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# Entanglement-induced collective many-body interference

Tommaso Faleo<EMAIL_ADDRESS>Institut für Experimentalphysik,
Universität Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria  Eric Brunner
Current address: Quantinuum, Partnership House, Carlisle Place, London SW1P
1BX, United Kingdom  Physikalisches Institut, Albert-Ludwigs-Universität
Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany  Jonathan W. Webb
Institute of Photonics and Quantum Sciences, School of Engineering and
Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK  Alexander
Pickston  Institute of Photonics and Quantum Sciences, School of Engineering
and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK  Joseph
Ho  Institute of Photonics and Quantum Sciences, School of Engineering and
Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK  Gregor
Weihs  Institut für Experimentalphysik, Universität Innsbruck, Technikerstr.
25, 6020 Innsbruck, Austria  Andreas Buchleitner  Physikalisches Institut,
Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg,
Germany  EUCOR Centre for Quantum Science and Quantum Computing, Albert-
Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
Christoph Dittel  Physikalisches Institut, Albert-Ludwigs-Universität
Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany  EUCOR Centre for
Quantum Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg,
Hermann-Herder-Straße 3, 79104 Freiburg, Germany  Freiburg Institute for
Advanced Studies, Albert-Ludwigs-Universität Freiburg, Albertstraße 19, 79104
Freiburg, Germany  Gabriel Dufour  Physikalisches Institut, Albert-Ludwigs-
Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany
Alessandro Fedrizzi  Institute of Photonics and Quantum Sciences, School of
Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14
4AS, UK  Robert Keil<EMAIL_ADDRESS>Institut für Experimentalphysik,
Universität Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria

(August 28, 2024)

###### Abstract

Entanglement and interference are both hallmark effects of quantum physics.
Particularly rich dynamics arise when multiple (at least partially)
indistinguishable particles are subjected to either of these phenomena. By
combining both entanglement and many-particle interference, we propose an
interferometric setting through which $N$-particle interference can be
observed, while any interference of lower orders is strictly suppressed. We
experimentally demonstrate this effect in a four-photon interferometer, where
the interference is nonlocal, in principle, as only pairs of photons interfere
at two separate and independent beam splitters. A joint detection of all four
photons identifies a high-visibility interference pattern varying as a
function of their collective four-particle phase, a genuine four-body
property.

††preprint: APS/123-QED

## introduction

Entanglement is arguably one of the most fascinating and powerful phenomena
arising in quantum physics. If two or more quantum objects are entangled they
can no longer be described as independent entities, instead, their properties
are interlinked through their associated degrees of freedom (dof), regardless
of how these dof are measured and how far apart the objects are from each
other [1]. This effect has been famously and unambiguously demonstrated in
Bell inequality tests [2, 3] and it enables a rich variety of applications
within the fields of quantum communication [4, 5, 6], quantum computation [7,
8, 9] and quantum simulation [10, 11].

A similarly fundamental effect is given by two-particle (2P) interference, as
first observed by Hong, Ou, and Mandel (HOM) via the absence of photon pair
coincidences at the output of a balanced beam splitter [12]. HOM interference
has emerged as one of the most widely employed quantum effects in the field of
photonics [13] and it requires indistinguishability of the involved particles.
Specifically, the particles should not be identifiable by their internal
states — the states of those dof which do not participate in the dynamics —
thus forbidding the retrieval of 2P which-way information [14, 15]. The HOM
experiment represents a starting point for the exploration of multi-particle
interference phenomena, and its generalisation to larger-scale $N$-particle
($N$P) systems [16, 17, 18] finds applications in fundamental tests of quantum
mechanics [19] and in the advancement of quantum technologies [20, 21].
However, $N$P interference for $N>2$ is inherently more complex than the HOM
effect. The rich spectrum of multi-particle interference terms stemming from
the various particle-exchange processes gives rise to nontrivial behaviours,
especially when partial distinguishability among the particles is introduced
[18, 22, 23]. Therefore, a substantial challenge is to discern genuine $N$P
interference, which has, in particular, been tackled by examining specific
signatures of this interference [24, 25, 26, 27, 28] and identifying instances
of totally destructive $N$P interference which generalise the HOM scenario
[17, 29, 30, 31, 32]. Recent experimental observations [33, 34] have,
furthermore, demonstrated the presence of genuine multi-particle interference
witnessed by its dependence on a collective geometric phase that is determined
by the overall internal quantum state of all of the particles. The emergence
of this collective $N$P phase can be attributed to carefully engineered
internal states of the particles as well as the specific structure of the
interference terms, involving cyclic permutations of all $N$ particles [35].

Here, we propose and experimentally demonstrate a novel interference effect to
realize genuine $N$P interference by combining the concepts of entanglement
and many-body interference. In particular, we show that by using entanglement
between a subset of particles at the input of separate interferometers, full
contrast interference fringes emerge for the $N$P correlator as a function of
a collective $N$P phase, while the signal of all lower-order correlators
remain independent of the involved phases. In contrast to the realisation of
$N$P phases of previous approaches [33, 35, 34], which necessitate the
engineering of an $N$-dimensional internal Hilbert space and the mixing of all
particles in an $N$-cyclic interferometer, here only a two-dimensional
internal Hilbert space and pairwise exchanges of particles at separate beam
splitters are needed to induce collective $N$P interference. By virtue of its
design, the entanglement-induced collective $N$P interference effect exhibits
full interference contrast for arbitrarily large numbers of particles.

The genuine $N$P interference effect presented here is by construction
independent of the spatial separation between the beam splitters, such that
the resulting collective many-body interference can be realised in a nonlocal
fashion. Multi-particle interference in disjoint interferometers is otherwise
only known from Franson-interference [36, 37], which relies on maximally
entangled states in the energy-time dof intrinsically linked to the photon
generation process. In contrast, the entanglement-induced collective phase
shown in this work can be imprinted on any bosonic state with a two-
dimensional internal Hilbert space.

##  I theory

Figure 1: Many-particle interference and collective phases. (a) In a fully
connected interferometer with single-particle unitary matrix $U$ (here 4x4),
multiple exchange processes among combinations of input-output channels
contribute to specific output events. The reduced 2P state associated with the
second (yellow) and fourth (red) particles contributes to the two-point
correlator $\braket{\hat{N}_{3}\hat{N}_{4}}$ via the $2$P transitions
indicated by the solid and dashed lines. The interference of these two $2$P
paths is encoded in the $2$P coherence at the bottom of the panel, which is
associated with the permutation $\pi=(2\,4)$ obtained by taking one $2$P path
in the forward direction followed by the other one in the backward direction
[38]. The matrix elements of $U$ weight these contributions, as well as the
contributions from all other 2P sets. Similar considerations apply to all
correlators from combinations of different output modes. (b) The four-particle
(4P) coherence corresponding to the depicted 4-cyclic permutation leads to a
genuine 4P interference that depends on a collective phase $\phi$. This
collective phase is set by the summation of phases $\phi_{ij}$ (see main text)
resulting from the overlaps of the particles’ internal states along a “circle-
dance” graph representative of the permutation process [34, 35], as depicted
at the bottom of the panel. (c) A genuine entanglement-induced 4P interference
can be achieved by interfering entangled particles (blue envelope) with
particles in separable states in two independent and separate beam splitters.
In this process, the entanglement induces a collective 4P phase term $\phi$,
set by the internal states of all particles, through the two 2P permutations
of $\pi=(1\,2)(3\,4)$. The collective phase $\phi$ only affects the four-point
correlator and introduces full contrast interference fringes:
$\braket{\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}}\propto\cos^{2}\left(\phi/2\right)$
(see main text).

Many-body interference and collective phases
In a general interference scenario among $N$ particles, as depicted in Fig.
1(a) for $N=4$ unentangled particles, each input mode $m_{i}=1\dots N$ of the
multiport interferometer is occupied by a single particle, which carries some
internal dof (illustrated by the particle’s colour in Fig. 1). Interference
effects across the multiport manifest through correlations of the particle
number at the output of the interferometer. We specifically consider
correlations between the occupations of $k\leq N$ distinct modes
$p_{1},\dots,p_{k}$. Denoting by $U$ the single-particle unitary matrix
describing the transformation between input and output modes (henceforth
termed external modes) implemented by the interferometer, the expectation
value of such a $k$-point correlator is given by

$\braket{\hat{N}_{p_{1}}\dots\hat{N}_{p_{k}}}=\sum_{\bm{m},\bm{n}}\prod_{i=1}^{k}U_{p_{i}m_{i}}U_{p_{i}n_{i}}^{\ast}\braket{\bm{m}}{\rho_{\text{ext}}^{(k)}}{\bm{n}}\,,$
(1)

where the external number operator $\hat{N}_{p}$ counts the number of
particles in output mode $p$ irrespective of their internal state. In terms of
the operator $\hat{a}_{p\alpha}$ annihilating a particle with internal state
$\ket{\alpha}$ in external mode $p$, the external number operator reads
$\hat{N}_{p}=\sum_{\alpha}\hat{a}_{p\alpha}^{{\dagger}}\hat{a}_{p\alpha}$,
where the sum runs over a basis of the internal Hilbert space.

Since it involves products of $k$ creation and $k$ annihilation operators, the
$k$-point correlator $\hat{N}_{p_{1}}\dots\hat{N}_{p_{k}}$ is an example of a
$k$-particle ($k$P) observable [39, 40, 27]. As a consequence, for $k<N$, it
only probes the reduced state $\rho_{\text{ext}}^{(k)}$, obtained by tracing
out the internal dof and $(N-k)$ particles from the input state [41, 42, 27].
Specifically, the matrix elements
$\braket{\bm{m}}{\rho_{\text{ext}}^{(k)}}{\bm{n}}$ appear in Eq. (1), with
$\ket{\bm{m}}=\ket{m_{1}}\otimes\dots\otimes\ket{m_{k}}$ the tensor product of
the external single-particle basis states $\ket{m_{i}}$, corresponding to the
particle occupying input port $m_{i}$ of the interferometer, and analogously
for $\ket{\bm{n}}$. The nonzero matrix elements
$\braket{\bm{m}}{\rho_{\text{ext}}^{(k)}}{\bm{n}}$ are indexed by tuples
$\bm{m}$ and $\bm{n}$ that are connected by a permutation
$\pi\in\mathrm{S}_{k}$ in the symmetric group $\mathrm{S}_{k}$ of their $k$
entries: $\bm{n}=\pi(\bm{m})=(m_{\pi^{-1}(1)},\dots,m_{\pi^{-1}(k)})$. The
off-diagonal matrix elements ($\bm{m}\neq\bm{n}$) are called $k$P coherences.

In the traditional scenario where each particle in mode $m_{i}$ is associated
with a particular internal quantum state $\ket{\xi_{m_{i}}}$, corresponding to
an input product state
$\ket{\Psi}=\hat{a}_{m_{1},\xi_{m_{1}}}^{{\dagger}}\dots\hat{a}_{m_{N},\xi_{m_{N}}}^{{\dagger}}\ket{0}$,
all matrix elements are set, up to a multiplicative constant, by the overlaps
between internal states of particles in the specific $k$ modes:

$\braket{\bm{m}}{\rho_{\text{ext}}^{(k)}}{\bm{n}}\propto\prod_{i=1}^{k}\braket{\xi_{m_{i}}}{\xi_{m_{\pi^{-1}(i)}}}.$
(2)

Therefore, these overlaps, i.e., the particles’ mutual
(in)distinguishabilities, determine all matrix elements. In general, the
overlaps are complex numbers $\braket{\xi_{i}}{\xi_{j}}=r_{ij}e^{i\phi_{ij}}$,
with real amplitude $r_{ij}$ and phase $\phi_{ij}$. In the two-point
correlator $\braket{\hat{N}_{3}\hat{N}_{4}}$ shown in Fig. 1(a), for example,
the indicated transition processes lead to HOM-type interference involving the
diagonal term
$\braket{2,4}{\,\rho_{\text{ext}}^{(2)}\,}{2,4}\propto\braket{\xi_{2}}{\xi_{2}}\braket{\xi_{4}}{\xi_{4}}=1$
and the $2$P coherence term
$\braket{2,4}{\,\rho_{\text{ext}}^{(2)}\,}{4,2}\propto\braket{\xi_{2}}{\xi_{4}}\braket{\xi_{4}}{\xi_{2}}=r_{2,4}^{2}$,
corresponding to the permutations (in cycle notation [43]) $\pi=(2)(4)$ and
$\pi=(2\,4)$, respectively (see bottom of Fig. 1(a)). As in HOM (i.e. 2P)
interference, no phase dependence arises and only the real amplitude
$r_{2,4}^{2}$, which expresses the particles’ pairwise indistinguishability,
determines $\braket{\hat{N}_{3}\hat{N}_{4}}$ [12].

In more complex multi-particle exchanges, the engineering of internal states
$\ket{\xi_{i}}$ via an $N$-dimensional Hilbert space allows to witness a phase
dependence that is set by the specific permutation $\pi$ and the overlaps
among particle pairs [33, 34]. Notably, the genuine $N$P interference
introduced in [35] is encoded in a $N$P coherence
$\braket{\bm{m}}{\rho_{\text{ext}}^{(N)}}{\pi(\bm{m})}$ associated with a
$N$-cyclic permutation $\pi=(1\,2\,3\dots N)$, as shown in Fig. 1(b), which
depends on the “circle-dance” collective $N$P phase
$\phi=\phi_{1,2}+\phi_{2,3}+\dots+\phi_{N,1}$. Engineering zero overlaps of
the internal states of non-neighbouring particles in the permutation —
particles in disconnected vertices of the permutation graph — ensures that no
other phase dependence appears in the signal. This, however, requires in the
traditional scenario the control over an (at least) $N$-dimensional internal
Hilbert space

Scheme for entanglement-induced collective interference
We introduce here a novel scheme, depicted for $N=4$ in Fig. 1(c), to witness
a collective $N$P phase using entanglement. In this scheme, the interferometer
consists of $N/2$ separate beam splitters, where the first one mixes modes 1
and 2, the second one modes 3 and 4, and so on. Pairwise 2P entanglement in
one of the internal dof (here polarisation) between particles in mode 2 and 3,
4 and 5, …, $N-2$ and $N-1$ can then be used to induce collective interference
across the separate beam splitters. For simplicity, we now restrict our
discussion to $N=4$ (as realised in our experiment), while the general case of
$N$ particles follows analogously. As input, we consider the combination of a
polarisation-entangled photon pair in modes 2 and 3 and an unentangled photon
pair in the remaining modes 1 and 4 (see Fig. 1(c)). This is described by the
four-particle ($4$P) state

$\ket{\psi}=\frac{1}{\sqrt{2}}\hat{a}_{1,\mathrm{S}}^{{\dagger}}(\varphi)(\hat{a}_{2,H}^{{\dagger}}\hat{a}_{3,V}^{{\dagger}}+e^{-i\chi}\hat{a}_{2,V}^{{\dagger}}\hat{a}_{3,H}^{{\dagger}})\hat{a}_{4,\mathrm{S}}^{{\dagger}}(\theta)\ket{0},$
(3)

where $\hat{a}^{\dagger}_{i,H(V)}$ creates a photon in external mode $i$ with
horizontal (vertical) polarisation internal state, and

$\hat{a}^{\dagger}_{i,\mathrm{S}}(\varphi)=\frac{1}{\sqrt{2}}(\hat{a}^{\dagger}_{i,H}+e^{i\varphi}\hat{a}^{\dagger}_{i,V})$
(4)

sets the polarisation to a balanced superposition in the H/V-basis with phase
$\varphi$, and analogously for $\hat{a}^{\dagger}_{i,S}(\theta)$. Note that
while Eq. (2) no longer holds for entangled input states, the expression of
the $k$-point correlator in Eq. (1) remains valid independently of the exact
state associated with $\rho_{\mathrm{ext}}^{(k)}$.

Due to the specific topology of the interferometer in Fig. 1(c) (defining the
single-particle unitary $U$), not all four modes mix, such that many terms in
Eq. (1) vanish. The only $k$P coherences that contribute are associated with
permutations $\pi=(1\,2)$, which exchanges modes 1 and 2, $\pi=(3\,4)$, which
exchanges modes 3 and 4, and their product $\pi=(1\,2)(3\,4)$. This
factorisation reflects the structure of the interferometer, with two separate
beam splitters, which can, in principle, be arbitrarily far apart. If we
assume the input to be given only by the first summation term in Eq. (3), the
contributions arising from particle exchanges according to the above three
permutations are given by $|\braket{\varphi}{H}|^{2}$,
$|\braket{V}{\theta}|^{2}$ and
$|\braket{\varphi}{H}|^{2}|\braket{V}{\theta}|^{2}$, respectively (see also
Eqs. (12) and (LABEL:eq:app1_helper2) in App. A for a more detailed picture) —
i.e., by the modulus square of the inner products of the exchanged particles’
internal states. These are independent of $\varphi,\theta$ and $\chi$.
Analogously, if only the second term of Eq. (3) was considered, no phase
contributions would be present in the four-point correlator (1). The phase
dependence results from the superposition of both $4$P states. The additional
cross terms are the projection of the first term in (3) onto the second term
exchanged according to $\pi=(1\,2)(3\,4)$ — i.e. the matrix element
$\braket{1,2,3,4}{\,\rho^{(4)}_{\text{ext}}\,}{2,1,4,3}$ in Eq. (1) — and its
complex conjugate (c.c.). This matrix element is proportional to
$e^{-i\chi}\braket{\varphi}{V}\braket{V}{\theta}\braket{\theta}{H}\braket{H}{\varphi}\propto
e^{-i(\chi+\varphi-\theta)}$. Note that the four bra-vectors in the scalar
product are given by a cyclic permutation of the four ket-vectors. Hence, the
particle exchange $\pi=(1\,2)(3\,4)$ in combination with the exchange of the
roles of modes $2$ and $3$ in the two terms of the entangled state
$\ket{\psi}$ (Eq. (3), see also Fig. 1, right panel) can be viewed as an
effective four-cycle acted on the particles internal states (see also Eq. (2))
which captures the dependence on the complex phases $\varphi,\chi,\theta$ (see
also App. A for a more detailed discussion). Overall we find, for $k=4$,

$\braket{\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}}=\frac{1}{8}\cos^{2}\left(\frac{\phi}{2}\right),$
(5)

with collective 4P phase $\phi=\chi+\varphi-\theta$. The full derivation is
given in App. A. Thus, the 4P correlator oscillates with full interference
contrast. One can show that this observation also holds for arbitrary particle
numbers $N>4$, without loss of contrast, by adding beam splitters in parallel
and entangled pairs between the unentangled particles, as discussed in the
beginning of this section.

In contrast, for $k<N$, the structure of the input state in Eq. (3) and the
specific interferometer topology ensure that all probed matrix elements
$\braket{\bm{m}}{\rho_{\text{ext}}^{(k)}}{\bm{n}}$ in the expectation value
(1) are independent of the phases $\varphi,\,\chi,\,\theta$. The idea is as
follows: the three-point correlator only depends on the matrix elements of
$\rho_{\text{ext}}^{(3)}$, see Eq. (1). Note that the considered
interferometer does not mix the modes $1,2$ with the modes $3,4$ and that the
only non-vanishing $k$P coherences
$\braket{\bm{m}}{\rho_{\text{ext}}^{(k)}}{\bm{n}}$ are indexed by $k$-tuples
$\bm{m}$ and $\bm{n}$ that contain the same modes, only differing by a
relative permutation $\bm{n}=\pi(\bm{m})$, as discussed after Eq. (1).
Therefore, the only contributing $3$P coherences in (1) for $k=3$ are — up to
complex conjugation and permuting both index tuples by the same permutation
111Note that the reduced state $\rho_{\text{ext}}^{(k)}$ commutes with the
action of the symmetric group,
$\pi\rho_{\text{ext}}^{(k)}=\rho_{\text{ext}}^{(k)}\pi$. Therefore, its matrix
elements are invariant under simultaneous permutations of both index-tuples
$\bm{m},\bm{n}$:
$\braket{\bm{m}}{\rho_{\text{ext}}^{(k)}}{\bm{n}}=\braket{\pi(\bm{m})}{\rho_{\text{ext}}^{(k)}}{\pi(\bm{n})}$
for all $\pi\in\mathrm{S}_{k}$ [41, 27]. —
$\braket{123}{\,\tilde{\rho}\,}{213}$, $\braket{124}{\,\tilde{\rho}\,}{214}$,
$\braket{134}{\,\tilde{\rho}\,}{143}$ and
$\braket{234}{\,\tilde{\rho}\,}{243}$. In all these terms only two particles
are exchanged, similar to the $2$P HOM interference case (see Fig. 1(a)),
which only involves squared moduli of the internal states’ overlaps. For
example,

$\braket{123}{\,\rho_{\text{ext}}^{(3)}\,}{213}\propto|\braket{\varphi}{H}|^{2}+|\braket{\varphi}{V}|^{2}\,.$
(6)

The first contribution comes from the first summation term of the initial
state, where the particle in mode $2$ has horizontal polarisation, such that
the exchange $\pi=(12)$ leads to $|\braket{\varphi}{H}|^{2}$. Correspondingly,
the second contribution stems from the second term, where the particle in mode
$2$ has vertical polarisation. Note that the cross contribution (which encodes
the complex phase dependence for the four-point correlator as discussed before
Eq. (5)) is zero since the particle in mode $3$ has orthogonal polarisation
states in both terms. Analogously, all other contributing matrix elements in
Eq. (1) are independent of the phases $\varphi,\theta,\chi$. The same holds
for $k=2$. As a consequence, the $N$P collective phase remains invisible in
all correlation orders lower than $N$.

Photonic experiments typically employ threshold-detectors and record
coincidence events involving $k$ detectors, referred to as $k$-fold
coincidences, where at least one photon is detected in each of the output
ports $p_{1},\dots,p_{k}$. In contrast, measurements of $k$-point correlators,
as in Eq. (1), are $k$P observables that require resolving the photon number
at each detector, which is only possible with photon-number-resolving
detectors [45]. In an $N$-particle scenario, the $N$-fold coincidence rate is
equivalent to the $N$-point correlator, as necessarily only one photon
impinges each detector. However, $k$-fold coincidence events with $k<N$ do not
constitute true $k$P observables (as multiple photons may impinge on a single
detector unnoticed) and are, in general, garnished by contributions from $l$P
terms, with $k<l\leq N$, thus exhibiting weak traces of the $N$P collective
phase $\phi$ (see App. B).

##  II Experiment

Figure 2: Experimental setup. (a) A $\qty{}{\pico}$-pulsed laser clocked at
$\qty{80}{\mega}$ is split via a balanced beam splitter (BS) to pump two SPDC
sources based on apodised KTP crystals (aKTP) embedded in Sagnac
interferometers (Source 1 and Source 2). Half-wave plates (HWP) and Glan-
Taylor polarisers (GT) are used to adjust the input power and set the pump
polarisation to obtain entangled photon pairs (Source 1), or separable photon
pairs (Source 2). Down-converted photons are separated from the laser light
via polarising beam splitters (PBSs), dichroic mirrors (DMs), long-pass
filters (LPFs), and collected through single-mode fibres. (b) Photons are
adjusted in polarisation via combinations of HWPs and quarter-wave plates
(QWP), or via fibre polarisation controllers (FPC), and guided to two
independent balanced beam splitters. At the output, photons are detected by
multiplexing each output channel (ch.A-ch.D) with two SNSPDs at the output
ports of a PBS, as shown in the inset. The single-photon detection events are
analysed at a field programmable gate array (FPGA) logic unit, and post-
processed to determine every two-fold, three-fold, and four-fold coincidence
event.

Experimental setup
We experimentally tested the collective-phase interference as predicted by Eq.
(5) by implementing the scheme in Fig. 1(c) and the input state in Eq. (3).
Figure 2 shows the experimental setup. We produce the four-photon input states
via two high-brightness, high-purity (spectral purity $\geq 98\%$), and high-
fidelity telecom (emission wavelength $1550\text{\,}\mathrm{n}\mathrm{m}$)
spontaneous parametric down-conversion (SPDC) sources with apodised crystals
[46, 47], as schematically represented in Fig. 2(a). In the employed Sagnac
interferometer configuration [48, 49, 50], we obtained experimental fidelities
of $98.56(2)\%$ and $98.33(2)\%$ when adjusting the phase $\chi$ of the
entangled photon pair in modes 2 and 3 of Eq. (3) to $\chi=0,\,\pi$
(representing the well-known Bell states $\ket{\psi^{+}}$ and
$\ket{\psi^{-}}$), respectively, as shown in App. C.

Before interfering the photons as shown in Fig. 2(b), we prepare the
polarisation states of the photons in modes 1 and 4 according to Eq. (4),
through combinations of quarter- and half-wave plates, and we set the phase
$\chi$ of the entangled pair via fibre polarisation controllers. More
precisely, we varied the collective phase of the input state by changing the
phase $\varphi$ of the photons in mode 1 and the phase $\chi$ of the entangled
pair to obtain the Bell state $\ket{\psi^{+}}$ or $\ket{\psi^{-}}$, but fixing
the phase $\theta=0$, corresponding to the setting of a diagonal polarisation
state $\ket{D}_{4}=\hat{a}^{\dagger}_{4,S}(0)\ket{0}$ of the photons in mode
4.

After interfering, the photons collected at the output ports of the beam
splitters are detected via superconducting nanowire single-photon detectors
(SNSPDs) with quantum efficiencies of $\geq 80\%$. To account for the
polarisation-dependent SNSPD detection efficiency, we project to the H/V-basis
and multiplex the outputs to eight SNSPD channels. The post-processing of the
data retrieves the aggregated single counts of each output channel and all
possible combinations of $k$-fold coincidences among these channels, with
$k=2,\,3,\,4$. The multiplexed eight-detector scheme has two additional
advantages: first, since SPDC sources are affected by multi-pair emission
contributions (more than two photon pairs produced), this scheme makes it
possible to reject coincidences of more than four photons, thus partially
cleaning the four-fold statistics from these contributions. Second, the
presence of additional beam splitters and (non-number-resolving) detectors
helps to provide a pseudo-photon-number resolution [51], which is required to
perform correlation measurements as in Eq. (1). Although a single multiplexing
layer is insufficient for a complete correlation measurement, it already
significantly reduces the visibility of the collective phase dependence of any
$k(<4)$-fold coincidence events (see example in App. B).

Experimental results
The specific arrangement of SPDC sources in our experimental scheme causes
double-pair emissions from an individual source to propagate through the setup
with the same probability as the desired two-source emissions. This results in
a four-fold coincidence background from each source with the same order of
magnitude as the coincidences from two-source emissions, thus reducing the
visibility compared to the expected theoretical prediction of Eq. (5).
However, our setup has the advantage of consisting of completely independent
beam splitters, such that photons from single-source double-pair emissions do
not interfere. The double-pair background is therefore independent of the
collective phase by construction, and can be subtracted from the entanglement-
induced four-photon collective interference phase signal. Specifically, by
blocking one of the two sources at a time, the four-photon background from the
other source can be measured independent of the signal and subtracted from the
raw data obtained in the main experiment [34].

We recorded photon events for combinations of the two phases $\chi=0,\,\pi$
and 31 settings of the phase $\varphi\in(-\pi/2,3\pi/2)$, thus testing the
collective phase within the $3\pi$ phase range $\phi\in(-\pi/2,5\pi/2)$. We
measured each setting of $\varphi$, at fixed $\chi$, over $\qty{60}{}$ and we
averaged the measurements over 20 repetitions of this phase scan. We
subsequently performed the measurement of the background for each photon
source by following the same procedure, but using a reduced set of 8 phase
scans per source to obtain a good trade-off between acquiring sufficient data
statistics and avoiding long-term drifts affecting the measurements.

Figure 3: Results of four-fold coincidence counts for entanglement-induced
collective $4$P interference. (a)(b) Four-fold background coincidences from
multi-pair emissions of individual sources. (c)(d) Background-subtracted four-
fold coincidences data (blue dots) fitted with a cosine function (red curve).
The red-shaded region shows the fit prediction interval at a confidence level
of one standard deviation. The graphs include the results of the multi-photon
interference simulations (black dashed curve). The visibility of the fit is
$69.2(2.5)\%$ and $85.8(4.5)\%$ for panels (c) and (d), respectively.
Simulations (App. E) predict a visibility of $70.2(1.2)\%$ and $88.1(1.7)\%$.
The integration time for each point of all panels is 60.

Figure 3 shows the results of both the four-fold background coincidences and
the background-subtracted four-fold coincidences when preparing the photon
pair in modes 2 and 3 in the two Bell states $\ket{\psi^{+}}$ ($\chi=0$) and
$\ket{\psi^{-}}$ ($\chi=\pi$). The background coincidences in Figs. 3(a) and
3(b) have negligible fluctuations with relative magnitude $\simeq 1.2\%$,
uncorrelated to the collective phase $\phi$. In contrast, the background-
subtracted four-fold coincidences in Figs. 3(c) and 3(d) exhibit pronounced
cosine oscillations, as predicted in Eq. (5), with high visibilities of
$69.2(2.5)\%$ and $85.8(4.5)\%$, respectively. This distinctive dependence of
four-fold coincidences on the collective phase cannot be attributed to
fluctuations of single counts or two-fold and three-fold coincidences, which
exhibit only weak fluctuations imputable to secondary effects, as detailed in
App. D, where we also present the raw (uncorrected) four-photon coincidence
data. Moreover, the experimental data are in good agreement with the numerical
simulations of the experiment, which provide visibilities of $70.2(1.2)\%$ and
$88.1(1.7)\%$, respectively. The numerical results are obtained by taking into
account input and output losses, unbalanced splitting ratios of the beam
splitters, multi-pair emissions of the sources up to six photons, and laser
power drifts (see App. E).

These numerical simulations also help to quantify the contribution of effects
causing the residual visibility reduction of four-fold coincidences in Fig.
3(d) and, in particular, Fig. 3(c). The primary factors affecting the
visibility are, in order of significance: laser power drifts, higher-order
($>3$) photon-pair emissions, and the imperfect implementation of the unitary
transformation. Deviations in the average laser power during the main
measurement and the two background measurements in Figs. 3(a) and 3(b) lead to
the subtraction of a background level that differs from the level affecting
the main measurement. The background four-fold coincidences in Fig. 3(a) are
slightly lower than those measured in Fig. 3(b). This causes the slightly
lower visibility in Fig. 3(c) with respect to the measurement in Fig. 3(d),
which is virtually unaffected by power drifts (see simulation in App. E.3 for
details). The second noise contribution is associated with higher-order
emissions with at least three photon pairs emitted by the two sources—two
pairs in one source, and one in the other. Our background subtraction
procedure cannot account for this process. Imperfections in the experimental
implementation of the unitary transformation are mainly caused by the slight
manufacturing deviations of beam splitters from a 50:50 splitting ratio.
Additional minor contributions include residual polarisation dependence of the
beam splitters’ splitting ratios, the imperfect spectral purity of the down-
converted photons or slight temporal indistinguishability at the beam
splitters, as well as imperfect state preparation.

## Conclusions

We have shown in theory and photonic experiments that the entanglement between
particles at the input ports of independent and separate beam splitters sets
the stage for a collective interference effect of all particles, which cannot
be traced back to the interference of smaller subsets. The peculiarity of this
phenomenon lies in the fact that a collective many-body dynamics arises
despite the setup allowing only for pairwise interference and some particles
occupying distinguishable internal states. The entanglement in the input
photon state bridges the gaps between the disjoint interferometers and,
therefore, acts as a mediator of the collective interference in a nonlocal
fashion. Compared to the conventional case of non-entangled photons [35, 34],
this effect has two fundamental advantages for scaling towards more particles:
first, the equivalent interference scenario with $N>4$ photons can be
implemented by adding beam splitters and pairs of entangled photons in
parallel, without the requirement of engineering additional internal dof as
$N$ increases, such that the single-particle Hilbert space dimension stays
constant at two [41]. Second, the visibility of the entanglement-induced
collective phase interference is ideally always one and does not diminish with
a growing number of particles. This will be instrumental in obtaining clear
signals in interference scenarios with a large number of particles involved.

Our experimental data and numerical simulations produce a high interference
visibility, which is mainly limited by the nature of the SPDC photon sources
and the errors introduced with the background subtraction. However, for $N>4$
input particles, the partial (input-output) connectivity of the interferometer
necessitates the participation of all $N/2$ sources to achieve an $N$-fold
coincidence, that is, background subtraction is no longer required to obtain
visibilities comparable with the highest visibility obtained here ($\sim
86\%$). The detrimental influence of multi-pair emissions could also be
completely overcome by using deterministic single-photon sources, such as
semiconductor quantum dots, if the required input state can be engineered by
appropriate excitation schemes [52, 53, 54] and high photon
indistinguishability between separate source can be achieved [55].

Our study gives another striking example of how entanglement can play a key
role in shaping interference phenomena [56, 57, 37]. It reveals a hitherto
unexplored interference effect that, by virtue of the partial
distinguishability of the involved particles, extends the complexity of
traditional many-body interference scenarios, potentially leading to
unexpected phenomena. For instance, a recent work has shown that by using
$N\geq 7$ partially distinguishable particles, indistinguishable bosons do not
maximise the probability of bunching in a subset of output modes [28]. This
prompts questions on whether and how a collective behaviour, as induced here
through entanglement between internal dof of a subset of particles, can affect
the bunching dynamics in similar many-body conditions. The interference scheme
presented here readily extends to larger entangled GHZ states while preserving
its properties. Employing such multi-partite entangled states offers various
advantages in quantum communication protocols [58, 59]. Consequently, it seems
worthy to further investigate the potential of entanglement-induced collective
interference with GHZ states for multiparty quantum communication.
Furthermore, the presence of an $N$P collective phase producing high
interference contrast could be exploited for quantum metrology purposes. In
particular, the possibility of scaling up the scheme by adding only Bell pairs
can be beneficial to avoid the typical complex preparation of NOON states and
their exposure to decoherence at large photon numbers $N$ [60]. Overall, the
combination of partial distinguishability and entanglement in many-body
interference could serve as a valuable tool for advancing quantum
technologies, or manifest as a subtle yet noteworthy side-effect.

## Acknowledgements

The authors acknowledge Stefan Frick for valuable discussions during the
preparation of the manuscript. T.F., G.W. and R.K. acknowledge the Austrian
Science Fund (FWF) projects FG 5, F 7114 and W1259 (DK-ALM) for financial
support. C.D. acknowledges the Georg H. Endress Foundation for support and the
Freiburg Institute for Advanced Studies for a FRIAS Junior Fellowship. A.F. is
supported by the EPSRC Quantum Technology Hub in Quantum Communication
(EP/T001011/1).

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##  Appendix A Four-point correlator and collective phase

In the following, we derive Eq. (5). The expectation value of the four-point
correlator can be evaluated as
$\braket{\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}}=\braket{\psi}{\mathcal{U}^{\dagger}\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}\mathcal{U}}{\psi}$,
with $\mathcal{U}=U^{\otimes N}$ and $U$ the single-particle unitary
transformation implemented by the array of beam splitters in Fig. 1(c):

$U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0\\\ 1&-1&0&0\\\ 0&0&1&1\\\
0&0&1&-1\end{pmatrix}.$ (7)

For convenience, we introduce 2P operators
$\hat{A}_{ij,kl}=\sum_{\alpha,\beta}\hat{a}^{\dagger}_{i\alpha}\hat{a}^{\dagger}_{j\beta}\hat{a}_{k\alpha}\hat{a}_{l\beta}$,
where the sum runs over a complete set of basis modes of the internal Hilbert
space, e.g. $\alpha,\,\beta\in\\{H,V\\}$, in our situation. Since the unitary
in Eq. (7) does not mix the modes $(1,2)$ with the modes $(3,4)$, we can look
at the measurements on both subsets of modes individually. We can write
$\mathcal{U}^{\dagger}\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}\mathcal{U}=\mathcal{U}^{\dagger}\hat{N}_{1}\hat{N}_{2}\mathcal{U}\mathcal{U}^{\dagger}\hat{N}_{3}\hat{N}_{4}\mathcal{U}$
and calculate

$\begin{split}&\mathcal{U}^{\dagger}\hat{N}_{1}\hat{N}_{2}\mathcal{U}\\\
&=\frac{1}{4}\left(\hat{A}_{12,12}-\hat{A}_{12,21}-\hat{A}_{21,12}+\hat{A}_{21,21}+\hat{R}_{12}\right)\end{split}$
(8)

where $\hat{R}_{12}$ collects those terms $\hat{A}_{ij,kl}$ with either
$i=j=1,2$ or $k=l=1,2$. These terms have zero overlap with the initial state
$\ket{\psi}$ given in Eq. (3). Analogously, we have

$\begin{split}&\mathcal{U}^{\dagger}\hat{N}_{3}\hat{N}_{4}\mathcal{U}\\\
&=\frac{1}{4}\left(\hat{A}_{34,34}-\hat{A}_{34,43}-\hat{A}_{43,34}+\hat{A}_{43,43}+\hat{R}_{34}\right)\,,\end{split}$
(9)

where, as before, $\hat{R}_{34}$ describes the terms that have no overlap with
$\ket{\psi}$. Because $\hat{A}_{ij,kl}$ contains a sum over internal indices,
it is symmetric under permuting the first and second tuples of indices
$\hat{A}_{ij,kl}=\hat{A}_{ji,lk}$. By omitting terms involving
$\hat{R}_{12},\,\hat{R}_{34}$, we have

$\begin{split}&\mathcal{U}^{\dagger}\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}\mathcal{U}\\\
&=\frac{1}{16}\left(2\hat{A}_{12,12}-2\hat{A}_{12,21})(2\hat{A}_{34,34}-2\hat{A}_{34,43})+\dots\right)\\\
&=\frac{1}{4}\Big{(}\hat{A}_{1234,1234}-\hat{A}_{1234,1243}\\\
&\qquad\qquad-\hat{A}_{1234,2134}+\hat{A}_{1234,2143}+\dots\Big{)}\,.\end{split}$
(10)

The 4P operators
$\hat{A}_{ijkl,mnop}=\sum_{\alpha,\beta,\gamma,\delta}\hat{a}^{\dagger}_{i\alpha}\hat{a}^{\dagger}_{j\beta}\hat{a}^{\dagger}_{k\gamma}\hat{a}^{\dagger}_{l\delta}\hat{a}_{m\alpha}\hat{a}_{n\beta}\hat{a}_{o\gamma}\hat{a}_{p\delta}$
are defined analogously to their 2P counterparts. The first operator results
in a contribution where no particles are exchanged and, hence, is independent
of the internal states of the particles. The second and third terms describe
the exchange of particles in modes $1,2$ and $3,4$, respectively. As we will
see below, these contributions depend only on the squared moduli of the
internal states’ overlaps and are, therefore, independent of any phase. Only
the fourth operator $\hat{A}_{1234,2143}$ (realising a simultaneous exchange
of particles in $1,2$ and in $3,4$) mediates the collective phase dependence.

As a next step, we need to evaluate all four non-zero contributions in the
initial state (3), which is a sum of two states

$\begin{split}&\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{\psi_{1}}+\ket{\psi_{2}})\\\
&\ket{\psi_{1}}=\hat{a}^{\dagger}_{1S}(\varphi)\hat{a}^{\dagger}_{2H}\hat{a}^{\dagger}_{3V}\hat{a}^{\dagger}_{4S}(\theta)\ket{0}\\\
&\ket{\psi_{2}}=e^{-i\chi}\hat{a}^{\dagger}_{1S}(\varphi)\hat{a}^{\dagger}_{2V}\hat{a}^{\dagger}_{3H}\hat{a}^{\dagger}_{4S}(\theta)\ket{0}\,.\end{split}$
(11)

We start with $A_{1234,2143}$:

$\begin{split}&\braket{\psi_{1}}{\hat{A}_{1234,2143}}{\psi_{1}}=\braket{\varphi}{H}\braket{H}{\varphi}\braket{V}{\theta}\braket{\theta}{V}=1/4\\\
&\braket{\psi_{2}}{\hat{A}_{1234,2143}}{\psi_{2}}=\braket{\varphi}{V}\braket{V}{\varphi}\braket{H}{\theta}\braket{\theta}{H}=1/4\\\
&\braket{\psi_{1}}{\hat{A}_{1234,2143}}{\psi_{2}}=e^{-i\chi}\braket{\varphi}{V}\braket{H}{\varphi}\braket{V}{\theta}\braket{\theta}{H}\\\
&=e^{-i\chi}\braket{\varphi}{V}\braket{V}{\theta}\braket{\theta}{H}\braket{H}{\varphi}=\frac{1}{4}e^{-i\chi-i\varphi+i\theta}\\\
&\braket{\psi_{2}}{\hat{A}_{1234,2143}}{\psi_{1}}=\braket{\psi_{1}}{\hat{A}_{1234,2143}}{\psi_{2}}^{\ast}\,.\end{split}$
(12)

The fourth line is a reordering of the terms of the third line that shows that
the four bra-vectors in the scalar products are a cyclic permutation of the
four ket-vectors. One can interpret this effective four-cycle permutation
stemming from the combination of exchanging particles in modes $(1,2)$ and
$(3,4)$ (the effect of the operator $\hat{A}_{1234,2143}$) and exchanging the
roles of modes $2$ and $3$ ($\ket{\psi_{1}}$ versus $\ket{\psi_{2}}$, see Eq.
(11)). The last line follows from
$\braket{\psi_{2}}{\hat{A}_{1234,2143}}{\psi_{1}}=\braket{\psi_{1}}{\hat{A}_{3412,4321}}{\psi_{2}}^{\ast}=\braket{\psi_{1}}{\hat{A}_{1234,2143}}{\psi_{2}}^{\ast}$,
where the second equality follows by permuting the first and second index
tuples by the permutation $\pi=(1\,3)(2\,4)$. Altogether we end up with

$\begin{split}&\braket{\psi}{\hat{A}_{1234,2143}}{\psi}=\frac{1}{4}\left(1+\cos(\theta-\chi-\varphi)\right)\\\
&=\frac{1}{4}\left(1+\cos(\chi+\varphi-\theta)\right)\,.\end{split}$ (13)

Next we calculate $\hat{A}_{1234,1243}$: here only the particles in modes $3$
and $4$ are exchanged. The expectation values in the states $\ket{\psi_{1}}$
and $\ket{\psi_{2}}$ are readily calculated to be

$\begin{split}&\braket{\psi_{1}}{\hat{A}_{1234,1243}}{\psi_{1}}=\braket{\varphi}{\varphi}\braket{H}{H}\braket{V}{\theta}\braket{\theta}{V}=\frac{1}{2}\\\
&\braket{\psi_{2}}{\hat{A}_{1234,1243}}{\psi_{2}}=\braket{\varphi}{\varphi}\braket{V}{V}\braket{H}{\theta}\braket{\theta}{H}=\frac{1}{2}\,.\end{split}$
(14)

Permuting only particles in mode $3,4$ leads to absolute squares of
$\braket{V}{\theta}$ and $\braket{H}{\theta}$, which do not depend on the
complex phase. Interestingly, the cross-terms vanish, since the permutation of
the particles in mode $3,4$ enforces an inner product of the orthogonal states
$V$ and $H$:

$\begin{split}&\braket{\psi_{1}}{\hat{A}_{1234,1243}}{\psi_{2}}=\frac{1}{2}e^{-i\chi}\braket{\varphi}{\varphi}\braket{H}{V}\braket{V}{\theta}\braket{\theta}{H}=0\\\
&\braket{\psi_{2}}{\hat{A}_{1234,1243}}{\psi_{1}}=\braket{\psi_{1}}{\hat{A}_{1234,1243}}{\psi_{2}}^{\ast}=0\,.\end{split}$
(15)

An analogous calculation for the exchange of particles in modes 1 and 2
through $\hat{A}_{1234,2134}$ leads to similar results. With this we end up
with

$\braket{\psi}{\hat{A}_{1234,1243}}{\psi}=\braket{\psi}{\hat{A}_{1234,2134}}{\psi}=\frac{1}{2}\left(\frac{1}{2}+\frac{1}{2}\right)=\frac{1}{2}\,.$
(16)

For the last contribution, $\hat{A}_{1234,1234}$, we reorder
$\hat{A}_{1234,1234}=\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}$ and find

$\braket{\psi}{\hat{A}_{1234,1234}}{\psi}=1\,,$ (17)

since each mode is occupied by exactly one particle.

We combine Eqs. (10), (LABEL:eq:deriv_app_fourpointcor1), (16) and (17) and
obtain the expectation value of the ideal four-point correlator:

$\begin{split}&\braket{\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}}=\braket{\psi}{\mathcal{U}^{\dagger}\hat{N}_{1}\hat{N}_{2}\hat{N}_{3}\hat{N}_{4}\mathcal{U}}{\psi}\\\
&=\frac{1}{4}\left(1-\frac{1}{2}-\frac{1}{2}+\frac{1}{4}\left(1+\cos(\chi+\varphi-\theta)\right)\right)\\\
&=\frac{1}{16}\left(1+\cos(\chi+\varphi-\theta)\right)=\frac{1}{8}\cos^{2}\left(\frac{\chi+\varphi-\theta}{2}\right),\end{split}$
(18)

as given in Eq. (5).

##  Appendix B Correlation versus coincidence measurements

The operator associated with the measurement of _at least one_ photon in
output mode $p$ is given by (disregarding internal degrees of freedom)

$\displaystyle\hat{M}_{p}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}(\hat{a}_{p}^{\dagger})^{n}\hat{a}_{p}^{n}$
(19)
$\displaystyle=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}\hat{N}_{p}(\hat{N}_{p}-1)\dots(\hat{N}_{p}-n+1).$
(20)

Indeed, the number state $\ket{m}$ with $m$ photons in mode $p$ satisfies

$\displaystyle(\mathbb{I}-\hat{M}_{p})\ket{m}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}(\hat{a}_{p}^{\dagger})^{n}\hat{a}_{p}^{n}\ket{m}$
(21) $\displaystyle=\sum_{n=0}^{m}\frac{m!}{n!(m-n)!}(-1)^{n}\ket{m}$ (22)
$\displaystyle=\delta_{m,0}\ket{m},$ (23)

where we have made use of the binomial formula. Therefore, $k$-fold
coincidence rates are obtained as the expectation value of

$\displaystyle\hat{M}_{p_{1}}\dots\hat{M}_{p_{k}}.$ (24)

The lowest-order term in the expansion of Eq. (24) is the $k$-point correlator
$\hat{N}_{p_{1}}\dots\hat{N}_{p_{k}}$, corresponding to taking terms with
$n=1$ in Eq. (19). This leading term is corrected by higher-order observables,
which are sensitive to interference from higher orders than $k$.

As an example, when considering a system of $N=4$ particles and neglecting
terms beyond the fourth order, the explicit expansion of Supplementary Eq.
(24) reads

$\displaystyle\hat{M}_{p_{1}}$
$\displaystyle=\hat{N}_{p_{1}}-\frac{1}{2}\hat{N}_{p_{1}}(\hat{N}_{p_{1}}-1)+\frac{1}{6}\hat{N}_{p_{1}}(\hat{N}_{p_{1}}-1)(\hat{N}_{p_{1}}-2)-\frac{1}{24}\bm{\hat{N}_{p_{1}}(\hat{N}_{p_{1}}-1)(\hat{N}_{p_{1}}-2)(\hat{N}_{p_{1}}-3)}$
(25) $\displaystyle\hat{M}_{p_{1}}\hat{M}_{p_{2}}$
$\displaystyle=\hat{N}_{p_{1}}\hat{N}_{p_{2}}-\frac{1}{2}\hat{N}_{p_{1}}(\hat{N}_{p_{1}}-1)\hat{N}_{p_{2}}-\frac{1}{2}\hat{N}_{p_{1}}\hat{N}_{p_{2}}(\hat{N}_{p_{2}}-1)+\frac{1}{4}\bm{\hat{N}_{p_{1}}(\hat{N}_{p_{1}}-1)\hat{N}_{p_{2}}(\hat{N}_{p_{2}}-1)}$
$\displaystyle\qquad+\frac{1}{6}\bm{\hat{N}_{p_{1}}(\hat{N}_{p_{1}}-1)(\hat{N}_{p_{1}}-2)\hat{N}_{p_{2}}}+\frac{1}{6}\bm{\hat{N}_{p_{1}}\hat{N}_{p_{2}}(\hat{N}_{p_{2}}-1)(\hat{N}_{p_{2}}-2)}$
(26) $\displaystyle\hat{M}_{p_{1}}\hat{M}_{p_{2}}\hat{M}_{p_{3}}$
$\displaystyle=\hat{N}_{p_{1}}\hat{N}_{p_{2}}\hat{N}_{p_{3}}-\frac{1}{2}\bm{\hat{N}_{p_{1}}(\hat{N}_{p_{1}}-1)\hat{N}_{p_{2}}\hat{N}_{p_{3}}}-\frac{1}{2}\bm{\hat{N}_{p_{1}}\hat{N}_{p_{2}}(\hat{N}_{p_{2}}-1)\hat{N}_{p_{3}}}-\frac{1}{2}\bm{\hat{N}_{p_{1}}\hat{N}_{p_{2}}\hat{N}_{p_{3}}(\hat{N}_{p_{3}}-1)}$
(27)
$\displaystyle\hat{M}_{p_{1}}\hat{M}_{p_{2}}\hat{M}_{p_{3}}\hat{M}_{p_{4}}$
$\displaystyle=\bm{\hat{N}_{p_{1}}\hat{N}_{p_{2}}\hat{N}_{p_{3}}\hat{N}_{p_{4}}},$
(28)

where fourth-order terms are highlighted.

Figure 4: Simulations of correlations and coincidence events. Correlations
show (a) a fully visible fringe pattern of the four-point correlator, whereas
(c) the three-point correlators, which depend on the coherence of the reduced
three-particle states, are all independent of the collective phase. We sampled
input and output transmission efficiencies from a normal distribution with a
mean value of 0.8 and a standard deviation of 0.1. Different total losses
displace the curves among each other, with higher losses resulting in lower
curves. Simulations of coincidence events show (b) four-fold coincidence with
same behaviour as the four-point correlation (see Eq. (28)), and (d) a
signature of the collective 4P phase also in the three-fold coincidences, as
these are not proper 3P measurements. The visibility of this signature
strongly depends on the optical losses, with lower values as losses increase.

The presence of fourth-order terms in Eq. (25)-(27) results in a residual
dependence of $k(<N)$-fold coincidences on the collective phase $\phi$, where
the magnitude of this influence increases as $k$ approaches $N$. The presence
of these higher-order contributions is illustrated in Fig. 4: numerical
simulations indeed show that three-fold coincidences depend on $\phi$, but
with much weaker visibility than for the four-fold coincidence.

To grasp the dependence of three-fold coincidences on the collective phase
$\phi$, consider the three-fold events among, for example, the output channels
A, C, and D in the lossless conditions where $\phi=\pi,\,0$. The output mode-
occupation lists (counting the number of particles in each output mode)
contributing to such events are $\mathbf{s_{1}}=(2,\,0,\,1,\,1)$ and
$\mathbf{s_{2}}=(1,\,1,\,1,\,1)$. When $\phi=\pi$, the absence of four-fold
coincidences indicates a bunching behaviour of photons, resulting in the
suppression of events corresponding to $\mathbf{s_{2}}$ and in a three-fold
coincidence probability solely given by $\mathbf{s_{1}}$, i.e.,
$p(\mathrm{ACD}|\phi=\pi)=p(\mathbf{s_{1}}|\phi=\pi)$. For symmetry reasons,
$\mathbf{s_{1}}$ must have the same probability as the output state
$\mathbf{s_{3}}=(0,\,2,\,1,\,1)$ (which does not produce a coincidence in A, C
and D). When $\phi=0$, all three states $\mathbf{s_{1}}$, $\mathbf{s_{2}}$,
and $\mathbf{s_{3}}$ are allowed, of which $\mathbf{s_{2}}$ can be achieved
via two distinct exchange processes at the beam splitter connecting modes A
and B (double reflection and double transmission), while $\mathbf{s_{1}}$ and
$\mathbf{s_{3}}$ can only be obtained via a single process each. Intuitively,
this halves the probability of $\mathbf{s_{1}}$ with respect to the
$\phi=\pi$-scenario, whereas the probability of $\mathbf{s_{2}}$ is double the
probability of $\mathbf{s_{1}}$, that is,
$p(\mathbf{s_{2}}|\phi=0)=p(\mathbf{s_{1}}|\phi=\pi)$. Therefore, the three-
fold coincidence probability becomes
$p(\mathrm{ACD}|\phi=0)=p(\mathbf{s_{1}}|\phi=0)+p(\mathbf{s_{2}}|\phi=0)=p(\mathbf{s_{1}}|\phi=\pi)(\frac{1}{2}+1)=\frac{3}{2}p(\phi=\pi)$.
When losses are considered, such as in Fig. 4, the factor $3/2$ is reduced
because three-fold coincidences from $\mathbf{s_{1}}$ are less affected by
losses with respect to $\mathbf{s_{2}}$ (one photon loss in channel A of
$\mathbf{s_{1}}$ still result in a coincidence).

Simulations of coincidences with the addition of one detector multiplexing
layer, as in Fig. 2, predict a further visibility reduction of a factor $\sim
2.3$ (influenced by the value of the total optical losses) in the three-fold
coincidences fluctuations of Fig. 4(d).

Figure 5: Tomography of the entangled photon pair. The panels in the upper and
lower rows are the results for the input states $\ket{\psi^{+}}$ and
$\ket{\psi^{-}}$ of the photons in modes 2 and 3, respectively. Panels (a) and
(d) show the dependence of the states’ fidelities on the laser pump power,
whereas the other panels show the real ((b) and (e)) and imaginary ((c) and
(f)) parts of the reconstructed density matrix $\rho$ for a representative
value of 150.

##  Appendix C State fidelity

We evaluated the state overlap, referred to as fidelity [61], of the entangled
state of the photons in modes 2 and 3 with respect to the Bell states
$\ket{\psi^{+}},\,\ket{\psi^{-}}$, corresponding to setting $\chi=0,\,\pi$ in
Eq. (3), by performing a maximum likelihood state tomography [62]. To this
end, the source outputs are connected to two polarisation-tomography stages
where we measure single counts and coincidence events. As mentioned in the
main text, the phase $\chi$ is adjusted via fibre polarisation controllers. We
performed six state tomographies for the average pump power range 20–250,
which includes the power used to perform the experiment
$\simeq\qty{174}{\milli}$, and we evaluated the fidelity of the resulting
density matrices with the corresponding states
$\ket{\psi^{+}},\,\ket{\psi^{-}}$.

The results of these measurements are shown in Fig. 5. From the fit of the
data, we find a fidelity at $\qty{174}{\milli}$ of 98.53(2)% and 98.33(2)%,
respectively, for $\ket{\psi^{+}}$ and $\ket{\psi^{-}}$.

##  Appendix D Raw data

Figure 6: Measurements and simulations of the single-source measurements for
$\chi=\pi$ in Eq. (3). Panels (a)-(d) report single counts and coincidence
events measured when blocking the source of separable photons (Source 1). The
panels also show the corresponding results achieved via simulations based on
the optimisation algorithm discussed in Appendix E. Panels (e)-(h) report the
same results obtained when blocking the source of entangled photons (Source
2). In all graphs, the integration time for each point is 60.

We present here the raw data of the background-subtracted measurement in Fig.
3. As described in the main text, we performed $\varphi$-phase scans at fixed
phases $\chi=0,\,\pi$, and $\theta=0$ in Eqs. (3)-(4). We first show the
results of single-source measurements (when only one source is pumped by the
laser) and, subsequently, the results of the experiment when background
subtraction is not employed.

Single-source measurements provide a useful comparison for the expected
fluctuations of single counts and coincidence events in the absence of
interference of photons from different sources. In addition, since photons
from one source are coupled to separate beam splitters (cf. Fig. 2), also no
interference effect between photons from the same source can take place, and
fluctuations can be purely associated with other contributions, such as
residual polarisation dependence of the system.

We report in Fig. 6 the measurements of single counts and coincidence events
(together with the simulations discussed in Appendix E) for the two separate
sources when $\chi=\pi$, but similar results are obtained for $\chi=0$.
Moreover, all four-fold background coincidence measurements (for both
$\chi=0,\,\pi$) can be found in Figs. 3(a) and 3(b). Single counts in Figs.
6(a) and 6(e) exhibit relative fluctuations of the order of $\simeq 0.2\%$.
Similarly to single counts, two-, three-, and four-fold coincidence events
have only marginal fluctuations, respectively of the order of $\simeq 0.3\%$,
$\simeq 0.5\%$ and $\simeq 1.2\%$. The increasing value in the fluctuations of
coincidence events is mainly given by the smaller photon counting statistics
of these cases.

Fig. 7 shows the data collected when both sources are employed, before
subtraction of the background, for both settings of the phase $\chi=0,\,\pi$
(with $\theta=0$).

The single counts in Figs. 7(a) and 7(e) show only small changes as we vary
the collective phase $\phi$, with relative fluctuations of $\simeq 0.1-0.2\%$.
This value is consistent with the corresponding fluctuations of single counts
measured when using only one source at a time, where the smaller sample size
produces slightly higher values.

Two-fold coincidences in Figs. 7(b) and 7(f) have relative fluctuations of the
order of $\simeq 1.7\%$ and $\simeq 2.4\%$, respectively. These values are
larger than the corresponding ones measured with the single sources ($\simeq
0.3\%$). Nevertheless, they are connected to interference effects between the
input states produced by the two sources [18] that cannot be attributed to a
dependence on the collective phase. In particular, these fluctuations do not
show the expected cosinusoidal dependence on the collective phase, but rather
exhibit a complex interference pattern that arises as a consequence of the
relative phase between the dominant single-pair emission contributions of the
two sources. In fact, two-fold coincidences of channels associated with the
same beam splitter — chs. A-B or chs. C-D in Fig. 2 —, which are not
influenced by these contributions, exhibit significantly weaker fluctuations
($\simeq 0.3-0.4\%$), as shown in the insets of Figs. 7(b) and 7(f), despite
their reduced photon counting statistic. Specifically, the partial
connectivity of the interferometer does not allow single-pair emissions from
one source to contribute to two-fold coincidences of chs. A-B or chs. C-D,
which necessitate two-pair emissions.

Figure 7: Measurements of single and coincidence counts for two settings of
the entangled state phase $\chi=0,\,\pi$ in Eq. (3) ($\ket{\psi^{+}}$ and
$\ket{\psi^{-}}$ in panels (a)-(d) and (e)-(h), respectively). (a)(e) Single
counts. (b)(f) Two-fold coincidences. (c)(g) Three-fold coincidences. (d)(h)
Four-fold coincidences data (blue dots) fitted with a cosine function (red
curve). The red-shaded region shows the fit prediction interval at a
confidence level of one standard deviation. The graphs include the results of
the multiphoton interference simulations (black dashed curve). The visibility
of the fitting cosine is $11.6(5)\%$ and $11.8(4)\%$ for insets (d) and (h),
whereas simulations predict a visibility of $12.0(1)\%$ and $11.9(1)\%$,
respectively. In all graphs, the integration time for each point is 60.

This requirement prevents two-fold coincidences between these channels from
only two particles in the system and ensures contributions from a wider
spectrum of interfering terms, originating from all exchange processes of at
least four particles at the two beam splitters, thus reducing the influence of
relative phase terms and, consequently, possible fluctuations. Therefore, the
coincidences between chs. A-B and chs. C-D are a good indicator for the
independence of two-fold coincidences from the collective 4P phase. Of these,
chs. C-D do not exhibit a clear dependence on $\phi$, as shown in the two
insets, whereas chs. A-B have slightly larger fluctuations with a specific
pattern displaying a maximum in the centre of the $\phi$ range (corresponding
to half the rotation range of the half-wave plate in input mode 1). In
contrast, a pure dependence on the collective phase $\phi$ would lead to a
$\pi$-shift, i.e., an inversion of the pattern between the two curves for
$\ket{\psi^{+}}$ and $\ket{\psi^{-}}$, which is clearly not the case.
Therefore, we conclude that this fluctuation pattern is dominated by artefacts
of the wave-plate rotation rather than the collective phase.

Three-fold coincidences in Figs. 7(c) and 7(g) have residual interference
effects similar to two-fold coincidences that appear as anti-correlated
patterns of coincidences comprising only one of the output channels A and B.
The influence of such interference effects on the three-fold coincidences is
already significantly weaker with respect to the two-fold case, with
fluctuations of the order of $\simeq 0.8\%$. Conversely, the largest
fluctuations are here connected to three-fold coincidences among chs. A-B-C,
which, however, show a pattern that, in both cases, can be correlated to the
two-fold coincidence data of chs. A-B (Pearson correlation coefficients
$\simeq 0.91$). Consequently, fluctuations are once more associated with
unwanted artefacts and residual interference effects, instead of a dependence
on the collective phase.

While two-fold and three-fold coincidences do not exhibit any obvious
dependence on the collective phase $\phi$, the four-fold coincidences reported
in Figs. 7(d) and 7(h) show a clear oscillation. In particular, the four-fold
coincidences are in excellent agreement with the cosine behaviour predicted by
Eq. (5), showing relatively high visibility compared to the previous cases of
$11.6(5)\%$ and $11.8(4)\%$, which excludes the possibility that the observed
four-fold oscillations are an artefact of the interference effects mentioned
above for two- and three-fold coincidences. Moreover, as also shown in the
main text, the experimental data exhibit a good agreement with numerical
simulations of the experiment (further details in Appendix E).

##  Appendix E Simulations

To further support the experimental results, we performed simulations of the
experiment based on the specific experimental scheme and the photon sources
used to witness the entanglement-induced collective interference.

For this purpose, we performed a characterisation of the two SPDC sources and
developed a model of the unitary transformation realised in Fig. 2(b). Source
characterisation is required to calculate the emission rates of single-pair
emissions and higher-order emissions from the SPDC processes, depending on the
laser pump power and pulse repetition rate. The unitary transformation model,
instead, allows us to evaluate the probability of all possible output events
for each input state corresponding to combinations of different emission
orders from the two sources. The reconstruction of the unitary transformation
is achieved via an optimisation algorithm based on measurements of single
count events — the detection events measured by each detector independently of
the measurements of all other detectors — and all $k$-fold coincidence rates,
with $k=2,\,3,\,4$, produced by the individual sources. As a result, we can
calculate the total single counts and coincidence rates by using the emission
rates of each input state and the relative output probabilities.

The simulations take into account a few fundamental approximations. First, the
two sources are considered independent of each other. This is strictly true
only when we use single counts and coincidence rates from single source
emissions, i.e., the data exploited for the reconstruction of the unitary
transformation. In fact, as already mentioned in Sec. D, the presence of phase
terms among the input states of the combined two-source SPDC process can give
rise to interference in the coincidence rates. Second, it is assumed that the
down-conversion process of the pump photons results in spectrally pure down-
converted photon pairs, corresponding to fully separable output photon states
— the production of entangled pairs is only associated with the geometry of
the Sagnac interferometers [48]. This approximation is well justified due to
the high spectral purity ($\geq 98\%$) of the SPDC sources used in the
experiment, as extensively studied in [47]. Partial distinguishability of the
input particles (see Eqs. (3)-(4)) is obtained in terms of polarisation
states, which, following the formalism developed in [63], are considered as
internal degrees of freedom of the input state. While this is computationally
efficient, the unitary transformation is unable to act on the photons’
transformation depending on their polarisation state, thus excluding any
modelling of residual polarisation dependence of losses and beam splitters’
splitting ratios. However, this limitation does not significantly influence
the results to a first approximation.

###  E.1 Photon sources

The photon sources in Fig. 2(a) are modeled with two independent SPDC
processes, where we expressed the resulting photon states in terms of two-mode
squeezed vacuum states in the photon number basis $n$:

$\ket{\varphi}=\sqrt{1-\gamma^{2}}\sum_{n=0}^{\infty}\gamma^{n}\ket{n,n}_{s,i},$
(29)

with $s$ and $i$ indicating the signal and idler modes, and $\gamma$ the
squeezing parameter. This photon state represents a good approximation for
calculating the emission rates even for the bidirectionally pumped crystal in
Fig. 2(a) (used to produce entangled pairs). The emission probability of a
specific photon number state $\ket{n,n}_{s,i}$ per laser pulse is associated
with the squeezing parameter by $P(n)=(1-\gamma^{2})\gamma^{2n}$. Moreover,
the squeezing parameter can be expressed as $\gamma=\sqrt{\tau p}$, in terms
of the pump power $p$ and a constant $\tau$ that quantifies the nonlinear
interaction within the aKTP crystals [64, 47]. Similarly to [64], we
calculated $\tau$ for both sources by measuring single counts and coincidence
rates and knowing the repetition rate of the laser pulses $f$
($\qty{80}{\mega}$). Specifically, by considering an input optical power of
$\qty{174}{\milli}$ and $\qty{168}{\milli}$ for Source 1 and Source 2,
respectively, we obtained squeezing parameters of $\gamma_{S_{1}}=0.102$ and
$\gamma_{S_{2}}=0.094$.

The evaluation of $\gamma$ allows us to determine the single photon pair rates
and the rates of higher-order pair productions depending on the pump power. To
show this, by using the polarisation basis, the overall state of the down-
converted photons collected at the output of the Sagnac interferometers can be
written as:

$\ket{\psi}=\ket{\varphi_{S_{1}}}\otimes\ket{\varphi_{S_{2}}}=\Big{(}\sqrt{1-\gamma_{S_{1}}^{2}}\sum_{n=0}^{\infty}\gamma_{S_{1}}^{n}\big{[}\frac{1}{\sqrt{2}}(\ket{H,V}_{2,3}+e^{-i\chi}\ket{V,H}_{2,3})\big{]}^{\otimes
n}\Big{)}\otimes\Big{(}\sqrt{1-\gamma_{S_{2}}^{2}}\sum_{m=0}^{\infty}\gamma_{S_{2}}^{m}\ket{H,V}_{1,4}^{\otimes
m}\Big{)},$ (30)

where $S_{1}$ and $S_{2}$ refer to the source of entangled photon states
(Source 1) and of separable photon states (Source 2), the indices
$1,\leavevmode\nobreak\ 2,\leavevmode\nobreak\ 3,\leavevmode\nobreak\ 4$
correspond to chs. 1-4 before the beam splitters in Fig. 2(b), and $\chi$ is
the usual phase of the entangled state as defined in Eq. (3). Starting from
Eq. (30), we can calculate the emission rates of each mode-occupation list
$\mathbf{R}=(r_{1},r_{2},r_{3},r_{4})$ corresponding to the occupation number
of each unitary input channel $1-4$ [63]. Given the nature of the SPDC process
(two-photon emissions) and the arrangement of the channels in Fig. 2, we have
$r_{1}=r_{4}:=r_{S_{2}}$ and $r_{2}=r_{3}:=r_{S_{1}}$, where $r_{S_{1}}$ and
$r_{S_{2}}$ are defined as the number of pair emissions of each source.
Therefore, each mode-occupation list $\mathbf{R}$ can be associated with an
input state similar to Eq. (3):

$\begin{split}&\ket{\xi}=\left[\frac{1}{\sqrt{2}}(\hat{a}_{2,H}^{{\dagger}}\hat{a}_{3,V}^{{\dagger}}+e^{-i\chi}\hat{a}_{2,V}^{{\dagger}}\hat{a}_{3,H}^{{\dagger}})\right]^{r_{S_{1}}}\cdot\\\
&\qquad\quad(\hat{a}_{1,\mathrm{S}}^{{\dagger}}(\varphi)\hat{a}_{4,\mathrm{S}}^{{\dagger}}(\theta))^{r_{S_{2}}}\ket{0}.\end{split}$
(31)

The state $\ket{\xi}$ is related to the contribution of a specific pair of
values $(m,n)$ in Eq. (30), from which one can evaluate the emission rate by
knowing $\gamma_{S_{1}}$, $\gamma_{S_{2}}$, and $f$.

For example, the input state in Eq. (3) is associated with
$r_{S_{1}}=r_{S_{2}}=1$, $\mathbf{R}=(1,1,1,1)$, i.e., a single firing of both
sources. The contribution to this state is related to the $m=n=1$ terms in Eq.
(30), whose emission rate is given by
$r_{1,1}=fP(\mathbf{R}_{1,1})=f(1-\gamma_{S_{1}}^{2})\gamma_{S_{1}}^{2}(1-\gamma_{S_{2}}^{2})\gamma_{S_{2}}^{2}$.
In a similar fashion, one can calculate the rates of all mode-occupation lists
resulting from all possible combinations of sources’ emissions. We considered
here contributions of up to three photon-pair productions, which can arise
from both single-source emissions ($r_{S_{1}}=3,\leavevmode\nobreak\
r_{S_{2}}=0$, or the opposite) or emissions from combinations of both sources
($r_{S_{1}}=2,\leavevmode\nobreak\ r_{S_{2}}=1$, or the opposite).

###  E.2 Unitary reconstruction

Each mode-occupation list contributes differently to the total single counts
and coincidence rates. In fact, each mode-occupation list has a different
emission rate and probability of giving rise to specific output events after
propagation through the interference setup. The calculation of these
probabilities is based on the formalism developed in [63], which, given a
certain unitary transformation, provides the input-output probabilities for
each input mode-occupation list $\mathbf{R}$.

We reconstructed the unitary transformation associated with Fig. 2(b), given
by the two independent beam splitters, to include the possibility of both
optical losses before and after the two beam splitters, and unbalanced
splitting ratios of the beam splitters themselves. Losses are modelled by
extending the four-mode unitary with additional unbalanced beam splitters and
external modes. We consider independent transmission factors $\eta_{j}$ for
each of the eight input and output modes of the beam splitters.

The multiplexed eight-detector scheme (inset of Fig. 2(b)) is modelled via
additional balanced beam splitters, randomly distributing the arriving photons
on the two detectors of each output channel A-D. This is a valid
simplification, as in the experiment the polarisation projection is only used
to fix the photon polarisation on individual detectors. All losses of this
second transformation stage and the detector efficiencies are incorporated
into the output channel losses of the main beam splitter unitary.

As a result, ten parameters are required to reconstruct the extended unitary:
eight transmission coefficients $\eta_{j}$ to model the input and output
losses and the two splitting ratios of the main beam splitters. We employed an
optimisation algorithm for these ten parameters, which aims to match the
average values of the simulated and measured single counts and coincidence
rates. More precisely, we carried out this parameter search by simultaneously
emulating both single-source measurements performed by blocking one source at
a time — thus matching eight single count rates, twelve two-fold coincidence
rates, eight three-fold coincidence rates, and two four-fold coincidence
rates. This procedure is used for both the two measurements with
$\chi=0,\,\pi$ to allow for marginal deviations in alignment between the two
experimental conditions.

The panels in Fig. 6 show the comparison between measurements and simulations
for the case $\chi=\pi$, and similar results are obtained for $\chi=0$.
Consistently, the transmission coefficients of the main beam splitters do not
differ between the two optimisations to all reasonable orders of
approximation, and variations of input and output losses are of the order of
$\pm 0.8\%$. Moreover, for the specific beam splitters used, we found
deviations from a balanced 50:50 splitting ratio that are within the $\pm 3\%$
tolerance interval given by the producer [65]. The panels associated with the
coincidence rates in Fig. 6 show good agreement with the measurements, with
relative deviations $<5\%$. This is only partially true for single count
rates, where channel C has a relative deviation of $\simeq 25\%$. Presumably,
this is an effect of correlated loss affecting channel C, e.g. due to small
errors in setting the FPGA unit’s logic delays, which is hardly noticeable
beforehand without applying this optimisation routine. This defect is modelled
by the algorithm with an effective higher loss of channel C.

###  E.3 Power drifts

As mentioned in Sec. II, drifts in laser power between the main measurements
in Fig. 7 and the background measurements in Fig. 6 lead to errors in the
background subtraction.

Therefore, we performed another optimisation routine based on the single
counts and all $k$-fold coincidence rates of the main measurements (again one
optimisation for each phase $\chi=0,\,\pi$ in Eq. (3)), but this time only
allowing for a variation of the input power affecting both photon sources in
the same manner. According to this procedure, the power is approximately
unchanged for the measurement of $\chi=\pi$, demonstrating that no significant
power fluctuation influences Fig. 3(d), while a relative power drift of
$\simeq 2.3\%$ affects the measurement of $\chi=0$, thus resulting in the
considerable visibility reduction obtained in Fig. 3(c).