Citation:

# Direct characterization of a nonlinear photonic circuit's wave function with laser light

• Light: Science & Applications  7, Article number: 3 (2018)
More Information
• Corresponding author:
M Lobino, Email: m.lobino@griffith.edu.au
• Received: 24 April 2017
Revised: 13 October 2017
Accepted: 15 October 2017
Published online: 12 January 2018
•  [1] Vergyris P, Meany T, Lunghi T, Sauder G, Downes J et al. On-chip generation of heralded photon-number states. Sci Rep 2016; 6: 35975. [2] Solntsev AS, Setzpfandt F, Clark AS, Wu CW, Collins MJ et al. Generation of nonclassical biphoton states through cascaded quantum walks on a nonlinear chip. Phys Rev X 2014; 4: 031007. [3] Silverstone JW, Bonneau D, Ohira K, Suzuki N, Yoshida H et al. On-chip quantum interference between silicon photon-pair sources. Nat Photonics 2014; 8: 104–108. [4] Solntsev AS, Sukhorukov AA. Path-entangled photon sources on nonlinear chips. Rev Phys 2017; 2: 19–31. [5] Latmiral L, Spagnolo N, Sciarrino F. Towards quantum supremacy with lossy scattershot boson sampling. New J Phys 2016; 18: 113008. [6] Zhang P, Aungskunsiri K, Martín-López E, Wabnig J, Lobino M et al. Reference-frame- independent quantum-key-distribution server with a telecom tether for an on-chip client. Phys Rev Lett 2014; 112: 130501. [7] Lobino M, Korystov D, Kupchak C, Figueroa E, Sanders BC et al. Complete characterization of quantum-optical processes. Science 2008; 322: 563–566. [8] Rahimi-Keshari S, Scherer A, Mann A, Rezakhani AT, Lvovsky AI et al. Quantum process tomography with coherent states. New J Phys 2011; 13: 013006. [9] Liscidini M, Sipe JE. Stimulated emission tomography. Phys Rev Lett 2013; 111: 193602. [10] Helt LG, Steel MJ. Effect of scattering loss on connections between classical and quantum processes in second-order nonlinear waveguides. Opt Lett 2015; 40: 1460–1463. [11] O'Brien JL, Furusawa A, Vučković J. Photonic quantum technologies. Nat Photon 2009; 3: 687–695. [12] Tanzilli S, Martin A, Kaiser F, De Micheli MP, Alibart O et al. On the genesis and evolution of Integrated Quantum Optics. Laser Photon Rev 2012; 6: 115–143. [13] Li HW, Przeslak S, Niskanen AO, Matthews JCF, Politi A et al. Reconfigurable controlled two-qubit operation on a quantum photonic chip. New J Phys 2011; 13: 115009. [14] Tanzilli S, De Riedmatten H, Tittel W, Zbinden H, Baldi P et al. Highly efficient photon-pair source using periodically poled lithium niobate waveguide. Electron Lett 2001; 37: 26–28. [15] Zhang Q, Xie XP, Takesue H, Nam SW, Langrock C et al. Correlated photon-pair generation in reverse-proton-exchange PPLN waveguides with integrated mode demultiplexer at 10 GHz clock. Opt Express 2007; 15: 10288–10293. [16] Martin A, Alibart O, De Micheli MP, Ostrowsky DB, Tanzilli S. A quantum relay chip based on telecommunication integrated optics technology. New J Phys 2012; 14: 025002. [17] Meany T, Ngah LA, Collins MJ, Clark AS, Williams RJ et al. Hybrid photonic circuit for multiplexed heralded single photons. Laser Photon Rev 2014; 8: L42–L46. [18] Jin H, Liu FM, Xu P, Xia JL, Zhong ML et al. On-chip generation and manipulation of entangled photons based on reconfigurable lithium-niobate waveguide circuits. Phys Rev Lett 2014; 113: 103601. [19] James DFV, Kwiat PG, Munro WJ, White AG. Measurement of qubits. Phys Rev A 2001; 64: 052312. [20] Eckstein A, Boucher G, Lemaître A, Filloux P, Favero I et al. High-resolution spectral characterization of two photon states via classical measurements. Laser Photon Rev 2014; 8: L76–L80. [21] Fang B, Cohen O, Liscidini M, Sipe JE, Lorenz VO. Fast and highly resolved capture of the joint spectral density of photon pairs. Optica 2014; 1: 281–284. [22] Jizan I, Helt LG, Xiong CL, Collins MJ, Choi DY et al. Bi-photon spectral correlation measurements from a silicon nanowire in the quantum and classical regimes. Sci Rep 2015; 5: 12557. [23] Grassani D, Simbula A, Pirotta S, Galli M, Menotti M et al. Energy correlations of photon pairs generated by a silicon microring resonator probed by Stimulated Four Wave Mixing. Sci Rep 2016; 6: 23564. [24] Rozema LA, Wang C, Mahler DH, Hayat A, Steinberg AM et al. Characterizing an entangled-photon source with classical detectors and measurements. Optica 2015; 2: 430–433. [25] Fang B, Liscidini M, Sipe JE, Lorenz VO. Multi-dimensional characterization of an entangled photon-pair source via stimulated emission tomography. Opt Express 2016; 24: 10013–10019. [26] Titchener JG, Solntsev AS, Sukhorukov AA. Generation of photons with all-optically-reconfigurable entanglement in integrated nonlinear waveguides. Phys Rev A 2015; 92: 033819. [27] Poddubny AN, Iorsh IV, Sukhorukov AA. Generation of photon-plasmon quantum states in nonlinear hyperbolic metamaterials. Phys Rev Lett 2016; 117: 123901. [28] Lenzini F, Kasture S, Haylock B, Lobino M. Anisotropic model for the fabrication of annealed and reverse proton exchanged waveguides in congruent lithium niobate. Opt Express 2015; 23: 1748–1756. [29] Korkishko YN, Fedorov VA, Morozova TM, Caccavale F, Gonella F et al. Reverse proton exchange for buried waveguides in LiNbO3. J Opt Soc Am A 1998; 15: 1838–1842. [30] Nielsen MA, Chuang IL. Quantum Computation and Quantum Information 10th ed.Cambridge: Cambridge University Press, 2011.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Figures(3)

## Article Metrics

Article views(45) PDF downloads(2) Citation(0) Citation counts are provided from Web of Science. The counts may vary by service, and are reliant on the availability of their data.

## Direct characterization of a nonlinear photonic circuit's wave function with laser light

• 1. Centre for Quantum Dynamics, Griffith University, Brisbane, QLD 4111, Australia
• 2. Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, ACT 2601, Australia
• 3. ITMO University, Saint Petersburg 197101, Russia
• 4. Ioffe Institute, Saint Petersburg 194021, Russia
• 5. School of Engineering, RMIT University, Melbourne, VIC 3000, Australia
• 6. School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, NSW 2007, Australia
• 7. Queensland Micro- and Nanotechnology Centre, Griffith University, Brisbane, QLD 4111, Australia
• ###### Corresponding author: Mirko Lobino, m.lobino@griffith.edu.au
• Integrated photonics is a leading platform for quantum technologies including nonclassical state generation1-4, demonstration of quantum computational complexity5 and secure quantum communications6. As photonic circuits grow in complexity, full quantum tomography becomes impractical, and therefore an efficient method for their characterization7, 8 is essential. Here we propose and demonstrate a fast, reliable method for reconstructing the two-photon state produced by an arbitrary quadratically nonlinear optical circuit. By establishing a rigorous correspondence between the generated quantum state and classical sum-frequency generation measurements from laser light, we overcome the limitations of previous approaches for lossy multi-mode devices9, 10. We applied this protocol to a multi-channel 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, large-scale, device production.

Practical applications of quantum photonic technologies11, 12 require the integration of linear and nonlinear waveguides on a single device, where photons can be generated1, 4 and manipulated13. Spontaneous parametric down-conversion (SPDC) and spontaneous four-wave 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 MHz14, 15. Monolithic integration of SPDC sources with multi-port optical circuits has been achieved in several contexts, with applications in quantum communication16, quantum metrology1, spatial multiplexing of heralded single-photon sources17, quantum state generation in nonlinear waveguide arrays2 and small-scale reconfigurable quantum photonic circuits18.

The near future of quantum photonics will involve an expansion in scale and applications of integrated circuits. However, the characterization of the two-photon state generated by a nonlinear waveguide network is a cumbersome experimental task19, 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 two-photon 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 large-scale wafer production.

A method based on stimulated emission tomography (SET) was proposed9 for predicting the two-photon wavefunction produced by a nonlinear device using the analogy between spontaneous nonlinear processes and their classical stimulated counterparts, that is, difference-frequency generation or stimulated four-wave mixing. This technique was demonstrated for spectral characterization of two-photon states20-23, and fast reconstruction of the density matrix of entangled-photon sources24, 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 structure26. 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 multi-step procedure prone to errors and not applicable to 'black-box' circuits. Additionally, SET is strictly valid only in the limit of zero propagation losses10, posing a fundamental limitation for the characterization of real optical circuits.

Characterization via sum-frequency 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 waveguides10, 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 sum-frequency field generated in the reverse direction of SPDC for any multimode non-linear device, overcoming the limitations of previous approaches. Our theoretical analysis is based on the rigorous use of the Green-function method27 (Supplementary Information), and holds for arbitrarily complex second-order nonlinear circuits, in the presence of any type of losses. More importantly, the SFG-SPDC analogy can be expressed in any measurement basis, providing a simple and fast experimental tool for the characterization of any 'black-box' χ2 -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 np at the input of the device. Photon-coincidence counting measurements between each pair of waveguides (ns, ni) at the output are used to measure photon-pair generation rates and relative absolute squared values of the wavefunction. (b) SFG: Laser light at signal and idler frequencies is injected into waveguides ns and ni in the reverse direction of SPDC. Absolute photon-pair generation rates and relative absolute squared values of the wavefunction can be predicted by direct optical power detection of the sum-frequency field emitted from waveguide np.

Multimode SFG characterization can reconstruct any degree of freedom of the photonic state including spatial mode, frequency, time-bin, and polarization. Here we illustrate its application to a 'black-box' device with N spatial modes of the same polarization, as schematically depicted in Figure 1. When a pump beam with frequency ωp is injected into waveguide np 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 ns(ni) 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 two-photon wavefunction20. In the classical SFG process shown in Figure 1b, two beams with signal frequency ωs and idler frequency ωi are injected into waveguides ns and ni from the SPDC output directions. The generated sum-frequency electric field $E_{{n_s}{n_i}}^{{n_p}}$ is detected from waveguide np.

We reveal that the sum-frequency field in the undepleted pump regime is directly proportional to the two-photon 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 sum-frequency power PSFG, and predict the absolute photon-pair 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, Pp is the pump beam power, dNpair/dωsdt is the photon-pair generation rate per unit signal frequency, and $\eta_{n_{s} n_{i}}^{\mathrm{SFG}} \equiv P_{\mathrm{SFG}} / P_{s} P_{i}$ is the sum-frequency 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 sum-frequency 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 technique28, 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 patterns26. Waveguides are fabricated on a lithium niobate substrate by reverse proton exchange (Supplementary Information)28, 29. (b) Measured classical sum-frequency 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 (λs=λi=2λp). (d) Time histogram for the photon coincidences between waveguides 2–3 and waveguides 1–2 for a 50 min acquisition time, a pump wavelength λp=775 nm, and a pump power Pp=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 λp=775 nm.

We performed the SFG measurements by coupling two frequency tunable lasers into the device and measuring sum-frequency generation from waveguide 1. Figure 2b shows the SFG efficiency ηSFG 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 λs=λi=2λp. Similar results are predicted for nondegenerate SPDC since the phase-matching 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 λp=775 nm pump is coupled into waveguide 1 in the reverse direction and the down-converted photon pairs pass through a 6 nm band-pass 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. Coincidence-to-accidental-ratio (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 NSFG=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 NSPDC=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 η1=8% and η2=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 sum-frequency 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 SFG-phase measurements are presented as a proof-of-concept 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 number30 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 V-groove array. An electro-optic phase modulator is used to generate an interference pattern between the sum-frequency fields generated from the combinations of signal and idler beams in waveguides 2–3 and waveguides 1-1. SFG and signal-idler beams are collected in free-space 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 wavelength-division 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 Ψ23, Ψ11. Solid red line is the theoretical fit (see Supplementary Information for details). (c) Relative phases between wavefunction elements measured for all the combinations of signal-idler beams in the three waveguides. Waveguide 1 is the fixed reference for all the phase measurements. Measurements are performed for a signal wavelength λs=1550.12 nm and an idler wavelength λi=1556.65 nm. The sample was heated up to T=108 ℃ to get a phase matching condition centred at $2\left(\lambda_{s}^{-1}+\lambda_{i}^{-1}\right)^{-1} \simeq 1553.3 \; \mathrm{nm}$. See Supplementary Information for a calculation of the error bars.

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 'black-box' second-order nonlinear device and supports the development of integrated photon sources and large-scale 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.

• Reference (30)

### Catalog

/

DownLoad:  Full-Size Img PowerPoint