# Interferometric sorting of temporal Hermite-Gauss modes via temporal Gouy
phase

Dmitri B. Horoshko<EMAIL_ADDRESS>Univ. Lille, CNRS, UMR 8523
- PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
Mikhail I. Kolobov  Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers
Atomes et Molécules, F-59000 Lille, France

###### Abstract

We propose a device consisting of $m$ Mach-Zehnder interferometers and
realizing sorting of first $2^{m}$ temporal Hermite-Gauss modes of light
passing though it by adjusting the accumulated temporal Gouy phase acquired by
every mode. This mode-order-dependent phase shift is achieved by a fractional
Fourier transform realized by a time lens in one of interferometer’s arms. We
consider application of such a sorter with just two interferometers to sorting
the Schmidt modes of a photon pair generated in spontaneous parametric
downconversion and find the theoretical lower bond on the cross-talk
probability of 5.5%.

##  I Introduction

Decomposition of the electromagnetic field into a set of orthogonal spatio-
temporal and polarization modes is a common practice in optics, plain
monochromatic waves [1] and resonator modes [2] being the most notorious
examples. Versatility of modal decompositions is determined by the fact that
any complete orthonormal set of functions of spatial, temporal and
polarization variables can be used as a modal basis for the field [3]. In the
last decade, much interest has been attracted to temporal modes of optical
pulses, which provide a reliable platform for quantum information encoding [4,
5, 6] for numerous applications such as linear optics quantum computation [7,
8], boson sampling [9, 10], quantum communications [11, 12] and quantum
sensing [13, 14]. A key problem when using any modal basis for information
encoding is an efficient mode sorting (demultiplexing), allowing one to
address individually every mode and the inverse process of mode combining
(multiplexing) for compact joint transmission.

Any polarization modal basis, intrinsically two-dimensional, can be rather
easily sorted by a set of birefringent elements and polarizing beam splitters.
Spatial modes are easily sorted only when they are non-overlapping in space
(pixels) or in wavevector space (plane waves). For sorting overlapping spatial
modes in free space, three groups of approaches can be distinguished. First,
overlapping modes can be mapped onto spatially separated pixels by interleaved
application of spatial phase modulation and diffractive propagation, a
technique known as multiple plane light conversion, which was successfully
applied to Hermite-Gauss (HG) modes [15, 16, 17, 18, 19]. Second, overlapping
modes can be mapped onto quasi-plane waves. For Orbital Angular Momentum (OAM)
modes it means a log-polar to Cartesian coordinate transform realized by
spatial light modulators [20] or refractive optical elements [21, 22, 23].
Third, different spatial modes can be sent to different outputs of an
interferometer when a mode-order-dependent phase shift is applied to the field
in one of its arms. This approach was applied to sorting OAM modes [24, 25,
26, 27], Laguerre–Gauss (LG) modes by radial index [28, 29, 30, 31], and to
free-space sorting of linear-polarized (LP) modes of few-mode fiber [32, 33].
The simplest variant of this approach, parity-sorting, is know in microscopy
as image-inversion interferometry [34, 35, 36]. Combinations of the second and
third approaches were used for full sorting of LG [37] and HG [38] modes by
both indexes. In addition to these free-space sorters, a variety of spatial
mode multiplexers exist for guided modes of optical fibers [39].

In temporal domain, modes are easily sorted if they are non-overlapping in
time (time-bins) or in frequency. Approaches to sorting overlapping temporal
modes may be divided into the same three groups as for the spatial modes with
the help of space-time analogy [40, 41, 42] based on the mathematical
equivalence of descriptions of quadratic dispersion and paraxial diffraction.
First, the overlapping modes can be mapped to temporally non-overlapping time
bins by means of sequential temporal phase modulation and dispersion, which
was shown for temporal HG modes [43, 44]. Second, mapping temporal modes to
non-overlapping spectral regions requires a nonlinear field transformation and
can be achieved for temporal HG modes with a mode-selective frequency
conversion by sum-frequency generation [45, 46, 47, 48, 49] or four-wave
mixing [50, 51]. Third, temporal modes can be parity sorted in a time-axis-
inversion interferometer [52].

In this work, we present a method of sorting temporal Hermite-Gauss modes by
means of arrays of interferometers with mode-order-dependent phase shift,
providing thus an extension of the parity sorting of Ref. [52] to
modulo-$2^{m}$ sorting, where $m$ is the number of used interferometers. Our
approach shares the general idea with the approaches of the third group of
spatial sorting, mentioned above.

In Sec. II, we review the general formalism for the description of the
temporal degree of freedom in optics and introduce a temporal counterpart of
the Gouy phase, which we call “the temporal Gouy phase”. We derive also the
temporal versions of the linear canonical transform and its particular case,
the fractional Fourier transform, both well studied in spatial domain [53]. In
Sec. III, we describe the proposed temporal mode sorter for parity or modulo 4
sorting and show how it can be extended to modulo $2^{m}$ sorting. In Sec. IV,
we consider an application of the proposed mode sorter to sorting the Schmidt
modes of two entangled photons generated in spontaneous parametric
downconversion (SPDC) and calculate analytically the cross-talk probability.
The results are summarized in the Conclusion, Sec. V.

##  II General temporal formalism

###  II.1 Temporal modes

Consider a discrete set of complex functions of time
$\\{\Psi_{n}(t),n=0,1,2...\\}$ satisfying the conditions of orthonormality and
completeness

$\displaystyle\int\Psi_{n}(t)\Psi_{m}^{*}(t)dt$ $\displaystyle=$
$\displaystyle\delta_{nm},$ (1)
$\displaystyle\sum_{n=0}^{\infty}\Psi_{n}(t)\Psi_{n}^{*}(t^{\prime})$
$\displaystyle=$ $\displaystyle\delta(t-t^{\prime}).$ (2)

Here and below integration from $-\infty$ to $+\infty$ is implied if not
otherwise specified. The set of Fourier-transformed functions

$\psi_{n}(\Omega)=\int\Psi_{n}(t)e^{i\Omega t}dt$ (3)

satisfies the conditions of orthonormality and completeness similar to Eqs.
(1) and (2) with replacements $\Psi_{n}(t)\to\psi_{n}(\Omega)$, $dt\to
d\Omega/2\pi$, $\delta(t-t^{\prime})\to 2\pi\delta(\Omega-\Omega^{\prime})$.

The functions $\Psi_{n}(t)$ and $\psi_{n}(\Omega)$ can be considered as modal
functions of electromagnetic field in time and frequency domains respectively.
To show it, we consider propagation of an optical wave with the carrier
frequency $\omega_{0}$ and carrier wavevector $k_{0}$ along the $z$ axis in a
linear medium, where the wavevector can be approximated by a second-order
polynomial of the frequency detuning $\Omega=\omega-\omega_{0}$,
$k(\Omega)=k_{0}+k_{0}^{\prime}\Omega+\frac{1}{2}k_{0}^{\prime\prime}\Omega^{2}$,
$k_{0}^{\prime}$ and $k_{0}^{\prime\prime}$ being the first and second
derivatives of $k(\Omega)$ at $\Omega=0$. In such a medium, we write the
positive-frequency part of electric field [54] as

$E^{(+)}(z,t)=A(z,t-z/v_{g})e^{ik_{0}z-i\omega_{0}t},$ (4)

where $A(z,t-z/v_{g})$ is the envelope function at delayed time $t-z/v_{g}$,
$v_{g}=1/k^{\prime}_{0}$ being the group velocity. When a wave passes through
multiple media, the temporal argument should be delayed by the total group
delay in all media [41, 42]. Here, we do not consider the transverse
coordinates of the optical wave implying they are decoupled from $z$ and $t$
during propagation through achromatic optical elements. The units for the
electric field are chosen so that $E^{(-)}(z,t)E^{(+)}(z,t)$ represents the
photon flux. Making a Fourier decomposition

$A(z,t)=\int a(z,\Omega)e^{-i\Omega t}\frac{d\Omega}{2\pi},$ (5)

we obtain $a(z,\Omega)$, the annihilation operator for a photon with frequency
$\omega_{0}+\Omega$ at point $z$, satisfying the canonical equal-space
commutation relations
$[a(z,\Omega),a^{\dagger}(z,\Omega^{\prime})]=2\pi\delta(\Omega-\Omega^{\prime})$
[55, 56]. In a non-dispersive medium $k^{\prime\prime}_{0}=0$ and the field
propagation is described by a simple delay:
$E^{(+)}(z,t)=E^{(+)}(z_{0},t-[z-z_{0}]/v)$, where $v=\omega_{0}/k_{0}$ is the
phase velocity, coinciding with the group one. It means that the time-delayed
envelope remains invariant, $A(z,t)=A(z_{0},t)$. Thus, it is sufficient to
make a modal decomposition at some point $z_{0}$, to know it at any other
point. Multiplying both sides of Eq. (2) by $A(z_{0},t^{\prime})$ and
integrating over $t^{\prime}$, we obtain

$A(z_{0},t)=\sum_{n=0}^{\infty}A_{n}\Psi_{n}(t),$ (6)

where

$A_{n}=\int A(z_{0},t)\Psi_{n}^{*}(t)dt=\int
a(z_{0},\Omega)\psi^{*}_{n}(\Omega)\frac{d\Omega}{2\pi}$ (7)

is the annihilation operator of a photon in $n$th mode, satisfying
$[A_{n},A_{m}^{\dagger}]=\delta_{nm}$.

###  II.2 Dispersive medium and time lens

In a dispersive medium, where $k^{\prime\prime}_{0}\neq 0$, the frequency-
domain operator acquires a phase shift when propagating over a distance $l$,
$a(z+l,\Omega)=a(z,\Omega)\exp[ik^{\prime\prime}_{0}\Omega^{2}l/2]$, which
means in the time domain [42]

$A(z+l,t)=\int A(z,t^{\prime})\frac{e^{-i(t-t^{\prime})^{2}/2D}}{\sqrt{-2\pi
iD}}dt^{\prime},$ (8)

where $D=k_{0}^{\prime\prime}l$ is the group delay dispersion (GDD) of the
medium. This transformation is a temporal counterpart of the Huygens-Fresnel
integral for paraxial diffraction [2]. The temporal modal functions undergo a
similar integral transformation.

Another temporal counterpart of a spatial optical element is a time lens [41,
57], providing a quadratic temporal phase modulation of the signal pulse
passing from $z^{\prime}$ to $z$:

$A(z,t)=e^{it^{2}/2D_{\mathrm{f}}}A(z^{\prime},t),$ (9)

where $D_{\mathrm{f}}$ is the focal GDD of the time lens. Time lenses are
realized by electrooptical phase modulation or by nonlinear parametric
processes such as sum-frequency generation or four-wave-mixing, see Refs. [58,
59] for reviews. Effective processing of quantum states requires that a time
lens is lossless, which, for a parametric lens, means perfect phase matching
and unit quantum conversion efficiency [60, 61, 62, 63]. Single photons were
processed by time lenses based on sum-frequency generation [64], four-wave
mixing [44], electrooptical phase modulation [65, 66, 67, 68], and atomic-
cloud-based quantum memory [52].

###  II.3 Linear canonical transform

Similar to how it is done in spatial domain, the field transformations,
described by Eqs. (8) and (9) can be viewed as particular cases of a class of
linear canonical tansforms [53] having the form

$A(z,t)=\int K_{\mathbf{T}}(t,t^{\prime})A(z^{\prime},t^{\prime})dt^{\prime},$
(10)

where the kernel is

$K_{\mathbf{T}}(t,t^{\prime})=\frac{1}{\sqrt{2\pi
ib}}\exp\left(\frac{i}{2b}\left[dt^{2}-2tt^{\prime}+a{t^{\prime}}^{2}\right]\right),$
(11)

which represents a normalized phase shift by a quadratic form of the variables
$t$ and $t^{\prime}$. The parametrization of this quadratic form by three real
numbers $a$, $b$ and $d$ has a remarkable property: When they are supplemented
by the fourth real number $c$ such that $ad-bc=1$, and written in the matrix
form

$\mathbf{T}=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right),$ (12)

the transform has the property of cascadability, i.e., two consecutive
transforms with kernels $K_{\mathbf{T}_{1}}(t,t^{\prime})$ and
$K_{\mathbf{T}_{2}}(t,t^{\prime})$ create a transform of the same class with a
kernel

$K_{\mathbf{T}}(t,t^{\prime})=\int
K_{\mathbf{T}_{2}}(t,t^{\prime\prime})K_{\mathbf{T}_{1}}(t^{\prime\prime},t^{\prime})dt^{\prime\prime},$
(13)

where $\mathbf{T}=\mathbf{T}_{2}\mathbf{T}_{1}$. In spatial domain, matrix
$\mathbf{T}$ is a scaled version of the ray matrix, widely used in geometrical
optics [2]. For this reason, we call it “temporal ray matrix”. Comparing Eqs.
(8) and to Eq. (11), we find that dispersive propagation corresponds to the
choice

$\mathbf{T}_{\mathrm{prop}}(D)=\left(\begin{array}[]{cc}1&-D\\\
0&1\end{array}\right).$ (14)

In the limit $b\to 0$, we consider $a$ and $c$ as independent variables and
write $d=1/a+bc/a$. Substituting this into Eq. (11) and taking the limit, we
obtain

$K_{\mathbf{T}}(t,t^{\prime})=\sqrt{a}e^{ict^{2}/2a}\delta(t-at^{\prime}),$
(15)

where we have used the relation
$(i\pi\epsilon)^{-1/2}\exp\left(it^{2}/\epsilon\right)\to\delta(t)$ at
$\epsilon\to 0$, which can be verified by taking Fourier transforms of both
its sides. Comparing Eq. (15) to Eq. (9) we conclude that a time lens
corresponds to

$\mathbf{T}_{\mathrm{lens}}(D_{\mathrm{f}})=\left(\begin{array}[]{cc}1&0\\\
1/D_{\mathrm{f}}&1\end{array}\right).$ (16)

Due to cascadability, any sequence of dispersive media and time lenses can be
described by the product of the corresponding temporal ray matrices.

###  II.4 Temporal Hermite-Gauss modes

Among infinitely many choices for modal functions, we choose
$\Psi_{n}(t)=h_{n}(t/\tau)/\sqrt{\tau}$, where
$h_{n}(x)=\left(2^{n}n!\sqrt{\pi}\right)^{-\frac{1}{2}}H_{n}(x)e^{-x^{2}/2}$
is the Hermite-Gauss function, $H_{n}(x)$ being the Hermite polynomial. The
temporal width of the modal function is determined by the parameter $\tau$,
having the dimension of time. These modes are a good approximation for the
Schmidt modes of photon pairs generated in spontaneous parametric
downconversion (SPDC), one of the key sources of quantum light in modern
quantum photonics [4, 69].

Transformation of $\Psi_{n}(t)$ when propagating through an optical system
with arbitrary temporal ray matrix $\mathbf{T}$ can be obtained by adopting
the dimensionless formula of Ref. [53] to temporal domain:

$\frac{1}{\sqrt{\tau}}h_{n}\left(\frac{t}{\tau}\right)\to\frac{e^{-i\gamma(n+1/2)}}{\sqrt{\tau\beta}}h_{n}\left(\frac{t}{\tau\beta}\right)e^{-\alpha(t/\tau)^{2}/2},$
(17)

where $\alpha=\left[(d-ic\tau^{2})(a-ib/\tau^{2})-1\right]/\beta^{2}$,
$\beta=|a+ib/\tau^{2}|$, $\gamma=\arg(a+ib/\tau^{2})$. In particular,
propagation through a dispersive medium with the temporal ray matrix (14)
increases the temporal width to
$\tau_{D}=\tau\beta=\tau\sqrt{1+D^{2}/\tau^{4}}$.

###  II.5 Accumulated temporal Gouy phase

The phase $\gamma$ may be rewritten in the general case as

$\gamma=\arctan\left(\frac{b}{a\tau^{2}}\right)+\pi\operatorname{sgn}(b)\theta(-a)$
(18)

with $\theta(x)$ the step function and $\arctan(x)\in[-\pi/2,\pi/2]$. This
phase is known in spatial domain as accumulated Gouy phase shift [70]. To
justify this name in temporal domain, let us consider the temporal counterpart
of the Gouy phase shift.

In spatial domain, a monochromatic field with a wavevector $k$ directed along
the $z$ axis and a Gaussian transverse distribution at plane $z=0$,
$E^{(+)}(x,y,0)=E_{0}\exp\left[-(x^{2}+y^{2})/w_{0}^{2}\right]$ propagates
diffractively so that at any other plane $z>0$ its transverse distribution, in
the paraxial approximation, is [2]

$\displaystyle E^{(+)}(x,y,z)$ $\displaystyle=$ $\displaystyle
E_{0}\frac{w_{0}}{w(z)}\exp\left(i[kz-\Phi(z)]\right)$ $\displaystyle\times$
$\displaystyle\exp\left(-\frac{x^{2}+y^{2}}{w^{2}(z)}+ik\frac{x^{2}+y^{2}}{2R(z)}\right),$

where $w(z)=w_{0}\sqrt{1+4z^{2}/(k^{2}w_{0}^{4})}$ is the spot size,
$R(z)=z[1+k^{2}w_{0}^{4}/(4z^{2})]$ is the radius of curvature, and
$\Phi(z)=\arctan\left(2z/kw_{0}^{2}\right)$ is the axial phase shift. Equation
(II.5) describes a divergent Gaussian beam. This equation is also valid for
$z<0$ and describes a convergent Gaussian beam: If a curved wavefront is
created for a Gaussian beam at a plane $z<0$, e.g. by a lens, the beam
converges to its focus at $z=0$. A beam passing through the focus acquires
between the planes $-z_{0}$ and $z_{0}$, where $z_{0}\gg kw_{0}^{2}/2$, a
total axial ($x=y=0$) phase shift with respect to the phase of a plane wave
$\Phi_{\text{Gouy}}=\Phi(z_{0})-\Phi(-z_{0})=\pi$, which is known as Gouy
phase shift [2].

Figure 1: (a) Focusing of a Gaussian beam. Thick solid lines show the beam
radius $\pm w(z)$ at different planes, dashed lines show the wavefronts. (b)
Temporal focusing of a Gaussian pulse in a medium with a positive dispersion
($k_{0}^{\prime\prime}>0$). In a negatively-chirped pulse, the blue part
advances the red part. Since the group velocity for the blue part is lower
than that for the red part, all spectral parts arrive at the temporal focus
simultaneously. Then the red part advances the blue one resulting in a
positively-chirped pulse.

In temporal domain, a plane wave with central frequency $\omega_{0}$ and a
Gaussian temporal distribution at point $z=0$ of its envelope,
$A(0,t)=E_{0}\exp\left[-t^{2}/2\tau_{0}^{2}\right]$, propagates in a
dispersive medium with the dispersion law $k(\Omega)$ so that at any other
plane $z>0$ its temporal distribution, in the quadratic-dispersion
approximation, is

$\displaystyle A(z,t)$ $\displaystyle=$ $\displaystyle
E_{0}\sqrt{\frac{\tau_{0}}{\tau(z)}}\exp\left[-i\Phi^{\text{temp}}(z)\right]$
$\displaystyle\times$
$\displaystyle\exp\left(-\frac{t^{2}}{2\tau^{2}(z)}-i\frac{t^{2}}{2R^{\text{temp}}(z)}\right)$

where $\tau(z)=\tau_{0}\sqrt{1+D^{2}/\tau_{0}^{4}}$ is the temporal standard
deviation, $R^{\text{temp}}(z)=D[1+\tau_{0}^{4}/D^{2}]$ is the temporal chirp
rate, $\Phi^{\text{temp}}(z)=-\frac{1}{2}\arctan\left(D/\tau_{0}^{2}\right)$
is the temporal-axial phase shift, and $D=k_{0}^{\prime\prime}z$ is the GDD
between the two planes with $k_{0}^{\prime\prime}>0$. Equation (II.5) is
obtained by substituting the Gaussian envelope $A(0,t)$ specified above into
Eq. (8) and describes a temporally spreading Gaussian pulse. This equation is
also valid for $z<0$ and describes a temporally focused Gaussian pulse: If a
negative chirp rate is created for a Gaussian pulse at a plane $z<0$, e.g. by
a time lens, the pulse temporally focuses to its transform-limited shape at
$z=0$. A pulse passing through the temporal focus acquires between the planes
$-z_{0}$ and $z_{0}$, where $k_{0}^{\prime\prime}z_{0}\gg\tau_{0}^{2}$, a
total temporal-axial ($t=0$) phase shift
$\Phi_{\text{Gouy}}^{\text{temp}}=\Phi^{\text{temp}}(z_{0})-\Phi^{\text{temp}}(-z_{0})=-\pi/2$,
which can be called “temporal Gouy phase shift.”

It should be noted that the temporal axis at plane $z$ means $t=0$ for the
time-delayed envelope, and $t=z/v_{g}$ for the full field amplitude in accord
with Eq. (4). It means that the full phase of the field at the temporal axis
at plane $z$ is $k_{0}z-\omega_{0}z/v_{g}-\Phi^{\text{temp}}(z)$ and the
temporal-axial phase shift is defined with respect to the phase of a plane
monochromatic wave delayed by the group velocity delay time $z/v_{g}$. This is
a slight but important difference with the definition of Gouy phase shift in
spatial domain.

We see that the temporal Gouy phase shift differs from its spatial counterpart
in two aspects: It has the opposite sign and its modulus is twice less. The
sign inversion is explained by different signs for spatial and temporal
coordinates in the argument of a retarded-potential solution of the wave
equation. The twice higher modulus in spatial domain is explained by the
focusing happening in two dimensions $x$ and $y$, while the temporal focusing
is one-dimensional.

Propagation of a general Hermite-Gauss mode in a dispersive medium is
described by Eq. (17) with $\tau=\tau_{0}$ and the parameters of the temporal
ray matrix taken from Eq. (14), i.e.,
$\alpha=|1-iD/\tau_{0}^{2}|=\tau(z)/\tau_{0}$, $\beta=iD/\tau^{2}(z)$, and
$\gamma=-\arctan(D/\tau_{0}^{2})$. For a Gaussian mode, $n=0$, we identify
$\gamma/2$ as the temporal-axis phase shift discussed above, leading to the
temporal Gouy phase shift in the temporal far-field limit. For higher-order
modes the corresponding phase shift is $(n+\frac{1}{2})\gamma$. Note that for
propagation in a system with a negative parameter $a$, $\gamma$ can take
values in $[-\pi,\pi]$ outside the range $[-\pi/2,\pi/2]$ given by the
$\arctan(x)$ function. Similar to the spatial domain terminology, the phase
shift $(n+\frac{1}{2})\gamma$ will be called “accumulated temporal Gouy
phase.”

###  II.6 Fractional Fourier transform

We see from Eq. (17) that Hermite-Gauss functions are eigenfunctions of a
kernel with $\alpha=0$ and $\beta=1$, which is equivalent to two conditions
$d=a$, $c\tau^{2}=-b/\tau^{2}$, wherefrom it follows that $|a+ib/\tau^{2}|=1$
(we recall that for any temporal ray matrix $ad-bc=1$). Such a kernel,
additionally multiplied by $\exp(i\gamma/2)$, corresponds to a transformation
known as fractional Fourier transform [53] realized recently in temporal
domain [71, 72]. It is characterized by just one parameter, for which we
choose $\gamma$ known in this context as “fractional angle”. Substituting the
required values of the parameters into Eq. (12) we obtain the temporal ray
matrix

$\mathbf{T}=\left(\begin{array}[]{cc}\cos\gamma&\tau^{2}\sin\gamma\\\
-\frac{1}{\tau^{2}}\sin\gamma&\cos\gamma\end{array}\right),$ (21)

which corresponds to fractional Fourier transform of order $2\gamma/\pi$ for a
pulse of width $\tau$. The order shows the fraction of the ordinary Fourier
transform which is realized: order 1 corresponds to the ordinary Fourier
transform. Note, that in quantum optics, one decomposes the positive-frequency
part of the field as function of time $t$ and position $z$ into plane waves of
the form $\exp\left(ikz-i\omega t\right)$, where $k$ and $\omega$ are the wave
vector and circular frequency respectively, which is considered as an inverse
Fourier transform in space and time [54]. Thus, mathematically, the Fourier
transform in time is similar to the inverse Fourier transform in space, hence
the minus sign in the definition of the fractional order with respect to Ref.
[53].

Fractional Fourier transform defined by Eq. (21) does not change the temporal
width of the Hermite-Gauss mode if it is exactly $\tau$. A more general
fractional Fourier transform can include a magnification, as the ordinary
Fourier transform does, but in the method described below, the transformed
pulse will interfere with the initial pulse, and for a perfect constructive or
destructive interference, their widths should be the same. The field
transformation by the fractional Fourier transform is given by Eq. (10) with
the kernel

$\tilde{K}_{\mathbf{T}}(t,t^{\prime})=\frac{e^{i\gamma/2+i\left(t^{2}\cos\gamma-2tt^{\prime}+{t^{\prime}}^{2}\cos\gamma\right)/\left(2\tau^{2}\sin\gamma\right)}}{\sqrt{2\pi
i\tau^{2}\sin\gamma}},$ (22)

where the tilde denotes a kernel phase-shifted by $\gamma/2$ with respect to
one defined by the matrix $\mathbf{T}$. In the limit $\sin\gamma\to 0$, which
happens at $\gamma=\pi m$ with an integer $m$, this kernel, according to Eq.
(15), tends to

$\tilde{K}_{\mathbf{T}}(t,t^{\prime})=e^{i\gamma/2}\sqrt{(-1)^{m}}\delta[t-(-1)^{m}t^{\prime}],$
(23)

where we have substituted $\cos(\pi m)=(-1)^{m}$.

Type-1 canonical decomposition of linear canonical transform [53] suggests
that a fractional Fourier transform can be realized by just one time lens of
focal GDD $D_{\mathrm{f}}$ preceded and followed by dispersive elements of GDD
$D$ each, supplemented by a phase shift $\gamma/2$. Such a system is described
by the temporal ray matrix

$\displaystyle\mathbf{T}_{\mathrm{prop}}(D)\mathbf{T}_{\mathrm{lens}}(D_{\mathrm{f}})\mathbf{T}_{\mathrm{prop}}(D)\phantom{TTTTTTTTTTTTT}$
(24)
$\displaystyle\phantom{TTTTTTTT}=\left(\begin{array}[]{cc}1-D/D_{\mathrm{f}}&-D(2-D/D_{\mathrm{f}})\\\
1/D_{\mathrm{f}}&1-D/D_{\mathrm{f}}\end{array}\right).$ (27)

In order that this matrix coincides with that of Eq. (21), two conditions
should be satisfied

$\displaystyle D=D_{\mathrm{f}}(1-\cos\gamma),$ (28) $\displaystyle
D_{\mathrm{f}}\sin\gamma=-\tau^{2}.$ (29)

Equation (17) takes for a fractional Fourier transform a simple form

$\frac{1}{\sqrt{\tau}}h_{n}\left(\frac{t}{\tau}\right)\to e^{-i\gamma
n}\frac{1}{\sqrt{\tau}}h_{n}\left(\frac{t}{\tau}\right),$ (30)

i.e. a Hermite-Gauss function of order $n$ is an eigenfunction of the
fractional Fourier transform with the eigenvalue $e^{-i\gamma n}$. When the
input pulse contains exactly one photon, its quantum state can be considered
as a superposition of an infinite number of basis states, corresponding to
photon occupying one of the Hermite-Gauss modes. In the Hilbert space spanned
by such states, the fractional Fourier transform of fractional angle
$\gamma=-2\pi/k$, $k=2,3,...$, realizes a gate, multiplying the $n$th basis
state by $e^{i2\pi n/k}$, which is known as $Z_{k}$ gate [73].

We see from Eq. (30), that the fractional Fourier transform does not change
the bandwidth of a Hermite-Gauss mode with the temporal scale $\tau$. At first
glance, this might seem unusual, because the quadratic phase modulation
realized by the time lens changes the instantaneous frequency and, in general,
the bandwidth of the field. However, in a fractional Fourier transform,
satisfying Eqs. (28) and (29) the interplay of dispersion and phase modulation
results in preservation of the bandwidth for any fractional angle. This can be
seen explicitly for the zeroth-order mode (a Gaussian pulse) by calculating
its bandwidth after a dispersive medium and a time lens and verifying that it
is preserved when conditions (28) and (29) are satisfied (Appendix A).

##  III Temporal mode sorter

The proposed temporal mode sorter is shown in Fig. 2. It represents a cascade
of Mach-Zehnder interferometers numbered by $\ell=1,2,...$, each containing in
one of its arms a $Z_{2^{\ell}}$ gate realized by a single-time-lens
fractional Fourier transform discussed above. The first beam splitter in every
interferometer is symmetric and provides a linear transformation of its input
fields written as a column vector
$\mathbf{A}_{\ell}^{\text{in}}=[A_{\ell}^{\text{in}}(t),B_{\ell}^{\text{in}}(t)]^{T}$
to its output fields (inside the interferometer)
$\mathbf{A}_{\ell}^{\prime}=[A_{\ell}^{\prime}(t),B_{\ell}^{\prime}(t)]^{T}$
so that $\mathbf{A}^{\prime}_{\ell}=\mathbb{U}\mathbf{A}_{\ell}^{\text{in}}$,
where

$\mathbb{U}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right)$ (31)

is the unitary transformation matrix of a symmetric beam splitter.

Figure 2: Schematic of interferometric temporal mode sorter including just two
interferometers $\ell=1$ and $\ell=2$ for sorting four first temporal Hermite-
Gauss modes. Two arms of each interferometer are denoted by A (single line)
and B (double line), which corresponds also to the notation of beams outside
the interferometers. The arm A of each interferometer contains two dispersive
elements (GDD), a time lens, and a phase shifter. At the sorter output, the
modes of orders 0 and 1 appear in the beam A separated by time $\Delta t$
surpassing their durations. The modes of orders 2 and 3 appear in the beam B
separated by the same time interval.

Both optical paths between the beam splitters in an interferometer should have
(i) the same total optical phase shift at the carrier frequency $\omega_{0}$
and (ii) the same group delay of the field envelope, in order to produce
constructive and destructive interference at the second beam splitter’s
output. These two conditions cannot be satisfied simultaneously by varying
just one parameter, the length difference of the interferometer arms. Both
conditions can be satisfied, e.g., by placing a dispersive element with the
dispersion law $k_{b}(\Omega)$ into the arm B of the interferometer. If the
total phase shift at the carrier frequency and the total group delay in the
arm A are $\phi_{a}$ and $\tau_{a}$ respectively, than the length of the air
part of the arm B $l_{1}$ and the length of the dispersive element $l_{2}$
should satisfy two equations: $\omega_{0}l_{1}/c+k_{b}l_{2}=\phi_{a}$ and
$l_{1}/c+k_{b}^{\prime}l_{2}=\tau_{a}$, where $k_{b}$ and $k_{b}^{\prime}$ are
coefficients of decomposition $k_{b}(\Omega)\approx
k_{b}+k_{b}^{\prime}\Omega$.

The second beam splitter in every interferometer will be described by the
matrix $\mathbb{U}^{\dagger}$ so that the total field transformation in an
empty interferometer is identity: $\mathbb{U}^{\dagger}\mathbb{U}=\mathbb{I}$,
where $\mathbb{I}$ is the unit $2\times 2$ matrix. We denote the action of a
$Z_{k}$ gate by operator $\mathcal{Z}_{k}$ so that the input field $A(t)$ is
transformed into $\mathcal{Z}_{k}A(t)$ and write the field at the
interferometer output as

$\displaystyle\mathbf{A}_{\ell}^{\text{out}}$ $\displaystyle=$
$\displaystyle\mathbb{U}^{\dagger}\left(\begin{array}[]{cc}e^{i\theta_{\ell}}\mathcal{Z}_{2^{\ell}}&0\\\
0&1\end{array}\right)\mathbb{U}\mathbf{A}_{\ell}^{\text{in}}$ (34)
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}e^{i\theta_{\ell}}\mathcal{Z}_{2^{\ell}}+1&e^{i\theta_{\ell}}\mathcal{Z}_{2^{\ell}}-1\\\
e^{i\theta_{\ell}}\mathcal{Z}_{2^{\ell}}-1&e^{i\theta_{\ell}}\mathcal{Z}_{2^{\ell}}+1\end{array}\right)\mathbf{A}_{\ell}^{\text{in}},$
(37)

where $\theta_{\ell}$ is an additional phase shift necessary for reaching
constructive and destructive interference at the interferometer output for the
targeted temporal modes.

###  III.1 Image inversion interferometer

The purpose of the first interferometer is to separate the even and odd order
modes at two its outputs. For this purpose, the interferometer is equipped
with a $Z_{2}$ gate (image inverter) in its arm A, a device realising a
temporal inversion of the input waveform. For this device the fractional angle
$\gamma\to-\pi$ and from Eqs. (28) and (29) we have $D=2D_{\mathrm{f}}$ and
$\tau^{2}/D_{\mathrm{f}}\ll 1$, which corresponds to a temporal counterpart of
the “2f-2f” spatial imaging system, having the magnification $-1$.
Substituting these values to Eq. (24), we obtain $a=d=-1$, $b=0$,
$c=1/D_{\mathrm{f}}$ and from Eq. (15) we have
$A(z,t)=iA(z^{\prime},-t)\exp(-it^{2}/2D_{\mathrm{f}})$, where the time
inversion is accompanied by a chirp and a $\pi/2$ phase shift (see also Ref.
[42] for field transformation in a system with magnification $-1$). In the
temporal far-field limit, $D\gg\tau^{2}$, required by the second condition,
the chirp is negligible. The $\pi/2$ phase shift is compensated by the
$\gamma/2=-\pi/2$ phase shift, which one needs to add to a linear canonical
transform in order to obtain a fractional Fourier transform. The overall field
transformation is

$A(z,t)=\mathcal{Z}_{2}A(z^{\prime},t)=A(z^{\prime},-t).$ (38)

If a long dispersive medium is not desirable, the chirp can be removed by a
second time lens of focal GDD $D_{\mathrm{f}}$ placed after the second
dispersive medium, which would create a time telescope with magnification -1
[63].

Additional phase shift is not necessary in this interferometer and we set
$\theta_{1}=0$. The even modes $\Psi_{0}$ and $\Psi_{2}$ are not affected by
the action of the image inverter, $\mathcal{Z}_{2}A(t)=A(t)$, the
transformation matrix in Eq. (34) is a unit matrix for these modes, in other
words, they experience a constructive interference with the field passing
through the arm B of the interferometer and go “right” after the second beam
splitter of the fist interferometer (Fig. 2). The odd modes $\Psi_{1}$ and
$\Psi_{3}$ acquire a $\pi$ phase shift in the image inverter,
$\mathcal{Z}_{2}A(t)=-A(t)$, and, as a consequence, the transformation matrix
in Eq. (34) is antidiagonal for these modes, which means that they experience
a destructive interference at the second beam splitter, and go “down” after
this beam splitter.

###  III.2 $Z_{4}$ interferometer

The second interferometer separates the zeroth and second order modes and
later the first and third order modes. This interferometer is equipped with a
$Z_{4}$ gate in its arm A. For this device, $\gamma=-\pi/2$ and from Eqs. (28)
and (29) we have $D=D_{\mathrm{f}}=\tau^{2}$, which corresponds to a temporal
Fourier processor. The field transformation in the temporal Fourier processor
is given by Eq. (10) with $a=d=0$ and $b=-1/c=-\tau^{2}$, supplemented by a
$\gamma/2=-\pi/4$ phase shift:

$A(z,t)=\mathcal{Z}_{4}A(z^{\prime},t)=\frac{1}{\sqrt{2\pi\tau^{2}}}\int
e^{itt^{\prime}/\tau^{2}}A(z^{\prime},t^{\prime})dt^{\prime}.$ (39)

This $Z_{4}$ gate, according to Eq. (30), realizes a transformation
$\Psi_{n}(t)\to e^{i\pi n/2}\Psi_{n}(t)$. Thus, the mode $\Psi_{0}$ is
unaffected by this gate and goes “right” after the output beam splitter of the
second interferometer (Fig. 2). The mode $\Psi_{2}$ acquires a $\pi$ phase
shift and goes “down”. For these two modes we set $\theta_{2}=0$.

The modes $\Psi_{1}$ and $\Psi_{3}$ are delayed by $\Delta t\gg\tau$ and enter
the second interferometer through the other port of its input beam splitter.
During their passage an additional phase shift $\theta_{2}=\pi/2$ is added to
the $Z_{4}$ gate, the overall phase shift being $\gamma/2+\theta_{2}=+\pi/4$.
For this reason, the phase shifter in the second interferometer is denoted as
“variable” in Fig. 2. As a result, the mode $\Psi_{1}$ acquires a
$\pi/2+\theta_{2}=\pi$ phase shift, experiences a destructive interference at
the output beam splitter of the second interferometer and goes “right” after
it. In contrast, the mode $\Psi_{3}$ acquires a $3\pi/2+\theta_{2}=2\pi$ phase
shift, i.e. is unaffected, experiences a constructive interference at the
output beam splitter of the second interferometer and goes “down” after it.

###  III.3 Extension to higher dimensions

The cascade of two interferometers shown in Fig. 2 acts identically on modes
of orders $n$ and $n+4$ and, therefore, can be called a “modulo 4 sorter”. A
third interferometer may be added to separate the zeroth and fourth order
modes and later other pairs of modes of orders $n$ and $n+4$. This
interferometer is equipped with a $Z_{8}$ gate in its arm A. For this device
we have the fractional angle $\gamma=-\pi/4$ and, consequently, from Eqs. (28)
and (29), $D_{\mathrm{f}}=\sqrt{2}\tau^{2}$ and
$D=D_{\mathrm{f}}(\sqrt{2}-1)/\sqrt{2}$, which corresponds to a temporal
fractional Fourier processor of order $\frac{1}{2}$. The field transformation
in this processor is given by Eq. (10) with $a=d=\sqrt{2}/2$ and
$b=-\sqrt{2}\pi\tau^{2}$, supplemented by a $-\pi/8$ phase shift:

$A(z,t)=\frac{e^{i\pi/8}}{2^{1/4}\sqrt{\pi\tau^{2}}}\int
e^{i\left(2^{-1/2}tt^{\prime}-t^{2}-{t^{\prime}}^{2}\right)/\tau^{2}}A(z^{\prime},t^{\prime})dt^{\prime}.$
(40)

This $Z_{8}$ gate, according to Eq. (30), realizes a transformation
$\Psi_{n}(t)\to e^{i\pi n/4}\Psi_{n}(t)$. The mode $\Psi_{0}$ passes it
unaffected and exits “right” the third interferometer. The mode $\Psi_{4}$
propagating together with $\Psi_{0}$ acquires a phase $\pi$ and exits it
“down”. After the time interval $\Delta t$ the modes $\Psi_{1}$ and $\Psi_{5}$
enter the third interferometer. During their passage, an additional phase
shift $\theta_{3}=-\pi/4$ should be applied to the variable phase shifter in
this interferometer, so that they acquire phase shifts $0$ and $\pi$
respectively and go in different directions after the output beam splitter of
this interferometer.

The modes $\Psi_{2}$ and $\Psi_{3}$ should enter the third interferometer upon
passing a delay line introducing the delay $2\Delta t$. They are separated
from the accompanying modes $\Psi_{6}$ and $\Psi_{7}$ in a similar way.

##  IV Sorting the temporal modes of SPDC

As an application of the proposed temporal mode sorter, we consider sorting of
temporal modes of two entangled photons (a biphoton) generated in SPDC.
Entanglement of photons means that a set of discrete spatial, temporal and
polarization modes (Schmidt modes) can be defined for each photon, which are
perfectly correlated: If one photon is found in a given Schmidt mode, the
other photon is found in its corresponding Schmidt mode [74]. The total number
of entangled modes is characterized by Schmidt number, which can be very high
for the general case of spatio-temporal entanglement [75, 76]. Here, we
consider a regime of type-II collinear and frequency-degenerate SPDC where the
generated photons have orthogonal polarizations, occupy the same spatial mode
and are entangled in their temporal degree of freedom only. In addition, the
condition of symmetric group-velocity matching can be achieved by means of
quasi-phase matching at a given pump wavelength [69] providing two benefits:
The generated photons have identical temporal Schmidt modes and the number of
these modes can be varied by tuning the pump bandwidth. The Schmidt modes of
one photon can be sorted and, subsequently, those of the other. Since the
central frequencies and modal shapes coincide for both photons, the same mode
sorter can be applied to both of them. The scheme of the proposed experiment
is shown in Fig. 3.

Figure 3: Schematic of proposed experiment, where photon pairs are generated
in frequency-degenerate collinear type-II SPDC with symmetric group velocity
matching. The ordinary and extraordinary photons are separated on a polarizing
beam splitter (PBS). The ordinary photon enters the sorter in beam A (single
line). The extraordinary photon has its polarization rotated by a $\lambda/2$
plate and is delayed by $\Delta T$, then enters the sorter in beam B (double
line). At the interferometer output, the even modes of the ordinary photon
exit in beam A, while its odd modes exit in beam B. For the extraordinary
photon, the exits are switched, the even modes exit in beam B, while the odd
ones in beam A.

###  IV.1 Parametric downconversion

We consider a nonlinear crystal with type-II frequency-degenerate phase
matching. The plain-wave pump with the central frequency $\omega_{p}$
propagates along the $z$ axis and is polarized as either ordinary or
extraordinary wave. As a result of the nonlinear transformation of the pump
field in the crystal, two subharmonic waves with the central frequency
$\omega_{0}=\omega_{p}/2$ emerge, polarized as ordinary and extraordinary
waves. The interaction of these waves is most easily described in the
Heisenberg picture in the spectral domain. The positive-frequency part of the
field of each wave (in photon-flux units) is

$E_{\mu}^{(+)}(z,t)=\int\epsilon_{\mu}(z,\Omega)e^{ik_{\mu}(\Omega)z-i(\omega_{0}+\Omega)t}\frac{d\Omega}{2\pi},$
(41)

where $\mu$ takes values $\\{p,o,e\\}$ for the pump, ordinary and
extraordinary subharmonic waves respectively, $t$ is time, $\Omega$ denotes
the frequency detuning from the carrier frequency, and $k_{\mu}(\Omega)$ is
the wave vector of the corresponding wave at frequency $\omega_{\mu}+\Omega$.
The spectral amplitude of the pump $\epsilon_{p}(\Omega,z)$ is independent of
$z$ since the pump is assumed to be undepleted. We assume also that it is a
Gaussian transform-limited pulse with full width at half maximum (FWHM)
duration $\tau_{p}$, whose peak passes the position $z=0$ (crystal input face)
at time $t=0$, and write
$\epsilon_{p}(z,\Omega)=\alpha(\Omega)=\alpha_{0}\exp(-\Omega^{2}/4\Omega_{p}^{2})$,
where $\Omega_{p}=\sqrt{2\ln 2}/\tau_{p}$.

The spectral amplitude of the ordinary or extraordinary subharmonic wave,
$\epsilon_{\mu}(z,\Omega)$, is the annihilation operator of a photon at
position $z$ with the frequency $\omega_{\mu}+\Omega$ and the corresponding
polarization, satisfying the canonical equal-space commutation relations [55,
56, 77]
$\left[\epsilon_{\mu}(z,\Omega),\epsilon_{\nu}^{\dagger}(z,\Omega^{\prime})\right]=2\pi\delta_{\mu\nu}\delta(\Omega-\Omega^{\prime})$.
The evolution of this operator along the crystal is described by the spatial
Heisenberg equation [78]

$\frac{d}{dz}\epsilon_{\mu}(z,\Omega)=\frac{i}{\hbar}\left[\epsilon_{\mu}(z,\Omega),G(z)\right],$
(42)

where the spatial Hamiltonian $G(z)$ is given by the momentum transferred
through the plane $z$ [77] and equals

$G(z)=\chi(z)\int\limits_{-\infty}^{+\infty}E^{(-)}_{p}(z,t)\hat{E}^{(+)}_{o}(z,t)\hat{E}^{(+)}_{e}(z,t)dt+\mathrm{H.c.},$
(43)

where $\chi(z)$ is the nonlinear coupling coefficient and
$E^{(-)}_{p}(z,t)=E^{(+)*}_{p}(z,t)$ is the negative-frequency part of the
field. In a periodically poled crystal, $\chi(z)$ changes its sign every
distance of $\Lambda/2$, where $\Lambda$ is the poling period. i.e.,
represents a meander function. This function can be decomposed into Fourier
series, where only the term of the order $-1$ affects the phase matching [79].
Thus, we write $\chi(z)\approx\chi_{0}\exp(-2\pi iz/\Lambda)$. Substituting
Eqs. (41) and (43) into Eq. (42), performing the integration, and applying the
commutation relations, we obtain the spatial evolution equations

$\begin{split}\frac{d\epsilon_{o}(z,\Omega)}{dz}&=\kappa\int\alpha(\Omega+\Omega^{\prime})\epsilon^{\dagger}_{e}(z,\Omega^{\prime})e^{i\Delta(\Omega,\Omega^{\prime})z}d\Omega^{\prime},\\\
\frac{d\epsilon_{e}(z,\Omega)}{dz}&=\kappa\int\alpha(\Omega+\Omega^{\prime})\epsilon^{\dagger}_{o}(z,\Omega^{\prime})e^{i\Delta(\Omega^{\prime},\Omega)z}d\Omega^{\prime},\end{split}$
(44)

where $\kappa=i\chi_{0}^{*}/2\pi\hbar$ is the new coupling constant and
$\Delta(\Omega,\Omega^{\prime})=k_{p}(\Omega+\Omega^{\prime})-k_{o}(\Omega)-k_{e}(\Omega^{\prime})+2\pi/\Lambda$
is the phase mismatch for the three interacting waves.

In the low-gain regime, where the pump amplitude is sufficiently small, Eq.
(44) can be solved perturbatively, by substituting
$\epsilon^{\dagger}_{\mu}(z,\Omega^{\prime})\to\epsilon^{\dagger}_{\mu}(0,\Omega^{\prime})$
under the integral. In this way, we obtain the fields at the output of a
crystal of length $L$

$\begin{split}\epsilon_{o}(L,\Omega)&=\epsilon_{o}(0,\Omega)+\int
J(\Omega,\Omega^{\prime})\epsilon^{\dagger}_{e}(0,\Omega^{\prime})\frac{d\Omega^{\prime}}{2\pi},\\\
\epsilon_{e}(L,\Omega)&=\epsilon_{e}(0,\Omega)+\int
J(\Omega^{\prime},\Omega)\epsilon^{\dagger}_{o}(0,\Omega^{\prime})\frac{d\Omega^{\prime}}{2\pi},\end{split}$
(45)

where $J(\Omega,\Omega^{\prime})=2\pi\kappa
L\alpha(\Omega+\Omega^{\prime})\Phi(\Omega,\Omega^{\prime})$ is the joint
spectral amplitude (JSA) of two generated photons and
$\Phi(\Omega,\Omega^{\prime})=e^{i\Delta(\Omega,\Omega^{\prime})L/2}\operatorname{sinc}[\Delta(\Omega,\Omega^{\prime})L/2]$
is the phase-matching function.

A typical approximation in type-II SPDC consists in linearizing the dispersion
law: $k_{\mu}(\Omega)\approx k_{\mu}^{0}+k_{\mu}^{\prime}\Omega$, where
$k_{\mu}^{0}=k_{\mu}(0)$ and $k_{\mu}^{\prime}$ is the derivative of $k_{\mu}$
with respect to $\Omega$ at $\Omega=0$, having the meaning of inverse group
velocity. In this approximation, the phase mismatch angle takes the form
$\Delta(\Omega,\Omega^{\prime})L/2\approx\tau_{o}\Omega+\tau_{e}\Omega^{\prime}$,
where $\tau_{\mu}=(k_{p}^{\prime}-k_{\mu}^{\prime})L/2$ and we assume that the
poling period $\Lambda$ is chosen so that the condition of phase matching at
degeneracy $k_{p}^{0}-k_{o}^{0}-k_{e}^{0}+2\pi/\Lambda=0$ is satisfied. The
time $\tau_{\mu}$ has the meaning of the interval by which the peak of the
corresponding subharmonic pulse advances the peak of the pump pulse at the
crystal output: The peak of the subharmonic is generated when the pump pulse
is in the center of the crystal, the travel time for the pump and subharmonic
peaks through the second half of the crystal are $k_{p}^{\prime}L/2$ and
$k_{\mu}^{\prime}L/2$ respectively, hence the meaning of $\tau_{\mu}$ as the
difference of these times. The JSA in this approximation reads

$\displaystyle J(\Omega,\Omega^{\prime})$ $\displaystyle=$ $\displaystyle
2\pi\kappa
L\alpha_{0}e^{-\frac{(\Omega+\Omega^{\prime})^{2}}{4\Omega_{p}^{2}}+i(\tau_{o}\Omega+\tau_{e}\Omega^{\prime})}$
$\displaystyle\times\operatorname{sinc}(\tau_{o}\Omega+\tau_{e}\Omega^{\prime}).$

Substituting the solutions, Eq. (45), into Eq. (41), we obtain the field
transformation from the crystal input face to its output one. To write this
transformation in a compact form, we define the group-delayed envelopes in the
spirit of Eq. (4): $E_{\mu}^{(+)}(0,t)=A_{\mu,\text{vac}}(t)e^{-i\omega_{0}t}$
and
$E_{\mu}^{(+)}(L,t)=A_{\mu}(t-k_{\mu}^{\prime}L)e^{ik_{\mu}^{0}L-i\omega_{0}t}$.
In this notation, the field transformation in the crystal has the form

$\displaystyle A_{o}(t)$ $\displaystyle=$ $\displaystyle
A_{o,\text{vac}}(t)+\int\tilde{J}(t,t^{\prime})A_{e,\text{vac}}^{\dagger}(t^{\prime})dt^{\prime},$
(47) $\displaystyle A_{e}(t)$ $\displaystyle=$ $\displaystyle
A_{e,\text{vac}}(t)+\int\tilde{J}(t^{\prime},t)A_{o,\text{vac}}^{\dagger}(t^{\prime})dt^{\prime},$
(48)

where the joint temporal amplitude (JTA) of two generated photons is the
double Fourier transform of their JSA:

$\tilde{J}(t,t^{\prime})=\int\int J(\Omega,\Omega^{\prime})e^{-i\Omega
t-i\Omega^{\prime}t^{\prime}}\frac{d\Omega d\Omega^{\prime}}{(2\pi)^{2}}.$
(49)

Substituting Eq. (IV.1) into Eq. (49), we obtain

$\tilde{J}(t,t^{\prime})=J_{N}\Pi\left(\frac{t-t^{\prime}}{2\tau_{-}}-\frac{1}{2}\right)e^{-\left(t\tau_{e}-t^{\prime}\tau_{o}\right)^{2}\Omega_{p}^{2}/\tau_{-}^{2}},$
(50)

where $J_{N}=\sqrt{\pi}\kappa L\alpha_{0}\Omega_{p}/|\tau_{-}|$, $\Pi(x)$ is
the rectangle function equal to 1 for $|x|<\frac{1}{2}$ and 0 otherwise, and
$\tau_{-}=\tau_{o}-\tau_{e}$. The same JTA was derived by considering the
temporal evolution of the fields [80], instead of the spatial evolution
considered here.

###  IV.2 Gaussian modeling

Both JSA and JTA can be significantly simplified by approximating them with
double Gaussian functions. For this purpose, we replace the
$\operatorname{sinc}(x)$ function in Eq. (IV.1) with the Gaussian function
$e^{-x^{2}/2\sigma_{s}^{2}}$ having the same width at half maximum for
$\sigma_{s}=1.61$ [81, 82]. Upon this replacement, JSA becomes a double
Gaussian function. Substituting it into Eq. (49) and applying the
multidimentional Gaussian integration technique [62, 63], we obtain JTA also
in the form of a double Gaussian function

$\tilde{J}(t,t^{\prime})=J_{1}e^{-M_{11}(t-\tau_{o})^{2}-M_{22}(t^{\prime}-\tau_{e})^{2}-2M_{12}(t-\tau_{o})(t^{\prime}-\tau_{e})},$
(51)

where $J_{1}=2\kappa L\alpha_{0}\Omega_{p}^{2}/|T_{o}-T_{e}|$,
$T_{\mu}=\sqrt{2}\Omega_{p}\tau_{\mu}/\sigma_{s}$, and $M_{ij}$ are elements
of the matrix

$\mathbf{M}=\frac{\Omega_{p}^{2}}{(T_{o}-T_{e})^{2}}\left(\begin{array}[]{cc}1+T_{e}^{2}&-1-T_{o}T_{e}\\\
-1-T_{o}T_{e}&1+T_{o}^{2}\end{array}\right).$ (52)

A double Gaussian JTA can be represented in a form of Schmidt decomposition by
means of the Mehler’s formula for Hermite polynomials (see Appendix B)

$\tilde{J}(t,t^{\prime})=\sqrt{P_{b}}\sum\limits_{n=0}^{\infty}\sqrt{\lambda_{n}}\psi_{n}(t)\varphi_{n}(t^{\prime}),$
(53)

where $P_{b}=2\pi(\kappa L\alpha_{0}\Omega_{p})^{2}/|T_{o}-T_{e}|$ is the
probability of biphoton generation, $\psi_{n}(t)$ and
$\varphi_{n}(t^{\prime})$ are the Schmidt modal functions of the ordinary and
extraordinary photons respectively, defined as

$\displaystyle\psi_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\tau_{1}}}h_{n}\left(\frac{t-\tau_{o}}{\tau_{1}}\right),$
(54) $\displaystyle\varphi_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\tau_{2}}}h_{n}\left(\frac{t-\tau_{e}}{\tau_{2}}\right),$
(55)

$\lambda_{n}$ are the Schmidt coefficients

$\lambda_{n}=\frac{2}{K+1}\left(\frac{K-1}{K+1}\right)^{n}$ (56)

normalized to unity, $\sum_{n}\lambda_{n}=1$, and

$K=\frac{1}{\sum_{n}\lambda_{n}^{2}}=\frac{\sqrt{(1+T_{o}^{2})(1+T_{e}^{2})}}{|T_{o}-T_{e}|}$
(57)

is the Schmidt number. In the Schrödinger picture, the JTA gives the wave
function of two photons, which are entangled when the JTA is not separable in
its arguments. The Schmidt number $K$ shows the effective number of entangled
modes for each photon and as such is a measure of the degree of entanglement
[83, 76, 75]. The temporal widths of the Schmidt modes are

$\tau_{1,2}=\frac{\sqrt{|T_{o}-T_{e}|}}{\sqrt{2}\Omega_{p}}\left(\frac{1+T_{o,e}^{2}}{1+T_{e,o}^{2}}\right)^{\frac{1}{4}}.$
(58)

We see from Eqs. (54) and (55) that the modes with polarization $\mu=o,e$ are
delayed by $\tau_{\mu}$. Indeed, in accord with the general formalism of Sec.
II, the field envelopes at position $z=L$ are delayed by the group delay time
$k_{\mu}^{\prime}L$. However, the peak of the zeroth-order mode exits the
crystal at time $t=k_{\mu}^{\prime}L+\tau_{\mu}$ if the peak of the pump pulse
entered it at time $t=0$: The peak of subharmonic is generated when the pump
pulse is in the center of the crystal, which happens at time
$t=k_{p}^{\prime}L/2$, it passes the second half of the crystal during time
$k_{\mu}^{\prime}L/2$, and exits the crystal at time
$t=k_{p}^{\prime}L/2+k_{\mu}^{\prime}L/2=k_{\mu}^{\prime}L+\tau_{\mu}$. We
see, that the group delays for fields _generated_ in dispersive media are
different from those for fields _passing_ through them.

Another observation from Eqs. (54) and (55) is that the temporal widths of the
Schmidt modes are, in general, different for two photons. It means that the
task of sorting these modes requires two different mode sorters, each tuned to
a given temporal width $\tau_{1}$ or $\tau_{2}$. However, for applications
like quantum cryptography or quantum computing with temporal modes [4], it is
highly desirable to have one mode sorter applied sequentially to the first and
then to the second photon of the pair. The next section shows how this mode
symmetry can be reached.

###  IV.3 Symmetric group velocity matching

We see from Eq. (58) that the practically interesting regime with
$\tau_{1}=\tau_{2}$ can be reached when $\tau_{o}=\pm\tau_{e}$. In the case
$\tau_{o}=\tau_{e}$, the kernels Eqs. (IV.1) and (50) are not square-
integrable and the linear approximation for the dispersion is invalid. The
case $\tau_{o}=-\tau_{e}$ is more practical and is known as symmetric group
velocity matching [80, 69] or extended phase matching [84]. It was first
engineered in a crystal of periodically-poled potassium titanyl phosphate
(ppKTP) [85], which we consider here as an example. The pump propagates along
the $X$ axis of this biaxial crystal and is polarized along the $Y$ axis, i.e.
represents an ordinary wave. The two subharmonic waves are polarized along the
$Y$ (ordinary wave) and $Z$ (extraordinary wave) axes of the crystal. The
latter should not be confused with the $z$ axis of the reference frame, the
direction of pump propagation.

The refractive indices for the $Y$ and $Z$ polarized waves are obtained from
the Sellmeier equations for KTP [85] and, for a crystal of length $L=40$ mm
pumped at wavelength $\lambda_{p}=2\pi c/\omega_{p}=791.5$ nm, we obtain
$\tau_{o}=-\tau_{e}=2.95$ ps. The poling period required for reaching the
phase matching at degeneracy is $\Lambda=47.6$ $\mu$m. Substituting
$\tau_{o}=-\tau_{e}>0$ into Eq. (58), we obtain

$\tau_{1}=\tau_{2}=\tau=\sqrt{\frac{\tau_{o}\tau_{p}}{\sigma_{s}\sqrt{\ln
2}}}\approx 0.86\sqrt{\tau_{o}\tau_{p}}.$ (59)

The Schmidt number, Eq. (57), takes the form

$K=\frac{1}{2}\left(T_{o}+\frac{1}{T_{o}}\right)=\frac{1}{2}\left(\frac{\delta_{s}\tau_{o}}{\tau_{p}}+\frac{\tau_{p}}{\delta_{s}\tau_{o}}\right),$
(60)

where $\delta_{s}=2\sqrt{\ln 2}/\sigma_{s}\approx 1.03$. The minimal value of
$K=1$ is reached at the pump pulse duration $\tau_{p}=\delta_{s}\tau_{0}$. In
this case the photons are disentangled, each occupying just one Gaussian
temporal mode, as required for some applications. We, however, are interested
in a longer pump pulse, resulting in a moderate number of entangled modes.

When choosing the pump pulse duration $\tau_{p}$, we need to remember, that
Eq. (9) is valid only under the condition that the temporal aperture of a time
lens surpasses the duration of the stretched pulse passing through it [41],
otherwise including integrals over temporal point-spread functions [42]. The
temporal aperture of a parametric time lens is
$T_{A}=D_{\mathrm{f}}\Omega_{m}$, where $\Omega_{m}$ is the FWHM of the
intensity spectrum (in circular frequency units) of the pump pulse in a sum-
frequency generation based time lens [57, 60] and that divided by $\sqrt{2}$
in a four-wave mixing based time lens [59]. However, a parametric time lens
has typically a non-unit frequency conversion efficiency, which means a
probabilistic action for single photons. A great advantage of a time lens
based on electro-optical phase modulation (EOPM) consists in the deterministic
nature of the electrooptical effect, providing an almost lossless operation
[65, 68]. Its main shortcoming is a rather short temporal aperture
$T_{A}=\sqrt{AD_{\mathrm{f}}}$ for the given focal GDD $D_{\mathrm{f}}$ and
phase modulation amplitude $A$ [41, 59]. This shortcoming is removed in the
recently proposed Fresnel time lens [86, 67], successfully applied to
bandwidth compression of single photons [68]. The driving voltage in such a
time lens is not sinusoidal, as in ordinary modulators, but represents a
wrapped parabola, created by an arbitrary waveform generator. To create a
phase shift $\phi=t^{2}/2D_{\mathrm{f}}$ within the temporal aperture $T_{A}$,
the modulator should realize the instantaneous circular frequency
$\partial\phi/\partial t=t/D_{\mathrm{f}}$ at times $t=\pm T_{A}/2$, which is
limited by the electronic bandwidth of the arbitrary waveform generator
$f_{\mathrm{RF}}$, i.e. $T_{A}/2D_{\mathrm{f}}=2\pi f_{\mathrm{RF}}$. Thus,
for a Fresnel time lens, the temporal aperture scales linearly with the focal
GDD: $T_{A}=D_{\mathrm{f}}\Omega_{m}$, where $\Omega_{m}=4\pi
f_{\mathrm{RF}}$. For a parametric or electro-optical Fresnel time lens, we
write the condition on the time aperture as
$T_{1}^{F}<D_{\mathrm{f}}\Omega_{m}$, where $T_{1}^{F}$ is the FWHM duration
of the stretched pulse entering the time lens. Considering the zeroth-order
Hermit-Gauss mode (a Gaussian pulse) of temporal scale $\tau$ undergoing a
fractional Fourier transform, we arrive at the condition (Appendix A)
$\tau>\tau_{G}$, where $\tau_{G}=4\sqrt{\ln 2}/\Omega_{m}$. The spectral width
of Hermite-Gauss modes scales approximately as $\sqrt{n}$ with the mode order
$n$ [87, 88] and we can estimate the duration of the stretched pulse by the
same scaling factor. In order that $K$ modes fit into the temporal aperture,
we need to satisfy the condition $\tau>\sqrt{K}\tau_{G}$. Substituting Eqs.
(59) and (60) (in the limit $\tau_{p}\gg\tau_{o}$) into this inequality, we
obtain a limitation on the time lens bandwidth

$\Omega_{m}>\frac{c_{m}}{\tau_{o}},$ (61)

where $c_{m}=2\sqrt{\ln 2}\sigma_{s}\approx 2.68$. For an electrooptical
Fresnel time lens, we obtain $f_{\mathrm{RF}}>72$ GHz, which is within the
reach of modern technology, especially for integrated solutions [89]. Choosing
$f_{\mathrm{RF}}=75$ GHz and $K=4$, we obtain $\tau_{G}=3.5$ ps, $\tau_{p}=24$
ps, and $\tau=7.3$ ps.

###  IV.4 Modulo $2$ (parity) sorting

Let us consider the field transformation in the temporal-image-inverting
interferometer shown in Fig. 3. In the group-delayed reference frame of the
ordinary wave, the field at the interferometer entrance is non-vacuum around
time $\tau_{o}$, when the ordinary wave arrives, and around time
$\tau_{e}+\Delta T$, when the extraordinary wave does. We have thus two
temporal windows of width $\Delta T$ each, the first centered at the peak of
zeroth-order mode of the ordinary wave, the second at the peak of zeroth-order
mode of the extraordinary one. Note, that in the reference frame of the
extraordinary wave, the group delay $\Delta T$ is included into the definition
of the field envelope. Below, the window number will be indicated in square
brackets in the upper index. We recall that the lower index is the
interferometer number.

In the first window, we shift the time axis to the peak of the zeroth-order
mode and define $A_{1}^{[1]\text{in}}(t)=A_{o}(t+\tau_{o})$. The field at the
other input is just vacuum
$B_{1}^{[1]\text{in}}(t)=B_{1,\text{vac}}^{[1]}(t)$. The fields at the
interferometer output are given by Eq. (34), where $\ell=1$, $\theta_{1}=0$,
and the action of operator $\mathcal{Z}_{2}$ is described by Eq. (38). The
time lens in the interferometer is synchronized with the time axis, i.e. the
vertex of the parabolic phase coincides with the passage of the peak of the
zeroth-order mode; the effects of desynchronization can be found in Ref. [62].
At the interferometer output, we identify the detector of beam A as measuring
parity $j=+1$ and define the field on its surface as
$F_{+1}^{[1]}(t)=A_{1}^{[1]\text{out}}(t)$ and the detector of beam B as
measuring parity $j=-1$ and define $F_{-1}^{[1]}(t)=B_{1}^{[1]\text{out}}(t)$.
Applying the field transformation in the interferometer, Eq. (34), we obtain
(for $j=\pm 1$)

$F_{j}^{[1]}(t)=\frac{\mathcal{Z}_{2}+j}{2}A_{1}^{[1]\text{in}}(t)+\frac{\mathcal{Z}_{2}-j}{2}B_{1,\text{vac}}^{[1]}(t).$
(62)

In the second window, we shift, in a similar way, the time axis to the peak of
the zeroth-order mode and define $B_{1}^{[2]\text{in}}(t)=A_{e}(t+\tau_{e})$.
The field at the other input is just vacuum
$A_{1}^{[2]\text{in}}(t)=A_{1,\text{vac}}^{[2]}(t)$. At the interferometer
output, we identify the detector of beam B as measuring parity $k=+1$ and
define $F_{+1}^{[2]}(t)=B_{1}^{[2]\text{out}}(t)$ the detector of beam A as
measuring parity $k=-1$ and define $F_{-1}^{[2]}(t)=A_{1}^{[1]\text{out}}(t)$.
Applying the field transformation in the interferometer, as above, we obtain
(for $k=\pm 1$)

$F_{k}^{[2]}(t)=\frac{\mathcal{Z}_{2}+k}{2}B_{1}^{[2]\text{in}}(t)+\frac{\mathcal{Z}_{2}-k}{2}A_{1,\text{vac}}^{[2]}(t).$
(63)

The probability to observe, in the same pumping cycle, one photon on detector
$j=\pm 1$ in the first window and another photon on detector $k=\pm 1$ in the
second window is

$P_{jk}=\int\limits_{-\Delta T/2}^{+\Delta T/2}dt\int\limits_{-\Delta
T/2}^{+\Delta T/2}dt^{\prime}\left|C_{jk}(t,t^{\prime})\right|^{2},$ (64)

where $C_{jk}(t,t^{\prime})=\langle
F_{j}^{[1]}(t)F_{k}^{[2]}(t^{\prime})\rangle$ is the anomalous correlator of
the fields on the detectors. Here, we have used the Gaussian moment theorem
[54] and kept only the term of the lowest order in the smallness parameter of
the low-gain regime of SPDC $\kappa L\alpha_{0}\Omega_{p}\ll 1$ (see Appendix
B). Substituting Eqs. (47), (48), (62), and (63) into the anomalous
correlator, applying commutation relations
$[A(t),A^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$, and equating to zero
all normally ordered correlators of vacuum fields, we obtain

$C_{jk}(t,t^{\prime})=\frac{1}{4}\left(\mathcal{Z}_{2}+j\right)\left(\mathcal{Z}_{2}^{\prime}+k\right)\tilde{J}_{0}(t,t^{\prime}),$
(65)

where $\tilde{J}_{0}(t,t^{\prime})=\tilde{J}(t+\tau_{o},t^{\prime}+\tau_{e})$
is the zero-centered JTA of two generated photons and prime on the operator
$\mathcal{Z}_{2}^{\prime}$ means that it acts on a function of time
$t^{\prime}$. The zero-centered versions of the exact JTA, Eq. (50) and its
Gaussian approximation, Eq. (51), for $\tau_{o}=-\tau_{e}>0$, are

$\displaystyle\tilde{J}_{0}^{E}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\left(\frac{\pi}{2}\right)^{\frac{1}{4}}\sqrt{\frac{P_{b}^{\prime}\Omega_{p}}{4\tau_{0}}}\Pi\left(\frac{t-t^{\prime}}{4\tau_{o}}\right)e^{-(t+t^{\prime})^{2}\Omega_{p}^{2}/4},$
(66) $\displaystyle\tilde{J}_{0}^{G}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{P_{b}\Omega_{p}^{2}}{\pi
T_{0}}}e^{-M_{11}(t^{2}+t^{\prime 2})-2M_{12}tt^{\prime}},$ (67)

where $P_{b}^{\prime}=2(\pi/2)^{3/2}(\kappa
L\alpha_{0})^{2}\Omega_{p}/\tau_{o}$ is the biphoton generation probability
calculated from the exact JTA, and the elements of the matrix $\mathbf{M}$ are
$M_{11}=\Omega_{p}^{2}(T_{o}^{2}+1)/4T_{o}^{2}$ and
$M_{12}=\Omega_{p}^{2}(T_{o}^{2}-1)/4T_{o}^{2}$. These functions are double
Fourier transforms of the phase-shifted JSA,
$J_{0}(\Omega,\Omega^{\prime})=J(\Omega,\Omega^{\prime})e^{-i(\tau_{o}\Omega+\tau_{e}\Omega^{\prime})}$,
and are shown in Fig. 4.

Figure 4: (a) JSA of two photons generated in a 40-mm-long ppKTP crystal
pumped at wavelength 791.5 nm by pulses of duration 24 ps. The JSA (color map)
lies on the intersection of the pump-limited area and phase-matched area. The
red dashed line shows the points where the Gaussian approximation is equal to
0.1. (b) JTA of the same photons lying on the intersection of the same two
areas in temporal domain. The red dashed line shows the points where the
Gaussian approximation is equal to 0.1.

To simplify the calculations, we consider any square-integrable complex
function $g(t,t^{\prime})$ as a vector in the space of square-integrable
functions of $t$ and $t^{\prime}$ and denote it by $|g)$. Similar to as it was
done for ket and bra vectors by Dirac [1], we introduce the conjugated space
of vectors $(f|$ with a scalar product

$(f|g)=\int\limits_{-\infty}^{+\infty}dt\int\limits_{-\infty}^{+\infty}dt^{\prime}f^{*}(t,t^{\prime})g(t,t^{\prime}).$
(68)

For any operator $\mathcal{Q}$, its conjugate $\mathcal{Q}^{\dagger}$ is
defind by $(f|\mathcal{Q}^{\dagger}|g)=(g|\mathcal{Q}|f)^{*}$. In these
notations and in the limit of sufficiently long $\Delta T$, we rewrite Eq.
(64) as $P_{jk}=(\tilde{J}_{0}|\mathcal{O}_{jk}|\tilde{J}_{0})$, where

$\mathcal{O}_{jk}=\frac{1}{16}\left(\mathcal{Z}_{2}^{\dagger}+j\right)\left(\mathcal{Z}_{2}^{{}^{\prime}\dagger}+k\right)\left(\mathcal{Z}_{2}+j\right)\left(\mathcal{Z}_{2}^{\prime}+k\right).$
(69)

Physically, this corresponds to calculation of outcome probabilities as
averages of the elements of a positive operator-valued measure (POVM)
$\mathcal{O}_{jk}$ in a specific interaction picture, where
$\tilde{J}_{0}(t,t^{\prime})$ is the temporal wave function of the biphoton.
In this picture, the temporal dependence of the wave function includes the
interaction of the waves in the source, but all interaction in the sorter is
described by the evolution of the field operators.

The time inversion operator $\mathcal{Z}_{2}$ is unitary,
$\mathcal{Z}_{2}^{\dagger}\mathcal{Z}_{2}=1$, and Hermitian,
$\mathcal{Z}_{2}^{\dagger}=\mathcal{Z}_{2}$, which results in all
$\mathcal{O}_{jk}$ summing up to unity, as should be for a POVM. In addition,
in the subspace of central-symmetric functions having the property
$g(-t,-t^{\prime})=g(t,t^{\prime})$, to which $\tilde{J}_{0}(t,t^{\prime})$
belongs, the following equalities hold:
$\mathcal{Z}_{2}\mathcal{Z}_{2}^{\prime}|g)=|g)$ and
$\mathcal{Z}_{2}|g)=\mathcal{Z}_{2}^{\prime}|g)$. From these properties and
Eq. (69), we find $\mathcal{O}_{\pm 1\mp 1}=0$ and $\mathcal{O}_{\pm 1\pm
1}=(1\pm\mathcal{Z}_{2})/2$. Finally, we obtain the probabilities of outcomes
as

$P_{jk}=\frac{P_{b}}{2}\left(1+j\frac{(\tilde{J}_{0}|\mathcal{Z}_{2}|\tilde{J}_{0})}{P_{b}}\right)\delta_{jk}.$
(70)

The probability of detecting the photons with different parities is zero
identically, even for the exact JTA. Thus, the theoretical cross-talk of the
parity sorter is zero, because the exact zero-centered JTA is central-
symmetric on the $(t,t^{\prime})$ plane. For the Gaussian model, we obtain
from Eq. (67) $(\tilde{J}_{0}^{G}|\mathcal{Z}_{2}|\tilde{J}_{0}^{G})=P_{b}/K$.
The probability of detecting an even mode, $(1+K^{-1})/2$ is higher than that
of detecting an odd mode, $(1-K^{-1})/2$, however, with the growing number of
modes, both probabilities approach $P_{b}/2$.

###  IV.5 Modulo $4$ sorting

Now, we consider a sorter including two interferometers $\ell=1$ and $\ell=2$,
as shown in Fig. 2. The delay time between the interferometers is $\Delta
t=\Delta T/2$, and therefore each time window of duration $\Delta T$ discussed
in the preceding section is split into two sub-windows of durations $\Delta t$
so that the field is non-vacuum in the centers of these four sub-windows only.
The sub-window number will be indicated in parenthesis in the upper index of
the field envelope.

In the first sub-window of the first window, the input of the second
interferometer is $A_{2}^{[1](1)\text{in}}(t)=A_{1}^{[1]\text{out}}(t)$ and
$B_{2}^{[1](1)\text{in}}(t)=B_{2,\text{vac}}^{[1](1)}(t)$, where the latter
field is vacuum. Its output fields are given, as usual, by Eq. (34). The
fields on the detectors will be denoted by two sets of indices. Indices $u$
(for the first window) and $v$ (for the second window) run from 0 to 3 and
denote the order of the mode arriving on this detector. Indices $j$, $k$, $l$,
and $m$ take values $\pm 1$ and denote the constructive (+1) or destructive
(-1) interference in the first interferometer ($j$ for the first window, $k$
for the second one) and the second interferometer ($l$ for the first window,
$m$ for the second one) with respect to the field arriving to the given
detector in the given window. The interference is constructive if the photon
remains in the same beam and destructive otherwise. Note, that $j$ and $k$
have the same meanings (mode parity) as in the preceding section. Thus, we
write the field on the detector of beam A as
$F_{0}^{[1]}(t)=F_{+1+1}^{[1]}(t)=A_{2}^{[1](1)\text{out}}(t)$ and that on the
detector of beam B as
$F_{2}^{[1]}(t)=F_{+1-1}^{[1]}(t)=B_{2}^{[1](1)\text{out}}(t)$.

In the second sub-window of the first window, the input of the second
interferometer is $A_{2}^{[1](2)\text{in}}(t)=A_{2,\text{vac}}^{[1](2)}(t)$
and $B_{2}^{[1](2)\text{in}}(t)=B_{1}^{[1]\text{out}}(t)$, where the former
field is vacuum. In the output, we write the field on the detector of beam A
as $F_{1}^{[1]}(t)=F_{-1-1}^{[1]}(t)=A_{2}^{[1](2)\text{out}}(t)$ and that on
the detector of beam B as
$F_{3}^{[1]}(t)=F_{-1+1}^{[1]}(t)=B_{2}^{[1](2)\text{out}}(t)$.

In the first sub-window of the second window, the input of the second
interferometer is $A_{2}^{[2](1)\text{in}}(t)=A_{1}^{[2]\text{out}}(t)$ and
$B_{2}^{[2](1)\text{in}}(t)=B_{2,\text{vac}}^{[2](1)}(t)$, where the latter
field is vacuum. In the output, we write the field on the detector of beam A
as $F_{3}^{[2]}(t)=F_{-1+1}^{[2]}(t)=A_{2}^{[2](1)\text{out}}(t)$ and that on
the detector of beam B as
$F_{1}^{[2]}(t)=F_{-1-1}^{[2]}(t)=B_{2}^{[2](1)\text{out}}(t)$.

Finally, in the second sub-window of the second window, the input of the
second interferometer is
$A_{2}^{[2](2)\text{in}}(t)=A_{2,\text{vac}}^{[2](2)}(t)$ and
$B_{2}^{[2](2)\text{in}}(t)=B_{1}^{[2]\text{out}}(t)$, where the former field
is vacuum. In the output, we write the field on the detector of beam A as
$F_{2}^{[2]}(t)=F_{+1-1}^{[2]}(t)=A_{2}^{[2](2)\text{out}}(t)$ and that on the
detector of beam B as
$F_{0}^{[2]}(t)=F_{+1+1}^{[2]}(t)=B_{2}^{[2](2)\text{out}}(t)$. We see, that
the values of $u=(0,1,2,3)$ correspond to values of
$jl=($+1+1$,-1-1,+1-1,-1+1)$ respectively, with a similar relation between $v$
and $km$.

The probability of observing one photon in mode $u$ in the first window and
another photon in mode $v$ in the second window is given by Eq. (64), where
the indices should be changed as $jk\to uv=jl,km$. Substituting Eqs. (47),
(48), and (34) into the anomalous correlator, applying the commutation
relations, and equating to zero all normally ordered correlators of vacuum
fields, as above, we write this probability as
$P_{uv}=(\tilde{J}_{0}|\mathcal{O}_{uv}|\tilde{J}_{0})$, where
$\mathcal{O}_{uv}=\mathcal{Q}_{uv}^{\dagger}\mathcal{Q}_{uv}$ and

$\mathcal{Q}_{uv}=\frac{1}{16}\left(\mathcal{Z}_{2}+j\right)\left(\mathcal{Z}_{2}^{\prime}+k\right)\left(\rho_{j}\mathcal{Z}_{4}+l\right)\left(\rho_{k}\mathcal{Z}_{4}^{\prime}+m\right),$
(71)

where the variable $\rho_{j}$ describes the additional phase shift
$\theta_{2}=\pi/2$ applied to the odd modes and takes the values
$\rho_{+1}=1$, $\rho_{-1}=i$.

We see from the property $(\mathcal{Z}_{2}\pm 1)(\mathcal{Z}_{2}^{\prime}\mp
1)|\tilde{J}_{0})=0$ and Eq. (71), that $P_{uv}=0$ when the parities of $u$
and $v$ differ, which is a consequence of errorless parity sorting in the
first interferometer, as found in the preceding section. The only non-zero
off-diagonal elements of the matrix $P_{uv}$ are $P_{02}=P_{20}$ and
$P_{13}=P_{31}$, which describe the cross-talk between the corresponding modes
of the same parity. To calculate these probabilities, we apply the properties
of the operator $\mathcal{Z}_{2}$, established in the preceding section, and
the following properties of the operator $\mathcal{Z}_{4}$: unitarity,
$\mathcal{Z}_{4}^{\dagger}\mathcal{Z}_{4}=1$, commutativity with
$\mathcal{Z}_{2}$,
$[\mathcal{Z}_{2},\mathcal{Z}_{4}]=[\mathcal{Z}_{2}^{\dagger},\mathcal{Z}_{4}]=0$,
and the property $\mathcal{Z}_{4}^{\dagger}=\mathcal{Z}_{2}\mathcal{Z}_{4}$.
These properties are naturally obtained when $\mathcal{Z}_{2}$ and
$\mathcal{Z}_{4}$ are represented as rotations by angles $\pi$ and $\pi/2$
respectively in the chronocyclic space of harmonic oscillator [6]. In
addition, we employ the symmetry of the zero-centered JTA with respect to the
exchange of its arguments,
$\tilde{J}_{0}(t^{\prime},t)=\tilde{J}_{0}(t,t^{\prime})$, which gives
$(\tilde{J}_{0}|\mathcal{Z}_{4}|\tilde{J}_{0})=(\tilde{J}_{0}|\mathcal{Z}_{4}^{\prime}|\tilde{J}_{0})$.
From all these properties and Eq. (71), we find
$P_{02}=P_{20}=P_{b}^{\prime}(p_{1}+p_{2})$ and
$P_{13}=P_{31}=P_{b}^{\prime}(p_{1}-p_{2})$, where

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{8P_{b}^{\prime}}(\tilde{J}_{0}|1-\mathcal{Z}_{2}\mathcal{Z}_{4}\mathcal{Z}_{4}^{\prime}|\tilde{J}_{0}),$
(72) $\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8P_{b}^{\prime}}(\tilde{J}_{0}|\mathcal{Z}_{2}-\mathcal{Z}_{4}\mathcal{Z}_{4}^{\prime}|\tilde{J}_{0}).$
(73)

For the Gaussian JTA given by Eq. (67), $p_{1}=p_{2}=0$ and the sorter is
errorless. Physically, it means that a Fourier-transformed mode overlaps
perfectly with itself at the output beam splitter of the second
interferometer, since a Hermite-Gauss function is an eigenfunction of the
Fourier transform. For the exact JTA given by Eq. (66), however, these
probabilities are non-zero, because the Schmidt modes are slightly different
from Hermite-Gauss modes and are not eigenfunctions of the Fourier transform
exactly. Substituting Eq. (66) into Eqs. (72) and (73) and performing the
integration, we find

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[1-\sqrt{\pi\ln
2}\frac{\tau^{2}}{\tau_{o}\tau_{p}}\operatorname{erf}^{2}\left(\frac{\tau_{o}\tau_{p}}{\sqrt{2\ln
2}\tau^{2}}\right)\right],$ (74) $\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[\sqrt{\frac{\pi}{4\ln
2}}\frac{\tau_{p}}{\tau_{o}}\operatorname{erf}^{2}\left(\frac{\sqrt{2\ln
2}\tau_{o}}{\tau_{p}}\right)\right.$ $\displaystyle-\left.\sqrt{\frac{4\ln
2}{\pi}}\frac{\tau^{2}}{\tau_{o}\tau_{p}\sqrt{1+(2\ln
2)^{2}\tau_{o}^{4}/\tau_{p}^{4}}}\operatorname{Si}\left(\frac{2\tau_{o}^{2}}{\tau^{2}}\right)\right],$

where $\operatorname{erf}(x)$ and $\operatorname{Si}(x)$ are the error
function and the sine integral, respectively.

The total conditional probability of error under condition that a biphoton is
generated is
$p_{\text{tot}}=(P_{02}+P_{20}+P_{13}+P_{31})/P_{b}^{\prime}=4p_{1}$. We see
from Eq. (74) that this probability is a function of just one argument,
$\tau/\sqrt{\tau_{o}\tau_{p}}$. This functional dependence is shown in Fig. 5.

Figure 5: Total probability of error $p_{\text{tot}}$ as a function of the
sorter temporal scale $\tau$.

The minimal value of $p_{\text{tot}}=0.055$ is reached at
$\tau=0.93\sqrt{\tau_{o}\tau_{p}}$, which is slightly higher than the optimal
value of $\tau$ given by Eq. (59) for the Gaussian model. Consequently, for
$\tau_{o}=2.95$ ps and $\tau_{p}=24$ ps, we need to choose $\tau=7.8$ ps and
the GDD for the Fourier transformer $D=\tau^{2}=61$ ps2. For such an optimal
modulo 4 sorter, the theoretical total cross-talk probability is 5.5%, which
value is determined by the maximal overlap of a rectangle function with an
exponential.

Substituting the optimal value of the temporal scale $\tau$ into Eq. (IV.5),
we obtain $p_{2}$ as a function of the ratio $\tau_{p}/\tau_{o}$. This allows
us to calculate conditional probabilities of errors in even modes
$p_{\text{even}}=(P_{02}+P_{20})/P_{b}^{\prime}=2(p_{1}+p_{2})$ and in odd
modes $p_{\text{odd}}=(P_{13}+P_{31})/P_{b}^{\prime}=2(p_{1}-p_{2})$, which
sum up to the total error probability (Fig. 6). We see that for
$\tau_{p}\approx 8\tau_{o}$, as required for the choice $K=4$, $p_{2}$ tends
to zero and the probability of error is almost equally distributed between the
even and odd modes.

Figure 6: Probabilities of error for even and odd modes as functions of the
pump pulse duration $\tau_{p}$.

##  V Conclusion

We have proposed a sorter of Hermite-Gauss temporal modes of light based on a
cascade of Mach-Zahnder interferometers, each containing a time lens in one of
its arms. We have shown how the temporal counterpart of the Gouy phase, which
we call “the temporal Gouy phase”, allows one to obtain constructive or
destructive interference of modes of different order, and, as a consequence,
direct them to different interferometer outputs. We have considered in detail
the operation of a parity sorter and a modulo 4 sorter, applied to mode
sorting of two entangled photons generated by SPDC in a nonlinear crystal with
symmetric group velocity matching. For the parity sorter, we have shown that
its theoretical cross-talk probability is zero, even when the eigenmodes of
the source are not Hermite-Gauss modes exactly; it is sufficient that the JTA
is central-symmetric. For the modulo 4 sorter, we have found the theoretical
total probability of error on the level of 5.5%. If this value of error is too
high for a quantum network, e.g. for quantum error correction in linear
quantum computation, then apodization should be applied to the SPDC crystal,
which allows one to obtain a Gaussian phase-matching function [90].

The proposed mode sorter can be applied to classical fields for manipulating
the temporal modal structure of light pulses. It can also be applied to
fragile quantum states of light, especially if realized on a single photonic
chip with low loss [89]. Such a realization will open up bright perspectives
for quantum information processing and quantum sensing with high-dimensional
quantum encoding based on temporal modes of light [4, 5, 6].

## Acknowledgments

D. B. H. thanks Nicolas Treps for an insightful discussion. This work is
supported by the network QuantERA of the European Union’s Horizon 2020
research and innovation programme under project “Quantum information and
communication with high-dimensional encoding” (QuICHE) and funded by Agence
Nationale de la Recherche, France, grant ANR-19-QUANT-0001.

##  Appendix A Passage of a Gaussian pulse through a dispersive medium and a
time lens

We consider a classical field with the envelope
$Y_{0}(t)=E_{0}e^{-t^{2}/4\Delta t_{0}^{2}}$, where $E_{0}$ is the peak
amplitude and $\Delta t_{0}$ is the intensity standard deviation. In frequency
domain, we have

$\tilde{Y}_{0}(\Omega)=\int Y_{0}(t)e^{i\Omega t}dt=2\sqrt{\pi}\Delta
t_{0}E_{0}e^{-\Delta t_{0}^{2}\Omega^{2}},$ (76)

so that the standard deviation of the intensity spectrum
$S_{0}(\Omega)=|\tilde{Y}_{0}(\Omega)|^{2}$ is $\Delta\Omega_{0}=1/2\Delta
t_{0}$. For the FWHM temporal and spectral widths
$T_{0}^{\mathrm{F}}=2\sqrt{2\ln 2}\Delta t_{0}$ and
$\Omega_{0}^{\mathrm{F}}=2\sqrt{2\ln 2}\Delta\Omega_{0}$ respectively, we find
the time-bandwidth product $T_{0}^{\mathrm{F}}\Omega_{0}^{\mathrm{F}}=4\ln
2\approx 2\pi\times 0.44$, as expected for a Fourier-limited Gaussian pulse.

This pulse passes through a dispersive medium of GDD $D_{\mathrm{in}}$. In the
group-delayed reference frame, defined by Eq. (4), the field at the output is
$\tilde{Y}_{1}(\Omega)=\tilde{Y}_{0}(\Omega)e^{iD_{\mathrm{in}}\Omega^{2}/2}$,
which gives in time domain

$Y_{1}(t)=\int\tilde{Y}_{1}(\Omega)e^{-i\Omega
t}\frac{d\Omega}{2\pi}=E_{1}e^{-t^{2}/4\Delta t_{1}^{2}-it^{2}/2C_{1}},$ (77)

where $E_{1}=E_{0}/\sqrt{1-iD_{\mathrm{in}}/2\Delta t_{0}^{2}}$ is the new
peak amplitude, $C_{1}=D_{\mathrm{in}}+4\Delta t_{0}^{4}/D_{\mathrm{in}}$ is
the chirp coefficient of the output pulse, and

$\Delta t_{1}=\sqrt{\Delta t_{0}^{2}+D_{\mathrm{in}}^{2}/4\Delta t_{0}^{2}}$
(78)

is the intensity standard deviation of the temporal distribution of the
outgoing pulse. The spectral distribution is unchanged
$|\tilde{Y}_{1}(\Omega)|^{2}=|\tilde{Y}_{0}(\Omega)|^{2}$ and the spectral
standard deviation is $\Delta\Omega_{1}=\Delta\Omega_{0}$.

Then, the stretched pulse passes through a time lens and its envelope becomes
$Y_{1}(t)=e^{it^{2}/2D_{\mathrm{f}}}Y_{1}(t)$ in accord with Eq. (9). In
frequency domain, it gives

$\tilde{Y}_{2}(\Omega)=\int Y_{2}(t)e^{i\Omega
t}dt=\tilde{E}_{2}e^{-\Omega^{2}/4\Delta\Omega_{2}^{2}+iD_{2}\Omega^{2}/2},$
(79)

where $\tilde{E}_{2}=2\sqrt{\pi}\Delta t_{1}E_{1}[1+2i\Delta
t_{1}^{2}/C_{2}]^{-1/2}$ is the spectral peak amplitude,
$C_{2}^{-1}=C_{1}^{-1}-D_{\mathrm{f}}^{-1}$, $D_{2}=[C_{2}/4\Delta
t_{1}^{4}+1/C_{2}]^{-1}$ and

$\Delta\Omega_{2}=\Delta\Omega_{0}\sqrt{(1-R)^{2}+r^{2}R^{2}}$ (80)

is the spectral standard deviation with $R=D_{\mathrm{in}}/D_{\mathrm{f}}$ and
$r=2\Delta t_{0}^{2}/D_{\mathrm{in}}$. For a fractional Fourier transform,
where conditions (28) and (29) are satisfied for $\tau=\sqrt{2}\Delta t_{0}$,
we substitute $R=1-\cos\gamma$, $r=-\sin\gamma/(1-\cos\gamma)$ and obtain
$\Delta\Omega_{2}=\Delta\Omega_{0}$.

The condition on the temporal aperture from Sec. IV.3 is $2\sqrt{2\ln 2}\Delta
t_{1}<D_{\mathrm{f}}\Omega_{m}$, which can be rewritten as

$4\ln 2\frac{R^{2}(1+r^{2})}{\tau^{2}}<\Omega_{m}^{2},$ (81)

or, upon substitutions for $R$ and $r$ as above,

$\tau>2\sqrt{2\ln 2}\frac{\sqrt{1-\cos\gamma}}{\Omega_{m}}.$ (82)

The most stringent condition on the mode scale is imposed by the first gate
with $\gamma=-\pi$.

##  Appendix B Temporal Schmidt decomposition for type-II biphoton

Probability density for detecting at the crystal output an ordinary photon at
time $t$ and an extraordinary one at time $t^{\prime}$ is $\langle
E_{e}^{(-)}(L,t^{\prime})E_{o}^{(-)}(L,t)E_{o}^{(+)}(L,t)E_{e}^{(+)}(L,t^{\prime})\rangle\approx|\langle
E_{o}^{(+)}(L,t)E_{e}^{(+)}(L,t^{\prime})\rangle|^{2}=|\langle
A_{o}(t)A_{e}(t^{\prime})\rangle|^{2}$, where we have used the fact that the
statistics of SPDC radiation is Gaussian [82], applied the Gaussian moment
theorem [54] and kept only the term of the lowest order in the smallness
parameter of the low-gain regime of SPDC $\kappa L\alpha_{0}\Omega_{p}\ll 1$.
Substituting Eqs. (47) and (48) into this probability density and integrating
it over both times, we arrive at the total probability of biphoton generation

$P_{b}=\int\int|\tilde{J}(t,t^{\prime})|^{2}dtdt^{\prime}=\pi
J_{1}^{2}\frac{|T_{o}-T_{e}|}{\Omega_{p}^{2}}$ (83)

where we have used the JTA defined by Eq. (51).

The spectral decomposition of a real symmetric double-Gaussian kernel is given
by multiplying both sides of the Mehler’s formula for Hermite polynomials by
$e^{-x^{2}/2-y^{2}/2}$ [81, 82]

$\frac{1}{\sqrt{\pi}}e^{-\frac{1+q^{2}}{2(1-q^{2})}\left(x^{2}+y^{2}\right)+\frac{2q}{1-q^{2}}xy}=\sum_{n=0}^{\infty}pq^{n}h_{n}(x)h_{n}(y),$
(84)

where $-1<q<1$, $p=\sqrt{1-q^{2}}$, and $h_{n}(x)$ are Hermite-Gauss functions
introduced in Sec. II.4. Identifying $x=(t-\tau_{o})/\tau_{1}$,
$y=(t^{\prime}-\tau_{e})/\tau_{2}$, $q=\sqrt{(K-1)/(K+1)}$, and equalizing (up
to a constant factor) the right-hand side of Eq. (51) to the left-hand side of
Eq. (84), we find the following system of equations:

$\displaystyle\frac{K}{\tau_{1}^{2}}$ $\displaystyle=$ $\displaystyle
2\Omega_{p}^{2}\frac{1+T_{e}^{2}}{(T_{o}-T_{e})^{2}},$ (85)
$\displaystyle\frac{K}{\tau_{2}^{2}}$ $\displaystyle=$ $\displaystyle
2\Omega_{p}^{2}\frac{1+T_{o}^{2}}{(T_{o}-T_{e})^{2}},$ (86)
$\displaystyle\frac{\sqrt{K^{2}-1}}{\tau_{1}\tau_{2}}$ $\displaystyle=$
$\displaystyle 2\Omega_{p}^{2}\frac{1+T_{o}T_{e}}{(T_{o}-T_{e})^{2}}.$ (87)

Solving this system for $K$, $\tau_{1}$ and $\tau_{2}$, we obtain Eqs. (57)
and (58). We imply here that $1+T_{o}T_{e}>0$, which is the regime of our
interest in Sec. IV.

The singular value (Schmidt) decomposition of JTA can be easily obtained from
its spectral decomposition, especially in the case of a positive $q$,
considered here. Since all eigenvalues are positive, they are also the
singular values. The corresponding singular functions are given by the
eigenfunctions, Eqs. (54) and (55).

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