1. Introduction

Quantum entangled photon-pairs are key element to realize various applications using quantum information and communication technologies (QICTs) such as quantum key distribution for secure communications. Parametric fluorescence by using spontaneous parametric down conversion (SPDC) in a second-order (χ⁽²⁾) optical nonlinear material and spontaneous four-wave mixing (SFWM) in a third-order (χ⁽³⁾) optical nonlinear material is useful and realistic tool to generate the entangled photon-pairs. It has been used in a lot of experimental studies concerning the QICT applications using the entangled photon-pairs.

Maximum entangled state, Bell state, of the polarization entanglement is expressed by one of the following four quantum states.

Most of the experimental studies in the polarization-entangled photon-pair sources have focused on the realization of one of the Bell states. For example, photon-pair sources based on the SFWM in optical fibers [1–4] and on the SPDC in type-I quasi-phase-matched (QPM) –χ⁽²⁾ devices using LiNbO₃, KTP and etc. [5–11] are generally used to achieve the $|{\Phi }^{(\pm )}〉$, whereas the photon-pair sources using type-II QPM-devices [10, 11] are used to generate the $|{\Psi }^{(\pm )}〉$.

Although there are a lot of studies in the realization of the polarization-entangled photon-pair sources, there are few reports dealing with the Bell states engineering, especially the realization of all Bell states in a single photon-pair source. Lee et al. have reported the entangled photon-pair source with variable Bell states using the SFWM in optical fibers [12]. This was achieved with help of external polarization controllers. A polarization controller was used to adjust the optical phase difference between the ${|H〉}_s{|H〉}_i$and the ${|V〉}_s{|V〉}_i$ states by changing the optical phase difference between two orthogonally polarized pump photons. This makes it possible to exchange between the $|{\Phi }^{(+)}〉$ and the $|{\Phi }^{(-)}〉$. Inserting a half-wave plate (HWP) at either output of the signal photons or the idler photons makes it possible to rotate the polarization of the signal (idler) photons by 90 degree, resulting in exchange between the $|{\Phi }^{(\pm )}〉$ and the $|{\Psi }^{(\pm )}〉$ states.

Although all-fiber-based entangled photon-pair source has a merit that it can be achieved by using reliable and low-cost optical resources compatible to classical fiber-optic telecommunications, it has a serious drawback concerning noise photons by spontaneous Raman scattering. Coincidence-to-accidental ratio (CAR), which is a powerful tool to estimate how many uncorrelated noise photons are generated together with the needed correlated photon-pairs [3], is typically much less than 100 in the SFWM-based system using optical fibers. The CAR can be dramatically improved when the fiber was cooled down to liquid nitrogen temperature [3], and sometimes to liquid helium temperature (4 K) [4]. Such a need for extremely low temperature operation is quite non-realistic for practical uses, while the CAR of multi wavelength-band system using the SPDC could greatly exceed 1000 even at room temperature operation [8, 10].

Recently we have reported generation of high-purity polarization-entangled photon-pairs using cascaded χ ⁽²⁾ processes, the second harmonic generation (SHG) and the following SPDC (c-SHG/SPDC process), in a periodically poled LiNbO₃ (PPLN) ridge-waveguide device [13]. Although the c-SHG/SPDC process mimics the SFWM in a pure χ ⁽³⁾ material, the efficiency is greatly effective than the pure SFWM process. This leads to very low contribution of the Raman noise photons, even though the pump photon, the Raman photon, and the photon-pairs are in the same wavelength-band. We have reported the CAR of higher than 4000 and the visibilities of higher than 99% (without subtraction of accidental counts) by using the c-SHG/SPDC-based polarization entangled photon-pair source [13].

In this paper we report the polarization entangled photon-pair source with variable Bell states using the c-SHG/SPDC in a PPLN ridge-waveguide device. The system basically consisted of a Sagnac loop configuration including the PPLN device, the optical phase bias compensator (OPBC) to achieve the maximum entangle states, and a half-wave plate outside the loop for polarization conversion. We observed clear two-photon interference fringes with visibilities of 96% in all four Bell states. Quantum state tomography and violation of Bell inquiry were also tested. Fidelity and S parameter in inequality of Clauser, Horne, Shimony, and Holt (CHSH) [2, 14] was 97.1% and 2.71, respectively. These values agreed well with the theoretical ones assuming a Poisson distribution as the photon-number statics. Detailed experimental studies on the dependence of the mean number of the photon-pairs revealed that the quantum states were well understood as the Werner state.

2. Operation principle

Figure 1 schematically draws the polarization-entangled photon-pair source used in this work. The basic setup consisted of a Sagnac loop configuration with polarization beam splitter/combiner (PBSC) including the PPLN device and the optical phase bias compensator (OPBC) [13]. All the setup consisted of polarization-maintaining-fiber (PMF)-pigtailed optical components for standard telecommunication uses. The PM pigtail fibers of the optical modules were 90 ° -spliced at the center position to roughly compensate birefringent of the PMFs. Two counterpropagating pump pulses excited the ${|H〉}_s{|H〉}_i$ and the ${|V〉}_s{|V〉}_i$ photon-pairs, respectively, in the fiber loop. The two biphoton states were mixed at the output port of the PBSC, and then the polarization entanglement was achieved.

 

Fig. 1 Polarization-entangled photon-pair source with variable Bell states. PPLN: PPLN device (module). PBSC: polarization beam splitter/combiner. BSC: Babinet-Soleil compensator. FR: Faraday rotator. OPBC: optical phase-bias compensator. The position of the BSC could be precisely shifted by a micrometer.

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The OPBC was installed to adjust optical phase difference between two counterpropagating photon-pairs, ${|H〉}_s{|H〉}_i$ and ${|V〉}_s{|V〉}_i$, in the Sagnac loop. The OPBC consisted of two Faraday rotators (FRs) and a quartz Babinet-Soleil compensator (BSC) located between the Faraday rotators. In the OPBC, the two counterpropagating lights propagating the OPBC undergo orthogonal polarization states only at the BSC. This implies that the two lights undergo a phase difference only given by the BSC. By using the OPBC we can give any optical phase difference between the two counterpropagating photon-pairs. This makes it possible to convert the entanglement states between $|{\Phi }^{(+)}〉$ and $|{\Phi }^{(-)}〉$ (and also between $|{\Psi }^{(+)}〉$ and $|{\Psi }^{(-)}〉$), simultaneously compensating the residual optical phase difference due to residual birefringent of the PMF-pigtailed optical components and the difference of the optical path lengths.

In this work the optical phase shift in the BSC could be accurately tuned by using a micrometer so as to adapt to any optical phase difference. The absolute value of the optical phase shift could be identified by the scale of the micrometer. Mechanical resolution of the micrometer was 0.01 mm, corresponding to approximately 0.004π of the optical phase shift in this work. This value was enough to control the optical phase difference precisely.

After the WDM filter to separate spatially the signal and the idler photons, a HWP (HWP#2) was installed in either output port of the signal photons or the idler photons. The HWP#2 converts the polarization of the idler (signal) from $|H〉$ to $|V〉$ (also $|V〉$ to $|H〉$), leading to conversion from the $|{\Phi }^{(\pm )}〉$ states to the $|{\Psi }^{(\pm )}〉$ states.

Probabilities of excitation of the ${|H〉}_s{|H〉}_i$ and the ${|V〉}_s{|V〉}_i$ states can be tuned by adding another HWP (HWP#1) for the pump pulses before the Sagnac loop and rotating its optical axes (θ_p). This is usable to compensate the polarization-dependent loss of the optical setup.

One of the unique features of this polarization-entangled photon-pair source is using the c-SHG/SPDC process in the PPLN device for the photon-pair generation [13]. Figure 2 schematically illustrates the c-SHG/SPDC process. In the c-SHG/SPDC process, an external pump light first excites second-harmonic generation (SHG) inside the PPLN device, and then the second-harmonic (SH) photon is down-converted into a photon-pair through the following SPDC process via another χ ⁽²⁾ process in the same PPLN device. According to this process, a short-wavelength pump-photon, which should be a seed of the SPDC process, can be also created inside the PPLN device without an external short-wavelength pump light. Since the SH photon is needed only inside the PPLN device, this photon-pair source never needs optical coupling of the short-wavelength lights into and out of the PPLN device, just like the SFWM-based system..

 

Fig. 2 Cascaded SHG/SPDC (c-SHG/SPDC) process in a PPLN device used in this study.

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3. Experimental setup

Figure 3 shows experimental setup. The home-made PPLN device used here was a MgO-doped, LiNbO₃ ridge-waveguide device [15]. Details of the device fabrication can be referred to [15]. The period of the domain inversion was 19.1 μm, corresponding to approximately 1548.6 nm of the QPM wavelength at 25.3 °C in this work.

 

Fig. 3 Experimental setup of polarization entanglement with variable Bell states.

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The PPLN device was then packaged in a fiber-pigtailed optical module with a thermistor, a thermoelectric cooler, and two polarization-maintaining optical fibers (PMFs) for standard telecommunication uses. The device length was 60 mm in this work. The insertion loss of the module was estimated to be approximately 3.0 dB at the 1.5-μm band, including coupling losses to the pigtail fibers and transmission loss of the PPLN waveguide.

The SHG conversion efficiency was estimated to be approximately 700%/W under the QPM condition. The 3-dB bandwidth of the SHG curve was approximately 0.2 nm in terms of the pump wavelength.

All the setup of the polarization entangled photon-pair source consisted of polarization-maintaining-fiber (PMF)-pigtailed optical components for standard telecommunication uses. No special optical components, for example, the PBSC operating both at short and long wavelengths, were needed in this system, verifying the merit of the c-SHG/SPDC-based system considering practical uses.

The pump pulses were generated using a wavelength-tunable external-cavity laser diode and a LiNbO₃ intensity modulator. The pulse repetition rate, the pulse width, and the center wavelength of the pump pulses were 240 MHz, 120 ps, and 1548.66 nm (corresponding to the QPM wavelength), respectively. After amplification by a polarization-maintaining erbium-doped fiber amplifier (PM-EDFA), residual amplified spontaneous emission (ASE) was eliminated by using a narrow-band optical bandpass filter (OBF#1). The pump pulses then passed the HWP#1, the WDM#1, the PBSC, and finally excited the PPLN module bidirectionally. Ratio of the probabilities of the excitation between the ${|H〉}_s{|H〉}_i$ photon-pairs and the ${|V〉}_s{|V〉}_i$ photon-pairs was controlled by rotating the optical axis of the HWP#1 (θ_p). Ideally for the maximum entanglement, the θ_p should be set at 22.5 °. When the θ_p was set at 0 ° (45 °), only one of the polarization states (${|H〉}_s{|H〉}_i$ or ${|V〉}_s{|V〉}_i$) can be selectively excited. This was usable to adjust optical alignment of the system, especially compensation of the polarization rotation during the fiber propagation.

The signal and the idler photons by the c-SHG/SPDC from the fiber loop passed the optical filtering system consisting of an optical low pass filter (LPF) and three-step WDM filters (WDM#1, #2, and #3). The LPF, the WDM#1, and the WDM#2 were mainly used to eliminate the pump light and the SHG light. Then the signal/idler photons were spatially separated by WDM#3 and two optical bandpass filters (OBF#2 and OBF#3). Polarization controllers consisting of a HWP and a quarter-wave plate (QWP) were installed before the photon detectors (D1 and D2) in order to compensate polarization rotation originating in misalignment of the optical axes of the PMF-pigtailed optical components. The HWP#2 for exchange between the $|{\Phi }^{(\pm )}〉$ and the $|{\Psi }^{(\pm )}〉$ states was installed in the output port of the idler photons.

The signal/idler photons were then photon-counted by using InGaAs-APD based single photon detectors (Princeton lightwave benchtop receiver PGA-600HSU) (D1, D2). Two detectors were synchronized to the pump pulses and the detection rate was 40 MHz. The detection efficiencies of both the APDs were estimated to be approximately 20%. The gate width was 1 ns. The dark count rates were approximately 2x10⁻⁶ per pulse for both detectors.

In this work we set the center wavelengths of the signal photons and the idler photons to 1538.8 nm and 1558.66 nm, respectively. Wavelength detuning from the pump wavelength was approximately 10 nm. The peak transmittance was −9.4 dB for the signal photons and −7.9 dB for the idler photons, respectively, including the insertion losses of the polarization controllers and polarizers. The 3-dB bandwidths of the transmittance window were approximately 0.65 nm for both the signal photons and the idler photons.

At the front of the single photon detectors, rotatable polarizers were installed for the polarization correlation measurements of the generated photon-pairs such as two-photon interference fringes, S parameters in inequality of CHSH, and quantum state tomography. Additional sets of a QWP and a HWP were also installed at the front of the polarizers for the quantum state tomography measurements.

4. Experimental results

4.1 Measurements of the $|{\Phi }^{(+)}〉$ state

We characterized the quantum state of our polarization-entangled photon-pair source with variable Bell states by using some quantitative measurements. We first investigated two-photon interference fringes by using polarizers at the front of the single photon detectors (see Fig. 3). We also measured the S parameter of inequality of CHSH [14] and quantum state tomography [16]. The experimental values were compared and discussed.

In the measurement of the two-photon interference fringes, we fixed the polarizer angle of the signal polarizer (θ_s), and measured the coincidence count rates while rotating the polarizer angle of the idler polarizer (θ_i). The mean number of the photon-pair per pulse was fixed to be approximately 0.04 during all the experiments. This was estimated from the single count rate and the total transmittance including the detection efficiency of the photon detectors and the insertion losses of the optical components (see below for details). The averaged pump power (P_ave) was approximately −5.5 dBm per end facet of the PPLN module (totally −2.5 dBm for both facets). The pulse peak power was estimated to be + 9.1 dBm from the experimental conditions.

Figure 4 show the coincidence counts as a function of θ_i at the H/V basis and the diagonal basis (θ_s = 0 ° and 45 °, respectively) when the photon-pair source was set to be the $|{\Phi }^{(+)}〉$. The acquisition time was 15 seconds. The optical phase difference in the OPBC was adjusted so that the coincidence counts were minimized when θ_s = 45 ° and θ_i = −45 °. The setting value of the OPBC remained constant during the experiments. All the data were raw data without subtracting the accidental counts.

 

Fig. 4 Two-photon interference fringes of $|{\Phi }^{(+)}〉$ state at the H/V basis (black) and the diagonal basis (red). Polarizer angle of the signal polarizer (θ_s) were 0 ° (black) and 45 ° (red), respectively. Closed circles are experimental data. The solid curves in the figure are fitting curves assuming ${\cos }^2\left({\theta }_s-{\theta }_i\right)$.

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Clear two-photon interference fringes were measured. The interference fringes were well fitted with ${\cos }^2\left({\theta }_s-{\theta }_i\right)$curves (solid curves in Fig. 4), indicating that the $|{\Phi }^{(+)}〉$ was actually realized in these experiments. The visibilities of the fitted curves were estimated to be 95.82 ± 0.22% in the H/V basis (θ_s = 0 °) and 95.90 ± 0.43% in the diagonal basis (θ_s = 45 °).

Assuming Poisson distribution as photon-number statics, the visibility in the two-photon interference of the polarization entanglement, V, is given by [17],

From comparison between the signal count rates and the coincidence count rates, the total transmittance was estimated to be −19.5 dB for the signal photons (α_s) and −18.9 dB for the idler photons (α_i), respectively. The loss assignment was as follows; −9.4 dB (signal) and −7.9 dB (idler) from the transmittance of the optical filters, the polarization controllers, and the polarizers, −7 dB (20%) from the detection efficiency of the APD, and −1.5 dB from the half of the insertion loss of the PPLN module. The residual loss, approximately −2 dB, was thought to come from filtering effect of the SPDC spectra. While the SPDC spectra from the PPLN module initially had a broad bandwidth exceeding 30nm [13], they were spliced to 0.65 nm by using the OBFs in this work. The d_x was approximately 2x10⁻⁶ per pulse for both detectors.

From the experimental values of the transmittance and the dark counts, the V was estimated to be approximately 96.09% when the mean number of the photon-pairs (µ_c) was 0.04. This value agreed well with the experimental results. This indicates that counting of unwanted photons due to noise photons by spontaneous Raman scattering and also due to misalignment of the measurement system was negligibly small. This also implies again that the OPBC can fully compensate the optical phase difference between the ${|H〉}_s{|H〉}_i$ and the ${|V〉}_s{|V〉}_i$ photon-pairs. The peak coincidence counts were approximately 2500 counts per 15 seconds (170 counts per second). This value was almost the same for both the H/V basis and the diagonal basis.

We also evaluated violation of Bell inequality. S parameter in inequality of CHSH [14] was investigated by measuring the coincidence counts while rotating both the signal and the idler polarizers.

The S parameters are expressed in slightly different forms depending on literature. A typical form of the S parameter in [2] is given by,

To evaluate the S parameters in different Bell states, we should in general use different settings of the polarizer angles for different Bell states. For experimental convenience, we slightly modify the S parameter as,

The measured S parameter was estimated to be 2.705 ± 0.012. Theoretically, the S parameter is related to the visibility (V) by,

Then we measured the quantum state tomography and reconstructed the density matrix. The density matrix was reconstructed from data of 24 combinations in polarization correlation measurements by using maximum likelihood estimation method [16]. This set of data included all the possible cases when the coincidence counts were theoretically maximized and minimized, i.e., combinations of polarization correlations corresponding to $|HH〉$, $|VV〉$, $|DD〉$, $|\overline{D}\overline{D}〉$, $|RL〉$, and $|LR〉$ for the maximum coincidence counts (in contrast, $|HV〉$, $|VH〉$, $|D\overline{D}〉$, $|\overline{D}D〉$, $|RR〉$, and $|LL〉$ for the minimum cases).

The results of the reconstructed density matrix are shown in Fig. 5 . The values of the diagonal elements ($|HH〉〈HH|$and $|VV〉〈VV|$) and offdiagonal elements ($|HH〉〈VV|$ and $|VV〉〈HH|$) were close to 0.5 and they have almost the same values, while those of the other elements are almost zero (less than 0.02) in both the real and the imaginary parts. Thus, we observed that the experimentally reconstructed density matrix closely resembled the expected one, $|{\Phi }^{(+)}〉$.

 

Fig. 5 Reconstructed density matrix of the $|{\Phi }^{(+)}〉$ state. (a) Real part. (b) Imaginary part.

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However, there were some deviations in the reconstructed density matrix from the ideal one. This is mainly because the quantum state of the polarization entangled photon-pair source based on the SPDC is actually not expressed by the Bell states in Eq. (1) and Eq. (2), but by a mixture with some degrees of fully mixed state, the Werner state. This is because the SPDC process always has probability of multi-photon-pair generation. Detailed discussion about this is given in the paragraph 4.3.

From the reconstructed density matrix, the fidelity (F), defined by $F\equiv 〈{\Phi }^{(+)}|\rho |{\Phi }^{(+)}〉$, was estimated to be approximately 97.06%. The purity (P), defined by $P\equiv Tr\left({\rho }^2\right)$, was approximately 94.43%. As discussed later, these values were reasonable considering the probability of the multi-photon-pair generation inherent to the SPDC-based photon-pair source.

4.2 Results of the other Bell states

We carried out a series of the experiments for the other Bell states ($|{\Phi }^{(-)}〉$ and $|{\Psi }^{(\pm )}〉$). The pump power, hence the mean number of the photon-pairs, remained the same during the experiments.

Figure 6 show the results of the two-photon interference fringes. Clear two-photon interference fringes were also measured. The interference fringes were well fitted with ${\cos }^2\left({\theta }_s\mp {\theta }_i+\Delta \theta \right)$curves (solid curves in Fig. 6), where Δθ=0 for the $|{\Phi }^{(\pm )}〉$ and Δθ=π/2 for the $|{\Psi }^{(\pm )}〉$, with almost the same maximum and minimum coincidence counts. This indicates that all the four states were close to the maximum entangled states.

 

Fig. 6 Two-photon interference fringes of (a) $|{\Phi }^{(-)}〉$, (b) $|{\Psi }^{(+)}〉$, and (c) $|{\Psi }^{(-)}〉$ state, respectively. Black: H/V basis. Red: diagonal basis. Polarizer angle of the signal polarizer (θ_s) were 0 ° (black) and 45 ° (red), respectively. Closed circles are experimental data. The solid curves in the figure are fitting curves assuming sinusoidal curves.

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Table 1 shows summary of the experimental results. The measurement values were almost identical for all the Bell states, implying that our polarization-entangled photon-pair source could achieve any Bell states with the similar quality by a simple scheme.

Table 1. Summary of the Experimental Results

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By changing the system parameters, especially the θ_p of the HWP#1 and the optical phase shift in the BSC, a variety of the quantum states can be achieved. Figure 7 shows an example of the two-photon interference fringes when the optical phase difference was set at 0.5π and the quantum state was expressed by $\left({|H〉}_s{|H〉}_i+\exp \left(i\pi /2\right){|V〉}_s{|V〉}_i\right)/\sqrt{2}$. In this case, although a clear fringe curve was measured in the H/V basis, the coincidence counts had almost unchanged values in the diagonal basis while rotating the idler polarizer. This is well understood because the above quantum state is also expressed by $\left({|D〉}_s{|R〉}_i+{|\overline{D}〉}_s{|L〉}_i\right)/\sqrt{2}$. This implies that the coincidence counts should be constant in the diagonal basis, agreeing well with the experimental results.

 

Fig. 7 Two-photon interference fringes when the system was set to be $\left({|H〉}_s{|H〉}_i+\exp \left(i\pi /2\right){|V〉}_s{|V〉}_i\right)/\sqrt{2}$ state. Polarizer angle of the signal polarizer (θ_s) were 0 ° (black) and 45 ° (red), respectively.

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4.3 Detailed investigation in the Werner state

The experimental results in the earlier sections verified that our polarization-entangled photon-pair source can generate the entangled states very close to ideal maximum entangled states. However, there are still some deviations from the ideal ones. A part of this should be owing to some experimental errors from changes in state of polarization of signal/idler photons, appearance of residual optical phase offset, and etc. during the experiments. But more essential origin is that there is probability of multi-photon-pair generation inherent to the SPDC-based photon-pair source. Considering this, quantum state of this photon-pair source should be more properly expressed by the Werner state.

The Werner state is a mixture of a pure entangled state and fully mixed state [18]. The density matrix of the Werner state, ρ_W, of a $\left(1-a\right):a$statistical mixture of the $|{\Phi }^{(+)}〉$ and the fully mixed state is given by [18],

The parameter $a$is simply related to the visibility (V) in the two-photon interference fringe as,

A main feature of the density matrix of the Werner state is that the amounts of diagonal elements$|HH〉〈HH|$and $|VV〉〈VV|$ and offdiagonal elements $|HH〉〈VV|$ and $|VV〉〈HH|$ become smaller while the $|HV〉〈HV|$and $|VH〉〈VH|$ elements appear and become larger, as the visibility decreases. This indicates that fidelity (F) and the other quantitative values to evaluate the degree of the entanglement is dependent on the visibility, hence the mean number of the photon-pairs, in the SPDC-based entangled photon-pair source.

Figure 8 show the reconstructed density matrices of the $|{\Phi }^{(+)}〉$ states as a function of the mean number of the photon-pair (μ_c). The corresponding averaged pump powers per end facet of the PPLN module were also indicated in the figures. The results clearly showed that the $|HV〉〈HV|$and $|VH〉〈VH|$ elements became larger as the mean number of the photon-pairs increased, while the other unwanted offdiagonal elements remained close to zero (less than 0.024) in despite of the changes of the μ_c.

 

Fig. 8 Reconstructed density matrices of the$|{\Phi }^{(+)}〉$ states as a function of the mean number of the photon-pair (μ_c). (a), (e) μ_c=0.00854/pulse. (b), (f) μ_c=0.1098/pulse. (c), (g) μ_c=0.2309/pulse. (d), (h) μ_c=0.4623/pulse. (a)-(d) Real parts. (e)-(h) Imaginary parts.

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We estimated the quantitative values to evaluate the degree of the entanglement (visibility (V), purity (P), and fidelity (F)) from the reconstructed density matrices, and compared them with the theoretical values assuming the Werner state.

From Eq. (8) and Eq. (9), the P and the F are simply related to the V in the Werner state as,

Figure 9 shows the results of the comparison between the experimental values and the theoretical values. The experimental values agreed very well the theoretical values. In the figure, we also showed the fidelity to the Werner state (F_W), defined as $F_W\equiv {\left(Tr\sqrt{\sqrt{{\rho }_W}\rho \sqrt{{\rho }_W}}\right)}^2$ [18], as gray closed squares. The F_W was almost unchanged and showed the values higher than 99% even when the μ_c was changed, while the F to the Bell state was monotonically decreased as the μ_c increased. This experimentally verified that our photon-pair source was actually close to the Werner state.

 

Fig. 9 Dependences of the visibility (V), the purity (P), and the fidelity (F) on the mean number of the photon-pair (μ_c). Black: visibility. Red: purity. Blue: fidelity. Open circles: experimental values from reconstructed density matrices. Solid curves: calculations assuming Werner states. Gray closed squares are fidelities to the Werner state (F_W).

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5. Conclusion

In summary we have achieved the generation of the polarization-entangled photon-pairs with variable Bell states using the c-SHG/SPDC in a PPLN ridge-waveguide device. The performance was evaluated from several quantitative measurements by using the two-photon interference, the inequality of CHSH, and quantum state tomography measurements. The results showed almost the same values in despite of exchange of the quantum states among four different Bell states ($|{\Phi }^{(\pm )}〉$ and $|{\Psi }^{(\pm )}〉$), also remaining the unchanged peak coincidence counts in the two-photon interference measurements. We have obtained high visibilities of 96%, fidelities of 97%, 2.71 of the S parameters in inequality of CHSH, whose values were in good agreement with the theoretical assumption from the mean number of the photon-pairs, 0.04. Detailed experimental investigation in quantum state tomography revealed that the achieved quantum states were well explained by the Werner states. The experimental achievement in this work verified advantage of the c-SHG/SPDC-based system as high-purity photon-pair source with reliable and controllable performance and negligibly small noise photons. This indicates that the c-SHG/SPDC-based system can provide noise-photon-free, high-purity entangled photon-pair source for quantum information and communication technologies.