## Abstract

We proposed and demonstrated a simple but deterministic scheme for generating polarization-entangled photon pairs at telecommunication wavelengths with type-II quasi-phase-matched spontaneous parametric down-conversion (QPM-SPDC) having two poling periods. We fabricated a LiNbO_{3} crystal having two poling periods so as to generate entangled photons at two wavelengths, i.e., 1506 nm and 1594 nm. We characterized the two-photon polarization state with state tomography and confirmed that the state was highly entangled.

© 2012 Optical Society of America

## 1. Introduction

Entanglement is a key resource for quantum information technology, including quantum communication and quantum computation. Photons are often used to conduct proof-of-principle experiments on quantum information such as Bell’s inequality violation [1], quantum telepor-tation [2], and linear optics quantum gate [3], by virtue of their robustness against decoherence. Especially, polarized photons are used as qubits because photon polarization provides an ideal two-level quantum system that is easy to deal with in practical experiments. Therefore, it is desirable to establish efficient methods for the generation and detection of polarization-entangled photon pairs for practical applications in quantum information processing. Spontaneous parametric down-conversion (SPDC) is the main method so far used to generate entangled photon pairs [4]. Quasi-phase matching (QPM) is a fascinating method that can generate entangled photon pairs with high efficiency [5]. To date, a number of methods to generate polarization-entangled photon pairs with type-II QPM have been reported [6–8]. However, some of them are not deterministic, while others require complex optical systems. In this paper, we report a simple but deterministic scheme for generating polarization-entangled photon pairs with type-II quasi-phase-matched SPDC (QPM-SPDC) having two poling periods.

## 2. Methods

In the SPDC process, when pump photons of frequency *ω _{p}* illuminate a nonlinear optical crystal, these photons are converted into twin (signal and idler) photons of frequency

*ω*and

_{s}*ω*according to the conservation of photon energy

_{i}*ω*=

_{p}*ω*+

_{s}*ω*. The conversion efficiency is highest when their momenta are also conserved, i.e., under the phase-matching condition

_{i}*k*,

_{p}*k*, and

_{s}*k*are the wave vectors of the pump, signal, and idler photons, respectively. QPM is a technique that compensates for a phase mismatch with the help of periodic modulation of nonlinear susceptibility

_{i}*χ*

^{(2)}. In ferroelectric crystals such as LiNbO

_{3},

*χ*

^{(2)}can be modulated by periodic domain inversion, i.e., periodic poling. The QPM condition involves an additional term that depends on the poling period Λ to the phase mismatch Δ

*k*;

Periodically poled LiNbO_{3} (PPLN) is the most major device used for QPM. Figure 1 shows the calculated tuning curves, i.e., signal and idler wavelengths where the phase-matching condition (2) is satisfied, for type-II collinear QPM-SPDC in PPLN (pump wavelength *λ _{p}*=775 nm, temperature

*T*=166°C) as a function of the poling period Λ. The refractive indices of LiNbO

_{3}were calculated using the Sellmeier equations [9, 10]. As the figure shows, when Λ = Λ

*= 9.25*

_{a}*μ*m, the ordinary ray (o-ray) at

*λ*

_{1}= 2

*πc/ω*

_{1}= 1590 nm and the extraordinary ray (e-ray) at

*λ*

_{2}= 2

*πc/ω*

_{2}= 1510 nm are emitted. On the other hand, when Λ = Λ

*= 9.50*

_{b}*μ*m, the e-ray at

*ω*

_{1}and the o-ray at

*ω*

_{2}are emitted. We designed a PPLN crystal consisting of two adjacent areas (

*a*and

*b*) having two different poling periods (Λ

*and Λ*

_{a}*), as shown in Fig. 2. These two areas have almost the same lengths:*

_{b}*L*≃

_{a}*L*, so that both equally contribute to the down-conversion. In the following, we consider a crystal orientation in which the o-ray and e-ray have horizontal and vertical polarizations, respectively. The state of the superposed emission from the PPLN crystal can be written as

_{b}*H*(

*V*),

*ω*

_{1}(

*ω*

_{2})〉 indicates the single-photon state having horizontal (vertical) polarization and frequency

*ω*

_{1}(

*ω*

_{2}). The twin photons emitted from the poled areas

*a*and

*b*are associated by the two-photon wave packets |

*ϕ*〉 and |

_{a}*ϕ*〉, respectively. The difference between |

_{b}*ϕ*〉 and |

_{a}*ϕ*〉 arises from the longitudinal walk-off, i.e., the temporal separation between orthogonally polarized photon wave packets, and produces distinguishability between the photons emitted from the two areas [11, 12]. When the twin photons emerge from the crystal, the relative phase

_{b}*φ*in (3) is given by From the state (3), we can split the photon pair deterministically in terms of their frequencies [13] to paths 1 and 2, so that

*HV*〉 (|

*VH*〉) corresponds to the state where one photon, in horizontal (vertical) polarization, is in path 1 and the other, in vertical (horizontal) polarization, is in path 2. The density matrix

*ρ*

_{(p)}of the polarization state is obtained by taking a partial trace of the state (5) with respect to |

*ϕ*〉 and |

_{a}*ϕ*〉 as

_{b}*ν*| = 1, i.e., |

*ϕ*〉 = |

_{a}*ϕ*〉. Therefore a pure polarization-entangled state requires indistinguishability between the two-photon wave packets. In other words, partial distinguishability between |

_{b}*ϕ*〉 and |

_{a}*ϕ*〉 degrades the purity and entanglement in the state (6). In our case, as we will discuss later, we expect |

_{b}*v*| ≤ 0.954.

The experimental setup is shown in Fig. 3. A PPLN crystal (0.5 mm thick, *L _{a}* ≃

*L*≃ 20 mm), having the two poling periods Λ

_{b}*= 9.25*

_{a}*μ*m and Λ

*= 9.50*

_{b}*μ*m, was fabricated from a congruent z-cut LiNbO

_{3}crystal by the full-cover electrode method [14–16]. To produce SPDC, the crystal was illuminated by the pump pulses at 775 nm, which is the second harmonics of an amplified external cavity laser diode. The pulse width, repetition rate, and average power of the pump were 2.5 ns, 4 MHz, and 9 mW, respectively. The crystal temperature was controlled by an oven with the stability of ± 0.01°C. In order to observe the parametric emission spectra, we used a grating spectrometer followed by an InGaAs linear photodiode array (not shown in the figure). A polarization-division Michelson interferometer was used in order to compensate for the group velocity difference between the photons having orthogonal polarizations and thus to maximize the visibility |

*ν*| in Eq. (7). The relative phase

*φ*was also adjusted by the interferometer;

*l*=

*l*−

_{V}*l*is the pass-length difference of the interferometer having arm lengths of

_{H}*l*and

_{H}*l*for the horizontal and vertical polarizations, respectively. Then, a dichroic mirror split the photon pair in a deterministic way in terms of their frequencies to paths 1 and 2. Then each photon passed through a polarization analyzer (PA), which consisted of a half-wave plate and a polarizing beam splitter, and was detected by a single photon detector (id Quantique, id201). The coincidence signal between the two detectors was collected by a time-interval analyzer (EG&G, 9308).

_{V}## 3. Results

We measured the temperature dependence of the parametric emission spectra generated from the PPLN crystal having two poling periods. Figure 4(a) shows the emission spectra at temperatures from 120°C to 140°C. Note that we have two wavelength regions for the parametric emission: *λ*_{1} ∼ 1510 nm and *λ*_{2} ∼ 1590 nm. The decreased intensity in the wavelength region longer than 1600 nm was caused by the decreased sensitivity of the detector. Also note that these unpolarized spectra contain both o-ray and e-ray emissions. Two spectral peaks, one corresponding to the o-ray and the other to the e-ray, were observed for each spectral region at temperatures other than 120°C. The two peaks merged into a single peak at 120°C, having center wavelengths at 1506 nm and 1594 nm and a full width at half maximum of 1.4 nm. Figure 4(b) presents the temperature dependence of the peak positions shown in Fig. 4(a). The solid and dashed curves are the calculated values for the ordinary and extraordinary rays, respectively. As observed, the spectral peak positions for the o-ray and e-ray become identical at 120°C. Note that this spectral identity was expected at 166°C, as shown in Fig. 1. In the calculation for Fig. 4(b), the temperature was shifted so that the calculated curves fit the experimental values. The disagreement would originate from a slight difference between the actual refractive indices of our congruent LiNbO_{3} crystal and those calculated from the Sellmeier equations [9, 10]. Slight changes in the actual polling periods from the designed values are also possible reasons. It is noteworthy that, despite these possible deviations from the designed values, one can achieve the spectral identity at a certain temperature, even if the temperature and wavelengths differ somewhat from the original design.

In order to characterize the polarization state of the photon pair, we examined their polarization correlation. In the experiment, the crystal was kept at 120.0°C where the o-ray and e-ray have identical emission spectra, and the coincidence signals from the two detectors were collected as a function of *θ*_{1} and *θ*_{2}, which represent the directions of linear polarizations transmitted by the polarization analyzers (PA) shown in Fig. 3. We adjusted the path-length difference Δ*l* in Eq. (8) so as to obtain *φ* = 0 and the maximum fringe visibility of the polarization correlation. Figures 5(a) and (b) present the observed polarization correlation as a function of *θ*_{2}, while *θ*_{1} was fixed either at 0°, 90°, 45°, or −45°. Assuming that the polarization state is in the state (6), the probability *p*(*θ*_{1}*,θ*_{2}) to find the photons in the linear polarizations of *θ*_{1} and *θ*_{2} is given by

*θ*

_{1}=0° or 90°, we obtain For

*θ*

_{1}=45° or −45°,

*θ*

_{1}=0° or 90°, we observed the fringe visibility (

*V*) of almost unity, as expected from the theory. In the case of

*θ*

_{1}=45° or −45°, however, it is expected from Eq. (11) that

*V*= Re(

*ν*). In practice, we obtained a degraded visibility

*V*= 0.90 from Fig. 5(b).

To obtain the density matrix of the two-photon polarization state, we carried out the correlation measurements of 16 polarization combinations including not only linear but also circular polarizations. Then we reconstructed the density matrix applying the maximum-likelihood estimation [17] to the measured data. Figure 6 shows the density matrix of the two-photon polarization state thus obtained. The fidelity *F* of the reconstructed density matrix *ρ* to the ideal Bell state

*C*) [18, 19] and the entanglement of formation (

*E*) [19,20] obtained from

_{F}*ρ*were 0.93 and 0.90, respectively, demonstrating that the obtained state had a high degree of entanglement with slight degradation. Thus, our source demonstrated the generation of a highly entangled polarization state of the photon pairs.

## 4. Discussion

Although we have obtained highly entangled photons as described above, it is valuable to discuss the theoretical degree of entanglement expected from our scheme. As given in Eqs. (6) and (7), the polarization state we generated would be the maximally entangled state only if |*ϕ _{a}*〉 = |

*ϕ*〉. However, the longitudinal walk-off, i.e., the temporal separation between orthogonally polarized photon wave packets, is inherent to all SPDC schemes using Type-II phase matching. Since the amount of walk-off is proportional to the photons’ travel length in the crystal, it labels the position where the twin photons are created. This effect produces the change in |

_{b}*ϕ*〉 and |

_{a}*ϕ*〉, and thus establishes distinguishability between the photons created in the poling areas

_{b}*a*and

*b*in our crystal. Compensation for this effect is essential for good entanglement. Thus far, two methods have been used to compensate for the longitudinal walk-off effect. One is to use a polarization-division Michelson interferometer [21], as in the present experiment. The other is to use a crystal compensator, which has half the length and a 90° rotation of the crystal orientation with respect to those of the SPDC crystal [4, 6].

We first examine the interferometer method by evaluating *v* in Eq. (7). The phase *φ* is given in Eq. (8) and 〈*ϕ _{a}*|

*ϕ*〉 can be calculated as follows [11, 12]. In our calculation, we assumed a continuous pump because the pump pulse width we used was sufficiently longer than the photons’ traveling time across the crystal. In this assumption, 〈

_{b}*ϕ*|

_{a}*ϕ*〉 is given by

_{b}*t*

_{−}=

*t*

_{1}−

*t*

_{2}, and

*t*

_{1}(

*t*

_{2}) is the arrival time of the photon at detector D1 (D2). The wave packet functions

*ϕ*(

_{a}*t*

_{−}) and

*ϕ*(

_{b}*t*

_{−}) are given by

*t*′ = Δ

*l/c*. The parameters

*D*and

_{a}*D*are defined as

_{b}*u*(

_{o}*ω*) and

*u*(

_{e}*ω*) are the group velocities of the o-ray and e-ray having frequency

*ω*, respectively. In our case, both

*D*and

_{a}*D*have positive values and 0 <

_{b}*D*<

_{a}*D*. Figure 7 is the plot of 〈

_{b}*ϕ*|

_{a}*ϕ*〉 as a function of

_{b}*t*′ when

*L*=

_{a}*L*=

_{b}*L*. We see that 〈

*ϕ*|

_{a}*ϕ*〉 exhibits the highest value within the range 2

_{b}*D*≤ 2

_{a}L*t*′ ≤ (

*D*+

_{a}*D*)

_{b}*L*. We can adjust Δ

*l*of the interferometer in order to achieve this condition. Note that 〈

*ϕ*|

_{a}*ϕ*〉

_{b}_{max}= 1 when

*ω*

_{1}=

*ω*

_{2}. In our case, since

*ω*

_{1}≠

*ω*

_{2}, we obtain 〈

*ϕ*|

_{a}*ϕ*〉

_{b}_{max}= 0.954. This is the expected upper limit of the visibility of our scheme. From this value, we expect the fidelity of our state to be

*F*= 0.977. The observed fringe visibility (

*V*= 0.90) and fidelity (

*F*= 0.94) were slightly lower than the expected values. This degradation might originate from a slight misalignment of the interferometer and from a slight deviation of the phase

*φ*from 0.

A crystal compensator is another powerful and commonly used device for obtaining desirable entanglement when signal and idler photons are degenerate in frequency [4, 6]. When the photons are non-degenerate, as in our case, the compensator crystal adds a phase change between the two terms in the state (3) as

*L*) of the compensator crystal. However, unlike the case of the interferometer we used, it is difficult to adjust the phase by using the crystal compensator alone. The wave-packet functions

_{c}*ϕ*(

_{a}*t*

_{−}) and

*ϕ*(

_{b}*t*

_{−}) are given by

*L*=

_{a}*L*=

_{b}*L*, we again obtain Eq. (19) for the range 2

*D*≤ (

_{a}L*D*+

_{a}*D*)

_{b}*L*≤ (

_{c}*D*+

_{a}*D*)

_{b}*L*. Thus, when

*L*=

_{c}*L*, we can achieve the same degrees of visibility and fidelity as in the case of the interferometer we used. In applications where the phase-control ability is not important, the use of a crystal compensator would be a good option.

## 5. Conclusion

We proposed a simple but deterministic scheme for generating polarization-entangled photons using type-II two-period QPM-SPDC that produces non-degenerate, entangled photon pairs at a telecommunication wavelength. Using a PPLN crystal having two poling periods, we demonstrate a scheme in which highly entangled photon pairs are generated. We showed that the generated state had high visibility in the polarization correlation measurement and high fidelity to the ideally entangled state, although they were perfect in neither theory nor practice. We note that our two-period device is fabricated on a single crystal chip. Thus the scheme we proposed is readily applicable to waveguide devices, from which we expect to obtain the much higher generation rates required for practical applications.

## Acknowledgments

This work was supported by a Grant-in-Aid for Creative Scientific Research (17GS1204) from the Japan Society for the Promotion of Science.

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