1. Introduction

Quantum interference [1,2] rests on the concept of superposition of probability amplitudes. The interference patterns involving single photon or the light intensity does not reveal strictly nonclassical phenomena. However, the quantum description and the classical one for this interference differ drastically from each other [3]. More counterintuitive effects emerge with two or more photon interference, most of which are completely incomprehensible with any classical concepts.

The interference of two photons is usually known as Hong-Ou-Mandel (HOM) interference [4]. Typically this interference with a visibility exceeding 0.5 can only be explained by quantum interference of the probability amplitudes of the two-photon events [3,5,6]. HOM interference can be applied in various quantum information processing applications, such as quantum teleportation [7], quantum repeaters [8,9], quantum entanglement swapping [10,11], quantum networks [12], quantum computation [13], and measurement-device-independent quantum key distribution [14]. In most of these applications, interference of photons transmitted through independent channels is required.

The optical fiber is a good candidate, for the spatial mode of the photons can remain relatively stable during transmission in fibers. Indeed, fiber-based repeaterless quantum key distribution (QKD) has been demonstrated on a distance of hundreds of kilometers [15–18]. Several small-scale fiber-based field QKD networks have also been tested under real-world environments [19–21]. However, beyond this length scale, the global-scale quantum communication becomes extremely challenging since the quantum signal in QKD cannot be noiselessly amplified, owing to the quantum non-cloning theorem [22].

Satellite-based quantum communication with free-space links is a more promising solution since the much lower transmission loss and negligible decoherence in space. It has achieved lots of progress very recently [23–33]. Therefore, it is urgent to develop quantum interference in atmospheric channels, such as HOM interference. The main challenge for free-space-based quantum interference is caused by atmospheric turbulence. Specifically, due to atmospheric turbulence, the spatial mode of photons transmitted through a free-space channel will be significantly disturbed, resulting in beam wandering and scintillation in the receiving aperture. Thus, a mixed state in spatial mode will be obtained in the receiver, resulting in degradation in spatial mode indistinguishability.

Here, we report an experimental implementation of HOM interference using free-space channels. Specifically, HOM interference of correlated single-photon pair generated in SPDC process is implemented using a real-world unstable 220 m free-space channel. A visibility of $0.744 \pm 0.013$ is achieved using the SPDC source, well beyond the classical limit of 0.5. Therefore, our results show that the indistinguishability of photons remains even after propagation through free-space channels, making it possible to perform further quantum information processing in realistic situations. The HOM interferometer can be applied to precision ranging. In this demonstration, a time measurement precision of 8.8 fs is obtained, corresponding a ranging precision of 2.6 $\mu$m. Furthermore, considering practical applications, HOM interference between weak coherent states is also demonstrated, and a visibility of $0.428 \pm 0.006$ is observed. With a pulse width of 2 ns, a time measurement precision of 8.6 ps is achieved, and this precision can be improved further with narrower pulses.

2. Experimental implementations

Theoretically, coincidence probability for HOM interference of two ideal single photons at the dip vanishes [4], thus the visibility can exceed the classical limit of 0.5. Generally genuine single photons can be extracted from a quantum dot, or from single photon-pair generated in the SPDC process in a second-order nonlinear crystal. Here, single-photon pairs generated from SPDC process are applied in HOM interference demonstration.

The experimental setup is depicted as Fig. 1. A type-II nonlinear KTiOPO4 (PPKTP) crystal of length 11 mm is pumped by an ultra-violet photon of 405 nm with a linewidth of less than 100 MHz. Then a spectrally degenerated single-photon pair with a central wavelength of 810 nm and a spectral full-width at half-maximum (FWHM) of approximately 1.2 nm, is generated in the SPDC process in PPKTP. The spectral spread of the pumping photon is far below the spectral FWHM of the down-converted photon pair, thus according to energy conservation law, the frequencies of the photon pair are approximately anti-correlated. So that to ensure good frequency indistinguishability between the photon pair, temperature control to the PPKTP crystal with a precision of 0.015 K is applied to achieve spectral degeneracy between the photon pair. Next, the photon pair is separated by a polarization beam splitter.

 

Fig. 1. Experimental setup for HOM interference using single photons. Wavelength-degenerated photon pairs are generated in the SPDC process in a periodically poled KTiOPO4 (PPKTP) crystal. Signal photons are sent through a free-space channel of 220 m, and idler photons are kept local. HOM Interference of the photon pair occurs on the beam splitter. PBS: polarized beam splitter; FPC, fiber polarization controller; BS, beam splitter; TDC, time digital converter.

Download Full Size | PDF

The signal photon is then sent through the real-world free-space channel. In free-space transmission, the signal photon firstly flies 110 m to a flat mirror, and is then reflected back in a slightly different direction to the receiver, thus achieving a total free-space transmission distance of about 220 m. The idler photon is transmitted in the local lab by an optical fiber. After transmission, the signal photon then interferes with the local idler photon on a 50:50 beam splitter, and coincidence measurement is applied with two single photon detectors, and recorded with a time-to-digital converter (TDC).

To achieve high quality HOM interference, it is well known that the interfering photons should be indistinguishable. Specifically, distinguishability in four degrees of freedom of the photons mostly affect the interference visibility, that is, the frequency, the polarization, the spatial mode, and the arrival time. As mentioned above, the frequency indistinguishability is achieved by degenerating the spectrum of the photon pair. As for the polarization degree of freedom, two fiber polarization controllers (FPCs) are applied to manage each photon’s polarization state, and by this method, the polarization visibility can reach over 98% for at least hours. Besides, as the atmospheric turbulence could affect the spatial mode of the signal photons, thus single-mode-fiber (SMF) coupling is applied at the receiver as a spatial mode filter. The core size of the single-mode fiber is narrow enough that only the spatial base mode is collected, thus can filter out the higher-mode components, ensuring indistinguishability in this degree of freedom between the photons.

The last key degree of freedom is the arrival time of the photon pair. To manage this degree of freedom, two steps are adopted. Firstly, the length of the local optical fiber is tuned to about 170 m, roughly match the signal photon’s flight time in free space. Besides, the idler photon is directed to a prism, which is mounted on a translation stage with a range of 200 mm, and is then coupled into the local single mode fiber after reflection from the prism. The translation stage can reach a minimum step length of 2 $\mu$m, which is almost two orders of magnitude lower than the coherence length of the photon pair. By using the tunable translation stage, not only precise arrival time synchronization between the photon pair can be achieved, but also HOM interference dip could be observed by scanning the position of the stage, which results in an arrival time scanning of the idler photon.

3. Results and discussions

3.1 Main results of HOM interference of correlated photons

To obtain the HOM interference dip, an arrival time scanning of the idler photon is applied. It takes 2 minutes to finish one scanning process, and to suppress the effect of accidental coincidence, a total of 25 times of scanning is applied. In each separate scanning process, the position of dip center is recorded and the visibility is calculated, as shown in Fig. 2. Figure 2(a) shows the visibility of each scanning, and Fig. 2(b) shows the corresponding dip center position of each scanning. Actually this figure shows the stability of our HOM interference system. From Fig. 2(a) it is clear that each scanning process achieves a visibility well beyond the classical limit of 0.5, implying that our system has a very good performance of quantum interference. From Fig. 2(b) it can be seen that the dip center position of each scanning varies with time, and a slow drift is observed.

 

Fig. 2. Stability performance of the HOM interference system of single photons. (a) Visibility of each separate scanning. Classical limit of 0.5 is denoted in red dotted line. A high visibility exceeding classical limit is clearly observed for each scanning. (b) Position of dip center of each separate scanning. The overall average dip positionis denoted in green dotted line. Slow drift of dip center position is observed during the whole scanning process. Errorbar of one standard deviation is represented in both (a) and (b). The horizontal axis denotes the relative time for each separate scanning.

Download Full Size | PDF

The dip center drift is caused by two factors. First, in the whole scanning process, a temperature drift of about 0.9 K is observed, inducing about several micrometers of channel length variation as a result of variation of refraction index of air. On the other hand, temperature fluctuations have a much severe effect on the transmitter and the receiver. A slight deformation of their supporting framework will cause a different pointing direction, thus inducing a great variation of channel length. The observed dip center drift is about 120 $\mu$m, mainly induced by the temperature drift. So if temperature control is applied to the transmitter and receiver, the dip center drift could be expected much lower.

Having finished all the scanning process, a summed scanning HOM dip curve could be observed. Normalized coincidence counts for HOM interference is illustrated in Fig. 3. Each scanning process takes a small step of 2 $\mu$m near the dip, and a much larger step of 20 $\mu$m far away from the dip. As expected, a HOM interference dip between the coincidence and the arrival time difference between the two photons is clearly observed. The blue circle dots with error bars of one standard deviation represent the experimental data, and the red line is a corresponding Gaussian fitting curve. By Gaussian fitting, a visibility of $0.744 \pm 0.013$ is observed, demonstrating a violation of classical limit of 0.5, indicating that good quantum interference can still be obtained even after photons have been transmitted through unstable free space. Furthermore, the fitting curve gives a FWHM of about 0.9 ps, indicating a sub-picosecond correlation time between the photon pair, fairly match with the spectral width of 1.2 nm of the photon pair. It should be noted that by summing all 25 scanning data, the statistical error of the visibility is effectively suppressed to only about 1/5 of that in one scanning.

 

Fig. 3. Normalized coincidence counts between single photons. Blue circle dots with errorbars of one standard deviation denote the experimental data. Gaussian fitting is applied, denoted in red line. A visibility of $0.744 \pm 0.013$ and a FWHM of about 0.9 ps are observed. The FWHM implies the correlation time between the photon pair, fairly match with the spectral width of 1.2 nm.

Download Full Size | PDF

Turbulence in our experiment is relatively weak, as the experiment is accomplished at night from about 11 PM to 6 AM, and the distance is only 220 m. Taking the horizontal channel length, the wind speed, and the receiving aperture into consideration, according to Ref. [34–36], the construct function of turbulence $C_n^2$ in our experiment is about $10^{-15}-10^{-14} m^{-2/3}$. Turbulence of this strength typically results in about 30% coupling efficiency, and this fairly match with our experimental data. However, in long-range transmission and much stronger turbulence situations, for example in daylight free-space HOM interference, the effect of turbulence may be much more severe. Strong turbulence will result in a much lower coupling efficiency and much larger coupling efficiency fluctuations, making it much tougher to implement quantum interference. In these situations, another key technique could be considered in, i.e., adaptive optics (AO) [37], to achieve higher and more stable coupling efficiency. Typically, free-space channel efficiency could be improved greater than 6 dB in moderate turbulence strength with the help of AO, as reported in recent demonstrations [38,39].

As a measurement of time interval, the HOM dip curve could be applied in HOM-interferometer-based high precision ranging [40,41]. Specifically, the least-square fitting calculates the optimal HOM dip center position, and its uncertainty. This uncertainty actually represents the arrival time uncertainty of the free-space photon. Thus here we define the arrival time uncertainty as the time measurement precision. By this definition, the time measurement precision of HOM interference of correlated photons in our demonstration is calculated to be 8.8 fs, corresponding a ranging precision of 2.6 $\mu$m. The precision could be further improved using photon pair with a broader spectrum.

3.2 Extension to HOM interference of weak coherent pulses

In practical applications, another typical quantum single photon source is also broadly used, i.e., weak coherent state. In fact, weak coherent state could be much more easily prepared, and is the most frequently used photon sources in practice. Considering this and to test the performance of HOM interference of coherent states in precision ranging applications, we further implement a HOM interference experiment using weak coherent pulses under a much longer free-space channel of length 1.4 km, obtaining the scanning HOM dip curve. In this further demonstration, key techniques and devices are reused from that in our previous work in Ref. [42]. Nonetheless, the focus, the intention, and the results are different from our previous work, thus here our demonstration of HOM interference dip of weak coherent states is not a trivial extender of our previous work.

Experimental setup for HOM interference of weak coherent states is shown in Fig. 4(a). Independent coherent pulse sources are prepared into pulse trains with a repetition of 25 MHz and a width of 2 ns, which corresponds a spectral width of about $0.004$ nm. The pulse train prepared in the remote lab is transmitted through a free-space channel of length 1.4 km, and then collected using a SMF coupling technique. The collected remote pulse interferes with the local pulse transmitted in single mode fibers on a 50:50 fiber beam splitter. Lastly, coincidence measurement is applied and recorded with a TDC. During the free-space transmission, the geometric attenuation and coupling efficiency are about 10 dB and 30% respectively, resulting a total channel efficiency of about −15 dB. Besides, at the measurement station, the average photon number of the two interfering pulses is about $10^{-3}$, thus the probability for multi-photon emission is less than $10^{-3}$.

 

Fig. 4. Experimental setup and result for HOM interference of weak coherent states. (a) Pulse from remote lab is transmitted through a free-space channel of length 1.4 km, and then collected to interfere with the local pulse transmitted by optical fibers on a 50:50 polarization maintaining beam splitter. 10% of each pulse is sampled to detect the pulse’s arrival time for arrival time synchronization. LD, laser diode; AM, amplitude modulator; AWG, arbitrary waveform generator, ATT, attenuator; FL, frequency locking; USO, ultra-stable oscillator; TB, time board; PBS, polarized beam splitter; PMBS, polarization maintaining beam splitter; SNSPD, superconducting nanowire single photon detector; TDC, time-digital converter. (b) Experimental data is denoted in circle dots with errorbars of one standard deviation. Gaussian fitting is implemented. A visibility of $0.428 \pm 0.006$ is observed, approaching the classical limit of 0.5.

Download Full Size | PDF

Normalized coincidence count rate for HOM interference using weak coherent states is illustrated in Fig. 4(b). The scanning process takes a small step of 0.2 ns near the dip, and a bigger step of 0.5 ns far away from the dip. Circle dots with error bars and the orange line represent the experimental data and it’s Gaussian fitting separately. By Gaussian fitting, a visibility of $0.428 \pm 0.006$ is observed, approaching the classical limit of 0.5. Besides, a time measurement precision in this demonstration of 8.6 ps is achieved, corresponding a ranging precision of 2.6 mm. Notice that the width of weak coherent pulses is 2 ns, so that if pulses with much narrower width is applied, the ranging precision could be improved.

4. Conclusions

In this experiment, HOM interference between single-photon pair generated in SPDC process using a free-space channel is demonstrated. Notice that, as far as we know, our demonstration is the first real-world free-space HOM interference experiment of single-photon pair. The high visibility indicates that quantum property of single photons can still remain even after free-space transmission, and that it is feasible to implement quantum interference, for example HOM interference, using free-space channels. Even more complex quantum information processing missions involving quantum interference could also be possible over free-space channels, such as quantum repeater [8,9], MDI-QKD [15,42,43], TF-QKD [17,44,45], and so on. Besides, the scanning HOM curve could be directly applied to high precision ranging, and HOM interference of another typical photon source , i.e., the weak coherent states, is also implemented to test its performance in practical ranging applications. In the future, long-distance free-space-based quantum information processing, such as high precision ranging, could be possible with the aid of adaptive optics.