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 a 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) [11,12], and through atomic-cloud-based quantum memory [13–15]. Single-time-lens imaging systems were successfully applied to spectral bandwidth compression of light at a single-photon level using SFG [16], EOPM [17,18], and FWM [19,20] and to temporal transformations of one [21] or both [22] photons of an entangled photon pair. Conditions for noiseless temporal imaging of broadband squeezed [23,24] or antibunched [25] light were established.

The temporal imaging technique was historically inspired by the space-time analogy [26] 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 [27,28]. 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 [29] 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 [30–33], all of them being inverting, where the output pulse (temporal image) is the inverted in time and scaled replica of the input pulse (temporal object). Such an inverting time telescope has a spatial counterpart – Keplerian telescope [34,35]. 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 telescope with 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: $k(\Omega ) \approx k_0+k_0'\Omega + k_0''\Omega ^2/2$, where $k_0=k(0)$, $k_0' = (d k/d \Omega )_{\Omega =0}$ is the inverse group velocity, and $k_0'' = (d^2 k/d \Omega ^2)_{\Omega =0}$ 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 $\tau _\mathrm {g}=k_0'l$, 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

 

Fig. 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 $D_\mathrm {out}+D_\mathrm {in}'=D_\mathrm {inter}$.

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The second-order term of the series is responsible for the spreading of the pulse, which is characterized by the GDD $D=k_0''l$. The input and output dispersive media of the temporal imaging system have GDDs $D_\mathrm {in}$ and $D_\mathrm {out}$ respectively, and the group-delayed envelopes are transformed in these media as $\tilde A_1(\Omega )=\tilde A_0(\Omega )\exp \left (iD_\mathrm {in}\Omega ^2/2\right )$ and $\tilde A_3(\Omega )=\tilde A_2(\Omega )\exp \left (iD_\mathrm {out}\Omega ^2/2\right )$, where $\tilde A_n(\Omega )$ is the Fourier transform of $A_n(t)$. A time lens is a device realizing a quadratic-in-time phase modulation of the passing pulse and is characterized by its focal GDD $D_\mathrm {f}$ [4,24], an ideal time lens providing a transformation $A_2(t)=A_1(t)\exp \left (it^2/2D_\mathrm {f}\right )$. A realistic time lens, based on EOPM or a parametric process, can provide such a modulation in a limited time window only, which is known as the temporal aperture of the time lens. In order for the above ideal transformation to be valid, it is necessary that the dispersed temporal object is shorter than the temporal aperture. A parametric lens, where the fields $A_2(t)$ and $A_3(t)$ have typically different carrier frequencies, should, in addition, have a frequency conversion efficiency close to unity within the temporal aperture. When all these conditions and the temporal imaging condition

The chirp described by Eq. (3) may be highly undesirable in a quantum interconnect, preventing the achievement of indistinguishability of two waveforms from different sources [29]. This chirp can be compensated by an additional dispersive medium when the input waveform is a Gaussian pulse. In the general case of an arbitrary input waveform, a second time lens is necessary for this purpose.

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 $D_\mathrm {f}'$, 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 $D_\mathrm {out}'$. 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 $D_\mathrm {in}'=D_\mathrm {inter}-D_\mathrm {out}$. The field between these sections $A_\mathrm {out}(t)=A_\mathrm {in}'(t)$ 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

Substituting Eq. (3) into Eq. (4) and taking into account the equality $A_\mathrm {out}(t)=A_\mathrm {in}'(t)$, we obtain the total field transformation in a two-time-lens imaging system

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 $M=mm'$ and an additional chirp. This chirp is zero when $D_\mathrm {f}'=-MD_\mathrm {f}$, under which condition Eq. (5) reduces to

Expressing $m$ and $m'$ through $D_\mathrm {f}$, $D_\mathrm {f}'$, $D_\mathrm {in}$ 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

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 input GDD for the object field. The image is formed after the output GDD

We also find $D_\mathrm {f} = D_\mathrm {inter}/(1-M)$ and $D_\mathrm {f}' = -MD_\mathrm {inter}/(1-M)$.

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 the 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 $D_\mathrm {f}'>D_\mathrm {f}>0$, 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 $D_\mathrm {f}'<D_\mathrm {f}<0$, 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 $D_\mathrm {f}>D_\mathrm {f}'>0$, 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 [34,35]. Compression in space corresponds to magnification in angular size, which is the aim of 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 $D_\mathrm {f}<D_\mathrm {f}'<0$, which has no spatial counterpart.

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

For a positive $D_\mathrm {inter}$, we have $D_\mathrm {f}>0$, $D_\mathrm {f}'<0$, $D_\mathrm {f}>|D_\mathrm {f}'|$, 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 [34,35]. 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 $D_\mathrm {f}<0$, $D_\mathrm {f}'>0$, $|D_\mathrm {f}|>D_\mathrm {f}'$, which has no spatial counterpart.

2.3.4 Erecting magnifying time telescope, $M>1$

For a positive $D_\mathrm {inter}$, we have $D_\mathrm {f}<0$, $D_\mathrm {f}'>0$, $|D_\mathrm {f}|<D_\mathrm {f}'$, 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 [35], a Galilean telescope turned by the other side to the object.

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

2.4 Time telescopes studied in the literature

All time telescopes described in the literature up to date [30–33] have spatial counterparts. The time telescope realized experimentally by Gaeta group [30–32] is inverting compressing, it was applied to temporal compression of digital optical signals [30], temporal pulse inversion [31], and spectral magnification [32]. It is a special case of the inverting compressing time telescope with $D_\mathrm {in}=D_\mathrm {f}$, and, as follows from Eq. (8), $D_\mathrm {out}'=D_\mathrm {f}'$. 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 [33] 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 $D_\mathrm {f}'=-mD_\mathrm {f}$, 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 $D_\mathrm {out}=D_\mathrm {f}(1-m) = D_\mathrm {f}+D_\mathrm {f}'$, that is, this system satisfies the telescopic condition (7). This approach is equivalent to the limiting case $D_\mathrm {in}'=\epsilon \to 0$, $D_\mathrm {out}'=-\epsilon /(1-\epsilon /D_\mathrm {f})\to 0$, $m'\to 1$. This is a special case of application of the inverting magnifying time telescope, where the object is placed at the input GDD $D_\mathrm {in}=-D_\mathrm {inter}/M$ before the telescope, so that, according to Eq. (8), the image is created at $D_\mathrm {out}'=0$, which has an advantage of requiring one dispersive medium less.

Note, that though the paper of Christov [36] 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. [36] 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 $D_\mathrm {in}$, $D_\mathrm {out}'$ is always negative. In temporal imaging, however, the image is real, because the output GDD can be made negative.

 

Fig. 2. Galilean telescope and erecting compressing time telescope with the same magnification $M=1/3$. (a) Galilean telescope is composed of two lenses with focal lengths $f>0$ and $f'<0$. When an object is placed at a distance $l_\mathrm {in}>f$ before the objective lens, its virtual image is formed at the output distance $l_\mathrm {out}<0$. The geometrical light rays (red lines) show transformations of individual spatial frequency components in the imaging system. (b) 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 $D_\mathrm {f}'<0$, where the condition $D_\mathrm {f}>|D_\mathrm {f}'|$ is satisfied. The time rays (red lines) show transformations of individual frequency components in the imaging system: vertical position corresponds to time while the direction of the time ray corresponds to frequency [9]. 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.

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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 $D_\mathrm {out}'$ can be made equal to zero. If we put $D_\mathrm {in}=0$, we have from Eq. (8) $D_\mathrm {out}'=-MD_\mathrm {inter}$, 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. [33], we have from Eq. (8) $D_\mathrm {out}'=0$, 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 preferable. Thus, we arrive at a configuration shown in Fig. 3.

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 $D_\mathrm {inter}>0$.

 

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

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A great advantage of an EOPM-based time lens consists of the deterministic nature of the electrooptical effect, providing an almost lossless operation [17,18]. 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 $A$ [4,7]. This shortcoming is removed in the recently proposed Fresnel time lens [37], successfully applied to bandwidth compression of single photons [18]. 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 $t=\pm T_A/2$, i.e. its driving voltage 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 $2\pi$ [37]. 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 [24] similar to spatial imaging, where a point-spread function describes a limited resolution due to a finite lens aperture. 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: $T_0\le D_\mathrm {f}\Omega _m$ and $\sqrt {2\ln 2}\Delta t_2\le MD_\mathrm {f}\Omega _m'$, where $\Omega _m$ and $\Omega _m'$ 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 $\Omega _m'\ge \Omega _0/M$, 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 $\Omega _m'=\Omega _0/M$, we obtain the minimal focal GDD of the first time lens as

Substituting this value into the first inequality, we obtain $\Omega _m\ge \Omega _0\sqrt {2/M}$, 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. [18] with $\Omega _m'/2\pi =70$ GHz, we obtain the minimal output duration $MT_0=4\ln 2/\Omega _m'=6.3$ 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 $D_\mathrm {f}=61700$ ps$^2$, and, as a consequence, $D_\mathrm {f}'=-185$ ps$^2$. 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 [16–18,37]. 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 [18,37]: $M_\Omega = D_\mathrm {f}\Omega _m^{'2}/4\ln 2=4300$, 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. Similar to our previous work [29], the quantum description of the field transformation is done in the Heisenberg picture, instead of the Schrödinger picture traditionally employed for such schemes [38–42], because the time-lens formalism is developed in the Heisenberg picture [23,24]. In addition, the Heisenberg picture formalism provides a natural extension to the high-gain regime of parametric downconversion, where multiple photon pairs are created at once.

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 [17,41], 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

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 [43]

In the low-gain regime of SPDC, where at most one pair is generated per excitation, Eqs. (13) can be solved perturbatively, by replacing $\epsilon ^\dagger _\mu (\Omega ',x)\to \epsilon ^\dagger _\mu (\Omega ',0)$ in their right-hand sides. We denote the envelope of the ordinary wave at the crystal output face at $x=L$ by $A_1(t)=E_o^{(+)}(t,L)e^{i(\omega _ot-k^0_oL)}$, while that of the extraordinary wave by $B_1(t)=E_e^{(+)}(t,L)e^{i(\omega _et-k_e^0L)}$, where $k_\mu ^0=k_\mu (0)$. In these notations, the field transformation in the crystal has the form of an integral Bogoliubov transformation

Here, $J(\Omega,\Omega ')= \kappa L\alpha (\Omega +\Omega ')\Phi (\Omega,\Omega ')$ is the joint spectral amplitude (JSA) of the two generated photons, where $\Phi (\Omega,\Omega ')=e^{i\Delta (\Omega,\Omega ')L/2}\textrm {sinc}[\Delta (\Omega,\Omega ')L/2]$ 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 [46] 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: $k_p'=k_o'$, where $k_\mu '$ is the derivative of $k_\mu (\Omega )$ at $\Omega =0$, which is known as asymmetric group velocity matching [47]. 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 $\tau _e=(k_p'-k_e')L/2=360$ fs, for a crystal of length $L=5$ mm. Similar to Ref. [41], 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. We see from Fig. 4, that when the sidelobes are filtered out, the JSA becomes separable and can be approximately written as

 

Fig. 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.

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The spectra of the ordinary and extraordinary photons $S_\mu (\Omega )$ can be obtained by integrating $|J(\Omega,\Omega ')|^2$ over the frequency of the other photon [29]. 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 [29] $I_\mu (t) \propto \exp (-(t-t_\mu )^2/2\Delta t_\mu ^2)$, 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. Since we are considering a frequency-degenerate source of single photons, it is important that the parametric time telescope has the same output carrier frequency as its input one: If the first time lens shifts the frequency, which is typical for SFG- or FWM-based time lenses, then the second time lens should shift it back. After the ordinary photon is stretched, it becomes indistinguishable from the ordinary one, and their Hong-Ou-Mandel interference (HOMI) [38] can be observed in an experiment, depicted in Fig. 5.

 

Fig. 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.

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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 [29,39]

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:

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.

 

Fig. 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 is low without a time telescope. However, a time telescope applied to one of the photons can make the photons indistinguishable and achieve the unit HOMI visibility.

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3.2 Single-photon source

As the second example, we consider two photons generated by single-photon sources, such as quantum dots [49,50], 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 [25,50]

Let us suppose that the second source has a longer radiative 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 $A_1'(t)= A_1(t/M)/\sqrt {M}$ described by Eq. (6).

 

Fig. 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.

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Similar to the preceding section, we write the field at the detector $D_\pm$ as $C_\pm (t)=[A_1'(t)\pm A_2(t+\delta \tau )]/\sqrt {2}$, and the coincidence rate as

Substituting Eq. (26) into Eq. (27) and applying the commutation relations $[A_i(t),A_i^\dagger (t')]=\delta (t-t')$, 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

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

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.

 

Fig. 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 ($M=-3$).

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

The maximal value of this function is reached at

Thus, the visibility is

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

The coincidence rate calculated from Eq. (22) with $p_\mathrm {int}(\delta \tau )$ calculated obtained 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 emitted 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 realized on an integrated photonic chip with very low losses [51], which opens up bright perspectives for applications of time telescopy to realizations of quantum photonic-photonic interconnects.