# Erecting time telescope for photonic quantum networks

Shivang Srivastava  * Dmitri B. Horoshko  and Mikhail I. Kolobov  Univ. Lille,
CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000
Lille, France<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

A time lens allows one to stretch or compress optical waveforms in time,
similar to the conventional lens in space. However, a single-time-lens imaging
system always imparts a residual temporal chirp on the image, which may be
detrimental for quantum networks, where the temporal image interacts with
other fields. We show that a two-time-lens imaging system satisfying the
telescopic condition, a time telescope, is necessary and sufficient for
creating a chirpless image. We develop a general theory of a time telescope,
find the conditions for loss minimization, and show how an erecting time
telescope creating a real image of a temporal object can be constructed. We
consider several applications of such a telescope to making indistinguishable
the photons generated by spontaneous parametric downconversion or single
emitters such as quantum dots.

##  1 Introduction

Quantum technologies are expected to revolutionize many aspects of human life,
bringing numerous scientific and societal benefits. A global quantum network
is envisaged by many scientists, where quantum processors, sensors, and
memories at local nodes will be interconnected by optical fields in highly
entangled states [1, 2]. One of the key elements of such networks are
photonic-photonic interconnects [2], transforming the physical parameters of
the optical carrier while leaving the encoded quantum information untouched,
e.g. converting picosecond-scale pulses in the telecommunication band, optimal
for high-rate fiber transmission, to nanosecond-scale pulses in the visible
range processed by quantum memories. An efficient tool for such
transformations is provided by the technique of temporal imaging [3, 4]
allowing one to stretch and compress optical waveforms in time. This technique
requires a device imprinting quadratic in time phase modulation on the input
waveform, known as the time lens [5, 6]. Optical time lenses are engineered
through electro-optic phase modulation (EOPM) [7], through parametric
processes like cross-phase modulation [8], sum-frequency generation (SFG) [9,
10], or four-wave mixing (FWM)[11], and through atomic-cloud-based quantum
memory [12, 13, 14]. Single-time-lens imaging systems were successfully
applied to spectral bandwidth compression of light at a single-photon level
using SFG [15], EOPM [16, 17], and FWM [18] and to temporal transformations of
one [19] or both [20] photons of an entangled photon pair. Conditions for
noiseless temporal imaging of broadband squeezed [21, 22] or antibunched [23]
light were established.

The temporal imaging technique was historically inspired by the space-time
analogy [24] following from the mathematical equivalence of descriptions of
paraxial diffraction and quadratic dispersion. This analogy helps to
understand the fundamental property of a single-time-lens imaging system: It
never provides a pure scaling of the waveform, and a residual chirp is always
present in the output, similar to the wavefront curvature imposed on the image
by a single lens in spatial imaging. In the experiments, where the intensity
of the output waveform is measured, this chirp is insignificant. However, when
the output waveform is intended for further optical processing, as in a
quantum network, its time-dependent phase may be of great importance. It is
known, in particular, that even a slight distinguishability of photons used in
boson sampling leads to classical simulability of the experimental results,
destroying the quantum advantage [25, 26]. Recently, we have studied the
effect of the residual chirp on indistinguishability of two single-photon
pulses whose durations are made equal by a single time lens [27] and found
that the residual chirp can be disregarded at rather high values of the focal
group delay dispersion (GDD) of the time lens, which may be impractical,
especially for an EOPM-based time lens. Thus, a truly noiseless temporal
scaling of quantum fields requires the application of a two-time-lens system.
As is shown below, the no-chirp condition implies a _telescopic condition_ for
a two-time-lens imaging system, i.e. the imaging system should be a _time
telescope_.

Several particular cases of time telescope are studied in the literature [28,
29, 30, 31], all of them being _inverting_ , where the output pulse (temporal
image) is inverted in time and scaled replica of the input pulse (temporal
object). Such an inverting time telescope has a spatial counterpart –
Keplerian telescope [32, 33]. Surprisingly, the general case of a two-lens
temporal imaging system was never considered. In this work, we fill this gap
by developing a general theory of time telescope. In particular, we show that
a time telescope with a real erect (non-inverted) image can be easily built by
choosing the negative sign for the output GDD. Such a time telescope has no
spatial counterpart because the space-time analogy is incomplete: The
dispersion can be positive or negative, while the diffraction is always
positive. An erecting time telescope provides a pure temporal scaling of the
input waveform, exactly what is necessary for a photonic-photonic interconnect
in a quantum network.

A general two-lens temporal imaging system is studied in Sec. 2, a no-chirp
condition is deduced, possible types of time telescopes are classified, and
the conditions are found for a time telscope with the minimal loss. In Sec. 3,
we show how an erecting time telescope succeeds in making two photons,
initially of very different durations, indistinguishable, by calculating the
coincidence rate at the output of a Hong-Ou-Mandel interferometer. We
summarize the results in Sec. 4.

##  2 General two-time-lens temporal imaging system

###  2.1 Single time lens

A single-lens temporal imaging system consists of an input dispersive medium,
a time lens, and an output dispersive medium, as shown in Fig. 1(a). Let us
consider the propagation of an optical pulse through these elements. We denote
the positive-frequency part of its electric field by $E^{(+)}(t,x)$, where $t$
is time and $x$ is the direction of propagation, and denote the carrier
wavevector and frequency of the pulse by $k_{0}$ and $\omega_{0}$
respectively. Passing through a transparent medium, such a pulse experiences
dispersion characterized by the dependence of its wave vector $k(\Omega)$ on
the frequency $\omega$, which we decompose around the carrier frequency in
powers of $\Omega=\omega-\omega_{0}$ and limit the Taylor series to the first
three terms: $k(\Omega)\approx
k_{0}+k_{0}^{\prime}\Omega+k_{0}^{\prime\prime}\Omega^{2}/2$, where
$k_{0}=k(0)$, $k_{0}^{\prime}=(dk/d\Omega)_{\Omega=0}$ is the inverse group
velocity, and $k_{0}^{\prime\prime}=(d^{2}k/d\Omega^{2})_{\Omega=0}$ is the
group velocity dispersion of the medium at the carrier frequency $\omega_{0}$.
The effect of the first-order term of this series is the group delay by the
time $\tau_{\mathrm{g}}=k_{0}^{\prime}l$, where $l$ is the length of the
medium. This delay is traditionally made implicit by defining the group-
delayed envelope $A_{n}(t)$ at point $x_{n}$ after the $n$th dispersive medium
by the relation

$E^{(+)}(t,x_{n})=A_{n}(t-\tau_{n})e^{ik_{0}x_{n}-i\omega_{0}t},$ (1)

where $\tau_{n}$ is the total group delay in $n$ media.

The second-order term of the series is responsible for the spreading of the
pulse, which is characterized by the GDD $D=k_{0}^{\prime\prime}l$. The input
and output dispersive media of the temporal imaging system have GDDs
$D_{\mathrm{in}}$ and $D_{\mathrm{out}}$ respectively. A time lens is a device
realizing a quadratic-in-time phase modulation of the passing pulse and is
characterized by its focal GDD $D_{\mathrm{f}}$ [4, 22]. The condition of
single-lens temporal imaging reads

$\frac{1}{D_{\mathrm{in}}}+\frac{1}{D_{\mathrm{out}}}=\frac{1}{D_{\mathrm{f}}}$
(2)

and is similar to the thin lens equation. When this condition is satisfied,
the output waveform is a magnified version of the input one with a
magnification $m=-D_{\mathrm{out}}/D_{\mathrm{in}}$. Transformation of the
field envelope operator in the single-time-lens temporal imaging system can be
found in Ref. [22], (see also Ref. [7] for its classical form) and for a
sufficiently large temporal aperture reads

$A_{\mathrm{out}}(t)=-\frac{1}{\sqrt{m}}e^{it^{2}/2mD_{\mathrm{f}}}A_{\mathrm{in}}(t/m).$
(3)

As we see, the output field is a temporally scaled copy of the input field
with an additional chirp described by the term $e^{it^{2}/2mD_{\mathrm{f}}}$.

Figure 1: (a) Single-time-lens imaging system. Yellow boxes denote dispersive
media with GDDs $D_{\mathrm{in}}$ and $D_{\mathrm{out}}$, while the blue box
denotes a time lens with the focal GDD $D_{\mathrm{f}}$. (b) General two-time-
lens imaging system with three dispersive elements. (c) Equivalent scheme of a
general two-time-lens imaging system represented as a combination of two
single-time-lens imaging systems with
$D_{\mathrm{out}}+D_{\mathrm{in}}^{\prime}=D_{\mathrm{inter}}$.

###  2.2 Two time lenses

Now we consider a general two-time-lens imaging system, shown in Fig. 1(b). It
consists of two time lenses of focal GDDs $D_{\mathrm{f}}$ and
$D_{\mathrm{f}}^{\prime}$, separated by a dispersive medium of GDD
$D_{\mathrm{inter}}$; the first time lens is preceded by a dispersive medium
of GDD $D_{\mathrm{in}}$, while the second one is followed by a dispersive
medium of GDD $D_{\mathrm{out}}^{\prime}$. It is easy to see, that this system
is equivalent to one shown in Fig. 1(c). Indeed, let us find
$D_{\mathrm{out}}$ from Eq. (2) and split the dispersive medium between the
lenses into two sections with GDDs $D_{\mathrm{out}}$ and
$D_{\mathrm{in}}^{\prime}=D_{\mathrm{inter}}-D_{\mathrm{out}}$. The field
between these sections $A_{\mathrm{out}}(t)=A_{\mathrm{in}}^{\prime}(t)$ can
be considered as the temporal image of the input field created by the first
time lens and, at the same time, as the object for the second time lens.
Similarly to Eq. (3), we write the field transformation by the second time
lens as

$A_{\mathrm{out}}^{\prime}(t)=-\frac{1}{\sqrt{m^{\prime}}}e^{it^{2}/2m^{\prime}D_{\mathrm{f}}^{\prime}}A_{\mathrm{in}}^{\prime}(t/m^{\prime}),$
(4)

where $m^{\prime}=-D_{\mathrm{out}}^{\prime}/D_{\mathrm{in}}^{\prime}$ is the
magnification of the second time lens and the imaging condition is implied for
the second lens similar to Eq. (2) but with primed GDDs.

Substituting Eq. (4) into Eq. (3) and taking into account the equality
$A_{\mathrm{out}}(t)=A_{\mathrm{in}}^{\prime}(t)$, we obtain the total field
transformation in a two-time-lens imaging system

$A_{\mathrm{out}}^{\prime}(t)=\frac{1}{\sqrt{mm^{\prime}}}\text{e}^{i\frac{t^{2}(mm^{\prime}D_{\mathrm{f}}+D_{\mathrm{f}}^{\prime})}{2mm^{\prime
2}D_{\mathrm{f}}D_{\mathrm{f}}^{\prime}}}A_{\mathrm{in}}\left(\frac{t}{mm^{\prime}}\right).$
(5)

We see, that the field at the output of a two-time-lens imaging system is a
scaled version of its input field with a magnification $M=mm^{\prime}$ and an
additional chirp. This chirp is zero when
$D_{\mathrm{f}}^{\prime}=-MD_{\mathrm{f}}$, under which condition Eq. (5)
reduces to

$A_{\mathrm{out}}^{\prime}(t)=\frac{1}{\sqrt{M}}A_{\mathrm{in}}(t/M).$ (6)

Expressing $m$ and $m^{\prime}$ through $D_{\mathrm{f}}$,
$D_{\mathrm{f}}^{\prime}$, $D_{\mathrm{in}}$ and $D_{\mathrm{inter}}$ via the
imaging conditions of two time lenses, substituting the results into the no-
chirp condition found above, and solving the obtained equation for
$D_{\mathrm{inter}}$, we obtain

$D_{\mathrm{inter}}=D_{\mathrm{f}}+D_{\mathrm{f}}^{\prime},$ (7)

which is equivalent to the telescopic (or afocal) condition in spatial imaging
[32]. Thus, in order that a two-time-lens imaging system is chirpless, it is
necessary and sufficient that it is a time telescope.

A single-time-lens imaging system is characterized by two parameters, e.g. its
input and focal GDDs, while the output GDD can be found from the imaging
condition (2). Two such systems have four parameters, one of which can be
found from the telescopic condition (7). Thus, a time telescope is determined
by three real parameters, which can be chosen as $M$, $D_{\mathrm{inter}}$ and
$D_{\mathrm{in}}$. The first two parameters determine the type of telescope,
while the third parameter determines the temporal distance at which the object
is placed. The image is formed at the temporal distance

$D_{\mathrm{out}}^{\prime}=-M^{2}D_{\mathrm{in}}-MD_{\mathrm{inter}}$ (8)

after the telescope. We also find $D_{\mathrm{f}}=D_{\mathrm{inter}}/(1-M)$
and $D_{\mathrm{f}}^{\prime}=-MD_{\mathrm{inter}}/(1-M)$.

###  2.3 Classification of time telescopes

Various types of time telescopes can be distinguished by the magnification $M$
they provide. Some of these time telescopes have spatial counterparts and some
do not because the space-time analogy is incomplete: In contrast to the
diffraction, which is always positive, the dispersion can be positive or
negative. We accept a convention, that a time lens with a positive (negative)
focal GDD corresponds to a convergent (divergent) spatial lens. As a
consequence, a dispersive medium with a positive GDD corresponds to
diffraction, which is always positive. A positive (negative) input or output
GDD corresponds to a real (virtual) object or image respectively. The only
parameter having no spatial counterpart is a negative $D_{\mathrm{inter}}$,
since the latter parameter corresponds to the telescope length, which cannot
be negative.

####  2.3.1 Inverting magnifying time telescope, $M<-1$

For a positive $D_{\mathrm{inter}}$, we have
$D_{\mathrm{f}}^{\prime}>D_{\mathrm{f}}>0$, which corresponds to a combination
of two convergent lenses, where the second one has a higher focal length. In
spatial imaging, such an imaging system is known as _beam expander_.

For a negative $D_{\mathrm{inter}}$, we have
$D_{\mathrm{f}}^{\prime}<D_{\mathrm{f}}<0$, which has no spatial counterpart.
This would be a combination of two divergent lenses, which are unable to
create a chirpless image for any distance between them.

####  2.3.2 Inverting compressing time telescope, $-1<M<0$

For a positive $D_{\mathrm{inter}}$, we have
$D_{\mathrm{f}}>D_{\mathrm{f}}^{\prime}>0$, which corresponds to a combination
of two convergent lenses, where the first one has a higher focal length. In
spatial imaging, such an imaging system is known as _Keplerian_ or
_astronomical telescope_ [32, 33]. Compression in space corresponds to
magnification in angular size, which is the aim in an astronomical
observation. Similarly, a time telescope of this type can be used for spectral
magnification. The image is inverted in this type of time telescope, which may
be undesirable for quantum networks, as indicated above.

For a negative $D_{\mathrm{inter}}$, we have
$D_{\mathrm{f}}<D_{\mathrm{f}}^{\prime}<0$, which has no spatial counterpart.

####  2.3.3 Erecting compressing time telescope, $0<M<1$

For a positive $D_{\mathrm{inter}}$, we have $D_{\mathrm{f}}>0$,
$D_{\mathrm{f}}^{\prime}<0$, $D_{\mathrm{f}}>|D_{\mathrm{f}}^{\prime}|$, which
corresponds to a combination of a convergent and divergent lens, where the
first one has a longer focal length. In spatial imaging, such an imaging
system is known as _Galilean_ or _terrestrial telescope_ [32, 33]. As in the
previous case, compression in space corresponds to magnification in angular
size. The image is erect in this type of time telescope, which makes it
promising for photonic quantum networks.

For a negative $D_{\mathrm{inter}}$, we have $D_{\mathrm{f}}<0$,
$D_{\mathrm{f}}^{\prime}>0$, $|D_{\mathrm{f}}|>D_{\mathrm{f}}^{\prime}$, which
has no spatial counterpart.

####  2.3.4 Erecting magnifying time telescope, $M>1$

For a positive $D_{\mathrm{inter}}$, we have $D_{\mathrm{f}}<0$,
$D_{\mathrm{f}}^{\prime}>0$, $|D_{\mathrm{f}}|<D_{\mathrm{f}}^{\prime}$, which
corresponds to a combination of a divergent and convergent lens, where the
second one has a longer focal length. In spatial imaging, such an imaging
system is known as _inverted Galilean telescope_ [33], a Galilean telescope
turned by the other side to the object.

For a negative $D_{\mathrm{inter}}$, we have $D_{\mathrm{f}}>0$,
$D_{\mathrm{f}}^{\prime}<0$, $D_{\mathrm{f}}<|D_{\mathrm{f}}^{\prime}|$, which
has no spatial counterpart.

###  2.4 Time telescopes studied in the literature

All time telescopes described in the literature up to date [28, 29, 30, 31]
have spatial counterparts. The time telescope realized experimentally by Gaeta
group [28, 29, 30] is inverting compressing, it was applied to temporal
compression of digital optical signals [28], temporal pulse inversion [29],
and spectral magnification [30]. It is a special case of the inverting
compressing time telescope with $D_{\mathrm{in}}=D_{\mathrm{f}}$, and, as
follows from Eq. (8), $D_{\mathrm{out}}^{\prime}=D_{\mathrm{f}}^{\prime}$.
Such a time telescope creates a Fourier-transformed field between the time
lenses and may be useful if a spectral manipulation of the signal is
necessary. For pure temporal scaling, however, this time telescope is
suboptimal, because it uses three dispersive media, while just two are enough,
as is shown below.

The two-time-lens system studied analytically and numerically by Gauthier
group [31] is based on the idea of using a temporal equivalent of the field
lens: After the first single-time-lens system with positive $D_{\mathrm{in}}$
and $D_{\mathrm{f}}$, one places the second time lens with
$D_{\mathrm{f}}^{\prime}=-mD_{\mathrm{f}}$, which dechirps the image. Since
the second time lens is placed exactly where the first image is created, we
have $D_{\mathrm{inter}}=D_{\mathrm{out}}$. It is easy to see that
$D_{\mathrm{out}}=D_{\mathrm{f}}(1-m)=D_{\mathrm{f}}+D_{\mathrm{f}}^{\prime}$,
that is, this system satisfies the telescopic condition (7). This approach is
equivalent to the limiting case $D_{\mathrm{in}}^{\prime}=\epsilon\to 0$,
$D_{\mathrm{out}}^{\prime}=-\epsilon/(1-\epsilon/D_{\mathrm{f}})\to 0$,
$m^{\prime}\to 1$. This is a special case of application of the inverting
magnifying time telescope, where the object is placed at the temporal distance
$D_{\mathrm{in}}=-D_{\mathrm{inter}}/M$ before the telescope, so that,
according to Eq. (8), the image is created at $D_{\mathrm{out}}^{\prime}=0$,
which has an advantage of requiring one dispersive medium less.

Note, that though the paper of Christov [34] is titled _Theory of a ‘time
telescope’_ , it considers a combination of a dispersive medium, represented
by a pair of gratings, a parametric process with dispersed pump, and another
dispersive medium. Such a combination can be considered as a two-time-lens
system if the frequency is regarded as the counterpart of the transverse
position. However, this system does not satisfy the telescopic condition and
imparts a residual chirp on the output field. In the modern understanding of
temporal imaging, where time is the counterpart of transverse position, the
system of Ref. [34] is a single-parametric-time-lens imaging system.

###  2.5 Time telescope for quantum networks

A new type of time telescope, erecting compressing, i.e. having $0<M<1$, is
proposed in this work. It is a temporal variant of the Galilean telescope and
its scheme is shown in Fig. 2. The principal difference with the Galilean
telescope is that the latter creates a virtual image of a real object: We see
from Eq. (8), that for a positive $D_{\mathrm{in}}$,
$D_{\mathrm{out}}^{\prime}$ is always negative. In temporal imaging, however,
the image is real, because the output GDD can be made negative.

Figure 2: Geometrical optics representation of the erecting compressing time
telescope. The objective time lens has a focal GDD $D_{\mathrm{f}}>0$, while
the eyepiece time lens has a focal GDD $D_{\mathrm{f}}^{\prime}<0$, where the
condition $D_{\mathrm{f}}>|D_{\mathrm{f}}^{\prime}|$ is satisfied. The time
rays (red lines) show transformations of individual frequency components in
the imaging system: vertical position corresponds to time while direction of
the ray corresponds to frequency [9, 10]. The grey area shows a dispersive
medium with a negative GDD, resulting in the creation of a real image at the
output. This element is not possible in the spatial domain because negative
diffraction does not exist while negative dispersion does.

In contrast to classical applications of temporal imaging, where high losses
are typically tolerable, the losses in photonic quantum networks destroy the
quantum coherence and should be minimized. For this purpose, one can consider
a time telescope configuration with (i) a minimal number of dispersive media,
which would minimize the insertion loss, and (ii) a minimal total modulus of
GDD, which is typically connected to the total length of the dispersive media,
and consequently, to total losses in them.

The minimal number of dispersive elements is obviously two: either
$D_{\mathrm{in}}$ or $D_{\mathrm{out}}^{\prime}$ can be made equal to zero. If
we put $D_{\mathrm{in}}=0$, we have from Eq. (8)
$D_{\mathrm{out}}^{\prime}=-MD_{\mathrm{inter}}$, and the total dispersion
modulus is $(1+M)|D_{\mathrm{inter}}|$. If we put
$D_{\mathrm{in}}=-D_{\mathrm{inter}}/M$, as in the case of field lens of Ref.
[31], we have from Eq. (8) $D_{\mathrm{out}}^{\prime}=0$, and the total
dispersion modulus is $(1+1/M)|D_{\mathrm{inter}}|$. For an erecting
compressing telescope with $0<M<1$, considered here, the first choice is
obviously preferable. Thus, we arrive at a configuration shown in Fig. 3.

Figure 3: Erecting compressing time telescope with no input dispersive medium.
Blue rectangles are time lenses, yellow ones are dispersive elements. The
signs of $D_{\mathrm{f}}$ and $D_{\mathrm{f}}^{\prime}=-MD_{\mathrm{f}}$ are
opposite. The signs of
$D_{\mathrm{inter}}=D_{\mathrm{f}}+D_{\mathrm{f}}^{\prime}$ and
$D_{\mathrm{out}}^{\prime}=-MD_{\mathrm{inter}}$ are also opposite.

For a given $M$, the value $|D_{\mathrm{inter}}|=(1-M)|D_{\mathrm{f}}|$ can be
minimized by choosing $|D_{\mathrm{f}}|$ as small as possible. We provide this
optimization for an EOPM-based time telescope, assuming for definiteness
$D_{\mathrm{inter}}>0$.

A great advantage of an EOPM-based time lens consists of the deterministic
nature of the electrooptical effect, providing an almost lossless operation
[16, 17]. Its main shortcoming is rather short temporal aperture
$T_{A}=\sqrt{AD_{\mathrm{f}}}$ for given focal GDD $D_{\mathrm{f}}$ and phase
modulation amplitude $A$ [7, 4]. This shortcoming is removed in the recently
proposed Fresnel time lens [35], successfully applied to bandwidth compression
of single photons [17]. The driving current 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 $t=\pm T_{A}/2$, i.e. its
driving current should have a bandwidth $\Omega_{m}=T_{A}/D_{\mathrm{f}}$.
Thus, for a Fresnel time lens, the temporal aperture scales linearly with the
focal GDD: $T_{A}=D_{\mathrm{f}}\Omega_{m}$, while the phase modulation
amplitude is limited to $2\pi$ [35]. Now, if we wish to compress a pulse of
full width at half maximum (FWHM) duration $T_{0}$ to duration $MT_{0}$, we
need to choose $D_{\mathrm{f}}$ big enough so that the temporal apertures of
both time lenses surpass the pulse durations at their positions. We recall
that Eqs. (3) and (4) are valid only under these conditions [6], otherwise
including integrals over point-spread functions [22]. Assuming a transform-
limited Gaussian shape of the pulse, we calculate in the Appendix the temporal
standard deviation $\Delta t_{2}$ of the pulse at the second time lens. Thus,
we obtain the following inequalities: $T_{0}\leq D_{\mathrm{f}}\Omega_{m}$ and
$\sqrt{2\ln 2}\Delta t_{2}\leq MD_{\mathrm{f}}\Omega_{m}^{\prime}$, where
$\Omega_{m}$ and $\Omega_{m}^{\prime}$ are bandwidths of the first and second
Fresnel time lenses respectively. The second of these inequalities can be
satisfied (for practically interesting values $M\ll 1$) only for
$\Omega_{m}^{\prime}\geq\Omega_{0}/M$, where $\Omega_{0}$ is FWHM spectral
width of the input pulse, which has a simple physical explanation: the
bandwidth of the second time lens should be at least equal to the bandwidth of
the output pulse. Assuming $\Omega_{m}^{\prime}=\Omega_{0}/M$, we obtain the
minimal focal GDD of the first time lens as

$D_{\mathrm{f}}=\frac{\sqrt{2M}T_{0}^{2}}{8\ln 2}.$ (9)

Substituting this value into the first inequality, we obtain
$\Omega_{m}\geq\Omega_{0}\sqrt{2/M}$, which again has a simple physical
explanation: the bandwidth of the first time lens should be at least equal to
the bandwidth of the signal after it, which is exactly $2\sqrt{2\ln
2}\Delta\Omega_{1}=\Omega_{0}\sqrt{2/M}$ (see Appendix).

Taking as an example the Fresnel time lens of Ref. [17] with
$\Omega_{m}^{\prime}/2\pi=70$ GHz, we obtain the minimal output duration
$MT_{0}=4\ln 2/\Omega_{m}^{\prime}=6.3$ ps. Let us consider a time telescope
with $M=0.003$ and a pulse of duration $T_{0}=2.1$ ns at the input. From Eq.
(9) we obtain the value $D_{\mathrm{f}}=61700$ ps2, and, as a consequence,
$D_{\mathrm{f}}^{\prime}=-185$ ps2. The input bandwidth is
$\Omega_{0}/2\pi=210$ MHz and, therefore, the minimal bandwidth of the first
time lens is $\Omega_{m}/2\pi=5.4$ GHz.

When the input pulse has a Gaussian temporal profile, a similar effect,
temporal compression or spectral magnification, can be obtained by a much
simpler system composed of one time lens and one dispersive medium of the same
GDD [15, 16, 35, 17]. Such a system represents a temporal Fourier processor
and can be used for stretching or compressing Gaussian pulses since they are
eigenfunctions of the Fourier transform. For comparison, we calculate the
bandwidth magnification ratio, achievable with the same resources by a Fourier
transform based on a Fresnel time lens [35, 17]:
$M_{\Omega}=D_{\mathrm{f}}\Omega_{m}^{{}^{\prime}2}/4\ln 2=4300$, which is
much higher than $1/M=333$ in our example. However, in contrast to a Fourier
processor, a time telescope provides temporal scaling for any shape of the
input waveform.

Concluding this section, we notice, that an erecting magnifying time telescope
is obtained by using the erecting compressing one, considered here, in the
reverse order.

##  3 Applications to making single photons indistinguishable

In this section, we describe possible applications of the erecting time
telescope to reach the indistinguishability of single photons coming from
different sources and having initially different temporal sizes. We consider
the two most important sources of single photons: spontaneous parametric
downconversion (SPDC) and a single quantum emitter such as a quantum dot.

###  3.1 Parametric downconversion

As the first example, we consider an SPDC source of unentangled photon pairs.
As the model crystal, we choose potassium-dihydrogen phosphate (KDP) crystal
pumped at $\lambda_{p}=415$ nm [36, 16], where a generation of separable
photons was first demonstrated. Three waves propagate collinearly in the
crystal, the strong undepleted extraordinary pump wave at the carrier
frequency $\omega_{p}=2\pi c/\lambda_{p}$ and two signal waves at the same
carrier frequency $\omega_{o}=\omega_{e}=\omega_{p}/2$ generated spontaneously
in the crystal: the ordinary and extraordinary. The interaction of these waves
is most easily described in the Heisenberg picture in the spectral domain. We
direct the $x$ axis along the crystal and decompose the positive-frequency
part of the field of each wave as

$E^{(+)}_{\mu}(t,x)=\int\limits_{-\infty}^{+\infty}\epsilon_{\mu}(\Omega,x)e^{ik_{\mu}(\Omega)x-i(\omega_{\mu}+\Omega)t}\frac{d\Omega}{2\pi},$
(10)

where $\mu$ takes values $\\{p,o,e\\}$ for the pump, ordinary and
extraordinary 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 $\epsilon_{p}(\Omega,x)$ is independent of $x$ 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 through the position $x=0$ (crystal input face)
at a time $t=0$, and write
$\epsilon_{p}(\Omega,x)=\alpha(\Omega)=\alpha_{0}\exp(-\Omega^{2}/4\Omega_{p}^{2})$,
where $\Omega_{p}=\sqrt{2\ln 2}/\tau_{p}$.

The spectral amplitude of the ordinary or extraordinary wave,
$\epsilon_{\mu}(\Omega,x)$, is the annihilation operator of a photon at
position $x$ with the frequency $\omega_{\mu}+\Omega$ and the corresponding
polarization. The evolution of this operator along the crystal is described by
the spatial Heisenberg equation [37]

$\frac{\partial}{\partial
x}\epsilon_{\mu}(\Omega,x)=\frac{i}{\hbar}\left[\epsilon_{\mu}(\Omega,x),G(x)\right],$
(11)

where the spatial Hamiltonian $G(x)$ is given by the momentum transferred
through the plane $x$ [38] and equals

$G(x)=\chi\int\limits_{-\infty}^{+\infty}E^{(-)}_{p}(t,x)E^{(+)}_{o}(t,x)E^{(+)}_{e}(t,x)dt+\mathrm{H.c.},$
(12)

where $\chi$ is the nonlinear coupling constant and
$E^{(-)}_{p}(t,x)=E^{(+)*}_{p}(t,x)$ is the negative-frequency part of the
field. Substituting Eqs. (10) and (12) into Eq. (11), performing the
integration, and using the canonical equal-space commutation relations [39,
38]
$\left[\epsilon_{\mu}(\Omega,x),\epsilon_{\nu}^{\dagger}(\Omega^{\prime},x)\right]=2\pi\delta_{\mu\nu}\delta(\Omega-\Omega^{\prime})$,
we obtain the spatial evolution equations

$\begin{split}\frac{\partial\epsilon_{o}(\Omega,x)}{\partial
x}&=\kappa\int\alpha(\Omega+\Omega^{\prime})\epsilon^{\dagger}_{e}(\Omega^{\prime},x)e^{i\Delta(\Omega,\Omega^{\prime})x}d\Omega^{\prime},\\\
\frac{\partial\epsilon_{e}(\Omega,x)}{\partial
x}&=\kappa\int\alpha(\Omega+\Omega^{\prime})\epsilon^{\dagger}_{o}(\Omega^{\prime},x)e^{i\Delta(\Omega^{\prime},\Omega)x}d\Omega^{\prime},\end{split}$
(13)

where $\kappa=i\chi/2\pi\hbar$ is the new coupling constant,
$\Delta(\Omega,\Omega^{\prime})=k_{p}(\Omega+\Omega^{\prime})-k_{o}(\Omega)-k_{e}(\Omega^{\prime})$
is the phase mismatch for the three interacting waves, and the limits of
integration are infinite.

In the low-gain regime of SPDC, where at most one pair is generated per
excitation, Eqs. (13) can be solved perturbatively, by replacing
$\epsilon^{\dagger}_{\mu}(\Omega^{\prime},x)\to\epsilon^{\dagger}_{\mu}(\Omega^{\prime},0)$
in their right-hand sides. We denote the envelope of the ordinary wave at the
crystal output face at $x=L$ by
$A_{1}(t)=E_{o}^{(+)}(t,L)e^{i(\omega_{o}t-k^{0}_{o}L)}$, while that of the
extraordinary wave by
$B_{1}(t)=E_{e}^{(+)}(t,L)e^{i(\omega_{e}t-k_{e}^{0}L)}$, where
$k_{\mu}^{0}=k_{\mu}(0)$. In these notations, the field transformation in the
crystal has the form of an integral Bogoliubov transformation

$\displaystyle A_{1}(t)$ $\displaystyle=$ $\displaystyle\int
U_{A}(t,t^{\prime})A_{0}(t^{\prime})dt^{\prime}+\int
V_{A}(t,t^{\prime})B_{0}^{\dagger}(t^{\prime})dt^{\prime},$ (14)
$\displaystyle B_{1}(t)$ $\displaystyle=$ $\displaystyle\int
U_{B}(t,t^{\prime})B_{0}(t^{\prime})dt^{\prime}+\int
V_{B}(t,t^{\prime})A_{0}^{\dagger}(t^{\prime})dt^{\prime},$ (15)

where $A_{0}(t)=E_{o}^{(+)}(t,0)e^{i\omega_{o}t}$,
$B_{0}(t)=E_{e}^{(+)}(t,0)e^{i\omega_{e}t}$, and the Bogoliubov kernels are

$\displaystyle U_{A}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle\int
e^{i[k_{o}(\Omega)-k_{o}^{0}]L+i\Omega(t^{\prime}-t)}\frac{d\Omega}{2\pi},$
(16) $\displaystyle U_{B}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle\int
e^{i[k_{e}(\Omega)-k_{e}^{0}]L+i\Omega(t^{\prime}-t)}\frac{d\Omega}{2\pi},$
(17) $\displaystyle V_{A}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle\int
e^{i[k_{o}(\Omega)-k_{o}^{0}]L-i(\Omega^{\prime}t^{\prime}+\Omega
t)}J(\Omega,\Omega^{\prime})\frac{d\Omega d\Omega^{\prime}}{2\pi},$ (18)
$\displaystyle V_{B}(t,t^{\prime})$ $\displaystyle=$ $\displaystyle\int
e^{i[k_{e}(\Omega^{\prime})-k_{e}^{0}]L-i(\Omega^{\prime}t+\Omega
t^{\prime})}J(\Omega,\Omega^{\prime})\frac{d\Omega d\Omega^{\prime}}{2\pi}.$
(19)

Here, $J(\Omega,\Omega^{\prime})=\kappa
L\alpha(\Omega+\Omega^{\prime})\Phi(\Omega,\Omega^{\prime})$ is the joint
spectral amplitude (JSA) of the two generated photons, where
$\Phi(\Omega,\Omega^{\prime})=e^{i\Delta(\Omega,\Omega^{\prime})L/2}\operatorname{sinc}[\Delta(\Omega,\Omega^{\prime})L/2]$
is the phase-matching function. These functions can be calculated numerically
using Sellmeier equations for the ordinary $n_{o}(\omega)$ and extraordinary
$n_{e}(\omega)$ refractive indices of KDP [40] and writing
$k_{\mu}(\Omega)=n_{\mu}(\omega_{\mu}+\Omega)(\omega_{\mu}+\Omega)/c$. In this
way, we find the angle between the propagation direction and the optical axis
of the crystal $\theta_{p}=67.8^{\circ}$, corresponding to the collinear
degenerate type-II phase matching with $k_{p}^{0}-k_{o}^{0}-k_{e}^{0}=0$. The
inverse group velocities of the pump and the ordinary wave coincide in this
configuration: $k_{p}^{\prime}=k_{o}^{\prime}$, where $k_{\mu}^{\prime}$ is
the derivative of $k_{\mu}(\Omega)$ at $\Omega=0$, which is known as
asymmetric group velocity matching [41]. This means that the pump pulse and
the ordinary photon exit the crystal simultaneously. The extraordinary photon,
however, has a higher group velocity and advances them by the time
$\tau_{e}=(k_{p}^{\prime}-k_{e}^{\prime})L/2=360$ fs, for a crystal of length
$L=5$ mm. Similar to Ref. [36], we consider a pump of spectral width 4.1 nm,
which corresponds to $\tau_{p}=62$ fs and $\Omega_{p}=19$ rad/ps. The
calculated JSA is shown in Fig. 4.

Figure 4: JSA of a photon pair generated by SPDC in a 5-mm-long KDP crystal
pumped at 415 nm by pulses of spectral width 4 nm. The dashed lines delimit
the pump envelope region. When the sidelobes are filtered out by a filter
shown by the dot-dashed lines, the JSA becomes separable in two of its
arguments.

We see from Fig. 4, that when the sidelobes are filtered out, the JSA becomes
separable and can be approximately written as

$J(\Omega,\Omega^{\prime})=\alpha_{0}e^{-\Omega^{2}/4\Omega_{p}^{2}}e^{-\tau_{e}^{2}\Omega^{{}^{\prime}2}/2\sigma_{s}^{2}+i\tau_{e}\Omega^{\prime}},$
(20)

where we have made three approximations. The first one is the approximation of
linear dispersion in the crystal [42, 43]: $k_{\mu}(\Omega)\approx
k_{\mu}^{0}+k_{\mu}^{\prime}\Omega$, so that
$\Delta(\Omega,\Omega^{\prime})\approx\tau_{e}\Omega^{\prime}$. The second one
is the assumption that the width of the function
$\operatorname{sinc}(\tau_{e}\Omega^{\prime})$ is so small compared to the
pump bandwidth, $\pi/\tau_{e}\ll\Omega_{p}$, that the dependence of the pump
spectrum on $\Omega^{\prime}$ can be disregarded,
$\alpha(\Omega+\Omega^{\prime})\approx\alpha(\Omega)$. The third approximation
is the replacement of the $\operatorname{sinc}(x)$ function by a Gaussian
function having the same width at half maximum, $e^{-x^{2}/2\sigma_{s}^{2}}$,
where $\sigma_{s}=1.61$ [43, 44].

The spectra of the ordinary and extraordinary photons $S_{\mu}(\Omega)$ can be
obtained by integrating $|J(\Omega,\Omega^{\prime})|^{2}$ over the frequency
of the other photon [27]. Using the approximate JSA, Eq. (20), we obtain
$S_{\mu}(\Omega)\propto\exp(-\Omega^{2}/2\sigma_{\mu}^{2})$, where the
spectral standard deviations are $\sigma_{o}=\Omega_{p}=19$ rad/ps,
$\sigma_{e}=\sigma_{s}/\sqrt{2}\tau_{e}=3.15$ rad/ps.

The temporal shapes of the photons are given by their average intensities (in
photon flux units)
$I_{\mu}(t)=\langle\hat{E}_{\mu}^{(-)}(t,L)\hat{E}_{\mu}^{(+)}(t,L)\rangle$.
With the help of the same approximations as above, we find [27]
$I_{\mu}(t)\propto\exp(-(t-t_{\mu})^{2}/2\Delta t_{\mu}^{2})$, where $t_{\mu}$
is some delay and the temporal standard deviations are $\Delta
t_{o}=1/2\Omega_{p}=26$ fs, $\Delta t_{e}=\tau_{e}/\sqrt{2}\sigma_{s}=158$ fs,
which corresponds to FWHM widths $\Delta T_{o}=2\sqrt{2\ln 2}\Delta t_{o}=62$
fs and $\Delta T_{e}=2\sqrt{2\ln 2}\Delta t_{e}=373$ fs. We see that the
ordinary photon has the same duration as the pump pulse; this is a consequence
of the group velocity matching between these two waves. The ratio of the
temporal widths of the extraordinary and ordinary photons is equal to the
inverse of that of their spectral widths: $K=\Delta t_{e}/\Delta
t_{o}=\sigma_{o}/\sigma_{e}\approx 6$.

The duration of the ordinary photon can be made equal to that of the
extraordinary one by an erecting magnifying time telescope, similar to the one
described in the preceding section, but used in the reverse order. For this
purpose, one needs to choose $M=6$ and the input bandwidth corresponding to a
62-fs-long pulse. The latter is out of reach of any EOPM-based time telescope
and a parametric one should be used. After the ordinary photon is stretched,
it becomes indistinguishable from the ordinary one, and their Hong-Ou-Mandel
interference (HOMI) [45] can be observed in an experiment, depicted in Fig. 5.

Figure 5: Scheme of the proposed experiment for observing a two-photon
interference using a time telescope when the photons have different durations.
A polarization beam splitter (PBS) is used to separate the ordinary and
extraordinary photons produced in type-II SPDC with asymmetric group velocity
matching. The ordinary photon is temporally stretched by the time telescope
and then interferes with the extraordinary one (delayed by the time which can
be adjusted by $\delta\tau$) on the beam splitter (BS), whose outputs are
monitored by detectors $D_{+}$ and $D_{-}$. A coincidence is registered if
both detectors click in the same excitation cycle.

The field at the detector $D_{\pm}$ is
$E^{(+)}_{\pm}(t)=[E^{(+)}_{o}(t,x_{2})\pm
E^{(+)}_{e}(t+\delta\tau,x_{2})]/\sqrt{2}$, where $x_{2}$ denotes the point
just after the time telescope for the ordinary photon and just before the
adjustable mirror for the extraordinary one. The average coincidence rate is
given by [42, 27]

$R_{c}(\delta\tau)=\frac{1}{T}\int\limits_{0}^{T}\int\limits_{0}^{T}dt_{1}dt_{2}\left|\langle
E^{(+)}_{-}(t_{2})E^{(+)}_{+}(t_{1})\rangle\right|^{2},$ (21)

where $T$ is the coincidence detection time. Expressing the fields through the
group-delayed envelopes via Eq. (1), rewriting the time telescope
transformation (6) as $A_{2}(t)=A_{1}(t/M)/\sqrt{M}$, assuming
$B_{2}(t)=B_{1}(t-t_{d})$, where $t_{d}$ is the delay time in the
extraordinary arm, substituting the solutions (14) and (15), applying the
commutation relations
$[A_{0}(t),A_{0}^{\dagger}(t^{\prime})]=[B_{0}(t),B_{0}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$,
and equating to zero all normally ordered averages at $x_{0}=0$, we obtain

$R_{c}(\delta\tau)=\frac{P_{b}}{2T}\left[1-p_{\mathrm{int}}(\delta\tau)\right],$
(22)

where

$p_{\mathrm{int}}(\delta\tau)=\frac{|M|}{P_{b}}\int\int
J_{\mathrm{out}}(M\Omega,\Omega^{\prime})J_{\mathrm{out}}^{*}\left(M\Omega^{\prime},\Omega\right)e^{i(\Omega-\Omega^{\prime})(\delta\tau-
t_{d})}d\Omega d\Omega^{\prime}$ (23)

is the conditional probability of destructive interference of two photons
under the condition that they are generated and
$P_{b}=\int\left|J(\Omega,\Omega^{\prime})\right|^{2}d\Omega d\Omega^{\prime}$
is the probability of biphoton generation per pump pulse. By destructive
interference we mean, in the spirit of Ref. [45], the case where both photons
go to the same beam-splitter output. The JSA at the crystal output
$J_{\mathrm{out}}(\Omega,\Omega^{\prime})$ differs from the JSA at its input
by a phase (relative to the carrier wave) acquired by the photons during the
dispersive propagation in the crystal:
$J_{\mathrm{out}}(\Omega,\Omega^{\prime})=J(\Omega,\Omega^{\prime})e^{i[k_{o}(\Omega)-k_{o}^{0}]L+i[k_{e}(\Omega^{\prime})-k_{e}^{0}]L}$.
The probability of destructive interference can be calculated analytically
from the approximate JSA, Eq. (20), which gives

$p_{\mathrm{int}}(\delta\tau)=\frac{2K|M|}{K^{2}+M^{2}}\exp\left(-\frac{2\Omega_{p}^{2}\delta\tau^{2}}{K^{2}+M^{2}}\right),$
(24)

where we have put the time delay in the extraordinary arm to
$t_{d}=[(2M-1)k_{p}^{\prime}-k_{e}^{\prime}]L/2$, which provides the maximal
interference probability at $\delta\tau=0$.

As Eqs. (22) and (24) show, the coincidence rate has a dip at $\delta\tau=0$.
The HOMI visibility is defined as the ratio of the dip to the coincidence rate
at a high delay:

$V=\frac{R_{c}(\infty)-R_{c}(0)}{R_{c}(\infty)}=p_{\mathrm{int}}(0).$ (25)

This function is shown in Fig. 6. It reaches the maximum value
$V_{\mathrm{max}}=1$ when $M=\pm K$, i.e. when the magnification modulus is
equal to the ratio of the temporal standard deviations of the extraordinary
and ordinary photons. Note, that the sign of magnification is irrelevant in
the case of symmetric in time pulses: the inverting time telescope is as good
as the erecting one.

Figure 6: (a) HOMI visibility of two unentangled photons generated in a 5-mm-
long KDP crystal pumped by pulses of 62 fs at 415 nm as a function of the time
telescope magnification $M$. The visibility reaches the maximum at the optimal
magnification $|M|=K=\Delta t_{e}/\Delta t_{o}$ for both the erect and
inverted images. (b) HOMI coincidence rate. Since the photons have highly
different durations, the HOMI visibility without time telescope is low.
However, a time telescope applied to one of the photons can make the photons
indistinguishable and achieve the unit HOMI visibility.

###  3.2 Single-photon source

As the second example, we consider two photons generated by single-photon
sources, such as quantum dots [46, 47], with different radiative lifetimes and
show how they can be made indistinguishable by a time telescope. The state of
light emitted by a quantum dot with a radiative lifetime $\tau_{i}$ and a
brightness $\mu_{i}$ ($i=1,2$) can be written as [47, 23]

$\rho_{i}=\int dt_{1}\int
dt_{2}\psi_{i}(t_{1})\psi_{i}^{*}(t_{2})A_{i}^{\dagger}(t_{1})|0\rangle\langle
0|A_{i}(t_{2})+(1-\mu_{i})|0\rangle\langle 0|,$ (26)

where $A_{i}(t)$ is the annihilation operator for a photon at time $t$, while
$\psi_{i}(t)$ defines the temporal mode of the emitted photon and is assumed
to be a decaying exponential
$\psi_{i}(t)=\sqrt{\mu_{i}/\tau_{i}}\exp(-t/2\tau_{i})\theta(t)$, where
$\theta(t)$ is the Heaviside step function. The limits of integration can be
understood as infinite.

Let us suppose that the second source has a longer radiation lifetime,
$\tau_{2}>\tau_{1}$. Stretching the first photon by means of an erecting
magnifying time telescope with a magnification $M$, we could expect that the
photons become indistinguishable at $M=\tau_{2}/\tau_{1}$. In this case, they
should result in a unit HOMI visibility in the setup depicted in Fig. 7. Let
us calculate this visibility assuming that the field emitted by the first
source undergoes a transformation $A_{1}^{\prime}(t)=A_{1}(t/M)/\sqrt{M}$
described by Eq. (6).

Figure 7: Schematic representation of the generation of two photons from
individual sources and their second-order interference. Different radiative
lifetimes $\tau_{2}=3\tau_{1}$ correspond to different shapes of the pulses
produced by spontaneous emission. An erecting magnifying time telescope is
used to stretch the first photon and observe the second-order interference.

Similar to the preceding section, we write the field at the detector $D_{\pm}$
as $C_{\pm}(t)=[A_{1}^{\prime}(t)\pm A_{2}(t+\delta\tau)]/\sqrt{2}$, and the
coincidence rate as

$R_{c}(\delta\tau)=\frac{1}{T}\int dt\int
dt^{\prime}\text{Tr}\left[C_{+}^{\dagger}(t)C_{-}^{\dagger}(t^{\prime})C_{-}(t^{\prime})C_{+}(t)\rho_{1}\rho_{2}\right].$
(27)

Substituting Eq. (26) into Eq. (27) and applying the commutation relations
$[A_{i}(t),A_{i}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$, we obtain the
coincidence rate in the form of Eq. (22) with the probability of photon pair
generation $P_{b}=\mu_{1}\mu_{2}$ and the conditional probability of
destructive interference
$p_{\mathrm{int}}(\delta\tau)=|c(\delta\tau)|^{2}/P_{b}$, where

$c(\delta\tau)=\frac{1}{\sqrt{M}}\int\psi_{1}(t/M)\psi_{2}^{*}(t+\delta\tau)dt$
(28)

is the temporal overlap of the stretched first and delayed second photons.

In the case of a positive magnification, $M>0$, we obtain by substituting the
explicit forms of the modal functions

$p_{\mathrm{int}}(\delta\tau)=\begin{cases}\frac{4M\tau_{1}\tau_{2}}{(M\tau_{1}+\tau_{2})^{2}}e^{-\delta\tau/\tau_{2}},&\text{if
}\delta\tau\geq 0,\\\ \\\
\frac{4M\tau_{1}\tau_{2}}{(M\tau_{1}+\tau_{2})^{2}}e^{\delta\tau/M\tau_{1}},&\text{if
}\delta\tau\leq 0,\end{cases}$ (29)

and the visibility $V=p_{\mathrm{int}}(0)$ is

$V_{M>0}=\frac{4M\tau_{1}\tau_{2}}{(M\tau_{1}+\tau_{2})^{2}}.$ (30)

This function (see Fig. 8(a)) reaches its maximum value 1 at
$M=\tau_{2}/\tau_{1}$, which means that the optimal magnification is given by
the ratio of the lifetimes, as anticipated above.

Figure 8: (a) HOMI visibility as a function of applied magnification. The
inverting time telescope ($M<0$) fails to reach the unit visibility, while the
erecting one ($M>0$) succeeds at $M=\tau_{2}/\tau_{1}=3$. (b) Normalized
coincidence count rate plotted against the relative delay for different values
of magnification. The erecting time telescope ($M=3$) performs much better
than the inverting one ($M=-3$).

In the case of negative magnification, $M<0$, we obtain

$p_{\mathrm{int}}(\delta\tau)=\frac{4|M|\tau_{1}\tau_{2}}{(|M|\tau_{1}-\tau_{2})^{2}}\biggl{(}e^{-\delta\tau/2|M|\tau_{1}}-e^{-\delta\tau/2\tau_{2}}\biggr{)}^{2}\theta(\delta\tau).$
(31)

The maximal value of this function is reached at

$\delta\tau_{\mathrm{min}}=\frac{2|M|\tau_{1}\tau_{2}}{|M|\tau_{1}-\tau_{2}}\ln\left(|M|\tau_{1}/\tau_{2}\right).$
(32)

Thus, the visibility is

$V_{M<0}=p_{\mathrm{int}}(\delta\tau_{\mathrm{min}})=4\left(\frac{\tau_{2}}{|M|\tau_{1}}\right)^{\frac{|M|\tau_{1}+\tau_{2}}{|M|\tau_{1}-\tau_{2}}}.$
(33)

This function (see Fig. 8(a)) reaches its maximum at $M=-\tau_{2}/\tau_{1}$,
where its value is $4e^{-2}\approx 0.54$. Substituting the optimal value of
magnification into Eq. (31), we obtain

$p_{\mathrm{int}}(\delta\tau)=\left(\frac{\delta\tau}{\tau_{2}}\right)^{2}e^{-\delta\tau/\tau_{2}}\theta(\delta\tau).$
(34)

The coincidence rate calculated from Eq. (22) with
$p_{\mathrm{int}}(\delta\tau)$ calculated above is shown in Fig. 8(b). We see
that the erecting time telescope outperforms significantly the inverting one,
when applied to non-symmetric single-photon pulses, such as decaying
exponentials typical to single-photon emitters. The maximal HOMI visibility
reachable with an inverting time telescope is limited to $4e^{-2}\approx 0.54$
for any ratio of the emitters’ lifetimes, because the shape of the emiited
pulses is not time symmetric. In contrast, the erecting time telescope
provides unit visibility at optimal magnification.

##  4 Conclusion

In this work, we have developed a general theory of a time telescope created
by two time lenses and shown that minimal losses are reached when just two
dispersive media are used. We have also shown that an erecting time telescope
can be built by using two time lenses with opposite signs of their focal GDDs.
In contrast to the spatial Galilean telescope, this device creates a real
erect image and can be used in photonic quantum networks for pure temporal
scaling of optical fields, not accompanied by residual temporal chirp.

We have considered two examples of the application of such a time telescope to
distinguishable photons produced by SPDC and single emitters such as quantum
dots and shown that in both cases the photons can be made perfectly
indistinguishable by means of an erecting time telescope. Recent research
shows that both time lenses and dispersive elements can be realised on an
integrated photonic chip with very low losses [48], which opens up bright
perspectives for applications of time telescopy to realizations of quantum
photonic-photonic interconnects.

Funding This work was supported by the network QuantERA of the European
Union’s Horizon 2020 research and innovation program under the project
“Quantum information and communication with high-dimensional encoding”
(QuICHE), the French part of which is funded by Agence Nationale de la
Recherche via grant ANR-19-QUAN-0001.

## Disclosures

The authors declare no conflicts of interest.

## Appendix

### Gaussian pulse passage through a time telescope

We consider a classical field with the positive-frequency part
$E^{(+)}(t)=Y_{0}(t)e^{-i\omega_{0}t}$, where $\omega_{0}$ is the carrier
frequency and $Y_{0}(t)$ is the envelope amplitude, which we assume to have a
Gaussian distribution with the (intensity) standard deviation $\Delta t_{0}$,

$Y_{0}(t)=E_{0}e^{-t^{2}/4\Delta t_{0}^{2}},$ (35)

where $E_{0}$ is the peak amplitude. In the frequency domain, we have

$\tilde{Y}_{0}(\Omega)=\int Y_{0}(t)e^{i\Omega t}dt=2\sqrt{\pi}\Delta
t_{0}E_{0}e^{-\Delta t_{0}^{2}\Omega^{2}},$ (36)

so 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 $T_{0}=2\sqrt{2\ln 2}\Delta
t_{0}$ and $\Omega_{0}=2\sqrt{2\ln 2}\Delta\Omega_{0}$ respectively, we find
the time-bandwidth product $T_{0}\Omega_{0}=4\ln 2\approx 2\pi\times 0.44$, as
expected for a Fourier-limited Gaussian pulse.

We consider the passage of this pulse through the time telescope shown in Fig.
3. Every time lens changes the pulse bandwidth, leaving its duration intact.
In contrast, every dispersive medium changes the pulse duration, leaving its
bandwidth intact. After the first time lens the field becomes
$Y_{1}(t)=Y_{0}(t)\exp(it^{2}/2D_{\mathrm{f}})$. By taking the Fourier
transform, we find the standard deviation of its spectrum,

$\Delta\Omega_{1}=\Delta\Omega_{0}\sqrt{1+D_{\mathrm{f}}^{-2}/4\Delta\Omega_{0}^{4}}.$
(37)

After passing through the dispersive medium with GDD
$D_{\mathrm{inter}}=D_{\mathrm{f}}(1-M)$, the field becomes [22]

$Y_{2}(t)\propto\int
Y_{1}(t^{\prime})e^{-i(t-t^{\prime})^{2}/2D_{\mathrm{inter}}}dt,$ (38)

which gives a chirped Gaussian pulse with the (intensity) standard deviation

$\Delta t_{2}=M\Delta t_{0}\sqrt{1+D_{\mathrm{f}}^{2}(1-M)^{2}/4\Delta
t_{0}^{4}M^{2}}.$ (39)

After the second time lens with the focal GDD
$D_{\mathrm{f}}^{\prime}=-MD_{\mathrm{f}}$, the field is
$Y_{3}(t)=Y_{2}(t)\exp(it^{2}/2D_{\mathrm{f}}^{\prime})$ and the standard
deviation of its spectrum is $\Delta\Omega_{3}=\Delta\Omega_{0}/M$. Passage
through the last dispersive medium compresses the pulse to duration $\Delta
t_{4}=M\Delta t_{0}$.

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