
Integrated photonics is a leading platform for quantum technologies including nonclassical state generation^{14}, demonstration of quantum computational complexity^{5} and secure quantum communications^{6}. As photonic circuits grow in complexity, full quantum tomography becomes impractical, and therefore an efficient method for their characterization^{7, 8} is essential. Here we propose and demonstrate a fast, reliable method for reconstructing the twophoton state produced by an arbitrary quadratically nonlinear optical circuit. By establishing a rigorous correspondence between the generated quantum state and classical sumfrequency generation measurements from laser light, we overcome the limitations of previous approaches for lossy multimode devices^{9, 10}. We applied this protocol to a multichannel nonlinear waveguide network and measured a 99.28±0.31% fidelity between classical and quantum characterization. This technique enables fast and precise evaluation of nonlinear quantum photonic networks, a crucial step towards complex, largescale, device production.
Practical applications of quantum photonic technologies^{11, 12} require the integration of linear and nonlinear waveguides on a single device, where photons can be generated^{1, 4} and manipulated^{13}. Spontaneous parametric downconversion (SPDC) and spontaneous fourwave mixing are the two most common processes used for photon generation on chip with the former being the most efficient by far, needing only a few microwatts of pump power for generation rates exceeding several MHz^{14, 15}. Monolithic integration of SPDC sources with multiport optical circuits has been achieved in several contexts, with applications in quantum communication^{16}, quantum metrology^{1}, spatial multiplexing of heralded singlephoton sources^{17}, quantum state generation in nonlinear waveguide arrays^{2} and smallscale reconfigurable quantum photonic circuits^{18}.
The near future of quantum photonics will involve an expansion in scale and applications of integrated circuits. However, the characterization of the twophoton state generated by a nonlinear waveguide network is a cumbersome experimental task^{19}, requiring the collection of statistics from coincidence counts and a quadratically increasing number of measurements with system size. Here we propose and demonstrate a practical method for the characterization of the twophoton wavefunction generated by an arbitrary device with quadratic nonlinearity that uses only laser probes and power measurements. This technique fully reconstructs the spectral and spatial properties of the generated photon pairs from the measurements of bright optical beams and, with optimized hardware, it performs the same number of measurements at least four orders of magnitude faster than the corresponding quantum characterization. Our protocol is of both fundamental and practical importance for the development of integrated quantum photonics technologies including characterization of largescale wafer production.
A method based on stimulated emission tomography (SET) was proposed^{9} for predicting the twophoton wavefunction produced by a nonlinear device using the analogy between spontaneous nonlinear processes and their classical stimulated counterparts, that is, differencefrequency generation or stimulated fourwave mixing. This technique was demonstrated for spectral characterization of twophoton states^{2023}, and fast reconstruction of the density matrix of entangledphoton sources^{24, 25}.
However, SET has never been realized on multimode optical networks since it requires injection of the seed beam into the individual supermodes supported by the structure^{26}. A possible workaround is to inject the seed beam into each single channel individually then perform a transformation through supermode decomposition to obtain quantum predictions. Regardless, complete knowledge of the linear light dynamics inside the whole structure is required, making SET a multistep procedure prone to errors and not applicable to 'blackbox' circuits. Additionally, SET is strictly valid only in the limit of zero propagation losses^{10}, posing a fundamental limitation for the characterization of real optical circuits.
Characterization via sumfrequency generation (SFG), the reverse process of SPDC, gives exact results in the presence of any type of losses. This approach was previously formulated only for single, homogeneous waveguides^{10}, posing a stringent restriction for the characterization of more complex devices. In this work, we uncover a fundamentally important equivalence between the biphoton wavefunction and the classical sumfrequency field generated in the reverse direction of SPDC for any multimode nonlinear device, overcoming the limitations of previous approaches. Our theoretical analysis is based on the rigorous use of the Greenfunction method^{27} (Supplementary Information), and holds for arbitrarily complex secondorder nonlinear circuits, in the presence of any type of losses. More importantly, the SFGSPDC analogy can be expressed in any measurement basis, providing a simple and fast experimental tool for the characterization of any 'blackbox' χ^{2} nonlinear process (Figure 1).
Fig. 1
Scheme for the characterization of the biphoton state produced by an array of N waveguides with an arbitrary χ^{(2)}nonlinear process. (a) SPDC: a pump beam is injected into waveguide n_{p} at the input of the device. Photoncoincidence counting measurements between each pair of waveguides (n_{s}, n_{i}) at the output are used to measure photonpair generation rates and relative absolute squared values of the wavefunction. (b) SFG: Laser light at signal and idler frequencies is injected into waveguides n_{s} and n_{i} in the reverse direction of SPDC. Absolute photonpair generation rates and relative absolute squared values of the wavefunction can be predicted by direct optical power detection of the sumfrequency field emitted from waveguide n_{p}.Multimode SFG characterization can reconstruct any degree of freedom of the photonic state including spatial mode, frequency, timebin, and polarization. Here we illustrate its application to a 'blackbox' device with N spatial modes of the same polarization, as schematically depicted in Figure 1. When a pump beam with frequency ω_{p} is injected into waveguide n_{p} at the input of the device it produces, by SPDC, the biphoton state (Figure 1a)
$$ \Psi\rangle_{\text {pair }}=\iint\limits_{0}^{\infty} \mathrm{d} \omega_{s} \mathrm{d} \omega_{i} \sum\limits_{n_{s}, n_{i}=1}^{N} \Psi_{n_{s} n_{i}}^{n_{p}}\left(\omega_{s}, \omega_{i}\right) \hat{a}_{n_{s}}^{\dagger}\left(\omega_{s}\right) \hat{a}_{n_{i}}^{\dagger}\left(\omega_{i}\right)0\rangle $$ (1) where n_{s}(n_{i}) is the index for signal(idler) output waveguide number, $\hat a_n^\dagger \left(\omega \right)$ is the photon creation operator in the waveguide n with the frequency ω, and $\Psi _{{n_s}{n_i}}^{{n_p}}\left({{\omega _s}, {\omega _i}} \right)$ is the twophoton wavefunction^{20}. In the classical SFG process shown in Figure 1b, two beams with signal frequency ω_{s} and idler frequency ω_{i} are injected into waveguides n_{s} and n_{i} from the SPDC output directions. The generated sumfrequency electric field $E_{{n_s}{n_i}}^{{n_p}}$ is detected from waveguide n_{p}.
We reveal that the sumfrequency field in the undepleted pump regime is directly proportional to the twophoton wavefunction $\Psi _{{n_s}{n_i}}^{{n_p}}\left({{\omega _s}, {\omega _i}} \right)$ (Supplementary Information). From this correspondence we infer the squared amplitudes of the wavefunction elements by direct optical measurements of the sumfrequency power P_{SFG}, and predict the absolute photonpair generation rates for SPDC through the relation:
$$ \frac{1 \mathrm{d} N_{p a i r}}{P_{p} \mathrm{d} \omega_{s} \mathrm{d} t}=\frac{\omega_{i} \omega_{s}}{2 \pi \omega_{p}^{2}} \eta_{n_{s} n_{i}}^{\mathrm{SFG}}\left(\omega_{s}, \omega_{i}\right) $$ (2) Here, P_{p} is the pump beam power, dN_{pair}/dω_{s}dt is the photonpair generation rate per unit signal frequency, and $\eta_{n_{s} n_{i}}^{\mathrm{SFG}} \equiv P_{\mathrm{SFG}} / P_{s} P_{i}$ is the sumfrequency conversion efficiency. Full spectral characterization of the biphoton state is obtained by scanning the signal and idler wavelengths, with an accuracy that is limited only by the spectral resolution of the laser source. In addition we can characterize the relative phases of the wavefunction components by classical interferometric measurements of the generated sumfrequency field.
The validity of the SFG protocol for multimode and inhomogeneous circuits was experimentally verified on an array of three evanescently coupled nonlinear waveguides schematically depicted in Figure 2a. The device was fabricated in lithium niobate by the use of the Reverse Proton Exchange technique^{28, 29} and heated to T=84 ℃ to obtain phase matching at λ=1550 nm. The waveguides have an inhomogeneous and asymmetric poling pattern along the propagation direction in order to test the generality of the method where laser light propagates in the opposite direction of the SPDC process (Supplementary Information).
Fig. 2
Comparison between SFG and SPDC measurements. (a) Schematic of the device used for biphoton state generation. The device is made of three coupled waveguides with five defects in the poling pattern introduced by translating the poled domains by half a poling period Λ (inset). This design is based on the recently developed concept for quantum state engineering with specialized poling patterns^{26}. Waveguides are fabricated on a lithium niobate substrate by reverse proton exchange (Supplementary Information)^{28, 29}. (b) Measured classical sumfrequency conversion efficiency from waveguide 1 as a function of signal and idler wavelengths coupled to waveguides 2 and 3. (c) Predicted squared relative amplitudes of the biphoton wavefunction for a pump injected into waveguide 1, proportional to the SFG signal for different combinations of signal and idler in coupled waveguides vs. the pump wavelength in the degenerate regime (λ_{s}=λ_{i}=2λ_{p}). (d) Time histogram for the photon coincidences between waveguides 2–3 and waveguides 1–2 for a 50 min acquisition time, a pump wavelength λ_{p}=775 nm, and a pump power P_{p}=32±5 μW. Time bin width is 82 ps. Complete data sets are in Supplementary Fig. S2. (e) Normalized biphoton wavefunctions predicted by SFG (left) and measured by SPDC (right) for λ_{p}=775 nm.We performed the SFG measurements by coupling two frequency tunable lasers into the device and measuring sumfrequency generation from waveguide 1. Figure 2b shows the SFG efficiency η_{SFG} as a function of signal and idler wavelengths coupled to the waveguides 2 and 3, respectively. Similar data were taken for all input combinations (Supplementary Fig. S1).
Figure 2c shows the probabilities ${\left {\Psi _{{{\rm{n}}_{\rm{s}}}{{\rm{n}}_{\rm{i}}}}^{{\rm{SFG}}}} \right^2}$ predicted from SFG efficiencies as a function of the SPDC pump wavelength for the degenerate case λ_{s}=λ_{i}=2λ_{p}. Similar results are predicted for nondegenerate SPDC since the phasematching bandwidth of the device is larger than the tuning range of our laser (Figure 2b).
We verified our characterization results by measuring the biphoton state generated when a λ_{p}=775 nm pump is coupled into waveguide 1 in the reverse direction and the downconverted photon pairs pass through a 6 nm bandpass filter (Supplementary Fig. S2). Figure 2d shows two characteristic time histograms of photon coincidences for outputs from the waveguides 2–3 and 1–2 acquired by two avalanche photodiodes and a time tagging module. Coincidencetoaccidentalratio (CAR) is ~24.5.
Figure 2e shows the squared amplitudes of the wavefunction elements predicted by SFG and those directly measured through normalization of SPDC coincidences (see complete data set and speed up analysis in Supplementary Information, and Materials and Methods for details on the calculation). SFG predictions are obtained by integrating the measured conversion efficiencies over a bandwidth of 6 nm along the diagonal $\left(\lambda_{\mathrm{s}}^{1}+\lambda_{\mathrm{i}}^{1}\right)^{1}=775 \; \mathrm{nm}$. The two matrices have a fidelity $\mathrm{F}=\sum\limits_{\mathrm{n}_{\mathrm{s}} \mathrm{n}_{\mathrm{i}}} \sqrt{\left\Psi_{\mathrm{n}_{\mathrm{s}} \mathrm{n}_{\mathrm{i}}}^{\mathrm{SFG}}\right^{2}\left\Psi_{\mathrm{n}_{\mathrm{s}} \mathrm{n}_{\mathrm{i}}}^{\mathrm{SPD}}\right^{2}}=99.28 \pm 0.31 \%$. From equation (2), using the SFG measurements, we calculated a photon pair generation rate N_{SFG}=2.36±0.14 MHz, which is the sum of the rates from all 6 output combinations. Direct measurement of this rate from SPDC data gives N_{SPDC}=1.67±0.15 MHz, showing a good qualitative agreement between the two values. We believe that an overestimation of the detector efficiencies from the η_{1}=8% and η_{2}=10% provided by the manufacturer, and not measurable with our current setup, introduces a systematic error that underestimates the measured SPDC rate.
Our method allows direct characterization of the phases between the wavefunction elements, by performing interferometric detection of the generated sumfrequency field. Verification of the generated state by quantum state tomography would be experimentally difficult due to phase fluctuations between the different paths introduced by thermal and mechanical instabilities. Hence, the SFGphase measurements are presented as a proofofconcept and not directly verified by SPDC measurements.
Figure 3a shows the phase measurements setup used for input into waveguides 2 and 3. This procedure allows us to infer the relative phases between wavefunction elements $\theta_{n_{s} n_{i}}$ up to the phases of signal and idler beams $\left(\theta_{n_{s}}^{s}+\theta_{n_{i}}^{i}\right)$ from the output of waveguide 1 (Figure 3b). The predicted wavefunction phases are shown in Figure 3c. Since the unknown phase multiplier $\exp \left[i\left(\theta_{n_{s}}^{s}+\theta_{n_{i}}^{i}\right)\right]$ does not alter the degree of entanglement of the biphoton state, we calculated a Schmidt number^{30} S=1.59. (Supplementary Information), which precisely characterizes the degree of spatial entanglement and cannot be obtained with only photon correlations.
Fig. 3
Measurement of the relative phases between wavefunction elements by SFG. (a) Schematic of the experimental setup for inputs into waveguides 2 and 3. Signal and idler beams are split and recombined with a network of 50:50 fibre couplers and injected into the three waveguides with a fibre Vgroove array. An electrooptic phase modulator is used to generate an interference pattern between the sumfrequency fields generated from the combinations of signal and idler beams in waveguides 2–3 and waveguides 11. SFG and signalidler beams are collected in freespace at the output of waveguide 1 with a lens of 0.5 NA (not shown in the figure) and separated with a dichroic mirror. A wavelengthdivision multiplexer (not shown in the figure) is used to separate signal and idler wavelengths. (b) Oscilloscope traces obtained by collecting the beams with three different photodiodes for a modulation frequency f=500 kHz. The three traces are used to measure the relative phase between the wavefunction elements Ψ_{23}, Ψ_{11}. Solid red line is the theoretical fit (see Supplementary Information for details). (c) Relative phases between wavefunction elements measured for all the combinations of signalidler beams in the three waveguides. Waveguide 1 is the fixed reference for all the phase measurements. Measurements are performed for a signal wavelength λ_{s}=1550.12 nm and an idler wavelength λ_{i}=1556.65 nm. The sample was heated up to T=108 ℃ to get a phase matching condition centred at . See Supplementary Information for a calculation of the error bars.$2\left(\lambda_{s}^{1}+\lambda_{i}^{1}\right)^{1} \simeq 1553.3 \; \mathrm{nm}$ The SFG characterization method proposed here provides a practical path for characterization and development of monolithically integrated networks that for devices similar to ours can be four orders of magnitude faster than the equivalent quantum measurements and with two orders of magnitude greater accuracy (Supplementary Information). This technique can be applied to any arbitrary 'blackbox' secondorder nonlinear device and supports the development of integrated photon sources and largescale quantum photonics technologies. In the future it will be of interest to explore how the SFG analogy can be applied to larger photon number states generated through SPDC.
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Signal and idler beams, generated by two tunable laser diodes with 100 kHz linewidth, were injected into each pair of waveguides with a fibre Vgroove array. All the beams were collected in freespace at the output of the waveguides with a lens with 0.5 NA. SFG and signalidler wavelengths were separated with a dichroic mirror. SFG power from the output of waveguide 1 and signalidler powers from the outputs of all three waveguides were then measured with two standard power meters. The measured powers were corrected for Fresnel losses at the chip interface and used to calculate the SFG conversion efficiency at the output of the array. SFG conversion efficiencies for the single channel inputs were measured by combining signal and idler beams with a 50:50 fibre coupler. The measurement process was automated with Labview.

A pump beam with 775 nm wavelength and 100 kHz linewidth was generated by secondharmonic generation in a periodically poled lithium niobate waveguide and injected into waveguide 1 with a lens of 0.5 NA. The three outputs were collected with a fibre Vgroove array, and photon coincidences between each pair of waveguides were measured with two gated InGaAs avalanche photodiodes and a timetagging module. A filtering stage in freespace, made from a set of 5 longpass filters and a bandpass filter, was used to attenuate the pump beam by 150 dB. Photon pairs were filtered with a 6 nm bandpass filter centred at λ_{c}=1550 nm to restrict the SPDC emission bandwidth to the range measured by SFG. Photon coincidences from the single channels were measured by splitting signalidler photons with a 50:50 fibre coupler.

For each pair of waveguides n_{s}, n_{i} the signal wavelength was scanned in steps of Δλ=0.25 nm in a 6 nm bandwidth centered at 1550 nm. At each step j the idler wavelength was set to $\left(\lambda_{i}\right)_{j}=\left(\lambda_{p}^{1}\left(\lambda_{s}\right)_{j}^{1}\right)^{1}$, where λ_{p}=775 nm is the pump wavelength for SPDC. Absolute photon pair generation rates were calculated by discretization of equation (2) through the relation
$$ \frac{1 \mathrm{d} N_{\text {pair }}}{P_{\mathrm{p}} \mathrm{d} t}=\sum\limits_{j} \eta_{j}^{\mathrm{SFG}} \frac{\lambda_{p}^{2}}{\left(\lambda_{s}\right)_{j}\left(\lambda_{i}\right)_{j}\left[\left(\lambda_{s}\right)_{j}\right]^{2}} $$ where $\eta _j^{{\rm{SFG}}}$ is the normalized sumfrequency conversion efficiency measured at each step j. The pump power P_{p} was measured during the SPDC characterization from the first output of the fibre array. Relative squared amplitudes of the wavefunction elements were calculated as
$$ {\left {\Psi _{{n_s}{n_i}}^{{\rm{SFG}}}} \right^2} = \frac{{{{\left( {\sum_j {\eta _j^{{\rm{SFG}}}} } \right)}_{{n_s}{n_i}}}}}{{\sum_{{n_s}{n_i}} {{{\left( {\sum_j {\eta _j^{{\rm{SFG}}}} } \right)}_{{n_s}{n_i}}}} }} $$ 
The error in the fidelity between the correlation matrices predicted by SFG and measured by SPDC was calculated with an iterative numerical algorithm with N=10^{6} cycles. At each step we assigned to the two correlation matrices a random value calculated from a normal distribution with a sigma given by the error in the measurements. Average value and error in the fidelity were finally calculated from the simulated distribution.

For SFGpower measurements secondharmonic generation (SHG) contributions were first measured by inputting signal and idler beams into each channel individually. SHG powers were then subtracted from SFGpower measurements. The procedure was repeated and automated with Labview. For SFGphase measurements, SFG and SHG contributions were separated at the output of the array with the aid of a diffraction grating.