# Observation of nonlocal quantum interference
between the origins of a four-photon state in a silicon chip

Lan-Tian Feng  CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, 230026, People’s Republic of China.
CAS Synergetic Innovation Center of Quantum Information $\&$ Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China.
Ming Zhang  State Key Laboratory for Modern Optical Instrumentation, Centre
for Optical and Electromagnetic Research, Zhejiang Provincial Key Laboratory
for Sensing Technologies, Zhejiang University, Zijingang Campus, Hangzhou
310058, China.  Di Liu  Yu-Jie Cheng  Guo-Ping Guo  CAS Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei,
230026, People’s Republic of China.  CAS Synergetic Innovation Center of
Quantum Information $\&$ Quantum Physics, University of Science and Technology
of China, Hefei, Anhui 230026, China.  Dao-Xin Dai  State Key Laboratory for
Modern Optical Instrumentation, Centre for Optical and Electromagnetic
Research, Zhejiang Provincial Key Laboratory for Sensing Technologies,
Zhejiang University, Zijingang Campus, Hangzhou 310058, China.  Guang-Can Guo
CAS Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei, 230026, People’s Republic of China.  CAS
Synergetic Innovation Center of Quantum Information $\&$ Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China.
Mario<EMAIL_ADDRESS>Department of Chemistry $\&$ Computer
Science, University of Toronto, Toronto, Canada.  Vector Institute for
Artificial Intelligence, Toronto, Canada.  Xi-Feng<EMAIL_ADDRESS>CAS
Key Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, 230026, People’s Republic of China.  CAS Synergetic Innovation
Center of Quantum Information $\&$ Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China.

###### Abstract

Quantum mechanically, multiple particles can jointly be in a coherent
superposition of two or more different states at the same time. This property
is called quantum entanglement, and gives rise to characteristic nonlocal
interference and stays at the heart of quantum information process. Here,
rather than interference of different intrinsic properties of particles, we
experimentally demonstrated coherent superposition of two different
birthplaces of a four-photon state. The quantum state is created in four
probabilistic photon-pair sources, two combinations of which can create photon
quadruplets. Coherent elimination and revival of distributed 4-photons can be
fully controlled by tuning a phase. The stringent coherence requirements are
met by using a silicon-based integrated photonic chip that contains four
spiral waveguides for producing photon pairs via spontaneous four-wave mixing.
The experiment gives rise to peculiar nonlocal phenomena without any obvious
involvement of entanglement. Besides several potential applications that
exploit the new on-chip technology, it opens up the possibility for
fundamental studies on nonlocality with spatially separated locations.

## 1\. INTRODUCTION

In 1994, Herzog, Rarity, Weinfurter and Zeilinger demonstrated a remarkable
quantum interference effect herzog1994frustrated . They overlapped the paths
of emerging photon pairs from two probabilistic photon-pair sources in such a
way that it cannot be distinguished – not even in principle – whether the pair
has been created in the first or second source. As a result, the photons are
in a coherent superposition of being created in either of the two sources.
This generalization of superpositions of properties of photons to the mere
origin of their creation has exciting consequences. They now tune a phase
between the two crystals, and thereby manipulate the total amount of emerging
photon pairs. Imagine for a moment, that we set the phase to $\pi$, such that
the two creation processes destructively interfere. In that case, even if two
sources individually would create photon pairs, together they do not.
Importantly, in contrast to a quantum eraser, here the photon pair has never
been born in the first place. This effect has been called frustrated down-
conversion chekhova2016nonlinear ; hochrainer2021quantum .

Now, 27 years later, we experimentally demonstrate for the first time the
multi-particle generalization of this mind-boggling quantum effect, which was
theoretically proposed only 2 years ago via a detour to graph theory
krenn2017quantum ; gu2019quantum . In our four-photon experiment, the paths of
the photons are overlapped in such a way that there are exactly two different
possibilities of how each of the four detectors sees a single photon. Taking
advantage of high quality integrated optics at a silicon chip, we align the
photon paths in such a way that it cannot be distinguished whether the pairs
have been created in the first or second possibility. In that way, the origin
of the four photons is in a coherent position and we can observe constructive
and destructive interference of 4-photon systems that cannot be observed in
the photon pairs themselves.

The multi-partite generalization of frustrated down-conversion has non-trivial
consequences. In contrast to the two-photon case, this time the origin of the
four-photon state consists of two locations that could be spatially separated.
This opens a myriad of exciting foundational experiments, for example,
investigating time delays of the interference effects Ma2016quantum .
Furthermore, by introducing the ability to change state properties in a
distributed, remote way, it could also conceptually advance quantum
technologies that are based on the idea of path identity wang1991induced ;
greenberger1993multiparticle . Examples involve quantum metrology
kalashnikov2016infrared ; paterova2020nonlinear , quantum imaging
lemos2014quantum , quantum microscopy kviatkovsky2020microscopy ;
paterova2020hyperspectral , photon-pair shaping su2019versatile  and
generation of complex multi-photon states krenn2017entanglement ;
krenn2020conceptual . Furthermore, the effect is the basis of new quantum
computing schemes gu2019quantum .

## 2\. RESULTS

###  .1 Physical interpretation of the four-photon interference

Let’s first look at the experiment by Herzog et al. herzog1994frustrated ,
depicted in Fig.1a. Each of the two nonlinear crystals can probabilistically
create a photon pair. If all properties of the photon pair, including their
path, are identical, there is no way to distinguish from which crystal a pair
was created. In that case, the two-photon state is

$\displaystyle\left|\psi\right\rangle=g\Big{(}\overbrace{\left|a,b\right\rangle}^{\text{crystal
II}}+\overbrace{e^{i\varphi}\left|a,b\right\rangle}^{\text{crystal
I}}\Big{)}=g\left(1+e^{i\varphi}\right)\left|a,b\right\rangle,$ (1)

where $\left|a,b\right\rangle$ stands for a photon in path $a$ and one in path
$b$, and $g$ is related with the pair creation probability. For clarity, we
ignore the vacuum and higher-order terms.

The probability to detect a photon pair in $a$ and $b$ is $P_{a,b}\approx
2g^{2}\left(1+\cos(\varphi)\right)$. With $\varphi=0$, constructive
interference increases the number of photon pairs by a factor of 4 compared to
the number of pair generated by a single crystal. In the regime of destructive
interference, for $\varphi=\pi$, the two possibilities for pair generation
cancel each other. That means, even though the crystals would produce photons
independently when two crystals are set up in this way, no photon pair is
created at all.

In our system, we use four photon-pair sources, I-IV and a phase $\theta$
between the two layers, as shown in Fig.1b. Their paths are overlapped in such
a way that detectors A-D only see photon quadruples if the source I & II or
III & IV created a photon pair simultaneously. For small photon pair creation
rate $g$, we can write the state as,

$\displaystyle|\psi\rangle$
$\displaystyle=g\Big{(}\overbrace{|a,b\rangle}^{\text{crystal
III}}+\overbrace{|c,d\rangle}^{\text{crystal
IV}}+\overbrace{|a,c\rangle}^{\text{crystal
I}}+\overbrace{e^{i\theta}|b,d\rangle}^{\text{crystal II}}\Big{)}$ (2)
$\displaystyle+g^{2}\Big{(}\underbrace{|a,b,c,d\rangle}_{\text{crystal
III\&IV}}+\underbrace{e^{i\theta}|a,b,c,d\rangle}_{\text{crystal
I\&II}}\Big{)}+\dots$
$\displaystyle=g\Big{(}|a,b\rangle+|c,d\rangle+|a,c\rangle+e^{i\theta}|b,d\rangle\Big{)}$
$\displaystyle+g^{2}\left(1+e^{i\theta}\right)|a,b,c,d\rangle+\dots$

where $\left|a,b\right\rangle$ stands for a photon pair in paths $a$ and $b$,
and $\left|a,b,c,d\right\rangle$ stands for a photon quadruple in the
detectors A-D. No other combinations of two simultaneous photon-pair creations
create a photon in each of the four detectors (see Methods). By careful
design, the two possibilities for photon quadruplets can be created in a
coherent superposition. In this way, we create a 4-photon quantum state that
originates from either of two locations. Again, by introducing a phase
$\theta$ between the origins, we can observe constructively and destructively
interfere of the four-photons creation process. For example, if the phase is
$\theta=\pi$, no four-photon state can be observed in the detectors A-D, while
each of the individual crystals continue to create photon pairs.

We would like to argue how our experiment is related to nonlocality in quantum
mechanics, in a stronger way than its two-photon analogy herzog1994frustrated
; herzog1994herzog2 . There is no nonlocality in the sense of Bell’s
inequality or GHZ violation because the final four-photon term is not
entangled. Yet, we suggest that our experiment shows nonlocality of a
different form. This can be inferred by the fact that changing the phase in
one path does not change any single-photon or two-photon count rates from the
four crystals, but modulates only the four-fold count rates, theoretically
with perfect visibility. This has a surprising consequence, as described in
hochrainer2021quantum : Imagine for a moment that the crystals III and IV in
Fig.1b are separated by a large distance. This configuration still satisfies
all coherence requirements. Now if we modify the phase $\theta$ (which we
assume to be close to crystal IV), the single or two-photon counts in paths
$a$ and $b$, which are separated by a large distance from the phase-shifter
$\theta$, do not change. If they would change, we could send messages faster
than the speed of light. A computation that goes beyond the approximations in
eq.(2) of the process confirms that (see SI). If we set the phase
$\theta=\pi$, no four-fold coincidences will emerge because of destructive
four-photon interference. If now our detectors C and D both see a photon, we
know that there cannot be one photon in each detector A and B simultaneously.
That is because otherwise, we would have a four-fold event that does not occur
in this setting. Consequently, we can infer that crystal III has not created a
photon pair in this situation. Likewise, if $\theta=0$ and the detectors in C
and D click, crystal III appears to produce four times as much photon pairs as
in an incoherent situation. This conclusion is quite surprising because
crystal III could be far away from the phase-setting $\theta$, and it produces
photons via an entirely spontaneous process. It appears that we can make
conclusions about random processes that could be space-like separated, without
entanglement being involved in any obvious way.

Figure 1: Quantum interference by indistinguishable origins. (a) Two photon-
pair sources are aligned such that it cannot be distinguished in which of the
two crystal a photon pair is created. By changing the phase $\varphi$, the
resulting photon pairs can be enhanced and suppressed, which is denoted as
frustrated two-photon creation herzog1994frustrated . (b) A multi-photon
generalization. In this case, four photons in detectors A-D are created either
in crystal I and II or in crystals III and IV. By varying the phase $\theta$,
the four-photon count rate oscillates. Interestingly, the four crystals can be
separated by large distances, which allows for interesting investigations of
nonlocal influences.

###  .2 Integrated quantum photonic chip

Figure 2: Experimental setup. (a) Microscopic photograph of the integrated
silicon photonic chip. The total size of the chip is 3.8$\times$0.8 mm2. (b)
Experimental setup to observe four-photon interference in a silicon photonic
chip. An external pump laser is coupled to the chip. There, four spiral
waveguide sources produce photon quadruples (Source I and Source II can
produce a four-photon state as well as Source III and IV). The two
possibilities for creating four photons are made indistinguishable by
identifying all of their properties, including the photons’ paths. A relative
phase between the two possibilities is introduced via the pump laser in the
”Delay line” (at position marked $\theta$). A tunable MZI (MZI2) is used to
change from two-photon interference (which is used for alignment and
characterization) to four-photon interference. MZI, Mach-Zehnder
interferometer; UMZI, unbalanced Mach-Zehnder interferometer; WDMs:
wavelength-division multiplexers.

We explore this multi-photon interference effect using an integrated silicon
photonic chip. Integrated optics is considered a scalable platform for quantum
information processing. In recent years we have seen huge progress in
integrating all necessary components including lasers, photon-pair sources,
tunable linear optics, and detectors into a single chip silverstone2016silicon
; ono2019observation ; elshaari2020hybrid ; feng2020progress . Particularly
important for our experiment is high phase-stability. In short, we pump four
integrated photon-pair sources coherently and observe high-quality multi-
photon interference over a time span of more than 10 hours.

A photograph of the chip and a conceptual layout is shown in Fig.2. The device
was manufactured in a standard silicon photonics foundry. All the processes
required for state preparation, manipulation, and measurement are achieved
through thermo-optical phase shifters and Mach-Zehnder interferometers. All
phase shifters are controlled by one multichannel DC power supply and are
calibrated with one continuous-wave laser.

In our experiment, a 200 GHz bandwidth pulsed laser centred at 1550.11 nm acts
as the pump source. It is coupled into the chip using grating couplers and
split coherently into two paths using a Mach-Zehnder interferometer configured
to act as a 50:50 beam splitter (MZI1). The upper beam is split again at a
50:50 beam splitter (BS1) and acts as the pump for sources I and II, while the
lower beam is delayed and subsequently pumps sources III and IV. Each photon-
pair source (I-IV) is a 5 mm single-mode spiral silicon waveguide, with a high
$\chi^{(3)}$ optical nonlinearity. In these silicon waveguides, two pump
photons are annihilated, and signal-idler photon pairs can be generated with a
spontaneously four-wave mixing process (SFWM). The signal and idler photons
have a wavelength of 1545.32 nm and 1554.94 nm, respectively. After the
sources I and II, an unbalanced Mach-Zehnder interferometer (UMZI1) is used to
filter out the pump beam. Subsequently, the signal and idler photons in both
paths are then separated using another unbalanced Mach-Zehnder interferometer
(UMZI2). We estimated that the average error introduced by phase shifters in
(U)MZIs was less than 1%. The two-arm length difference of UMZI1 is set to 80
$\upmu$m, which shows a free spectral range of approximately 4.8 nm. The
interferometer UMZI2 has two arms with a 40 $\upmu$m length difference and a
free spectral range of approximately 9.6 nm. The extinction ratio of both
filters was estimated exceeding 20 dB (see SI for more details). In this way,
we can achieve four photons in four paths as $|a\rangle|b\rangle$ and
$|c\rangle|d\rangle$.

In paths $b$ and $c$, we place an MZI to act as a beam splitter with a tunable
splitting ratio (MZI2). We use it for two functions (see inset). First, we can
set the splitting ratio to zero, such that the photons remain in their paths.
This allows us to observe two-photon frustrated pair-creation, as shown in
Fig.1a. We use this to calibrate our chip and confirm the coherent creation of
two-photon pairs, as demonstrated by Herzog et al. in bulk optics
herzog1994frustrated  and Ono et al. on-chip ono2019observation . The second
setting of the MZI2 swaps the paths of the photons in $b$ and $c$, resembling
the configuration in Fig.1b. After the tunable MZI, we use another pair of
UMZI2 to combine signal and idler photons into one path.

The second, lower pump (in the Delay Line region, depicted in violet) is
combined with the photons created in sources I and II using two UMZI1s (in the
Combination region, depicted in red). The overlap is temporally controlled
such that it enters into source III and IV at the same time as the photon
pairs created in sources I and II. This is necessary to ensure the coherence
criteria, such that it cannot be distinguished whether the photons are created
in the sources I and II or in III and IV. Note that in contrast to Fig.1, we
introduce the phases $\varphi$ and $\theta$ into the laser which pumps source
III and IV, rather than into the generated single photons. In this way, we can
observe multiple oscillation cycles in one phase cycle (0 to 2$\pi$). It can
also be considered that we have changed the phase $\varphi$ or $\theta$
depicted in Fig. 1 with multiple times (details in SI).

The signal and idler photons are subsequently coupled out from the chip to a
single-mode fibre array through two end couplers. For each output, we first
use a broadband filter to filter out the pump light, and the signal and idler
photons are divided into two optical fibres by using wavelength-division
multiplexers (WDMs) with a bandwidth of 100 GHz. The signal and idler photons
were delivered into four off-chip superconducting nano-wire single-photon
detectors for detection and further time correlation analysis (see SI for more
details).

###  .3 Experimental results

Figure 3: Interference fringe for the two-photon quantum state. (a) The x-axis
shows the phase $\varphi$, while the y-axis shows the number of photon pairs
in two detectors. Each phase setting was measured for 1 second. The visibility
is close to 100%, which confirms high-quality coherence between the two
origins of the photon pairs. (b) Interference fringe of single-photon counts.
Blue: Signal Photons, red: Idler Photons. Partial distinguishability, photon
loss, unequal count rates and nonlinear noisy photons from the sources and
dark counts in the detectors lead to a decrease in visibility. Figure 4:
Interference of four photons created in two separate locations. (a) The
interference fringe has a visibility of 78.3$\pm$11.6% which clearly
demonstrates that the four photons are indeed generated, with high quality, in
two different locations. Here, each data collection took 30 minutes. (b)
Fluctuation results of two-photon coincidence counts. As expected, the two-
photon counts are nearly constant, and cannot explain the high visibility of
the four-photon state, thus demonstrating genuine four-photon interference.

The full controllability and high stability of the device make it possible for
the practical observation of quantum interference between different creation
processes, as shown in Fig.1a and b. To confirm the quality of the device, we
first demonstrate the two-photon frustrated pair generation process of Fig.1a
herzog1994frustrated . We set MZI2 in the chip to the configuration where the
photons stay in their path. At this time, our device can be regarded as two
separated two-photon nonlinear interferometers, and we can modulate the phase
in any of the interferometers. Considering to measure photons corresponding to
the term $|c\rangle|d\rangle$, the resulting quantum state up to first order
of SFWM is

$\displaystyle|\psi\rangle_{2}=2|a\rangle|b\rangle+(1+e^{i\varphi})|c\rangle|d\rangle.$
(3)

Note that the amplitude before term $|a\rangle|b\rangle$ is meaningless. In
Fig.3, we show two-photon constructive and destructive interference results
with each data point accumulated for 1 sec. To minimize multiphoton events,
only 300 $\upmu$W of the pulsed laser was coupled into the chip. Fig.3a shows
the coincidence counts of signal and idler photons by varying $\varphi$. The
interference fringes are fitted with $1+V\sin[\pi(\varphi-\varphi_{c})/T]$,
where $V$ is the fringe visibility, $\varphi_{c}$ is the initial phase, and
$T$ is the oscillation period. The fringe visibility $V$ is defined as
$V=(d_{\rm{max}}-d_{\rm{min}})/(d_{\rm{max}}+d_{\rm{min}})$, where
$d_{\rm{max}}$ and $d_{\rm{min}}$ are the maximum and minimum of the fitted
data, respectively. The visibility of the coincidence fringe was estimated as
100.0$\pm$5.1% with no background subtraction (the same below). Fig.3b shows
single counts of signal (solid blue line) and idler photons (solid red line).
As expected, they also vary depending on $\varphi$ with the same period of the
fringe obtained from coincidence counts. Visibilities of them are
43.0%$\pm$4.0% (solid blue line) and 52.3%$\pm$3.1% (solid red line),
respectively.

In the final step, we aim to observe the interference depicted in Fig.1b. To
do so, we set the MZI2 into the setting where the incoming photons swap their
paths. We get a two-photon state, which is expressed as

$\displaystyle|\psi\rangle_{2}=$
$\displaystyle|a\rangle|c\rangle+|b\rangle|d\rangle$ (4)
$\displaystyle+e^{i\theta/2}|a\rangle|b\rangle+e^{i\theta/2}|c\rangle|d\rangle.$

As we introduce the phase in the pump laser, it emerges in both terms created
by crystals III and IV. Correspondingly, the four-photon state is expressed as

$|\psi\rangle_{4}=\left(1+e^{i\theta}\right)|a\rangle|b\rangle|c\rangle|d\rangle.$
(5)

From these equations, if we change the phase $\theta$, four-photon coincidence
counts will increase or decrease, while two-photon coincidence counts will
keep unchanged. This is a multiphotonic generalization of two-photon
frustrated down-conversion.

We experimentally record four-photon coincidence counts when varying the phase
$\theta$, as shown in Fig.4a. To reduce the impact of multiphoton noise, we
set the pump power as 980 $\upmu$W. The integral time of each point is 30 min.
The four-photon coincidence fringe shows a four-fold oscillation frequency,
which is perfectly consistent with the theoretical prediction. The visibility
of the four-photon coincidence fringe was estimated as 78.3$\pm$11.6%.

Two-photon coincidence counts with linear fitting (ab, ac, bd, cd) are given
in Fig.4b, which hardly changed with the phase. The small fluctuation of two-
photon coincidence counts is mainly due to the experimental measurement
errors, and cannot explain the very large oscillation shown in Fig. 4a.
Besides phases $\varphi$ and $\theta$, we also adjust other phases on the
silicon chip and have observed similar frustrated interference phenomenon,
which is given in Supplemental Information. These results prove that we
observe a new type of nonlocal multi-photon quantum interference, which cannot
be understood by the behaviour of local properties such as individual photon
pair productions.

## 3\. DISCUSSION AND CONCLUSION

We have shown experimentally that it is possible to create a four-photon
quantum state coherently in two separate locations. As a consequence, we were
able to observe genuine four-photon interference which does not occur in
single-photon or two-photon states from the crystals. This was made possible
by a highly stable, low-loss integrated photonics chip with four nonlinear
photon-pair sources that are pumped coherently.

We would like to compare and contrast the multi-particle generalizations of
the famous Hong-Ou-Mandel (HOM) interference-effect. There, multiple photons
are mixed in multiports. If the output path combinations created by the
multiport are indistinguishable, the different possibilities
interferetillmann2015generalized . This is even true if a subset of the
photons is distinguishable menssen2017distinguishability ;
jones2020multiparticle . This interference effect is at the heart of
BosonSampling experiments wang2019boson . In contrast to that, rather than
superposing intrinsic photonic properties such as the path of the photons, in
our experiment, the two origins of the multi-photon state are
indistinguishable and therefore in a coherent superposition. This introduces
the notion of space in two different ways. First, the two locations of the
four-photon creation process can be far apart. Second, the creation of a
single four-photon event uses two crystals that can be at a large distance
from each other. This opens entirely new experimental ways to study non-
locality via multiphoton interference.

Beyond the immediate physical interest, our work also suggests advances in the
context of quantum technology applications. First and foremost, for the
photonic quantum system continuing to be scaled up, the quality of photonic
sources and system losses need to be further improved paesani2019generation ;
zhang2019generation ; wang2020integrated . This involves the high-level
control of on-chip photon-pair sources paesani2019generation ; feng2019chip ;
paesani2020near ; liu2020high ; feng2020progress . Our experiment demonstrates
a new level of control, by coherently creating and overlapping highly-
distinguishable photon pairs from different on-chip sources. Thereby it opens
the possibility of more complicated on-chip multi-photon interference, which
is one requirement for a different type of special-purpose quantum computing
scheme gu2019quantum . Also, destructive multi-photon interference in the
exact way as demonstrated here is the basis of several new proposals for the
generation of important photonic quantum states. This includes efficient new
ways to generate heralded multi-photon and high-dimensional entangled states
krenn2020conceptual . The new interference effect can readily be studied in
connection to quantum metrology polino2020photonic . Furthermore, numerous
proof-of-concepts applications have used the idea of path identity to enhance
quantum imaging, microscopy, or spectroscopy hochrainer2021quantum . The main
advantage of these applications is the ability to probe objects with one
wavelength for which detects are not available, and detecting the result in a
wavelength that can easily be observed lemos2014quantum ;
kalashnikov2016infrared ; kviatkovsky2020microscopy . The very same advantage
can be utilized in applications exploiting the new interference effect.

To conclude, we want to emphasize again that this effect might be best
understood as interference between two possible ways to create photon
quadruplets, rather than interference of the photons themselves. As the four
locations of the photon creations can be spatially separated in a number of
ways, variations of this experiment will allow for investigations of quantum
nonlocality that does not rely in any obvious way on quantum entanglement.

## MATERIALS AND METHODS

Theory for photon pair generation. The interaction Hamiltonian for the photon
pair generation by SFWM in the silicon waveguide is described as

$\hat{H}_{2}=i\hbar\chi(a^{\dagger}b^{\dagger}-ab),$ (6)

where $\hbar$ is the reduced Planck constant, $\chi$ is proportional to the
third-order nonlinear susceptibility $\chi^{3}$ and the amplitude of the pump,
and $a^{\dagger}(a)$ and $b^{\dagger}(b)$ are the creation (annihilation)
operators in paths $a$ and $b$ feng2019generation . Therefore, the time
evolution of the quantum state is given by

$|\Psi\rangle=e^{-i(H/\hbar)}|vac\rangle,$ (7)

where $|vac\rangle$ is the vacuum state. In Fig. 1(b) of the main text, the
interaction Hamiltonian is described as

$\displaystyle\hat{H}_{4}=$
$\displaystyle\frac{1}{2}i\hbar\chi(a^{\dagger}b^{\dagger}+c^{\dagger}d^{\dagger}+a^{\dagger}c^{\dagger}+e^{i\theta}b^{\dagger}d^{\dagger}$
(8) $\displaystyle-ab-cd-ac-e^{i\theta}bd),$

We assume that the pump light is strong and can be treated as a classical
oscillator, and leading to the full state in eq.2. The state of four fold term
is

$\displaystyle|\psi\rangle=$ $\displaystyle
g^{2}(|2a,2b,0,0\rangle+|2a,0,2c,0\rangle+e^{2i\theta}|0,2b,0,2d\rangle$ (9)
$\displaystyle+|0,0,2c,2d\rangle)+\sqrt{2}g^{2}(|2a,b,c,0\rangle+e^{i\theta}|a,2b,0,d\rangle$
$\displaystyle+|a,0,2c,d\rangle+e^{i\theta}|0,b,c,2d\rangle)+\color[rgb]{1,0,0}{g^{2}(1+e^{i\theta})|a,b,c,d\rangle}.$

We can see that exactly only one term exists when post-selecting one photon in
each path (depicted in red), which stands for four-photon interferences (i.e.
its amplitude changes when the phase $\theta$ changes).

Loss. Based on measurements on test structures on the same chip, we estimate
losses as: 5.5 dB per grating coupler, 4 dB per end coupler, 2 dB cm-1 of
spiral waveguide propagation, 0.1 dB per cross and 0.2 dB per 2$\times$2
multimode interferometer BS. All measurements are done at wavelength of
1550.11 nm. The associated channel efficiencies, $\eta_{s}$ and $\eta_{i}$ are
estimated via $\eta_{s(i)}=\frac{C}{N_{i(s)})}$, where $C$, $N_{s}$ and
$N_{i}$ are, respectively, the coincidence, signal and idler channel counts.
Because of the interference between photonic sources, we set the device as two
separate two-photon nonlinear interferometers first and record two-photon
interference counts. Then we select the maximum values to estimate channel
efficiencies, and they are $\eta_{a}=-13.2$ dB, $\eta_{b}=-13.2$ dB,
$\eta_{c}=-14.7$ dB and $\eta_{d}=-17.7$ dB, respectively.

Acknowledgements. The authors thank Alba Cervera-Lierta for useful comments on
the manuscript. M.K. thanks Armin Hochrainer, Mayukh Lahiri, Manuel Erhard and
Anton Zeilinger for valuable discussions on the concept of path identity. This
work was supported by the National Natural Science Foundation of China (NSFC)
(Nos. 61590932, 11774333, 62061160487, 12004373), the Anhui Initiative in
Quantum Information Technologies (No. AHY130300), the Strategic Priority
Research Program of the Chinese Academy of Sciences (No. XDB24030601), the
National Key R & D program (No. 2016YFA0301700), the Postdoctoral Science
Foundation of China (No. 2020M671860) and the Fundamental Research Funds for
the Central Universities. This work was partially carried out at the USTC
Center for Micro and Nanoscale Research and Fabrication. M.K. acknowledges
support from the Austrian Science Fund (FWF) through the Erwin Schrödinger
fellowship No. J4309.

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Author contributions: All authors contributed extensively to the work
presented in this paper. L.T.F, M.Z. Y.J.C and D.X.D. prepared the samples.
L.T.F., Y.J.C, M. K. and X.F.R. performed the measurements, data analyses and
discussions. D.L., G.P.G., G.C.G. and M.K. provided support in experiment and
conducted theoretical analysis. L.T.F., M.K. and X.F.R. wrote the manuscript,
and M. K. and X.F.R. supervised the project. Competing interests: The authors
declare no competing financial interests. Date and materials availability:
Supplementary information is available in the online version of the paper.
Correspondence and requests for materials should be addressed to M.K. or
X.F.R..