Generalized quantum interference of correlated photon pairs

Superposition and indistinguishablility between probability amplitudes have played an essential role in observing quantum interference effects of correlated photons. The Hong-Ou-Mandel interference and interferences of the path-entangled photon number state are of special interest in the field of quantum information technologies. However, a fully generalized two-photon quantum interferometric scheme accounting for the Hong-Ou-Mandel scheme and path-entangled photon number states has not yet been proposed. Here we report the experimental demonstrations of the generalized two-photon interferometry with both the interferometric properties of the Hong-Ou-Mandel effect and the fully unfolded version of the path-entangled photon number state using photon-pair sources, which are independently generated by spontaneous parametric down-conversion. Our experimental scheme explains two-photon interference fringes revealing single- and two-photon coherence properties in a single interferometer setup. Using the proposed interferometric measurement, it is possible to directly estimate the joint spectral intensity of a photon pair source.

measurement of photons has to be performed under strict local condition (nonlocal interference) 14 . (b) The two-photon amplitudes are kept separate to observe phase-sensitive and/or phase-insensitive interference fringes 15 . (c) Constituent photons consisting N00N-like two-photon state are not bunched up in the same spatial mode (space-like and time-like separation) 13 . (d) Single-photon and two-photon coherence properties are simultaneously observed in a single interferometer setup. In particular, our two-photon interferometric scheme can provide a new interferometric method to measure the joint spectral intensity of the two-photon source via the inverse Fourier cosine transform of the full interferogram [16][17][18] .

Results
Schemes for two-photon interferometer. To observe the nonlocal two-photon interference effect in an interferometer, it is required that the two interfering photons (two correlated photons) do not meet at the same BS simultaneously. Rather, the two photons have to be separated sufficiently far from each other to satisfy the Einstein locality condition, i.e., the travel time difference between the two photons must be longer than the coincidence time window 14 . However, the two-photon state has to be in a superposition state regardless. Figure 1a shows a typical HOM-type two-photon interferometer with a large time delay between two input photons, but the input photons are in a symmetrically superposed state in space and time 3 . The gray bars represent the correlated photons in a pair. As is well known, the HOM interference effect is interpreted as the destructive interference of two two-photon probability amplitudes corresponding to the coincidence detection events of transmitted-transmitted (TT) and reflected-reflected (RR) cases 1 . In general, the coincidence probability in the HOM experiment is described as where V denotes the visibility of the coincidence fringe and dt the time delay between two input ports of the BS and Dv the angular frequency difference of two input photons. The filter function f dt ð Þis related to the spectral shape of the detected photons. The oscillation cos Dvdt ð Þcan be observed when the two incoming photons are in a frequency-entangled state 7,19 , and cos h is a continuous modulation term if there is a relative phase difference h between the TT and RR amplitudes. Almost all the experiments performed thus far have not recorded the phase-sensitive HOM interferogram resulting from Eq. (1), because in these experiments, the TT and RR amplitudes of the two-photon states were overlapped in the same interferometer setup 15 . When the BS is separated to control the two amplitudes individually, a phase-sensitive HOM interference fringe can be observed. Figure 1b depicts an interferometer involving the N00N-like twophoton state (a fully unfolded version of the path-entangled N00N state) that leads to neither bunching of two photons in the same spatial mode nor overlapping at the same BS. Unlike in the case of the conventional path-entangled state (all identical photons are in the same spatial mode at a time), spatially separated two photons do not arrive at the same BS at the same time in this configuration. Nevertheless, indistinguishable two two-photon probability amplitudes leading to coincidence detection can cause two-photon interference. It has been known that an unbalanced arrival time between two photons is not the critical restriction to observe the two-photon interference effect 3,20,21 . Indeed, the separation time (or distance) between the two correlated photons does not affect the interference fringe 13 . If the BS in Fig. 1a is separated into two BSs (BS1 and BS2), then we can construct the most generalized two-photon interferometry including the HOM scheme and N00N-like two-photon state in a single interferometer setup. Furthermore, in this case, we cannot distinguish between configuration of the HOM scheme and the interferometer involving the N00N-like two-photon state. In the case that one of the four paths is controlled individually, the phase-sensitive HOM fringe in Eq. (1) can be obtained in coincidence detection with two detectors (for example, D1 and D3). On the other hand, if the two paths are adjusted simultaneously, then the two-photon interference fringe reveals two-photon coherence property with phase superresolution 22 . Although the HOM is an explicit two-photon interference phenomenon that originates from interference between two two-photon probability amplitudes, an examination of the full fringe pattern reveals the single-photon coherence property irrespective of the phase sensitivity or phase insensitivity of the fringe pattern only when there is no frequency entanglement between two photons in a pair. On the other hand, in the case of N00N-like two-photon interference, the full interferogram exhibits pump beam coherence or the two-photon coherence property, whether or not the two constituent photons arrive at the same BS or not. Now, we propose a conceptual schematic to explain the generalized nonlocal two-photon interference between separate photon pair sources, as shown in Fig. 1c. Let us consider only the case of single photon pair generation from a separate source W A or W B . If these two emission processes are coherent, then the quantum state of the total system can be described as Here w represents the relative phase between the two processes. S i and L i are short path and long path of the source W i . v 1 and v 2 are angular frequencies of the photons in S i and ±L i paths, respectively. The two correlated single photons (v 1 and v 2 ) from a pair do not meet at the same BS at the same time. Thus, the local behavior of the individual photons at each BS is completely independent. Ou-Mandel interferometer with a long time delay between two photons. Two photons in a pair incident from different input ports do not arrive at the beamsplitter (BS) simultaneously. However, the total state of two photons in two input ports is in a symmetrically superposed state in space and time. (b) Interferometer involving the N00N-like two-photon state. In this scheme, the two photons do not arrive at the same BS simultaneously. Unfolded two two-photon probability amplitudes can arise from the independent pair sources. (c) Conceptual schematic of our nonlocal quantum interference experiment with two independently generated pairs of photons under Einstein's locality condition. Two correlated photons with angular frequencies v 1 and v 2 originate from the separate sources W A or W B , respectively. In the interferometer, one photon travels along the short path S A (S B ) while the other photon travel along the long path L A (L B ), and subsequently each photon arrives at different BSs (BS1 and BS2) to undergo nonlocal two-photon interference. However, the simultaneous behavior of two photons in a pair is mutually correlated, because the two possible amplitudes leading to coincidence detection are superposed 22 . If we detect the coincident photons with two single-photon detectors D1 and D3, then the final state just before detection can be written as Since the phase factor w depends on the relative path-length difference between the short and long paths, it can be adjusted by varying the path-length difference, L A {L B j j , when the short path is fixed (S A~SB ). Eq. (2) does not contain any constraint on the emission/travel time difference between the two photons. It is worth noting from Eq. (2) that traveling time of one photon (v 2 ) in a pair can be considerably longer than that of the other photon (v 1 ) when compared with their coherence time (single-and two-photon coherence time). Moreover, this time can also be considerably longer than the coincidence time window 14 .
Experimental setup. The experimental setup to demonstrate the proposed two-photon interference is shown in Fig. 2. Correlated photon pairs at 1.5 mm telecommunication wavelengths are generated through the quasi-phase-matched spontaneous parametric down-conversion (QPM-SPDC) process in type-0 periodically-poled lithium niobate (PPLN) crystals. We use a mode-locked picosecond fibre laser (PriTel, FFL-20-HP-PRR and SHG-AF-200) as a pumping source for QPM-SPDC, whose pulse duration is 3.5 ps at the center wavelength of 775 nm with a repetition rate of 20 MHz. Pump pulses amplified by an erbiumdoped fibre amplifier (EDFA) are mode-locked in an intra-cavity and fed into a 1 mm-long PPLN crystal for frequency doubling. The second harmonic pulses are divided by the BS, and the outputs are used to pump the two 10 mm-long PPLN crystals. In our experiments, the average pump power was set as 20 mW for each crystal. Nondegenerate (1530 nm and 1570 nm) photon pairs are emitted with the full-opening angle of 4.6u in the noncollinear regime (see Methods for details). Broadband down-converted photons are coupled to a single-mode fibre (SMF) whose coupling bandwidth was measured as about 194 nm via HOM dip measurement without spectral filtering.
Our experimental scheme is composed of two fibre interferometers as depicted in Fig. 2. A nonlocal interference under the Einstein locality condition can be observed even when the total traveling time of the two photons is considerably longer than the coincidence resolving time (10 ns in our experiment). To satisfy this condition, an additional fibre delay line (6-m SMF) is inserted in the outer interferometer. The fibre length of the inner interferometer is about 4 m. Optical delay lines (ODL1 and ODL2) are inserted at each interferometer to adjust the path-length differences within single-and two-photon coherence lengths. With fine adjustment of ODL2, we observed phase-sensitive two-photon interference fringes, which are depicted in Fig. 3b.
Correlated photons generated from a PPLN crystal do not overlap at the same BS. Individual photons from the two sources  are combined at two spatially separated fibre BSs (FBS1 and FBS2) to interfere nonlocally. To obtain the full interferogram of the singlephoton and the two-photon wave packet, ODL 2 is scanned in the range of single-photon coherence length for various positions of ODL1 within the two-photon coherence length, as shown in Fig. 4a. Single photons are detected after they pass through coarse wavelength-division multiplexing (CWDM) filters via four InGaAs/ InP single-photon avalanche photodiodes (APDs) operated in the gated mode. Electronic trigger signals are sent from the pump to the gate of the detectors by way of electric delay lines. The trigger signals are lowered from 20 MHz to 4 MHz to achieve the maximum external trigger rate of the four detectors. The quantum efficiency of detectors is set as 15%.

Discussion
For alignment of the interferometer, the HOM two-photon interference between independent photons from the two PPLN crystals was observed in the two-fold coincidence measurements with detectors D1 (D3) and D2 (D4) as a function of the path-length difference Dx 1 (Dx 2 ) while varying the path-length difference with ODL1 (ODL2). Figure 3a shows the two-fold raw coincidence counts between D1 and D2. Each point represents the averaged raw data of two-fold coincidence counts per 10 s, and the error bars represent the counting uncertainties over 10 trials. Next, to observe nonlocal two-photon interference between the photon-pair amplitudes from two independent sources, it is required to match all of the four paths within their coherence length, as shown in Fig. 2. We measured the two-fold coincidences with detectors D1 and D3 by scanning Dx 2 (step size of 1 mm) under the condition of Dx 1~0 . The coincidence detection probability according to the path-length difference Dx 2 has the form where V represents the fringe visibility and s x the fringe width related to the single-photon spectral bandwidth and l the single-photon wavelength. Figure 3b shows the measured interference pattern, which is the most general HOM type two-photon interference fringe including the relative phase difference between two amplitudes in Eq.
(2) 15,21 . Spatially separated photons of a pair exhibit nonlocal behavior with the given interferometer because the two-photon amplitudes are superposed (Fig. 1c). The phase-sensitive full fringe pattern (ps x~0 :141mm) reveals the single-photon coherence property (l 2 =dl~0:137mm). To show the phase sensitivity, we measured the fringe at Dx 2 <0 (step size of 150 nm) as shown in Fig. 3c. The error bars represent the standard deviation assuming a Poissonian distribution for the counting statistics. Accidental coincidences are subtracted from the raw data, which is simply calculated from the equation N acc:~N1 N 3 =f trig: , where N 1 and N 3 the single photon counting rates are 95 kHz and 91 kHz, respectively, and f trig: the trigger frequency for each detector is 4 MHz in our experiment. The gray area in Fig. 3b corresponds to the interference fringe obtainable by using rectangular-shaped spectral filters, corresponding to Eq. (3). High visibility of the two-photon interference fringe is observed even though the two photons in a pair do not meet at the same BS simultaneously. Furthermore, the difference between the travel times of two photons is longer than the coincidence time window. Upon introducing an unbalanced path-length difference using Dx 1 and Dx 2 (Fig. 2), we can observe the full interferogram of the N00N-like two-photon state including single-photon and twophoton coherence properties. In this case, the fringe visibility is strongly dependent on the symmetry between the two amplitudes in Eq. (2), and the two-photon interference fringes are functions of the path-length differences Dx 1 and Dx 2 , as shown in Fig. 4a. As expected, the which-path information originating from the geometric asymmetry degrades visibility within the range of the two-photon coherence length. Figure 4b shows the fringe visibilities as a function of Dx 1 . Maximum visibility is observed when individual photons from different pairs are exactly matched at each BS within the length of the single-photon wave packet (Dx 1~0 and Dx 2~0 ). The FWHM of the Gaussian fitted data is 1.17 6 0.03 mm. This value closely agrees with the coherence length of the two-photon source (, 1.26 mm), estimated from the pump duration and the group velocity dispersion (GVD).  In particular, the analysis of the two-photon interference fringes obtained from the proposed interferometric measurement using the theoretical description of the joint spectral intensity of the photon pair source (see Methods for details) can yield the asymmetric biphoton wave functions. Figure 5 shows the bi-photon wave functions and their coincidence count rates for both the frequency separable and entangled cases. We assume that t 1~D x 1 =c,t 2~{ Dx 2 =c and that the profiles of the bi-photon wave functions are Gaussian, while the bi-photon wave functions are filtered by CWDMs in the actual experimental setup shown in Fig. 2. The interference fringes shown in Fig. 5b,d are asymmetric since the center frequencies v c,1 and v c,2 of W v 1 ,v 2 ð Þare different. This indicates that the period of the interference pattern of t i is v c,i when t j is fixed. The interference pattern obtained by varying t~t 1~{ t 2 is a HOM type (phase-insensitive) interference, and it exhibits oscillations related to the spatial beating effect represented by cos Dvdt ð Þterm as written in Eq. (1). The experimental results in Fig. 4a correspond to the discrete version of those shown in Fig. 5d for specific values of t 1~D x 1 =c. The position correlation of the interference fringe between Dx 1 and Dx 2 shows that the measured bi-photon wave function has frequency entanglement. Since the bi-photon wave function is filtered by the CWDM (square type), the interference patterns in Fig. 4a are the form of a sinc function. In our experiment, the two-photon coherence length, which is the width of interference pattern for Dx~Dx 1~{ Dx 2 , is estimated by the visibilities of the patterns shown in Fig. 4b. In the actual experiment, the interference pattern is very dense, and therefore, we consider simple examples in Fig. 5.
In conclusion, we experimentally demonstrated nonlocal twophoton interference between pairs of photons in a fully unfolded and the most general form of the two-photon interferometer including the HOM scheme with non-overlapping two-photon amplitudes and involving spatially separated N00N-like two-photon states, which was carried out by employing two separate but identical photon pair sources. Nonlocal interference between photon pairs was observed even when two photons of a pair did not arrive at the same BS at the same time; this situation cannot be treated using the classical wave theory, and therefore, the situation has no classical analogy. The observed interference fringes simultaneously showed single-photon and two-photon coherence properties in a single interferometer setup. Using the proposed interferometric measurement, we can obtain the joint spectral intensity via the inverse Fourier cosine transform of the full interferogram. Quantum interferences between correlated photons can be utilized to explore quantum communication technologies such as entanglement-based quantum communication, single-photon quantum switching, and single-photon quantum routing by virtue of distributed nonlocal quantum correlations [23][24][25] .

Methods
Characterization of photon pair source. The full characterization of the photon pair source in our experiment can be obtained by simply measuring the single and coincidence counting rates as a function of the applied pump power. For an average pump power of 20 mW, the coincidence efficiency (coincidence to single ratio) was measured to be about 4.7% (the fibre coupling efficiency including optical loss was around 31.3%), and the coincidence to accidental coincidence ratio was about 2.68. The pair production probability per pulse is obtained by dividing the accidental coincidence by the actual coincidence ratio, which was estimated to be 0.37 per pulse in our experiment. For the pulse-pumped SPDC photons accidental coincidences are mainly caused by redundant multiple pair events per pulse.
HOM interference between two independent photons. Since the two incoming photons have no time correlations, the HOM effect is rarely observed when the two photons arrive from each source simultaneously. In the experiments, when the two individual photons arrived at FBS1 (FBS2) at the same time, the coincidence counting rate between two detectors was reduced to 0.33 in raw visibility under perfect experimental conditions 11 . From Fig. 3a, the two-fold raw visibility and dip width was measured to be 4.48 6 0.17% and 0.95 6 0.05 mm, respectively. The visibility was degraded, and the dip width was broadened due to timing jitter between the independent photon-pair sources 12 . The timing jitter can be caused by the duration time of the pump pulse (3.5 ps) and by the GVD in the SPDC crystals. In our experiment, the timing uncertainty by GVD in a 10 mm-long PPLN crystal was estimated to be about 2.34 ps, which corresponds to 3.42 nm in terms of spectral bandwidth. Unlike the HOM fringe shown in Fig. 3a, the single-photon timing jitter does not affect the interference between photon pairs (Fig. 3b).
Theoretical description of experimental results and joint spectral intensity. The quantum state of photon pairs from separate source W A and W B (Fig. 1c) has the superposed form of where W v i ,v j À Á denotes the bi-photon wave function, a { i v i ð Þthe creation operator of frequency v i at path i, and 0 j i the vacuum state. (4) is the same as the Eq. (2). However, in general, the bi-photon wave function of photon pairs from SPDC is not separable. The two-photon coincidence counting rate between D1 and D3 is proportional to the time-averaged value of the photon detection probability that is defined as whereÊ z ð Þ Di t i ð Þ denotes the positive part of the electric field operator at t i in D i and is superposed form of electric field operators of each input modes (S A , S B ) or (L A , L B ) 1 . If we assume that each path has its own optical delay lines t SA , t SB , t LA , and t LB , then the coincidence counting rate of D1 and D3 is given by where dt S~tSA {t SB and dt L~tLA {t LB . If the two-photon sources are identical, i.e., Þdenotes the Fourier transform of the joint spectral intensity, i.e., the absolute square of the bi-photon wave function. Figure 5 shows two examples of bi-photon wave functions W v i ,v j À Á (a: separable, c: nonseparable) and their coincidence count rates G 2 ð Þ 13 dt S ,dt L ð Þ . The experimental results in Fig. 4a correspond to the discrete version of Fig. 5d for specific values of dt s . This result is similar to Eq. (6) in Ref. 15, but not identical. The result of Ref. 15 is the Fourier transform of the symmetrized bi-photon wave function, which is given as ð Þ , and therefore the scheme in Ref. 15 cannot be applied to asymmetric bi-photon wave functions. The expression for C dt S ,dt L ð Þclearly indicates that the two-photon coherence length is