Supplementary Information Parametric down-conversion photon-pair source on a nanophotonic chip

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1 Supplementary Information Parametric own-conversion photon-pair source on a nanophotonic chip Xiang Guo, Chang-Ling Zou, Carsten Schuck, Hojoong Jung, Risheng Cheng, an Hong Tang Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA Contents I. Wrap-aroun waveguie for higher orer moe excitation S1 II. Waveguie couple superconucting single photon etector S1 III. χ (2) in the microring resonator S3 A. Basics of χ (2) process S3 B. Secon harmonic generation in microring resonator S4 C. Degenerate SPDC S5 D. Non-egenerate SPDC S7 E. Experiment estimation of SPDC efficiency S8 F. Raw ata of coincience measurement S8 IV. Dispersion an banwith of the generate own-conversion photon combs S9 V. The secon orer correlation function g (2) (τ) S9 A. Photon correlation function S9 B. Detector jitter an bin-with s broaening effect on the g (2) (τ) function S10 C. Cross correlation for other two groups of photon pairs S10 VI. Aiabatic on-chip wavelength ivision multiplexer (WDM) S10

2 S1 I. WRAP-AROUND WAVEGUIDE FOR HIGHER ORDER MODE EXCITATION High frequency (visible wavelength) TM 2 moe an low frequency (IR wavelength) TM 0 moe are co-existing in the same microring resonator, as illustrate in Fig. S1a. We tune the visible pump laser into phase-matche resonance to excite corresponing TM 2 moe in microring resonator. Due to cavity enhance χ (2) effect, visible photons in TM 2 moe will be own converte to IR photon pairs in TM 0 moe. Critical coupling for visible TM 2 moe is esire to maximize pump power insie microring cavity an hence the photon pair generation efficiency (Supplementary section III). To maintain the high Q for IR moes, however, the visible light coupling waveguie cannot be too close to the microring. We therefore aopt a very narrow wrap-aroun waveguie for visible light coupling. Efficient TM 2 moe excitation is achieve when the effective refractive inex of funamental TM 0 moe in wrap-aroun waveguie matches that of TM 2 moe in the microring resonator. Visible light TM 2 moe s effective refractive inex is n TM2 = with w = 1.10 µm, as shown in Fig. 1b in main context. Consiering the raius ifference, the phase match conition requires the targete effective inex of the wrap-aroun waveguie shoul be n TM0 = n TM2 Rmicroring R bus = = The waveguie with is etermine to be w = 70 nm accoring to numerical simulation. In reality, however, this phase match conition is not strict because critical coupling only requires small coupling efficiency (on the orer of 1%). We use a tapere wrap-aroun waveguie esign, e.g. from 150 nm to 100 nm, for better fabrication robustness. We simulate coupling region taking into account bening effect ue to the microring curvature (Fig. S1b). It is clear that the TM 2 moe can be effectively excite with tapere narrow waveguie. Coupling efficiency with ifferent bus waveguie withs an ifferent gap values is simulate with coupling length fixe to be 80 µm, as shown in Fig. S1c. We fin that critical coupling gap can be quite large when choosing proper wrap-aroun waveguie with. For example, critical coupling coul be achieve with a gap of 0.5 µm with a wrap-aroun waveguie tapering from 150 nm to 100 nm. The evice shown in the main context is slightly uner-couple with relative small resonance extinction. A near critically-couple transmission spectrum is shown in Fig. S1e. The resonances fitte with re lines correspon to TM 2 moe. A critically couple Q for TM 2 moe near 775 nm is measure to be The existence of wrap-aroun waveguie may inuce extra scattering for IR photons an eteriorate Q for IR moes. We numerically calculate scattering Q inuce by the wrap-aroun waveguie assuming a microring raius of 30 µm, as shown in Fig. S1. The scattering Q is aroun with a gap of 0.5 µm, much bigger than the critically couple IR Q of Hence we conclue that the existence of wrap-aroun waveguie will not egrae the Q of IR moes. II. WAVEGUIDE COUPLED SUPERCONDUCTING SINGLE PHOTON DETECTOR We use superconucting single-photon etectors (SSPD) to measure the statistical properties of the light fiel an extract correlations between photon etection events. The SSPDs are fabricate in traveling wave geometry on top of SiN waveguie-evices on a separate silicon chip insie a cryostat at 1.7 K. Each SSPD is realize as a U-shape 40 µm long, 70 nm narrow an 8 nm thin NbTiN nanowire. This SSPD esign realizes more than 95% photon absorption efficiency, milli-hz ark count rate an high timing accuracy. The on-chip etection efficiency for a representative evice is shown in Fig. S2c as a function of SSPD bias current for photons of 1550 nm wavelength. For all SSPDs use here we observe aroun 14% on-chip etection efficiency when biase close to the critical current, I c. We measure a system etection efficiency of 0.7% consiering insertion loss from fiber to chip an calibration outputs. The grating couplers use for guiing light from the output of an optical fiber into the waveguies on chip show some wavelength epenency, as seen from Fig. S2. The transmission spectrum is centere at 1550 nm an translates irectly to system etection efficiency as the intrinsic quantum efficiency of an SSPDs only varies negligibly for small wavelength changes aroun 1550 nm. We use the evice shown in Fig S2a to measure self-correlations of egenerate own conversion an the iler of the nearest non-egenerate own conversion. Photons are couple onto the chip via one of the central two input grating couplers, split ranomly at a 50/50 irectional coupler (center) an are etecte at its outputs. To measure cross-correlations between signal an iler photons in the non-egenerate case we use the evice shown in Fig S2b. Here photons are guie to the SSPDs via inepenent optical waveguies from the two outermost grating couplers.

3 S2 a c b TM0 TM2 e FIG. S1: Triply-resonant parametric own conversion process in a microring resonator an wrap-aroun narrow waveguie for TM 2 moe excitation. a, A schematic showing the parametric own conversion process in a triply resonant microring resonator. The high energy photon (blue) are couple into TM 2 moe through wrap-aroun narrow waveguie an own converte to low frequency photon pairs (re) in TM 0 moe. The generate photon pairs are finally couple out by the point-contact bus waveguie. The noes number in the figure is arbitrary. For the real evice, noes number is aroun 484 for TM 2 moe near 775 nm. b, Intensity profile of visible light (λ = 775 nm) in the wrap-aroun waveguie coupling region. c, Simulate coupling efficiency for ifferent bus waveguie with an gap values, with fixe coupling length to be 80 µm. The bening effect has been taken into consieration by efining effective curvature in the moe solver., Simulate scattering Q for IR moes inuce by the scattering of wrap-aroun waveguie. e, A critically-couple visible light transmission spectrum. The resonances fitte with re lines are TM 2 moes.

4 S3 a b a c FIG. S2: Waveguie couple superconucting single photon etectors (SSPDs). a, SSPDs with 50/50 irectional coupler. b, SSPDs without irectional coupler. c, On-chip an off-chip etection efficiency uner ifferent bias current. During measurement the bias current is set to be 95% of the critical current I c., Transmission spectrum of the evice. III. χ (2) IN THE MICRORING RESONATOR A. Basics of χ (2) process In general, the secon orer nonlinear optics (χ (2) ) process can be escribe by the Hamiltonian H = 2g(a bc + ab c ), (S.1) where a, b an c enote the Bosonic operator of three optical moes which satisfy the energy an momentum conservation relation. The factor of 2 is ue to the non-egenerate process (b c). Here g is the nonlinear coupling

5 strength proportional to material s secon orer nonlinear coefficient (χ (2) ) as well as the moe overlap, with expression ωa ω b ω c rzχ (2) u a (r, z)u b g = (r, z)u c(r, z) ε0 2πR ( rzn 2 a(r, z)u a(r, z)u a (r, z)) 1 2 ( rzn 2 b (r, z)u b (r, z)u b(r, z)) 1 2 ( rzn 2 c(r, z)u c(r, z)u c (r, z)) 1 2 (S.2) where ω x is the angular frequency, n x is refractive inex an u x is moe profile of moe x, x can be a, b, c. R is the raius of microring resonator. For ifferent experiment with ifferent pump an signal moes, we have a general table for χ (2) processes [Table. I]. Therefore, we know that these processes share the same coupling strength g an are relate in experiments. In our experiment, we mainly focus on the egenerate an non-egenerate SPDC processes, while using the secon harmonic an ifference frequency generation experiments to characterize the χ (2) process an estimate the coupling strength g. Name Hamiltonian Description Sum-frequency H = 2g(a bc + ab c ) 2g b c a + h.c. pump at b an c, generate a Secon harmonic generation H = g[a b 2 + a(b ) 2 ] g b 2 a + h.c. b = c egenerate, pump at b Difference frequency generation H = 2g[a bc + ab c ] 2g a b c + h.c. pump at a an b, generate c Frequency Conversion H = 2g[a bc + ab c ] 2g b a c + h.c. pump at b, signal is a or c Non-egenerate SPDC H = 2g[a bc + ab c ] 2g a b c + h.c. pump at a Degenerate SPDC H = g[a b 2 + a(b ) 2 ] g a (b ) 2 + h.c. pump at a TABLE I: Different χ (2) nonlinear processes. S4 B. Secon harmonic generation in microring resonator For secon harmonic generation process insie a microring resonator, the Hamiltonian [Table. I] can be written as H = ω a a a + ω b b b + g[(a ) 2 b + a 2 b ] + iɛ p ( ae iωpt + a e iωpt ). P Here, ω a,b are frequency of the IR an visible moes, ɛ p = 2κ p a,1 ω p correspons to external pump laser with frequency ω p an power P p. κ a,1 is the external coupling coefficient for the IR moe. The total amplitue ecay rate of microring moe consists of intrinsic loss an external coupling as κ a,tot = κ a,0 + κ a,1, corresponing to a Q-factor as Q = ωa 2κ a,tot. Similar parameters are use for visible moe by replacing the subscript a to b. In the rotating frame of A = ω p a a + 2ω p b b, the Hamiltonian becomes (S.3) H = δ a a a + δ b b b + g((a ) 2 b + a 2 b ) + iɛ p ( a + a ), (S.4) where δ a = ω a ω p, δ b = ω b 2ω p. Then, the ynamics of a can be solve by the Heisenberg equation t a = i[a, H] κ a,tota = ( iδ a κ a,tot )a + ɛ p ig2a b Uner the non-epletion approximation, we omit last term. At steay state a/t = 0, we have a = α = ɛ p 2κ a,1 P p iδ a κ a,tot eiθ = δa 2 + κ 2 e iθ. a,tot ω p (S.5) (S.6) Here, θ is a phase factor epening on the riving laser. Treat a as complex constant α, an ignore the phase of α (as which can be combine into the phase of b), the Hamiltonian becomes H = δ b b b + g α 2 (b + b ) (S.7) Then, we can solve b as t b = ( iδ b κ b,tot )b ig α 2. (S.8)

6 S5 At steay state, we have the generate secon harmonic photon Due to the input-output relation, the output is b = ig α 2 iδ b κ b,tot. (S.9) The corresponing power of secon harmonic output is b out = 2κ b1 b = i 2κ b1 g α 2 iδ b κ b,tot. (S.10) 2κ b1 P SHG = b out 2 ω b = g 2 δb 2 + κ2 b,tot ( 2κa,1 The total secon harmonic generation efficiency can be calculate as δa 2 + κ 2 a,tot ω p ) 2 ( ) 2 Pp ω b. (S.11) η SHG = P SHG P 2 p ( ) 2 ( ) 2 = g 2 2κ b1 2κa,1 1 δb 2 + κ2 b,tot δa 2 + κ 2 ω b (S.12) a,tot ω p The momentum conservation (azimuthal quantum number conservation) is containe in the non-zero coupling strength g. The energy conservation conition is inicate by the efficiency that maximum efficiency can be achieve when δ b = δ a = 0, which requires ω b = 2ω a. C. Degenerate SPDC Now we consier the SPDC insie the microring resonator an start with the egenerate case. From Table. I, the Hamiltonian for egenerate SPDC is the same with the SHG, while with ifferent pump laser frequency H = ω a a a + ω b b b + g[(a ) 2 b + a 2 b ] + iɛ s ( be iωst + b e iωst ). (S.13) P Here, ω a,b are frequency of the IR an visible moes, ɛ s = 2κ s b1 ω s correspons to external pump laser with frequency ω s an power P s. Similar to the SHG case, we solve the pump fiel uner non-epletion an steay state approximations as 2κb1 Ps b = β = κ 2 b,tot + e iθ, (S.14) δ2 b ω s where δ b = ω b ω s is the pump etuning of the moe b an θ is the phase epening on the riving. Treat b as complex constant β, an ignore the phase of β (as which can be combine into the phase of a), the Hamiltonian becomes H = δ a a a + g β [(a ) 2 + a 2 ], (S.15) with δ a = ω a ωs 2. For the SPDC, the vacuum noise plays great role to inuce the spontaneous photon pair generation. Therefore, we solve the ynamics of IR photons with vacuum noise. The Langevin equation reas t a = ( iδ a κ a,tot )a i2g β a i 2κ a,tot a in, where a in is vacuum noise incluing all channels. Introucing the Fourier transforme operators as a(ω) = 1 te iωt a(t), 2π a ( ω) = 1 te iωt a (t), 2π (S.16) (S.17) (S.18)

7 S6 we have Then, we can solve 0 = [ i(δ a + ω) κ a,tot ]a(ω) i2g β a ( ω) i 2κ a,tot a in (ω). (S.19) a(ω) = i 2κ a,tot [i(δ a ω) κ a,tot ]a in (ω) + 2g β 2κ a,tot a in ( ω) [i(δ a ω) κ a,tot ] [ i(δ a + ω) κ a,tot ] 4g 2 β 2. (S.20) a ( ω) = i 2κ a,tot [ i(δ a + ω) κ a,tot ]a in ( ω) + 2g β 2κ a,tot a in (ω) [ i(δ a + ω) κ a,tot ] [i(δ a ω) κ a,tot ] 4g 2 β 2. (S.21) The output photon number spectrum ensity aroun moe a is: Due to the noise spectrum an we finally obtain S a,out (ω, ω ) = 2κ a,1 a (ω)a(ω ). (S.22) a in (t)a in(t ) = 0, (S.23) a in (t)a in (t ) = δ(t t ), (S.24) a in (ω)a in(ω ) = 0, (S.25) a in (ω)a in ( ω ) = 1 2π δ(ω + ω ), (S.26) S a,out (ω, ω ) = 1 2π 2κ a,1 δ(ω 2g β 2κ a,tot ω) {[ i(δ a ω) κ a,tot ][i(δ a + ω) κ a,tot ] 4g 2 β 2} 2g β 2κ a,tot (S.27) {[i(δ a ω ) κ a,tot ][ i(δ a + ω ) κ a,tot ] 4g 2 2} β When input pump power below the threshol of optical parametric oscillator, we have 4g2 β 2 1, then S a,out (ω, ω ) The output photon rate is δ 2 a +κ2 a,tot 1 2π 2κ a,1(2g β 2κ a,tot ) 2 δ(ω ω). {[ i(δ a ω) κ a,tot ][i(δ a + ω) κ a,tot ]} {[i(δ a ω ) κ a,tot ][ i(δ a + ω ) κ a,tot ]} N a,out = 2κ a,1 a (t)a(t) = ωω S a,out (ω, ω )e i(ω ω)t = 1 2π 2κ a,1 (2g β 2κ a,tot ) 2 ω [ i(δ a ω) κ a,tot ][i(δ a + ω) κ a,tot ] 2 (S.28) The total photon generation rate is = 4g2 κ a,1 κ 2 a,tot + δ 2 a 2κ b1 κ 2 b,tot + δ2 b P s ω s. N a = 2κ a,tot a (t)a(t) = 4g2 κ a,tot κ 2 a,tot + δ 2 a 2κ b1 κ 2 b,tot + δ2 b In terms of photon pair, the total photon pair generation rate is R = N a /2 = 2g2 κ a,tot κ 2 a,tot + δ 2 a 2κ b1 κ 2 b,tot + δ2 b P s ω s. P s ω s. (S.29) (S.30) (S.31)

8 S7 D. Non-egenerate SPDC For non-egenerate case, we introuce IR signal an ile moes as a an c, then the Hamiltonian is H = ω a a a + ω b b b + ω c c c + 2g(a bc + ab c) +iɛ s ( be iωst + b e iωst ). (S.32) With other symbols similar to previous section, an uner the non-epletion an steay state approximations, the Hamiltonian is simplifie as H = δ a a a + δ c c c + 2g β (a c + ac), with δ b = ω b ω s, δ a + δ c = ω a + ω c ω s. The Langevin equations of a, c are t a = ( iδ a κ a,tot )a i2g β c i 2κ a,tot a in, t c = ( iδ c κ c,tot )c i2g β a i 2κ c,tot c in. By Fourier transformation, we solve the equations an obtain Then, the output spectrum ensities are (S.33) (S.34) (S.35) 0 = [ i(δ a + ω) κ a,tot ]a(ω) i2g β c ( ω) i 2κ a,tot a in (ω), (S.36) 0 = [ i(δ c + ω) κ c,tot ]c(ω) i2g β a ( ω) i 2κ c,tot c in (ω). (S.37) a(ω) = i 2κ a,tot [i(δ c ω) κ c,tot ]a in (ω) + 2g β 2κ c,tot c in ( ω) [ i(δ a + ω) κ a,tot ][i(δ c ω) κ c,tot ] 4g 2 β 2, (S.38) c ( ω) = i 2κ c,tot [ i(δ a + ω) κ a,tot ]c in ( ω) + 2g β 2κ a,tot a in (ω) [ i(δ a + ω) κ a,tot ][i(δ c ω) κ c,tot ] 4g 2 β 2. (S.39) S a,out (ω, ω ) = 1 2π 2κ a,1 δ(ω 2g β 2κ c,tot ω) {[i(δ a + ω) κ a,tot ][ i(δ c ω) κ c,tot ] 4g 2 β 2} 2g β 2κ c,tot (S.40) {[ i(δ a + ω ) κ a,tot ][i(δ c ω ) κ c,tot ] 4g 2 2} β S c,out (ω, ω ) = 1 2π 2κ c,1 δ(ω 2g β 2κ a,tot ω) {[i(δ a + ω) κ a,tot ][ i(δ c ω) κ c,tot ] 4g 2 β 2} 2g β 2κ a,tot (S.41) {[ i(δ a + ω ) κ a,tot ][i(δ c ω ) κ c,tot ] 4g 2 2} β 4g Below the threshol of optical parametric oscillator, we have 2 β 2 ( iδ a κ a,tot)(iδ c κ c,tot) 1, then we have the total photon pair generation rate to be R = The output photon rate for moe a an c are 8g 2 (κ a,tot + κ c,tot ) 2κ b1 P s (κ a,tot + κ c,tot ) 2 + (ω a + ω c ω s ) 2 κ 2 b,tot + (ω b ω s ) 2. ω s N a,out = κ a,1 κ a,tot R, N c,out = κ c,1 κ c,tot R, (S.42) (S.43) (S.44) Comparing Eq. (S.42) with Eq. (S.31), we conclue that non-egenerate SPDC rate is two times more efficient than egenerate SPDC, assuming the phase match conition are both fulfille an quality factors of IR moes are the same.

9 S8 E. Experiment estimation of SPDC efficiency Assuming the phase match conition is perfectly fulfille (δ a = δ b = 0), together with Eq. (S.12), we can estimate the egenerate photon pair generation rate from SHG efficiency as R = 2g2 κ a,tot κ 2 a,tot 2κ b1 P s κ 2 b,tot ω s = κ3 a,tot 8κ 2 η SHG P s. a,1 (S.45) Therefore, the SHG efficiency η SHG can be use as a figure of merit to optimize the evice performance. Strong SHG will occur when IR pump is tune into phase matche resonance. A typical IR transmission aroun the phase-matche resonance an the corresponing SHG are shown in Fig. S3a an b. With mw IR pump power on-chip, the maximum SHG is 24.8 nw. This correspons to a SHG efficiency to be: η SHG = P SHG P = 1.16 (W 1 ). p 2 By tuning the temperature, IR an visible resonance teeth will be shifte respectively. Experimentally a re shifting with increasing temperature is observe for both IR an visible resonances, with ifferent coefficient. For IR moe, λ IR T = (nm/ C), while for visible moe λvis T = (nm/ C). The fact that λ IR T 2 λvis T = (nm/ C) 0 means that the two resonance teeth will have relative shift with the change of temperature. Uner a certain temperature, when the corresponing resonances are aligne (λ N = 2λ 2N ), maximum SHG will occur. The temperature epenence of SHG efficiency η SHG is shown in Fig. S3c. Theoretically, with IR moe to be critically couple (κ a,tot = 2κ a,1 ) an a loae Q = 197 K, an a maximum SHG efficiency to be η SHG = 1.16 (W 1 ), the R egenerate own conversion photon pair generation rate per unit pump power is calculate to be: P s = 1 2 κ a,totη SHG = ω a 4Q η SHG = 1.8 (MHz/mW). a c b FIG. S3: Secon harmonic generation (SHG) in microring resonator. a, IR transmission aroun phase-matche resonance. b, Measure SHG uner ifferent IR pump wavelength. c, SHG efficiency η SHG uner ifferent temperature. F. Raw ata of coincience measurement The raw ata of coincience measurement for the nearest non-egenerate own conversion is shown in Fig. S4, with on-chip pump power of 1.9 mw. A coincience time winow of 128 ps is use to get the coincience histogram. Due to the small jitter time of our SSPDs compare to the photon life time, the coincience peak can be fully resolve. By summing up the coincience uner the peak, the total measure coincience rate is aroun 80 Hz.

10 S9 25 Coincience rate (Hz) Delay time (ns) FIG. S4: Raw ata of coincience measurement. Here the binwith is chosen to be 128 ps. The total measure coincience rate is aroun 80 Hz. IV. DISPERSION AND BANDWIDTH OF THE GENERATED DOWN-CONVERSION PHOTON COMBS The waveguie with in our evice is engineere so that phase match conition between TM 0 moe in IR an TM 2 moe in visible ban will be fulfille. The group velocity ispersion (GVD) for the IR moe is not optimize. Due to the non-zero GVD, the generate photon frequency comb will have a limite banwith. For our waveguie size, effective refractive inex of TM 0 moe uner ifferent wavelength has been simulate. The fitting equation is n eff = λ λ 2 (unit of λ is µm), corresponing to a GVD of D = λ 2 n ps C λ = km nm aroun 1550 nm. Consiering that the temperature is optimize so that the resonance teeth is aligne, the phase match conition for egenerate own conversion is perfectly fulfille. Then we nee to calculate how much etuning there is for the non-egenerate photon pairs whose wavelength is δλ away from the egenerate own conversion photon wavelength. The etuning ω can be expresse with D as ω = ω a + ω c ω b = ω2 (δλ) 2 2πn g D (S.46) Let ω = κ a,tot + κ c,tot (Consiering IR moes are critically couple with loae Q = 197 K) ω = ω2 (δλ) 2 2πn g D = κ a,tot + κ c,tot = ω Q (S.47) δλ = 2πng = 20.2 nm ωq D (S.48) From Eq. (S.42), when ω = κ a,tot + κ c,tot, the SPDC generation rate will rop to half. Hence the full with at half maximum (FWHM) of the SPDC photon comb is 2 δλ = 40.4 nm. V. THE SECOND ORDER CORRELATION FUNCTION g (2) (τ) A. Photon correlation function For non-egenerate own conversion cross correlation, g (2) (τ) can be fitte by a ouble exponential function g (2) cross(τ) = Rτ c e τ /τc, (S.49)

11 For egenerate own conversion self correlation, photons will be split ranomly by the 50/50 irectional coupler. There are 50% of chance that both photons go to the same etector an coincience cannot be etecte. So the peak value will rop by a factor of 2 an the g (2) (τ) function can be fitte as S10 g (2) self (τ) = Rτ c e τ /τc, (S.50) B. Detector jitter an bin-with s broaening effect on the g (2) (τ) function The etector s jitter means that the time recore by the time-tagging electronics is not exactly the photon arrival time. This gives error on the time tagging ata, thus affects the shape of the measure coincience histogram. Another factor that will influence the with of coincience histogram is the coincience time winow (bin-with) that is chosen to process the time tagging ata. The bigger bin-with will give more uncertainty, resulting in a broaer coincience histogram. Uncertainty of arrival time ue to photon lifetime together with the bin-with an etector jitter etermine the with of the coincience histogram istribution. To calibrate the etector jitter s effect, a ownconversion source base on short PPKTP waveguie is teste by our integrate photon correlation measurement system. Due to the broaban nature of the own-converte photons from short crystal waveguie, the intrinsic photon arrival time ifference is within several ps, much shorter than the etector jitter. Thus the istribution of the coincience histogram (Fig. S5) is almost fully ue to the jitter an bin-with. Here the bin-with is chosen to be τ b = 10 ps. The Gaussian fitting of the histogram gives the stanar eviation to be τ w = 26 ps. With the equation of ( τ w = τb ) τ 2 j, the etector jitter is calculate to be τ j = 18 ps. Due to jitter an bin-with, a sharp coincience peak is transforme into a Gaussian istribution. For photons generate from the microring resonator whose photon life time is comparable to the jitter an bin-with, the measure g (2) function is actually the convolution between original ouble exponential function ue to the source property an the Gaussian istribution where f ± (τ) = [ 1 erf g (2) 1 1 self (τ) = [1 + e τ /τc ] [ e τ 2 /2τ 2 w ] 4Rτ c 2πτw = e τ 2 w /2τ 2 c [f+ (τ) + f (τ)], (S.51) 8Rτ c g cross(τ) (2) = [ e τ /τc ] [ e τ 2 /2τ 2 w ] 2Rτ c 2πτw ( )] τ±τ 2 w /τc 2τw e ±τ/τc. = e τ 2 w /2τ 2 c [f+ (τ) + f (τ)], (S.52) 4Rτ c C. Cross correlation for other two groups of photon pairs Cross correlation of the other two neighboring photon-pair group has also been measure. The measurement result is shown in Fig. S6. A 50/50 fiber-coupler is use to ranomly split the photon flux into two arms. Tunable filters are inserte in both arms. By tuning the passban of the each banpass filter, the signal an iler coul be selecte an the coincience is measure by two separate waveguie couple SSPDs. VI. ADIABATIC ON-CHIP WAVELENGTH DIVISION MULTIPLEXER (WDM) The esigne on-chip WDM is shown in the Fig. 5 in the main context, which consists of two tapere couple waveguies. Shown in Fig. S7a is the cross section of such two waveguies, one with with w 1 = w 0 w an the other with w 2 = w 0 + w. Here 2w is the with ifference of the two waveguies. We linearly tapere the two waveguies, with one waveguie tapere to be narrower, the other to be wier, with expressions: w = (z l0 2 ) a. Here z is the coorinate along the tapering irection, l 0 is the total length of the WDM evice an a is the tapering spee. The gap between the two waveguies is fixe in the entire tapere structure, an the esigne central waveguie with is w 0 = 0.7 µm.

12 S11 FIG. S5: Secon orer correlation function of PPKTP waveguie own conversion source measure by on-chip SSPDs a b FIG. S6: Secon orer cross correlation function for the secon an thir nearest non-egenerate own conversion photon pairs. a, The secon nearest non-egenerate photon pairs with λ signal = nm, λ iler = nm b, The thir nearest non-egenerate photon pairs with λ signal = nm, λ iler = nm. The binwith is chosen to be 200 ps. In the couple waveguies, new eigenmoes forme as combination of eigenmoes in separate waveguies. The relationship between the new eigenmoes effective refractive inices an the waveguie with ifference 2w is shown in Fig. S7b an c. For wavelength of λ = 1550 nm, The big avoie-crossing gap (Fig. S7b) inicates strong moe coupling for IR light. While for λ = 775 nm, the avoie-crossing gap (Fig. S7c) is much smaller ue to better light confinement in the waveguie an inicates weak coupling. In this tapere couple waveguie, the aiabatic moe coupling efficiency can be estimate by Lauer-Zener formula as ζ = 1 e πg2 kl 0 n, (S.53) where g is the coupling strength, k = 2π λ is the wave vector, an n is the refractive inex ifference between the wie (w wie = 0.8 µm) an narrow (w narrow = 0.6 µm) waveguie. From the couple moe theory, the coupling strength g between two waveguies can be euce from the avoie-crossing gap as g equals to the half of the minimum

13 S12 a w 1 =w 0 -w w 2 =w 0 +w gap e b f c g FIG. S7: Aiabatic on-chip WDM. a, Cross section of the WDM structure. The narrow waveguie with is w 0 w an wie waveguie w 0 + w. The gap is fixe throughout the structure. b, Effective refractive inex for IR light (λ = 1550 nm) symmetric an anti-symmetric TM 0 moes when w varies from 0.1 µm to 0.1 µm. c, Effective refractive inex of visible light (λ = 775 nm) symmetric an anti-symmetric TM 0 moe when w varies from 0.1 µm to 0.1 µm., Coupling strength g for both IR an visible light uner ifferent gap. Waveguie with w 0 is 0.7 µm. e, Coupling strength ratio g IR g vis between IR an visible light. f, The optimize WDM efficiency uner ifferent gap values. WDM efficiency ζ W DM stans for cross port coupling efficiency for IR, an rop port coupling efficiency for visible. g, The coupling length neee to achieve best WDM performance. refractive inex ifference. The big ifference in coupling strength g leas to ifferent performances of the same structure uner ifference wavelengths. With IR light passing through the structure, aiabatic moe coupling will happen in the avoie-crossing region an light s energy is couple to the other waveguie graually. With visible light input, the photons goes through the avoie-crossing region non-aiabatically an most of photons will still remain in the same waveguie after the coupling region. Mathematically, for IR moes whose πg2 kl 0 n 1, ζ will approach unity with the increase of l 0. For visible moe whose πg2 kl 0 n 1, ζ πg2 kl 0 n will be a small value close to 0 an linearly epenent on l 0. These explaine the two curves in Fig. 5c in the main context. We esign the structure as an efficient on-chip WDM between IR an visible light. Because we want to make IR s cross coupling to be as high as possible, while keeping visible light s cross coupling to be as low as possible, the figure of merit that nees to be optimize is the ratio between IR coupling strength g IR an visible light coupling g strength g visible : IR g visible. As coupling between waveguies are weaker with bigger gap, both IR an visible light coupling strength (g IR an g vis ) will ecrease with the gap (Fig. S7). However, the coupling strength ratio g IR g visible

14 S13 a b FIG. S8: Wavelength epenent coupling efficiency for on-chip WDM. a, Cross port coupling efficiency for IR light. For IR light whose wavelength λ 1510 nm, cross port coupling efficiency is higher than 99%. b, Drop port coupling efficiency for visible light. For visible light whose wavelength λ 816 nm, rop port coupling efficiency is higher than 99%. will increase monotonically with the gap as shown in Fig. S7e. Let IR s crossing coupling to be equal with visible s rop coupling, with equation: ζ IR = 1 ζ visible, we can calculate the optimize coupling length l 0 an corresponing WDM efficiency ζ W DM (= ζ IR = 1 ζ visible ). The WDM efficiency ζ will increase with gap an approach unity (Fig. S7f). However, the corresponing coupling length l 0 will also increase with gap (Fig. S7g). Consiering the footprint an the propagation loss in the waveguie, the WDM s length l 0 cannot be arbitrarily long. WDM efficiency ζ W DM is greater than 99.9% with 0.55 µm gap an 2 mm coupling length. This performance correspons to 30 B isolation of visible light with B insertion loss for IR light for each WDM. We also expects a broa-ban operation from the robust performance of the on-chip WDM. The wavelength epenence of coupling efficiency is shown in Fig. S8 with the following esign parameters: w wie = 0.8 µm, w narrow = 0.6 µm, gap = 0.4 µm, l 0 = 260 µm. For IR light with wavelength λ 1510 nm, cross port coupling efficiency is more than 99%. For visible light with wavelength λ 816 nm, rop port coupling efficiency is more than 99%.