1. Introduction

Quantum entanglement [1] is a useful resource that when shared among distant parties can be exploited for quantum communication applications such as multi-party quantum cryptography [2] and distributed quantum computing [3]. It is also required for the transfer of quantum information via quantum teleportation [4]. However, it has been found that shared entanglement cannot be increased by local operations and classical communications [5], and this implies that entanglement must be physically distributed to application users for sharing.

A source of high quality entangled photon-pairs in the 1550-nm telecom-band would enable distribution of entanglement over low-loss optical fiber. While the use of time-binentangled photon-pairs is gaining popularity [6, 7] due to their tolerance to polarization drifts in optical fiber, the use of polarization-entangled photon-pairs has the important advantage that the quantum state can be easily manipulated using common and inexpensive optical elements such as wave plates and polarization beam-splitters. Active polarization stabilization methods such as those demonstrated in [8, 9] could alleviate the polarization-drift problem during entanglement distribution over optical fiber.

Spontaneous four-wave mixing in a dispersion-shifted fiber (DSF) can be used to realize a source of telecom-band polarization-entangled photon-pairs [10, 11], but a low operating temperature (typically liquid nitrogen temperature) is required to reduce noise photons due to spontaneous Raman-scattering that occurs in the DSF [12, 13]. Quasi-phase-matched nonlinear optical crystals or waveguides such as periodically-poled lithium niobate (PPLN) [14, 15] and periodically-poled potassium titanyl phosphate (PPKTP) [16] have also been used as highly efficient sources of polarization-entangled photon-pairs. Several recent experiments have employed a two-crystal-geometry for producing polarization-entangled photon-pairs [17, 18] but this method requires careful selection of two nearly identical nonlinear crystals. Several-mm-long or even cm-long QPM crystals or waveguides would give a high photon-pair generation rate for moderate pumping power, but very stringent temperature control, to an accuracy of better than 0.1 K in some cases, is necessary to keep a stable phase difference between orthogonal polarization components so as to maintain high quality polarization entanglement.

In this work, we do away with temperature control altogether and demonstrate a stable pulse-pumped source by using a single 1-mm-long PPLN waveguide placed at the center of a polarization-diversity fiber-loop. In contrast to recent experiments [17, 18], we have not used any group-velocity compensation crystal at the source, and this simplifies the setup. All photons are generated within the 1550-nm telecom-band, and so our source can be placed at a service provider who distributes entanglement to application users via optical fiber transmission lines according to demand. Quantum state tomographic measurement performed on the photon-pairs using commercially-available InGaAs single-photon detectors and without subtracting accidental coincidences has revealed a very high state purity of 0.94 and an entanglement fidelity exceeding 0.96 in the low-generation-rate regime. At higher rates, entanglement quality of the photon-pairs degrades as expected. Using a new theoretical model, we show that this degradation is largely due to an increased number of accidental coincidence counts caused by double-pair and triple-pair emissions.

This paper is organized as follows. In Section 2, we give the principle of the proposed source. In Section 3, we describe our experiment. Experimental results are given and discussed in Section 4. Section 5 concludes this paper. The theoretical model that we use to account for the degradation of entanglement quality is provided in an appendix.

2. Principle

 

Fig. 1. (color online) Schematic of the proposed source. Red arrows show input pump pulses having a center wavelength of 776 nm. Black arrows show telecom-band photon-pairs created in the waveguide. ATT: attenuator, HWP: half-wave plate, PBS: polarization beam-splitter, PMF: polarization-maintaining fiber, PPLN: periodically-poled lithium niobate waveguide, QWP: quarter-wave plate, SMF: single-mode fiber

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Figure 1 shows a schematic of the proposed source. It is pumped with optical pulses (pulse-width about 300 fs) from a Ti:Sapphire femtosecond laser operating at a center wavelength of 776 nm. The first polarization beam-splitter (PBS) is used for setting the pump to linear polarization and for fine adjustment of the polarization angle. Ideally, pump pulses that are incident on the second PBS should be diagonally-polarized. The second PBS separates a diagonally-polarized pump pulse into vertically- and horizontally-polarized components, which are fiber-coupled into a 1-mm-long MgO-doped type-0 PPLN waveguide (HC Photonics) from opposite directions. The loop is formed using polarization-maintaining fibers (PMFs) having the same length (21 cm) for each arm. A 90-degree-twist of one of the PMF’s principal axes [19] allows bi-directional usage of the same polarization mode of the PPLN waveguide for photon-pair production [20]. The clockwise-propagating pump component produces vertically-polarized photon-pairs of the state |V〉_s|V〉_i, while the anti-clockwisepropagating pump component produces horizontally-polarized photon-pairs of the state |H〉_s|H〉_i. The subscripts s and i denote signal and idler, respectively. Automatic combination of counter-propagating photon-pairs at the second PBS produces the entangled state, |H〉_s|H〉_i+e^iθ|V〉_s|V〉_i, where θ is an unknown but constant phase. The second PBS has been specially designed to function at 775 nm as well as at the telecom-band. The filter at the exit of the second PBS is a piece of mirror that is coated highly reflective (HR) for 1550 nm and anti-reflective (AR) for 775 nm. Since the signal frequency and idler frequency must add up to give the pump frequency (due to energy conservation), signal and idler photons are frequency correlated and so they can be separated by using a dichroic mirror having a dividing wavelength at 1550 nm. The phase-matching condition of the short waveguide is satisfied over a broad temperature range. In addition, the polarization-maintaining loop configuration makes this source phase-stable against temperature fluctuations and external perturbations and therefore it can be operated for long hours with absolutely no temperature control. For type-II phase-matching, a free-space polarization Sagnac interferometer configuration has been demonstrated to produce good quality polarization-entangled photon-pairs at visible wavelengths [21].

3. Experiment

 

Fig. 2. (color online) Schematic of the experiment. BPF: band-pass filter, DM: dichroic mirror, PC: polarization controller, POL: polarizer, SPCM: single-photon counter module

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To fully characterize the source, we have performed quantum state tomography (QST) on the photon-pairs. Figure 2 shows the experimental setup. At the receiving side, we filtered the signal and idler photons with two 9-nm-bandwidth filters (center wavelengths 1542 and 1562 nm, respectively). Pump photons were further suppressed using two interference filters. A quarter-wave plate followed by a polarizer placed before InGaAs single-photon counter modules (SPCMs, id Quantique) at both channels allowed the sixteen polarization settings necessary for QST. Letting H, V, D, R denote horizontal, vertical, +45 degrees diagonal and right circular polarizations, respectively, the sixteen polarization settings are HH, HV, HD, HR, VH, VV, VD, VR, DH, DV, DD, DR, RH, RV, RD, and RR. We have placed a half-wave plate (HWP) before the quarter-wave plate (QWP) in the signal channel for phase compensation. By carefully optimizing the HWP’s rotation angle, the phase difference between horizontal and vertical polarization components at the signal channel is adjusted such that we obtain the maximally-entangled state |Φ⁺〉≡|H〉_s|H〉_i+|V〉_s|V〉_i. The SPCMs were gated at 4.06 MHz, and the gating widths were 2.5 ns. A dead-time of 10µs was imposed to reduce after-pulses. Dark-count rates were 250 and 700 Hz, respectively. Each coincidence count measurement was taken over 100 seconds and so one full QST measurement requiring sixteen measurements took about 30 minutes. Density matrix of the two-photon-state was then reconstructed using the maximum likelihood method described in [22].

4. Results and discussions

 

Fig. 3. (color online) Density matrix reconstructed from quantum state tomographic measurements without subtracting accidental coincidences.

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Figure 3 shows the reconstructed density matrix for a mean photon-pair generation rate of 0.019 per pump pulse. From the reconstructed density matrix ρ, we can calculate the state purity from P≡Tr(ρ ²), and the entanglement fidelity from F≡〈Φ⁺|ρ|Φ⁺〉. The state’s purity and entanglement fidelity are found to be 0.942 and 0.968, respectively. If we subtract away accidental coincidence counts due to detector dark counts, the values improve to 0.964 and 0.979, respectively. This shows that our source produces very high quality polarization-entangled photon-pairs at the low photon-pair generation rate regime.

 

Fig. 4. Experiment results showing how purity (black circles) and entanglement fidelity (white circles) decrease with increasing photon-pair generation rate. The number of coincidence counts observed in 1 second for the HH polarization setting (squares) is also shown. Solid line is linear fit.

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We repeated the quantum state tomographic measurement for higher pump powers under the same experimental condition. The pump power was varied using an optical attenuator. Figure 4 shows how the state purity (black circles) and entanglement fidelity (white circles), both calculated from reconstructed density matrices, decrease with increasing photon-pair generation rate. The number of coincidence counts observed in 1 second for the HH polarization setting (squares) is also shown.

 

Fig. 5. Decrease of two-photon interference fringe visibility with increasing photon-pair generation rate for the H/V basis. Black circles are experimental results. The broken curve is theoretical result that includes double-pair contributions but omits triple-pairs. The solid curve includes triple-pair contributions.

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Figure 5 shows how the two-photon interference fringe visibility in the H/V basis (average value calculated from HH, HV, VH and VV coincidence count rates) drops with increasing mean photon-pair generation rate. For diagonal basis, the values are slightly lower (not shown), with a maximum deviation of only 2.1 percent. To better understand the effect of multiple-pair emission on entanglement quality, we have developed a theoretical model and obtained formulas that take into account double-pair and triple-pair emissions (see Appendix A), assuming a Poissonian photon-pair number distribution (this assumption is valid for low photon-pair generation rates as in our experiment). The theoretical results are also shown in the figure. The broken line is theoretical result that includes only double-pair contributions, while the solid line includes triple-pair contributions as well. For triple-pairs, we have neglected in our model detection of more than three photons due to the low probability of occurrence. Good agreement of experimental data with the solid line confirms that the degradation in entanglement quality at high pump powers is largely due to accidental coincidences caused by the emission of double- and triple-pairs. To obtain more accurate results, quadruple-pair contributions should also be included into the model. We caution that for photon-pair generation rates higher than 0.2 per pump pulse, the photon-pair number distribution cannot simply be assumed to be Poissonian anymore because it could contain a thermally distributed proportion whose impact would become significant at higher rates. The current model must be suitably extended to be able to cover this regime.

5. Conclusion

We have demonstrated a source of high quality telecom-band polarization-entangled photonpairs based on a single, pulse-pumped, short PPLN waveguide. The use of a dual-band PBS and a phase compensating HWP in the current setup improves the entanglement quality of the photon-pairs tremendously as compared to previous work [20]. At the low-generation-rate regime, we have observed a very high state purity of 0.94 and entanglement fidelity exceeding 0.96 using InGaAs single-photon detectors and without subtracting accidental coincidences. We have also shown using a new theoretical model that at higher generation rates, entanglement quality degrades due to multiple-pair emissions. Good agreement of experimental data with the model confirms the high quality of the source. We thus expect our source to play an important role in future entanglement distribution fiber networks.

Appendix A: Theoretical model

In this appendix, we outline a theoretical model that takes into account double-pair and triple-pair emissions. The model requires one to assume a photon-pair number distribution for the source. Since the photon-pair generation rates in our experiment are lower than 0.2, we have assumed a Poisson distribution

(1)$$P(\mu ,n)=\frac{{\mu }^n{e}^{-\mu }}{n!},$$

where P(µ, n) is the probability of emitting n photon-pairs when the mean photon-pair generation rate is µ. We note that a similar model for type-II spontaneous parametric down-conversion (SPDC) has been considered in a recent paper [21]. The model presented here is specifically for our proposed source that is based on type-0 SPDC from a single PPLN waveguide. Let us call the signal and idler channels, channel 1 and channel 2, respectively. Since the single-photon detectors that we use are not photon-number-resolving detectors, the single-channel count rates s_i (i=1, 2) can be expressed simply as [23]

(2)$$s_i=\left(1-e^{\frac{{-p}_i\mu }{2}}\right)F+d_i,$$

where F is the clock rate, d_i is the detector dark count rate, and p_i is the photon collection and detection efficiency, which includes both channel transmittivity and quantum efficiency of the detector, but excludes the 3 dB loss due to the presence of a polarizer. Instead, we include a factor of 1/2 to account for the polarizer. For illustrative purpose, we assume here that p ₁=p ₂≡p and that the mean photon-pair generation rate is independent of polarization. In actual fact, Eq. (2) depends on polarization, and so one needs to define parameters such as s_iH, s_iV, p_iH, and p_iV for horizontal and vertical polarizations. The mean photon-pair generation rate must also be defined for horizontal and vertical polarizations, denoted by μ_H and μ_V, respectively. However, to avoid presenting lengthy equations in this appendix, we assume here that μ_H=μ_V≡μ/2. Of course, in our calculation leading to the theoretical curves in Fig. 5, we do not make these simplifying assumptions.

 

Fig. 6. (color online) Diagram for counting the double-pair contribution to coincidence count rates when the polarizers of both channels are aligned parallel (red ovals encircling HH) and perpendicular (blue ovals encircling HV). The leftmost set of squares indicates the various ways that photons could be lost. Double-headed arrows indicate polarization-entanglement. It should be noted that although we illustrate the counting method here for H/V basis, it is in principle valid for any other bases.

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The double-pair contribution to maximum and minimum coincidence count rates, denoted by C _max and C _min, can be obtained using the diagram shown in Fig. 6. Let us look at the first row. In the absence of polarizers, the probability that all four photons survive and get detected is p ⁴. However, this leads to a contribution to C _max only if both detectors detect one or more horizontally-polarized photons when both polarizers are in the H orientation. This happens 3 out of 4 times, as indicated by red ovals encircling HH in the figure. On the other hand, there is a contribution to C _min only 2 out of 4 times, since two polarizers are orthogonally orientated in this case, as indicated by blue ovals encircling HV in the figure. This method thus accounts for the coincidence detection probability for parallel and orthogonal polarizer orientations accurately. For the second row, one photon is lost, and the contribution to C _max is 8 out of 16, while the contribution to C _min is 4 out of 16. The overall double-pair contribution to C _max is thus the sum $\frac{3}{4}{p}^4+\frac{8}{16}4p^3\left(1-p\right)+\frac{6}{24}6p^2{\left(1-p\right)}^2$ , which simplifies to $\frac{p^2}{4}\left(6-4p+p^2\right)$ . Similarly, one finds that the double-pair contribution to C _min is given by $\frac{2}{4}{p}^4+\frac{4}{16}4p^3\left(1-p\right)+\frac{2}{24}6p^2{\left(1-p\right)}^2=\frac{1}{2}{p}^2$ .

A similar diagram can also be drawn to find triple-pair contributions. For triple-pairs, we have neglected detection of more than three photons because of the low probability of occurrence. This procedure leads to the following expressions,

(3)$$C_{max}=\frac{p^2}{2}P(\mu ,1)F+\frac{p^2}{4}\left(6-4p+p^2\right)P(\mu ,2)F+\left(3+\frac{33}{4}\frac{p}{1-p}\right)p^2{\left(1-p\right)}^{4}P(\mu ,3)F+C_{\mathrm{acc}}$$

(4)$$C_{min}=\frac{p^2}{2}P(\mu ,2)F+\left(\frac{3}{2}+\frac{21}{4}\frac{p}{1-p}\right)p^2{\left(1-p\right)}^{4}P(\mu ,3)F+C_{\mathrm{acc}}$$

where C _acc is the accidental coincidence count rate due to detector dark counts, given as

(5)$$C_{\mathrm{acc}}=\frac{s_1{d}_2+s_2{d}_1-d_1{d}_2}{F}.$$

With Eqs. (3) and (4), the two-photon interference fringe visibility can be easily calculated from

(6)$$V=\frac{C_{max}-C_{min}}{C_{max}+C_{min}}$$

as a function of p and µ. Next we show how to obtain p from experiment. At low pump power, multiple-pair emission is not that significant, and so we can approximately write

(7)$$C_{max}\approx \frac{p^2}{2}P(\mu ,1)F+\frac{3p^2}{2}P(\mu ,2)F+C_{\mathrm{acc}},$$

(8)$$s_i\approx \frac{p\mu }{2}F+d_i.$$

These equations lead to the relation

(9)$$\frac{\mu }{1+\frac{3}{2}\mu }{e}^{\mu }\approx \frac{2\left(s_1-d_1\right)\left(s_2-d_2\right)}{F\left(C_{max}-C_{\mathrm{acc}}\right)},$$

where the right-hand-side contains only experimentally observable parameters. We can thus estimate µ at low pump power by solving Eq. (9). Once we have found µ at low pump power, we can use Eq. (2) to find p, which is independent of pump power. As the pump power is raised, we can substitute the observed single-channel count rates into Eq. (2) to find µ for the various pump powers.

Finally, we assert that in order to obtain the theoretical curves given in Fig. 5, one needs to go one step further to include polarization dependence. Consequently, Eq. (9) must be replaced by the following set of simultaneous equations,

(10)$$\frac{{\mu }_H}{1+{\mu }_V+2{\mu }_H}{e}^{{\mu }_H+{\mu }_V}\approx \frac{\left(s_{1H}-d_1\right)\left(s_{2H}-d_2\right)}{F\left(C_{\mathrm{HH}}-C_{\mathrm{acc}\mathrm{HH}}\right)},$$

(11)$$\frac{{\mu }_V}{1+{\mu }_H+2{\mu }_V}{e}^{{\mu }_H+{\mu }_V}\approx \frac{\left(s_{1V}-d_1\right)\left(s_{2V}-d_2\right)}{F\left(C_{\mathrm{VV}}-C_{\mathrm{acc}\mathrm{VV}}\right)},$$

and instead of C _max and C _min, one needs to find C_HH, C_VV, C_HV and C_VH. If one only includes double-pair contributions, the expressions would become

(12)$$C_{HH}=p_{1H}{p}_{2H}{e}^{-\left({\mu }_H+{\mu }_V\right)}{\mu }_H\left[1+{\mu }_V+\left(p_{1H}-2\right)\left(p_{2H}-2\right)\frac{{\mu }_H}{2}\right]F+C_{\mathrm{acc}\mathrm{HH}},$$

(13)$$C_{HV}=p_{1H}{p}_{2V}{e}^{-\left({\mu }_H+{\mu }_V\right)}{\mu }_H{\mu }_{V}F+C_{\mathrm{acc}HV},$$

where the accidental coincidence count rates are given by

(14)$$C_{\mathrm{acc}HH}=\frac{\left(s_{1H}{d}_2+s_{2H}{d}_1-d_1{d}_2\right)}{F},$$

(15)$$C_{\mathrm{acc}HV}=\frac{\left(s_{1H}{d}_2+s_{2V}{d}_1-d_1{d}_2\right)}{F}.$$

Expressions including triple-pair contributions would be far too lengthy to be included in this paper.

Acknowledgments

The authors thank S. Odate (Nikon) and Y. L. Lim (DSO) for discussions. H. C. Lim was on a postgraduate scholarship awarded by DSO National Laboratories.

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