# Time-frequency as quantum continuous variables

Nicolas<EMAIL_ADDRESS>Camille Noûs2, Arne Keller3,4 and Pérola
Milman3 1Departamento de Óptica, Facultad de Física, Universidad Complutense,
28040 Madrid, Spain  2 Laboratoire Cogitamus  3Université de Paris, CNRS,
Laboratoire Matériaux et Phénomènes Quantiques, 75013 Paris, France  4
Department de Physique, Université Paris-Saclay, 91405 Orsay Cedex, France

###### Abstract

We present a second quantization description of frequency-based continuous
variables quantum computation in the subspace of single photons. For this, we
define frequency and time operators using the free field Hamiltonian and its
Fourier transform, and show that these observables, when restricted to the one
photon per mode subspace, reproduce the canonical position-momentum
commutation relations. As a consequence, frequency and time operators can be
used to define a universal set of gates in this particular subspace. We
discuss the physical implementation of these gates as well as their effect on
single photon states, and show that frequency and time variables can also be
used to implement continuous variables quantum information protocols, in the
same way than polarization is currently used as a two-dimensional quantum
variable.

##  I Introduction

Since its beginnings and more than ever nowadays, the perspectives opened by
using quantum mechanics in processing information and securing communication
have excited the scientific community. The quantum computer - whatever meaning
one gives to it -, that about fifteen years ago appeared as a distant mirage
and an almost imaginary device, seems closer and closer to reality,
materializing and capitalizing in all the senses of the word, in a scalable
way, the superior performance of some quantum algorithms. Apart from the
advertised scientific and non-scientific implications of quantum information
research, it seems reasonable to think that information processing machines
must make a different use of quantum physics - and consequently, technologies
- than the old-fashioned computer we are using to type these words. Also, in
spite of the relevance, one thinks such technological breakthroughs have to
fundamental research, we can safely state that quantum information changed the
way one approaches not only information theory, but also quantum physics in
general and, in particular, quantum optics.

Quantum field statistics, the field intensity, or its modal decomposition - in
summary, all possible ways of measuring, manipulating and extracting
information from radiation - are the basic tools of different quantum
computation, and communication architectures based in quantum optics. Starting
from a generic perspective, information can be extracted from, carried or
processed by quantum fields by describing them essentially in two distinct
ways: using observables with a continuous or with a discrete spectrum. Of
course, both regimes can also be combined into hybrid devices van Loock et al.
(2008), which gather the advantages and drawbacks of continuous and discrete
variables.

In quantum optics, the traditional examples of continuous variables are the
field’s quadratures - linear combinations of the electric and magnetic field’s
amplitudes -, whose measurement using, for instance, homodyne detection, can
provide complete information about the quantum state of radiation, encoded in
the field’s statistics. Alternatively, one can perform measurements that count
Sridhar et al. (2014); Zhong et al. (2021); Hamilton et al. (2017) and
correlate photons. The photon number is often associated to encoding
information in discrete variables, represented by the photon number itself or
the presence or absence of photons. It is also possible to use single photons
degrees of freedom - as the polarization or the orbital angular momentum - as
discrete variables, and this is particularly useful in quantum communication
and in testing fundamental aspects of quantum mechanics Aspect et al. (1982);
Giustina et al. (2015); Shalm et al. (2015).

Single photon’s continuous degrees of freedom, as their longitudinal momentum
or transverse position and momentum can be associated to continuous variables,
also leading to an alternative implementation of quantum computation Tasca et
al. (2011), quantum communication Walborn et al. (2007) and fundamental tests
of quantum mechanics Borges et al. (2012). In Tasca et al. (2011), in
particular, the isomorphism between the propagation equation of single photons
transverse variables and the Schrödinger equation of a massive particle was
used to define continuous variables universal logical gates manipulating the
transverse momentum and position of the photon’s wave function with simple
linear quantum optical devices or spatial light modulators (SLM).

In the present paper, we provide a comprehensive second quantization based
description of frequency and time variables of single photons, showing how to
manipulate and use these degrees of freedom as quantum continuous variables.
For such, we introduce universal quantum operators acting on a subspace
consisting of single photons which occupy distinct modes. These gates are
perfectly analogous to the ones introduced in Lloyd and Braunstein (1999-02);
Braunstein and van Loock (2005) for generic position and momentum-like
variables. Frequency and time variables do not obey, in the general case, the
same commutation relation as position and momentum. Nevertheless, in the
subspace where each mode is at most occupied by one photon, such commutation
relations can be retrieved from the bosonic commutation relations and provided
that we consider frequency modes in experimentally reasonable ranges, avoiding
for instance the issue of negative frequencies.

Once these commutation relations between operators associated to time and
frequency measurements of single photons are obtained, they can be used to
define a set of universal gates in the same form as the ones manipulating
momentum and position variables or the field’s quadratures. This is the main
scope of the present paper. We’ll then introduce the physical contexts where
such a universal set can be implemented, as well as its properties and ways to
represent and detect quantum information encoded in frequency and time. In
particular, we’ll see that even though highly non-linear frequency and time
gates are required, it is possible to define a quantum information processing
architecture both in a measurement-based and in the circuit model with current
technology. Finally, we’ll discuss the phase space representation of frequency
and time as quantum continuous variables, which is a useful tool to provide
intuition on some non-classical aspects of states. We’ll see however that the
non-classical picture given by the constructed phase space is different from
the one of quadratures.

The paper is organized as follows. In Sec. II, we define the spectral and
temporal properties of single photons through a Hilbert space description. We
underline the possibility of defining time and frequency operators and the
importance of being in the single-photon subspace. In Sec. III, the universal
set of gates for time and frequency continuous variables is defined based on
the mathematical analogy with quadrature position-momentum variables.
Experimental implementations are also discussed. In Sec. IV, we recall the
principles of the chronocyclic phase space description of single photons and
consider in details the physical interpretation of such a phase space.

##  II States, operators and commutation relations

###  II.1 Preliminary tools

A single photon pure state at mode $i$ with frequency $\omega$ is described by
the application of the creation operator to the vacuum state:
$\hat{a}_{i}^{\dagger}(\omega)\ket{\text{vac}}=\ket{\omega}_{i}$. The label
$i$ can be polarization, a spatial mode - as the transverse propagation
direction -, or any other combination of modes that plays the role of an
ancillary mode that creates distinguishability between each photon. We can of
course also define the annihilation operator such that
$\hat{a}_{i}(\omega)\ket{\omega^{\prime}}_{i}=\delta(\omega-\omega^{\prime})\ket{\text{vac}}$.
In addition, the commutation relation between creation and annihilation
operator can be written as:

$[\hat{a}_{\alpha}(\omega),\hat{a}_{\beta}^{\dagger}(\omega^{\prime})]=\delta(\omega-\omega^{\prime})\delta_{\alpha\beta}\mathds{I},$
(1)

where $\alpha$ and $\beta$ are auxiliary modes. We also have that
$[\hat{a}_{\alpha}(\omega),\hat{a}_{\beta}(\omega^{\prime})]=0$ and
$[\hat{a}^{\dagger}_{\alpha}(\omega),\hat{a}^{\dagger}_{\beta}(\omega^{\prime})]=0$.

If we consider to be in the narrow-band approximation Fedorov et al. (2005),
so that the central frequency of the spectral distribution is much larger than
its spectral width, integrals can be extended over the whole frequency
spectrum, and the Fourier transform of the annihilation operator is the
annihilation operator at the arrival time $t$:

$\hat{a}(t)=\frac{1}{\sqrt{2\pi}}\int_{\mathds{R}}d\omega\hat{a}(\omega)e^{-i\omega
t},$ (2)

the same being valid for the creation operation, of course. We also have that

$[\hat{a}_{\alpha}(t),\hat{a}_{\beta}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})\delta_{\alpha\beta}\mathds{I},$
(3)

where $\alpha$ and $\beta$ are auxiliary modes, and
$[\hat{a}_{\alpha}(t),\hat{a}_{\beta}(t^{\prime})]=0$ and
$[\hat{a}^{\dagger}_{\alpha}(t),\hat{a}^{\dagger}_{\beta}(t^{\prime})]=0$.

We stress that in the present description time is seen not as a parameter but
as a degree of freedom associated to the arrival time of photons in a
detector.

###  II.2 States

A general single photon pure state can be decomposed in the time basis or,
equivalently, in the spectral basis as,

$\ket{\psi}=\int_{\mathds{R}}d\omega
S(\omega)\hat{a}^{\dagger}(\omega)\ket{\text{vac}}.$ (4)

The spectrum $S(\omega)$ is the Fourier transform of the time of arrival
distribution and
$\absolutevalue{S(\omega)}^{2}=\absolutevalue{\bra{\omega}\ket{\psi}}^{2}$
denotes the probability density of detecting a photon with frequency $\omega$.
We can, of course, also construct from this principles general mixed single
photon states described by a density matrix.

The space of states we consider in the present contribution consists of a
collection of $n$ single photon states in $n$ different ancillary modes. This
space will be called from now on ${\cal S}_{n}$, where $n$ is the number of
distinguishable modes and also the number of photons. It means that only cases
where there is at most one photon per mode are considered.

A general pure state in ${\cal S}_{n}$ can be written as

$\ket{\psi}=\int d\omega_{1}...\int
d\omega_{n}F(\omega_{1},...,\omega_{n})\hat{a}_{1}^{\dagger}(\omega_{1})...\hat{a}_{n}^{\dagger}(\omega_{n})\ket{0},$
(5)

where the spectral function $F(\omega_{1},...,\omega_{n})$ is normalized to
one : $\int d\omega_{1}...\int
d\omega_{n}|F(\omega_{1},...,\omega_{n})|^{2}=1$. One of the goals of defining
a set of universal gates is to show that states as (5) can be approached with
arbitrary precision, from an arbitrary initial state using a finite set of
operations.

Of course, for $n\geq 2$, we’ll have to consider, in addition to separable
states, frequency and time entangled states. In order to introduce the basics
of entanglement in ${\cal S}_{n}$, let’s discuss one of the simplest and most
currently processes used to produce pairs of entangled states, which is
spontaneous parametric down conversion (SPDC). In this case, we have that the
two-photon output state can be written as :

$\ket{\psi}=\iint
d\omega_{s}d\omega_{i}\text{JSA}(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_{i})\ket{\text{vac}},$
(6)

where we considered that the creation operators of signal and idler photons
are distinguishable and associated to different auxiliary modes labeled $s$
and $i$. The JSA acronym denotes “joint spectral amplitude”. One can also
define its Fourier transform, the joint temporal amplitude, or JTA. The JSA
contains all the information about the two photon frequency state.

Of particular interest for analyzing entanglement in state Eq. (6) is its
decomposition in a set of orthogonal functions, as introduced in Law et al.
(2000); Walborn and Pimentel (2012); Lamata and Leon (2005). One can show that

$\ket{\psi}=\sum_{n=1}^{\infty}\Lambda_{n}\ket{f_{n}}_{a}\otimes\ket{g_{n}}_{b}.$
(7)

An effective Schmidt rank $K_{N}$ can be defined as
$1/K_{N}=\sum_{n=1}^{N}\Lambda_{n}^{2}$. If there is an $N$ such that
$K_{N}>1$, we can conclude that the state $\ket{\psi}$ is entangled.

We can study in details the particular case of a Gaussian shaped JSA,
corresponding to the Joint Spectral Intensity (JSI) shown in Fig. 1.The JSI
has no phase information, but displays the essential information for analyzing
correlations - and entanglement, when provided with phase information -
between two photons. In the context of SPDC, the JSI’s shape, and the
existence of different degrees of frequency correlation between signal and
idler photons, depend on the energy and momentum conservation of the non-
linear interaction generating the photon pair, in one hand, and the pump laser
spectral properties on the other hand. More specifically, the energy
conservation condition is related to the sum of frequencies of signal and
idler photons while the phase-matching condition is related to the difference
of frequencies Orieux et al. (2011).

Fig. 1 (a) displays a situation where the spectrum of signal and idler photons
are uncorrelated, so the Schmidt decomposition has only one term,
$\Lambda_{1}\ket{f_{1}}_{a}\ket{g_{1}}_{b}$. It corresponds to a separable
state of two photons, and this can be seen from the circular symmetry of the
JSI, which means that it can always be decomposed as a product of two
functions of any pair of variables related to $\omega_{s}$ and $\omega_{i}$ by
an unitary transformation: the circle has no privileged principal axis.

Notice that this situation is different from the one represented in Fig. 1
(b), where the JSI displays anti-correlation between variables $\omega_{s}$
and $\omega_{i}$. As a matter of fact, the JSI in Fig. 1 (b), is an ellipsoid
with principal axis $\omega_{\pm}=(\omega_{s}\pm\omega_{i})/\sqrt{2}$, which
means that the JSI can be written as product of two independent functions of
these two variables. By defining $\delta_{\pm}$ as the width of the
distribution in the $\omega_{\pm}=(\omega_{s}\pm\omega_{i})/\sqrt{2}$ axis, we
have that the ratio $r=\delta_{+}/\delta_{-}<1$. It’s a known result that the
degree of squeezing of the JSA, and consequently of the JSI, is proportional
to the degree of mode entanglement for pure quantum states Lamata and Leon
(2005): the Schmidt rank $K_{N}$ (or alternatively, the degree of mode
entanglement with the chosen mode partition) increases with decreasing $r$.

From the discussion above, we see that the JSA associated to the JSI in Fig. 1
(b) cannot be separated in a product of functions of variables $\omega_{s}$
and $\omega_{i}$ only, and these variables are correlated. Nevertheless, a
change of variables in the JSA to $\omega_{\pm}$ instead of $\omega_{s}$ and
$\omega_{i}$ transforms it into a separable function. If one could, in
addition to that, associate signal and idler photons as well to frequencies
$\omega_{+}$ and $\omega_{-}$ instead of $\omega_{s}$ and $\omega_{i}$, then
the Schmidt decomposition would have a single term and would correspond to a
product state. This is an important point for the proposed quantum computation
set-up that will be discussed in details in Sec. III. Before this, we
introduce time and frequency operators which will be shown to be analogous to
the quadratures of the electromagnetic field, expressed here in the simplified
dimensionless form $\hat{p}=i(\hat{a}^{\dagger}-\hat{a})/2$ and
$\hat{x}=(\hat{a}^{\dagger}+\hat{a})/2$. In this sense, frequency and time can
also be considered as continuous variables to encode quantum information.

###  II.3 Time and frequency operators

The time and frequency operators are defined as:

$\displaystyle\hat{t}_{a}=\int_{\mathds{R}}t\hat{a}^{\dagger}(t)\hat{a}(t)dt,$
(8)
$\displaystyle\hat{\omega}_{a}=\int_{\mathds{R}}\omega\hat{a}^{\dagger}(\omega)\hat{a}(\omega)d\omega.$
(9)

When applied to single photons states, these operators fulfill the eigenvalues
equation : $\hat{t}_{a}\ket{t}_{a}=t\ket{t}_{a}$ and
$\hat{\omega}_{a}\ket{\omega}_{a}=\omega\ket{\omega}_{a}$. The frequency
operator is proportional to the free Hamiltonian
$\hat{E}_{a}=\hbar\hat{\omega}_{a}$. As previously, we considered the narrow
band approximation of a photon with central frequency far from origin.
Consequently, the integration over the frequency can safely be considered as
covering all $\mathds{R}$. As for the time variable, it corresponds to the
Fourier transform of frequency for all practical purposes and is physically
associated to the time of detection conditioned to the fact that a detection
has indeed happened Giovannetti et al. (2015); Maccone and Sacha (2020).

Using Eq. (1), we can see that time and frequency operators do not commute in
the single photon (single mode) regime (see Appendix A):

$[\hat{\omega}_{a},\hat{t}_{a}]=i\mathds{I}.$ (10)

They form, together with the identity operator $\mathds{I}$, a three-
dimensional Heisenberg algebra in perfect analogy with the position and
momentum operators. This fact, not true in general for modes occupied by more
than one photon (see Appendix A), is essential for building a set of universal
gates which manipulate frequency and time as the universal gates defined for
position and momentum manipulate states defined in these basis. The present
manuscript is thus entirely based on considering
$\hat{x}\rightarrow\hat{\omega}$ and $\hat{p}\rightarrow\hat{t}$.

Also, from Eq. (10), we can use the Robertson’s inequality to derive a time-
energy Heisenberg inequality:

$\Delta\hat{E}_{a}\Delta\hat{t}_{a}\geq\hbar/2.$ (11)

Rather than expressing a classical Fourier transform relation, Eq. (11)
provides an operator description of the time and the frequency bandwidth at
the single photon level as quantum noise Fadel and Maccone (2021).

##  III Universal operations

We have introduced in the previous section the basic tools which will be
useful for the definition of a universal set of quantum operations using time
and frequency as continuous variables.

Before moving to its description, we recall the basics of continuous variables
universality, as introduced in Braunstein and van Loock (2005). The authors
show that it’s possible to define an universal set of unitary operators using
observables $\hat{x}$ and $\hat{p}$, and central to their result is the fact
that such observables obey the commutation relation
$\left[\hat{x},\hat{p}\right]=i\mathds{I}$ (where we have used dimensionless
operators for convenience). Such operators can be associated either to the
position and momentum variables of a particle or, in the context of quantum
optics, to the electromagnetic field’s quadrature.

A universal set of unitary operators is defined as a finite set that, when
combined, approaches with arbitrary precision any unitary operation which is a
polynomial in $\hat{x}$ and $\hat{p}$. There are of course many possible sets,
but we’ll focus on the following one :

${\cal
U}=\\{e^{i\hat{x}s},e^{i\frac{\pi}{4}(\hat{x}^{2}+\hat{p}^{2})},e^{i\hat{x}^{2}s},e^{i\hat{x}_{i}\otimes\hat{x}_{j}},e^{i\hat{x}^{3}s}\\}.$
(12)

We briefly explain the action and properties of each one of the unitary
operators in ${\cal U}$ : $e^{i\hat{x}s}$ is the displacement operator that
displaces momentum of an amount of $s$.
$e^{i\frac{\pi}{4}(\hat{x}^{2}+\hat{p}^{2})}$ is the Fourier transform, and
$e^{i\hat{x}^{2}s}$ is called a shear operation (governed by the parameter
$s$), that compresses one quadrature while also implementing a quadrature
rotation. As for $e^{i\hat{x}_{i}\otimes\hat{x}_{j}}$, it involves two
distinct modes, $i$ and $j$, and is an entangling operation that generates
displacements in one mode according to the state of the system in the other
mode. Finally, $e^{i\hat{x}^{3}s}$ is a non-Gaussian operation that is
essential to generate polynomials of order higher than $2$ using the canonical
commutation relation.

A first comment is that the set ${\cal U}$ acts on what one calls qumodes,
which are, in the case of the field’s quadratures, frequency, momentum or
polarization distinct modes. Each operation is thus defined independently for
each mode, except for $e^{i\hat{x}_{i}\otimes\hat{x}_{j}}$ that couples two
different modes.

As mentioned, universality of the set ${\cal U}$ is proven using the fact that
$\hat{x}$ and $\hat{p}$ obey the canonical commutation relation. Any other
pairs of observables obeying the same commutation relation can be used to
define a universal set of operations in the same form as the ones in ${\cal
U}$. Consequently, using such operators, it’s possible to generate a complete
computational space, provided that one identifies a set-up where such
operators make a physical sense and can be implemented.

From these observations, we move to the time and frequency variables
restricted to the single photons subspace. Using Eq. (10), we deduce that with
the set :

${\cal
W}=\\{e^{i\hat{\omega}s},e^{i\frac{\pi}{4}(\hat{\omega}^{2}+\hat{t}^{2})},e^{i\hat{\omega}^{2}s},e^{i\hat{\omega}_{i}\otimes\hat{\omega}_{j}},e^{i\hat{\omega}^{3}s}\\}$
(13)

we are capable of approaching, with arbitrary precision, any operation within
the $n$ single photons subspace ${\cal S}_{n}$ (in analogy with the $n$
qumodes one), i.e., we are capable of constructing unitary operators which
approach with arbitrary precision any polynomial in $\hat{\omega}$ and
$\hat{t}$ in ${\cal S}_{n}$.

The set ${\cal W}$ is then a universal one, in the sense of Braunstein and van
Loock (2005), in ${\cal S}_{n}$, for time and frequency operations.

A first comment is that all the operators in ${\cal W}$ are non-linear in
$\hat{a}(\omega),\hat{a}^{\dagger}(\omega)$, the annihilation and creation
operator in a given mode $\omega$. $\hat{a}^{\dagger}(\omega)\hat{a}(\omega)$
is the field’s intensity in this mode, which is itself a polynomial in
$\hat{x}(\omega)=\frac{\hat{a}(\omega)^{\dagger}+\hat{a}(\omega)}{\sqrt{2}}$
and
$\hat{p}(\omega)=i\frac{\hat{a}(\omega)^{\dagger}-\hat{a}(\omega)}{\sqrt{2}}$
and, for this reason, can be built from ${\cal U}$, as is the case for all
operators in ${\cal W}$.

With this in mind, that defines the computational space, we now briefly
discuss the operators in ${\cal W}$, their physical meaning and different ways
to implement them.

###  III.1 The displacement operator and the Fourier transform

The operator $e^{i\hat{\omega}_{a}s}$ is the generator of displacement in time
of a single photon in mode $a$, and it can be associated to the free
evolution. Restricted to the ${\cal S}_{n}$ subspace, the expansion of the
exponential leads to (see Appendix A and Fabre et al. (2020a))

${\cal{\hat{D}}}_{\hat{\omega}_{a}}(s)=e^{i\hat{\omega}_{a}s}=\int\hat{a}^{\dagger}(t+s)\hat{a}(t)dt,$
(14)

which corresponds to the annihilation of one photon at a given time and the
creation of a photon at a displaced time, as one would expect. Operator
$\hat{F}=e^{i\frac{\pi}{4}(\hat{\omega}_{a}^{2}+\hat{t}_{a}^{2})}$ acts as a
Fourier transform so that

${\cal{\hat{D}}}_{\hat{t}_{a}}(\mu)=\hat{F}{\cal{\hat{D}}}_{\hat{\omega}_{a}}(s)\hat{F}^{\dagger}=\int\hat{a}^{\dagger}(\omega+\mu)\hat{a}(\omega)d\omega.$
(15)

Operators ${\cal{\hat{D}}}_{\hat{t}_{a}}(\mu)$ and
${\cal{\hat{D}}}_{\hat{\omega}_{a}}(s)$ verify the Weyl’s algebra,

${\cal{\hat{D}}}_{\hat{\omega}_{a}}(s)\hat{D}_{\hat{t}_{a}}(\mu)=e^{is\mu/2}\hat{D}_{\hat{t}_{a}}(\mu){\cal{\hat{D}}}_{\hat{\omega}_{a}}(s),$
(16)

where we have again used the bosonic non-commutation relation Eq. (1) Not .

We now discuss the optical implementation of the introduced operators. Time
displacements in the single photon regime can be realized using the free
evolution operator. As a matter of fact, since time is defined in practice by
conditioning it to detection Maccone and Sacha (2020), we are most of the time
interested in time differences, which are associated to optical path
differences that reveal the photonic temporal profile, as for instance in
interferometers. As for frequency displacements, they require non-linear
optics, as in Hu et al. (2021). This is an important difference from the
implementation point of view with the previously mentioned quadrature-based
continuous variable quantum computation scheme, where all displacement
operators require using linear optics only.

The Fourier transform $\hat{F}$ has a quadratic form in frequency and time.
Such an operator, in the quadrature or transverse position/momentum based
encoding, is proportional to the system’s free Hamiltonian: the
electromagnetic field’s energy, in the quadrature case, or the the free
propagation in the transverse position/momentum one. Thus, in this last set-
up, the free propagation can be used to implement either the fractional or the
complete Fourier transform, according to the propagation distance Walborn et
al. (2010). In the present frequency/time encoding, as discussed, the free
Hamiltonian creates time displacements and is linear in $\hat{\omega}$.
Nevertheless, propagation inside a linear media leads to group velocity
dispersion, which is a currently used technique to implement the Fourier
transform from frequency to time variables and vice-versa. The fractional
Fourier transform can be implemented using the same technique by controlling
the propagation distance. Finally, another possible way to implement both
fractional and the full Fourier transform is using a $4f$-like configuration
for frequency and time, as detailed in Fabre (2020), or by using an optical
quantum memory Mazelanik et al. (2020).

###  III.2 Squeezing-like and non-Gaussian operations

The remaining single photon operators in ${\cal W}$ consist of frequency and
time dependent quadratic and cubic phases applied to the single photon states.
They can be re-expressed as follows (see, for technical details, Appendix A):

$\displaystyle e^{is\hat{\omega}_{a}^{2}}=\int
e^{is\omega^{2}}\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm d}\omega,$
$\displaystyle e^{i\gamma\hat{\omega}_{a}^{3}}=\int
e^{i\gamma\omega^{3}}\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm d}\omega.$
(17)

These transformations, either in the frequency or in the time basis, can be
realized using spatial light modulators after mapping frequency to transverse
position Mazzotta et al. (2016) or, in a measurement-based model, by pump
engineering Boucher et al. (2015-08-04). As a consequence, it’s possible to
implement them without using non-linearity at the single photon level. We’ll
leave the discussion about the meaning of squeezing and non-Gaussianity in the
present encoding to Sec. IV.

###  III.3 Two-photon gate

We now move to operator $e^{i\hat{\omega}_{i}\otimes\hat{\omega}_{j}}$, which
manipulates in a conditional way the frequency of two photons in distinct
auxiliary modes. In order to build a more intuitive picture, we’ll rather
discuss the Fourier transform in mode $j$ of this operator, i.e.,
$e^{i\hat{\omega}_{i}\otimes\hat{t}_{j}}=\hat{F}_{j}e^{i\hat{\omega}_{i}\otimes\hat{\omega}_{j}}\hat{F}_{j}^{\dagger}$.
This operator acts in the state $\ket{\omega,\omega^{\prime}}_{ij}$ as

$e^{i\hat{\omega}_{i}\otimes\hat{t}_{j}}\ket{\omega,\omega^{\prime}}_{ij}=e^{i\omega\hat{t}_{j}}\ket{\omega,\omega^{\prime}}_{ij}=\ket{\omega,\omega^{\prime}-\omega}_{ij}.$
(18)

It is, indeed, a conditional operation that displaces the frequency of one
photon of an amount that depends on the frequency of the other one, or a
continuous variable version of the discrete CNOT gate.

By combining this operator and other ones from the Gaussian set in ${\cal W}$
\- as, again, the Fourier transform and the quadratic gate - one can build

$\displaystyle
e^{i\frac{\pi}{4}(\hat{\omega}_{i}\otimes\hat{t}_{j}-\hat{t}_{i}\otimes\hat{\omega}_{j}})=$
(19)
$\displaystyle\iint\hat{a}^{\dagger}(\frac{\omega_{i}+\omega_{j}}{\sqrt{2}})\hat{a}^{\dagger}(\frac{\omega_{i}-\omega_{j}}{\sqrt{2}})\hat{a}(\omega_{i})\hat{a}(\omega_{j}){\rm
d}\omega_{i}{\rm d}\omega_{j}.$

The applications of this operator are numerous Fabre et al. (2020a); Fabre
(2021) and present interesting analogies with the beam-splitter operator
defined for continuous variables representation using quadratures. In the
quadrature case, a balanced beam-splitter combines two modes of the
electromagnetic field and performs a conditional operation that produces as an
output, two modes which are the sum and the difference between two input
modes. Still in the quadrature case, if we consider Gaussian states it’s a
known result that an optical circuit consisting of beam-splitters and passive
linear optical elements manipulating modes Fabre and Treps (2020) can be used
to physically implement the Bloch-Messiah decomposition Braunstein (2005) and
transform mode entangled squeezed states into separable ones in the minimal
mode representation of the considered state. In this scenario, beam-splitters
play a central role, since they are two-mode operations that entangle and
disentangle states by performing mode basis change.

The transformation Eq. (19) implements the same type of operation as beam-
splitters perform to spatial modes and to the field’s amplitude to the
frequency degree of freedom. The frequencies of two input photons are
transformed in a conditional way into the sum and difference of each photons’
frequencies. It’s natural to ask wether Eq. (19) can play a similar role. For
this, we refer again to the case of photon pairs generated by SPDC from a pump
with a Gaussian spectrum, as in Sec. II.

We have seen that in SPDC the frequency of the two generated photons can be
correlated or anti-correlated according to the energy and momentum
conservation functions of the pump Francesconi et al. (2020) and that the
degree of mode entanglement of the photon pair depends on the parameter $r$.
We’ll show now that this spectral distribution is the perfect analog of a two-
mode squeezed state in the quadrature basis and that it can be separated into
a product state using a Bloch-Messiah like decomposition.

This fact is not entirely obvious at first sight. It’s true that for many
relevant experimental set-ups, the interaction generating multi-mode entangled
quadrature states in optical parametric oscillators (OPO) is the same as the
one that generates photon pairs from SPDC. The main difference between these
two physical situations is the intensity regime: while in OPOs multi-mode
squeezed states are generated, in SPDC the interaction Hamiltonian is expanded
until first order in the creation and annihilation operator, and only the
photon pair generation process is considered. The produced state is then post-
selected by detection, which ensures its non-Gaussian character and, with high
probability, its single photon nature. Nevertheless, such intensity
considerations do not change the properties of the JSA, and its expansion in a
set of pairs of modes, as was done Eq. (7) and Law et al. (2000), is exactly
the same in both intensity regimes. In the case of OPOs, it leads to the mode
decomposition that describes the state as a separable one (Bloch-Messiah
decomposition) while in the case of SPDC it leads to the Schmidt
decomposition. These two states, which are the result of the same mathematical
operation in two different intensity regimes are, of course, essentially
different.

Nevertheless, they have the same JSA and if we consider it to be in the
Gaussian form of Fig. 1 (b), it is a particular case of a function that can be
expressed as a product of functions of only two different variables,
$\omega_{+}$ and $\omega_{-}$. This fact can be expressed in the SPDC regime
as:

$\displaystyle\iint{\rm d}\omega_{s}{\rm d}\omega_{i}$
$\displaystyle\text{JSA}(\omega_{s},\omega_{i})\ket{\omega_{s},\omega_{i}}=\iint{\rm
d}\omega_{+}{\rm d}\omega_{-}f(\omega_{+})g(\omega_{-})$ $\displaystyle\times$
$\displaystyle\ket{(\omega_{+}+\omega_{-})/\sqrt{2},(\omega_{+}-\omega_{-})/\sqrt{2}},$
(20)

which is an entangled state. Now, it’s clear that by applying Eq. (19) to Eq.
(III.3), we obtain:

$\displaystyle
e^{i\frac{\pi}{4}(\hat{\omega}_{i}\otimes\hat{t}_{j}-\hat{t}_{i}\otimes\hat{\omega}_{j}})\iint{\rm
d}\omega_{s}{\rm
d}\omega_{s}\text{JSA}(\omega_{s},\omega_{i})\ket{\omega_{s},\omega_{i}}=$
$\displaystyle\int{\rm d}\omega_{+}f(\omega_{+})\ket{\omega_{+}}\int{\rm
d}\omega_{-}g(\omega_{-})\ket{\omega_{-}}.$ (21)

which is a separable state.

So, we can conclude that in the single photon regime considered in the present
contribution, the Bloch-Messiah decomposition also makes sense and transforms
spectrally entangled Gaussian states into separable ones instead of
transforming Gaussian entangled states (many-modes squeezed states) into
separable ones, as is the case in the quadrature representation. Also, in the
single photon regime considered here, the physical and mathematical
implementation of the frequency beam-splitter like operator Eq. (19) involve
using a non-linear photon-photon interaction.This may sound unpractical, but
this is not exactly the case: a particularly interesting consequence of this
fact is that one can thus re-interpret the non-linear interaction that
generates the photon pair as the application of gate Eq. (19) to an initially
separable state in variables $\omega_{s}$ and $\omega_{i}$, suggesting that a
measurement based (or hybrid) model of computation is probably the most
suitable for the proposed configuration. This can be schematized as in the
quantum circuit represented in Fig. 1(c), and which is compared with the
quadrature case Fig. 1(d), (e), (f). We should finally mention that another
possible way to experimentally implement gate (19) gate is by using an
auxiliary optical active medium such as atomic system Oehri et al. (2015) or
by using split-ring resonators Kues et al. (2019); Yang et al. (2021); Lu et
al. (2019); Grassani et al. (2015).

Figure 1: Illustration of Bloch-Messiah decomposition for time-frequency (a)
and quadrature (b) encoding. In the quadrature encoding, the beam-splitter
interaction between two one-mode squeezed state leads to a two-mode squeezed
state. In the time-frequency encoding, the analogous is the interaction of two
single photons with a Gaussian spectrum through the beam-splitter -like
operator Eq. (19). The frequency unit have been rescaled with respect to a
single photon state chosen as a reference. The ratio between the major and
minor axis of the ellipse in (b) and (e) is here $1/10$.

As a conclusion, the Gaussian-shaped JSA can be factorized by the application
of a beam-splitter like gate as Eq. (19), in analogy to what happens in the
Bloch-Messiah transformations for Gaussian states in quadratures. Of course,
the non-linear operations we defined modify, as in the quadrature case, the
number of relevant modes used to express the state, since an entangled state
involving many terms of the Schmidt decomposition is reduced to a single
separable state. We recall that the nature of the physical implementation in
the discussed example - a non-linear two-photon interaction - and the type of
variables involved - the spectrum of single photons - are completely different
from the quadrature case.

The presented scenario consists of an original way to analyse and interpret
mode transformations that is not only relevant from the quantum information
perspective but can also bring new applications of frequency-based quantum
optics in the regime of individual photons.

Finally, the case of a Gaussian-shaped function discussed here is by no means
the only example of a state that can be separable in two variables, the
presented techniques can be used for different distributions.

###  III.4 Encoding a qubit with the continuous variables of single photons

Universal quantum computing has to be supplemented with error-correction codes
to provide fault-tolerant quantum computation. In the case of quadrature
continuous variables encoding, different types of qubits have been defined,
and as we see things today, the leading experimental and practical candidates
are the so-called “cat-code” Cochrane et al. (1999) and the Gottesman-Kitaev-
Preskill (GKP) code Gottesman et al. (2001). Each one of these codes is built
so as to be resilient against different types of errors. Cat codes are robust
against photon losses while GKP codes resist to small position and momentum
displacements. A detailed study of performance for these two codes can be
found in Albert et al. (2018).

In the time-frequency encoding of single photons, similar codes can be
defined. Time-frequency cat codes Fabre et al. (2020b), defined as the linear
superposition of two frequencies, are then built to be resilient against time
displacement operations which is a second-order dispersive effect. Time-
frequency GKP codes Fabre et al. (2020a) are protected against both small
shifts in time and in frequency, and error correction procedures can be
implemented using for instance the Steane protocol or teleportation based-
error correction (see Ref. Fukui et al. (2021) in the quadrature case, which
is the perfect analogous of the present one). Also, in certain optical fibers,
the frequency-polarization coupling Gordon and Kogelnik (2000); Poon and Law
(2008); Antonelli and Mecozzi (2005) can induce frequency broadening. It’s
important to notice that the time-frequency encoding of single photons is
conditioned to photon detection, so that photon losses do not affect the
encoded information, but rather its existence contrary to the quadrature case.

##  IV The chronocyclic phase space

In order to provide a full continuous variable description of frequency and
time degrees of freedom of single photons, it’s useful to introduce a phase
space picture and interpret it. To begin with, it is indeed possible to define
the chronocyclic phase space in the single photon subspace, as well as the
chronocyclic Wigner distribution Brecht and Silberhorn (2013); Fabre et al.
(2020a). As for the quadrature based phase space, this distribution has a one-
to-one correspondance with the spectral density matrix.

We consider continuous real variables, so the phase space is rectangular and
we can apply the construction procedure relying on the Stratonovich-Weyl rules
(see Ref. Tilma et al. (2016) for a reminder of these rules). It starts from
defining the parity operator

$\hat{\Pi}=\int\hat{a}^{\dagger}(\omega)\hat{a}(-\omega)d\omega$ (22)

with respect to the central frequency of a reference state that here was set
to zero. The displaced parity operator can also be defined from Eq. (22) as

$\hat{\Pi}(\mu,\tau)=\hat{D}(\mu,\tau)\hat{\Pi}\hat{D}^{\dagger}(\mu,\tau),$
(23)

where
$\hat{D}(\mu,\tau)=e^{-i\mu\tau/2}\hat{D}_{t}(\mu)\hat{D}_{\omega}(\tau)$ is a
combined arbitrary time/frequency displacement. The expectation value of the
displaced parity operator is the chronocyclic Wigner distribution:

$W_{\hat{\rho}}(\mu,\tau)=\langle\hat{\Pi}(\mu,\tau)\rangle=\int d\omega
e^{2i\omega\tau}\bra{\mu-\omega}\hat{\rho}\ket{\mu+\omega}$ (24)

The chronocyclic Wigner function can be measured using a modal decomposition
Gil-Lopez et al. (2021-05-26) or a frequency beam-splitter operation Fabre
(2021), for instance. For a single photon pure state
$\hat{\rho}=\ket{\psi}\bra{\psi}$ and $\ket{\psi}=\int
S(\omega)d\omega\ket{\omega}$ , the chronocyclic Wigner distribution can be
written as

$W_{\hat{\rho}}(\mu,\tau)=\int d\omega
e^{2i\omega\tau}S(\mu-\omega)S^{*}(\mu+\omega),$ (25)

and this distribution can be negative. In particular, it’s always negative for
pure non-Gaussian states. The marginals of the chronocyclic Wigner
distribution correspond to the spectral and time-of arrival distributions,
which can be measured directly experimentally :

$\displaystyle\int d\mu
W_{\hat{\rho}}(\mu,\tau)=\absolutevalue{\tilde{S}(\tau)}^{2},$
$\displaystyle\int d\tau W_{\hat{\rho}}(\mu,\tau)=\absolutevalue{S(\mu)}^{2}.$

Considering displacements in the present case is essential, since there is no
natural absolute reference state as the physical vacuum (the absence of
photons): using a “zero frequency” state as a reference doesn’t make real
sense since it’s physically out of reach and doesn’t correspond to energy
scales that can be attained experimentally. We can thus re-scale the space
with respect to a given reference state, with a given average energy and a
finite Gaussian width in time and frequency. The origin of the phase space and
the reference state is then related to the physical situation of interest.
Consequently, squeezing and relative displacements do not have a meaning per
se and are defined with respect to a reference state. Also, symmetries will be
relative to some chosen axis in phase space, or equivalently, energy, that in
all practical situations will be far from the zero frequency state.

We can notice that the distribution Eq. (25) is quadratic in the spectral
function of interest and its description is identical to the Wigner-Ville
distribution for classical field Dorrer and Walmsley (2005). Based on these
similarities, some may then conclude that the presented Wigner function does
not display quantum properties of the photon. Nevertheless, this naïve
inductive conclusion is wrong, and this can be shown using different types of
arguments. The first one is also inductive, and compares frequency to
polarization. Polarization is also a property that is well defined for
classical fields (or any field statistics). Nevertheless, it is isomorphic to
a two-dimensional Hilbert space when associated to single photons, and this is
a fundamental ingredient to demonstrate fundamental aspects of quantum
mechanics using single photon polarization detection, as for instance in non-
locality tests Aspect et al. (1982). Frequency is the continuous analogous of
polarization, and even if well defined for classical fields, it is a quantum
(continuous) variable when associated to single photons. In spite of that, in
the literature, frequency is usually discretized into modes or bins Maltese et
al. (2020); Brecht et al. (2015).

Given these facts, the difference between Gaussian and non-Gaussian
distributions deserves some attention. Non-Gaussianity is known to be an
essential property for continuous variables quantum computation, since it’s
essential to prove quantum advantage over classical protocols. Gaussian only
quantum protocols can be efficiently simulated using classical resources. In
the case of quadratures, it turns out that implementing the non-Gaussian
operation in ${\cal U}$ is challenging. For instance, the quadrature cubic
phase gate requires non-linearity and the production of ancillary non-Gaussian
states which are hard to implement Miyata et al. (2016). However, this is not
the case here: in the case of pure frequency-time states, the non-Gaussianity
of the time-frequency distributions is only related to some particular mode
engineering, which leads to distributions with specific parity properties. For
instance, all pure non-Gaussian states have negativities, or equivalently,
their displaced parity is negative at some regions. This simply means that the
spectrum of single photons in non-Gaussian states can be odd at some points,
and such states are a particular type of superposition states which are odd
with respect to some displaced parity. One example is the state
$\ket{\psi}=\frac{1}{\sqrt{2}}(\ket{\omega_{1}}-\ket{\omega_{2}})$) which is
odd with respect to the axis $\omega_{+}=(\omega_{1}+\omega_{2})/\sqrt{2}$.

Of course, non-Gaussian states and operations in the present context are, as
well, essential for universality, and the non-Gaussian operator in ${\cal W}$
plays the same role as in ${\cal U}$, which is transforming Gaussian operators
into non-Gaussian ones. We recall that this is also the case when one uses
polarization encoding, where non-Clifford gates can be implemented using
linear operations of the same nature of any other local gate in the universal
set. It’s rather a particularity of quadrature-based continuous variables
systems to have non-Gaussianity associated to a challenging experimental
process. Instead of coming from a particular hard to implement physical
operation, the fact that the present quantum computing architecture cannot be
classically simulated comes from two independent facts. From the physics side,
we have that we are dealing with single photons which obey to particular
commutation relations. As a consequence, single photons prepared in different
modes inherit quantum properties from these commutation relations, and this is
the essential ingredient to manipulate, in a quantum way, every operator
defined in ${\cal S}_{n}$. In particular, Gaussian and non-Gaussian operators.
Thus, using single photon measurements and manipulations, ensures that we are
dealing with a quantum system and that some of the resulting operations cannot
be efficiently simulated with classical ressources. Frequency, or
polarization, or any other degree of freedom associated to single photons are
used as ways to manipulate them, encoding and decoding quantum information and
exploiting their fundamental symmetries.

Before concluding, we’ll discuss the two-photon case so as to provide more
intuition about entanglement and the phase space. The two-photon chronocyclic
Wigner distribution can also be defined for pairs of single photon (${\cal
S}_{2}$) and it provides information on both the amplitude and phase of the
photon pair Brecht and Silberhorn (2013):

$W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=\iint
e^{2i\omega^{\prime}_{s}t_{s}}e^{2i\omega^{\prime}_{i}t_{i}}d\omega^{\prime}_{s}d\omega^{\prime}_{i}\\\
\crossproduct\bra{\omega_{s}-\omega^{\prime}_{s},\omega_{i}-\omega^{\prime}_{i}}\hat{\rho}\ket{\omega_{s}+\omega^{\prime}_{s},\omega_{i}+\omega^{\prime}_{i}}.$
(26)

Their marginals correspond to the Joint spectral intensity (JSI), the joint
temporal intensity (JTI) or cross-marginal (JTSI):

$\displaystyle\iint
d\omega_{s}d\omega_{i}W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=\text{JTI}(t_{s},t_{i}),$
$\displaystyle\iint
dt_{s}dt_{i}W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=\text{JSI}(\omega_{s},\omega_{i}),$
$\displaystyle\iint
d\omega_{s}dt_{i}W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=\text{JTSI}(t_{s},\omega_{i})$
$\displaystyle\iint
d\omega_{i}dt_{s}W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=\text{JTSI}(t_{i},\omega_{s}).$

These four marginals were measured in MacLean et al. (2019). Also in the two-
photon case, the issue of Gaussian and non-Gaussian states appear. It was
shown in Douce et al. (2013) that the biphoton Wigner function associated to
variable $\omega_{-}$ can be directly measured using the Hong-Ou-Mandel
experiment, and also that non-Gaussianity - or, equivalently, a coincidence
detection probability greater than $1/2$ \- is a proof of frequency
entanglement between in the produced photon pair Eckstein and Silberhorn
(2008).

A final issue to be discussed concerns the efficiency of production and
detection of single photons. We can mention deterministic and bright single
photon sources based on quantum dots Nowak et al. (2014); He et al. (2017);
Somaschi et al. (2016-05) that can be integrated on chip Uppu et al. (2020) or
probabilistic ones, as the one produced by SPDC Belhassen et al. (2018); Guo
et al. (2017) or by four-wave mixing Paesani et al. (2020). The state-of-the-
art single photon detectors in the 1550 nm range, for instance, can reach 90 %
for superconducting nanowire single-photon detector (SNSPD) composed of
materials which can be integrated on chip You (2020). While the SNSPD used for
detecting 925 nm single photons produced by micro-pillar sources Somaschi et
al. (2016-05) reach an efficiency of 60-82 % in the boson sampling experiment
Wang et al. (2019).

Also, the CV measurement and manipulation process, which is necessary, for
instance, in teleportation or in the measurement-based protocol, leads to
deformation of the spectral or temporal shape of the single photon. This can
be associated to encoding errors that reduce the fidelity of the CV
operations. Nevertheless, this is expected, and intrinsic to any type of CV
encoding Pirandola and Mancini (2006).

##  V Discussion and conclusion

We have described in detail how to encode quantum information in time and
frequency of single photons, formally treated as quantum continuous variables.
For this, we built an analogy between these variables and the position and
momentum of a particle or the field’s quadratures. This analogy was possible
because, in the single photon regime, time and frequency operators obey the
canonical commutation relations of the Heisenberg algebra, and can be formally
manipulated as position and momentum are. As a consequence, one can define a
finite set of universal gates that when combined lead to unitary operations
that are arbitrary polynomials in time and frequency variables.

The set of universal operations defined using time and frequency consists of
highly non-linear transformations at the single photon level. While the single
photon operations in the circuit model can be implemented with current
technology, the two-photon interaction is, for the time being, easier to
implement using the natural energy and momentum conservation relations
observed in some currently used sources of photon pairs. One can also benefit
from a complete measurement-based strategy for devices where pump engineering
is possible.

We also briefly discussed the equivalent to the dynamics of the single photons
evolving under the analogous of a Hamiltonian defined in frequency and time
variables. It corresponds to the propagation in a dispersive linear device.
Finally, we showed that it is possible to represent and interpret non-
classical and symmetry aspects of quantum states using phase space, even
though some properties as non-Gaussianity play a less distinctive role than in
quadrature-based set-ups in what concerns their physical implementation.

Our results are relevant from a fundamental point of view, pointing out an
original way to interpret, manipulate and use frequency and time state of
single photons. They may help developing novel ways to encode quantum
information and implement quantum communication and quantum metrology
protocols in practice.

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## Appendix A

###  A.1 Canonical Commutation Relation

We now provide the technical details for the calculation of the commutation
relations between different unitary operators, polynomials in $\hat{\omega}$
or $\hat{t}$.

To begin with, we compute the commutation relation
$\left[\hat{\omega},\hat{t}\right]$ using that :

$\displaystyle\hat{\omega}=\int\omega\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm
d}\omega$ $\displaystyle\hat{t}=\int t\hat{a}^{\dagger}(t)\hat{a}(t){\rm d}t,$
(27)

with $\hat{a}(t)=\int\hat{a}(\omega)e^{-i\omega t}{\rm d}\omega$. For that,
we’ll use the fact that we are in the single photon regime, which is an
intrinsically single mode state. It can thus be expressed in some mode
$\hat{b}_{n}=\int U_{n}(\omega)\hat{a}(\omega){\rm d}\omega$, so that
$\hat{b}_{n}=\iint U_{n}(\omega)\hat{a}(t)e^{i\omega t}{\rm d}\omega{\rm
d}t=\int\tilde{U}_{n}(t)\hat{a}(t){\rm d}t$, where $\tilde{U}_{n}(t)=\int
U_{n}(\omega)e^{i\omega t}{\rm d}\omega$ and $U_{n}(\omega)$ is an isometric
linear mode transformation fulfilling :
$\sum_{n}U(\omega)^{\dagger}U_{n}(\omega^{\prime})=\delta(\omega-\omega^{\prime})$.
We can then re-write Eq. (A.1) in terms of these modes as:

$\displaystyle\hat{\omega}=\sum_{n,m}\int\omega\left(U_{n}(\omega)\hat{b}^{\dagger}_{n}\right)\left(U^{*}_{m}(\omega)\hat{b}_{m}\right){\rm
d}\omega$ $\displaystyle\hat{t}=\sum_{n,m}\int
t\left(\tilde{U}_{n}(t)\hat{b}^{\dagger}_{n}\right)\left(\tilde{U}^{*}_{m}(t)\hat{b}_{m}\right){\rm
d}t,$

so that

$\displaystyle\left[\hat{\omega},\hat{t}\right]=\sum_{n,m,n^{\prime},m^{\prime}}\iint\omega
tU_{n}(\omega)U^{*}_{m}(\omega)\tilde{U}_{n^{\prime}}(t)\tilde{U}^{*}_{m^{\prime}}(t)\times$
$\displaystyle(\hat{b}^{\dagger}_{n}\hat{b}_{m}b^{\dagger}_{n^{\prime}}\hat{b}_{m^{\prime}}-b^{\dagger}_{n^{\prime}}\hat{b}_{m^{\prime}}\hat{b}^{\dagger}_{n}\hat{b}_{m}){\rm
d}t{\rm d}\omega.$ (28)

We can change this sum into

$\displaystyle\left[\hat{\omega},\hat{t}\right]=$
$\displaystyle\sum_{n,m,n^{\prime},m^{\prime}}\iint\omega
t\left(U_{n}(\omega)U^{*}_{m}(\omega)\tilde{U}_{n^{\prime}}(t)\tilde{U}^{*}_{m^{\prime}}(t)-\right.$
$\displaystyle\left.U_{n^{\prime}}(\omega)U^{*}_{m^{\prime}}(\omega)\tilde{U}_{n}(t)\tilde{U}^{*}_{m}(t)\right)\hat{b}^{\dagger}_{n}\hat{b}_{m}b^{\dagger}_{n^{\prime}}\hat{b}_{m^{\prime}}{\rm
d}t{\rm d}\omega.$ (29)

Since
$\hat{b}^{\dagger}_{n}\hat{b}_{m}b^{\dagger}_{n^{\prime}}\hat{b}_{m^{\prime}}=\hat{b}^{\dagger}_{n}\hat{b}_{m^{\prime}}\delta_{m,n^{\prime}}+\hat{b}^{\dagger}_{n}b^{\dagger}_{n^{\prime}}\hat{b}_{m}\hat{b}_{m^{\prime}}$,
Eq. (A.1) becomes

$\displaystyle\left[\hat{\omega},\hat{t}\right]=$ (30)
$\displaystyle\sum_{n,m,m^{\prime}}\iint\omega
t\left(U_{n}(\omega)U^{*}_{m}(\omega)\tilde{U}_{m}(t)\tilde{U}^{*}_{m^{\prime}}(t)-\right.$
$\displaystyle\left.U_{m}(\omega)U^{*}_{m^{\prime}}(\omega)\tilde{U}_{n}(t)\tilde{U}^{*}_{m}(t)\right)\hat{b}^{\dagger}_{n}\hat{b}_{m^{\prime}}{\rm
d}t{\rm d}\omega+$ $\displaystyle\sum_{n,m,n^{\prime},m^{\prime}}\iint\omega
t\left(U_{n}(\omega)U^{*}_{m}(\omega)\tilde{U}_{n^{\prime}}(t)\tilde{U}^{*}_{m^{\prime}}(t)-\right.$
$\displaystyle\left.U_{n^{\prime}}(\omega)U^{*}_{m^{\prime}}(\omega)\tilde{U}_{n}(t)\tilde{U}^{*}_{m}(t)\right)\hat{b}^{\dagger}_{n}b^{\dagger}_{n^{\prime}}\hat{b}_{m}\hat{b}_{m^{\prime}}{\rm
d}t{\rm d}\omega.$ (31)

In the single mode situation, we have that $n=m^{\prime}=m=n^{\prime}=k$, so
it’s clear that the second sum in Eq. (30) is equal to zero. We will then
focus on the first one. After writing the $\tilde{U}_{m}(t)$ as a function of
$U_{m}(\omega^{\prime})$, the integral over $\omega^{\prime}$ in the first
term of the sum can be integrated by parts. In addition, we use that $\int
te^{-i(\omega^{\prime}-\omega^{\prime\prime})t}{\rm d}t=i\frac{\rm d}{\rm
d\omega^{\prime}}\delta(\omega^{\prime}-\omega^{\prime\prime})$. It leads to

$i\sum_{n}\int\int\omega U_{k}(\omega)(\frac{\rm d}{\rm
d\omega^{\prime}}U_{k}^{*}(\omega^{\prime}))U_{n}^{*}(\omega)U_{n}(\omega^{\prime}){\rm
d}\omega{\rm d}\omega^{\prime}.$ (32)

We now use the orthogonality condition:
$\sum_{n}U_{n}^{*}(\omega)U_{n}(\omega^{\prime})=\delta(\omega-\omega^{\prime})$,
leading to

$i\int\omega U_{k}(\omega)(\frac{\rm d}{\rm d\omega}U_{k}^{*}(\omega)){\rm
d}\omega,$ (33)

which can again be integrated by parts to obtain

$i+i\int\omega U_{k}^{*}(\omega)(\frac{\rm d}{\rm d\omega}U_{k}(\omega)){\rm
d}\omega.$ (34)

We now analyse the second part of the first term of Eq. (30) and proceed to an
integration by parts as in Eq. (32), which leads to

$i\sum_{n}\int\int\omega U_{k}^{*}(\omega)(\frac{\rm d}{\rm
d\omega^{\prime}}U_{k}(\omega^{\prime}))U_{n}(\omega)U_{n}^{*}(\omega^{\prime}){\rm
d}\omega{\rm d}\omega^{\prime}.$ (35)

We can again perform the sum in $n$ to reach

$i\int\omega U^{*}_{k}(\omega)(\frac{\rm d}{\rm d\omega}U_{k}(\omega)){\rm
d}\omega,$ (36)

that will cancel the last term in Eq. (34). We have then

$\left[\hat{\omega},\hat{t}\right]=i\hat{b}^{\dagger}_{k}\hat{b}_{k}=i\mathds{I},$
(37)

since $\hat{b}^{\dagger}_{k}\hat{b}_{k}$ is the photon number operator in the
single photon mode denoted here as $k$. This operator is equal to the identity
in the single photon subspace, since there is only one single photon in each
mode. Notice that no assumption on the state was made. These are general
properties of one photon per mode states.

###  A.2 Unitary operators

A given unitary operator which is generated by a hermitian operator which is a
polynomial $\hat{\omega}$ (or, equivalently, $\hat{t}$), can be expressed in
the single photon subspace ${\cal S}_{1}$ as
$e^{i\hat{\omega}_{a}^{n}\alpha}$. Since

$e^{i\hat{\omega}_{a}^{n}\alpha}=\sum_{k}^{\infty}\frac{(i\alpha)^{k}}{k!}\hat{\omega}^{nk},$
(38)

we will focus on the term $\hat{\omega}^{nk}\equiv\hat{\omega}^{s}$. Since
$\hat{\omega}^{s}=\hat{\omega}^{2}\hat{\omega}^{(s-2)}$, we have that :

$\hat{\omega}_{a}^{2}=\left(\int\omega\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm
d}\omega\right)^{2}.$ (39)

The r.h.s. of Eq. (39) can be easily computed, leading to
$\iint\omega\omega^{\prime}\hat{a}^{\dagger}(\omega)\hat{a}(\omega)\hat{a}^{\dagger}(\omega^{\prime})\hat{a}(\omega^{\prime}){\rm
d}\omega{\rm d}\omega^{\prime}$. Using that
$\hat{a}(\omega)\hat{a}^{\dagger}(\omega^{\prime})=\hat{a}^{\dagger}(\omega^{\prime})\hat{a}(\omega)+\delta(\omega-\omega^{\prime})$,
we obtain

$\displaystyle\hat{\omega}_{a}^{2}=\iint\omega\omega^{\prime}\hat{a}^{\dagger}(\omega)\hat{a}^{\dagger}(\omega^{\prime})\hat{a}(\omega)\hat{a}(\omega^{\prime}){\rm
d}\omega{\rm d}\omega^{\prime}$
$\displaystyle+\int\omega^{2}\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm
d}\omega.$ (40)

However, the first term in the r.h.s. of Eq. (A.2) is equal to zero in the
single photon subspace. We have thus :

$\hat{\omega}_{a}^{2}=\int\omega^{2}\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm
d}\omega.$ (41)

It’s thus clear that
$\hat{\omega}_{a}^{s}=\int\omega^{s}\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm
d}\omega$ and, consequently,

$e^{i\hat{\omega}_{a}^{n}\alpha}=\int
e^{i\omega^{n}\alpha}\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm d}\omega.$
(42)

Using Eq. (42) and the commutation relation between creation and annihilation
operators, we can prove all the commutation relations appearing in this
manuscript. Of course, the same reasoning is valid for unitary operators
involving the time operator.

The particular case of $n=1$ is relevant for the definition of the
displacement operators. In this case, Eq. (42) becomes :

$\displaystyle e^{i\hat{\omega}_{a}\alpha}=\int
e^{i\omega\alpha}\hat{a}^{\dagger}(\omega)\hat{a}(\omega){\rm d}\omega=$ (43)
$\displaystyle\iiint e^{i\omega\alpha}\hat{a}^{\dagger}(t)e^{-i\omega
t}\hat{a}(t^{\prime})e^{i\omega t^{\prime}}{\rm d}\omega{\rm d}t{\rm
d}t^{\prime},$

which, after integration in $\omega$ leads to :

$e^{i\hat{\omega}_{a}\alpha}=\int\hat{a}^{\dagger}(t+\alpha)\hat{a}(t){\rm
d}t.$ (44)