1. Introduction

Entangled photons are very important resources in quantum information systems such as quantum key distribution (QKD) [1][2] and quantum repeaters [3]. Spontaneous four-wave mixing (SFWM) in optical fiber is a promising candidate for a quantum correlated photon-pair source [4]–[10], because photons generated from such sources can be coupled into transmission fiber very efficiently. For long-distance quantum communication over optical fiber networks, a photon-pair source for the 1.5-μm band is especially important. Photon pairs can be generated in the 1.5 μm band by using SFWM in a dispersion shifted fiber (DSF) with the pump wavelength set close to the zero dispersion wavelength of the DSF [4][5][7][9]. This technique has been used for the successful generation of polarization [11][12] and time-bin entanglement [13] in the 1.5-μm band. Furthermore, the transmission of these entangled photons over standard single-mode fiber (SMF) has also been reported [11][13][14], which means that the generated photons were not seriously affected by the chromatic dispersion of the SMF. This is because the photon-pair generation efficiency per bandwidth was so high with these photon-pair sources that it was possible to set the bandwidth of the photon pairs at a very small value.

Although these fiber-based photon-pair sources have several merits as described above, a serious problem has been reported, namely noise photons generated by spontaneous Raman scattering [5][7][9][11][12][13]. The pump light for the SFWM also works as a pump for spontaneous Raman scattering, which generates Stokes and anti-Stokes photons that have longer and shorter wavelengths than the pump wavelength, respectively. Consequently, accidental coincidences are caused by the Stokes and anti-Stokes photons whose wavelengths coincide with those of the idler and signal channels of photon pairs generated by SFWM. Such accidental coincidences degrade the correlation characteristics of the photon-pair source. Although the use of a polarization filter to eliminate noise photons whose polarization is orthogonal to those of the photon pairs can improve the correlation characteristics, the ratio of true coincidence to accidental coincidence was at most 10 with an average photo-pair number of 0.04 per pulse [7]. These noise photons pose serious problems for quantum communication systems. For example, accidental coincidences cause bit errors in a QKD system using entangled photons. Therefore, it is very important to suppress the noise photons generated by spontaneous Raman scattering.

In this paper, we report the quantum correlation characteristics of photon pairs generated in a DSF cooled with liquid nitrogen. With this technique, noise photons were significantly suppressed and the ratio of true to accidental coincidence was improved to ~30. The organization of this paper is as follows: section 2 provides a brief explanation of the theoretical background, section 3 details the experiment, and section 4 forms the conclusion.

2. Theory

2.1. Spontaneous four-wave mixing

SFWM is a third-order optical nonlinear process in which two pump photons are annihilated and a signal-idler photon pair is created at the same time. We consider a partially degenerated case, where the optical frequencies of the two pump photons are the same, as in the case of the experiment described in section 3. Then, the pump photon, ω_p, signal photon, ω_s, and idler photon, ω_i frequencies satisfy the following relationship.

Here, a generated photon with a larger frequency is called a signal photon and that with a smaller frequency is called an idler photon (i.e. ω_s > ω_i). When the pump power is small and the power-dependent change in the refractive index is negligible, photon pairs are generated efficiently when the following phase-matching condition is satisfied:

Here, k_p, k_s and k_i are the wave numbers of the pump, signal and idler photons, respectively. We can achieve this phase-matching condition in the 1.5-μm band by using a DSF and setting the pump wavelength at the zero-dispersion wavelength.

2.2. Spontaneous Raman scattering

Spontaneous Raman scattering is a process in which spontaneous photons are generated by a nonlinear interaction between pump photons and phonons [15]. Spontaneous photons whose frequencies are smaller (larger) than those of the pump photons are called Stokes (anti-Stokes) photons.

When pump depletion is negligible, the number of Stokes photons generated by spontaneous Raman scattering in a fiber with a loss coefficient α obeys the following differential equation [15][16].

where P ₀ and P ₁ are the probabilities of finding a phonon in the ground state and excited states, respectively. g is a gain coefficient that is proportional to (P ₀ - P ₁)n_p, where n_p denotes the number of pump photons at the fiber input.

For anti-Stokes photon number n_as, the equation becomes

In a thermal equilibrium, P ₁/P ₀ follows the Boltzman distribution as

where h, v, k_B and T are Planck constant, phonon frequency, Boltzman constant, and temperature, respectively. From Eqs. (3)–(5), we obtain ns and nas as functions of T, which are given by

where L denotes the fiber length. If the fiber loss coefficient is independent of temperature, the ratio of the Stokes photon numbers at liquid nitrogen (77 K) and room temperature (293 K) at the same pump power is calculated to be n_s(77)/n_s(293) = 0.29, using Eqs. (6) and (7). Similarly, that of anti-Stokes photon numbers is n_as(77)/n_as(293) = 0.24. Here, we assumed ν = 400 GHz, which corresponds to the frequency separation between pump and signal/idler channels in the experiment. Thus, we can suppress the number of noise photons by a factor of ~3-4 by cooling the fiber with liquid nitrogen.

In the experiment, we observed a 0.9-dB increase in fiber loss when we cooled the fiber. Eqs. (6) and (7) imply that the experimentally observed ratio will be the product of the above calculated ratio and the loss increase. Therefore, we can expect that n_s(77)/n_s(293) = 0.23 and n_as(77)/n_as(293) = 0.19 will be observed experimentally.

2.3. Time correlation measurement of noisy photon pairs with time interval analyzer

Here we develop a theory related to the time correlation measurement of photon pairs with noise photons. We assume that entangled photon pairs are generated using a pulsed pump. In time interval analysis, we measure the coincidence rates at matched and un-matched slots, which we refer to as R_m and R_um, respectively. A coincidence in a matched slot is a “true coincidence”, which we define as a coincidence caused by photons generated by the same pump pulse. Please note that our definition means that “true coincidences” include accidental coincidences such as those caused by photons from two different pairs generated with the same pump pulse or by noise photons generated with the same pump pulse. A coincidence in an un-matched slot is an accidental coincidence caused by photons in different pulses and it appears at different time instances. Therefore, the ratio C = R_m/R_um gives the ratio of true coincidences to accidental coincidences, which is a good index with which to evaluate the quality of a time correlation: C > 1 implies the existence of a time correlation.

If there are both correlated photon pairs and noise photons in the pulse, the average count rates per pulse for signal (c_s) and idler channels (c_i) are expressed as

where μ_c, μ_xn, α_x and d_x are the average number of quantum correlated photon pairs per pulse, the average number of noise photons per pulse for channel x, the transmittance for channel x, and the dark count rate for channel x, with x = s (signal) or i (idler), respectively. If we use signal and idler clicks as start and stop events, respectively, the ratio of the counts caused by correlated photons in start events, p_c is given by

The coincidence rate in the matched slot R_m is given by

The first and second terms in parenthesis correspond to a coincidence resulting from correlated photon pairs and an accidental coincidence, respectively. On the other hand, the coincidence rate caused by accidental coincidence in the un-matched slot R_um is expressed as

Then, the ratio of true coincidence to accidental coincidence C is given by

From experimentally obtained C, c_s, c_i, α_s and α_i values, we can calculate μ_c, μ_sn and μ_m using Eqs. (8), (9) and (13), so that we can determine the portions of correlated and noise photons.

C provides a good index for estimating the performance of a photon-pair source in quantum communication systems. For example, the bit error rate ε of a QKD system using entangled photon pairs is approximated as

The above equation is accurate when errors that arise due to imperfections in the optical instruments used for the QKD system are negligible. Equation (14) shows that, for example, C = 10 is needed in order to obtain a 5 % error rate.

3. Experiment

 

Fig. 1. Experimental setup. PC: polarization controller, D: photon counter.

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Figure 1 shows our experimental setup. A continuous lightwave at a wavelength of 1551.1 nm was modulated into pulses with a 100-ps width and a 100-MHz repetition frequency using a LiNbO₃ intensity modulator (IM). The pulses were amplified by an erbium-doped fiber amplifier (EDFA), filtered to eliminate amplified spontaneous emission noise from the EDFA, and then launched into a Fujikura DSF placed in a styrofoam container, after polarization adjustment. Liquid nitrogen was poured into the styrofoam container when cooling the DSF. The length and zero-dispersion wavelength of the DSF were 500 m and 1551.1 nm, respectively. To reduce the bending stress, the DSF was formed into a loose coil of ~30-cm diameter without using a bobbin. In addition, the DSF was treated with talcum powder so that the fiber coatings did not stick to each other. The photons from the DSF were input into a polarizer to eliminate spontaneous Raman photons whose polarization was orthogonal to those of correlated photon pairs [7]. The photons were then input into fiber-Bragg gratings (FBG) to suppress the pump photons, and subsequently input into an arrayed waveguide grating (AWG) to separate signal and idler photons. The frequency separation of the signal/idler channels and pump was ±400 GHz. The signal and idler photons were input into optical bandpass filters (BPF) to further reduce the pump photons, and detected by photon counters that used InGaAs APDs operated in a gated mode with a 4-MHz frequency. The photon counters for the signal and idler channels were used as start and stop pulses for a time interval analyzer (TIA), respectively. The losses of the setup and the characteristics of the photon counters are summarized in Table I. The DSF loss, which includes ~0.5 dB connector loss, was measured and found to be 0.6 dB and 1.5 dB at room and liquid nitrogen temperatures, respectively. This 0.9-dB increase in the DSF loss may have been caused by micro-bending of the fiber resulting from the difference between the thermal expansion coefficients of the fiber and its coating material.

We changed the peak power of the pump pulse and measured the C value of the photon pairs generated using DSF at room temperature and that in liquid nitrogen. Figure 2 shows C as a function of average number of idler photons μ_i{= μ_c + μ_in). It is clear that the C value was improved significantly by cooling the DSF. At room temperature, C was at most ~9 with μ_i ≃ 0.01. On the other hand, C reached 28 with μ_i ≃ 0.02 when the fiber was cooled. The dotted curve in Fig. 2 shows the C value calculated on the assumption that there are no noise photons: i.e. μ_s = μ_i = μ_c and μ_sn = μ_in = 0. This result suggests that C for the cooled fiber almost coincides with that for ideal photon pairs if μ_i ≥ 0.15, which means that the number of noise photons is very small compared with the number of correlated photons in this regime.

Table I. Loss after DSF and characteristics of photon counters.

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Fig. 2. C value as a function of number of idler photons per pulse. Squares: cooled, x symbols: uncooled. The dotted line shows C when there are no noise photons.

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We then calculated the average number of correlated photons and noise photons for each peak pump power using Eq. (13) and the C values obtained in the experiment. The average number of correlated photon pairs per pulse as a function of peak pump power is shown in Fig. 3. Thus, the numbers of correlated photons from both the cooled and uncooled DSF increased quadratically with the pump power, as expected. The loss increase induced by cooling the fiber probably resulted in a ~1-dB decrease in the number of photon pairs from the cooled fiber compared with that from the uncooled fiber.

Figures 4(a) and (b) show the average number of noise photons as a function of peak pump power. In these plots, we limit the peak pump power to 0.2 W, because the correlated photons are dominant for peak pump powers that are larger than 0.2 W and so large fluctuations were observed in the number of noise photons calculated from Eqs. (8), (9) and (13) in this regime. From the linear fitting in these figures, the average numbers of Stokes and anti-Stokes noise photons from cooled DSF were estimated to be about 0.26 and 0.24 of those from uncooled DSF, respectively. This result agreed well with the calculated results shown in the previous section.

 

Fig. 3. Number of quantum correlated photon pairs per pulse as a function of pump peak power. Squares: cooled, x symbols: uncooled.

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Fig. 4. Number of noise photons per pulse as a function of pump peak power for (a) Stokes photons (idler channel) and (b) anti-Stokes photons (signal channel). Squares: cooled, x symbols: uncooled.

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4. Conclusion

We described the characteristics of quantum correlated photon pairs generated from a DSF cooled by liquid nitrogen. With a cooled DSF, the average numbers of noise photons generated through the spontaneous Raman scattering process were reduced by a factor of ~3-4 compared with the numbers generated with an uncooled DSF. As a result, the ratio of true coincidence counts to accidental coincidence counts reached ~30. The obtained result shows that a photon-pair source using SFWM in a cooled DSF can be useful for quantum communication systems such as entanglement-based QKD.