# The Hong-Ou-Mandel experiment: from photon indistinguishability to
continuous variables quantum computing

N. Fabre 1, M. Amanti 2, F. Baboux 2, A. Keller 2,3, S. Ducci 2 and P. Milman
2 1 Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040
Madrid, Spain  2Université Paris Cité, CNRS, Laboratoire Matériaux et
Phénomènes Quantiques, 75013 Paris, France  3Univ. Paris-Sud 11, Institut de
Sciences Moléculaires d’Orsay (CNRS), Bâtiment 350–Campus d’Orsay, 91405 Orsay
Cedex, France

###### Abstract

We extensively discuss the Hong-Ou-Mandel experiment taking an original phase-
space-based perspective. For this, we analyze time and frequency variables as
quantum continuous variables in perfect analogy with position and momentum of
massive particles or with the electromagnetic field’s quadratures. We discuss
how this experiment can be used to directly measure the time-frequency Wigner
function and implement logical gates in these variables. We also briefly
discuss the quantum/classical aspects of this experiment providing a general
expression for intensity correlations that explicitize the differences between
a classical Hong-Ou-Mandel like dip and a quantum one. Throughout the
manuscript we will often focus and refer to a particular system based on
AlGaAs waveguides emitting photon pairs via spontaneous parametric down
conversion, but our results can be extended to other analogous experimental
systems and to different degrees of freedom.

##  I Introduction

One of the greatest challenges when describing a scientific revolution is to
structure, from the future, a linear and coherent narrative of how ideas
emerged, and how they were discussed, combined, proved and disproved so as to
finally converge into a theory. This difficulty is particularly pronounced in
what concerns quantum physics. Not only there is no complete consensus about
its interpretation, but also it’s not clear wether it is indeed possible to
have a single only interpretation of it. In addition, and for some people,
most importantly, many of the existing interpretations require completely
abandoning some principles of physics which are known as “classical”. In spite
of all that, we keep on doing research, predicting and confirming physical
phenomena that would not have been imaginable if we hadn’t agreed at some
point that controversial as it is, using whatever interpretation we chose,
quantum physics is the most accurate theory to describe and predict physical
phenomena in a given scale.

Nevertheless, and as a consequence of this debate, a recurrent question arises
when observing and studying quantum phenomena: what is so quantum about all
that ? Is this really quantum ? Indeed, it’s a hard to answer question, that
becomes even harder when followed by its twin sibling: “what’s the classical
counterpart of the situation I am studying? ”, “what should I compare my
results to ?” “Is it possible to compare quantum and classical behavior ?”.

It’s clear that quantum computing and quantum information science brought some
light into these problems, by defining quantum advantage, or supremacy, as the
power of some quantum algorithms and protocols to move problems from one
complexity class to another. This is an objective method to separate quantum
from non-quantum phenomena. Putting all this into work and using quantum
mechanics to solve practical problems by designing quantum machines is what A.
Aspect, termed the second quantum revolution.

From the physics and quantum optics point of view, the quantum/classical
frontier and questions about how to define it and place it seems less
explicit, but we cannot avoid plunging into them when we thumb through Mr. A.
Aspect’s lectures, in particular Aspect (2019), seeking for inspiration for
writing the introduction of a paper in his honor. What does he mean when he
says something is quantum or classical ? Does it make sense ?

Starting from these questions, we concentrate on some interesting aspects of
one of the experiments Mr. A. Aspect uses to illustrate the differences
between quantum and classical physics in his lectures, the Hong-Ou-Mandel
(HOM) experiment Hong et al. (1987). We’ll see how this experiment, that we
briefly describe and discuss in Section II, followed by its classical analog
(Section II.1) can be associated to the time-frequency phase space (Section
III), and finally provide some applications of our results (Sections IV).

##  II The Hong-Ou-Mandel experiment

Quantum phenomena are usually, if not always, associated to interference
effects. The Hong-Ou-Mandel experiment, depicted in Fig. 1 is an example of a
clever way to use an interferometer to probe photon indistinguishability using
a consequence of the bosonic commutation relations. Indeed the experiment
shows that indistinguishable photons (even generated by independent sources
with no a priori phase relation) interfere in such way that they both exit the
same output port of the interferometer. We call this behavior “bunching” and
its experimental signature is the absence of coincidence detections (the
famous HOM dip). To understand this behavior we consider two photons in
different spatial modes, each one of them associated to one input arm of the
interferometer. We define the following notation:
$\hat{a}_{i}^{\dagger}(\omega_{i})\ket{0}=\ket{1_{\omega_{i}}}=\ket{\omega_{i}}$,
so that the corresponding state can be described in the general form

$\ket{\psi}=\int\int f(\omega_{1},\omega_{2})\ket{\omega_{1},\omega_{2}}{\rm
d}\omega_{1}{\rm d}\omega_{2},$ (1)

where the indices $i=1,2$ denote different spatial modes and $\omega$ is an
arbitrary photonic degree of freedom, as the transverse position and momentum
or frequency. In the present contribution, we consider pure states for
simplifying reasons (even though general expressions are given in Section
II.1) and we focus on the frequency degree of freedom, since we will take as
case study an experimental system consisting of semiconductor waveguides
emitting frequency and/or polarization correlated photon pairs by spontaneous
parametric down conversion Orieux et al. (2011a); Ducci et al. (2021);
Autebert et al. (2015). A more detailed description of the experimental device
can be found in Section IV.

In the particular case of photons generated from independent sources we have
that $f(\omega_{1},\omega_{2})=g(\omega_{1})g(\omega_{2})e^{i\phi}$ in Eq.
(1), which means that the spectral amplitude of each photon is the same up to
an arbitrary and random phase factor (a consequence from the fact that photons
are independent).This is the original context of the first HOM experiment Hong
et al. (1987).

The two photons are sent into the two input arms of an interferometer, as
shown in Fig. 1. After the $50/50$ beam-splitter, the input spatial modes are
combined in the following way:
$\hat{a}_{A}(\omega)=(\hat{a}_{1}(\omega)+\hat{a}_{2}(\omega))/\sqrt{2}$ and
$\hat{a}_{B}(\omega)=(\hat{a}_{1}(\omega)-\hat{a}_{2}(\omega))/\sqrt{2}$,
where $A$ and $B$ are the output spatial modes where photons are detected in
coincidence after leaving the beam-splitter. Thus, the state described by Eq.
(1) is transformed into

$\displaystyle\ket{\psi}=$ (2) $\displaystyle\frac{1}{2}\int\int
g(\omega_{1})g(\omega_{2})e^{i\phi}(\hat{a}_{A}^{\dagger}(\omega_{1})\hat{a}_{B}^{\dagger}(\omega_{2})-\hat{a}_{A}^{\dagger}(\omega_{2})\hat{a}_{B}^{\dagger}(\omega_{1})+$
$\displaystyle\hat{a}_{A}^{\dagger}(\omega_{1})\hat{a}_{A}^{\dagger}(\omega_{2})-\hat{a}_{B}^{\dagger}(\omega_{1})\hat{a}_{B}^{\dagger}(\omega_{2}))\ket{0}{\rm
d}\omega_{1}{\rm d}\omega_{2}.$

Two single photon detectors are used to measure temporal correlations between
the photon exiting the two output ports. We now express the coincidence
detection probability. The terms in the third line of Eq. (2) correspond to
photon bunching and do not lead to coincidence detections, since both photons
leave the interferometer in the same spatial mode. They are thus ignored from
now on and we focus on the two first terms in the r.h.s. of Eq. (2). The
coincidence detection probability ${\cal C}$ between photons in modes $A$ and
$B$ is given by the square of the absolute value of this term. We have thus:

${\cal
C}=\frac{1}{2}-\frac{1}{2}\int\int|g(\omega_{1})|^{2}|g(\omega_{2})|^{2}{\rm
d}\omega_{1}{\rm d}\omega_{2},$ (3)

and using the normalization condition
$\int\int|g(\omega_{1})|^{2}|g(\omega_{2})|^{2}{\rm d}\omega_{1}{\rm
d}\omega_{2}=1$, we obtain that ${\cal C}=0$ irrespectively of the phase
$\phi$. Consequently, we can conclude that single indistinguishable photons
generated by independent sources sent in a balanced interferometer - i.e.,
where both photons go through the same optical path but in different spatial
modes - always bunch and exit the interferometer from the same port.

This important result can be made even more interesting if we add some spice
to it, in the form of a controllable optical path difference between the input
photons. In this case, the initial state right before impinging the beam-
splitter in Fig. (1) is characterized by the wave-function
$f(\omega_{1},\omega_{2})=g(\omega_{1})g(\omega_{2})e^{i\omega_{1}\tau}$,
where we dropped the phase $\phi$, since it doesn’t play a role in the
coincidence detection, and added a time and frequency-dependent phase that
comes from the path difference between the two arms. It’s easy to verify that
in this case

$\displaystyle{\cal
C}(\tau)=\frac{1}{2}-\frac{1}{2}\int\int|g(\omega_{1})|^{2}|g(\omega_{2})|^{2}e^{i(\omega_{1}-\omega_{2})\tau}{\rm
d}\omega_{1}{\rm d}\omega_{2}$
$\displaystyle=\frac{1}{2}\left(1-|\int\tilde{g}(t)\tilde{g}^{*}(t-\tau){\rm
d}t|^{2}\right)$ (4)

where $\tilde{g}(t)$ is the Fourier transform of $g(\omega)$ at point $t$ .
Eq. (II) is in general different from zero. If the overlap between the Fourier
transforms calculated at the two points appearing in Eq. (II) is different
from one, we say that the two incoming photons are not completely
indistinguishable, and in the limit of complete distinguishability, ${\cal
C}(\tau)=1/2$. This result is the same one would expect to find if photons
were billiard balls arriving into a path bifurcation in pairs and are randomly
sent throught one output path or the other, $A$ or $B$. This is why we refer
to this situation as the classical one.

In spite of having been initially thought to probe the indistinguishability of
photons coming from independent sources, the first experimental demonstrations
of the Hong-Ou-Mandel effect used photon pairs generated by spontaneous down
conversion (SPDC) processes Hong et al. (1987). The reason for that is the
experimental difficulty at that time to built independent single photon
sources, a difficulty overcame in Kaltenbaek et al. (2006). Nevertheless, as
we will see throughout the present contribution, the HOM effect presents many
other facets than the one it was initially designed for. Its applications are
numerous, and for presenting some of them it’s important to provide, for the
general case of (1), the coincidence probability. Using the same manipulations
to obtain (3) we have that

${\cal C}(\tau)=\frac{1}{2}-\frac{1}{2}\int\int
f(\omega_{1},\omega_{2})f^{*}(\omega_{2},\omega_{1})e^{i(\omega_{1}-\omega_{2})\tau}{\rm
d}\omega_{1}{\rm d}\omega_{2},$ (5)

which is the central equation of the present manuscript. Simple as it seems to
be, Eq. (5) can provide information about entanglement Eckstein and Silberhorn
(2008), time-frequency phase space representation Douce et al. (2013), and be
used to characterize resources for quantum simulation Francesconi et al.
(2020, 2021), metrology Chen et al. (2019) and quantum error correction Fabre
et al. (2020a).

Figure 1: Modified HOM experiment including a time delay $\tau$ in one input
arm and frequency displacements of $\mu$ in the other input arms. As described
in Sections IV and IV.3, the experiment can also be seen as composed of a
preparation step (for state engineering and for an alternative to implement
frequency displacements for quantum state measurement, red box), a
manipulation step (that can be used to implement logical gates, as described
in IV.3 or for quantum state measurement as well, greed box) and a detection
step based on coincidence measurements (blue box).

###  II.1 On the “quantumness” of the HOM experiment

Before moving to the central point of the present contribution we’ll discuss a
little further the interpretation of the HOM experiment as probing quantum
properties of light. In Sadana et al. (2019), an experiment involving
intensity correlations between classical fields with a fixed, but random phase
reference showed that it is indeed possible to reproduce the HOM dip with
close to $100\%$ visibility. While the theoretical model presented in Sadana
et al. (2019) uses a classical description of light, we provide here a
theoretical description using intensity correlations between coherent states
which are sent at the two input ports of a HOM-like interferometer and where
intensity correlations are calculated between the signals recorded at the two
output ports, as in Sadana et al. (2019).

Using the same spatial modes labelling given in the introduction, we have
that, after the introduction of a time delay $t$ in one arm (say, arm 2) of
the HOM interferometer and recombination in the beam-splitter, the normalized
intensity correlation between modes $A$ and $B$ can be written as

${\cal
C}(t)=\frac{\langle\hat{N}_{A}\hat{N}_{B}\rangle}{\langle\hat{N}_{A}\rangle\langle\hat{N}_{B}\rangle},$
(6)

where

$\hat{N}_{i}=\int\hat{a}_{i}^{\dagger}(\omega)\hat{a}_{i}(\omega){\rm
d}\omega,$ (7)

with $i=A,B$, and where the average is taken both over classical parameters
(as the phase $\phi)$ and on the quantum state. Eq. (6) can be rewritten using
:

$\displaystyle\hat{N}_{A(B)}=\frac{1}{2}\int\left(\hat{a}_{1}^{\dagger}(\omega)e^{-i\omega
t}\pm\hat{a}_{2}^{\dagger}(\omega)\right)\times$
$\displaystyle\left(\hat{a}_{1}(\omega)e^{i\omega
t}\pm\hat{a}_{2}(\omega)\right){\rm d}\omega$
$\displaystyle=\frac{1}{2}\left[\hat{N}_{1}+\hat{N}_{2}\pm(\hat{\cal
I}_{1,2}(t)+\hat{\cal I}_{1,2}^{\dagger}(t))\right],$ (8)

with $\hat{\cal
I}_{1,2}(t)=\int\hat{a}_{1}^{\dagger}(\omega)\hat{a}_{2}(\omega)e^{i\omega
t}{\rm d}\omega$ and
$\hat{N}_{1(2)}=\int\hat{a}_{1(2)}^{\dagger}(\omega)\hat{a}_{1(2)}(\omega){\rm
d}\omega$. Finally, ${\cal C}(t)$ can be written as:

${\cal C}(t)=\frac{\langle\hat{N}^{2}\rangle-\langle(\hat{\cal
I}_{1,2}(t)+\hat{\cal
I}_{1,2}^{\dagger}(t))^{2}\rangle}{(\langle\hat{N}\rangle^{2}-\langle\hat{\cal
I}_{1,2}(t)+\hat{\cal I}_{1,2}^{\dagger}(t)\rangle^{2})},$ (9)

where $\hat{N}=\hat{N}_{1}+\hat{N}_{2}$ is related to the total field
intensity. Expression (II.1) is general and can be used also for non-pure
states Ollivier et al. (2021), and a similar expression was obtained in
Woolley et al. (2013) expressed as temporal correlations for microwave fields.
As a matter of fact, it is easy to verify that this expression reduces to the
coincidence probability if we consider as input state a biphoton by noticing
that in this case

$\displaystyle\hat{\cal I}_{1,2}(t)\hat{\cal
I}_{1,2}^{\dagger}(t)=\hat{N}_{1}+$
$\displaystyle\int\int\hat{a}_{1}^{\dagger}(\omega_{A})\hat{a}_{2}^{\dagger}(\omega_{B})\hat{a}_{1}(\omega_{B})\hat{a}_{2}(\omega_{A})e^{i(\omega_{A}-\omega_{B})t}{\rm
d}\omega_{A}{\rm d}\omega_{B},$

and

$\displaystyle\hat{\cal I}_{1,2}^{\dagger}(t)\hat{\cal
I}_{1,2}(t)=\hat{N}_{2}+$ (11)
$\displaystyle\int\int\hat{a}_{1}^{\dagger}(\omega_{B})\hat{a}_{2}^{\dagger}(\omega_{A})\hat{a}_{1}(\omega_{A})\hat{a}_{2}(\omega_{B})e^{i(\omega_{B}-\omega_{A})t}{\rm
d}\omega_{A}{\rm d}\omega_{B},$

and $\langle\hat{\cal I}_{1,2}^{\dagger}(t)\rangle=\langle\hat{\cal
I}_{1,2}(t)\rangle=\langle\hat{\cal
I}_{1,2}^{\dagger}(t)^{2}\rangle=\langle\hat{\cal I}_{1,2}(t)^{2}\rangle=0$.

###  II.2 Application to coherent states

We now compute (II.1) in the case where
$\ket{\psi}=\ket{\alpha}_{1}\ket{\beta}_{2}$, with $\beta=|\alpha|e^{i\phi}$
and $\alpha=|\alpha|$, in analogy to the calculations in Sadana et al. (2019)
done using classical fields. We have that

$\displaystyle{\cal C}(t)=\frac{1}{\langle\hat{N}\rangle^{2}_{\gamma}-4({\rm
Re}\left[\int|\alpha(\omega)|^{2}e^{i\phi}e^{i\omega t}{\rm
d}\omega\right])^{2}}\times$
$\displaystyle(\langle\hat{N}\rangle^{2}_{\gamma}-2{\rm
Re}\left[(\int|\alpha(\omega)|^{2}e^{i\phi}e^{i\omega t}{\rm
d}\omega)^{2}\right]-$ (12) $\displaystyle 2{\rm
Re}\left[\int\int|\alpha(\omega_{A})|^{2}|\alpha(\omega_{B})|^{2}e^{i(\omega_{A}-\omega_{B})t}{\rm
d}\omega_{A}{\rm d}\omega_{B}\right]),$

where $\langle\hat{N}\rangle_{\gamma}=2\int|\alpha(\omega)|^{2}{\rm d}\omega$.
Note that we haven’t averaged over $\phi$ yet: in order to do so, let’s take a
look at expression (II.2) for $t=0$. In this case, we have that ${\cal
C}(0)=\frac{\langle\sin^{2}{\phi}\rangle_{\phi}}{1-\langle\cos{\phi}\rangle_{\phi}^{2}}$.
From this expression, we see that if the phase $\phi$ is uniformly distributed
in the continuous interval ranging from $0$ to $2\pi$, we have ${\cal
C}(0)=1/2$ with a visibility of $1/2$. If the phase can take only two values
with equal probability and $\phi\in\\{0,\pi\\}$, we have that ${\cal C}(0)=0$
with a theoretical visibility of $1$, which reproduces the two-photon
coincidence case in the classical regime. Finally, if the two possible and
equally probable phases are such that $\phi\in\\{\pi/2,3\pi/2\\}$, ${\cal
C}(0)=1$ and the visibility is zero. These results are in accordance with the
ones in Sadana et al. (2019), and the authors use them to argue that observing
a near $100\%$ visibility dip in intensity correlations is not necessarily a
quantum signature. Accordingly, the authors suggest that by adding a wave-
particle duality-like experiment in the output of the HOM interferometer one
could then obtain observable differences between quantum and classical
properties, which is of course also true.

The conclusion that it is possible to mimic the HOM dip with classical fields
should be put into perpective by inspecting Eqs. (II.2) and (9). We can see
that the average zero correlation effect is caused by the $\phi$ dependent
term, which is a first order interference term. One can thus easily identify
and isolate it, and argue that quantum effects are associated to second-order
correlation terms only Bouchard et al. (2020). By keeping only these terms, it
is clear that the HOM dip cannot be reproduced with classical fields. Another
way to argue in this sense is by noticing that the $\phi$ dependent term
presents rapid oscillations that are averaged to zero depending on the
detection sensibility to fluctuations in a given frequency range. This is
precisely the situation of the Hanbury Brown and Twiss experiment, for
instance, where only purely second order correlation terms remain and such
first order phase-dependent effects do not play a role.

From our point of view, it is interesting to notice that the spectral
intensity overlap between classical fields in Eq. (II.2) doesn’t lead to
$100\%$ visibility of the intensity correlations while being always $\leq 0$,
so that in the absence of the first order interference term intensity
correlations are always $1/2\leq{\cal C}(t)\leq 1/4$, with visibility $1/2$.

As a conclusion, even though Sadana et al. (2019) presents an interesting
classical situation analogous to the HOM dip, it is not clear in our view how
to compare both situations and wether this classical result challenges the
usual interpretation of the HOM experiment.

##  III The Wigner function

We now turn back once again to the biphoton situation. Those working in
optics, and used to interference effects, are familiar to expressions as (5),
where two amplitudes overlap. The relation between the two overlapping
amplitudes in (5) is such that it may ring a bell in those who are also
familiar with the phase space representations of quantum mechanics. As a
matter of fact, Eq. (5) is pretty similar to the Wigner function of a massive
particle, or the Wigner function associated to the quadrature state of a
single mode electromagnetic field, which is given by:

$W(x,p)=\int\bra{x-\frac{q}{2}}\hat{\rho}\ket{x+\frac{q}{2}}e^{ips}{\rm dq}.$
(13)

For a pure state, Eq. (13) reduces to

$W(x,p)=\int\psi^{*}(x-\frac{q}{2})\psi(x+\frac{q}{2})e^{ipq}{\rm dq},$ (14)

where $x$ and $p$ are the position and momentum variables or, equivalently,
two orthogonal quadratures of the electromagnetic field, and $\psi(x)$ is the
wave-function in, say, the position basis. We see that Eqs. (14) and (5) are
indeed very similar but still, not identical.

Indeed, when comparing (14) to (5) many differences appear: in first place, it
seems that we have the equivalent to a two particle system in Eq. (5), since
we have a two-variable integral (in $\omega_{1}$ and $\omega_{2}$ variables).
Also, there is no displacement in the frequency variables, as there is in the
position one in (14). Finally, (14) describes the quantum state of a particle
or of the electromagnetic field, but what about (5) ? We will handle these
issues one by one to conclude that, indeed, (5) and (14) are pretty much the
same, or at least provide the same type of information about the quantum state
of a system in the case one limits the discussion to the case where each mode
is occupied by one photon only.

To start with, and for pedagogical reasons, we perform a change of variables
in (5) such that: $\omega_{\pm}=(\omega_{1}\pm\omega_{2})/2$. In this case,
$f(\omega_{1},\omega_{2})=f^{\prime}(\omega_{+},\omega_{-})$ and
$f(\omega_{2},\omega_{1})=f^{\prime}(\omega_{+},-\omega_{-})$. Then, following
Douce et al. (2013), we define $g(\omega_{-})g^{*}(-\omega_{-})=\int
f^{\prime}(\omega_{+},\omega_{-})f^{\prime*}(\omega_{+},-\omega_{-}){\rm
d}\omega_{+}$. In the case where $f^{\prime}(\omega_{+},\omega_{-})$ is a
separable function in the $\omega_{\pm}$ variables, $g(\omega_{-})$ is the
wavefunction associated to the $\omega_{-}$ variable and provides all the
information about this collective mode. In the general case, we’re considering
a marginal of the wave-function. In any case, using the previously introduced
definition, we have that:

${\cal C}(\tau)=\frac{1}{2}-\frac{1}{2}{\rm Re}[\int
g(\omega_{-})g^{*}(-\omega_{-})e^{i\omega_{-}\tau}{\rm d}\omega_{-}],$ (15)

where we have transformed the double integral of Eq. (5) into a single one, in
the form of (14).

Now we deal with the issue of the equivalent to the position displacement
appearing in Eq. (14), which in Eq. (15), a displacement in variable
$\omega_{-}$ (while momentum displacement in (14) is equivalent to the
$\omega_{-}$ dependent time displacement appearing in (15)). It’s clear that
transformations of the type $\pm\omega_{-}\rightarrow\mu\pm\omega_{-}$ would
transform (15) into the exact mathematical analog of (14) with the
replacements $\tau\rightarrow p$, $\omega_{-}\rightarrow 2q$ and
$x\rightarrow\mu$. By analyzing the interferometer in Fig. 1 we see that a
frequency displacement of $\mu$ in frequency in one of the interferometer’s
arms would modify the photonic state and its wave-function in such a way that
the coincidence detection at the outputs of the interferometer would give

${\cal C}(\mu,\tau)=\frac{1}{2}-\frac{1}{2}{\rm Re}[\int
g(\mu+\omega_{-})g^{*}(\mu-\omega_{-})e^{i\omega_{-}\tau}{\rm d}\omega_{-}].$
(16)

We leave the discussion about possible ways to experimentally implement such
displacements to Section IV. The integral in Eq. (16) has now the same form as
the one in (14). So, what’s the meaning of all that ? Is this really a Wigner
function, as (14) ? And if it’s the case, what type of information can it
provide about the state, or at least about the part of the state associated to
variable $\omega_{-}$ ? These issues were addressed in Douce et al. (2013);
Boucher et al. (2015-08-04); Fabre et al. (2020b), leading to the conclusion
that indeed, the coincidence detection of the output of a HOM interferometer
can be expressed as:

${\cal C}(\mu,\tau)=\frac{1}{2}-\frac{1}{2}W_{-}(\mu,\tau),$ (17)

where $W_{-}(\mu,\tau)=\int
g(\mu+\omega_{-})g^{*}(\mu-\omega_{-})e^{2i\omega_{-}\tau}{\rm d}\omega_{-}$
is the chronocyclic Wigner function associated to the variable $\omega_{-}$ of
the photon pair. In the case of a biphoton wavefunction which is separable in
the variables $\omega_{-}$ and $\omega_{+}$, this information completely
characterizes the quantum state of the biphoton associated to the variable
$\omega_{-}$. In particular, we can notice that the usual shape of the HOM dip
is nothing but a cut in the phase space of varying $\tau$ and at $\mu=0$ of a
Gaussian spectral function.

An important point to address is the fact that we’re dealing here with the
quantum description of the photon pair detected in coincidence, so the Wigner
function describes the photonic quantum state in frequency. The quantum
statistics of the field it refers to give to $W_{-}(\mu,\tau)$ specific
properties having a quantum interpretation. For instance, in Eckstein and
Silberhorn (2008), it was shown that if ${\cal C}(0,\tau)>1/2$, then the
photon pair is frequency entangled. We can show that having ${\cal
C}(\mu,\tau)>1/2$ is also an entanglement witness, and this corresponds
precisely to saying that the existence of negative points of $W_{-}(\mu,\tau)$
is a proof that the photon pair is frequency entangled.

Interpreting the integral in (16) as a Wigner function contributes to
providing a physical picture of frequency and time of single photons as
quantum continuous variables. Even though it’s known that such degrees of
freedom are continuous, in most protocols and applications they are either
discretized Maltese et al. (2020); Marcikic et al. (2002); Lu et al. (2019) or
distributed in basis of discrete modes Lamata and Leon (2005); Law et al.
(2000) so that their genuine continuous character is not fully exploited. As
shown in Fabre et al. (2020a), the single photon time and frequency variables
can also be used for quantum information, computing and communication
protocols analogously to “usual” quantum continuous variables in quantum
optics - as the vibrational state of trapped ions or the quadratures of the
electromagnetic field. This fact opens the perspective of broadening even more
the applications of single photon-based quantum information related protocols,
since frequency and time variables are shown to be the continuous version of
polarization degrees of freedom, for instance.

As a concluding remark for this part, we have seen that the coincidence
probability in a HOM experiment is a direct measurement of the Wigner function
associated to the $\omega_{-}$ variable of a photon pair. This may seem
restrictive, but it is only a consequence of the choice of the experimental
system and its relevant degrees of freedom. We have shown in Douce et al.
(2013) that the HOM provides a way to characterize other photonic degrees of
freedom as well, as for instance the Wigner function associated to the
transverse position and momentum coordinates. For these degrees of freedom, it
is easier to perform spatial rotations of the field’s profile so that one can
directly access different combinations of variables, as for instance the one
associated not only to the difference but also to the sum of the transverse
coordinates of the photons.

It is also possible to adapt the ideas proposed in Douce et al. (2013) so as
to access the full Wigner function of a single photon. In this case, instead
of seeing the photon pair produced by a SPDC process as produced by the
physical device, we can interpret it as the product of a series of logical
gates that are applied to a separable state. For this, it was shown in Fabre
(2022); Fabre et al. (2022), that the frequency beam-splitter operation
defined as

$\hat{U}\ket{\omega_{1},\omega_{2}}=\ket{\omega_{+},\omega_{-}}.$ (18)

plays the role of the entangling interaction that naturally occurs in the SPDC
process, which are physically associated to the phase matching condition and
energy conversation. Thus, the produced entangled state that outputs a SPDC
device can formally be obtained by starting from two initially separable
single photons described by the wavefunction $\ket{\psi}=\iint
d\omega_{1}d\omega_{2}f(\omega_{1})g(\omega_{2})\ket{\omega_{1},\omega_{2}}$
which are then entangled by a frequency beam-splitter operation.

As a result of this interpretation we can interpret the function $f$ (resp.
$g$) of the separable initial photon pair as being the same as the function
modeling the energy conservation (resp. the phase-matching conservation) of a
SPDC process. Thus, the coincidence measurement corresponds to measuring the
spectral function of an initial single photon that has been entangled (changed
variables) by the device. In Ref. Fabre (2022), we also proposed a direct way
for implementing such an operation with non-linear crystals in cascade.
However, more efficient ways could be investigated within light-matter
interaction using split-ring resonators for instance.

Finally, if a separable two-photon state enters into the HOM interferometer,
the coincidence probability corresponds to the spectrogram:

$S(\mu,\tau)=\frac{1}{2}(1-\absolutevalue{\int
f^{*}(\omega_{1})g(\omega_{1}-\mu)e^{i\omega_{1}\tau}{\rm d}\omega_{1}}^{2}).$
(19)

The spetrogram is the absolute value of the windowed Fourier transform. If one
spectral function is known, let us say $g$ which can be considered as a window
function, a phase of retrieval algorithm allows reconstructing the amplitude
and phase of the spectral function $f$.

As a conclusion, the HOM interferometer provides various ways for measuring
the amplitude and phase of the spectral function of single and two-photon
states.

##  IV Quantum state manipulation and engineering

We have seen that interpreting the HOM experiment as a direct measurement of
the Wigner function of the biphoton associated to the variable $\omega_{-}$
requires being capable of implementing displacements in the time-frequency
phase space. While time displacement are in practice easy to implement in
quantum optics by temporal delay lines, frequency displacement require more
involved experimental techniques at the single photon level, some of them
based on non-linear optic. Nevertheless, frequency displacements are necessary
not only to have access to the whole chronocyclic phase phase space and
characterize the single photon state, but also for quantum state engineering.
A possibility to implement frequency shifts is by using electro-optic
modulators which are already available, working at the single photon level.
Such devices are used to perform the spectral tomography of single photons
Davis et al. (2018) and the direct signature of such frequency shifts was
observed through the decrease of the visibility of the dip in HOM
interferometry in Chen et al. (2021).

In this section we take a different direction and provide solutions to
implement phase space displacements using techniques as pump engineering and
post-selection.

To start with, we provide the details of a specific experimental system to
illustrate our ideas.

###  IV.1 An experimental context

Counterpropagating phase-matching offers a high versatility to engineer the
spectral wavefunction and the exchange statistics of photon pairs, as probed
in the HOM experiment. A schematics of our semiconductor chip-integrated
source of photons pairs is shown in Fig. 2a, and a SEM image is shown in Fig.
2b. The source is a Bragg ridge microcavity made of a stacking of AlGaAs
layers with alternating aluminum concentrations Orieux et al. (2011b, 2013).
It is based on a transverse pump scheme, where a pulsed laser beam impinging
on top of the waveguide (with an incidence angle $\theta$) generates pairs of
counterpropagating and orthogonally polarized photons (signal and idler)
through SPDC. Along the epitaxial direction momentum conservation is
implemented through a quasiphase-matching structure of AlGaAs layers (shown in
orange in Fig. 2a) with alternating high and low values of the effective
nonlinear coefficient. The Bragg mirrors provide both a vertical microcavity
to enhance the pump field and a cladding for the twin-photon modes.

Figure 2:  (a) Working principle and (b) SEM image of an AlGaAs ridge
microcavity generating frequency-entangled photon pairs by SPDC in a
transverse pump geometry.

Two nonlinear interactions can occur simultaneously in the device Orieux et
al. (2013); we consider here the one that generates a TE-polarized signal
photon (propagating along $z>0$, as sketched in Fig. 2a) and a TM-polarized
idler photon (propagating along $z<0$). When the photons are generated close
to degeneracy and for a narrow pump spectrum, the biphoton state can be
written in the following form:

$\ket{\psi}=\iint
d\omega_{s}d\omega_{i}f_{+}(\omega_{s}+\omega_{i})f_{-}(\omega_{s}-\omega_{i})\ket{\omega_{s},\omega_{i}}$
(20)

The joint spectral amplitude (JSA)
$\phi(\omega_{s},\omega_{i})=f_{+}(\omega_{+})f_{-}(\omega_{-})$, where
$\omega_{\pm}=\omega_{s}\pm\omega_{i}$, gives the probability amplitude to
measure a signal photon at frequency $\omega_{s}$ and an idler photon at
frequency $\omega_{i}$. The function $f_{+}$, reflecting the condition of
energy conservation, is given by the spectrum of the pump beam, while the
function $f_{-}$, corresponding to the phase-matching condition, is determined
by the spatial properties of the pump beam:

$f_{-}(\omega_{-})=\int_{-L/2}^{L/2}dz\,\Phi(z)e^{-i(k_{\rm
deg}+\omega_{-}/\bar{v}_{\rm g})z}$ (21)

Here, $\Phi(z)$ is the pump amplitude profile along the waveguide direction,
$L$ is length of the waveguide, $\bar{v}_{g}$ is the harmonic mean of the
group velocities of the twin-photon modes and $k_{\rm deg}=\omega_{p}\text{\rm
sin}(\theta_{\rm deg})/c$, with $\omega_{p}$ the pump central frequency, $c$
the velocity of light and $\theta_{\rm deg}$ the pump incidence angle
corresponding to the production of frequency-degenerate photons.

###  IV.2 Pump beam engineering

Measurements in quantum physics can be performed in different ways. Usually,
experimental measurement apparatus can only measure quantum states in a given
specific basis. Thus, a solution to circumvent this and obtain full
information about the state is to manipulate the state so that a measurement
in a given state means that, in fact, the system was previously in another
state which is considered to be the measured one.

One can criticize such a technique, since the measured state is not the state
that was actually produced, and the limit between state production and
measurement is not clearly defined. In the present section, we take a loose
position on this debate, and propose quantum state measurement techniques
based on quantum state engineering.

In the previous subsection we have seen that the JSA of the produced photon
pair can be controlled by modifying the pump laser beam properties, as its
angle and position of incidence. In Boucher et al. (2015), we have exploited
this fact to propose methods not only to engineer exotic and useful quantum
states of the photon pair, but also to measure them by implementing
displacements in different directions of the phase space.

Using the results of the previous section, if the dimensions of the waveguide
are large with respect to the pump waist, i.e. in the limit where
$L\rightarrow\infty$, $f_{-}$ can be approximated as the Fourier Transform of
the spatial profile of the pump beam:

$f_{-}(\omega_{-})\approx\tilde{\varphi}\left(\frac{\omega_{-}}{\bar{v}_{g}}\right)$
(22)

We start by considering the situation depicted in Fig. 3 where a gaussian pump
beam with waist $w_{p}$ impinges onto the source at an angle $\theta$ and
position $z_{0}$. The field distribution along the $z$ axis is $\Phi(z)\propto
e^{-{(z-z_{0})^{2}\cos^{2}{\theta}}/{w_{p}^{2}}}e^{i(k\sin\theta)z}$ and
therefore $f_{-}$ reads:

$f_{-}(\omega_{-})\propto
e^{-i\omega_{-}\tau_{0}}e^{-\tfrac{\left(\omega_{-}-\omega_{-}^{(0)}\right)^{2}}{\Delta\omega^{2}}}$
(23)

with $\tau_{0}=z_{0}/\bar{v}_{g}$, $\Delta\omega=\bar{v}_{g}\cos{\theta}/\pi
w_{p}$ and $\omega_{-}^{(0)}=(k\sin\theta-k_{\mathrm{deg}})\bar{v}_{g}$.

From a general complex amplitude representation describing a pure state
$f_{-}(\omega_{-})$, the Wigner function $W(\tau,\mu)$ at points $\tau,\mu$ of
the phase space is given by:

$W(\tau,\mu)=\int_{-\infty}^{\infty}d\omega_{-}f_{-}(\mu-\frac{\omega_{-}}{2})f_{-}^{*}(\mu+\frac{\omega_{-}}{2})e^{i\tau\omega_{-}}.$
(24)

Using for instance the expression obtained in (23), it is easy to see that
this corresponds to a gaussian Wigner function centered at point
$\tau=\tau_{0}$, $\mu=\omega_{-}^{(0)}$. In this situation, shifting the
pumping spot $z_{0}$ is equivalent to realizing displacements in the $\tau$
axis of the phase space while changing the angle of incidence $\theta$ of the
pump beam corresponds to shifting the state along the $\mu$ axis.

More complex states can be obtained by engineering the pump beam. Indeed two
identical beams impinging at $z_{a}$ and $z_{b}$ generate a superposition of 2
coherent states displaced along the $\tau$ axis, which is a state analogous to
a Schrödinger cat in position and momentum phase space. The orthogonality
between the two Gaussian states is ensured if
$\frac{(z_{a}-z_{b})}{\bar{v}_{g}}\gg\frac{1}{\Delta\omega}$. A superposition
of two Gaussian packets can also be obtained along axis $\mu$ by using 2
different angles of incidence $\theta_{a}$ and $\theta_{b}$ impinging at the
same point $z_{0}$ (see Fig. 3), and this type of state was shown to be a
resource for quantum metrology in Chen et al. (2019); Fabre and Felicetti
(2021). We can generalize even more these Schrödinger cats to arbitrary
configuration of pump beams, for instance a compass state, which is a
superposition of four coherent states whose utility for metrology has been
pointed out elsewhereToscano et al. (2006) and studied in the context of SPDC
in Fabre and Felicetti (2021). To obtain it, a set of 4 beams is required: 2
pairs of beams impinging at 2 different points separated by a distance $\Delta
z$, each pair consisting of 2 beams symmetrically tilted with respect to the
degeneracy angle.

Figure 3:  Pump engineering for producing a Schrödinger cat-like state in
frequency. A divided pump that impinges on the medium at different angles lead
to different phase matching conditions which are angle dependent and center
the state’s frequency distribution at different values. By changing both
pump’s angles while keeping their angular distance constant one displaces the
center of both frequency distributions in the same direction and by the same
amount. This is equivalent to performing frequency displacements in phase
space, a strategy that can be used both for quantum state engineering and for
measuring the Wigner function using the HOM interferometer at different points
of the time frequency phase space.

As a conclusion, we can see that pump engineering is a valuable tool not only
to design interesting frequency-time entangled state but also to implement
displacements in phase space thus enabling quantum state measurement.

###  IV.3 Exploiting cavity effects

In the following, we describe the possibilities of quantum state manipulation
and engineering enabled by the the interplay between cavity effects and
temporal delay between photons of a pair. The method can be adapted and
applied to a large variety of systems, either bulk or integrated, thus
increasing their flexibility and the richness of the generated states. Here we
illustrate these concepts by taking the example of the biphoton state
generation in an AlGaAs waveguide based on a modal phase matching scheme in a
collinear geometry. In this case, the phase-velocity mismatch among the three
interacting beams is compensated by a multimode waveguide dispersion
engineering by confining the fields using Bragg reflectors Maltese et al.
(2020).

The phase matching condition depends not only on the properties of the pump,
as detailed in the previous section, but also on the properties of the device.
Then, another possibility for quantum state engineering is using (and
modifying, but this is a more difficult issue) the latter. Usually SPDC based
devices have a phase-matching condition which depends on the temperature, for
instance. Using this effect, frequency displacements enabling the direct
measurement of the Wigner function $W_{-}(\mu,\tau)$ were implemented Tischler
et al. (2015).

Other ways to engineer the non-linear susceptibility of a non-linear crystal
can be found in Dosseva et al. (2016) for the single photon and in Drago and
Brańczyk (2022) for the two-mode squeezed regime.

In the following, we describe some interesting properties of the AlGaAs source
that can be used for quantum state engineering which are not based on
temperature variation effects and turn out to be extremely interesting for
quantum state engineering.

In first place, the facets of the non-linear waveguide are reflective, due to
the refractive index contrast between semiconductor and air. For this reason,
a cavity effect occurs, which produces a temporal grating that generates a
frequency comb in the spectrum of the generated photons. The peaks of the comb
are spaced by $\bar{\tau}/2$, where $\bar{\tau}$ is the time a photon takes to
make a roundtrip in the cavity, and in Maltese et al. (2020) we have shown
that by changing the pump frequency we can engineer different phase profiles
for the comb, with different symmetry properties. An example can be seen in
Fig. 4, that shows as well the dependency of the peak visibility with the
reflectivity of the cavity, and the expected HOM profile for the existing
device, for which the reflectivity $R\approx 0.3$. This cavity effect enables
for instance the discretization of such frequency combs, which are rising a
considerable interest in the community, since they can be used as qudits
Maltese et al. (2020), that can have applications in quantum key distribution,
for instance Autebert et al. (2016).

(a) Horizontal axis: time spacing between peaks in units of $\bar{\tau}/2$.
Vertical axis: cavity reflectivity $R$. In the existing experimental device,
used in Maltese et al. (2020), $R=0.3$. The color code refers to the
coincidence probability that oscillates between $0$ and $1$ according to the
time delay. $P=1$ for an anti-symmetric state and $P=0$ for a symmetric one.
The symmetry of the produced state is related to the resonance or anti-
resonance of the pump beam with the cavity frequency and also to the produced
states. On the left, the pump beam frequency is resonant with the cavity
frequency: peaks are spaced of $\bar{\tau}/2$, half the time taken for a
photon to make a roundtrip in the cavity, and even and odd multiples of
$\bar{\tau}/2$ peaks have the same phase. On the right, the pump beam is anti-
resonant with the cavity frequency and peaks with different parity have
different phases. The corresponding produced states can be associated to GKP-
like states in different basis.

(b) Simulated HOM using as the cavity reflectivity $R=0.3$ Maltese et al.
(2020), as indicated in the Figure.

Figure 4: Signatures in the HOM interference profile of the cavity induced
time-frequency comb structure of the phase matching associated function of the
generated photon pair.

From the continuous variables perspective, large superpositions of numerous
highly localized states can be used to encode error-protected qubits. Such
states present redundancy and translational symmetry in the ideal case of an
infinite superposition of infinitely localized states. These properties are
interesting for quantum error correction, as first proposed by Gottesman-
Kitaev and Preskill (GKP) Gottesman et al. (2001). The so-called GKP states
are robust against small displacements in phase space (small with respect to
the comb spacing in position and momentum representation), and they can be
used not only to encode quantum information but also as a ressource to correct
displacement errors that affect different states defined in continuous
variables Baragiola et al. (2019). In addition, it has been shown that it is a
sufficient non-Gaussian element to complete Gaussian-based quantum computation
and turn it universal Baragiola et al. (2019).

In spite of its several applications and fundamental interest for quantum
computing with continuous variables, the experimental production of GKP state
is extremely challenging in the quadrature representation, since it involves
creating highly non-Gaussian states. Nevertheless, some experimental results
exist in the microwave range Campagne-Ibarcq et al. (2020) and using the
motional state of trapped ions e Neeve et al. (2022). In Barros et al. (2017),
it was shown that spatial gratings can be used to engineer GKP states using
the transverse position and momentum degrees of freedom of single photons. As
for time and frequency variables, we can say for once that we are lucky: the
cavity structure of AlGaAs semiconducting device naturally creates a comb
structure, as mentioned. We have shown in Fabre et al. (2020a) that such comb
can, indeed, be seen as an entangled GKP state, and measuring time or
frequency of one the photons enables correcting the errors of the other, in a
reminiscence of a measurement-based quantum computing scheme.

The main idea behind our results is the fact that the effective non-linear
interaction producing the photon pair in a cavity can be seen as a combination
of universal gates, as defined in Fabre et al. (2022), acting on a separable
ideal pair of GKP states. The effect of such gates is to add noise to the
ideal state, under the form of a classical distribution of displacement
operators, and entangle both states. Such operations can be seen as a small
quantum circuit that produces entangled GKP states which were proven to be a
resource for quantum error correction.

Additionally, frequency encoded GKP states can also be manipulated thanks to
the HOM interferometer. As a matter of fact, in the subspace formed by GKP
states, time and frequency displacements of fixed amounts, corresponding to
the combs’ interspacing, act as Pauli matrices do on qubits. Thus, by fixing
the time delay in one of the arms of the HOM interferometer, one obtains the
equivalent to the application of a quantum gate analogous to the Pauli matrix
$\sigma_{x}$ to a GKP state, as was shown in Fabre et al. (2020a).

Of course one can use different techniques to generate frequency comb
structures, as placing a cavity in one arm of the HOM interferometer, as done
in Lu et al. (2003). Nevertheless, using the natural cavity-like structure in
the device studied in Fabre et al. (2020a) is interesting since it avoids
extra sources of losses.

##  V Discussion and conclusion

We have revisited the HOM experiment, a benchmark in quantum optics and
quantum physics, taking a phase space perspective which enables interpreting
time and frequency degrees of freedom of single photons as genuine quantum
continuous variables, in perfect analogy with position and momentum. We also
discussed the quantum nature of this experiment and how classical results
reproducing the ones due to photon statistics may deserve to be interpreted
under a different perspective.

We have focused on a specific experimental set-up and shown how exotic states
can be engineered in such a device by manipulating a classical pump beam.
Also, we identified the “natural” output of such a device to a entangled state
with quantum properties which were shown to be useful for quantum error
correction. The presented original approach to the HOM experiment opens the
perspective to novel applications of single photon based protocols and shows
that quantum phenomena haven’t stopped surprising us.

As an interesting perspective, we should mention the extension of the
presented results to the atomic version of the HOM experiment Lopes et al.
(2015), performed by Alain Aspect and co-workers.

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