1. INTRODUCTION

Interference of two photons has been widely studied because it provides important information about the optical field, such as the properties of photon statistics. Since Hong–Ou–Mandel (HOM) interferometry was first presented in 1987 ^[1], it has been used in many areas such as testing the violation of Bell’s inequality ^[2,3], dispersion cancellation ^{[4–7" target="_self" style="display: inline;">–7]}, quantum computing ^[8,9], quantum communication ^{[7,1012" target="_self" style="display: inline;">–12]}, quantum metrology ^[13], and quantum imaging ^[5,14,15].

Usually, HOM interference experiments are carried out with two incident photons at the same frequencies. However, quantum beats will arise when the two photons have different frequencies ^{[16–20" target="_self" style="display: inline;">–20]}. This information can be used to study the nondegenerate spontaneous parametric downconversion (SPDC), which is very useful for quantum communications ^{[2123" target="_self" style="display: inline;">–23]}. In this paper we will investigate the interference effect of quantum beats when the HOM interferometer is combined with a Franson-type interferometer ^{[2426" target="_self" style="display: inline;">–26]}. With this combination, we can show that photons with different colors can not only interfere within their coherence lengths but also interfere beyond their coherence lengths. In this case, we can realize the measurement in three scales, i.e., the coherence time of the pump photons, the coherence time of downconverted photons, and a much smaller time interval shown in the beat, which can improve the measurement sensitivity in experiments.

2. MODEL AND ANALYTICAL SOLUTION

Our proposed scheme is sketched in Fig. 1. A type II degenerate nonlinear crystal is pumped by a continuous-wave (CW) laser ^[27] and generates pairs of frequency anticorrelated photons, referred to as the signal and the idler. The photon pairs are sent into an HOM interferometer. In each arm, there is an unbalanced Mach–Zehnder (MZ) interferometer, so that both the signal and the idler arms are divided into two paths. Before the MZ interferometer in the signal arm, we introduce a tunable time delay τ1 through which we can control the fourth-order interference. The lengths of the shorter (longer) paths in the signal and the idler arms have the same value when τ1=0. The difference between the longer path τ2 and the shorter path τ3 is much greater than the coherence time of the downconversion photon pairs τc, i.e., τ2–τ3≫τc. Two filters IF1 and IF2 with different central frequencies are placed in front of detectors D1 and D2, respectively.

Fig. 1. Schematic diagram of the scheme. Frequency anticorrelated photon pairs are generated from the spontaneous parametric downconversion source [nonlinear crystal (NLC)]. The signal and the idler photons are sent into an unbalanced MZ interferometer. In the signal arm, a tunable time delay τ1 is introduced outside the MZ interferometer. Photon pairs are combined at the last beam splitter (BS), and we can observe the interference of quantum beats by observing the coincidence count rates between detectors D1 and D2. IF1 and IF2 are filters with different central frequencies set in front of the detectors. M represents the reflecting mirrors.

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The biphoton state that is generated from the SPDC process can be given by ^[28,29] where Φ(ωs,ωi) is the biphoton spectral function, which is determined by the phase-matching conditions. As we introduce a tunable time delay τ1 in the signal arm and an MZ interferometer in each arm, it generates a phase shift, if we assume the lengths of the shorter paths τ3 in each arm have a value of 0. Then the biphoton state that interferes on the beam splitter should be rewritten as

The positive electrical field operators at detectors D1 and D2 are defined by respectively, where g1(ω1)=exp[−((ω1−ωa)2/2σ12)], g2(ω2)=exp[−((ω2−ωb)2/2σ22)] are optical spectral functions of filters in front of detectors D1 and D2, with their central frequencies at ωa and ωb, respectively. For simplicity, we set the bandwidth of each filter as σ1=σ2=σ in the following. With the state in Eq. (3) and the field operators in Eqs. (4) and (5), we can calculate the detection amplitude: Then the coincidence count rate between the two detectors is

For the frequency anticorrelated photon pairs, if the central frequencies of the degenerated photons are ω0, the frequencies of the signal and idler photons are ωs=ω0+ω, ωi=ω0−ω, respectively. In this case, the biphoton spectral function Φ(ωs,ωi) can be replaced by f(ω)=(sin(DLω/2)/DLω/2) for the type II SPDC ^[30,31] process, with D and L denoting the inverse group velocity difference for the biphoton and the length of the crystal, respectively. Then Eq. (7) can be rewritten as

As DLω≪1, the analytical results can be approximately given as

3. RESULTS AND THEORETICAL EXPLANATION

We then numerically calculate the coincidence count rate with feasible experimental parameters. A CW laser with a central wavelength of 406 nm is used to pump a type II degenerate beta-barium borate crystal. In order to observe the quantum beats, the central wavelengths of two filters are set at 800 and 824 nm. The fixed time delay τ2=6ps is much greater than the coherence time of the downconverted fields, which is typically 0.1–1 ps ^[32].

The simulated results are shown in Fig. 2. Three quantum beats emerge in different regions as we adjust the time delay τ1 continuously. Two quantum beats with 50% visibility are seen in the two side regions, while a quantum beat with 100% visibility is seen in the middle. This result can be understood by analyzing all the different paths that the biphotons choose to take during the measurement of coincidence events between D1 and D2. There are three stages occurring along with the increased time delay: (1)

First, as illustrated in Fig. 3(a), when we scan τ1 into the region |τ1|≈0ps≪τc, there are two alternative paths, the longer path and the shorter path, for the photon pairs to choose to take. Besides, as biphotons arrive at the last beam splitter, we cannot tell whether the photons are both reflected or transmitted. In this sense, this interferometer is the combined form of the Franson and the HOM interferometer. A quantum beat arises whether the photon pairs choose the longer path or the shorter one. As we cannot distinguish which paths the photon pairs choose to follow, quantum beats interfere with each other with 100% visibility.

Fig. 2. Normalized coincidence count rate, which shows three quantum beats with the same interval of τ2=6ps when the two filters in front of the detectors have different central frequencies. The three central dips are at the position of τ1=−6ps, τ1=0ps, and τ1=6ps.

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Fig. 3. Feynman’s path diagrams in different regions of . (a) , where each photon has two alternatives before arriving at the beam splitter; (b), where each photon only has one choice before arriving at the beam splitter in order to produce interference.

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4. DISCUSSION AND CONCLUSION

From what we have described above, we find that the three interference fringes in Fig. 2 are caused by both the HOM and the Franson-type interference; this indicates that although photons with different colors are distinguishable, the interference effect can also take place in the region far beyond the coherence time of the downconverted fields through the combination of these two kinds of interferometers. If we set limitations on the bandwidth of the downconverted field, the longer the coherence time of the pump laser is, the broader the middle envelope will be. If the coherence time of the single photon is long enough, the middle envelope will cover the other two envelopes. So through the quantum beats generated in the combined form of the HOM and the Franson-type interferometer, we can realize the measurement of time intervals on the scale of coherence time of the pump field, which is far beyond the single photon’s coherence time determined by the band filters, and improve the measurement sensitivity via the beats, which could be measured according to the frequency difference of the two photons.

Moreover, for comparison, in Fig. 4 we also show the simulated result in the situation where the two filters in front of the two detectors have the same central frequencies. The three dips shown in the normalized coincidence count rate are spaced by the same interval of τ2=6ps and located around τ1=−6ps, τ1=0ps, and τ1=6ps. In addition, it should be noted that if the tunable time delay in this scheme is moved into one of the longer paths, i.e., the shorter paths of the two MZ interferometers are of equal value, one longer path is fixed, and the other longer path becomes tunable, the interference fringes will be very complex and both second- and fourth-order interference effects will emerge ^[33].

Fig. 4. Normalized coincidence count rate when the two filters have the same central frequencies. It shows three dips with the same interval of τ2=6ps. The three central dips are at the positions of τ1=−6ps, τ1=0ps, and τ1=6ps.

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In conclusion, we have demonstrated a new scheme in which we can observe the interference of quantum beats when we combine the Franson-type interferometer with the HOM interferometer. Usually we discuss the interference effect of photons with different colors in the HOM interferometer within the coherence time of downconverted photons, but with the combination of the Franson and the HOM interferometer we can realize interference effects in the region far beyond the coherence time of the downconverted fields. Moreover, we can also realize the measurements of time intervals in the three scales shown above.