## Abstract

The classical Wiener–Khinchin theorem (WKT), which can extract spectral information by classical interferometers through Fourier transform, is a fundamental theorem used in many disciplines. However, there is still a need for a quantum version of WKT, which could connect correlated biphoton spectral information by quantum interferometers. Here, we extend the classical WKT to its quantum counterpart [i.e., extended WKT (e-WKT)], which is based on two-photon quantum interferometry. According to the e-WKT, the difference–frequency distribution of the biphoton wavefunctions can be extracted by applying a Fourier transform on the time-domain Hong–Ou–Mandel interference (HOMI) patterns, while the sum-frequency distribution can be extracted by applying a Fourier transform on the time-domain NOON state interference (NOONI) patterns. We also experimentally verified the WKT and e-WKT in a Mach–Zehnder interference (MZI), a HOMI, and a NOONI. This theorem can be directly applied to quantum spectroscopy, where the spectral correlation information of biphotons can be obtained from time-domain quantum interferences by Fourier transform. This may open a new path for the study of light–matter interaction at the single photon level.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The Wiener–Khinchin theorem (WKT), which expresses the power spectrum in terms of the autocorrelation function by Fourier transformation, was proven by Wiener [1] and by Khintchine (an alternate spelling of Khinchin) [2] in the 1930s. The WKT is a fundamental theorem used in many disciplines, including statistics, signal analysis, and optics. Especially in modern optics, thanks to the Wiener–Khinchin theorem, the interferometric spectrometer technology (also called Fourier transform spectrometry) has been well established [3]. For example, it is possible to extract the spectral information of light by making a Fourier transform on its time-domain Mach–Zehnder interference (MZI) or Michelson interference (MI) patterns. Such interferometric spectrometers are especially useful for simultaneously collecting high spectral resolution data over a wide spectral range. This provides a significant advantage over a dispersive spectrometer, which measures intensity over a narrow range of wavelengths at a time. The Fourier transform infrared spectroscopy (FTIR) has been commercially used in applications such as chemical analysis, polymer testing, and pharmaceutical analysis. [4].

With the development of quantum optics in the last several decades, several new interferometries have been demonstrated, such as the Hong–Ou–Mandel interference (HOMI) [5] and the NOON state interference (NOONI) [6] using biphotons from spontaneous parametric down conversion (SPDC). The HOMI has been widely used in quantum optical coherence tomography [7], dispersion cancellation [8,9], tests of the indistinguishability of two incoming photons [10–16], measurement of the biphoton wave function [17], frequency conversion [18], and discrete frequency modes generation [19]. The NOONI has been widely used in quantum lithography [6,20], quantum high-precision measurement [21], quantum microscopy [22–24], and error correction [25]. These two kinds of biphoton interferometry are totally quantum effect [26], which is different from the classical one-photon MZI patterns. This naturally gives rise to the question: Is it possible to construct a quantum interferometric spectrometer based on quantum interference patterns? In other words, what kind of spectral information can be extracted from the time-domain biphoton HOMI and NOONI patterns?

To answer this question in this work, we first provide a multimode theory for the MZI, HOMI, and NOONI, with the model shown in Figs. 1(a)–1(c). Then, we expand the classical WKT based on MZI into an extended WKT (e-WKT) based on HOMI and NOONI. Using this e-WKT, it is possible to extract the difference- or sum-frequency information between the constituent photons from the time-domain HOMI and NOONI patterns. Finally, we verified our theory experimentally by measuring the MZI/HOMI/NOONI patterns and two-photon spectral intensity distribution.

## 2. THEORY

In this paper, we expanded the traditional WKT to its quantum version. First, let us consider the classical WKT in the scenario of an MZI, as shown in Fig. 1(a).

As calculated in Supplement 1, in an MZI, the one-photon detection probability is determined by

Next, we consider the quantum counterpart of WKT (i.e., the e-WKT), which is based on the HOMI shown in Fig. 1(b) and the NOONI shown in Fig. 1(c). As calculated in Supplement 1, the two-photon detection probability ${P}_{2}^{\pm}(\tau )$ is

## 3. EXPERIMENT AND RESULTS

Next, we experimentally compare the e-WKT in Eq. (4) with the WKT in Eq. (2). First, we carry out three types of interference experiments (i.e., MZI, HOMI, and NOONI) in the time domain. We perform Fourier transformation on the time domain data so as to obtain the spectral information, especially the spectral bandwidths. Second, we measure the two-photon spectral intensity (TSI) distribution of our biphotons from SPDC, and we project the TSI data onto the x-axis, the diagonal axis, and the antidiagonal axis, respectively, to obtain the spectral bandwidth on each axis. Finally, we verify the e-WKT and WKT by comparing the experimentally measured spectral bandwidths and those calculated using e-WKT or WKT.

The setups for measuring the MZI, HOMI, and NOONI are shown in Fig. 2, and are similar to the setups reported in previous studies [28,29]. Pulses of 120-fs in length at 792 nm are used to pump a 30-mm-long PPKTP crystal for a type-II collinear SPDC. The PPKTP crystal can satisfy the group-velocity-matching (GVM) condition at the telecom wavelength [30–35]. Thanks to the GVM condition, we can manipulate two-photon spectral distributions and generate biphotons with positive spectral correlation. In practice, the full-width-at-half maximum (FWHM) of ${F}_{2}^{+}$ is determined by the pump laser spectrum while that of ${F}_{2}^{-}$ is determined by the crystal length. The signal and idler photons generated from SPDC have degenerate wavelengths and orthogonal polarizations. To compensate for their different group velocities due to the birefringence of the nonlinear crystal, the downconverted biphotons pass through a timing compensator composed of a polarization beam splitter (PBS0), two quarter-wave plates (QWP, at 45°), and two mirrors. One of the mirrors is set on a stepping motor to prepare an optical path delay of $\mathrm{\Delta}{L}_{1}$. Then, the polarizations of biphotons are mixed at a half-wave plate (HWP1, at 0° for HOMI, or at 22.5° for NOONI) before they are input into a Michelson interferometer that has the same configuration as the timing compensator. After that, the polarizations of biphotons are mixed again at HWP2 (fixed at 22.5°) and separated at PBS2. Finally, all the photons are coupled into two single-mode fibers (SMF) and detected by two InGaAs avalanche photodiodes (APDs), which are connected to a coincidence counter. This setup is versatile: By keeping HWP1 at 0°, the setup is for HOMI; by rotating HWP1 to 22.5°, the setup can measure NOONI; and by blocking one arm of the delay line ($\mathrm{\Delta}{L}_{1}$), the setup is ready for a one-photon MZI. Therefore, this setup can realize all the models in Fig. 1.

The measured interference patterns are shown in Figs. 3(a1)–3(c1). The MZI pattern in Fig. 3(a1) is fitted by a Gaussian function with a FWHM of 405 fs and visibility of $97.5\pm 0.5\%$ for the upper and lower envelopes. The HOMI in Fig. 3(b1) has a triangle profile with an FWHM of 4 ps and a visibility of $94.8\pm 0.8\%$. The NOONI in Fig. 3(c1) is fitted by a triangle function with an FWHM of 202 fs and a visibility of $89.7\pm 2.4\%$. The uncertainties for the visibility were added by assuming Poissonian statistics of the coincidence counts. Although we can estimate a center frequency of the spectral peak from a fringe period, it is hard to determine the spectral peak position with high accuracy due to the instability of the interferometers over the long accumulation time in the photon-counting measurements. Thus, here we focus on extracting the spectral shape, and just adopt the envelope shape of the interference patterns.

Figures 3(a2)–3(c2) shows the corresponding frequency distribution, which is calculated from the interference patterns by the Fourier transformation. Figure 3(a2) shows the corresponding spectral information of Fig. 3(a1), with an FWHM of 2.2 THz in frequency. Figure 3(b2) has a ${\mathrm{sinc}}^{2}$ profile with an FWHM of 0.22 THz in frequency, which is determined by its Fourier transform pair [i.e., the triangle-profile data in Fig. 3(b1)]. Figure 3(c2) also has a Gaussian distribution with an FWHM of 4.4 THz.

Second, we measured the TSI in an experiment using the same setup as reported in previous studies [28,30]. The TSI is measured by using two center-wavelength-tunable bandpass filters (BPF), which have a filter function of Gaussian shape with an FWHM of 0.56 nm and a tunable central wavelength from 1560 nm to 1620 nm [28–30]. The two single photon detectors used in this measurement are two InGaAs avalanche photodiode (APD) detectors, which have a quantum efficiency of around 20% with a dark count around 2 kHz. To measure the TSI of the photon pairs, we scanned the central wavelength of the two BPFs, and recorded the coincidence counts. The two BPFs were moved 0.1 nm per step and 60 by 60 steps in all. The coincidence counts were accumulated for 5 s for each point. The measured TSI is shown in Fig. 4(a), and was obtained by scanning two center-wavelength-tunable bandpass filters. The projected spectral distribution onto the x-axis, antidiagonal direction, and diagonal direction are labeled in Fig. 4(a) and in Figs. 4(b)–4(d), respectively. The corresponding FWHM values are 18.2 nm (2.18 THz), 1.9 nm (0.23 THz), and 24.6 nm (2.95 THz), respectively.

Finally, we compared the spectrally measured FWHM values from the TSI data with the FWHM values calculated using e-WKT or WKT in Table 1. The first row in Table 1 shows the $\mathrm{\Delta}t$, which is the FWHM of the MZI/HOMI/NOONI patterns in Figs. 3(a1)–3(c1). The second row shows the corresponding frequency bandwidth in Figs. 3(a2)–3(c2), as calculated from interference patterns by Fourier transformation. The third row shows the FWHM of the projection distributions in Figs. 4(b)–4(d).

The classical WKT is well verified using the data in the first column of Table 1, since the 2.2 THz bandwidth from the MZI data corresponds well with the 2.18 THz bandwidth from the TSI data. The e-WKT values for differential frequency distribution are also well verified using data in the second column. The 0.22 THz bandwidth from the HOMI data corresponds well with the 2.23 THz bandwidth from the TSI data, proving the validity of our theory. The data in the third column also partially verified the e-WKT for the sum frequency distribution, since the 4.4 THz bandwidth from the NOONI data is a little bigger than the 2.95 THz using TSI data. This may have been due to the fact that the InGaAs APDs have a large dark count (around 2 kHz), a low detection efficiency (around 20%) and a strong wavelength dependency of the detection efficiency around 1600 nm. As a result, the large background counts decreased the FWHM along the diagonal-direction in the TSI measurement. The sum frequency bandwidth of 4.4 THz, obtained from e-WKT, is in good agreement with the theoretically expected value. This means that measurement through e-WKT provides accurate spectral information while the direct spectral measurements may suffer from the detector characteristics.

## 4. DISCUSSION

The e-WKT expressed in Eq. (4) and the traditional WKT [in Eq. (2)] are unified in form. Both the WKT and e-WKT correspond to a one-dimensional Fourier transform, which builds a bridge between the spectral distribution in intensity and time-domain interference patterns. However, the WKT deals with uncorrelated photons, while the e-WKT is used with correlated biphotons. In the e-WKT, the TSI (in intensity, not amplitude) is directly related to the time-domain interference patterns. This feature is of great importance, because there is no need to measure the amplitude information, which is usually phase-sensitive and difficult to measure experimentally.

It should be noted that, in the deduction of the e-WKT in Eq. (4), we assumed the SPDC source had a symmetric distribution [i.e., $f({\omega}_{s},{\omega}_{i})=f({\omega}_{i},{\omega}_{s})$]. Under this condition, the e-WKT has a simple, elegant form. If this condition is not satisfied, we can overcome the limitation by creating a superposition state, $F({\omega}_{s},{\omega}_{i})=f({\omega}_{s},{\omega}_{i})+f({\omega}_{i},{\omega}_{s})$, which satisfies the exchange symmetry condition. Experimentally, we can realize this condition by placing a nonlinear crystal inside an interferometer as reported [36]. Thus, the exchange symmetry condition for the e-WKT could be mitigated.

We can now answer the question posed in the introduction: It is possible to realize a quantum interferometric spectroscopy that can extract difference- or sum-frequency information between two photons from the time-domain HOMI and NOONI patterns? Based on the classical WKT, it is possible to reconstruct the spectral information of optical pulses by doing MZI. In other words, we built a classical interferometric spectroscopy technology based on WKT. Base in turn on this e-WKT, it is possible to establish quantum interferometric spectroscopy technology, and many promising applications become possible. One immediate application of the e-WKT is for nonlinear spectroscopy at the single photon level, such as for entangled photon generation using an excitonic system [37]. Although exciton physics has been well-studied by classical spectroscopy, a spectral entanglement of photons may contain rich information on excitonic properties, which never extract by classical spectroscopy and allow us to discuss a new type of light–matter interaction. The Fourier transform spectroscopy based on the e-WKT is expected to be a powerful tool for investigating nonlinear light–matter interactions at the single photon level. In the future, we may apply this technique not only for biphotons from condensed matter but also for faint emissions from biological samples.

## 5. CONCLUSION

We theoretically and experimentally demonstrated an extended Wiener–Khinchin theorem (e-WKT). Unlike classical WKT, which can bridge the time-domain autocorrelation function and frequency-domain spectral intensity by Fourier transform for the classical uncorrelated photons, this theorem, which, to our knowledge, is a new concept, can establish such a bridge for the quantum-correlated biphotons. In other words, the sum- or difference-frequency information between the constituent photons can be extracted from the time-domain HOMI or NOONI patterns. This theorem can be directly applied to quantum spectroscopy, in which the spectral correlation information of biphotons can be obtained from time-domain quantum interference by Fourier transform.

## Funding

Research Foundation for Opto-Science and Technology, Hamamatsu, Japan; Educational Department of Hubei Province, China (D20161504); National Natural Science Foundation of China (NSFC) (11704290); Matsuo Foundation, Tokyo, Japan.

## Acknowledgment

We thank Zhen-Yu Wang and Wenxian Zhang for helpful discussions.

See Supplement 1 for supporting content.

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