1. Introduction

Hong-Ou-Mandel(HOM) interference [1] between independent photon sources (HOMI-IPS) is at the heart of quantum information processing involving the quantum interference of single photons [2,3]. Scaling such systems to large dimensions requires many single-photons to interfere in a single mode [4,5]. This establishes a fundamental challenge since the outcome fidelity of the quantum processor depends on the interference visibility of all photons in all modes. To insure high output-fidelity the interference of at least two independent photons must approach unity.

A variety of HOMI-IPS experiments have been demonstrated at near-infrared wavelengths [6–13] or telecom wavelengths [14–20], and with heralded single-photon sources [6–8,18–20], weak coherent sources [9, 10], or even thermal sources [18, 21]. In all the previous HOMI-IPS experiments, the HOM dips are obtained by recording the coincidence counts as a function of the temporal delay. The knowledge of the spectral correlations of the interfering photons can reveal some important information about the interference visibility [22]. In the past, the spectral correlation content could not be observed, partially due to the long acquisition times required when measuring spectral correlations using two tunable bandpass filters.

Recently, a new technique for measuring spectral correlations in a HOM interference was presented by Gerrits, et al [23,24]. This technique has the merits of high speed and high spectral resolution, and therefore allows measuring and analyzing the spectral correlations from a HOM interference.

Here, we measure the spectral correlation between the two output ports of the HOMI-IPS beam splitter with two different photon sources (heralded single-photon sources, and thermal sources) using the method developed in [23]. This experiment can help deepen our understanding of HOMI-IPS from the viewpoint of spectral domain and allows improving the HOMI-IPS visibility by post-processing spectral filtering.

First, we present the theory and simulation for HOM interference between two heralded single-photon states, and between two thermal states. We then present our experimental results for these two kinds of HOMI-IPS experiments. The appendix of this manuscript contains our theoretical model for HOM interference between two heralded single-photon states, by considering the infinitely higher order components in the photon sources.

2. Theory and simulation

2.1. Theory of four-fold HOM interference between two heralded single-photon states

Two independent spontaneous parametric down conversion (SPDC) sources are used in the experiment. The signal and idler photons from the first and the second SPDC source have the joint spectral amplitude (JSA) of f₁(ω_s1,ω_i1) and f₂(ω_s2,ω_i2), where ω is the angular frequency. The subscripts 1 and 2 represent the first and the second SPDC, respectively; s and i denote the down-converted photons, i.e., the signal and idler. The signal photons (s1 and s2) are sent to a 50/50 beamsplitter for HOM interference, while the idler photons (i1 and i2) function as heralders, i.e., detected by single-photon detectors to claim the existence of s1 and s2. The four-fold coincidence probability P₄(τ) as a function of the time delay τ can be written as [22]:

The correlated spectral intensity (CSI), ${\tilde{I}}_4(\tau )$, for the two beam splitter output ports at time delay τ can be expressed as:

Figure 1 shows the simulated HOM dip and CSI using the experimental parameters of the setup: ~2 ps pump pulses at 792 nm, 30-mm-long periodically poled KTiOPO₄ (PPKTP) crystal [25]. As shown in Fig. 1(a), without any bandpass filtering, the visibility of the four-fold HOM dip is 82.0%, and the FWHM (full width at half maximum) is 4.91 ps (1.47 mm). The intrinsic spectral purity of this source is 0.82 [25, 26], which is also the upper bound of the four-fold HOM interference visibility in our case [7, 18]. The theoretical predictions in Figs. 1(b)–1(g) show fringes in the CSI at different delay positions. The fringe frequency increases with increasing distance from the center of the HOM dip.

 

Fig. 1 The simulation for the HOM interference between two heralded single-photon states. (a) The simulated HOM dip. (b–e) Simulated correlated spectral intensity (CSI) at different delay positions in the HOM dip.

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2.2. Theory of two-fold HOM interference between two thermal states

The two-fold coincidence probability P_t(τ) between signal1 and signal2 (without the gating of idler1 and idler2, signal1 and signal2 are thermal states) as a function of the time delay τ can be written as [27]

In fact, I₄(τ) in Eq. (2) can be expressed as [27]

Eq. (5) and Eq. (9) suggest the HOM interference between two thermal states is including the HOM interference between two heralded single-photon states $\mathcal{A}-ℰ(\tau )$, plus a constant background term of $\mathcal{A}+ℰ$. The correlated spectral intensity (CSI), ${\tilde{I}}_t(\tau )$, for the two beam splitter output ports at time delay τ can be expressed as:

With the same condition for Fig. 1, we simulate the two-fold HOM dip in Fig. 2(a), and ${\tilde{I}}_t(\tau )$ at different delay positions is shown in Figs. 2(b)–2(g). The visibility V of HOM interference between two thermal state has an upper bound of $\frac{1}{3}$ [27]:

 

Fig. 2 The simulation for the HOM interference between two thermal states. (a) The simulated two-fold HOM dip. (b-g) CSI at different delay positions in the HOM dip.

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3. Experiment and results

3.1. Experiment of four-fold HOM interference between two heralded single-photon states

The experimental setup is shown in Fig. 3. The experimental setup is the extension of our previous work [18]. We add two kinds of dispersion fibers to the four detection channels. Each output port of the fiber beam splitter (FBS) is connected to a dispersion compensation fiber module (DCFM), a 7.53-km-long single-mode fiber (SMF) with high dispersion designed to compensate the dispersion of a 50-km-long commercial single-mode fiber (SMF) at the L-band of telecom wavelengths. The DCFM has a dispersion of 125.0 ps/km/nm and an insertion loss of 5.2 dB. Due to the large dispersion, the DCFM converts the spectral distribution information of the photons into arrival time information, which can be determined with high accuracy using superconducting nanowire single-photon detectors (SNSPDs). Our SNSPDs have a system detection efficiency of around 70% with a dark count rate less than 1 kcps [28–31]. The DCFMs, SNSPDs and the time interval analyzer (TIA) constitute a fiber spectrometer [32, 33]. To ensure that the idler photons have the same optical path delay as the signal photons, we add two 7.53-km-long commercial SMFs (L-SMF in Fig. 3) to the idler channels. L-SMF has a dispersion of 27.3 ps/km/nm and an insertion loss of 1.6 dB. Assuming a system jitter of 100 ps, the resolution of our fiber spectrometer is approximately 0.11 nm.

 

Fig. 3 The experimental setup. Picosecond laser pulses (76 MHz, 792 nm, temporal duration ~ 2 ps) from a mode-locked Titanium sapphire laser (Mira900, Coherent Inc.) were divided by a polarizing beam splitter (PBS) into two paths, and pumped two 30-mm-long PPKTP crystals with a poling period of 46.1 µm for type-II SPDC. The downconverted signal (horizontal polarization) and idler (vertical polarization) photons with a degenerate wavelength of 1584 nm were divided by two PBSs, and then coupled into single-mode fibers (SMFC). The signal photons were sent to a 50/50 fiber beam splitter (FBS) and connected to two dispersion compensation fiber modules (DCFM), in order to generate large dispersion. The idler photons were connected to two long-distance single-mode fiber spools (L-SMF), to achieve the same path delay as the signal photons. Finally, all collected photons were sent to four superconducting nanowire single-photon detectors (SNSPDs), which were connected to four input channels of a time interval analyzer (TIA). Electrical signals recording the timing information of the pump pulses were sent to the TIA for synchronization (Sync). LPF = long pass filters, HWP = half-wave plate, QWP = quarter-wave plate, FPC = fiber polarization controller.

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We pump the PPKTP crystal sources with a Ti:Sapphire oscillator at 76 MHz, each with an average pump power of 100 mW. We obtain a singles count rate of ~1.6 Mcps, a two-fold coincidence count rate (2f-CC) of 350 kcps, and a four-fold coincidence count rate (4f-CC) of 880 cps. The mean photon number per pulse and the total efficiency of each channel are then estimated to be $\overline{n}=0.114$ and η = 0.19, respectively. After insertion of the DCFM and the L-SMF, the 2f-CC and 4f-CC are reduced to 35 kcps and 37.7 cps. First, we measure a HOM interference between signal1 and signal2, with idler1 and idler2 as heralders. The result is shown in Fig. 4(a). Without subtraction of any background counts, the raw visibility of the HOM dip in Fig. 4(a) is 68.9±1.3%, which is consistent with our previous results in [18, 34]. The degradation from the expected visibility of 82.0% (Fig. 1(a)) to 68.9 ± 1.3% is mainly caused by the multi-pair emission in the SPDC [35]. This is confirmed by the numerical simulation. Based on the theory in [35], we develop a theoretical model including losses, multi-pair photon emissions (up to infinitely higher order photons), and the realistic frequency mode structure, i.e., the joint spectral amplitude (see Appendix). For $\overline{n}=0.114$ and η = 0.19, the visibility is simulated to be 68.1% which agrees with the experimental result. Note that in principle, the visibility can be further improved by decreasing the pump power, but with the tradeoff of lower count rate. The FWHM of this HOM dip is 4.44 ps (1.33 mm), which corresponds well with the theoretical prediction in Fig. 1(a).

 

Fig. 4 (a) The measured four-fold HOM dip between signal1 and signal2, with idler1 and idler2 as herladers. (b–e) Measured correlated spectral intensity (CSI) at different delay positions in the HOM dip. Each CSI data was accumulated in 2 hours.

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Figures 4(b)–4(g) show the measurements of the CSI at different delay positions, 0 FWHM to 5 FWHMs away from the dip center. The fringe observed at the HOM dip is very weak (shown in Fig. 4(b)), as expected from Eq. (3). The CSI clearly shows fringes at delay positions from 1FWHM to 4FWHM in Figs. 4(c)–4(f). The shape of the CSI is almost circular at 5FWHM (Fig. 4(g)), and fringes are not observed due to the limited resolution of our fiber spectrometer. The observed CSIs from 0FWHM to 4FWHM agree well with our theoretical predictions in Figs. 1(b)–1(f).

3.2. Experiment of two-fold HOM interference between two thermal states

Thermal state is not only fundamentally important in quantum optics, but also practically useful for quantum computation [36] and quantum information processing [37]. Therefore, it is meaningful to investigate the spectral correlation between two thermal states. In the following we present our results when measuring the two-fold coincidences between signal1 and signal2, without the heralding of idler1 and idler2, i.e., two thermal states.

The measured HOM dip is shown in Fig. 5(a), with a visibility of 24.6 ± 0.3% and FWHM of 4.62 ps (1.39 mm). This obtained visibility is comparable to our previous results in [18]. The measured CSI data are shown in Figs. 5(b)–5(g), with accumulation time of 10 minutes for each data set. Surprisingly, the CSIs also show fringes at the different delay positions from 1FWHM to 4FWHM, even with an interference visibility as low as 24.6±0.3%. These measured results correspond well with our simulated results in Fig. 2. The degradation from the expected visibility of 29.1% (Fig. 2(a)) to 24.6±0.3% is mainly caused by the limited indistinguishability of the two different SPDC sources.

 

Fig. 5 (a) The measured two-fold HOM dip between signal1 and signal2. (b–e) Measured CSI at different delay positions in the HOM dip. Each data was accumulated in 10 minutes.

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4. Discussion and outlook

To further investigate the spectral properties of individual SPDC sources, we show the measured joint spectral intensity (JSI) of each PPKTP crystal in Fig. 6. Figure 6(a) is experimentally measured JSI of photon pairs from PPKTP1, i.e., |f₁(ω_s1,ω_i1)|². Figure 6(b) indicates the square root of the JSI, which is absolute value of the joint spectral amplitude (JSA), i.e., |f₁(ω_s1,ω_i1)|. To view the side lobes in Fig. 6(b) more clearly, Fig. 6(c) provides the plot of Fig. 6(b) in logarithmic scale. Similarly, Figs. 6(d)–6(f) show the JSI (|f₂(ω_s2,ω_i2)|²), the square root of JSI (|f₂(ω_s2,ω_i2)|) and its log plot for PPKTP2. It can be noticed that the spectral distributions of the photon pairs from PPKTP1 and PPKTP2 are slightly different, mainly due to the difference in fabrication of the crystals. The theoretical simulations of the corresponding data are shown in Figs. 6(g)–6(i). Both the measured spectral distributions in Fig. 6(a) and Fig. 6(d) correspond well with the theoretical value in Fig. 6(g).

 

Fig. 6 The experimentally measured joint spectral intensity (JSI), square root of JSI and the log plot for photon pairs from PPKTP1 (a, b, c), and from PPKTP2 (d, e, f). The corresponding theoretical simulated results (g, h, i).

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It is meaningful to compare the HOMI-IPS (using two SPDCs, e.g., the case in this experiment) to the HOM interference using a twin photon source (using only one SPDC, e.g., the case in [1]). In the latter case, the two-fold coincidence probability P₂(τ) as a function of the time delay τ in the HOM interference is given by [38–40]:

The CSI of the two beam splitter output ports I(τ) in this HOM interference can be expressed as:

To achieve perfect HOM interference, Eq. (12) requires exchange symmetry of the joint spectral distribution, i.e., f (ω_s,ω_i) = f (ω_i,ω_s), but doesn’t require high spectral purity. However, to achieve high visibility, Eq. (1) doesn’t require exchange symmetry, but requires a high spectral purity. In Eq. (1), to obtain 100% visibility, i.e., P₄(0) = 0, the condition of f₁(ω_s1,ω_i1) f₂(ω_s2,ω_i2) = f₁(ω_s2,ω_i1) f₂(ω_s1,ω_i2) must be satisfied. To achieve this condition for all ω, firstly the two SPDC sources must be equal (indistinguishable): f₁(ω_s,ω_i) = f₂(ω_s,ω_i) = f (ω_s,ω_i), and secondly the two sources must be factorable (spectrally pure): f (ω_s,ω_i) = g_s(ω_s)g_i(ω_i) [22].

Besides the observation of the CSI, the technique in this work can also be applied to improve the visibility of HOMI-IPS by post-processing spectral filtering. Here, we can improve the visibility from 68.9±1.3% (in Fig. 4(a)) to 75.7±1.3% (in Fig. 7(a)) and 77.5±1.5% (in Fig. 7(b)), by decreasing the coincidence window from 5 ns to 2 ns and 1 ns, corresponding to spectral widths from 2.66 nm to 1.06 nm and 0.53 nm, respectively. The HOM interference visibilities with different coincidence window widths are shown in Fig. 7(c), which clearly shows that the visibility increases as the width of the coincidence window decreases. Thus, narrowing the coincidence window acts as a spectral filter. When narrowing the coincidence window, the spectral purities of the sources are improved and the HOM visibility is improved. One can utilize temporal filtering to enhance the visibility of the HOMI-IPS, however, at the cost of lower photon flux.

 

Fig. 7 (a) Post-processed HOM dip with a coincidence window of 2 ns (1.06 nm). (b) Post-processed HOM dip with a coincidence window of 1 ns (0.53 nm). (c) Post-processed HOM dip visibility as a function of the coincidence window.

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HOM interference is a very fundamental phenomenon which has been experimentally observed with photons not only from parametric down conversion [1], but also from semiconductor quantum dots [41–43], atoms [44–46], plasma [47–49] and microwave system [50]. Therefore, the technique of spectral correlation measurement in HOMI-IPS in this work may also be applied to many other fields in physics.

5. Conclusion

In summary, we have theoretically simulated and experimentally demonstrated the correlated spectral intensity (CSI) in a four-fold HOM interference between two heralded single-photon sources, and also in a two-fold HOM interference between two thermal sources. The interference pattern in the CSI is clearly observed and matches our theoretical predictions well. This experiment can help deepen our understanding of HOMI-IPS from the viewpoint of spectral domain. Further, our work also presents a tool to test the theoretical predictions of HOMI-IPS using spectrally engineered sources.

Appendix: The theoretical model of the four-fold HOM interference

This appendix briefly describes our theoretical model of the four-fold HOM interference. The model includes losses in the setup, the multi-pair emission (up to infinitely higher order), and the frequency multimode structure of the pulsed photon-pair source. Note that in the theoretical model of heralded single-photon state for Eq. (1) and Eq. (12), only the one-pair component in SPDC is considered. While in the theoretical model of thermal state for Eq. (5), both the one-pair and two-pair components in SPDC are included.

An SPDC theory including the first two (losses and multi-pair emission) was developed in [35] where the quantum state emitted from the SPDC source is described as a Gaussian continuous variable state, more precisely two-mode squeezed vacuum, which is distributed in a single frequency mode. For pulsed SPDC sources, however, photon-pairs are emitted with more complicated frequency distribution and thus has a multi-mode structure in frequency. This is particularly important to simulate the experiments including the interference between the photons from independent sources [2–5]. Here we show how the frequency multimode structure can be incorporated into the theory in [35] (complete description of the theory will appear elsewhere).

The quantum state emitted from a pulsed SPDC source is described by [38],

(14)$$|\mathrm{\Psi }〉=\exp \left[r{\displaystyle \int d{\omega }_s}{\displaystyle \int d{\omega }_{i}f({\omega }_s,{\omega }_i){\widehat{a}}_s^{†}({\omega }_s){\widehat{a}}_i^{†}({\omega }_i)-h.c.}\right]|0〉,$$

where r is the squeezing parameter (pump power), ${\widehat{a}}_s^{†}({\omega }_s)$ and ${\widehat{a}}_i^{†}({\omega }_i)$ are creation operators for the signal and idler modes with frequencies ω_s and ω_i, respectively. f (ω_s,ω_i) is the joint spectral amplitude, which is a product of the pump distribution and the phase-matching function of the nonlinear crystal [7, 38, 51]. We can assume r to be real and positive without losing generality. This joint spectral amplitude can be decomposed by the Schmidt decomposition as [7,51]

(15)$$f({\omega }_s,{\omega }_i)={\displaystyle \sum _l\sqrt{{\lambda }_l}}{g}_l({\omega }_s)h_l({\omega }_i),$$

where λ_l is the Schmidt eigenvalue for the lth Schmidt mode. This decomposition allows us to describe the state in Eq. (14) as

(16)$$\begin{array}{l}|\mathrm{\Psi }〉=\exp \left[r{\displaystyle \sum _l\sqrt{{\lambda }_l}}{\widehat{b}}_l^{†}{\widehat{c}}_l^{†}-h.c.\right]|0〉\\ ={\displaystyle \prod _l\exp \left[r\sqrt{{\lambda }_l}{\widehat{b}}_l^{†}{\widehat{c}}_l^{†}-h.c.\right]}|0〉\\ =|\mathrm{\Psi }(r\sqrt{{\lambda }_1})〉|\mathrm{\Psi }(r\sqrt{{\lambda }_2})〉\cdots ,\end{array}$$

where

(17)$${\widehat{b}}_l^{†}={\displaystyle \int d{\omega }_s{g}_l({\omega }_s){\widehat{a}}_s^{†}({\omega }_s),}$$

(18)$${\widehat{c}}_l^{†}={\displaystyle \int d{\omega }_i{h}_l({\omega }_i){\widehat{a}}_i^{†}({\omega }_i).}$$

Note that ${\widehat{b}}_l$ and ĉ_l satisfy a standard bosonic commutation relation $\left[{\widehat{b}}_l,{\widehat{b}}_m^{†}\right]=[{\widehat{c}}_l,{\widehat{c}}_m^{†}]={\delta }_{lm}$. In the second line of Eq. (16), decomposition of the exponential term is possible since the Schmidt modes are orthonormal. The last line of Eq. (16) represents that in the Schmidt mode basis, the state is described by a tensor product of two-mode squeezed vacuum $|\mathrm{\Psi }(r\sqrt{{\lambda }_l})〉$:

(19)$$|\mathrm{\Psi }(r\sqrt{{\lambda }_l})〉=\frac{1}{\cosh r\sqrt{{\lambda }_l}}{\displaystyle \sum _n{\left(\tanh r\sqrt{{\lambda }_l}\right)}^n{|n〉}_{B_l}|n{〉}_{C_l},}$$

with the effective squeezing parameter $r\sqrt{{\lambda }_l}$. Therefore, to extend the single frequency mode based theory in [35], what need is simply consider a tensor product of multiple two-mode squeezed vacua (see Fig. 8).

 

Fig. 8 (a) single frequency mode SPDC source. (b) pulsed (multiple frequency mode) SPDC source in Schmidt mode expansion.

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Distribution of the Schmidt eigenvalues are theoretically calculated using the experimental parameters of our setup: ~ 2 ps pump pulses at 792 nm, 30-mm-long periodically poled KTiOPO₄ (PPKTP) crystal. Under this condition, it can be calculated that {λ₁, λ₂,…} = {0:9;0:025;0:025;0:01;0:009;0:004…} [25, 52–54]. In our numerical simulation, we include the first five Schmidt modes and truncate the rest.

To fit the simulation with the experiment, we also need to connect r with the experimentally observable parameter, for example, the photon-pair generation rate of the SPDC source. The latter corresponds to the probability that the source emit non-zero photons:

(20)$$p=1-{\displaystyle \prod _l{|〈00|\mathrm{\Psi }(r\sqrt{{\lambda }_l})〉|}^2}.$$

Pluging Eq. (19) into, Eq. (20), we obtain the relation

(21)$$p=1-{\displaystyle \prod _l{\left(\cosh r\sqrt{{\lambda }_l}\right)}^{-2}},$$

which allows us to derive r numerically from experimentally estimated p. Note that the photon pair generation rate p is related to the mean photon number per pulse $\overline{n}$ via $p=\overline{n}/(\overline{n}+1)$.

The calculation of the four-fold HOM dip is then carried out by extending the two-fold HOM dip analysis in [35]. Instead of a single TMSV in [35], we start from the covariance matrix of two TMSVs and then after applying beam splitting operation with mode matching factor ξ, we obtain the four-spatial-mode covariance matrix γ(r,ξ) where r is the squeezing parameter of the TMSVs (see [35] for the details of the covariance matrix approach). The HOM visibility is defined by

(22)$$V=\frac{P_{\text{mean}}-P_{\min }}{P_{\text{mean}}},$$

where P_mean and P_min are the four-fold coincidence count probabilities with and without optical path delay, respectively. These coincidence counts are then given by

(23)$$P_{\text{mean}}=1+{\displaystyle \sum _{m=1}^4{\displaystyle \sum _{C(4,m)}^{{}_4{C}_m}\frac{{\left(-2\right)}^m}{\sqrt{{\displaystyle \prod {}_i\det ({\gamma }_{C(4,m)}(r_i,0)+I)}}}}}.$$

P_min is also obtained by a similar expression with γ_C(4,m)(r_i,1) instead of γ_C(4,m)(r_i,0). γ_C(4,m)(r,ξ) is the submatrix of γ(r,ξ) taking modes C(4,m). C(4,m) is the m-detector combination from the four detectors (see Sec. 3.3 of [35]). $r_i=r\sqrt{{\lambda }_i}$ is the effective squeezing parameter for the i-th Schmidt mode where we include the first five modes. The HOM visibility is then obtained by plugging the above P_mean and P_min into Eq. (22).

Acknowledgments

The authors are grateful to S. W. Nam for helpful discussion, and to K. Yoshino and K. Wakui for assistance in experiment. This work is supported by MEXT Grant-in-Aid for Young Scientists(B) 15K17477, and the Program for Impulsing Paradigm Change through Disruptive Technologies (ImPACT).

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