## Abstract

We study theoretically the joint spectral properties of photon-pairs produced through spontaneous four-wave mixing (SFWM) with two spectrally distinct pump pulses in optical fibers. We show that, due to the group velocity difference between the pulses, the signature of the interaction can be significantly different from spontaneous parametric down-conversion or SFWM with a single pump pulse. Specifically, we study the case where temporal walk-off between the pumps enables a gradual turn-on and turn-off of the interaction. By utilizing this property, we develop a new approach towards tailoring the spectral correlations within the generated photon pairs, demonstrating the ability to produce factorable photon-pair states, and hence heralded single photons in a pure wave-packet. We show that the use of two pumps is advantageous over single-pump SFWM approaches towards this goal: the usage of the dual-pump configuration enables, in principle, the creation of completely factorable states without any spectral filtering, even in media for which single-pump SFWM tailoring techniques are unsatisfactory, such as standard polarization-maintaining fiber.

© 2013 Optical Society of America

## 1. Introduction

Generation of photon pairs through spontaneous parametric downconversion (SPDC) or spontaneous four-wave mixing (SFWM) is a powerful tool for the realizations of various quantum states of light, such as heralded single photons, polarization-entangled photon pairs, squeezed states and more. Such states are not only useful for fundamental studies of quantum mechanics and the quantum mechanical nature of light, but also for the implementation of quantum information processing protocols, including quantum cryptography [1] and quantum communication [2], teleportation [3], linear optics quantum computation [4], and so forth. Many of these schemes rely on a Hong-Ou-Mandel (HOM) type interference [5] between two photons coming from distinct sources, which requires the two interacting photons to be in pure states and indistinguishable [6,7]. However, without special treatment, the photon pairs produced in SPDC and SFWM are inevitably entangled in the spectral and spatial degrees of freedom due to energy and momentum conservation constraints; thus, the individual photons are in mixed states rather than pure wave-packets, as can be seen by considering the reduced density matrix of the (pure) two-photon state. In other words, upon detection of one photon, which heralds the existence of its twin, the heralded photon is in a mixture of wave-packets due to the correlations between the two photons. Such mixedness forbids high-visibility interference between single photons from different sources, even if the sources are strictly identical. In contrast, if the photons produced through SPDC or SFWM are completely uncorrelated, i.e. the photon-pair state is factorable, each photon can be described as a pure wave-packet and perfect interference visibility between photons from distinct sources can be achieved.

The traditional way to reduce the effect of correlations is through strong spatial and spectral filtering of the photons, projecting them onto a single mode. However, the filtering process reduces the probability of detecting the photons and harms the reliability of the pair-wise nature of the source: the detection of one photon does not guarantee that its twin passed through the filters. This becomes a major difficulty for scaling up quantum networks composed of multiple sources.

In order to overcome this problem and avoid the need for filtering, extensive efforts have been carried out to engineer photon-pair sources to yield uncorrelated photon pairs through the nonlinear interaction itself, rather than post-processing them. Spatial correlations can be tailored in bulk crystals by adequate shaping of the pump [8], or by executing the interaction in single-mode fibers or other waveguides, thus producing the photons in the singly supported spatial mode. Spectral correlations can be tailored by choosing special dispersion properties of the non-linear medium such that the group velocity of the pump lies between the group velocities of the two generated photons, together with proper choices of pump bandwidth and medium length [6,9]. While this technique greatly reduces the spectral correlations between the twin photons, perfectly factorable states can only be achieved when group velocity matching occurs between one of the generated photons and the pump [10], a requirement that is not always attainable as it relies on nontrivial medium dispersion.

Both SPDC and SFWM are nonlinear spontaneous processes that generate photon pairs. In SPDC, a single photon from the pump light is down-converted into two daughter photons. In SFWM, on the other hand, *two* pump photons are annihilated by a *χ*^{(3)} nonlinear medium to create the photon pair, called signal and idler. The two annihilated pump photons need not originate from the same laser source, and may differ spectrally. This gives an additional degree of freedom to SFWM over SPDC and thus additional capabilities for tuning the wavelengths of the signal and idler photons without the need to reengineer the nonlinear medium. The use of two pumps at different wavelengths has been employed to generate signal and idler photons at degenerate wavelengths through SFWM [11–13].

Despite the fact that SFWM can occur with two distinct pumps, to date, most of the efforts to tailor the photon-pair joint spectrum have concentrated on the degenerate pump regime, in which a single pump interacts with itself. Garay-Palmett *et al.*[9] studied the spectral correlations within photon pairs produced by the interaction of two pump pulses at distinct wavelengths in photonic-crystal fibers (PCFs). They concentrated on the case where one of the pulses is very long – semi-single-wavelength – where temporal walk-off between the two pulses is negligible.

In this paper we present a general study of the spectral correlations within photon pairs produced through SFWM in optical fibers using two distinct pump pulses. We take into account the group velocity difference between the two pulses, and show that the temporal walk-off can be employed to completely eliminate correlations – which we quantify as the purity of the individual photons – to achieve unit purity. First, in Section 2, we show how independent wavelength-tuning of two pump pulses allows high tunability of the wavelengths of signal and idler produced in standard silica polarization-maintaining fibers (PMFs). In Section 3 we develop a general theory for the joint spectrum of the signal and idler in fibers using the dual-pump configuration, with emphasis on the conditions for a factorable joint spectral amplitude (JSA). We show that the group velocity difference between the pump pulses can be employed to improve the factorability of the photon pairs compared to the single pump case. We consider the implementation of the developed technique to PMF in Section 4. Finally, in Section 5, we conclude this paper.

## 2. Phase-matching with two pumps

Given two pump pulses with angular frequencies *ω*_{p1} and *ω*_{p2}, the SFWM interaction in fibers generates signal and idler at central angular frequencies *ω _{s}* and

*ω*, respectively, that obey the phase-matching conditions

_{i}*k*(

_{μ}*ω*) (

*μ*=

*p*1,

*p*2,

*s*,

*i*for pump 1, pump 2, signal and idler, respectively) is the effective wavenumber of the mode

*μ*propagating in the fiber. Here and throughout this paper we assume that the pump powers are low such that self- and cross-phase modulation are negligible and thus the phase-matching conditions do not depend on the power [9].

We exemplify the dual pump phase-matching in a standard polarization-maintaining fiber (PMF), where we consider the case where both pump polarizations are aligned with one of the principal axes of the fiber, and look at the signal and idler produced with polarizations along the orthogonal axis. In this case, we can model the effective wavenumbers as [14]:

*k*(

*ω*) =

*n*(

*ω*)

*ω*/

*c*,

*n*(

*ω*) is the refractive index at the angular frequency

*ω*given by the Sellmeier equation of bulk silica, Δ

*n*is the fiber birefrengence and

*c*is the speed of light in free space. Δ

*n*> 0 means the pumps travel on the slow fiber axis (signal-idler on the fast) while Δ

*n*< 0 corresponds to pumps aligned with the fast axis (signal-idler on slow). This simple model is very useful when the involved fields are far-detuned from the zero dispersion wavelength (ZDW) of the fiber, as the waveguide geometry has little effect on the fiber dispersion compared with the bulk material of which it is composed [14]. In order to understand the conditions under which phase-matching is satisfied, it is instructive to invoke the Taylor expansion of the conditions in Eqs. (1) about the central frequency

*ω*

_{0}= (

*ω*

_{p}_{1}+

*ω*

_{p}_{2})/2, which yields, using Eqs. (2):

*β*

_{2}=

*d*

^{2}

*k*/

*dω*

^{2}|

_{ω0}, the pumps’ detuning is given by Ω

*=*

_{p}*ω*

_{p}_{1}−

*ω*

_{0}=

*ω*

_{0}−

*ω*

_{p}_{2}and the signal-idler detuning is Ω

*=*

_{si}*ω*−

_{s}*ω*

_{0}=

*ω*

_{0}−

*ω*. Neglecting the higher-order terms, it follows that for a given pump detuning the phase-matched signal-idler detuning is given by:

_{i}*= 0, and solutions exist only when Δ*

_{p}*n*> 0, i.e. pumps on the slow axis. However, for non-degenerate pumps, phase-matching solutions may also exist when the pumps propagate on the fast axis. Note that the above model relies on birefringence as a key component for satisfying the phase-matching conditions. This is in contrast with the traditional method used in PCFs [15], telecom fibers [16] or silicon waveguides [17] where phase-matching relies on centering the fields around the ZDW, meaning

*β*

_{2}is small and higher orders in the Taylor expansion become significant enough to allow phase-matching solutions. The use of birefringence phase-matching allows the pumps to be spectrally far from the ZDW, and hence tunable over a large spectrum [14]. Note also that this method does not require any special medium engineering, and any birefringent medium can exhibit phase-matching. Thus, the techniques that we develop here for PMFs can also be implemented in other birefringent media, such as bulk crystals or on-chip waveguides.

In Fig. 1, phase-matched signal and idler wavelengths are plotted as a function of the average of the pumps’ wavelengths *λ*_{0} = 2*πc*/*ω*_{0}, for various values of detuning Δ*λ* = *λ _{p}*

_{1}−

*λ*

_{0}, where

*λ*= 2

_{μ}*πc*/

*ω*(

_{μ}*μ*=

*p*1,

*p*2) and we define

*λ*

_{p}_{1}>

*λ*

_{p}_{2}. The calculations use |Δ

*n*| = 4 × 10

^{−4}, which is a typical value for commercially-available PMFs. As can be seen, the use of two non-degenerate pumps allows great flexibility in choosing the signal-idler wavelengths and the possibility for individual tuning to match with application requirements, such as specific atomic transitions or matching between different sources.

## 3. Tailoring the joint spectrum

Generally, the state of a photon pair can be written as:

*ω*,

_{s}*ω*〉 is a state with signal (idler) at angular frequency

_{i}*ω*(

_{s}*ω*) and the joint spectral amplitude (JSA) is given by [9]

_{i}*N*is a normalization factor,

*L*is the fiber length, the phase mismatch Δ

*k*is given by Eq. (1b) and we made use of the energy conservation constraint Eq. (1a) to eliminate one of the pumps (

*p*2) degrees of freedom. The two pumps are considered as Gaussian spectral envelopes centered at ${\omega}_{{p}_{1}}^{0}\left({\omega}_{{p}_{2}}^{0}\right)$ with bandwidth (half width at 1/

*e*maximum amplitude)

*σ*

_{1}(

*σ*

_{2}). We also

*p*

_{1}(

*ωp*

_{2}) assume that one of the pump pulses (

*p*1) is delayed relative to the other (

*p*2) by a temporal delay

*τ*prior to entering the fiber.

The degree to which the JSA is factorable is quantified by the purity of the individual signal (idler) photon, whose density matrix *π _{s}* (

*π*) is found by tracing the two-photon state matrix over the idler (signal) degrees of freedom:

_{i}*π*= Tr

_{s}_{i}(|Ψ〈 〉Ψ|),

*π*= Tr

_{i}_{s}(|Ψ〉 〈Ψ|). The purity is then evaluated as $P=\text{Tr}\left({\pi}_{\text{s}}^{2}\right)=\text{Tr}\left({\pi}_{\text{i}}^{2}\right)$, or, in terms of the JSA,

*N*accounts for normalization and a factorable spectral phase, the group delays are given by

*θ*= −arctan(

_{si}*T*/

_{s}*T*) needs to be in the range 0° <

_{i}*θ*< 90°, meaning that

_{si}The JSA given by Eqs. (8) is still somewhat complicated and does not generally provide an analytical means to investigate the correlations between signal and idler or the purity of the photons. For the special case in which
${\omega}_{s}^{0}={\omega}_{i}^{0}$, i.e. degenerate signal and idler, it follows that *T _{s}* =

*T*, which means that the JSA is solely a function of (

_{i}*ν*+

_{s}*ν*), resulting in a highly correlated state. Therefore, the scheme presented in this paper does not allow the generation of degenerate pure photons.

_{i}In the following, we concentrate on non-degenerate signal and idler photons. We set *τ* = −*τ _{p}*/2, meaning that the slow pump pulse is sent ahead of the fast pump pulse by a time |

*τ*/2| and the two pulses are maximally overlapped at the center of the fiber, resulting in maximal overall interaction. In order to provide means for comparison with the single-pump configuration (e.g., for estimating the generation efficiency), we define the effective interaction length

_{p}*I*

_{p1}(

*z*,

*t*) (

*I*

_{p2}(

*z*,

*t*)) is the intensity of pump

*p*1 (

*p*2) at point

*z*along the fiber and time

*t*. Note that for the single pump configuration where there is no temporal walk-off between the degenerate pumps, the interaction strength does not vary along the fiber and

*L*

_{eff}=

*L*.

We now investigate different regimes of interest in which the JSA can be further simplified.

#### 3.1. Negligible temporal walk-off

Here we consider the case where the temporal walk-off between the two pump pulses is negligible (or even zero). This regime, expressed as |*στ _{p}*| ≪ 1, is obtained when the fiber is short, or when at least one of the pump pulses is long, or when the group velocities of the two pumps are almost identical. This condition is strictly satisfied with

*στ*= 0 in the degenerate pump case as

_{p}*τ*= 0. In this negligible temporal walk-off regime, the interaction switches on abruptly when the pumps enter the fiber and turns off abruptly when they exit the fiber, and the effective interaction length is simply

_{p}*L*

_{eff}=

*L*. In this case, it can be shown that the phase-matching function in Eq. (8c) is reduced to a sinc function:

*L*) that result in factorability. However, unlike the Gaussian function, the oscillatory behavior of the sinc function carries sidelobes that result in a spectrally correlated spread of signal and idler photons, thus limiting the maximal factorability one can achieve [10].

#### 3.2. Complete temporal walk-off

In the opposite regime, expressed as |*στ _{p}*| ≫ 1, there is no interaction at the beginning of the fiber as the two pump pulses are temporally well-separated (the slow pump is sent ahead of the fast by a time |

*τ*/2|). The interaction strength gradually increases as the fast pump catches up to the slower pump and the two pump pulses begin to sweep across each other, with peak interaction strength reached at the center of the fiber when the two pumps are maximally overlapped. The interaction then gradually turns off and vanishes when the pumps separate towards the end of the fiber. This regime is obtained when the fiber is long, the pump pulse durations are short, or when the group-velocity difference between the two pumps is large. In this case, the effective interaction length is

_{p}*r*=

*σ*

_{1}/

*σ*

_{2}. Note that this purity depends on the ratios

*τ*/

_{s}*τ*and

_{p}*τ*/

_{i}*τ*, but not on fiber length. This is because the fiber is long enough such that the two pump pulses do not interact at the beginning or end of the fiber and thus extending the fiber length does not extend the interaction length. In addition, the bandwidths scale the JSA but do not affect the correlations within this scale. Unit purity is obtained when Since this regime seems promising for the generation of completely spectrally-uncorrelated photon pairs, in Section 4 we will consider its implementation.

_{p}We note that quasi-phase-matched SPDC in a custom-poled medium can also be used to create a gradual variation of the interaction, and specifically a near-Gaussian phase-matching function, by varying the poling periodicity [22]. However, this requires special medium engineering.

#### 3.3. Asymmetric case

The asymmetric case [9] refers to the case where *T _{s}* = 0 (or

*T*= 0), and the phase-matching function in Eq. (8c) becomes independent of

_{i}*ν*(or

_{s}*ν*). For large enough temporal walk-off of the idler (signal), meaning

_{i}*T*≫ (1,

_{i}σ*στ*) (or

_{p}*T*≫ (1,

_{s}σ*στ*)), the phase-matching function becomes very narrow and the JSA can be approximated as the factorable product

_{p}*L*→ ∞ [9, 10, 18], with the (pure) signal photon in a sinc-shaped spectral amplitude. If, on the other hand, the temporal walk-off between the pumps is complete, both signal and idler are produced in transform-limited wave-packets, provided that the temporal walk-off of the idler is much larger than that the walk-off between the pumps.

## 4. Tailoring the joint spectrum in PMF

The generation of factorable photon-pair states in PCFs has been investigated, in the single pump configuration, both theoretically [9] and experimentally [18–20] in the asymmetric case. Ref. [9] also studied the dual pump configuration, but concentrated on the negligible temporal walk-off regime. Single-pump SFWM photon-pair state tailoring has been investigated in silica PMFs as well [14, 21]. In the following, we study source design for factorable photon-pair generation using PMF, exploiting the different signatures of SFWM in the dual-pump configuration compared to the single pump case.

In order to understand the JSA produced through SFWM in PMF, we invoke the approximation

*β*

_{2}=

*d*

^{2}

*k/dω*

^{2}|

_{ω0}, ${\omega}_{0}=\left({\omega}_{p1}^{0}+{\omega}_{p2}^{0}\right)/2$ and

*k*(

*ω*) is the effective wavenumber as given by the dispersion in silica. Based on this approximation, we can see that

*T*/

_{s}*T*≈ −1 in the degenerate pump configuration (in which

_{i}*T*=

_{s}*τ*and

_{s}*T*=

_{i}*τ*), irrespective of the wavelength of the pump [14]. It follows then that the condition in Eq. (10) is satisfied for any pump wavelength (as long as it is far from the fiber ZDW), thus factorability is possible. On the other hand, it also means that asymmetric state tailoring is not possible in PMF, and hence one cannot obtain 100% purity with a single pump due to the sidelobes of the sinc function [10, 21]. Thus, the dual pump configuration in the complete temporal walk-off regime can enhance the achievable purity by eliminating the sidelobes.

_{i}Figure 2(a) presents the purity in the |*στ _{p}*| ≫ 1 regime (calculated using Eq. (15)), as a function of the wavelength detuning Δ

*λ*from the given central pump wavelength

*λ*

_{0}= 715nm. The group delays are derived using the PMF model employed earlier, in which the dispersion is given by the Sellemeier equation for bulk pure silica [14, 21]. For simplicity, we choose two pumps with equal bandwidths (

*r*= 1). We also plot the minimal detuning between the pumps and the idler photon, $\mathrm{\Delta}f=\text{min}\left(\left|{\omega}_{i}^{0}-{\omega}_{p1}^{0}\right|,\left|{\omega}_{i}^{0}-{\omega}_{p2}^{0}\right|\right)/2\pi $.

As Fig. 2(a) indicates, the larger the detuning, the higher the purity. In order to understand this, we refer to the approximation in Eqs. (18), which leads to the following deductions: first, the only way to control the behavior of the phase-matching function (Eq. (14)) is by varying the detuning of the phase-matched signal and idler. This can be accomplished only through variable detuning between the pumps. Second, varying the pumps’ detuning allows us to control the width of the phase-matching function but not its orientation (i.e., for *r* = 1, *T _{s}*/

*T*≈ −1 irrespective of the detuning). Third, the unit purity condition (Eq. (16)) requires ${\omega}_{s}^{0}\approx {\omega}_{p1}^{0}$ and ${\omega}_{i}^{0}\approx {\omega}_{p2}^{0}$, or ${\omega}_{s}^{0}\approx {\omega}_{p2}^{0}$ and ${\omega}_{i}^{0}\approx {\omega}_{p1}^{0}$ (this is true, within the approximation in Eqs. (18), for any bandwidths ratio

_{i}*r*). This is obviously an unsatisfactory situation, as in this case the only degree of freedom that distinguishes the generated photons from the pumps is polarization (rather than spectrum), thus it is impractical to separate the signal and idler from the (intense) pumps. Moreover, it is undesirable to have the signal and idler lying in the spectral vicinity of the pumps, as in this region the pumps produce, in addition to the SFWM photons, background contamination through the process of spontaneous Raman scattering [23,24]. Despite the above, as Fig. 2(a) suggests, the usage of two distinct pump pulses offers the ability to compromise between spectral proximity of the generated photons to the pumps and high purity, and hence the ability to choose the best settings for specific experimental requirements. Note that cooling the fiber has been shown to effectively suppress Raman background [25,26], thus allowing one to obtain highly factorable JSA with negligible Raman background. Also, in media where the Raman gain exhibits a narrowband spectral gain, as is the case in silicon waveguides [27], the Raman background can be avoided by choosing the detuning Δ

*λ*.

According to Fig. 2(a), within the complete temporal walk-off regime the dual-pump SFWM interaction in PMF generates photon pairs for which the purity of the individual photons approaches zero for small pump detunings Δ*λ*. However, for small detunings it becomes increasingly difficult to maintain the condition |*στ _{p}*| ≫ 1, and it is impossible in the degenerate pumps configuration since

*τ*= 0. Therefore, in the case where the pumps are aligned with the slow axis of the fiber, it is more instructive to compare the purity that can be achieved with two pumps to the single pump case without restricting ourselves to a certain walk-off regime. Figure 2(b) shows the maximal purity that can be obtained as a function of the detuning Δ

_{p}*λ*. We calculate numerically the JSA using Eqs. (8), from which the purity is evaluated using Eq. (7). We set the central wavelength

*λ*

_{0}= 715nm and fiber length

*L*= 9cm (similar to the parameters used in Ref [21]), and for each detuning Δ

*λ*we find the pump bandwidth (again, we consider the case of

*σ*

_{1}=

*σ*

_{2}) that maximizes the purity. We also plot the value of |

*στ*| associated with these parameters. As can be seen, at zero and small detunings (less than 25 nm), the optimized purity is achieved in the |

_{p}*στ*| ≪ 1 regime, where the purity is limited to

_{p}*P*< 84%. As the detuning increases, the value of |

*στ*| at which optimized purity is achieved increases, resulting in an overall higher achievable purity. As a result, for Δ

_{p}*λ*> 130nm,

*P*> 95%, and

*P*> 99% for Δ

*λ*> 255nm. Figure 3 shows the joint spectral amplitudes (|

*f*(

*ω*,

_{s}*ω*)|) that result in the maximized purities in Fig. 2(b) for three different regimes: no temporal walk-off between pumps, intermediate regime, and complete walk-off. As can be seen, lack of walk-off results in the highly correlated sidelobes, which are reduced in the intermediate regime and (practically) vanish when temporal walk-off between the pumps is large.

_{i}## 5. Conclusion

We have presented a new method to engineer a photon-pair source based on distinct pumps through SFWM in optical fibers. The use of two pumps provides more flexibility not only in the choice of phase-matched photon-pair wavelengths, but also for engineering their joint spectral properties. While highly factorable photon-pair states can be obtained in the wide range of negligible to complete temporal walk-off between the two pump pulses, we show that when the two pumps are sufficiently spectrally detuned from each other, the temporal walk-off ensures gradual turn-on and turn-off of the SFWM interaction, provided that a time delay is introduced for the fast pump before entering the fiber. This yields a Gaussian-shaped phase-matching function such that the resulting joint spectral amplitude has no sidelobes. The scheme therefore allows the generation of perfectly factorable photon pairs without filtering when Eq. (16) is fulfilled. We have shown that in standard PMF, higher purity can be obtained as the detuning between the two pumps’ wavelengths increases. A maximally factorable state only occurs when the wavelengths of the signal and idler are close to (possibly spectrally overlapping with) the pumps; thus, a compromise between purity and spectral proximity to the pumps (where the Raman gain in silica glass is high) needs to be considered.

We expect this study to provide new ways and stimulate further research into the generation of factorable photon-pair states using dual-pump configurations through SFWM.

## Acknowledgments

This work was supported in part by the NSF Physics Division, Grant No. 1205812.

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