High-performance quantum entanglement generation via cascaded second-order nonlinear processes

In this paper, we demonstrate the generation of high-performance entangled photon-pairs in different degrees of freedom from a single piece of fiber pigtailed periodically poled LiNbO$_3$ (PPLN) waveguide. We utilize cascaded second-order nonlinear optical processes, i.e. second-harmonic generation (SHG) and spontaneous parametric down conversion (SPDC), to generate photon-pairs. Previously, the performance of the photon pairs is contaminated by Raman noise photons from the fiber pigtails. Here by integrating the PPLN waveguide with noise rejecting filters, we obtain a coincidence-to-accidental ratio (CAR) higher than 52,600 with photon-pair generation and detection rate of 52.3 kHz and 3.5 kHz, respectively. Energy-time, frequency-bin and time-bin entanglement is prepared by coherently superposing correlated two-photon states in these degrees of freedom, respectively. The energy-time entangled two-photon states achieve the maximum value of CHSH-Bell inequality of S=2.708$\pm$0.024 with a two-photon interference visibility of 95.74$\pm$0.86%. The frequency-bin entangled two-photon states achieve fidelity of 97.56$\pm$1.79% with a spatial quantum beating visibility of 96.85$\pm$2.46%. The time-bin entangled two-photon states achieve the maximum value of CHSH-Bell inequality of S=2.595$\pm$0.037 and quantum tomographic fidelity of 89.07$\pm$4.35%. Our results provide a potential candidate for quantum light source in quantum photonics.

The SPDC in second-order nonlinear medium can generate bright photon-pairs, and therefore have been abundantly applied in telecom band quantum photonics. Nevertheless, the generation of photonpairs at 1.5 m via SPDC process is usually pumped with laser at 750 nm, thus requiring a sophisticated optical system able to manage light at quite different wavelength bands. The scheme of cascaded second-harmonic generation (SHG) and SPDC processes with two 26-28 or a single waveguide 29-31 has been developed to generate photon-pairs with all fiber-coupled deat 1.5 m. However, the performance of such quantum light sources is limited by the spontaneous Raman scattering (SpRS) noise. Although the SpRS noise has been experimentally investegated in those sources 29 , a quantum light source with low noise level is not yet demonstrated.
In this paper, we obtain high-performance entangled photon-pairs in different degrees of freedom by cascaded SHG/SPDC processes in a single piece of fiber pigtailed periodically poled LiNbO3 (PPLN) waveguide. The SpRS noise photons are effectively eliminated by integrating the waveguide waveguide, a bandstop filter (F2). Two fiber pigtails are used to connect the waveguide and the noise rejecting filters. The cascaded SHG and type-0 SPDC process could take place in the PPLN waveguide with a pump light to generate the correlated/entangled photon-pairs. The noise photons generated by the pump light before the input port of the module can be removed by F1 and the ones caused by the residual pump light output from the PPLN waveguide could be effectively eliminated by F2. In our design, the fiber pigtails connecting the PPLN waveguide and noise rejecting filters are both 20-cm long -limited by our fabrication process, in which few SpRS noise photons could be generated. In principle, these SpRS noise photons could be further reduced by getting rid of fiber pigtails in a fully integrated scheme, for instance we can directly integrate F1 and F2 on the both ends of the PPLN waveguide in the future. Table 1 Fig. 2c to characterize the energy-time entanglement. The signal and idler photons are first separated by DWDMs, and then pass through two unbalanced Mach-Zehnder interferometers (UMZIs, MINT, Kylia), respectively. The optical path difference between the long and short arms is 625 ps in both UMZIs, while an additional phase difference  or  between the two arms can be tuned by applying a voltage on a piezo actuator. The photons from each output port of the UMZIs are detected by SNSPDs and the corresponding photon counts are recorded by the TDC.
In our experiment, three peaks appear in the coincidence measurement between signal and idler photons output from the port A1 and B1, respectively, in the UMZIs shown in Fig. 2c. To observe the Franson interference 33 , we select the coincidence counts in the central peak with a time window of 300 ps when the phase differences  and  are scanned and fixed, respectively. As shown in With the setups shown in Fig. 2d, we coherently manipulate the energy-time entangled two-photon state by sending it into a single UMZI (see Supplementary Materials, Section 2). The coherent manipulation can prepare a superposition state of spatial bunched and anti-bunched path-entangled states, and the complex superposition coefficients of the two states can be fully manipulated by the additional phase φ in the UMZI. The spatial bunched path-entangled state is measured by the coincidence measurement between signal and idler photons from one output of the UMZI 37 , which is shown by the red circles in Fig. 3d. We can see that such coincidence shows cosinoidal oscillation with a visibility of 94.58±0.63% when the additional phase φ changes, indicating that the prepared state converts between the spatial bunched and anti-bunched path-entangled states. An attenuated CW laser is also injected into the UMZI shown in Fig. 2d and its single photon interference is observed, the fringe of which is shown by the rectangle curves in Fig. 3d and has a visibility of 98.10±0.01%. It is obvious that the period of the oscillation of coincidence between signal and idler photons is half that of the observed single photon interference. Such difference verifies that our coherent manipulation on energy-time entangled state is based on the interference of matter wave of the entangled two-photon state. Fig. 2e, the generated energy-time entangled photon-pairs are directly sent into a single UMZI, by setting the additional phase difference φ=π to prepare the photon-pairs to a spatial anti-bunched path-entangled state which is exactly a frequency-bin entangled state. The frequency-bin entanglement is characterized by the spatial quantum beating 37,39,40 . The two-photon state output from the UMZI is injected into a 50:50 BS with a relative arrival time delay τ between the two paths which is controlled by a VODL (variable optical delay line).   Figure   4d shows the three-fold coincidence counts in a time window of 300 ps between the synchronous electrical signal, and signal (A1 or A2) and idler (B1 or B2) photons when the phase α in one UMZI is fixed and β in another UMZI is scanned. The coincidence counts involving the port combinations of A1&B1, A1&B2, A2&B1, and A2&B2 all show remarkable interference fringes, and the raw visibilities of them are 94.59±2.43%, 92.12±2.51%, 90.30±2.36%, 94.05±2.39%, respectively.

Generation of frequency-bin entanglement. As shown in
With those Franson interference fringes, the violation of CHSH-Bell inequality can be observed 34 .
The correlation coefficient is defined as (2) where is the three-fold coincidence counts involving the port combinations Ai (i=1,2) and Bj The real and imaginary parts of this matrix are shown graphically in Figs. 4f and 4g, respectively, from which a fidelity of 89.70±4.35% is obtained with respect to the state ( and represent the qubit in early and late time-bins, respectively). The probable main causes of this limited fidelity includes the imperfection in double-pulsed pump laser, such as nonuniform pulse intensity, instable phase difference between double pulses, and limited extinction ratio, as well as the imperfection in the UMZIs, such as the unbalanced splitting ratio in BS and instability of phase differences.

Discussion
In this paper, we demonstrated a high-performance entangled photon source. To show the performance of our entangled photon source, we compare the CAR and the raw detected photon-pair rate (DPPR) of our photon source with previous works in which various nonlinear optical media are It is obvious that the off-diagonal elements in Eq. (4) are determined by V and φ obtained via Eq. (1).
Here, the diagonal element a has a meaning of the ratio of the state in the frequency-bin entangled state, and it can be estimated by measuring the counting rate of signal and idler photons from the UMZI. In our experiment, a=0.502±0.001 is obtained. The real and imaginary parts of the reconstructed density matrix are shown in Figs. 4b and 4c, respectively.

Quantum state tomography of time-bin entanglement.
With the theory given in Ref. [36], the projection measurements of a single time-bin qubit is implemented by passing through an UMZI with the time delay difference between the two arms equal to the time interval of time-bins. By injecting a time-bin qubit with state ( ) ( and represent the qubit in early and late time-bins, respectively) into the UMZI and detecting photons from one output port of the UMZI, the photon could be observed possibly in three time slots. Photon detection at the first (third) slot corresponds to a projection of state onto ( ), namely the "time basis". Reversely, detection at the middle slot corresponds to a projection of state onto , which is "energy basis" depending on the additional phase difference θ between the two arms.
In order to obtain the density matrix of the time-bin entangled photon-pairs, quantum state tomography must be implemented by 16 combinations of projection measurements between different bases ( ) for for signal and idler photons, where and . Figure 5f show some typical raw data in these projection measurements.
When the additional phase differences  and  in the UMZIs in Fig. 2c are both set at 0, two-fold coincidence between photons from the ports A1 and B1 in Fig. 2c are measured, and five distinguishable peaks appear in the coincidence histogram shown in Fig. 5f-(i) 34 , which correspond to the projection on single basis or the sum of projections on different bases. The three middle peaks in the Fig. 5f-(i) can further split into two or three peaks all corresponding to the projection on single basis, when threefold coincidence is implemented between the coincidence events in these peaks and the synchronous electrical signal, as shown in Fig. 5f-(ii, iii and iv). As a consquence, we obtain the two-photon projection measurements on the following bases simultaneously: and . In a similar way, we can perform the two-photon projection measurements on other bases by setting the  and  at 0&/2, /2&, and /2&/2.    d Results of two-photon interference (red circle) and single photon interference (blue rectangle).    is the temporal width of the coincidence window, set at 300 ps in our experiment.

Correlated photon-pairs
Since and ( ) Tdepend quadratically and linearly on the pump power, respectively, we can extractthe terms and in Eq. (S1) by selecting the quadratic and linear components in the polynomial fitting of the experimentally measured , respectively. Then we can calculate the and under different pump power via Eq. (S1), and the results are shown in Fig. S1c and S1d.

Coherent manipulation of the energy-time entangled two-photon state
The energy-time entangled two-photon state prepared in Fig. 2a can be approximated by 2 , where and are the creation operators of the signal and idler photons, respectively; is the joint spectral amplitude of the two-photon state, defined as the product of the pump envelope function and the phase matching function 3 ; is the photon flux spectral density 2 of the two-photon state. Since the first term in (S3), vacuum state , has no effect in measurement, we can only keep the second term in in the following calculations 4 . It is obvious that the states and are spatial bunched and anti-bunched path-entangled states, respectively. According to Eq. (S6), we can switch the two-photon state between and by setting at and respectively.
For the preparation of energy-time entangled two-photon state, CW pump laser is used and its coherence time is long sufficiently for us to approximate the envelope function in by . Moreover, the phase matching function in can be approximated by 1 The Eq. (S11) can describe the oscillation of coincidence counts shown by the red circles in Fig. 3d, except that the visibility in this figure is lower than 1 because of the noise photons, and the imperfect BSs and phase instability in UMZI. Also, it is obvious that ( ) is proportional to the projection probability of on the component of ( ) in ( ) shown in Eq. (S9), and therefore the oscillation of coincidence counts in Fig. 3d actually results from the oscillation of the coherently manipulated two photon state between the spatial bunched and anti-bunched path-entangled states. According to the Eq. Comparing to Eqs. (S16) and (S17), we can find that the component in has no contribution to the calculation of . Thus, Eq. (S16) evolves into . (S18) By substituting Eqs. (S15) and (S17) into Eq. (S18), we can finally obtain (S19) where is the spacing between the central angular frequencies of the two DWDMs in Fig. S2b. The Eq. (S19) gives a formula for us to fit the data of spatial quantum beating in Fig. 4a, except a visibility lower than 1 and a nonzero initial phase in the cosine function, resulting from the noise photons, the residual spatial path bunched photon-pairs, the imperfection of the BS used in Fig.   S2b, and the instability of the delay . We undertake four sets of projection measurements and the accumulation time of the coincidence counts is 10 s in all of the sets. The coincidence counts obtained in the four sets are summarized in Table S1. The 1 st column defines the number of the projection basis, with the states in each row in the 2 nd and 3 rd columns being the states of the signal and idler photons in the corresponding basis, respectively. Totally, 16 bases are used and the values in each row in the 4 th , 5 th , 6 th and 7 th columns are the coincidence counts when the two-photon state is projected on the corresponding basis. The additional phase differences  and  in the UMZIs in Fig. 2c are set at 0&0, 0&/2, /2&, and /2&/2 in the 4 th , 5 th , 6 th and 7 th columns, respectively. A dash (-) indicates that coincidence count is absent in the corresponding projection measurement. The ni value in each row in the rightmost column is sum of the values in this row from 4 th to 7 th columns, and it is finally used to reconsctruc the density matrix of the photon pairs 6 .

Coincidence counts in the quantum state tomography of time-bin entanglement
We use a set of SNSPD system with 6 channels for all measurements at single photon level. The

SNSPD devices are manufactured in the Shanghai Institute of Microsystem and Information
Technology (SIMIT). The entire detection system is developed by PHOTEC Corp., which including SNSPD devices, cryostat system and electronic control system. The SNSPDs operate at ∼2.2 K in the cryostat system with a dark counting rate of ∼150 Hz and a time jitter of ~100 ps. The dead time of all detectors are less than 30 ns and the efficiencies of the 6 channels at 1.5 μm band from 63% to 72% with an average of 67%. Each channel is characterized and the data is listed in Table S2.