1. Introduction

Entanglement between correlated multiparticle quantum systems is an important resource not only for studying essential properties of quantum systems but also for implementing real-world photonic quantum information technologies [1,2] including long-distance quantum teleportation [3] and entanglement swapping [4]. The practical implementation of photonic quantum information networks strongly rely on entanglement of distant photons that have never interacted or come from a common source [5–8]. Precise mode matching between two photons that have never correlated in any degrees of freedom is a key element in realizing entanglement swapping through the Bell-state measurement. The most fundamental physical concept behind these techniques is a simple two-photon quantum interference effect, the so-called Hong-Ou-Mandel (HOM) effect [9] that is observed with two independent single photons from distinct sources. In the case of a long-distance fiber-based quantum communication channel involving correlated single photons, the observation of the HOM fringe with high stability is difficult because a single photon of a two-photon wavepacket is considered to have a very short coherence length determined by the spectral bandwidth [10]. To overcome this issue, the optical path-lengths have to be actively stabilized within a single photon wavepacket [11–15]. Furthermore, multi-step control sequences should be employed to compensate the fiber path-length difference due to thermally induced high/low-speed longitudinal expansion of the single-mode fiber core. In this paper, we present a method to stabilize a fiber-optic HOM interferometer actively by employing two-step optical length-control sequences. Using this stabilization method we performed the HOM two-photon interference experiment to confirm that the applied method allows us to achieve a highly stable interference fringe in a fiber-optic interferometer with an arm length of 6 km.

2. Experimental setup

Figure 1 shows the experimental setup used for demonstrating the proposed two-step active stabilization method and performing a quantum interference experiment using correlated photons in the 6-km-long fiber-optic HOM interferometer. To generate correlated photons using spontaneous parametric down-conversion (SPDC), laser pulses with a wavelength of 775 nm, pulse duration of 1.6 ps, and repetition rate of 75 MHz are emitted by a mode-locked Ti:Sapphire laser and used to pump an 8-mm-thick type-I beta barium borate (BBO) crystal. The signal and idler photons with the same polarization are generated in well-defined spatiotemporal modes through type-I noncollinear phase matching. The positions of fiber-optic collimators are accurately aligned for the optimum collection of the down-converted light in the 1.5 μm wavelength band. The details of the two-photon generation and collection procedures are similar to those described in [16,17]. The signal and idler photons coupled with standard single-mode fibers are directed to a 50/50 fiber-optic beam splitter (BS). The BS removes spatiotemporal distinguishability by mixing the signal with idler photons. The temporal and polarization modes also have to be adjusted for the complete indistinguishability between two photons entering the BS. Optical delay lines are used to balance the optical path lengths between the crystal and the BS, and fiber optic polarization controllers (PCs) are used to match the polarization modes of the propagating photons. One of the optical delay lines is composed of a motorized actuator (ODL1) and a PZT-based fiber stretcher (FST) which is used for stabilizing the interferometer and scanning the optical path-length difference.

 

Fig. 1 Schematic of the experimental setup. BS, 50/50 fiber-optic beam splitter; L, spherical lens; FC, fiber collimator; PC, polarization controller; SMF, single-mode fiber; ODL1 and ODL2, optical delay lines; FST, fiber stretcher; WDM1 and WDM2, 1480/1550 nm wavelength-division multiplexing (WDM) filters; BPF, band pass filter (1 nm); D1 and D2, single photon detectors; D3, linear photo detector; &, coincidence electronics.

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An additional BS and PCs are used to configure a Mach-Zehnder interferometer, which is utilized for stabilizing the fiber-optic interferometer with long arms. The HOM and Mach-Zehnder interferometers are combined using wavelength division multiplexing (WDM) filters. Reference light with a wavelength of 1480 nm, which is emitted by a tunable laser source with a linewidth of 100 kHz, enters the Mach-Zehnder interferometer after the attenuation of its power up to −35.4 dBm using an optical variable attenuator. Filters WDM1 and WDM2 are 1480/1550 nm WDM filters used for (de-) multiplexing the reference light with the down-converted two photons. WDM1 is a single 1480/1550 nm WDM filter, and WDM2 is composed of two 1480/1550 nm WDM filters cascaded to obtain sufficient isolation between the reference and down-converted photons. Additionally, the down-converted photons are filtered using 1 nm band-pass filters (BPFs) centered at 1550 nm. After passing through the BPFs, both signal and idler photons are detected using Geiger-mode single-photon detectors D1 and D2. The single photon detectors are triggered at a rate of 4 MHz after lowering the pump pulse rate of 75 MHz.

The lengths of the interferometer arms are approximately 6 km. Therefore, the difference in the photon flight time, which can be represented by the effective path-length difference between the two arms, can change mainly because of variation in the refractive index and thermal expansion of the optical fiber caused by various environmental parameters such as air flow and temperature fluctuation. Thus, an interferometer stabilization technique is used to maintain a constant path-length difference during single-photon counting required for the analysis of two-photon interference. The two 6 km fiber spools were placed inside a polystyrene box to avoid fast thermal fluctuations caused by ambient air flows in our laboratory environment. Additional details of the stabilization technique are described in our previous work [12]. Contrary to the previous work, a continuous wave tunable laser operating at 1480 nm is used as a source of the reference light and an FPGA chip-based analog voltage in-out card (NI-PCI 7831R) is used in feedforward/feedback control systems operating at a loop rate of 40 kHz. The FST is operated at an applied voltage up to ± 10 V. Half-wavelength voltage $V_{\pi }$ for the reference light is 116.3 mV. Thus, the optical path delay range through the FST is approximately 87 μm, which is calculated considering the refractive index of a single-mode fiber.

In general, for the measurement of the HOM dip, coincidence counting should be carried out by scanning the optical path-length difference between the two interferometer arms. The time of coincidence counting at each position during the scanning should be long enough to obtain a sufficient signal-to-noise ratio, which is mainly dependent on the coincidence counting rate per unit time. In this research, the recording time at each position was measured out to be longer than 10 minutes considering the counting time of 200 s and six repetitions of the counting, thus the total time required for the full scanning in the dip measurement was expected to be at least a few hours. Additionally, the coherence length of the down-converted photons filtered through the 1 nm BPF was of the order of millimeters. Therefore, it was impossible to complete the stabilization and the position scanning FST only. To obtain long-time stabilization and a wide range of the optical path delay, a motorized optical delay line (ODL1) was used in this experiment. The optical delay range of ODL1 was 25 mm in free space, and the translation speed was set to 8 μm/s. When the FST was driven along with ODL1 for long-term stabilization and the voltage applied to the FST was about to break through specific voltage values near the voltage limits of ± 10 V, the delay position of ODL1 was changed so that the applied voltage would be near the center of the voltage range of the FST. Thus, the optical path-length difference was maintained in a constant state. Because the translation speed of ODL1 was very low compared to the speed of the optical fiber displacement caused by the FST, the stabilization was achieved by the combined use of the FST and ODL1.

3. Experimental results and analysis

Prior to the HOM dip measurement, the fluctuation of the relative optical path-length difference between the two 6-km-long Mach-Zehnder interferometer arms was evaluated as a function of time in our laboratory environment. Figure 2(a) shows the optical power monitored at the one output port of the Mach-Zehnder interferometer. The result indicates that the interferometer was well stabilized during drift monitoring path-length difference. Each data point was obtained by monitoring the voltage applied to the FST for stabilization with a sampling period of 50 ms. The applied voltage values were converted into the fiber displacement considering $V_{\pi }$ and the optical fiber refractive index as shown in Fig. 2(b). When this monitoring was carried out, the delay position of ODL1 was fixed without changing. Figure 2(c) shows the similar monitoring result obtained through active tracking, which was carried out for about 300 min. with a sampling period of 10 s using FST and ODL1 simultaneously. For long-term monitoring, the optical path-length difference was calculated from the voltage applied to the FST and the delay position of ODL1. As can be seen from the measurement results, the relative optical path-length difference can be generated to be of the order of several hundred micrometers without the stabilization system for 5 h in our laboratory environment, which was much longer than the delay range of the FST used in these works.

 

Fig. 2 Fluctuation of the relative optical path-length difference between the two interferometer arms with lengths of 6 km. (a) The optical power measured at the one output port of the Mach-Zehnder interferometer. (b) The fiber displacement for short-term and (c) long-term monitoring.

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Optical polarization alignment was also one of the important parameters in the HOM dip measurement for the interferometer with long arms because the mismatch of the polarization states degrades indistinguishability between two photons entering the BS of the HOM interferometer. Long-term polarization state fluctuation for one of the interferometer arms with lengths of 6 km was estimated in our laboratory environment using an amplified spontaneous emission (ASE) source, in-line polarizer, and polarizing beam splitter (PBS). The ASE source emission was polarized using the in-line polarizer. Then the source injected into one of the Mach-Zehnder input ports. One of the interferometer arms was blocked. Afterwards, one of the BS output ports located at the end of the interferometer was connected to the PBS input port. The two output ports of the PBS were connected to the two single-photon detectors for long-term monitoring of count rate fluctuation. For the polarization alignment, the count rate of the single-photon detector detecting vertically polarized photons was minimized possibly by adjusting the polarization controllers manually. A polarization extinction ratio of approximately 27 dB was obtained excluding the dark count rates of the single-photon detectors. The long-term monitoring result of the polarization alignment showed a slow drift of the polarization state along time, and the extinction ratio decreased to 20 dB after 8 h. However, it was concluded that the aforementioned polarization condition was sufficient for observing the HOM dip considering the coincidence counting rate and the measurement time required for the full scanning.

Finally, we performed the HOM two-photon interference experiment in a fiber-optic interferometer with arm lengths of 6 km employing our active stabilization method. For the HOM dip measurement, the scanning step was set to 240 μm, which was longer than the delay range of the FST. The scanning step was obtained by 12 repetitions of 20 μm (hereafter, this 20 μm translation is called ‘the unit step’ translation) translations. First, the voltage applied to the FST was increased by the voltage step corresponding to the unit step translation without the movement of ODL1 resulting in an instant increase in the the path-length difference of 20 μm. Secondly, the delay position of ODL1 was slowly changed until the voltage applied to the FST became near the center position. These two processes were repeated 12 times. The optical path-length difference was maintained by the stabilization system during the time in which the delay position of ODL1 was changed. To the contrary, the stabilization system was inactive during the instant time when the voltage applied to the FST led to the increase in the path-length difference by the unit step. However, this inactive time was insignificant because it was equal to a feedback loop period of 25 μs that was short enough to neglect optical path-length fluctuations. A small position error could occur when the stabilization changed from the inactive to active states after the unit step translation. According to our feedback control algorithm, the maximal position error due to the transition from the inactive to active states was 3/4 of the reference light wavelength. Therefore, the total 12 unit step translation could cause a maximal position error of approximately 9 μm. However, this error was believed not to be critical for our dip measurement because it was quite small compared to a scanning step of 240 μm. Figure 3 shows the measured coincidence count as a function of the optical path-length difference. Each data point was obtained by measuring the coincidence counts for 200 s and six repetitions. The circle and square symbols represent the measured average coincidences and accidentals, respectively. The random error bars at each data point represent the standard deviations for six repetitions. The coincidence counting probability is given by $P\propto 1-Vf(\delta \tau )$, where V is the fringe visibility and $\delta \tau$ is the optical time delay between the two paths from the crystal to the BS. Function $f(\delta \tau )$ is determined by the spectral properties of the detected photons, which correspond to the transmission spectrum of the used band-pass filter. The measured coincidence counts were fit to the equation. The raw visibility was found to be 65.4 ± 1%, while the net visibility obtained with subtracting the accidental coincidences was 90.7 ± 2%.

 

Fig. 3 Two-photon interference fringe measured as a function of the optical path-length difference. The circle and square symbols represent the measured coincidences and accidentals, respectively.

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When the reference light was used for the interferometer stabilization as in this measurement, the crosstalk due to Raman scattering could generate noisy counts of the single-photon detectors. Therefore, the Raman scattering effect was also estimated by comparing the count rates with and without the use of the reference light. When operated at a gate rate of 4 MHz with a detection efficiency of 10% and a gate width of 5 ns, the dark count rates of single-photon detectors D1 and D2 were 139 Hz and 136 Hz, respectively. The single-photon counting rates after the HOM interferometer with arm lengths of 6 km were measured as 579 Hz and 567 Hz without using the reference light. They were measured at the plateau position of the dip pattern to avoid the interference effect. When a similar measurement of the single count rates using the reference light was carried out, the counting rates of D1 and D2 were 589 Hz and 577 Hz, respectively. Although there was a slight increase in the single count rates measured using the reference light, the difference in the accidental coincidence counts calculated with and without the implementation of the reference light was less than 1 count for a recording time of 200 s. Thus the crosstalk due to the reference light was neglected.

4. Conclusion

In conclusion, we presented a two-step active stabilization method to observe a stable two-photon interference fringe using a fiber-optic HOM interferometer with arm lengths of 6 km. Large-scale feedforward and small-scale feedback active control sequences were applied to obtain a highly stable HOM interference fringe. For the HOM dip measurement, an interferometer stabilization technique using a PZT-based fiber stretcher was applied and environmental effects on interference were evaluated in our laboratory environment. Using our stabilization system, we successfully obtained two-photon interference fringes for a long armed fiber-optic HOM interferometer with raw and net visibilities of 65.4% and 90.7%, respectively.