# Quantum-limited determination of refractive index difference by means of
entanglement

M. Reisner1, F. Mazeas1, R. Dauliat2, B. Leconte2, D. Aktas1,†, R. Cannon1, P.
Roy2, R. Jamier2, G. Sauder1, F. Kaiser1,††, S. Tanzilli1, and L. Labonté1
1Université Côte d’Azur, CNRS, Institut de Physique de Nice, 06108 Nice Cedex
2, France.  2Université de Limoges, XLIM, UMR 7252, Limoges, France.  †Now at
Quantum Engineering Technology Labs, University of Bristol, UK  ††Now at
Center for Integrated Quantum Science and Technology, University of Stuttgart,
Germany.

###### Abstract

Shaping single-mode operation in high-power fibres requires a precise
knowledge of the gain-medium optical properties. This requires accurate
measurements of the refractive index differences ($\Delta n$) between the core
and the cladding of the fiber. We exploit a quantum optical method based on
low-coherence Hong-Ou-Mandel interferometry to perform practical measurements
of the refractive index difference using broadband energy-time entangled
photons. The precision enhancement reached with this method is benchmarked
with a classical method based on single photon interferometry. We show in
classical regime an improvement by an order of magnitude of the precision
compared to already reported classical methods. Strikingly, in the quantum
regime, we demonstrate an extra factor of 4 on the accuracy enhancement,
exhibiting a state-of-the-art $\Delta n$ precision of
$6\text{\times}{10}^{-7}$. This work sets the quantum photonics metrology as a
powerful characterization tool that should enable a faster and reliable design
of materials dedicated to light amplification.

##  I Introduction

Fiber light sources are among key-growth technologies in the field of
photonics owing to their outstanding performance in terms of high average
power, excellent beam quality, single- and multi-pass gain, and agility
Richardson:10 . They have revolutionized existing scientific and industrial
applications in the biomedical field, and industrial materials processing for
example, as well initiate new ones, as metrology and imaging
noauthor_extending_2012 ; noauthor_fibre_2013 . Fiber laser development relies
on a complementary approach between tailored waveguide design and low-loss
optical materials synthesis for enabling high-power propagation. Much effort
has been devoted to waveguide engineering, leading to speciality fibre
architectures such as microstructured very large-area-mode fibers (VLMA)
kalli_large_2015 ; Dauliat16 . Optical materials have also received a great
attention through dedicated engineering work dragic_materials_2018 ;
schuster_material_2014 . However, despite the progress made over the last
decade, an experimental method allowing precise characterization of optical
material properties is still missing. A striking example is that of VLMA
fibers. The cornerstone of their fabrication lies in the precise knowledge of
the refractive index difference $\Delta n$ between the two different materials
composing the core and the cladding of the waveguide, which has to be lower
than $10^{-5}$ to ensure single-mode operation within a large core fiber
Dauliat16 . The associated precision should be at least one order of magnitude
lower, i.e $\sim 10^{-6}$. Unfortunately, state-of-art precision achievements
based on optical coherence tomography (OCT) are limited to ${10}^{-4}$ Tan:16
; singh2006 ; Yablon:13 , mainly due to chromatic dispersion.
In this paper, we introduce an experimental method based on quantum OCT
allowing measurements of $\Delta n$ with a precision down to
$6\text{\times}{10}^{-7}$, corresponding to a 4-fold enhancement with respect
to classical methods. This consists in exploiting an Hong-Ou-Mandel (HOM)-type
interferometer fed with low-coherence energy-time entangled photons HOM . In
comparison to single-photon based experiments, exploiting quantum biphoton
states exhibits two main advantages abouraddy_quantum-optical_2002 : (i) the
instrument’s resolution is not affected by even-order dispersion in the sample
thanks to dispersion cancellation resulting from the energy correlation, and
(ii) an augmented robustness to the losses of the sample under test (SUT). In
addition to the increased precision, this approach is independent on the SUT,
leading to universal and versatile optical property measurements
kaiser_quantum_2018 .
HOM-interferometry stands as a fundamental concept in quantum optics HOM and
is of particular relevance for the measurement of indistinguishable photons
mcmillan_two-photon_2013 , that lies at the heart of quantum teleportation and
entanglement swapping halder_entangling_2007 ; dauria_universal_2020 .
Furthermore, the HOM effect has been exploited for generating path-entangled
two-photon N00N state Hua:21 , a class of states widely used in enhanced
phase-sensing based quantum-metrology. This includes microscopy
PhysRevLett.112.103604 , measurements of material properties
kaiser_quantum_2018 , as well as medical and biological sensing
crespi_measuring_2012 . The common concept in these applications lies in
determining relative time delays accurately, as required for precise $\Delta
n$ measurements. To date, the key ingredients for obtaining the highest
precision time-delay measurement using the HOM effect are : i) the common-path
geometry that significantly helps the stability of the interferometer while at
the same time limiting the application only to birefringent samples
PhysRevA.62.063808 ; dauler1999 , and ii) the use of very short samples that
do not exceed the coherence length of the single photons (less than 100
$\mu$m) attosecond ; biphoton-beat-note .
Here, we propose practical $\Delta n$ measurements based on QOCT in a dual-arm
configuration with a 50 cm-long sample. The method aims at measuring the time-
delay between two optical paths, each associated with a given material to be
characterized. It is worth noting that HOM interferometry is immune to
relative phase fluctuations between the two arms avoiding complex and
expensive stabilization systems, as typically experienced in classical
interferometry. Moreover, stringent conditions (identical length and
temperature) are set for the two materials thanks to a special two-core rod-
type fiber packaging.

##  II Results

Theory of two-photon interference. A brief overview of the evolution of a two-
photon state through a Mach-Zehnder interferometer (MZI) lopez-
mago_coherence_2012 that lies at the heart of our measurement method, is
depicted in Fig. 1.a.

Energy-time entangled photon-pairs generated by spontaneous parametric down
conversion (SPDC) out of a second-order nonlinear crystal are here considered.
Such 3-waves mixing process is ruled by energy and momentum conservation,
written as $\omega_{\rm{p}}=\omega_{\rm{i}}+\omega_{\rm{s}}$ and
$\vec{k_{\rm{p}}}=\vec{k_{\rm{i}}}+\vec{k_{\rm{s}}}$, respectively, where p,
i, s refer to the pump, idler and signal photon, respectively. Their state can
be written as

$\ket{\Psi_{in}}=\int\text{d}\omega_{\rm{i}}\text{d}\omega_{\rm{s}}G(\omega_{\rm{i}},\omega_{\rm{s}})a^{\dagger}_{\omega_{\rm{s}}}a^{\dagger}_{\omega_{\rm{i}}}\ket{0},$
(1)

where $a^{\dagger}_{\omega_{\rm{s}}}$ ($a^{\dagger}_{\omega_{\rm{i}}}$) is the
creation operator of a photon in input mode $a$ at frequency $\omega_{\rm{i}}$
($\omega_{\rm{s}}$). $G(\omega_{\rm{i}},\omega_{\rm{s}})$ and
$|G(\omega_{\rm{i}},\omega_{\rm{s}})|^{2}$ are the joint-spectral amplitude
and density, respectively. The latter corresponds to the probability of
detecting one photon at frequency $\omega_{\rm{i}}$ and the other one at
$\omega_{\rm{s}}$. One should note that
$\int\text{d}\omega_{\rm{i}}\text{d}\omega_{\rm{s}}|G(\omega_{\rm{i}},\omega_{\rm{s}})|^{2}=1$.
The joint spectrum amplitude of the generated biphotons in CW regime using a
laser at frequency $\omega_{p}$ is given by
$G(\omega_{\rm{i}},\omega_{\rm{s}})=g(\omega_{\rm{i}})g(\omega_{\rm{s}})\delta(\omega_{\rm{p}}-\omega_{\rm{i}}-\omega_{\rm{s}})$.
Both photons pass through the same bandpass filter which is generally added to
clean the photons spectrum from spurious frequency components (see
tanzilli_ppln_2002 for more details on experimental SPDC sources). The exact
form of $g(\omega)$ depends on both the phase-matching condition and the
transmission profile of the filter. In the case of a Gaussian shaped filter,
$g(\omega)=(2\pi\sigma)^{-1/4}\cdot\text{e}^{\frac{(\omega-\omega_{p}/2)^{2}}{4\sigma
2}}$, with bandwidth $\sigma$ centered around $\omega_{\rm{p}}/2$.

Sending such a state to an MZI (see Fig.1), the probability of detecting two-
photon coincidences between the two output ports of the device, as a function
of the ajustable delay $\tau$ between its two arms reads lopez-
mago_coherence_2012 :

$P_{c}(\tau)=P_{\rm{c}}(0)(2-\cos\omega_{\rm{p}}\tau-\alpha\text{e}^{\tau^{2}\sigma^{2}}),$
(2)

where $\alpha$ and $P_{\rm{c}}(0)$ represent the HOM-dip visibility and the
average probability of registering two-photon events, respectively.

Figure 1: a) Different case scenarios in two-photon interferometry using a MZ-
type device. The pair can travel along four different paths. Interference in
the coincidence counts can occur provided two paths are indistinguishable.
Note that the different colors for the two photons are for representation
purpose only. Ideally the paired photons are indistinguishable. b)
Interference pattern at the output of a balanced interferometer

Interference in coincidence counting occurs between probability amplitudes of
indistinguishable paths. In Eq. (2) one can identify three terms. The first
term is a constant, stemming from all of the possible distinguishable paths.
The second term is related to two-photon contribution in superposition of
travelling along the same path (case (i) and (ii) in Fig. 1.a, resulting in a
Franson-type oscillation due to the interference of the so-called N00N-state
with N=2 franson_bell_1989 . Such a two-photon state enhances the phase
sensitivity by a factor $N$(2), Heisenberg-limited in precision Dowling2008 .
This results in an interference pattern oscillating at the pump frequency
$\omega_{p}$, instead of the central frequency of the single photons as it
would be the case with classical light. The third term comes from two photons
experiencing different arms (case (iii) and (iv) in Fig. 1.a and the
interference of those two identical single modes (over all observables) at the
second beamsplitter of the interferometer. This is equivalent to the HOM
effect and results in a dip in the coincidence counts, which shape and width
depends on the spectral amplitude $g$ of the photons. As a result, the figure
of merit associated with Eq. 2 is a superimposition of a HOM-dip over Franson-
type interferograms, as shown by Fig. 1.b It must be emphasized that the
precision on the path difference measurement is directly related to the
spectral bandwidth of the photons. The broader they are, the narrower is the
HOM-dip, and therefore better is the precision.
In classical OCT, the intensity $I(\tau)$ at one of the output ports as a
function of the path difference $\tau$ reads:

$I(\tau)=I_{0}[1-V\cdot\cos(\omega_{\rm{c}}\tau)\cdot f(\tau)],$ (3)

where $I_{0}$ is the average intensity, $V$ the experimental visibility,
$\omega_{\rm{c}}$ the central frequency on the interferogram, and $f(\tau)$ an
envelope function which depends on the spectral width and shape. Without
dispersion, $f(\tau)$ reaches its maximum value of 1 for $\tau=0$ since all
frequency components within the SPDC spectrum arrive simultaneously at the
second beamsplitter and interfere. With dispersion, the different frequencies
arrive at different times, resulting in a reduced visibility and in a larger
envelope function $f(\tau)$. Dispersive effects therefore reduce the
achievable precision in determining optical path differences equality in OCT.
On the other hand, HOM-interferometry is insensitive to even order dispersion,
including the dominant term of chromatic dispersion mozzotta2016 ; okano2013 ;
okano2015 ; abouraddy_quantum-optical_2002 . The related experimental
visibility $\alpha$ (Eq. 2) only depends on the indistinguishability of the
two photons (in terms of time, polarization, and spatial mode). This also
induces a robust immunity against propagation losses added by the sample under
test, which is not the case in standard OCT. Hence, the quantum approach is
fully independent on the sample under test characteristics (chromatic
dispersion, losses) and thus permits reliable, practical, and high-precision
measurements with the perspective of addressing actual quantum metrology
scenarios.

Refractive index differences measurement. We aim at measuring $\Delta n$
between two materials constituting the core and the cladding of a VLMA fiber.
These two materials are embedded into a special two-core (one material for
each core) rod-type fiber. The measured interferogram, obtained thanks the
experimental setup represented in Fig. 2, for one core is shown in Fig. 3, for
both OCT and QOCT methods. Similar patterns are obtained for the second core
with an offset $\Delta\tau$. A detailed description of both the experimental
setup and the methodology is provided in the “Methods”section.

Figure 2: a) Experimental setup. A periodically poled lithium niobate
waveguide (PPLN-wg) is pumped at 780 nm (CW laser) to generate entangled
photon pairs. Those are spectrally bandpass filtered (BPF) and sent to a home-
made Mach-Zehnder interferometer. One arm is adjustable and the other contains
the two-core rod-type fiber sample. The two output modes are directed to two
superconducting nanowire single-photon detectors (SNSPD) that are connected to
a time-to-digital converter (TDC) to record the coincidence counts. b)
Measured SPDC spectrum with and without the $90\text{\,}\mathrm{n}\mathrm{m}$
passband filter. Both curves are normalized with respect to their maximum. We
find a FWHM of $44\text{\,}\mathrm{n}\mathrm{m}$. c) Cross-section of the
special two-core fiber. There is an low refractive index barrier preventing
between the two cores to avoid evanescent coupling between them. Each core has
a diameter of $\sim$10\text{\,}\mu\mathrm{m}$$ and they are separated by
$\sim$30\text{\,}\mu\mathrm{m}$$. Core 1 and Core 2 refers to the material
constituting the core and the cladding of an VLMA fiber, respectively. Figure
3: a) Measured photon counts at one output port as a function of the delay
$\tau$. b) Zoom of the central region allowing to resolve the phase fringes at
$\lambda=$1560\text{\,}\mathrm{n}\mathrm{m}$$. The fit permits to infer a
visibility of 0.5. c) Measured coincidence counts between the two output ports
as a function of the delay $\tau$. d) Zoom of the central region allowing to
resolve the Franson-type oscillation at
$\lambda=$780\text{\,}\mathrm{n}\mathrm{m}$$. The fit permits to infer a
visibility of 0.74. All data are measured with an acquisition-time of 1 second
per point.

Prior to the estimation of the precision with both methods, we evaluate the
expected enhancement with HOM-interferometry in comparison with the OCT. For
the OCT-interferogram measurement, we fit the oscillation of the single counts
shown in Fig. 3.a according to Eq. 3. A visibility and a FWHM equal to
$V_{OCT}=50\%$ and $134\text{\,}\mu\mathrm{m}$ are respectively inferred. A
zoom showing both the experimental and fitted OCT-interferograms is shown in
Fig. 3.b. This reduced visibility mainly comes from the propagation losses of
the two-core sample.
Similarly, we fit the experimental HOM-dip shown in Fig. 3.c using Eq. 2. The
corresponding raw visibility and FWHM of the HOM-dip are deduced from the
fitting curve (Fig. 3.d) and are equal to $V_{QOCT}=74\%$ and
$25.8\text{\,}\mu\mathrm{m}$, respectively. For the
$44\text{\,}\mathrm{n}\mathrm{m}$ wide spectrum of the entangled-photon pairs
one would expect a FWHM of $21.7\text{\,}\mu\mathrm{m}$. This enlargement of
19% comes from the third-order dispersion, resulting in a slightly asymmetry
but keeping its integral constant mozzotta2016 ; okano2013 . This broadening
hence limits the expected $V_{QOCT}$ by a similar amount, i.e to 74%. As a
consequence, the visibility can no longer be considered as a criterion for the
indistinguishability between two photons in the presence of odd higher-order
dispersion. In this case, we rather have to compare the theoretical integral
of the HOM-dip, which is obtained by Fourier transform of the Gaussian
bandpass filter, to that of the experimental HOM-dip. This ratio is equal to
94 % and corresponds to the raw equivalent HOM-dip visibility subtracting
third-order dispersion contribution. Furthermore, this non-unit ratio is
explained by a non-perfect mode matching between the two input photons and by
a slightly unbalanced beam-splitter.
The precision is mainly given by the coherence length of the source, that is
inversely proportional to its spectral bandwidth. The HOM-dip width is 5-times
shorter than the envelope function of the classical interferogram. The
chromatic dispersion broadens the classical interferogram, while the HOM-dip
stays essentially unaltered, since the visibility only depends on the
indistinguishability of the two photons. The robust behaviour of the quantum
approach lies at the heart of the enhancement attained through the use of
entangled photons instead of classical light.
The achievable precision of $\Delta n$ measurements mainly depends on the
width of the interferogram but also on its intensity fluctuations. There is a
factor 100 between the coincidence and the single-photon counts. This comes
from the overall losses of about 20 dB from the output of the dual-core fiber
to the output of the interferometer. The major contributions come from the
coupling from free space to fibers and the injection into the two-core fiber.
Since coincidence and single-photon counts follow a Poisson-statistic
HAYAT1999275 , there are $\sim 10$ times more fluctuations due to the shot-
noise in the quantum compared to the classical measurement. Consequently, one
can expect a little bit less then a 5-fold enhancement in precision between
the classical and quantum methods.
All measurements were repeated 70 times to infer the statistical accuracy of
both approaches. We switch every time between the two cores of the two-core
fiber in order to keep the same environmental conditions (essentially the
temperature) during the overall experiment. The results of the statistical
data analysis are shown in Fig. 4. We obtain
$\Delta\tau^{\text{OCT}}=$40.7(12)\text{\,}\mu\mathrm{m}$$, which outperforms
any classical measurement by one order of magnitude Tan:16 ; singh2006 and
$\Delta\tau^{\text{QOCT}}=$41.1(3)\text{\,}\mu\mathrm{m}$$ corresponding to
OCT and QOCT approaches, respectively. This corresponds to a $\Delta n$
precision equal to $\sigma_{\Delta n}^{\text{OCT}}=$24\text{\times}{10}^{-7}$$
and $\sigma_{\Delta n}^{\text{QOCT}}=$6\text{\times}{10}^{-7}$$. This
precision, standing as the highest achieved in terms of $\Delta n$ fits well
our expectation considering the interferogram width and fluctuations
associated with the counts statistic. This enhancement results as a clear
manifestation of the peculiar properties of energy-time entangled photon
pairs, allowing for chromatic dispersion cancellation abouraddy_quantum-
optical_2002 . This work therefore demonstrates that such quantum advantages
are of high interest for characterizing optical samples without having any
prior knowledge on their properties. This becomes even more interesting when
working with realistic or long samples.

Figure 4: Histogram of inferred index difference after 70 repetitions with the
same two-core fiber for both standard (red) and quantum-enhanced (blue)
measurements. Fits to the data assumed a normal distribution.

The origin of the standard deviations in Fig. 4 arises from several reasons.
Due to the switching-method between the two cores in order to measure index
differences in the same conditions, mechanical drifts cause ineluctably small
systematic errors in the optical length. In order to minimize these
fluctuations, the positions of input- and output-lenses are fixed, hence
keeping the same focus points throughout the overall duration of the
experiment. We prefer moving the fiber on both extremities in order to align
them within the focus point of the lenses. That way, the angle alignment error
is minimized. Furthermore, thermal fluctuations play an important role for all
kinds of interferometric methods, especially when long samples are involved.
Note that temperature variations of $\Delta T\sim$0.1\text{\,}\mathrm{K}$$
result in drifts on the order of one-phase fringe in the quantum measurement.
Since we are working in laboratory conditions and both cores are contained
within the same rod, we have verified that our system is more stable than 0.1
K within the recording time. The total duration for the overall measurement
takes 8 hours corresponding to all the necessary data for the histogram in
Fig. 4, which encompasses both the variance due to thermal and mechanical
fluctuations, and also from the estimation method (see the appendix).
A further enhancement in precision is possible, when using for example a
larger SPDC spectrum okano2015 or techniques exploiting a maximum-likehood
estimator, while pre-tuning the interferometer to the position that contains
the maximum information content attosecond . Those demanding methods require
ultra-precise active thermal stabilization, that imposes further technical
challenges. Our method stands as a trade-off between practicability and
precision, that does not require complex implementation of active
stabilization systems while still achieving high-precision and being user-
friendly.

##  III Discussion

In this paper we have implemented an experimental method based on two-photon
interference, referred to QOCT, to measure index difference between two
materials that are embedded within the same fiber. Using HOM-interferometry
and large frequency-entangled photon pairs we achieved unprecedented
precisions up to $\sigma^{\text{QOCT}}=$6\text{\times}{10}^{-7}$$. We compared
the QOCT and OCT approach. Even though we already achieved ultra precise
results using standard approach we still found a 4-fold enhancement in
precision for the QOCT measurement due to both even term dispersion
insensitivity and robustness to the loss in HOM-interferometry. Our precise
results will find use in various fields, notably for special large-mode-area
fibers that are crucial for the development of powerful fiber lasers in the
future.

##  IV Methods

Experimental setup The experimental setup is shown in Fig. 2.a. A continuous-
wave laser operating at $780\text{\,}\mathrm{n}\mathrm{m}$ pumps a type-0
periodically poled lithium niobate waveguide (PPLN-wg) that produces, via
SPDC, degenerated, broadband, energy-time entangled photon pairs. Fig. 2.b
shows the spectral density (corresponding to $|g(\omega)|^{2}$) at the output
of the PPLN-wg. The side peaks are discarded thanks a
$90\text{\,}\mathrm{n}\mathrm{m}$ passband filter, centered at
$1560\text{\,}\mathrm{n}\mathrm{m}$. The filtered spectrum can be fitted by a
Gaussian function, with a full width at half maximum (FWHM) of
$44\text{\,}\mathrm{n}\mathrm{m}$.

The choice for a type-0 phase-matched source is motivated by its natural
broadband SPDC-spectrum, i.e low-coherent photons. The generated photons are
then sent to a Mach-Zehnder interferometer. The reference is adjustable via a
nano-positioning stage, ranging from $0$ to $500\text{\,}\mu\mathrm{m}$ and
having a precision of $20\text{\,}\mathrm{n}\mathrm{m}$. The other arm
contains the sample, being a special two-core fiber. Piezo-actuators allow
fast switching from one core to the other in the tranverse plane to the
waveguide axis. The two cores are made of different materials corresponding to
those constituting the core and the cladding of the VLMA (see Fig. 2.c),
leading two optical paths. The cores are separated by an air-gap to prevent
any coupling between them. Note that the fiber is actually embedded into a
solid rod to avoid systematic errors arising from the fiber-curvature and/or
polarization drifts. A single-mode fiber beamsplitter recombines the signal
from both arms of the interferometer to guarantee the projection onto
identical spatial modes. Moreover, a polarization controller ($\lambda/2$-,
$\lambda/4$-, $\lambda/2$-waveplate) in the free-space arm is added in order
to ensure the indistinguishability of the polarization modes.
The experimental method consists in coupling quantum light in one of the
cores, performing a fast scan ($\sim$ min) of the HOM-dip, and then repeat
this procedure after switching to the other core. Since each core is made of a
different material, the centers of the two interferograms have an offset
$\Delta\tau$ that corresponds exactly to the optical-path difference between
the two cores. Knowing the exact physical length
$L=$50.00(1)\text{\,}\mathrm{c}\mathrm{m}$$ of the sample, one can deduce the
index difference between the two cores, given by $\Delta
n=\frac{\Delta\tau}{L}$. Advantageously, the precision measurement is shifted
from $\Delta n$ to our ability of determining an optical-path difference in th
etime domain $\Delta\tau$ with a high precision. The value of $\Delta n$ is
related to the group-index difference including both material and waveguide
contributions. The latter contribution can easily be evaluated and then
removed thanks to standard simulations in order to infer the index difference
between the two materials 1464-4258-8-11-001 .
Post-data analysis In the perspective of determining the optical-path
difference $\Delta\tau$ between the two cores, a Fourier-transform based
estimator Diddams:96 is implemented based on its property under translation
$\mathcal{FT}_{x}[f(x+t_{0})](\omega)=\mathcal{TF}_{x}[f(x)](\omega)\cdot\text{e}^{\text{i}\omega
t_{0}}$. As described in details in the supplementary information section, the
offset $\Delta\tau$, that corresponds to the delay between the two HOM-dips,
is deduced from a linear fit of the spectral phase of the interferogram at low
frequencies (corresponding to the HOM-dip). In order to fairly compare the
QOCT and OCT methods, the spectral bandwidth of both sources has to be
identical. We simultaneously exploit coincidence counts and single-photon at
one of the output ports for the QOCT and OCT approaches, respectively.
Furthermore, as for the quantum approach, we apply a similar Fourier-transform
based estimator, now fitting the phase around the central frequency of single
photons. A detailed description of both quantum and classical estimation
methods can be found in the appendix.
In order to fairly estimate the precision, we switch 70 times between the two
cores of the same sample, estimating each time $\Delta\tau$ via quantum and
classical methods to infer the statistical accuracy of both approaches.

## Acknowledgment

This work has been conducted within the framework of the project OPTIMAL
granted by the European Union by means of the Fond Européen de développement
regional (FEDER). The authors also acknowledge financial support from the
Agence Nationale de la Recherche (ANR) through the projects METROPOLIS, the
CNRS through its program “Mission interdisciplinairité” under project labeled
FINDER, and the French government through its Investments for the Future
programme under the Université Côte d’Azur UCA-JEDI project (Quantum@UCA)
managed by the ANR (ANR-15-IDEX-01).
The authors also acknowledge technical support from IDQ. We also thank Elie
Gouzien and Yann Bouret for assistance with data post-processing.

## Author information

M.R., F.M., D.A. and R.C. performed the experiments. R.D., B.L., P.R. and R.J.
were in charge of designing and fabricating the two-core fiber. M.R., F.K.
L.L. and S.T. designed the experiment. M.R., L.L. and S.T wrote the paper with
inputs from P.R. and R.J.

## Competing interests

The authors declare that there are no competing interests.

## Data Availability

Data are available from the authors on reasonable request.

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## Supplementary information

###  .1  Fourier-based estimators for the optical path difference between the
two cores

Since we measure each interferogram as a function of the motor-position
$p=\tau+p_{0}$ and not directly as a function of the optical path length
difference $\tau$ between the two arms of the interferometer, we have to find
$p_{0}$, the motor-position that corresponds to the optical-path equality with
the highest precision possible. Therefore we use an estimator that is taking
advantage of the property of the Fourier transform under translation. Let’s
suppose a function $u(\tau)$ and its Fourier transform with respect to $\tau$
is $\mathcal{FT}_{\tau}[u(\tau)]=\hat{u}(f)$, a function of the frequency $f$.
The Fourier transform of the translated (time-shifted) function
$v(\tau)=u(\tau-\tau_{0})$ can then be written as
$\hat{v}(f)=\hat{u}(f)\cdot\text{e}^{-2\pi\text{i}f\tau_{0}}$ where an extra
phase term is appearing that is evolving linear with the frequency $f$ with
slope of $-2\pi\tau_{0}$. For an even function $u(\tau)$ , $\hat{u}(f)$ is a
real function and the only phase term of
$\hat{v}(f)=\hat{u}(f)\cdot\text{e}^{-2\pi\text{i}f\tau_{0}}$ is $-2\pi
f\tau_{0}$. Linear-fitting the phase of $\hat{v}(f)$, one thus can easily
deduce $\tau_{0}$.

Due to small third-order (and higher-odd order) dispersive effects, the
interferograms (for quantum and classical measurement) are not perfectly
symmetric mozzotta2016 ; okano2013 ; Diddams:96 . The Fourier-transform of an
unsymmetrical function $u(\tau)$ is complex
$\hat{u}(f)=\left|\hat{u}(f)\right|\cdot\text{e}^{\text{i}\psi(f)}$, with a
phase term $\psi(f)$ that is varying with the frequency $f$. The phase of the
Fourier transform of the translated function
$\hat{v}(f)=\left|\hat{u}(f)\right|\cdot\text{e}^{\text{i}\psi(f)}\cdot\text{e}^{-2\pi\text{i}f\tau_{0}}$
is thus a superposition of $\psi(f)$ and the linear term $-2\pi f\tau_{0}$. A
simple linear-fit cannot determine $\tau_{0}$.

Fortunately we are mainly interested in the difference $\Delta\tau$ of the
optical path between the two cores and not the precise motor-position, that
corresponds to the path equality for each core. Calculating the Fourier
transform of the interferogram for each core, its phase for core 1 (core 2)
takes the form $\Psi^{(1)}=\psi^{(1)}_{\text{asym}}(f)-2\pi f\tau^{1}_{0}$
($\Psi^{(2)}=\psi^{(2)}_{\text{asym}}(f)-2\pi f\tau^{2}_{0}$), where
$\psi^{(1)}_{\text{asym}}(f)$ ( $\psi^{(2)}_{\text{asym}}(f)$) stands as the
phase terms resulting form the unsymmetrical interferogram for core 1 (core
2). We suppose that the interferograms for the two cores are only translated
in respect to each other, but the form of the dip/ envelope is the same,
resulting in
$\psi^{(1)}_{\text{asym}}(f))=\psi^{(2)}_{\text{asym}}(f)=\psi_{\text{asym}}(f))$.
This is a valid approximation since we use the same spectra. Calculating the
difference $\Psi^{(2)}-\Psi^{(1)}=\psi_{\text{asym}}(f)-2\pi
f\tau^{2}_{0}-(\psi_{\text{asym}}(f)-2\pi f\tau^{1}_{0})=2\pi
f(\tau_{1}-\tau_{2})$, the phase terms resulting form the unsymmetrical
interferograms cancels out and with a linear-fit we have directly access to
$|\tau_{1}-\tau_{2}|=\Delta\tau$.

Since the interferogram is a discrete set of data-points, that are non
uniformly distributed due to the variance of
$\sim$20\text{\,}\mathrm{n}\mathrm{m}$$ in the motor-position, we
mathematically calculate its Fourier-transform with a nonuniform discrete
Fourier transform (NDFT) algorithm (Fig. 6 and 5).

We perform prior simulations of the expected interferogram according to (2),
supposing Gaussian distributed fluctuations within the motor-position and a
Poisson distribution for the photon counts/coincidences in order to find the
optimal stepsize and total range of the HOM-dip scan. This leads to a trade-
off between the total number of points and the resulting precision of the
estimator, since the faster we can perform the scan the less we are sensible
to temperature fluctuations during consecutive measurements. We can reduce the
total number of points to 500, before the precision degrades. This gives us a
total range of $120\text{\,}\mu\mathrm{m}$, centered around the prior
estimated optical path equality, with a step-size of
$0.24\text{\,}\mu\mathrm{m}$. With an integration time of
$0.5\text{\,}\mathrm{s}$ per point this results in a measurement time of $\sim
5$ minutes per core.

Figure 5: a) Measured classical interferogram with a step-size of
$0.24\text{\,}\mu\mathrm{m}$ and an integration time of
$0.5\text{\,}\mathrm{s}$. b) amplitude and phase of the Fourier transform of
the classical interferogram. c) amplitude and phase of the Fourier transform,
zoomed-in around side peak at $\frac{1}{$1560\text{\,}\mathrm{n}\mathrm{m}$}$.
d) Differential phase $\Psi^{(2)}-\Psi^{(1)}$ of the Fourier transform for two
different measurements (each in a different core) with a linear fit to find
$\Delta\tau=$40.9(4)\text{\,}\mu\mathrm{m}$$. Figure 6: a) Measured quantum
interferogram with a step-size of $0.24\text{\,}\mu\mathrm{m}$ and an
integration time of $0.5\text{\,}\mathrm{s}$. b) amplitude and phase of the
Fourier transform of the quantum interferogram. c) amplitude and phase of the
Fourier transform, zoomed-in around central peak, corresponding to the HOM-
dip. d) Differential phase $\Psi^{(2)}-\Psi^{(1)}$ of the Fourier transform
for two different measurements (each in a different core) with a linear fit to
find $\Delta\tau=$41.3(3)\text{\,}\mu\mathrm{m}$$.

###  .2 Estimator for the HOM and classical white-light interferometry

A measured interferogram and its calculated Fourier transform can be seen in
Fig. 6 and 5, corresponding to the quantum and classical regime respectively.
Before calculating the Fourier transform, we subtract the average outside the
dip bracewell_fourier_2000 . In the quantum approach, we therefore filter the
Fourier transform around the low-frequencies that correspond to the HOM-dip.
On the other hand, in the classical regime, the envelop function is modulated
by the single-photon frequency, we thus filter the Fourier transform around
the side peak at $\frac{1}{$1560\text{\,}\mathrm{n}\mathrm{m}$}$ in order to
estimate $\Delta\tau$. Since the interferogram is a real function, its Fourier
transform is symmetric, we therefore do not gain any more information by using
the two side-peaks at $\frac{1}{$1560\text{\,}\mathrm{n}\mathrm{m}$}$ and
$-\frac{1}{$1560\text{\,}\mathrm{n}\mathrm{m}$}$.
In both cases, we find a good agreement between the differential phase
$\Psi^{(2)}-\Psi^{(1)}$ and the linear fit (Fig. 6 (c) and 5 (c)), which is as
well promising for the precision of the classical estimator.