Accessing the spectrum of a single-photon by the Hong-Ou-Mandel interference

Lifeng Duan; Aojie Xu; Lirong Wang; Lirong Wang; Lirong Wang; Yun Zhang; Yun Zhang

doi:10.1364/OE.510983

1. Introduction

Quantum communication and quantum computation rely on the preparation and manipulation of well-defined single-photon state [1]. It is employed to encode quantum bit with one of its mode properties, such as polarization, path modes, frequency, or arriving time. In recent years, the frequency degree of freedom offers large potential for the stable and controllable realization of photonic quantum processing applications as it gives access to robust high-dimensional states with a better noise resilience [2]. Therefore, it is crucial to characterize a single-photon state on spectrum for modern quantum information technology. Comparing with photon number, spatial, polarization and producing rate, it is more challenged to characterize the spectrum. The standard approach to quantify the indistinguishability of single-photon wavepackets is to perform Hong-Ou-Mandel (HOM) interference, originally applied to measure the temporal interval of spontaneous parametric down-conversion (SPDC) photons [3,4]. When two indistinguishable single-photons incident at each input of a 50:50 beam splitter, they will exit the beam splitter together, resulting in no two-photon coincidental detection events at both outputs. Partial indistinguishability of the input states leads to coincidence detection events at the outputs and reduces HOM interference visibility. This interference visibility can give direct access to the indistinguishability of two single-photons [5].

Recent years have seen a growing interest in the observation of two photon interference between independent single-photon sources since it allows the implementation of logical photon-photon gates for quantum computing as well as the development of quantum repeater for long distance secure quantum communications [6–13]. Experimental demonstrations include single-photon sources produced by various methods, for example single atoms, molecules, or quantum dot systems. To obtain a high visibility interference, it requires a sufficient amount of perfectly indistinguishable and pure single-photons. Here, the indistinguishability refers to the identical quantum properties of photons, while purity quantifies the degree to which a single-photon is isolated from the environment. A highly pure single-photon ensures its integrity as an independent quantum entity, exhibiting a stochastic properties for the observing variable. This potential application has also boosted efforts towards the experimental characterization of the single-photon state, especially on the freedom degree of frequency [14].

Single-photon state can also be created through the conditional heralded of photon pairs, which are generated by either SPDC or four-wave mixing. In this case the detection of the trigger photon indicates the existence of a single-photon. Usually, parametric sources of photon pairs possess strong correlations in terms of transverse space and frequency, introducing classical uncertainty in the spatial and spectral mode of the heralded single-photon, and reducing its purity of heralded single-photon state. Thus the utility of such a source is restricted. The most straightforward approach of eliminating the correlation between signal and idler photons is to employ spatial and spectral filtering techniques at the cost of producing rate [15,16]. Recent studies have also explored that domain-engineered crystals can effectively minimize both spectral and spatial correlations at a reasonable production rate. Domain-engineered crystal with Gaussian nonlinear responses are capable of producing single-photon state with a spectral purity of about 99 % [17,18]. This makes SPDC photon pairs as an exceptional platform for generating highly pure single-photon and developing the technique for characterizing them.

While a photon’s spectrum can be measured using a scanning monochromator and single-photon detector, it needs a long measuring time because of the low producing rate [19]. It has been demonstrated that spectral profile of photons can be measured based on HOM dip [20–22]. The advantages of this method are to determine the spectral phase when the single-photon pulse train is chirped [20]. It is also possible to quickly measure spectral mode using a bright classical light. This generally requires a reference whose spectrum is known completely. Similar interference results are also observed with independent photon sources [23].

In this paper, we build up these previous efforts and develop a method for characterizing the spectrum of single-photon state. It relies on the HOM interference between the measured single-photon state and a well-defined weak coherent state. The coincidence events from more than two photons in the weak coherent state are isolated by independent measurements [24,25], which allows it to be treated as a single-photon state. A heralded single-photon state and single-photon level thermal state are well characterized using the auxiliary weak coherent state. In addition, a detailed investigation of interference between SPDC photon pairs was also performed, indicating that the two photon interference can be understood by the superposition of joint spectral amplitudes. This innovative method provides a way for monitoring the single-photon emitters with different approaches in the future quantum network systems.

2. Hong-Ou-Mandel interference with SPDC photon pairs

We now consider a HOM interferometer, which involves a pair of photons ($E_a(t)$ and $E_b(t+\tau )$) entering on a 50:50 lossless beam splitter (BS) from two separate input ports. The fields at the output of BS are given by

(1)$$\begin{aligned} E_c^+(t,\tau)=\frac{1}{\sqrt{2}}(E_a^+(t)+E_b^+(t+\tau)),\\ E_d^+(t,\tau)=\frac{1}{\sqrt{2}}(E_a^+(t)-E_b^+(t+\tau)), \end{aligned}$$

where $\tau$ is the relative delay between the two photons and

(2)$$\begin{aligned} E_u^+(t)\propto \int d \omega_u a_u(\omega_u) e^{{-}i\omega_u t}, (u=a,b) \end{aligned}$$

is the Fourier transform of the photon spectrum. Here $\omega _u$ is the angular frequency of field and $a_u(\omega _u)$ is the spectral function. The probability of detecting one photon at $D_c$ at time $t_1$ and $D_d$ at time $t_2$ is the second order correlation function

(3)$$\begin{aligned} G^{(2)}(t_1,t_2,\tau) & =\langle E_{c}^{-}E_{d}^{-} E_{d}^+E_{c}^+ \rangle \\ & \propto\Bigg |\iint d\omega_a \omega_b f(\omega_a, \omega_b) e^{{-}i\omega_b \tau}[e^{{-}i(\omega_a t_2+\omega_b t_1)}-e^{{-}i(\omega_a t_1+\omega_b t_2)}]\bigg|^2, \end{aligned}$$

where $f(\omega _a, \omega _b)=a_a(\omega _a)a_b(\omega _b)$ is joint spectral amplitude of two input fields. It should be noted that the joint spectral amplitude is factorable for two independent single-photons. For SPDC photon pairs, it is unfactorable. The detectors are slow, so their resolution time T is significantly larger than the correlation time of detected field. Then average coincidence counting rate is given by

(4)$$\begin{aligned} R_c(\tau) \propto\frac{1}{T} \iint_T dt_1 dt_2 G^{(2)}(t_1,t_2,\tau). \end{aligned}$$

It can be calculated as when the integration time tends to infinity since the detection time is much longer than the interaction time,

(5)$$\begin{aligned} R_c(\tau) \propto\iint d & \omega_a d \omega_b \Big[|f(\omega_a, \omega_b)|^2 - f(\omega_a, \omega_b) f^{*}(\omega_b, \omega_a)e^{{-}i(\omega_a-\omega_b)\tau}\Big ]. \end{aligned}$$

It reveals that the interference patterns depends on both the spectral amplitude and phase of input fields. Experimental observation of this pattern is performed with two independent single-photons [23]. It is also applied to measure the joint spectral mode of photon pairs with a Fourier filtering technique [20]. Here, we explain the two-photon interference of SPDC photon pairs and access the spectrum of a single-photon when auxiliary weak coherent state is employed.

2.1 Interference of SPDC photon pairs

In the process of SPDC, one pump photon is split into two lower energy photons, the signal and the idler. The two-photon component in SPDC can be expressed as

(6)$$|\psi\rangle=\iint d\omega_s d\omega_i f(\omega_s, \omega_i)\hat{a}(\omega_s)\hat{a}(\omega_i)|0\rangle,$$

where $f(\omega _s,\omega _i)=\phi (\omega _s,\omega _i)\alpha (\omega _s,\omega _i)$ represents the joint spectral amplitude for SPDC photon pairs, $\phi (\omega _s,\omega _i)$ and $\alpha (\omega _s,\omega _i)$ are the phase matching amplitude and the pump envelope amplitude. $\hat {a}(\omega _s)(\hat {a}(\omega _i))$ is the creation operator and the subscripts $s$ and $i$ denote the signal and the idler photons, respectively. For simplicity, we assume the pump spectrum has a Gaussian distribution with a center frequency of $\bar {\omega }=(\omega _s+\omega _i)/2$ and a bandwidth of $\delta _p$. In the case of type-II nonlinear crystal, the phase matching intensity can be expressed using the formula of $|\phi (\omega _s,\omega _i)|^2=|\phi (\bar {\omega }+\Omega _s,\bar {\omega }+\Omega _i)|^2= \mathrm {sinc}^2 (\Delta k L/2)$ where $\Delta k= \Omega _s(k_s^{\prime }-k_p^{\prime })+\Omega _i(k_i^{\prime }-k_p^{\prime })$. $(k_u^{\prime } (u=p,s,i))$ is group velocity of pump field at frequency of $2\bar {\omega }$, signal and idler fields at $\bar {\omega }$ and $L$ is the length of crystal. Hence, we have the joint spectral amplitude as [26,27]

(7)$$\begin{aligned} f(\omega_s, \omega_i)\propto & \exp[-(\Omega_s+\Omega_i)^2/2\delta_p^2]\exp[{-}iL( (k_p^{\prime}-k_s^{\prime})\Omega_s+ (k_p^{\prime}-k_i^{\prime})\Omega_i)/2] \\ & \mathrm{sinc} [( (k_p^{\prime}-k_s^{\prime})\Omega_s+ (k_p^{\prime}-k_i^{\prime})\Omega_i)L/2]. \end{aligned}$$

And, Eq. (5) can be integrated as [26,27]

(8)$$\displaystyle R_c(\tau) = \begin {cases} \displaystyle \frac{\delta_p}{\sqrt{2\pi}\Omega_+}-\mathrm{erf}[\frac{\delta_p}{2 \sqrt{2} \Omega_+}(1-|\tau|\Omega_{-})] \hspace{1cm} \hspace{5mm} |\tau|< \frac{1}{\Omega_{-}}\\ \displaystyle \frac{\delta_p}{\sqrt{2\pi}\Omega_+} \hspace{3cm} \mathrm{otherwise}. \end{cases}$$

where $1/\Omega _{\pm }=L|(k_p^{\prime }-k_s^{\prime })\pm (k_p^{\prime }-k_i^{\prime })|$ and erf($x$) is the error function. The two-photon interference visibility can be calculated with a normalized coincidence rate of $1-R_c(0)/R_c ({\infty })$. As the pump bandwidth increases, the visibility decreases. In the limit of infinite pump bandwidth, the visibility is zero and the interference does not occur. On the other hand, $\delta _{p} \to 0$, the visibility is 1 and the interference dip function has the V-shape on the parameter of $\tau$ [28].

It is more general to measure the two-photon interference by filtering the signal and idler beams. The joint spectral amplitude in Eq. (6) becomes,

(9)$$\begin{aligned} f^{\prime}(\omega_1,\omega_2)=f_1(\omega_1)f_2(\omega_2)\alpha(\omega_s,\omega_i)\phi(\omega_s,\omega_i). \end{aligned}$$

Normally, filter function $f(\omega )$ is centered at some specific frequency $\omega _{1(2)}$ with a Gaussian shape of standard deviations $\delta _{1(2)}$. The product of two filters can be treated as one Gaussian shape ($\omega _s+\omega _i$) with bandwidth of $\sqrt {\delta _1^2 \delta _2^2/(\delta _1^2+ \delta _2^2)}$. It is noted that both the pump envelope and the filter exhibit symmetrial characteristics with respect to $\omega _s$ and $\omega _i$. So, the filters can be roughly considered as the reduction of bandwidth of pump field, although the spectral limitation is implemented on the down-conversed fields. In this mean, the spectral filtering photon pairs always improve the observing interference visibility at a cost to the production rate when two identical filters are employed. The phase matching condition $\phi (\omega _s,\omega _i)$, however, is not symmetric due to the crystal is birefringent, $k(\omega _s) \neq k(\omega _i)$ in general. Because of this asymmetry, the wave packets of describing the two photons produced in pulsed pumping are identical, even when they are degenerate with same center frequencies. Hence, the filters also have impact on the shape of joint spectral amplitude. To make further understand, numerical calculations becomes necessary to determine the shape of spectral distribution.

2.2 HOM interference of independent photons

In the case of bandwidth $\delta _i(i=1,2)$ of filters are much less than the spectrum of the SPDC, $f^{\prime }(\omega _s,\omega _i)$ is dominated by the filters. $f(\omega _s,\omega _i)$ simply varies slowly within $\delta _i (i=1,2)$ and can be treated as a constant. The joint spectral amplitude function of the signal and idler photons can be approximately treated as a factorable state such $f^\prime (\omega _s,\omega _i )=f(\omega _s)f(\omega _i )$. So, the second correlation function in Eq. (3) is calculated by the Fourier transform of the filters,

(10)$$R_c(\delta_1, \delta_2, \tau)=\frac{1}{2}-\frac{\delta_1 \delta_2}{\delta_1^2 + \delta_2 ^2}\exp(-\frac{\delta_1^2 \delta_2^2}{\delta_1^2+ \delta_2^2} \tau^2).$$

This result is equivalent to the interference of two independent single-photon states since the spectrum are filtered to be factorable [29]. Another important property of interference is its temporal characteristics. For two-photon interference with SPDC, Eq. (8) shows the width of interference dip relies on the pump field and nonlinear crystal. For a pulsed pump filed, the observed results show that the coherence time is significantly shorter than the pulse duration. It indicates that the time of emission a photon occurs randomly during the pulse duration. On the other hands, Eq. (10) implies that the interference profile is completely determined by the bandwidth when two independent single-photon states have Gaussian shape spectrum. It can be a promising model function for estimating the spectrum of single-photon state if a well-defined single-photon state is employed as reference. A perfect visibility of 1 and an interference dip width of $1/\delta$ can be obtained, when the two states are identical (bandwidth of $\delta$) in spectral property.

2.3 Method for characterizing the spectrum of single-photon state

Our aim is to access the spectrum of single-photon state. Our first step is to determine the spectral properties of the referred single-photon state. In our scheme, a weak coherent state is employed as reference. Hence, two photon interference between phase randomized weak coherent state is performed to obtain a referred interference profile [30]. Since The coincidence event of more than two photon state in coherent state is subtracted, a near perfect two photon interference is able to be expected [24,25]. Our second step is to measure two-photon interference profile between the measuring single-photon state and the referred weak coherent state. The coincidence events of more than two-photon states in coherent state are subtracted by another independent measurement. The spectral property of the single-photon state is obtained using a standard estimation method with the interference visibility and temporal profile, such as least square estimation. Here, we assume that the single-photon state has a Gaussian shaped spectrum. It is also possible to use the other distribution shapes [31].

3. Experimental example

The schematic diagram of the experimental setup for accessing the single-photon state spectrum is shown in Fig. 1. The setup can be divided into two parts: preparation and measurement. In the preparation, a referred weak coherent state, a measured single-photon state, and a thermal state are produced. A mode-locked Ti: Sapphire laser, operating at 798 nm wavelength with a pulse duration of 2 ps and a pulse repetition rate of 82 MHz, is prepared as the fundamental laser source. The most laser power (650 mW) is sent into a single-pass second harmonic generator (SHG), of which a 15-mm-long type-I LBO nonlinear crystal is used to produce the maximum 35 mW of the pump light with 399 nm for generating the orthogonal photon pairs by the SPDC process. Especially, a dichromatic mirror and two short-pass filters are used to filter out the remaining of the 798 nm laser source from SHG. In the SPDC process, a 5-mm-long type-II BBO nonlinear crystal is used to generate an orthogonal photon pairs, long-pass, 10 nm bandpass, and 1 nm bandpass filters are employed to eliminate the residual pump light and to shape the spectrum of SPDC photon pairs. After the polarizing beam splitter (PBS), photon pairs can be prepared as heralded single-photon state and thermal state. Furthermore, a smaller remaining portion of the laser is prepared as the weak coherent state after traveling a strong attenuator and phase modulator. A half wave plate (HWP) is used to control the polarization of field for observing two-photon state. In the measurement, a fiber coupled (coupler A) single-photon counter modules (Excelitas Technologies, SPCM-AQRH14) is prepared for heralded detection single-photon state. A single-mode fiber beam splitter (FBS, Thorlabs TW805R5F2) is used to observe the HOM-type of two-photon interference, thus ensuring excellent spatial overlap. Its input are connected to fiber coupler B and C. The outputs of FBS are connected to two single-photon counter modules and two-fold or three-fold coincidence events are recorded via an electronic coincidence circuit and counter. The path delay between two interference beams is adjusted by using a high-precision translation stage, which moves in steps corresponding to a time delay of 3.3 fs.

Fig. 1. Schematic diagram of the experimental setup. SHG: second harmonic generator; BF: bandpass filter; PBS: polarizing beam splitter; HWP: half-wave plate; PM: phase modulator; SPCM: single-photon counting modulus; FBS: fiber beam splitter.

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3.1 Two-photon interference of SPDC photon pairs

The two-photon interference of SPDC photon pairs can be observed by coupling PBS output to coupler B and C. A typical interference dip is shown in Fig. 2. It shows the experimentally measured data when different filters are placed in the signal and idler beams. The Gaussian-like shape of two-photon dip indicates that the bandwidth of pump field is much smaller than that of type-II parametric down-converted photon pairs [27]. When a long pass filter is used (only pump field is filtered), a visibility of 80% two photon interference was observed indicating the frequency anticorrelation of the signal and idler photons. The pump spectrum simply facilitates proportion of distinguishable down conversion, i.e. their spectra are not identical. If the measurement of coincidence events are performed only on the indistinguishable photon pairs, then the interference would be restored. It can be accomplished by adding bandpass filter in the beams. When a 1-nm filter, the visibility are improved to 97%. Another important properties is the temporal characteristics of the down-converted photons. Two quantities are of interest: the time of emission and the coherence times. Photon pairs are randomly emitted within the time of the pump pulse duration, but the absolute time of emission is not fixed. The coherent times (or temporal envelope) are related to the spectral bandwidth of the photon, a large bandwidths, consequently, short coherence times. Since filters have effectively restricted some of the low coherence times photon pairs, the coherence times can be improved by spectral filtering. This can be understood by examining the full width at half maximum (FWHM) of interference dips in Fig. 2, which gives the values of 0.051, 0.1 and 1 ps for long pass, 10 nm, and 1nm filters, respectively.

A more detailed characterization of two photon interference of SPDC photon pairs is summarized in Table 1. As the above-mentioned, the high visibility on the condition of two long pass filters indicates the original frequency anticorrelation of the photon pairs. The visibility and the interference dip width can be significantly improved by employing identical filters on both photons. It is worth to note that the interference visibility decreases when two different filters are placed in signal and idler photons. The reason can be understood that one of the frequency anticorrelated photons is filtered out by the narrow bandpass filter. The observed visibility for different filter is larger than that of when two independent single photons with the same spectral ratio is employed. The width of interference dip is significantly increased even when only one photon is filtered by narrow bandpass filter. This feature also reveals that a frequency correlated single-photon could not be employed to measure spectrum as a reference. The joint spectral of SPDC photon pairs is usually demonstrated by numerical calculations using phase matching and pump pulse width [32]. In the matter of fact, the bandwidths of our photon pairs taking the values of about 12.8 nm for signal and 14.1 nm for idler are obtained using the program with our experimental parameters. The calculated interference disabilities of SPDC are in a reasonable agreement with measured results in Table 1. It is useful to think of the joint spectral amplitude as the probability distribution for the photon pairs. That is the probability for producing photon pairs at frequency $\bar {\omega }+\Omega _s$ and $\bar {\omega }+\Omega _i$ is proportional to the value of $|f(\Omega _s, \Omega _i)|$. Hence, the symmetry of joint spectral amplitude ($|f(\Omega _s, \Omega _i)|=|f(\Omega _i, \Omega _s)|$) indicates indistinguishbility of the signal and idler photons.

We know that it is possible to generate a heralded single-photon state from both a frequency-uncorrelated source and a frequency-anticorrelated photon pairs [33]. Each method has its unique features and advantages. The spectral width of heralded single-photon is independent of the filter bandwidth placed on the heralding photon when the joint spectral amplitude of photon pairs is factorable [34,35]. On the other hand, many interesting heralded single-photon states are able to produce by engineering the unfactorable joint spectral amplitude. For example, a frequency qubit state is produced with a frequency anticorrelated and long coherence photon pairs. According to numerical calculations [32] and the results presents in Table 1, the joint spectral amplitude of photon pairs becomes factorable when filters with bandwidth of 1 nm is employed.

Fig. 2. Normalized coincidence count of two-photon interference with SPDC photon pairs. Long pass, 1 nm bandpass, and 10 nm bandpass filters are employed on the signal and idler photons.

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Table 1. Summary of measured visibility and width of SPDC photon pairs interference with different filters. The upper gives the visibility and the lower gives the interference dip width with a unit of ps.

View Table

3.2 Two-photon interference between phase randomized weak coherent states

To calibrate the referred single-photon state, two-photon interference between phase randomized weak coherent states is performed. It is realized by coupling two weak coherent states to coupler B and C and recording the two-fold coincidence events. To randomize the phase, one of beams is passed through a phase-modulator, which is modulated by a white noise. The blue dots in Fig. 3 show the normalized coincidence count for equal and low mean photon numbers as the optical delay $\tau$. As $\tau$ approaches zero, the coincidence probability decreases with a characteristic shape related to the auto-correlation of the temporal profile of weak coherent state. We fit the measured data with an inverted Gaussian shape that gives a interference visibility of $0.49\pm 0.02$. Red dots give the coincidence count causing by the two-photon state in weak coherent state when one of beams is blocked. The two-photon interference of independent single-photon state in weak coherent state is able to obtain by subtracting coincidence count from the two-photon state as shown in black dots in Fig. 3. A inverted Gaussian shape with visibility of $0.98\pm 0.02$ and FWHM of $2.01\pm 0.02$ ps well agrees with the measured data. Assuming they have the same bandwidth, we estimate the referred single-photon state has a spectral bandwidth of $0.22\pm 0.03$ THz or linewidth of $0.45\pm 0.03$ nm as determined by Eq. (10).

Fig. 3. Normalized coincidence count of two photon interference with two phase randomized weak coherent states.

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3.3 Spectrum measurement of single-photon state

To access the spectrum of the measured single-photon state, we interfere it with a coherent reference and record the two photon interference dip. In our case, three kinds of states are prepared for demonstrating our proposal. Two heralded single-photon states with different bandwidths are produced by applying 1.0 nm and 10 nm spectral filters on both signal and idler photons of SPDC photon pairs. One of the SPDC photons, named heralding photon, is injected into fiber coupler A. Another SPDC photon, named heralded single-photon state, after an optical delay is prepared for HOM-type two-photon interference with the reference weak coherent state. The heralded single-photon is injected into a fiber coupler B. The two-photon interference between the reference weak coherent state and heralded single-photon state is confirmed by three-fold coincidence. A thermal state, which is the signal photon of SPDC pairs without heralding, is also prepared by applying a 1.0 nm spectral filter [36]. In this case, the two photon interference is recorded by two-fold coincidence when coupler C is injected with reference and coupler B is injected with signal photon of SPDC pairs. The results of our measurement are presented in Fig. 4, where we show typical HOM dips observed by interfering different measuring state with the reference beam. The coincidence event of two-photon state from weak coherent is subtracted by the above-mentioned isolating method [24,25]. The average photon number of the weak coherent state is also optimized to obtain the maximum two-photon interference visibility [34,37].

Fig. 4. The measured HOM dip by interfering the reference field with: (left) heralded single-photon state with 1 nm filter; (center) heralded single-photon state with 10 nm filter; (right) the thermal single-photon state. The width of the dip and the corresponding visibility are shown in the plot

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By analyzing several similar sets of data we extract the following visibility and FWHM values: V$_{1 \textrm {nm}}$=0.73$\pm$0.03 and FWHM$_{1 \textrm {nm}}$ = 1.75$\pm$ 0.05 ps when a heralded single-photon state with a filter of 1 nm linewidth interfered with reference. The spectral width of the single-photon state is retrieved by applying Eq. (10) and the spectral parameters of reference. One finds that the spectral width of the heralded single-photon state is $0.48\pm 0.03$ THz, which is about 2.2 times of that of the reference photon. It is legitimate to assume that the spectral width of the heralded single-photon state is equal to the filter width since the unfiltered marginal spectrum is much broader. The measured linewidth of $0.99\pm 0.03$ nm is close to the bandwidth of filter. Figure 2 (b) gives the two-photon interference between two independent single-photon states with different bandwidths, in which one is the coherent state with linewidth of 0.45 nm and the other is the heralded single-photon state with linewidth of 10 nm. Comparing with the result of frequency anticorrelated SPDC photon pairs with the different spectrum, the interference visibility significantly drops. In general, the distinguishability of photons is employed to explain the two-photon interference. The HOM interference only occurs between indistinguishable photon pairs. Once again, it is worth to emphasize that the decreased visibility is attributed to the axial ($\Omega _s=-\Omega _c$) asymmetry of joint spectrum amplitude, which gives the probability amplitude of producing with the frequencies of $\bar {\omega }+\Omega _s$ and $\bar {\omega }+\Omega _c$. The perfect interference occurs at the condition of these probability amplitudes have opposite signs and equal magnitudes ($|f(\Omega _s, \Omega _c)|=|f(\Omega _c, \Omega _s)|$). The independent photons with different spectral bandwidth have an asymmetrical joint spectral amplitude. Hence, the observed interference visibility is restricted. The diminished visibility between a thermal and coherent state in Fig.4(c) can also be understood the asymmetry of the joint spectral amplitude, although they have the same spectral width. Usually, a thermal state has a Lorentz distribution differing from that of coherent state. However, the spectral width measured single-photon and thermal state can be accurately estimated even low interference visibility. This indicates the measurement is insensitive to the interference visibility similar as the reported by Fourier filtering technique [20]. We notice that the width of the two interference dip does not determine the coherence time of the generated single-photon state. The coherence time of heralded photon seems limited only by its spectral width. Nevertheless, it is able to interfere with the coherent state since it emits at a random time within the pulse duration. This properties of photon pairs from type-II parametric down conversion are also experimentally reported [38,39]. In this experiment, the drawback is that the observation of HOM interference relies on measuring the three-fold coincidences, which can lead to slow data acquisition with pair sources that have low heralding efficiency. For an arbitrary single-photon state, this will be not restricted and the data acquisition can be much quicker by using a high average photon weak coherent state. Furthermore, it is possible to develop a visualization tools basing on the present measuring method.

The spectral properties of the single-photon state is estimated basing on the HOM interference dip. It is important to obtain a stable and high quality two photon interference dip. Taking the system noise into account, there exists an optimized average photon number for a given producing rate to obtain the maximum interference visibility [34,37]. It is also known the observed maximum interference occurs when the two single-photon state have the same spectral width. So it is necessary to prepare a weak coherent state with spectral width that closely matches the measured single-photon state. The information about the spectral should still be available in the temporal width of the HOM dip when the coherent state emits from a different light source. Finally, extending the scheme to measure arbitrary single-photon state is possible by interference it with a coherent state. Furthermore, the phase of the spectrum should be also possible to obtain by a suitable mathematics method, such as Fourier filtering.

4. Conclusion

We demonstrated a method for determining the spectral characterization of single-photon state. The technique is based on the HOM interference between a well prepared weak coherent state and the measured single-photon state. The spectral properties of the single-photon state are estimated by fitting the measured interference dip with proposed model and least square method. The HOM interference is also explained by the joint spectral amplitude and the diminished visibility between independent single-photon sources is attributed to the asymmetry of the joint spectral amplitude. We hope our work will stimulate further investigation into more practical and robust techniques for quantifying the HOM interference with independent single-photon state. We believe that our ideas will boost the characterization and optimization of high-quality heralded single-photon state.

Funding

Fund program for the scientific activities of selected returned overseas professionals in Shanxi province (2023001); Japan Society for the Promotion of Science (21K04919, 21K04923).

Disclosures

The authors declare no conflicts of interest

Data availability

Data underlying the results presented in this paper are not publicly available at this time bu may be obtained from the authors upon reasonable request.

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Accessing the spectrum of a single-photon by the Hong-Ou-Mandel interference

Abstract

1. Introduction

2. Hong-Ou-Mandel interference with SPDC photon pairs

2.1 Interference of SPDC photon pairs

2.2 HOM interference of independent photons

2.3 Method for characterizing the spectrum of single-photon state

3. Experimental example

3.1 Two-photon interference of SPDC photon pairs

3.2 Two-photon interference between phase randomized weak coherent states

3.3 Spectrum measurement of single-photon state

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Tables (1)

Equations (10)

Optics Express