## Abstract

We analyze the generation of single spatial mode, spectrally uncorrelated photon pairs via type II spontaneous parametric down-conversion in a Potassium Titanyl Phosphate (KTP) waveguide using real experimental parameters. We show that this source can be used as an efficient, heralded, pure-state single-photon source.

©2010 Optical Society of America

## 1. Introduction

Photon pairs born as twins, but without intra-pair correlation [1], are often useful to linear optical quantum computation [2] and many other quantum information applications [3]. The success of these applications often requires high visibility interference of single photons from independent sources on a beam splitter. Unit visibility can only be achieved if the independent single photons are in a pure state and indistinguishable. This is never an easy task. The primary single photon sources in use are heralded single photon sources based on parametric processes either in the form of spontaneous parametric down-conversion [4,5] or four-wave mixing [6,7], in which photon pairs are created in a large number of spectral and/or spatial modes. The detection of a single photon often projects its twin in a mixture of these modes. Thus strong filtering is mandatory to approximate heralded single photons in a pure state. This leads to a low heralding rate resulting in a serious bottleneck for two-photon quantum interference-based applications. This obstacle can be overcome with the proposal of Grice *et al.* [1,8]. They showed that if photon pairs are produced such that the two-photon joint spectral amplitude is factorable, then the heralding of one photon does not influence the state of the other photon. The advantage is that because post-selection is not required, high visibility two-photon interference can be created at high rates, significantly improving the performance of many applications in quantum information science.

While the generation of a factorable two-photon state was proposed eight years ago by Grice *et al.* [1], it was only very recently that there were experimental demonstrations via parametric down-conversion in bulk crystals [9,10] or using four-wave mixing in optical fiber [11–13]. High visibility two-photon interference from independent, heralded, single-photon sources were observed at high rates.

To have a high purity heralded single photon source via spontaneous parametric down-conversion in a bulk crystal, one has to consider the impacts from many aspects of experimental parameters, including the pump focusing, the phase-matching in the crystal, the collection of photon pairs, and possible inhomogeneities in the pump beam. All of these result in significant experimental effort to maximize the state purity [10].

There is a growing interest of realizing parametric down-conversion in a single-crystal waveguide, in which photon pairs were produced in only a few spatial modes [14–21]. The phase-matching in the waveguide is achieved by periodically inverting the material nonlinear coefficient to improve the efficiency of the nonlinear conversion at the desired wavelengths [22]. It is, however, difficult in practice to keep a perfect periodic poling structure along the entire length of the waveguide. As a result, the photon pairs produced in a periodically poled waveguide are still in a mixture of a few highly correlated modes.

We present here a realistic analysis of generating single spatial mode, spectrally uncorrelated photon pairs via parametric down-conversion in a Potassium Titanyl Phosphate (KTiOPO_{4} or KTP) waveguide with both daughter photons in the telecom band. We want to emphasize that the phase-matching is satisfied without the need of periodic poling structure in the waveguide. Being produced in the single spatial mode, the pair of photons is also spatially uncorrelated. We show that the factorization level, as characterized by the Schmidt number *K*, of the photon pairs approaches to within 0.2% of unity, with spectral filtering preserving ≈90% of produced photon pairs. This photon-pair source may be a good choice for a heralded, single spatial mode, pure-state single-photon source for quantum information applications.

## 2. Theory of type II spontaneous parametric down-conversion in the waveguide

We briefly describe the theory of the generation of photon pairs via type II spontaneous parametric down-conversion in a waveguide. For more details see Ref. 16 and 21.

In the regime of spontaneous radiation and considering that only the first order non-vacuum mode is significant, the quantum state for photon pairs produced via parametric down-conversion in a waveguide is expressed as

*χ*is a constant and it is proportional to the nonlinear coupling of the three-wave mixing as

*d*

_{24}is the nonlinear optical coefficient and given by the 2nd order optical susceptibility as ${d}_{24}=\frac{1}{2}{\chi}_{yzy}^{(2)}$. ${U}_{\mu}^{j}(y,z)$ is the transverse spatial profile of the electric field in the waveguide with

*j*=

*y, z*(crystal axes) and

*μ*=

*s*(signal),

*i*(idler), and

*p*(pump). The polarizations of the signal, idler, and pump photons are along the crystal axes accordingly (see Fig. 1(a) for the definitions of crystal axes)).

The two-photon joint spectral amplitude $f({\omega}_{s},{\omega}_{i})$is a product of the pump spectrum $p({\omega}_{s},{\omega}_{i})$and the phase-matching function $\text{sinc}(\Delta \kappa L/2)$ for parametric down-conversion in the waveguide,

*m*π/Λ (

*m*is an integer) is introduced to take into account the periodically inverted nonlinear optical coefficient

*d*

_{24}(with spatial period Λ) of the waveguide. For example, in a period, the nonlinear coefficient is

*d*

_{24}for

*x*∈[0, Λ/2) and -

*d*

_{24}for

*x*∈[Λ/2, Λ). Such a periodically modulated material nonlinear response can be expressed as the superposition of a set of spatial harmonic frequencies, {

*e*

^{i2mπ/Λ}} [22]. The perfect phase-matching condition of $\Delta \kappa =0$may be satisfied for multiple sets of signal, idler and pump photons that are offset in wavelength, for a given poling period Λ. Denoting the frequency detuning from the center frequencies for the signal and idler photons, respectively, as${\xi}_{s}={\omega}_{s}-{\omega}_{s0}$, ${\xi}_{i}={\omega}_{i}-{\omega}_{i0}$, with ${\omega}_{s0}+{\omega}_{i0}={\omega}_{p0}$, which is the center frequency of the pump photon, the phase-mismatch in the waveguide is written as,

Typically, a coherent pump laser beam may be assumed to have a Gaussian spectral profile, $p({\omega}_{s},{\omega}_{i})={e}^{-{({\omega}_{s}+{\omega}_{i}-{\omega}_{p0})}^{2}/{\sigma}^{2}}$. The central lobe of the sinc-function can be approximated also by a Gaussian function, $\text{sinc}\left(\Delta \kappa L/2\right)\approx {e}^{-\gamma {(\Delta \kappa L/2)}^{2}}$ with γ ≈0.193. Neglecting the small contributions from side-lobes of the sinc-function, with Eq. (5), the two-photon joint spectral amplitude is written as

*τ*

_{coherence}) of the pump laser pulse asTo meet the requirement in Eq. (8), the group delays of the pump, signal and idler photons after propagating through the nonlinear medium need to satisfy

*τ*

_{s}

*τ*

_{i}≤ 0, or equivalently, their group velocities satisfy the relation

To characterize the degree-of-factorization of produced photon pairs, the Schmidt decomposition is applied to write the normalized two-photon joint spectrum as a superposition of a number of orthonormal modes [10,23,24],

*u*

_{j}(

*ω*

_{s})〉 and|

*v*

_{j}(

*ω*

_{i})〉 are the orthonormal modes,

*η*

_{j}is the weight of the two-photon intensity in mode

*j*and Σ

_{j}

*η*

_{j}=1. The two-photon spectral function is factorable if there is only one mode, with

*η*

_{0}=1. The photon pairs are in the mixed states when there is more than one mode, with

*η*

_{j}< 1. Accordingly the Schmidt number is defined to characterize the degree-of-factorization,When

*K*= 1, the two-photon spectral function can be expressed as a product state as given in Eq. (7).

## 3. Optical modes in the KTP waveguide

KTP waveguides have been widely used in the generation of photon pairs. They can be fabricated by applying the diffusion exchange of 100% Rb^{+} ions for K^{+} ions and by masking the surface of the KTP crystal. The refractive index of the KTP crystal is calculated using the Sellmeier equation [25]. The area with Rb-diffusion (RTP) has a bigger refractive index than the KTP substrate. Along the +*z*-direction, the refractive index difference of RTP relative to KTP is described as Δ*n* erfc(*z*/*z*0) according to the experimental analysis (dashed lines in Fig. 3
), with Δ*n* = 0.02 at the surface *z* = 0 [26]. Here we choose *z*
_{0} = 6 μm. The width and length (*L*) of the waveguide are set to be 3 μm × 10 mm. These parameters are used in the simulation through the paper. The schema for the KTP crystal containing a fabricated waveguide and the crystal axes is shown in Fig. 1(a).

We numerically compute the bound optical modes polarized along the crystal *z*-axis (with the commercial software COMSOL [27,28]). As shown in Fig. 1(b), the number of bound optical modes in the waveguide decreases with increasing wavelength. For this waveguide, we find that there are 14 bound optical modes at the wavelength of *λ* = 400 nm. The number of bound modes drops to 4 at around *λ* = 700 nm. The waveguide becomes a single spatial mode optical waveguide for *λ* ≥ 1200 nm.

We are interested in down-converting a pump photon into two daughter photons with wavelengths in the telecom band, for example, to down-convert a pump photon with *λ*
_{p}=764 nm into a signal photon at *λ*
_{s} = 1550 nm and an idler photon at *λ*
_{i} = 1506 nm. The mode profiles (transverse electric field distributions) for the pump and daughter photons are significantly different but all three mode fields are well confined as shown in Fig. 2
, where Figs. 2(d) and 2(e) are contour plots at the full width at half maximum (FWHM) of the amplitudes. (We only consider the fundamental modes in this paper. The pump may not be in a single spatial mode since its wavelength is necessarily no more than half of the maximum of the signal and idler wavelengths.) The effective mode-overlapping area is determined by the spatial distribution of the pump beam as shown in Figs. 2(d) and 2(e). The spatial distributions are symmetric with respect to the crystal-*z* axis but asymmetric with respect to the crystal-*y* axis: the mode field quickly becomes negligible after crossing the crystal-air interface into the air (*z* < 0) by a couple hundred nanometers which is due to the large refractive index contrast between the crystal (*n* ≈1.8) and air (*n* = 1); in the +*z* direction, because of the smooth transition from the Rb-diffused area to the pure KTP substrate, the mode field extends into the crystal substrate for a few micrometers. Also we note that the two daughter photons in the telecom band have similar spatial distributions (see Figs. 2(d) and 2(e)), which will yield similar photon collection efficiencies.

We calculate the effective refractive indices of the fundamental optical modes propagating in the waveguide for both *y*– and *z*– polarizations. The indices are smaller than that of the RTP as shown in Fig. 3(a) but above that of KTP. The difference becomes bigger for longer wavelength. Since at longer wavelength more of the mode field is distributed out of the waveguide to propagate in the substrate which has smaller refractive index, this makes the effective modal refractive index smaller.

We then calculate the group velocity of the signal and idler photons in the waveguide along both crystal–*y* and –*z* axes. They increase monotonically with increasing wavelength (Fig. 3(b)). Since type II down-conversion process has pump and idler photons polarized along crystal–*y* axis and signal photon polarized along crystal–*z* axis, many sets of pump, signal and idler wavelengths satisfy the group velocity condition in Eq. (10) for the signal-idler photon pairs to be spectrally factorable. A selected group of wavelengths with *λ*
_{p} = 764 nm, *λ*
_{s} = 1550 nm and *λ*
_{i} = 1506 nm are shown as an example in Fig. 3(b).

## 4. Phase-matching of parametric down-conversion in the KTP

In the spontaneous parametric down-conversion process, the pump photons are down-converted into pairs of photons if energy conservation, ${\omega}_{s}+{\omega}_{i}={\xi}_{p}$, is satisfied. However, the conversion efficiency is significant only when the process is phase-matched. The periodic poling method can be applied to the crystal waveguide to phase-match the down-conversion process at the desired group of wavelengths for the pump and daughter photons. They are shown by the contour plots in Fig. 4(a) . Each contour plot represents that the perfect phase-matched down-conversion process occurs with the assistance of a certain spatial harmonic frequency, $2m\pi /\Lambda =-({k}_{s}^{z}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{k}_{i}^{y}-{k}_{p}^{y})$. The contour profiles curve back, showing that, for a certain spatial harmonic frequency, there are two groups of wavelengths that can be perfectly phase-matched for efficient down-conversion.

To have photon pairs produced from the down-conversion process spectrally uncorrelated, the group velocity relationship between the pump and daughter photons given by Eq. (10) needs to be satisfied. Furthermore, the wavelengths of photons used in many quantum information applications are either in the telecom band (~1550 nm) or in the visible, thus we require the wavelength of the produced photon pairs to be shorter than 1600 nm. These two conditions plus the energy conservation condition enclose a spectral region of interest for the parametric down-conversion process in the waveguide as shown by the shaded area in Fig. 4(b).

Although high order spatial modes may introduce additional degrees-of-freedom which are useful in encoding information [20], it is hard to harness the photons in these modes in practice; photons created in a single-spatial mode simplify the operations. As discussed in Sect. 3, the waveguide becomes a single-spatial mode optical waveguide for *λ* ≥ 1200 nm. We also wish to generate photon pairs with wavelengths in the telecom band where the instrumentation and the infrastructure for information exchange are sophisticated. With two daughter photons in the telecom band, this puts the pump wavelength at around 780 nm which is a standard output wavelength of the titanium-sapphire laser (with typical output wavelength between 700 nm and 1100 nm). With these additional requirements, there are still a range of wavelengths that satisfy the phase-matched down-conversion in this specific waveguide as shown in Fig. 4(b).

The contour (solid line in Fig. 4(b)) for the perfect phase-matching with zero-spatial harmonic frequency crosses the enclosed spectral region. Considering the imperfection of the periodic poling structure in the waveguide, (which causes the photon pairs produced in mixed states and may produce small reflections at the interfaces of the regions,) it is preferable to have a phase-matched down-conversion process without poling the crystal. We arbitrarily choose *λ*
_{p} = 764 nm, *λ*
_{s} = 1550 nm and *λ*
_{i} = 1506 nm along the zero-spatial harmonic frequency contour line for our discussion. However, other allowed choices would be qualitatively similar. It is worth noting that *λ*
_{p} = 787.5 nm and *λ*
_{s} = *λ*
_{i} = 1575 nm is an allowed choice with the signal and idler differing only in their polarizations.

## 5. Single-spatial mode, spectrally uncorrelated photon pairs

The spectral intensity profiles of the pump, sinc-function, and the two-photon joint spectrum (with peak intensity normalized to one) for down-converting the pump photons at *λ*
_{p} = 764 nm into pairs of daughter photons at *λ*
_{s} = 1550 nm and *λ*
_{i} = 1506 nm in a 1 cm KTP waveguide are plotted in Fig. 5(a)-(c)
, where the pump bandwidth is optimized to have the highest degree-of-factorization for the produced photon pairs.

By applying the Schmidt decomposition, we show that the lowest Schmidt number obtained is *K* ≈1.2 at the optimized pump bandwidth (Fig. 5(d)). This means that the produced photon pairs are in mixed states of different modes. The histogram plot of the mode distribution is plotted in the inset of Fig. 5(d). About 91% of the photon pairs are in the first Schmidt mode.

This result is not surprising. Recall that we have approximated the phase-matching sinc-function by a Gaussian function in the derivation of two-photon factorization condition. However, in the real case, the contribution from the side-lobes of the sinc function is not negligible as shown in Fig. 5(c). These side-lobes are spectrally distinguishable from the central lobes, which allow us to apply hard-edge filters to strongly attenuate the contributions from them. As shown in Fig. 6(a)
, filtering either the signal or idler photons leads to a marked reduction in *K*, but filtering both is even more effective. By applying hard edge filters of 5 nm bandwidth to both signal and idler photons, the Schmidt number *K* is ≈ 1.002 for the transmitted photon pairs and the pair transmittance is close to 90% (Fig. 6(b)). Comparing these numbers with the results presented in Fig. 5(d) where about 90% of photon pairs are in the first mode in the Schmidt decomposition, it suggests that the side-lobes in the phase-matching sinc function are the main cause of the residue intra-correlation in the photon pairs.

Following previous works [16,21], the two-photon flux produced via the waveguide down-conversion process is estimated to be

*ε*

_{0}is the vacuum permittivity,

*c*is the speed of light in the vacuum,

*P*is the average pump power,

*su*and

*sd*are upper and bottom bounds for the signal spectral filter function

*filt*

_{s}(

*λ*

_{s}), so do

*iu*and

*id*for the idler spectral filter function

*filt*

_{i}(

*λ*

_{i}). We also assume the mode profile varies slowly for the range of wavelengths of interest.

For the pump beam with a bandwidth of 0.75 nm (see Fig. 5(d)) and *P* = 1 mW, *d*
_{24} ≈3.5 pm/V, the two-photon flux is estimated to be ≈6 × 10^{7}pairs per second after applying 5 nm hard-edge filters to both signal and idler arms. This rate is similar to previously reported results in the waveguide SPDC processes [16, 21].

Finally, to complete our realistic modeling, we discuss the phase-matching process for the pump light not in the fundamental mode (now referred as the 1st mode), as there is no mechanism to couple the pump beam into the waveguide only into the fundamental mode. The modal refractive index for the pump beam (*λ*
_{p} = 764 nm) in the 2nd waveguide mode is calculated to be *n*
_{2nd}= 1.757, which differs from the modal refractive index of the fundamental waveguide mode, *n*
_{1st} = 1.762 by a small but significant amount. Simulation shows that with *n*
_{2nd} = 1.757, the phase-matched daughter photon-pairs are produced at *λ*
_{s2} = 1741 nm and *λ*
_{i2} = 1361 nm; at *λ*
_{s} = 1550 nm and *λ*
_{i} = 1506 nm, the phase-mismatch is significant ($\text{sinc}\left(\Delta \kappa L/2\right)$=0.002). As a result, at this pair of wavelengths, the efficiency of the parametric down-conversion with the pump photons in the 2nd spatial mode is negligibly small compared to the efficiency of the parametric down-conversion with the pump photons in the 1st spatial mode and$\text{sinc}\left(\Delta \kappa L/2\right)=1$. In addition, when the pump photons are in higher spatial modes, the effective mode-overlapping area is reduced compared to when the pump photons are in the 1^{st} spatial mode, thus the gain for the pump photons in the 2^{n}d spatial mode to be down-converted into pairs of photons at the wavelengths of *λ*
_{s} = 1550 nm and *λ*
_{i} = 1506 nm is further reduced. Because the photon pairs that are down-converted from the pump photons in the 1st spatial mode differ spectrally by ≈200 nm from the photon pairs that are down-converted from the pump photons in the 2^{nd} spatial mode, the photon pairs down-converted from the pump photons in the 1^{st} spatial mode can be extracted with spectral filtering as discussed in the above. The phase mismatches when the pump photons are in higher spatial modes are even larger. Thus our consideration of only the fundamental waveguide modes is justified.

In conclusion, we have presented a realistic analysis of the generation of spectrally uncorrelated, single spatial mode photon pairs via type II spontaneous parametric down-conversion process in a KTP waveguide, with both daughter photons in the telecom band. Being produced in the single spatial mode, the pair of photons is also spatially uncorrelated. We show that the factorization level, as characterized by the Schmidt number *K*, of the photon pairs equals to unity to within 0.2% with spectral filtering which preserves about 90% of produced photon pairs. This configuration provides a simple solution to the requirement of a pure-state single-photon source in many quantum information applications.

## Acknowledgments

The authors would like to thank T. D. Roberts for numerous discussions on the KTP waveguide fabrication and for directing our attention to Ref. 26. This work has been supported in part by the Intelligence Advanced Research Projects Activity (IARPA) entangled photon source program.

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**28. **Certain trade names and company products are mentioned in the text or identified in an illustration in order to specify adequately the experimental procedure and equipment used. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it necessarily imply that the products are the best available for the purpose.