## Abstract

We theoretically and experimentally investigate the spectral tunability and purity of photon pairs generated from spontaneous parametric down conversion in periodically poled KTiOPO_{4} crystal with group-velocity matching condition. The numerical simulation predicts that the spectral purity can be kept higher than 0.81 when the wavelength is tuned from 1460 nm to 1675 nm, which covers the S-, C-, L-, and U-band in telecommunication wavelengths. We also experimentally measured the joint spectral intensity at 1565 nm, 1584 nm and 1565 nm, yielding Schmidt numbers of 1.01, 1.02 and 1.04, respectively. Such a photon source is useful for quantum information and communication systems.

© 2013 OSA

## 1. Introduction

As it is well known, classical widely tunable laser sources play an important role in many fields, for example, spectroscopy, remote sensing, metrology, and optical communications [1,2]. However, the quantum counterpart of the tunable laser, the widely tunable single photon source, have been less investigated. Wavelength tunable single photon sources have been generated from quantum dots, but the tunable ranges achieved until now have been narrow [3, 4].

In this paper, we present a widely tunable single photon source based on spontaneous parametric down conversion (SPDC) in a periodically poled KTiOPO_{4} (PPKTP) crystal with group-velocity matching condition. The concept of group-velocity matching in SPDC was introduced by Keller and Rubin [5], and by Grice and Walmsley [6] in 1997. PPKTP crystal with group-velocity matching condition was experimentally demonstrated by König and Wong [7] for second-harmonic generation in 2004. Later, this condition was applied in SPDC for intrinsically pure photon state generation. Under this condition, the signal and idler photons from SPDC have no spectral correlation, *i.e.*, they are intrinsically pure. Therefore, there is no need to employ bandpass filters to obtain high-purity heralded single photons. Consequently, such photon sources are much brighter than the traditional sources and have showed high brightness in experiments, as reported by Evans, *et al.*, for entangled photons [8]; by Eckstein, *et al.*, for two-mode squeezer [9]; and by Yabuno, *et al.*, for four-photon interferometry [10].

Besides the high purity and high brightness, in this research, we focus on another important merit of this source: the wide spectral tunability. From the numerical calculation, we find that the spectral purity can be kept higher than 0.81 when the wavelength is tuned from 1460 nm to 1675 nm. To verify this simulation, we directly measured the joint spectral intensity, and obtained Schmidt numbers of 1.011, 1.017 and 1.044 at 1565 nm, 1584 nm and 1565 nm, respectively.

## 2. Theory

In the process of SPDC, one pump photon is split into two lower energy photons, the signal and the idler. The two-photon component in SPDC can be expressed as [11]

*f*(

*ω*,

_{s}*ω*) =

_{i}*ϕ*(

*ω*,

_{s}*ω*)

_{i}*α*(

*ω*+

_{s}*ω*) is the joint spectral amplitude (JSA),

_{i}*ϕ*(

*ω*,

_{s}*ω*) and

_{i}*α*(

*ω*+

_{s}*ω*) are the phase matching amplitude and the pump envelope amplitude.

_{i}*ω*is the angular frequency,

*â*

^{†}is the creation operator and the subscripts

*s*and

*i*denote the signal and the idler photons, respectively. Assuming the spectrum of the pump laser has a Gaussian distribution with a bandwidth of

*σ*, the pump envelope intensity can be written as ${\left|\alpha \left({\omega}_{s}+{\omega}_{i}\right)\right|}^{2}=\text{exp}\left[-{\left(\frac{{\omega}_{s}+{\omega}_{i}-{\omega}_{p}}{{\sigma}_{p}}\right)}^{2}\right]$. Under the collinear condition, the phase matching intensity can be written in the form of ${\left|\varphi \left({\omega}_{s},{\omega}_{i}\right)\right|}^{2}={\left[\text{sinc}\left(\frac{\mathrm{\Delta}kL}{2}\right)\right]}^{2}$, where $\mathrm{\Delta}k={k}_{p}-{k}_{s}-{k}_{i}-\frac{2\pi}{\mathrm{\Lambda}}$ is the difference between the wave vector of the pump (

_{p}*k*), the signal (

_{p}*k*), the idler (

_{s}*k*) and an extra vector ( $\frac{2\pi}{\mathrm{\Lambda}}$) introduced by periodical poling of the crystal.

_{i}*L*and Λ are the length and poling period of the SPDC crystal. Figures 1(a)–1(c) shows examples of the pump envelope intensity |

*α*(

*ω*+

_{s}*ω*)|

_{i}^{2}, phase matching intensity |

*ϕ*(

*ω*,

_{s}*ω*)|

_{i}^{2}and joint spectral intensity (JSI) |

*f*(

*ω*,

_{s}*ω*)|

_{i}^{2}.

Considering the Taylor expansion of the three wave vectors *k _{μ}* =

*k*(

_{μ}*ω*

_{μ}_{0}) +

*k*′

*(*

_{μ}*ω*

_{μ}_{0})(

*ω*−

_{μ}*ω*

_{μ}_{0}) +

*o*((

*ω*−

_{μ}*ω*

_{μ}_{0})

^{2}), (

*μ*=

*p*,

*s*,

*i*), the difference of them can be approximated as Δ

*k*= Δ

*k*

^{(0)}+ Δ

*k*

^{(1)}+ .... In our experiment, the phase matching condition is satisfied at ${\omega}_{s}={\omega}_{i}={\omega}_{0}=\frac{{\omega}_{p}}{2}$ and thus the zero order contribution disappears:

As analyzed in [12], the slope of the phase matching condition is determined by the gradient of Δ*k*.

*V*is defined as ${V}_{g,\mu}=\frac{d\omega}{d{k}_{\mu}\left(\omega \right)}=\frac{1}{{k}_{\mu}^{\prime}\left(\omega \right)}$, (

_{g,}_{μ}*μ*=

*p*,

*s*,

*i*). From the view-point of the tilting angle,

*θ*,

*θ*is the angle between positive direction of horizontal axis and the ridge direction of the phase matching intensity, as shown in Fig. 1(b).

The purity, a parameter which describes the spectral correlation of the photon source, is defined as
$p=\mathit{Tr}\left({\widehat{\rho}}_{s}^{2}\right)=\mathit{Tr}\left({\widehat{\rho}}_{i}^{2}\right)$, where *ρ̂ _{s}* =

*Tr*(|

_{i}*ψ*〉 〈

_{si}*ψ*|) is the reduced density operator of the signal, and Tr

_{si}*represents the partial trace on the subsystem*

_{i}*i*. This purity is determined by the factorability of the JSA,

*f*(

*ω*,

_{s}*ω*), and can be calculated numerically using Schmidt decomposition [11, 13]. By applying Schmidt decomposition on the JSA, we can obtain

_{i}*ϕ*(

_{j}*ω*

_{1}) and

*φ*(

_{j}*ω*

_{2}) are two orthogonal basis sets of spectral functions, known as the Schmidt modes.

*c*is a set of real, non-negative, normalized weighting coefficients with ${\mathrm{\Sigma}}_{j}{c}_{j}^{2}=1$. The Schmidt number [14] is defined as

_{j}*K*indicates the number of Schmidt modes existing in the two-photon state and thus can be viewed as an indicator of entanglement. The spectral purity of the signal or the idler state equals to the inverse of the Schmidt number

*K*: In experiment, it is difficult to directly measure

*f*(

*ω*

_{1},

*ω*

_{2}), the JSA. What we usually measured is |

*f*(

*ω*,

_{s}*ω*)|

_{i}^{2}, the JSI, as have been widely demonstrated in previous experiments [8–11]. To analyze such experimental data in experiment, the Schmidt decomposition can be applied on |

*f*(

*ω*,

_{s}*ω*)|

_{i}^{2},

*K*

_{JSI}is useful in experiment to characterize the factorability of the JSI.

## 3. Numerical simulation

With the Sellmeier equation in [7], we calculated spectral purity *p*, Schmidt number *K* and *K*_{JSI} versus the wavelength, as shown in Figs. 2(a) and 2(b). The horizontal axis is the wavelength of the signal (or the idler, in degenerate case), which is two times of the pump wavelength. The bandwidth of the pump is set to match the length of the crystal, so as to achieve maximum spectral purity for *p* and minimum value for *K* and *K*_{JSI} at 1584 nm. The optimal bandwidths for *K* and *K*_{JSI} are slightly different. It is noteworthy that the spectral purity can be kept between 0.815 and 0.821 from 1460 nm to 1675 nm. The purity is relatively low in the regions around 1450 nm and 1700 nm. This is because the same bandwidth of the pump is used for all the wavelengths, but this bandwidth is optimum only for 1584 nm. If the bandwidth of the pump is optimized for 1450 nm or 1700 nm, we can also achieve higher values for those regions. Figures 3(a)–3(d) and 3(e)–3(h) show several examples of the JSA and JSI at 1500 nm, 1550 nm, 1600 nm and 1650 nm, respectively. The *K* ∼ 1.22 in JSA is larger than *K*_{JSI} ∼ 1.01 in JSI, mainly caused by the side lobes in the JSA, as shown in Figs. 3(a)–3(d).

## 4. Experiment and results

The experimental setup is shown in Fig. 4. Picosecond laser pulses (temporal duration ∼ 2 ps, central wavelength was tunable from 700 nm to 1000 nm) from a mode-locked Titanium sapphire laser (Coherent, Mira900) were used as the pump source for the SPDC. Pump pulses with power of 50 mW passed through a 30-mm-long PPKTP crystal with a poling period of 46.1 *μ*m for type-II SPDC. The temperature was maintained at 32°C for the PPKTP crystal. The down-converted photons, i.e., the signal (vertically polarized) and the idler (horizontally polarized) were separated by a polarizing beam splitter (PBS) and collected into two single-mode fibers (SMF). Then the photons were filtered by two bandpass filters (BPFs, Optoquest). The BPFs had a filter function of Gaussian shape with an FWHM of 0.56 nm and a tunable central wavelength from 1560 nm to 1620 nm. Finally, all the collected photons were sent to two InGaAs avalanche photodiode (APD) detectors (ID210, idQuantique) connected to a coincidence counter (Ortec 9353).

To measure the JSI of the photon pairs, we scanned the central wavelength of the BPF1 and BPF2, and recorded the coincidence counts. BPF1 and BPF2 were moved in 0.1 nm per step and 60 by 60 steps in all. The coincidence counts were accumulated for 10 seconds for each point. With the pump wavelength set at 782.5 nm, 792 nm and 807.5 nm, we measured the JSI at 1565 nm, 1584 nm and 1615 nm, respectively.

The measured *K*_{JSI} were 1.011, 1.017 and 1.044, respectively, as shown in Fig. 5. The maximum coincidence count rates decreased from 1075 cps at 1565 nm to 281 cps at 1615 nm, due to the decrease of the quantum efficiency of the APDs. As predicted in Figs. 2(a) and 2(b), the theoretical purities *p*, for 1565 nm, 1584 nm and 1615 nm were 0.820, 0.820 and 0.820, respectively, while the Schmidt numbers *K* and *K*_{JSI} for these wavelength were 1.220, 1219 and 1.219; 1.008, 1.008 and 1.009, respectively. In the simulation of the theoretical *K*_{JSI}, we assumed that the bandwidth of two BPFs was narrow enough. However, in experiment the bandwidth of our BPFs (0.56 nm) was comparable to the bandwidth of the signal and idler (1.1 nm). Therefore, it was necessary to consider the convolution effect of the two BPFs. We repeated the simulation with the real filter function and obtained the convolved *K*_{JSI} as 1.005, 1.005 and 1.005. The three measured *K*_{JSI} were very close to these convolved values, as shown in Table 1. The small disparities may have been caused by the the dark counts in the APDs, or small differences in the two BPFs.

It was noteworthy to compare the theoretically expected bandwidth and experimentally measured one. The measured spectrum was a convolution of the original spectrum and the filter function of the BPFs. The BPFs had a filter function of Gaussian shape with an FWHM of 0.56 nm, while the spectra of the signal and idler were also in a Gaussian shape. Thus we were able to calculate the convolution as
${\text{FWHM}}_{\text{con}}=\sqrt{{0.56}^{2}+{\text{FWHM}}_{\text{the}}^{2}}$, where FWHM_{the} was the theoretically calculated FWHM of the original JSI. The experimentally measured FWHM_{exp} can be obtained from the marginal distribution of the data in Fig. 5. We compared the theoretical bandwidth and the experimentally measured bandwidth in Table 2. The convolved FWHMs were consistent with the measured ones. The small difference may arise from that the FWHM of pump laser was slightly different from the optimal FWHM in the simulation. In addition, the FWHM of the pump laser and the FWHM of the two BPFs were slightly wavelength-dependent.

## 5. Discussion and future prospect

We need to clarify the difference between the JSA and JSI. There is no information on phase of the two-photon wave packet in JSI. Thus, we cannot directly obtain the purity of the heralded single photons from JSI. As shown in Fig. 2, *K* is usually larger than *K*_{JSI}, due to the side lobes in the JSA. The detrimental effect of the side lobes can not be observed in the visibility of two-photon quantum interference, because it requires the purity of a two-photon state and the symmetricity of the joint spectral distribution, with respect to exchanging the constituent photons in frequency degree of freedom [15, 16]. But the purity of each photon is not required. Several groups have reported high-visibility two-photon quantum interference experiments using such PPKTP crystals, e.g., visibility has achieved 94% in [8] and 95% in [17], but it is unnecessary for these experiments to have the high purity of the constituent photons. Meanwhile, in the four-photon quantum interference with such PPKTP crystals, e.g., in [10, 18], to obtain high visibility requires not only symmetric joint spectral distribution but also high purity of the constituent photons [16]. In the application to multi-photon experiments or heralded single-photon source, minimization of the side lobes effect is necessary.

To reduce the effect of side lobes in JSA, two new methods have been proposed recently to engineer the phase-matching function of PPKTP by modulating the poling-order [17] or the duty-cycle [19]. With these methods, the phase matching amplitude can be tailored to approximate a Gaussian function. Therefore, the side lobes can be significantly suppressed and the spectral purity *p* can be improved from 0.81 to 0.99 at 1576 nm [17] and to 0.97 at 1582nm [19]. In both methods, the material property of KTP was not changed. Therefore, in principle, widely tunable photon sources with higher purity may be generated with such custom-poled crystals. However, both methods increase the spectral purity by sacrificing the production rate of the photon pairs. Such irregular modulations of the poling period result in relatively lower efficiency compared to the traditional PPKTP crystals.

Alternatively, the purity can be improved by using two wide-band BPFs to filter out the side lobes. For example, in our simulation the purity can be improved to 0.98 or 0.99 by adding two Gaussian-shape BPFs with FWHM of 3 nm or 2.4 nm. By simply filtering out the side lobes, it might be possible to achieve high purity and the same or even higher photon production rate than that in [17, 19].

This spectrally pure photon source can be easily transformed into an entangled photon source by adding two calcite prisms as displacers [8, 20] or setting it into a Sagnac loop configuration [21, 22]. This photon source can be used for pulsed two-mode squeezed state generation by combining the signal and idler in the degenerate condition [23]. Recently, we noticed that PPKTP crystal had been used to generate widely tunable entangled photon pairs with a tuning range of 60 nm in experiment [24]. Another possible application for this photon source is wavelength-multiplexing based multiparty quantum communication system. If this PPKTP crystal is pumped by a laser frequency comb, e.g., with wavelengths distributed from 730 nm to 835 nm, we can generate serial pure photons distributed from 1460 nm to 1675 nm. These photons can be used for a wavelength-multiplexed system, which is useful for practical multiparty quantum communication systems. The classical broadly tunable laser source have showed a tremendous impact in many and diverse fields of science and technology [1]. As the quantum counterpart of classical broadly tunable laser source, our widely tunable single photon source also has the potential to play an important role in quantum information and communication systems, when the wavelength tunability are necessary.

## 6. Summary

In summary, we have demonstrated the generation of widely tunable and highly pure photons from PPKTP crystal. In the theoretical simulation, we found the wavelength can be tuned from 1460 nm to 1675 nm with spectral purity over 0.81. In experiment, we measured the JSI at 1565 nm, 1584 nm and 1565 nm, and achieved *K*_{JSI} of 1.011, 1.017 and 1.0448, respectively. This result was well consistent with our theoretical simulation. We also discussed the future application of this source.

## Acknowledgments

The authors are grateful to M. Yabuno, P. G. Evans and T. Gerrits for helpful discussions. This work was supported by the Founding Program for World-Leading Innovative R&D on Science and Technology (FIRST).

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