## Abstract

Quantum beats can be produced in fourth-order interference such as in a Hong–Ou–Mandel (HOM) interferometer by using photons with different frequencies. Here we present theoretically the appearance of interference of quantum beats when the HOM interferometer is combined with a Franson-type interferometer. This combination can make the interference effect of photons with different colors take place not only within the coherence time of downconverted fields but also in the region beyond that. We expect that it can provide a new method in quantum metrology, as it can realize the measurement of time intervals in three scales.

© 2015 Chinese Laser Press

## 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], quantum computing [8,9], quantum communication [7,10–12], 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]. This information can be used to study the nondegenerate spontaneous parametric downconversion (SPDC), which is very useful for quantum communications [21–23]. In this paper we will investigate the interference effect of quantum beats when the HOM interferometer is combined with a Franson-type interferometer [24–26]. 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 ${\tau}_{1}$ 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 ${\tau}_{1}=0$. The difference between the longer path ${\tau}_{2}$ and the shorter path ${\tau}_{3}$ is much greater than the coherence time of the downconversion photon pairs ${\tau}_{c}$, i.e., ${\tau}_{2}\u2013{\tau}_{3}\gg {\tau}_{c}$. Two filters IF1 and IF2 with different central frequencies are placed in front of detectors D1 and D2, respectively.

The biphoton state that is generated from the SPDC process can be given by [28,29]

The positive electrical field operators at detectors D1 and D2 are defined by

For the frequency anticorrelated photon pairs, if the central frequencies of the degenerated photons are ${\omega}_{0}$, the frequencies of the signal and idler photons are ${\omega}_{s}={\omega}_{0}+\omega $, ${\omega}_{i}={\omega}_{0}-\omega $, respectively. In this case, the biphoton spectral function $\mathrm{\Phi}({\omega}_{\mathrm{s}},{\omega}_{\mathrm{i}})$ can be replaced by $f(\omega )=(\mathrm{sin}(DL\omega /2)/DL\omega /2)$ 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 $DL\omega \ll 1$, 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 ${\tau}_{2}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$ 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 ${\tau}_{1}$ 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 ${\tau}_{1}$ into the region $|{\tau}_{1}|\approx 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}\ll {\tau}_{c}$, 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.
- (2) Second, as shown in Fig. 3(b), when $|{\tau}_{1}|$ is increased to $|{\tau}_{1}|\approx {\tau}_{2}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$, quantum beats arise under the condition where the signal photons choose the shorter path while the idler photons choose the longer one when ${\tau}_{1}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$, and the signal photons choose the longer path while the idler photons choose the shorter one when ${\tau}_{1}=-6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$. At this time, interference occurs, albeit with 50% visibility, at the positions ${\tau}_{1}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$ and ${\tau}_{1}=-6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$, because of the presence of the possibility that the idler photons take the other path, i.e., the idler photons take the shorter path when ${\tau}_{1}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$ and the longer path when ${\tau}_{1}=-6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$, which leads to a background coincidence rate independent of ${\tau}_{1}$.
- (3) Lastly, when $|{\tau}_{1}|$ reaches the region of $|{\tau}_{1}|\gg {\tau}_{2}$, photon pairs arriving at the beam splitter can be distinguished, and no interferences take place.
It should be noted that the interval ${\tau}_{2}$ is only limited by the coherence time of the pump field.

## 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 ${\tau}_{2}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$ and located around ${\tau}_{1}=-6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$, ${\tau}_{1}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$, and ${\tau}_{1}=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$. 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].

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.

## ACKNOWLEDGMENTS

This work was funded by the National Natural Science Foundation of China (Grant Nos. 10974192 and 61275122).

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