Massive-mode polarization entangled biphoton frequency comb

A frequency-multiplexed entangled photon pair and a high-dimensional hyperentangled photon pair are useful to realize a high-capacity quantum communication. A biphoton frequency comb (BFC) with entanglement can be used to prepare both states. We demonstrate polarization entangled BFCs with over 1400 frequency modes, which is approximately two orders of magnitude larger than those of earlier entangled BFCs, by placing a singly resonant periodically poled LiNbO3 waveguide resonator within a Sagnac loop. The BFCs are demonstrated by measuring the joint spectral intensity, cross-correlation, and autocorrelation. Moreover, the polarization entanglement at representative groups of frequency modes is verified by quantum state tomography, where each fidelity is over 0.7. The efficient generation of a massive-mode entangled BFC is expected to accelerate the increase of capacity in quantum communication.

We briefly summarize the theoretical analysis of photon pair generation by SPDC inside a cavity, called the cavityenhanced SPDC [1][2][3][4][5][6][7][8]. Then, we explain the fitting function used for the analysis of the cross-correlation function of the photon pairs and its relationship with the spectral properties of the generated BFCs. A photon pair generated by a cavity-enhanced SPDC is represented by (S1) f (ω i , ω s ) represents the joint spectral amplitude (JSA) of the signal and idler photons, normalized by dω i dω s |f (ω i , ω s )| 2 = 1. α p (ω i + ω s ) denotes the spectral amplitude of the pump light. In the CW pump regime, the linewidth of the CW pump laser is sufficiently narrow, and we approximate it to α p (ω i + ω s ) ∝ δ(ω p − ω i − ω s ). h(∆k) = Lsinc(L∆k/2) represents a function of the phase-matching condition, where ∆k(ω i , ω s ) = n(ω i + ω s )(ω i + ω s ) − n(ω i )ω i − n(ω s )ω s , L denotes the length of the nonlinear crystal, and n denotes the refractive index [9]. A(ω) is a function that represents the transmission spectrum of the cavity [10][11][12]; it is represented as for a symmetric cavity without internal losses, where ∆(ω) = πc n(ω)L . L denotes the length of the resonator; we assume that is equal to the length of the nonlinear crystal, and c represents the speed of light. The resonant frequencies ω m for integers m, which are the peaks of the observed spectrum, are determined by the solutions of the equation ω = m∆(ω). The FWHM is ∆f 1/2 = γ/π and the FSR is c 2ngL , where n g (λ 0 ) = n(λ 0 ) − λ 0 dn dλ0 and λ 0 represent the group index and the wavelength in vacuum, respectively.
The temperature-dependent Sellmeier equation described in Ref. [13] for the rough estimation of the refractive index n(λ 0 ) is used to calculate the phase-matching function |h(∆k)| 2 shown in Fig. S1a. For the doubly resonant configuration, where both signal and idler photons are confined by the cavity, photon pair generation in most frequency ranges is suppressed because of the frequency dependency of the FSR, which is known as the "cluster effect" [14]. Figure S1c shows the simulation of the spectrum of the biphoton frequency comb in the doubly resonant configuration from equation (S1), where the cluster effect appears clearly.
We realized a singly resonant configuration using mirrors whose reflectance is frequency-dependent, as shown in Fig. S1b. Using these values of the reflectance, we simulate the spectrum of the biphoton frequency comb in our experiment, as shown in Fig. S1d. The biphoton frequency comb is generated with a certain intensity in all frequency ranges. The cluster effect does not appear if the singly resonant configuration is completely realized. However, this is not realized around the degenerate point of 1560.48 nm in our experiments. The cluster effect around the degenerate point causes a nonuniform shape in the spectrum of the biphoton frequency comb. The raw joint spectral intensities in Fig. S1c and S1d are very sensitive to the frequency difference because of their fine comb shape. Thus, we need to integrate these in the bandwidth for the measurement of 0.3 nm for comparison with the measurement results of the JSI. Figs. S1e and S1f show the results after integrating Fig. S1c and S1d, in this range, respectively.  We further simplified JSA f (ω i , ω s ) to analyze the cross-correlation measurements. The frequency windows of these measurements in our experiment are up to 3 nm, which is sufficiently small to neglect the frequency dependency of n(ω) and γ(ω) and the cluster effect within its frequency windows. Therefore, we replace m∆ by a resonant frequency ω m . When we set the pump frequency to satisfy ω m + ω m ′ ≃ ω p and represent γ(ω s(i) ) by γ s(i) , we have In the last equality, only the term about m ′ that satisfies ω m ′ = ω p − ω m is left. The case γ i = γ s corresponds to the (equally) doubly resonant configuration, and the case γ i → ∞ corresponds to the singly resonant configuration, where only signal photons are confined.
In the measurement of the coincidence counts between the signal and idler photons with a finite temporal resolution, the observed cross-correlation C s,i is described by where is a temporal second-order cross-correlation function, and is a Gaussian function that approximates the finite temporal resolution of the two photon detectors used for coincidence measurement. [3]. From equations (S1), (S3), and (S4), h(τ ) represents the step function [7]. Equation (S8) becomes when we extract a single-frequency mode using BPFs: where erf(x) denotes an error function. When we extract many frequency modes enough to the approximation, where T 0 = 2π/FSR represents the round-trip time of the cavity. Equation (S8) then becomes Coincidence counts are represented by equation (S9) if the state is a statistical mixture of the photon pairs of the different frequency modes. The cross-correlation can be described by considering a mixture of equations (S9) and (S11) with a normalization factor with relative weight p as In our experiment, the best fit of this function in equation (S12) to the observed temporal second-order crosscorrelation in Fig. S4 leads to p ≃ 1.

Analysis of the state generated from the Sagnac interferometer and its fidelity
We considered only one photon-pair state generated by the SPDC. As shown in Fig. S2, owing to the cavity structure for the signal photons, H-polarized and V-polarized pump lights respectively generate photon pairs described by where where η i(j)l(r) represents the total loss in the optical circuit for the signal (idler) photons coming from the left (right) side of the PPLN/WR. By setting the polarization of the pump light to satisfy g H η ir t l η sr = g V η il t r η sl , we obtain where For ∆τ ≫ 0, we obtain the maximally polarization-entangled state |Φ⟩ ≡ 1/ √ 2 |H⟩ i |H⟩ s + e iϕ |V ⟩ i |V ⟩ s . However, in our experiment, there is an overlap between |V ⟩ i |H⟩ s and |Φ⟩, as indicated in Fig. S3. To estimate these effects on the fidelity, we consider the delay time dependency of the generated state as Here, we approximated the temporal shaping of the state by e −γτ . In our experiment, we selected the time windows from 0 to ∆τ , i.e., We redefined this as Further, the randomization of the phase of the component |V ⟩ i |H⟩ s leads to From the definition of fidelity, holds. From these results, we estimated the fidelity expected in the case of ∆τ ≫ 0 by multiplying F (ρ Ψ ) −1 with the experimentally measured fidelity (Fig. 5 in the manuscript). From the best fit to the data of the temporal crosscorrelation in every polarization and frequency range, we extracted γ and β H , which are estimated as the ratio of the amplitude between the VH and HH components.

Analysis of the generated polarization entangled BFC
In the manuscript, we discuss a simplified model for generating polarization-entangled BFCs. Here, we analyze the polarization-entangled BFCs generated in the experiment using a more realistic model. As well as equation (S1), the time evolution of the cavity-enhanced SPDC for arbitrary pump power is represented by Equation (S24) is reduced to approximately equation (S1) when |g| 2 ≪ 1 is satisfied. In the CW pump regime, the We assume that each frequency mode is well separated from the others, that is, The generation processes of each cavity mode are independent.
Modifying equation (S17) for an arbitrary pump power using the above results, the state generated by our experiments can be represented as where we ignored the component of the reflected signal photons. In the case of |g H c H,m | 2 + |g V c V,m | 2 ≪ 1, equation (S27) is approximately This state can be considered a frequency-multiplexed entangled photon pair, which is similar to the state in equation After removing the vacuum state, this state may correspond to that in equation (2) in the manuscript. The difference between these states discussed here and the simplified model in the manuscript is that the spectral correlation between the signal and idler photons remains in each mode in these states; this often has unfavorable effects on the interference between independent photon pairs.
We have to use a pulsed-pump regime to generate the exact frequency-bin entangled state, which is expressed as This is approximately attained in the doubly resonant configuration by making the FWHMs of the cavities for the signal and idler photons considerably smaller than the FWHM of the pump laser because the frequencies of the signal and idler photons are determined independently and almost exclusively by the FWHMs of the cavities, respectively. In the singly resonant configuration, the exact frequency-bin entangled state itis also approximately attained by making the FWHM of the cavity for the signal photon considerably smaller than the FWHM of the pump laser. Then, the frequencies of the signal and idler photons are determined almost exclusively by the FWHMs of the cavity and pump laser, respectively. In this case, the FWHMs of both photons would differ significantly from each other.

Theory of the autocorrelation function and the cavity mode in CW pump regime
Ref. [15] reported that a value g (2) relevant to g (2) (τ ) of the signal or idler photon of a photon pair in the pulse pump regime is related to the number of effective modes of a photon pair. The effective mode indicates the Schmidt mode of continuous variable Schmidt decomposition [16,17]. Under the assumption of a uniform spectral amplitude, the relationship between g (2) and the number of effective modes N is expressed as g (2) = 1 + 1 N . A similar discussion is held in the CW pump regime [7], wherein g (2) (0) ≃ 1 + 1 M is derived with the assumption that detectors have a finite temporal resolution, where M represents the number of cavity modes. These results yield the same formula and have been applied in many experiments using pulse lasers [18,19] and CW lasers laser [20][21][22]. Here, we present another formula for the CW pump regime, which is independent of the temporal resolution of the detectors.
From equation (S3), we define f (ω i ) and f m (ω i ) as f (ω i ) and f m (ω i ) are normalized. f m (ω i ) represents the normalized spectral amplitude of each cavity mode, and we assume that each cavity mode is orthogonal. We calculate the second-order autocorrelation function of the idler photons of the photon pairs as Using the second-order term in equation (S24) without polarization dependency as the lowest-order approximation for |Ψ⟩, we obtain where F |f (ω i )| 2 represents the Fourier transform of the spectral intensity of the idler photon. When we define the normalized autocorrelation function as g (2) i (S33) This result shows that g (2) i,i (0) = 2, which is independent of the spectral characteristics of photon pairs. In the single mode case [7], equation (S33) is calculated as g (2) i,i (τ ) = 1 + In the multimode case, g (2) i,i (τ ) becomes more complex, and therefore, we alternatively define ∆g (2) i,i ≡ dτ g (2) i,i (τ ) − 1 .
The range of the temporal integral is assumed to be sufficiently large. From equation (S33), Substituting equation (S30) to equation (S36), we obtain ∆g (2) i Therefore, if γ i and γ s are known, the number of cavity modes can be estimated using this formula. In addition to equation (S4), the actual value obtained from the measurement is expressed as After integration over time, we obtained ∆g Therefore, ∆g (2) i,i can be measured without depending on the temporal resolution of the detector. In our experiment, we estimate the value of G (1,1) i,i (τ → ∞) from the best fit to the data far from the point of τ = 0, as indicated by the orange lines in Fig. 4. We then calculated ∆g (2) i,i as 2.1 ns and 0.17 ns, respectively. Combining the values of γ s obtained in Table. S1, we estimate the number of modes M to be 1.2 and 14.9, respectively.  Figure S4 shows all the results of the fitting obtained using equation (S12) for the bandwidth of 3.00 nm and equation (S9) for the bandwidth of 0.03 nm. The fitting results indicate that p ≃ 1 in equation (S12), and the results of the fitting using equation (S12) are almost the same as the results of the fitting using equation (S11). The properties of the generated frequency comb, which are estimated by the best fit to the data, are summarized in Table S1. The FWHM depends on the frequency because it is related to the frequency-dependent reflectance of PPLN/WR [23].

Another method for generating the polarization entangled state using all polarization components
From equation (S17), we can utilize the entire state as the polarization-entangled state by setting the length of the arms of the Sagnac interferometer to the same values: τ r = τ l with the stabilization of the relative phase. The inner product of the two terms in equation (S17) satisfies where we used the relation r * l t r + t * l r r = 0. Therefore, in the symmetric loss case, that is, η sl = η sr , the state in equation (S17) represents a maximally entangled state. However, asymmetric losses degrade orthogonality and cause a decrease in the fidelity to the maximally entangled state. The lower bound of its fidelity is In the second equation, |r r | = |r l | ≡ |r|, |t r | = |t l | ≡ |t|, |r| 2 + |t| 2 = 1, and 1 − β H β V = 1 − r r r l /t r t l = |t| −2 .
This lower bound is almost 1 in the typical experiments, and this method works well. This approach has twice the generation rate and does not require a temporal distinction.