ket{1}_{H}\Ket{1}_{V}$, where H and V denote horizontal and vertical
polarization, respectively. A half-wave plate (HWP) oriented at
$22.5\text{\,}\mathrm{\SIUnitSymbolDegree}$ (with respect to the horizontal
axis) transforms this product state into the polarization-entangled two-photon
N00N state $(\Ket{2}_{H}\Ket{0}_{V}-\Ket{0}_{H}\Ket{2}_{V})/\sqrt{2}$, where
the cross terms cancel out due to the indistinguishability of the photons.
Subsequently, this state is converted into a path-entangled state at a
polarizing beamsplitter (PBS), which separates the H and V photons into
clockwise and counter-clockwise propagating modes. After passing through the
$2\text{\,}\mathrm{km}$ fiber loops, the clockwise-traveling photons pick up a
Sagnac phase shift $\phi_{s}$ induced by Earth’s rotation relative to the
counterclockwise-traveling ones. The same PBS then converts the state back
into the polarization-entangled state:

$\displaystyle\frac{1}{\sqrt{2}}(\Ket{2}_{H}\Ket{0}_{V}-e^{i2\phi_{s}}\Ket{0}_{H}\Ket{2}_{V})\,.$
(2)

Interference takes place again at the $22.5^{\circ}$ HWP, leading to the
output state:

$\displaystyle\frac{1}{\sqrt{2}}\sin\phi_{s}(\Ket{2}_{H}\Ket{0}_{V}+\Ket{0}_{H}\Ket{2}_{V})-i\cos\phi_{s}\Ket{1}_{H}\ket{1}_{V}.$

A set of waveplates is used to control the detection probabilities by
introducing a bias phase $\phi_{0}$. 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, and turns the state into:

$\displaystyle\frac{1}{\sqrt{2}}(\Ket{2}_{H}\Ket{0}_{V}-e^{i(2\phi_{0}+2\phi_{s})}\Ket{0}_{H}\Ket{2}_{V}).$
(3)

To perform a projective measurement onto the $\Ket{1}_{H}\Ket{1}_{V}$
component of the state, we analyze the two-fold coincidence probability
$P_{HV}$ by collecting photons in both output ports of the PBS before
detectors:

$\displaystyle P_{HV}=\frac{1}{2}\left[1+\cos(2\phi_{0}+2\phi_{s})\right].$
(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
$2\text{\,}\mathrm{km}$ fibers wound around a $1.4\text{\,}\mathrm{m}$ square
aluminum frame (yellow) (Fig. 1 b). Because the detectable Sagnac phase shift
induced by Earth’s rotation depends on the direction of the area vector
$\vec{A}$, 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 $\Theta$.

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 $1\text{\,}\mathrm{km}$ fiber segments (orange
and blue), which are connected by the four-port optical switch. As shown in
Fig. 1 c, 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 due to 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 equation (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 Appendix A). Figure 2 shows the data for the Sagnac phase shifts induced by
Earth’s rotation at $\Theta=2.5^{\circ}$. The data points are acquired for
one- and two-photon N00N states propagating through the interferometer. For
the two-photon entangled states, eleven 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 $0.1\text{\,}\mathrm{Hz}$,
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.

##  IV Results

Fig. 2 shows quantum interference fringes of the Sagnac interferometer at
$\Theta=2.5^{\circ}$. In the central figure, the red and orange marks
represent normalized two-photon coincidence counts $P_{H}$ and $P_{V}$
measured with the optical switch on and off, respectively. These data were
generated from eleven sets of 30-min contiguous integration periods. Each data
set was taken with a specific value of $\phi_{0}$, ranging from $-\pi/8$ to
$2\pi+\pi/8$ to cover a full interference fringe. The blue and green marks are
heralded single-photon measurements, with eleven (seven shown) 15-min
contiguous integration periods, ranging from $-\pi/4$ to $2\pi+\pi/4$, serving
as the reference measurement. The uncertainties for each data point are
represented by $\pm 1$ standard deviations, which were calculated from Monte
Carlo simulations using $10^{5}$ samples of Poisson-distributed photon
coincidence counts (see details in Appendix A for a comprehensive error
analysis).

For the two-photon measurements, the eleven 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). Based on equation (4), the data are fit with:

$\displaystyle N_{\text{switch off}}(\phi)$
$\displaystyle=N_{0}(1+\mathcal{V}\cos(2\phi))$ (5) $\displaystyle
N_{\text{switch on}}(\phi)$
$\displaystyle=N_{0}(1+\mathcal{V}\cos(2\phi+\phi_{s}^{(2)})),$ (6)

where ${N_{0}}$ is the amplitude of the photon interference, $\mathcal{V}$ is
the interference visibility, and $\phi_{s}^{(2)}$ is Earth’s rotation-induced
phase shift to be measured. The extracted phase difference between two
interference fringes is $\phi_{s}^{(2)}=$5.5(5)\text{\,}\mathrm{mrad}$$. A
similar fitting and phase extraction procedure is employed for single-photon
reference measurements (blue and green), resulting in
$\phi_{s}=$2.8(2)\text{\,}\mathrm{mrad}$$. In the two-photon measurement, the
phase shift is enhanced by a factor of two due to the presence of
entanglement.

Sagnac phase shift measurements at five additional frame angles $\Theta$ are
presented in Fig. 3. This plot explicitly shows two things: First, the Sagnac
phase shift induced by Earth’s rotation is proportional to $\cos(\Theta)$ as
expected from equation (1). Second, the two-photon measurements consistently
exhibit a doubled phase compared to the single-photon measurements for all the
different frame angles. For each value of $\Theta$, the Sagnac phase shift is
extracted by comparing the interference fringes with the optical switch on and
off, following a procedure identical to that used for $\Theta=2.5^{\circ}$.
The data for the five additional angles were acquired with a shorter
integration time compared to Fig. 2, resulting in correspondingly larger
statistical uncertainties. The red and blue curves are the least-squares fits
using equation (1). From these fits, the maximum Sagnac phase shift induced by
Earth’s rotation in the two-photon N00N state is
$5.5(4)\text{\,}\mathrm{mrad}$, which corresponds to an Earth’s rotation rate
of
$\Omega_{E}=$7.1(5)\text{\times}{10}^{-5}\text{\,}\mathrm{rad}\mathrm{/}\mathrm{s}$$,
compared with $2.8(1)\text{\,}\mathrm{mrad}$ or
$\Omega_{E}=$7.2(3)\text{\times}{10}^{-5}\text{\,}\mathrm{rad}\mathrm{/}\mathrm{s}$$
in the one-photon measurement. Both agree with the internationally-accepted
value $7.3\text{\times}{10}^{-5}\text{\,}\mathrm{rad}\mathrm{/}\mathrm{s}$
[40], leading to an enhancement factor of $1.97(16)\text{\,}$.

##  V Discussion and Conclusions:

We have demonstrated the largest and most precise quantum optical Sagnac
interferometer to date, exhibiting sufficient sensitivity to measure Earth’s
rotation rate using both single and two-photon entangled states. When
comparing the use of quantum entanglement to classical probes, we observed a
factor of two improvement in the measured phase value due to super-resolution.
Our approach 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 rate of the multi-photon states. The
achievable phase resolution is primarily hindered by the 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.

However, a major advantage of our scheme, including the effective area-
switching, is its compatibility with the current cutting-edge technology of
fiber-optic gyroscopes (FOGs). FOGs have reached phase resolutions of less
than a $\text{\,}\mathrm{nrad}$ translating to rotation rates below
$0.1\text{\,}\mathrm{n}\mathrm{rad}~{}\mathrm{s}^{-1}$, with a stable signal
over more than a month [41, 42].

Remarkably, an analysis suggests that pairing these gyroscopes with state-of-
the-art sources [43] of two-photon entangled states would lead to a four-
orders-of-magnitude enhancement in sensitivity over our present work.
Anticipating the improvements of future single-photon sources, a logical next
step will be the use of the recently proposed giant-FOG with an area of
$15.000\text{\,}\mathrm{m}^{2}$ [44] reaching a resolution of about
$20\text{\,}\mathrm{p}\mathrm{rad}~{}\mathrm{s}^{-1}$. However, for the
observation of general-relativistic effects that couple to two-photon
entanglement , such as frame-dragging and geodetic corrections [2] to the
Sagnac phase, a resolution below
$0.1\text{\,}\mathrm{p}\mathrm{rad}~{}\mathrm{s}^{-1}$ is needed. We
anticipate that a square fiber ring interferometer (GFRING) with an area on
the order of $20\text{\,}\mathrm{k}\mathrm{m}^{2}$ would be able to probe a
regime lower than the general relativistic rotation rate correction term
$\Omega_{GR}=10^{-9}\Omega_{E}$ (see Fig. 5). Although interferometers such as
ring laser gyroscopes hold the record as the most precise local rotation
sensors, already being at the $\Omega_{GR}$ threshold [45], the feasibility of
probing these ring cavities with entangled photon pairs is limited by the
requirement of generating single-photons with ultranarrow linewidth at high
efficiency. We therefore believe that fiber ring interferometers are the most
promising platform to investigate the same regimes with entangled photon
states. Our work paves the way for other technically challenging proposals,
such as directly probing gravitationally-induced phase shifts [46], rotational
and gravitational decoherence in the quantum interference of photons [47],
testing fundamental symmetries in quantum field theory [3], investigating
local Lorentz invariance violation [48], detecting exotic low-mass fields from
high-energy astrophysical events [49], and dark matter searches like axion-
photon coupling [3]. Indeed, it is predicted that two photons can transition
to axions in presence of an external magnetic field. In an optical Sagnac
interferometer where two orthogonal polarizations counter-propagate, the
component parallel to the magnetic field would then be retarded with respect
to the other leading to a non-reciprocal 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 the effect of Earth’s
rotation highlights the practical feasibility of large-scale optical fiber
interferometers with entangled photons. After a century from the first local
observation of Earth’s rotation-induced fringe shift with light in a Sagnac
interferometer [5], we have experimentally observed the same shift in a
quantum interference pattern using entangled photon pairs enabling superior
phase measurements that can, in principle, reach the Heisenberg limit.
Notably, the techniques we developed to control the coupling of Earth’s
rotation effect into our measurements are applicable to other photonics
experiments involving small and locally static natural quantities as
gravitational fields. This experiment not only shows the potential of quantum
photonics in precision measurements but opens up to the exploration of new
frontiers in both quantum technology and fundamental physics research.

## Data availability

All data pertaining to this study are available from the corresponding authors
upon reasonable request.

###### Acknowledgements.

The authors thank P. Schiansky, I. Agresti and L.A. Rozema for help with the
photon source and the photon detectors. The authors thank Z. Yin and H. Cao
for discussions. R.S. acknowledges support from Uni:Docs fellowship program,
hosted by the University of Vienna. H.Y. acknowledges support from the Marie-
Skłodowska Curie Postdoctoral Fellowship program, hosted by the Horizon
Europe. 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 in whole, or in
part, from the research network TURIS, and by the European Union (ERC,
GRAVITES, No 101071779). Views and opinions expressed are however those of the
author(s) 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. Further funding was received from
the Austrian Science Fund (FWF) through BeyondC (F7113) and Research Group 5
(FG5). For the purpose of open access, 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.

## Competing interests

The authors declare no competing interests.

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$\bullet$

## Appendix A

Characteristics of the experimental setup: A periodically poled KTiOPO4
crystal produces orthogonally-polarized photon pairs centered at
$1545.76\text{\,}\mathrm{nm}$ in a type-II spontaneous parametric down-
conversion (SPDC) process. The crystal is pumped by continuous wave (CW)
Ti:Sapphire laser (Coherent Mira HP) emitting at $772.88\text{\,}\mathrm{nm}$
[39]. The photon-source pump power is set to $145\text{\,}\mathrm{mW}$,
leading to a detected photon-pair coincidence rate of approximately
$400\text{\,}\mathrm{kHz}$. 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%
($10\text{\,}\mathrm{dB}$). The Sagnac loop introduces $5\text{\,}\mathrm{dB}$
of losses, out of which $1\text{\,}\mathrm{dB}$ is the optical switch
insertion loss, $1\text{\,}\mathrm{dB}$ from the $2\text{\,}\mathrm{km}$
polarization-maintaining (PM) fiber
($\approx$0.5\text{\,}\mathrm{d}\mathrm{B}\mathrm{/}\mathrm{k}\mathrm{m}$$),
and $3\text{\,}\mathrm{dB}$ from fiber connections, while the input and output
of the optical setup contribute the remaining $5\text{\,}\mathrm{dB}$. In
detection, photons from the output paths are coupled into single-mode fibers
connected to superconducting nanowire single-photon detectors, housed in a
$1\text{\,}\mathrm{K}$ cryostat, with a detection efficiency of roughly
$95\text{\,}\mathrm{\char 37\relax}$ and a dark-count rate around
$300\text{\,}\mathrm{Hz}$. Amplified detection signals are counted using a
time tagging module with a timing resolution of $156.25\text{\,}\mathrm{ps}$,
and two-photon coincidence events are extracted using a coincidence window of
$3.75\text{\,}\mathrm{ns}$. In the two-photon N00N state measurements, when
both photons are propagating through the interferometer, the detected photon
pair rate is around $4\text{\,}\mathrm{kHz}$, consistent with the expected
exponential fragility to losses of a two-photon N00N state
$1-\eta_{i}^{2}\approx 99\%$, where $\eta_{\text{i}}=0.1$ is the total
transmission efficiency of the interferometer. In the one-photon measurements,
one photon of the pair is used as a trigger while the other propagates through
the interferometer. The total heralded single-photon rate in the two detection
ports is around $20\text{\,}\mathrm{kHz}$, which is compatible with the
overall losses $1-\eta_{\text{t}}\eta_{\text{i}}\approx 95\%$, where
$\eta_{\text{t}}=0.5$ is the transmission coefficient of the trigger photon
fiber path.

Figure 4: Comparison of photon measurements and CW measurements as
calibration. Data of one-photon state (blue) and two-photon N00N state data
(red) are inherited from Fig. 3 in Main text. Green triangle marks are
obtained using a polarimeter with CW light at six different $\Theta$’s ranging
from -90° to +22.5°, evenly spaced by 22.5°. The green solid curve is the
least-squares fit of CW measurements with fitting function as
$\phi_{E}(\Theta)=S\Omega_{E}\cos(\Theta+\Theta_{0})$, This measurement allow
us to find the frame angle that maximizes the Sagnac phase ($\Theta$ = 0°).
The green dashed curve is plotted as $2S\Omega_{E}\cos(\Theta+\Theta_{0})$ to
compare with the two-photon N00N state measurements.

Interferometer calibration: In the laboratory, the axis normal to fiber spool
plane when vertically oriented with respect to the horizon is pointed north.
The 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 ($\Omega_{E}$) as a
known quantity, with $\phi_{s}=S\Omega_{E}$. A set of phase measurements are
performed with a CW light source at telecom wavelength at six different
angular positions $\Theta^{k}$ of the fiber loop frame spaced by
$22.5\text{\,}\mathrm{\SIUnitSymbolDegree}$ (see Fig. 4), allowing us to find
the frame angle that maximizes the Sagnac phase ($\Theta=0^{\circ}$).
H-polarized light is injected into the interferometer, which is converted into
diagonal polarization by a half-wave plate before entering the Sagnac
interferometer. Due to Earth’s rotation, the H and V components acquire a
relative Sagnac phase $\phi_{s}$, which is encoded in the polarization state
ellipticity angle $\chi$ [50], such that $\phi_{S}=2\chi$. A compact free-
space polarimeter is employed to fully characterize the polarization state
after the waveplates, which compensate first for the polarization rotation in
the output fiber circulator path. As in the measurements with quantum light,
the optical switch is driven by a $0.1\text{\,}\mathrm{Hz}$ square-wave. The
recorded time trace of $\chi$ is partitioned into two sets by demodulating it
using the driving signal. For each frame angle $\Theta^{k}$ the differential
average between the two traces
$\delta\overline{\chi}^{k}=\overline{\chi}_{on}^{k}-\overline{\chi}_{off}^{k}$
is used to calculate the Sagnac phase $\phi_{S}^{k}$ and its associated
uncertainty $\sigma^{k}$. As part of a Monte-Carlo simulation resampling the
phase values $\phi_{S}^{k}$ using their uncertainties, the data are fit to the
model function $\phi_{E}(\Theta)=S\Omega_{E}\cos(\Theta+\Theta_{0})$, where
$\Omega_{E}=$7.29\text{\times}{10}^{-5}\text{\,}\mathrm{rad}\text{\,}{\mathrm{s}}^{-1}$$
is the known value of Earth’s rotation rate, and $S$ and $\Theta_{0}$ are free
parameters. The Monte-Carlo simulation estimating these parameters
additionally samples the frame angles $\Theta^{k}$ from uniform probability
distributions $[\Theta^{k}-1,\Theta^{k}+1]$. The extracted fit parameters are
the scale factor $S=$38.8(1)\text{\,}\mathrm{s}$$, and the angular offset
$\Theta_{0}=$0.03(33)\text{\,}\deg$$. The CW measurements are compared with
the photon measurements from Fig. 3 of the main text.

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-
varying fluctuations of the fiber length and refractive index. This passive
isolation increases the scale factor stability in time by stabilizing the
enclosed interferometric area. In addition, the optical switching method is
also a fundamental and powerful tool in our experimental implementation. It
not only provides a reference where the rotation effect, manifested as a non-
reciprocal Sagnac phase, is absent, but also eliminates possible phase errors
resulting from laser intensity fluctuations, imperfections in the input photon
state, non-ideal 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 to mitigate slow frequency drifts in the
measured phase via post-processing, 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 of the main text are estimated using a
Monte-Carlo simulation accounting for photon counting noise and uncertainties
in the phase offset. In each round of the simulation the photon counts are
sampled using a Poisson distribution with mean and standard deviation of
$N_{k}^{s}$ and $\sqrt{N_{k}^{s}}$, respectively, where $N_{k}^{s}$ is the
number of recorded photon counts for the offset phase $\phi_{0}^{k}$ and
switch state $s\in\\{\text{on},\text{off}\\}$. Additionally, phase-offset
noise, correlated between the on and off states, is sampled from a
distribution derived from the waveplate motor repeatability and added to the
offsets $\phi_{0}^{k}$. For each sampled data set a least-squares fit is
performed, using the amplitude $N_{0}$, fringe visibility $\mathcal{V}$, and
phase shift $\phi$ as free parameters. Finally, the values and uncertainties
of these parameters are estimated using the mean and standard deviation,
respectively, taken over $10^{5}$ repetitions of the simulation.

Figure 5: Rotation rate resolutions and enclosed areas of existing and
predicted quantum optical Sagnac interferometers. The plot is divided into
three sensitivity regimes: sensitivity below Earth rotation $\Omega_{E}$
(white zone), sensitivity above $\Omega_{E}$ but below general relativistic
effects $\Omega_{GR}$ (blue zone), and sensitivity above $\Omega_{GR}$ (orange
zone). Diamond-shaped markers (

$\blacksquare$

, $\Diamond$) represent existing interferometric platforms, while star-shaped
markers ($\medstar$) are proposed platforms but yet to be realized. Solid
markers (

$\blacksquare$

) represent performed experiments with quantum states of light, while empty
markers ($\Diamond$, $\medstar$) represent experiments yet to be performed.
Bertocchi et al. [36]: $L_{f}=550~{}m,P=0.63~{}m$, Restuccia et al. [23]:
$L_{f}=100~{}m,P=2.85~{}m$, Fink et al. [1]: $L_{f}=270.5~{}m,P=0.49~{}m$,
this work: $L_{f}=2~{}km,P=5.6~{}m$, CFOG (Commercial Fiber-Optic Gyroscope
(FOG)) [41]: $L_{f}=3~{}km,P=0.63~{}m$, LFOG (Large FOG) [42]:
$L_{f}=8~{}km,P=2.15~{}m$, GFOG (Giant FOG) [44]: $L_{f}=15~{}km,P=12.57~{}m$,
GFRING (Giant square Fiber RING interferometer): $L_{f}=47.5~{}km,P=6~{}km$,
where $L_{f}$ is the fiber length and $P$ is the perimeter. The photon-pair
generation rate is $1\text{\,}\mathrm{GHz}$ for the CFOG and LFOG, and
$10\text{\,}\mathrm{GHz}$ for the GFOG and GFRING, with integration times on
the order of a month (for more details see Appendix B).

$\bullet$

## Appendix B

## Interferometer with classical light

Continuous-wave (CW) light source: The CW source employed is a broadband NIR-
wavelength Superluminescent Diode (SLD). In front, a hard-coated bandpass
filter with 12 nm full-width at half-maximum (FWHM) centered at 1545.5 nm is
placed.

Interferometer operation: The light enters horizontally polarized ($H$), is
coupled into a single-mode (SM) fiber-optic circulator, and turns diagonally
polarized ($+$) after a half-wave plate (HWP) oriented at 22.5° with respect
to $H$. A Wollaston prism (WP) serves as a high-extinction ratio polarizing
beam-splitter (PBS), which spatially separates the light into its horizontal
and vertical ($V$) components. The output beams of the WP are coupled into
polarization-maintaining (PM) fiber, aligned so that both polarization
components propagate in the slow axis of the fiber. This ensures that the
birefringence of the fiber does not contribute to the phase noise. A four-port
PM optical micro-electro-mechanical systems (MEMS) switch connects the two
equal fiber spools of 1 km length. When this switch is in the state “ON”, the
$H$ and $V$ components pick a relative non-reciprocal Sagnac phase shift
$\phi_{S}$. After light recombination at the PBS, the polarization ellipse
major axis turns to anti-diagonal (azimuth angle $\psi=-45^{\circ}$), while
being slightly elliptical (ellipticity angle $\chi=\phi_{S}/2$). When passing
the same HWP again, the polarization ellipse rotates by 45° with its major
axis back to horizontal while still maintaining its ellipticity
($\psi=0,\chi=\phi_{S}/2$), then undergoes the fiber circulator and is coupled
back to free space. It follows a set of 3 waveplates, respectively two
quarter-wave plates (QWP) and one HWP, with a second WP as detection PBS (see
Fig. 6) [51, 52].

Waveplates operation: A set of two quarter-wave and one half-wave plate can be
used to implement any polarization transformation [53]. Thus, such a set of
waveplates is able to perform several tasks at the same time.
1) Polarization state tomography: They can select any polarization basis for
the measurement, namely rectilinear $\\{H,V\\}$, diagonal $\\{+,-\\}$
($\\{D,A\\}$) and circular $\\{RL\\}$, with their corresponding unitary
operators to set the measurement basis being
$\\{I,\hat{\sigma}_{2},\hat{\sigma}_{3}\\}$.
2) Polarization compensation of the circulator fiber: Since the SM fiber of
the circulator introduces random birefringence, the polarization-state
rotation at the output needs to be compensated for. This can be achieved by
preparing the state before the fiber in two different known polarization
states, for instance with two separate $H$ and $+$ polarizers. If full
polarization tomography on the two output states $\ket{P_{H}}=U_{f}\ket{H}$,
$\ket{P_{+}}=U_{f}\ket{+}$ is performed, the fiber unitary $U_{f}$ can be
fully reconstructed. By configuring the waveplates to implement the inverse
matrix $U^{-1}_{f}$ one reduces the overall action of the circulator on the
polarization state to an identity transformation.
3) Selecting the working point: To set the working point of the interferometer
it is sufficient to apply a unitary
$U(\phi)=e^{-i\phi\hat{n}_{2}\cdot\vec{\sigma}/2}$ that rotates the
compensated polarization state along the diagonal axis $\hat{n}_{2}$ by an
angle $\phi$, which corresponds to the bias phase. The rotation operator can
then simply be written as a phase shift matrix

$U(\phi)=\begin{pmatrix}1&0\\\ 0&e^{i\phi}\end{pmatrix}$ (S1)

expressed in the diagonal basis

$U_{\text{bias}}(\phi)=U_{\text{HWP}}(-22.5^{\circ})U(\phi)U_{\text{HWP}}(-22.5^{\circ}),$
(S2)

where $U_{\text{HWP}}(-22.5^{\circ})$ is the unitary of an HWP oriented at
-22.5°. The operation physically corresponds to rotating the compensated $H$
polarization to $A$, applying a relative phase between the $H$ and $V$
components, and rotating back.
In the end, the waveplates are able to implement an overall unitary
$U=U_{\text{proj}}U_{\text{bias}}U^{-1}_{f}$, with
$U_{\text{proj}}\in\\{I,\hat{\sigma}_{2},\hat{\sigma}_{3}\\}$, by setting a
triplet of angles $(\theta_{1},\theta_{2},\theta_{3})$ such that

$U=U_{\text{HWP}}(\theta_{3})U_{\text{QWP}}(\theta_{2})U_{\text{QWP}}(\theta_{1}).$
(S3)

Phase extraction: To measure the Sagnac phase encoded in the polarization
state after the waveplates, a compact free-space polarimeter with a sampling
rate of $20\text{\,}\mathrm{Hz}$ is employed. The phase is estimated as
$\phi_{S}\approx 2\sqrt{\delta\overline{\chi}^{2}+\delta\overline{\psi}^{2}}$,
for small differential mean ellipticity $\delta\overline{\chi}$ and azimuth
$\delta\overline{\psi}$ angles, where $\delta$ denotes the difference across
the two switch states. Ideally, the Sagnac phase should only be encoded in the
ellipticity angle. However, due to imperfect polarization rotation from the
waveplates, the measurement bases will change, and a small part of the phase
signal couples into the azimuth. The employed expression represents then a
good approximation of the effective Sagnac phase value for small angular
shifts.

Optical switch operation: A square-wave modulation at
$0.1\text{\,}\mathrm{Hz}$ with a $50\text{\,}\mathrm{\char 37\relax}$ duty
cycle was employed for all the measurements (classical and quantum). Even
though in principle our four-port PM fiber optical MEMS switch could operate
at a modulation signal frequency of almost $1\text{\,}\mathrm{kHz}$, we
observed a noisy transition region of $20\text{\,}\mathrm{ms}$ duration
centered at each rising/falling edges of the square signal. For this reason,
in post-processing, photon counts within a $10\text{\,}\mathrm{ms}$ time
window from to the edges are discarded to make sure none of the time tags fall
in a noisy region, while for the CW measurement at least the two data samples
around the transitions are removed ($50\text{\,}\mathrm{ms}$ time window) due
to the limited sampling rate ($20\text{\,}\mathrm{Hz}$) of the polarimeter
(see Fig. 7). This procedure effectively limits the switching frequency to
less than $25\text{\,}\mathrm{Hz}$. In the end, a modulation at
$0.1\text{\,}\mathrm{Hz}$ was selected since it showed the best stability over
time. The optical switch also introduces a state-dependent power loss of
$10\text{\,}\mathrm{\char 37\relax}$ when is turned off, so that the
transmission in the “OFF” state is 0.9 of the “ON” state.

## Interferometer with quantum light

Single-photon state measurement: A HWP oriented at 45° is placed into the $V$
photon input path, the photon is thus transmitted by the PBS and directly
coupled into a single-mode fiber connected to a detector channel, serving as a
trigger, while the H photon is transmitted out of the other port and injected
into the interferometer (see Fig. 6). Two-photon coincidence counts between
the two output channels and the trigger are recorded, and for each phase bias
the total number of events of each switch state is
$N^{(\text{on}/\text{off})}_{H}$. The selected observable for the fringe scan
points is the total normalized $V$ counts

$n_{V}=\frac{N_{V}}{N_{H}+N_{V}},$ (S4)

where

$\displaystyle N_{H}$
$\displaystyle=\frac{A_{H}}{2}[1+\mathcal{V}_{H}\cos(\phi+\phi^{(1)})],$ (S5)
$\displaystyle N_{V}$
$\displaystyle=\frac{A_{V}}{2}[1-\mathcal{V}_{V}\cos(\phi-\phi^{(1)})]$ (S6)

with the constraints $\mathcal{V}_{H}=\mathcal{V}_{V}=\mathcal{V}$ and
$\phi^{(1)}_{H}=\phi^{(1)}_{V}=\phi^{(1)}$. The corresponding fitting function
is

$n_{V}(\phi)=a_{V}\frac{1-\mathcal{V}\cos(\phi+\phi^{(1)})}{1+\eta\mathcal{V}\cos(\phi+\phi^{(1)})},$
(S7)

where $a_{V}=A_{V}/(A_{H}+A_{V})$ and $\eta=(A_{H}-A_{V})/(A_{H}+A_{V})$. The
fringe visibility $\mathcal{V}$, phase offset $\phi^{(1)}$ as well as the
amplitudes $A_{H}$ and $A_{V}$ are taken as free parameters of the fit.

N00N state measurement: By removing or rotating the HWP to 0°, and adjusting
the relative temporal delay to make the H and V photons indistinguishable in
all other degrees of freedom, a $\ket{1_{H}}\ket{1_{V}}$ Fock state is
generated after the input PBS. The Hong-Ou-Mandel interference in polarization
takes place at the 22.5° HWP, generating the two-photon N00N state.
Coincidence photon counts between the two output channels are recorded, for
each phase bias, and the total number of photons in the two switch states
$N^{(\text{on}/\text{off})}_{HV}$ are counted. We select the latter quantity
as observable for the fringe scan. The fitting function is

$N_{HV}(\phi)=\frac{A_{HV}}{2}[1+\mathcal{V}\cos(2\phi+\phi^{(2)})],$ (S8)

with $A_{HV}$, $\mathcal{V}$, and $\phi^{(2)}$ as free parameters.

## Sensitivity analysis of proposed experiments:

For the analysis of the proposed experiments with two-photon NOON states
(empty markers), the Heisenberg phase precision scaling at the point of
maximum sensitivity is assumed $\delta\phi=1/2\sqrt{R_{out}T}$, where
$R_{out}$ is the detected photon pairs rate and T is the total integration
time of the measurement. The newly proposed switching technique is included
and $\delta\phi_{on}=\delta\phi_{off}=\delta\phi$ is imposed with $T/2$
integration time for each state, giving

$\delta\phi=\frac{1}{\sqrt{2R_{out}T}}.$ (S9)

The optical losses during propagation in fiber are exponential in its length,
thus the output rate is related to the input rate as $R_{out}=\eta R_{in}$,
with transmission coefficient $\eta=10^{-\alpha L_{f}/10}$, where $\alpha$ and
$L_{f}$ are the fiber attenuation coefficient and length, respectively. For a
general NOON state the losses are exponential in the number of photons N such
that $\eta\rightarrow\eta^{N}$, so the two-photon rate is

$R_{out}=10^{-2\frac{\alpha L_{f}}{10}}R_{in}.$ (S10)

In the end, the phase precision is given by

$\delta\phi=\frac{1}{\sqrt{2R_{in}T}}10^{\frac{\alpha L_{f}}{10}}.$ (S11)

The corresponding rotation rate precision is then given by the Sagnac phase
formula at maximum signal $\delta\Omega=\delta\phi/S$, with the scale factor
defined as $S=\frac{8\pi A}{\lambda c}$, where $\lambda$ is the light
wavelength and $A$ is the effective area of the interferometer. $A=A(L_{f})$
is also a function of the fiber length depending on the interferometer
geometry. In particular, for the proposed giant square ring interferometer
(GFRING) the effective area is $A=\frac{1}{n_{t}}(\frac{L_{f}}{4})^{2}$, where
$n_{t}$ are the number of fiber turns around the square frame. Given the
expected dimensions of the ring it is assumed that its frame plane would be
fixed parallel to the Earth surface at its location. The rotation rate
resolution can then be written as

$\delta\Omega=\sqrt{\frac{2}{{R_{in}T}}}\frac{\lambda
c}{\pi\sin{\theta_{L}}}n_{t}\frac{10^{\frac{\alpha L_{f}}{10}}}{{L_{f}^{2}}},$
(S12)

where $\theta_{L}$ is the latitude at the location of the ring. The
geometrical parameters of the GFRING are numerically extracted by requiring
the signal-to-noise ratio (SNR) to be $\Omega_{GR}/\delta\Omega=3$ and
maximizing $n_{t}$ to reduce the surface area of the ring as much as possible.
The calculation assumes the latitude angle $\theta_{L}=48.2^{\circ}$ of
Vienna, Austria. Finally, the optimal fiber length and number of turns result
in $L_{f}=47.5$ km and $n_{t}=8$, corresponding to a square side of 1.5 km and
a perimeter of 6 km. For each experiment an integration time of $T=2$ months
is considered (see Table 3).

Figure 6: Detailed experimental setup. Experimental scheme for the single-
photon and CW measurements (green insets). In the one-photon measurement a HWP
at $45\text{\,}\mathrm{\SIUnitSymbolDegree}$ before the input PBS is used to
send the trigger photon directly to the SNSPDs. For the CW measurement the
SNSPDs are replaced with a polarimeter placed after the waveplates. The
polarization states before/after the Sagnac are indicated with red arrows, and
the Earth rotation phase manifests as ellipticity in the output polarization
state. SPDC. Spontaneous Parametric Down-Conversion single photons source.
SLD. SuperLuminescent Diode. PBS. Polarizing Beam Splitter cube. PC. fiber
Polarization Controller. WP. Wollaston Prism. HWP. Half-wave plate. QWP.
Quarter-wave plate. SNSPDs. Superconducting Nanowire Single-Photon Detectors.
TTM. Time Tagging Module.  Figure 7: Ellipticity angle time signal. The
ellipticity angle measured with CW light by a polarimeter, with
$20\text{\,}\mathrm{Hz}$ sampling rate, is plotted over time. It is shown how
the square-wave voltage applied to the switch at $0.1\text{\,}\mathrm{Hz}$
modulates the measured signal (dark blue trace). The high/low levels
correspond to the switch on/off states (green square wave). Data samples in
the trace are cut around the transition points (light blue samples) and two
separate on/off time traces are extracted. $\delta\overline{\chi}$ is
calculated as the difference between the two respective averages
$\overline{\chi}_{on}-\overline{\chi}_{off}$ (green dashed horizontal lines).
The same analysis is performed on the azimuth time trace.

## Tables

Table 1: Single-photon state measurement extracted parameters.

$\Theta$ |  $\mathcal{V}_{\text{on}}$ |  $\mathcal{V}_{\text{off}}$ |  $\phi_{\text{on}}$ (mrad) |  $\phi_{\text{off}}$ (mrad) |  $\phi_{\text{E}}$ (mrad)
---|---|---|---|---|---
$-87.5^{\circ}$ |  99.66(12) % |  99.67(12) % |  -1.36(2.45) |  -1.13(2.45) |  0.23(21)
$-65^{\circ}$ |  99.68(12) % |  99.69(12) % |  -1.99(2.45) |  -0.98(2.45) |  1.00(21)
$-42.5^{\circ}$ |  99.69(12) % |  99.68(12) % |  -1.86(2.45) |  0.28(2.45) |  2.14(21)
$-20^{\circ}$ |  99.59(13) % |  99.60(13) % |  -7.80(2.45) |  -5.13(2.45) |  2.66(25)
$~{}~{}\,2.5^{\circ}$ |  99.69(13) % |  99.67(13) % |  -2.43(2.43) |  0.34(2.46) |  2.77(18)
$~{}~{}\,25^{\circ}$ |  99.66(12) % |  99.66(12) % |  -1.22(2.45) |  1.37(2.45) |  2.59(21)

00footnotetext: Interferometric visibility
$\mathcal{V}_{\text{on}/\text{off}}$ and phase offset
$\phi_{\text{on}/\text{off}}$ values extracted from the fit for each frame
orientation angle $\Theta$ both in the switch on and off states, with the
corresponding Earth rotation induced phases $\phi_{\text{E}}$ calculated as
$\phi_{\text{off}}-\phi_{\text{on}}$.

Table 2: Two-photon NOON state measurement extracted parameters. $\Theta$ |  $\mathcal{V}_{\text{on}}$ |  $\mathcal{V}_{\text{off}}$ |  $\phi_{\text{on}}$ (mrad) |  $\phi_{\text{off}}$ (mrad) |  $\phi_{\text{E}}$ (mrad)
---|---|---|---|---|---
$-87.5^{\circ}$ |  96.81(45) % |  96.79(45) % |  -28.53(4.94) |  -27.71(4.95) |  0.82(65)
$-65^{\circ}$ |  96.78(45) % |  96.73(45) % |  -21.76(4.93) |  -19.49(4.94) |  2.28(65)
$-42.5^{\circ}$ |  96.72(45) % |  96.56(45) % |  -9.87(4.94) |  -6.02(4.95) |  3.86(65)
$-20^{\circ}$ |  95.67(44) % |  95.56(44) % |  -5.00(4.95) |  -0.07(4.96) |  4.93(76)
$~{}~{}\,2.5^{\circ}$ |  97.14(45) % |  97.17(45) % |  -24.60(4.92) |  -19.09(4.92) |  5.51(54)
$~{}~{}\,25^{\circ}$ |  97.54(46) % |  97.51(46) % |  -10.53(4.94) |  -5.09(4.95) |  5.44(69)
Table 3: Quantum optical Sagnac interferometers specifications and resolutions
comparison.

|  |  $L_{f}$ |  $P$ |  $A$ |  $S~{}(\text{s}^{-1})$ |  $\delta\phi_{S}$ (rad) |  $\delta\Omega~{}(\text{rad s}^{-1})$
---|---|---|---|---|---|---|---
| Our work |  2 km |  5.55 m |  $715~{}\text{m}^{2}$ |  38.8 |  $1.79\cdot 10^{-4}$ |  $4.61\cdot 10^{-6}$
| CFOG |  3 km |  0.63 m |  $150~{}\text{m}^{2}$ |  8.1 |  $1.34\cdot 10^{-8}$ |  $1.95\cdot 10^{-9}$
| LFOG |  8 km |  2.15 m |  $1372~{}\text{m}^{2}$ |  74 |  $2.38\cdot 10^{-8}$ |  $3.21\cdot 10^{-10}$
| GFOG |  15 km |  12.57 m |  $15000~{}\text{m}^{2}$ |  811 |  $1.69\cdot 10^{-8}$ |  $2.11\cdot 10^{-11}$
| GFRING |  47.5 km |  6 km |  $17.6~{}\text{km}^{2}$ |  951320 |  $2.31\cdot 10^{-8}$ |  $2.43\cdot 10^{-14}$

00footnotetext: Square markers () represent square interferometer frames,
while () are circular frames. The selected optical wavelength $\lambda$ is
$1550\text{\,}\mathrm{nm}$ for each proposed experiment. The existing and
predicted FOGs platforms utilize PM fibers ($\alpha=0.5$ dB/km), while for the
GFRING a standard single-mode fiber ($\alpha=0.16$ dB/km) is proposed. The
single-photon pair generation rate $R_{in}$ is 1 GHz, while for the GFOG and
GFRING is 10 GHz. $L_{f}$ is the fiber length, $P$ is the frame perimeter, $A$
is the effective interferometer area, $S$ is the scale factor, while
$\delta\phi_{S}$ and $\delta\Omega$ are the Sagnac phase and rotation
resolutions.