Biphoton Statistics of Quantum Light Generated on a Silicon Chip

Article pubs.acs.org/journal/apchd5

Biphoton Statistics of Quantum Light Generated on a Silicon Chip Xiyuan Lu,†,‡ Wei C. Jiang,†,⊥ Jidong Zhang,§ and Qiang Lin*,†,⊥,§ †

Center for Coherence and Quantum Optics, ‡Department of Physics and Astronomy, ⊥Institute of Optics, and §Department of Electrical and Computer Engineering, University of Rochester, Rochester, New York 14627, United States ABSTRACT: We demonstrate a silicon-chip biphoton source with an unprecedented quantum cross-correlation up to 4 g(2) si (0) = (2.58 ± 0.16) × 10 . The emitted photons are within single whispering gallery mode with self-correlations of (2) g(2) ss (0) = 1.90 ± 0.05 and gii (0) = 1.87 ± 0.06 for signal and idler photons, respectively. We observe the waveform asymmetry of cross-correlation between signal and idler photons and reveal the identical and nonexponential nature of self-correlations of individual signal and idler photon modes, which is the nature of cavity-enhanced nonlinear optical processes. The high efficiency and high purity of the biphoton source allow us to herald single photons with a conditional self-correlation g(2) c (0) as low as 0.0059 ± 0.0014 at a pair flux of 1.95 × 105 pairs/s, which remains below 0.026 ± 0.001 for a biphoton flux up to 2.93 × 106 pairs/s with a tunable photon preparation efficiency in the single-mode fiber up to 51% and a raw heralding efficiency up to 3.4%. Our work unambiguously demonstrates that silicon photonic chips are superior material and device platforms for integrated quantum photonics. KEYWORDS: four-wave mixing, photon correlation, photon pair, photon statistics, silicon microresonator

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date.9−41,45−57 Each of the signal and idler photon modes is effectively in a single thermal mode with a photon self(2) correlation of g(2) ss (0) = 1.90 ± 0.05 and gii (0) = 1.87 ± 0.06, respectively. The demonstrated photon cross-correlation (2) violates the classical Schwarz inequality, g(2) si (0) ≤ [gss (0) 1/2 (2) gii (0)] , by 4 orders of magnitude. The biphoton production is extremely efficient with a spectral brightness of 1.06 × 109pairs/s/mW2/GHz, the highest value reported in current FWM-based devices.9−11,13−16,18−22,24−28,30,33−40,43 The strong correlation and the long lifetime of biphotons allow us to resolve the fine structure of biphoton statistics in the time domain. We not only observe the waveform asymmetry of the signal-idler cross-correlation but also experimentally reveal for the first time, to the best of our knowledge, the identical and nonexponential nature of selfcorrelations for the signal and idler photons. All of the experimental results agree very well with a theory developed for the cavity-enhanced FWM. In particular, the strong photon correlation and high efficiency of biphoton production enable heralding of high-quality single photons with conditional selfcorrelation g(2) c (0) as low as 0.0059 ± 0.0014 at a pair flux of 1.95 × 105 pairs/s. g(2) c (0) remains as low as 0.026 ± 0.001, even at a biphoton flux as large as 2.93 × 106 pairs/s, with a preparation efficiency up to 51% for the photons collected directly in the single-mode fiber. The recorded g(2) c (0) values are among the best that have ever been reported.12,13

ntegrated quantum photonic technology has made great advances in recent years with the demonstration of diverse quantum functionalities on chip,1,2 some of which, such as quantum simulation3 and photonic Boson sampling,4−7 may even go beyond the reach of classical computing. Integrated photonics enables significant structural and functional complexity and thus shows great promise for realizing future large-scale quantum photonic networks.8 One key element underlying a fully integrated quantum photonic interconnect is to generate nonclassical states of light on chip, which has attracted tremendous interest recently in demonstrating single photons,9−14 biphotons,15−41 and their quantum entanglements16,24,28,31,33,39,40 on chip-scale nonlinear optical devices from various material platforms. Among the platforms developed to date, silicon is particularly attractive for quantum light generation9,11,14−16,18−21,24−28,30,33−37,39−41 given its mature nanofabrication technology and CMOS compatibility for high-quality device fabrication and integration, its strong optical Kerr nonlinearity for efficient nonlinear optical processes, its significant refractive index for strong mode confinement that supports device miniaturization, and particularly its clean narrowband phonon spectrum free from the broadband Raman noise42 that is deleterious for quantum light sources based upon some other device platforms.10,13,22,43,44 In this article, we demonstrate the generation of nonclassical biphoton and single-photon states in a quantum silicon photonic device with extremely high purity. We take advantage of the dramatic cavity enhancement on four-wave mixing (FWM) in a high-quality silicon microdisk resonator to produce biphotons with a signal-idler cross-correlation up to 4 g(2) si (0) = (2.58 ± 0.16) × 10 , the highest value reported to © XXXX American Chemical Society

Received: March 21, 2016

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DOI: 10.1021/acsphotonics.6b00204 ACS Photonics XXXX, XXX, XXX−XXX

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Figure 1. Scheme of the photon source. (a) A schematic illustration of the photon-pair generation process. The photons are coupled into and out of the device by the same tapered single-mode fiber. (b) Frequency and (c) energy diagrams for the cavity-enhanced four-wave mixing process.

Figure 2. Experimental setups measuring (a) the cross-correlation between signal and idler and the self-correlations of individual (b) signal and (c) idler modes. Setup (d) characterizes the antibunching of heralded single photons by conditional self-correlation. SNSPD: niobium nitride (NbN) superconducting nanowire single-photon detector, working continuously with a quantum efficiency of 7%, dark counts around 100 counts per second, and a timing jitter of 70 ps. The NbN SNSPD has relative low quantum efficiency compared to that of other single photon detectors but results in the advantage of low detector timing jitter, which is crucial to resolve second-order photon correlations. InGaAs SPD: indium gallium arsenide single-photon detector with a quantum efficiency of 15%, a gate at 20 ns, a repetition rate of 2.5 MHz, a deadtime of 10 μs, and a timing jitter of approximately 250 ps.

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RESULTS Device and Setup. The device is a high-Q silicon microdisk resonator with a diameter of 9 μm and a thickness of 260 nm (Figure 1a) sitting on the silica pedestal. The employed device structure enables precise dispersion engineering37 to satisfy the frequency matching condition among three cavity modes involved in the FWM process. As a result, pumping at the central mode produces biphotons in signal and idler modes with frequencies located symmetrically around the pump, as shown in Figure 1b and c. The pump wave is coupled to the device by a tapered single-mode optical fiber, which also

delivers the produced biphotons out of the device. An excellent feature of such coupling scheme is that photons are delivered to the same single spatial mode of the coupling fiber for all the optical modes involved in the FWM process. Moreover, the fiber-device coupling scheme enables flexible engineering of the photon coupling efficiency simply by changing the device-taper distance. The fabricated device is tested with the experimental setup shown in Figure 2. For the laser noise to be filtered out, the pump laser passes through a coarse wavelength-division multiplexer before coupling into the device. The multiplexer B


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Figure 3. Cavity transmission and photoluminescence spectra. (a) Transmission spectrum of the passive cavity scanned by a tunable continuouswave laser. The insets show detailed transmission spectra with theoretical fitting shown in red. (b) Photoluminescence spectra of the generated biphotons. The colors indicate the individual spectra recorded at different transmission ports of the demultiplexer. The demultiplexer has transmission bands centered at 1491 and 1531 nm, respectively, with 3 dB widths of 18 nm to filter out the residue pump photons for generated biphotons. The photoluminescence spectra clearly show that there are no other noise photon spectral lines. The spectrometer has a resolution of ∼16 GHz.

3b). The spectra show no noise mode at all, demonstrating an excellent spectral cleanness of the cavity-enhanced FWM process in silicon. The amplitude difference between the two photon modes is due to different external couplings of cavity modes to the tapered fiber. To characterize the biphoton source, we record the biphoton flux when pumping the device with a continuous-wave (CW) laser, as shown in Figure 4. The device is operating under a condition close to critical coupling for optimized power efficiency with loaded Q factors of 2.23 × 105, 2.97 × 105, and 2.59 × 105 for idler, pump, and signal modes, respectively. The biphoton flux depends quadratically on the pump power, which is a characteristic of the degenerate FWM process. A small pump power of 53.8 μW dropped into the cavity generates a biphoton flux of 1.61× 106 pairs/s, clearly demonstrating the high efficiency of the device. Such a high efficiency results from the significant Purcell enhancement on the triply resonant degenerate FWM process inside the cavity37 with a biphoton emission rate of Rsi = 2 (gNp)2ΓesΓei/(Γ̅ ΓtsΓti) where Np represents the pump photons inside the cavity, Γej and Γtj are the external coupling rate and photon decay rate of

has a 3-dB bandwidth of 17 nm for each of its transmission bands whose center wavelengths are separated by 20 nm with a band isolation over 120 dB. The biphotons generated from the device are then separated into individual photon modes by a demultiplexer, which is identical to the multiplexer used at the input end. The photoluminescence spectra of the biphotons are recorded at each transmission port of the demultiplexer to suppress the pump wave. Tunable bandpass filters with a 3 dB bandwidth of 1.2 nm are used in front of the single-photon detectors to cut the Raman noises produced in the delivery silica fibers.37 The quantum correlation properties of the photons are characterized by single photon detectors, and coincidence counting setups are shown in insets a−d in Figure 2. Biphoton Generation. The laser-scanned transmission spectrum of the passive cavity (Figure 3a) shows that a constant mode spacing of 2.437 THz is achieved for the cavity modes located at 1497.1, 1515.6, and 1534.4 nm with intrinsic Q factors of 3.33 × 105, 5.09 × 105, and 4.68 × 105, respectively. Consequently, a pair of clean photon modes is produced by pumping the central mode at 1515.6 nm (Figure C


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counts and accidental coincidence counts, and the accidental coincidence counts, which are closely related to coincidence-toaccidental ratio (CAR) by g(2) si (0) = CAR(0) + 1. For the cavityenhanced FWM process, detailed analysis shows that the emitted biphotons exhibit the following cross-correlation (see Methods) ⎧ ΓtsΓti −Γ τ e ts + 1(τ ≥ 0), ⎪ 4(gNp)2 ⎪ gsi(2)(τ ) = ⎨ ⎪ ΓtsΓti Γtiτ ⎪ 4(gN )2 e + 1(τ < 0) p ⎩

(2)

2 with a peak value of g(2) si (0) = ΓtsΓti/[4(gNp) ]+ 1, which scales inversely with the power square of the pumping level. When the pump power is weak (gNp ≪ Γts,ti), the generated biphotons are in the single-photon regime. When the pump power is strong (gNp ≫ Γts,ti), the source falls into the regime dominated by multiphoton generation. In the experiment, the cross-correlation g(2) si (τ) is characR si(τ ) . Figure 4 shows terized by coincident flux and singlet flux,

Figure 4. Photon pair flux (blue) and the cross-correlation (red) as a function of optical drop power. Optical drop power is the power dropped inside the cavity, i.e., optical input power × (1 − transmissivity). The experimental data are shown in dots, and the solid curves are theoretical predictions (without fitting parameters). The dashed red line takes into account the detector dark counts and the Raman noises produced in the delivery fibers. g(2) si (0)

NsNi

the power dependence of cross-correlation (g(2) si (0)), biphoton flux (Rsi), and their theoretical predictions. The produced biphotons exhibit a high cross-correlation at a large photon flux 6 with g(2) si (0) = 103 ± 0.2 at a biphoton flux of 1.61 × 10 pairs/ (2) s. For an ideal photon source, gsi (0) would keep increasing with decreasing pair flux, as shown in the red solid curve, due to the decreased probability of accidental multipair events. In reality, however, the cross-correlation is also affected by the Raman noise (from the silica fibers37) and the detector dark counts, which is predicted by the red dashed curve. The equation for the dashed red curve is given by R si , where Rsi, Ns, and Ni are calculated

the loaded cavity with subscripts j = s,i denoting the signal and idler modes, respectively, and Γ̅ = (Γts + Γti)/2; g represents the vacuum coupling rate of the FWM process (see Methods). FWM creates biphotons inside the cavity at a rate of 2 (gNp)2/ Γ̅ , which are delivered to the coupling waveguide with a pair extraction efficiency of ΓesΓei/(ΓtsΓti). As shown in Figure 4, the theoretical prediction of biphoton flux agrees closely with the experimental observation. The generation efficiency of biphotons is characterized by spectral brightness, the biphoton flux per unit spectral width per unit pump power square, because degenerate FWM depends quadratically on the pump power. Detailed analysis shows that the emitted signal and idler photons exhibit an identical emission spectra profile centering around cavity resonance frequencies (see Methods) Sj(ω) ∝

(Ns − NR − ND)(Ni − NR − ND)

from theory, and NR and ND are the flux of Raman noise and detector dark counts, which are measured by registering the detector flux while shifting the laser outside of the resonance and turning off the laser, respectively. Only the effects to singlets are considered here because (NR + ND) are much larger

g 2Np2 [(ω − ωj)2 + (Γts/2)2 ][(ω − ωj)2 + (Γti/2)2 ]

Ns/i

(1)

1 , CAR

where j = s,i stands for signal or idler mode, and ωj represents the center frequency of the cavity resonance. Eq 1 shows that the biphoton emission spectra exhibit a spectral width narrower than that of the passive cavity modes. The loaded cavity linewidths of signal and idler modes are 0.755 and 0.899 GHz, respectively, which lead to a spectral linewidth of 0.527 GHz for the emitted biphotons. The spectral brightness of the emitted biphotons is thus inferred to be 1.06× 109 pairs/s/mW2/GHz. This value is the highest in FWM-based photon-pair sources reported to date.9−11,13−16,18−22,24−28,30,33−40,43 It is 1 order of magnitude higher than our previous device37 because the current device exhibits clean singlet cavity resonances (Figure 3a) that are free from the photon backscattering. Cross-Correlation. Cross-correlation between signal and idler photons is characterized by gsi(2)(τ ) ≡

1

and the contribution to cross-correlation is ∼ CAR . than The measured cross-correlation agrees with this prediction quite well with a peak value of (2.58 ± 0.16) × 104 at a biphoton flux of 5.25 × 103 pairs/s. This quantum crosscorrelation suggests that, among over 25,000 detected photon pairs, only one pair is not correlated with each other. This value is the highest one reported to date among all biphoton sources.9−41,45−57 As biphoton flux decreases, the recorded g(2) si (0) deviates from the theoretical prediction, which is mainly due to the detector dark counts and the Raman noise from the optical fibers. Taking into account these two factors, the prediction of the dashed red line agrees with experimental data quite well (Figure 4). The long cavity lifetime of the device allows us to temporally resolve the quantum cross-correlation. Two examples are shown in Figure 5a,b with pump powers of 1.2 and 53.8 μW, respectively. The temporal waveforms of cross-correlations are consistent with the theoretical predictions. Note that g(2) si (τ) becomes asymmetric with respect to τ when the signal and idler photons exhibit different lifetimes in the cavity. Although biphotons are created simultaneously, individual signal (or idler) photons stay inside the cavity over their photon lifetimes

⟨E i†(t )Es†(t + τ )Es(t + τ )E i(t )⟩ ⟨E i†(t )E i(t )⟩⟨Es†(t )Es(t )⟩

where Ej(t) (j = s,i) is the field operator for the signal and idler modes.58 Cross-correlation is a unitless parameter describing the ratio of the total counts, i.e., the sum of true coincidence D


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Figure 5. Cross-correlations between signal and idler photons. g(2) si (τ) at optical drop powers of (a) 1.2 and (b) 53.8 μW with experimental data in blue and the theoretical prediction in red. (c) Normalized (2) (2) (2) cross-correlations, g(2) si (τ)/gsi (0) and gis (τ)/gis (0), with experimental data shown in dots and theory in solid curves. The loaded cavity qualities are 1.21 × 105 and 1.99 × 105 for signal and idler modes, respectively. The gray dots show the recorded instrument response of the detection system, which is dominated by the timing jitters of the two SSPDs. The cross-correlations are recorded with a time bin of 4 ps.

Figure 6. Self-correlation functions of (a) signal and (b) idler photons with experimental data shown in blue and theory in red. (c) Direct (2) comparison between g(2) ss (τ) (blue) and gii (τ) (red) together with the theory in black. The red and blue data points are from a and b, respectively. The theoretical curves in a−c are the same, which are obtained from eq 3 convolved by the instrument response (Figure 5c, gray curve). The self-correlation functions are recorded with a time bin size of 8 ps.

58 g(2) Our analysis shows that the signal and idler jj (0) = 2. photons produced from the cavity enhanced FWM process exhibit the following self-correlation (Methods)

before transmitting out of the cavity. Consequently, the signal (or idler) photon transmitted later is correlated with the idler (or signal) photon transmitted earlier within the lifetime of the signal (or idler) photon. The asymmetric exponential decay is a fingerprint of the cavity-enhanced nonlinear optical process,51 which is distinct from the waveguide-based biphoton sources.15,16,18−20,22−24,26,28,33,35 This feature is clearly observed in Figure 5c. We measured (2) g(2) si (τ) and gis (τ) by switching the signal and idler paths before the SSPDs in Figure 2a. The tapered fiber coupling scheme leads to different photon lifetimes of the two modes and thus enables observation of the asymmetric waveform of g(2) si (τ). Figure 5c shows a clear asymmetric g(2) si (τ) resulting from different lifetimes of 98.5 and 158 ps for the signal and idler photons, respectively. To the best of our knowledge, this is the first time observing such an asymmetric cross-correlation for a chip-scale photon-pair source.9−16,18−37,39,40 Self-Correlations. Self-correlations of the signal and idler photons are characterized by the expression58 g jj(2)(τ )

≡

⟨Ej†(t )Ej†(t + τ )Ej(t + τ )Ej(t )⟩ ⟨Ej†(t )Ej(t )⟩2

g jj(2)(τ )

⎡ Γ e−Γti |τ| /2 − Γ e−Γts|τ| /2 ⎤2 ti ⎥ +1 = ⎢ ts ⎢⎣ ⎥⎦ Γts − Γti

(3)

where j = s,i stands for the signal and idler photons, respectively. Eq 3 shows g(2) jj (0) = 2. In practice, the finite instrument response of the detection system (Figure 5c, gray curve) will slightly broaden the self-correlation function, leading to a slightly reduced peak value of 1.98, as shown by the solid curve in Figure 6. Panels a and b in Figure 6 show the recorded self-correlation functions (blue dots) for signal and idler photons, respectively, which agree well with the theoretical prediction (red curve). (2) These measurements show g(2) ss (0) = 1.90 ± 0.05 and gii (0) = 1.87 ± 0.06 for the signal and idler, respectively, which are very close to the theoretical value of 1.98 and reveal the single-mode nature of the interacting whispering-gallery modes. This singlemode nature of the photons results from the high finesse of the device, which is different from the cavity-enhanced SPDC in a large microresonator that generally produces photons in multimodes and requires external filtering.57 Moreover, the (2) (2) recorded values of g(2) ss (0), gii (0), and gsi (0) violate the 1/2 (2) (2) classical Schwarz inequality,58 g(2) (0) ≤ [g si ss (0) gii (0)] , by

(j = s, i)

In general, g(2) jj (0) is a measure of the mode nature of the produced photons. For an ideal single-mode thermal source, E


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Figure 7. Schematic of the identical spectra profile of signal and idler photons. The spectra heights of pump, signal, and idler modes are normalized to the same height for a better view of the spectra profiles. The gray Lorentzian functions represent cavity modes of involved optical modes. The pump laser has a narrow linewidth indicated by the green line. Although the signal mode has a broader loaded cavity linewidth than that of idler mode, the emitted signal (red) and idler (blue) photons share the same spectra profile due to energy conservation.

Figure 8. Heralded single photons and conditional self-correlations under various fiber device coupling conditions. (a) Schematic illustration of the photon extraction efficiency of the generated photons, where photons lost in the cavity cannot be heralded. (b) By tuning the fiber device distance, the external taper coupling rate is increased to improve the photon extraction efficiency. The yellow curve is the theoretical curve deduced from cavity transmission traces. (c) The heralded single photon source is characterized by the Hanbury−Brown−Twiss experiment. The data points have the coupling conditions described in b and labeled by different colors. The solid circles represent the raw data without any corrections. The empty circles represent the processed data with detector dark counts subtracted.

more than 4 orders of magnitude, which clearly shows the nonclassical nature of the biphotons generated by the device. Remarkably, eq 3 shows that the signal and idler photons exhibit identical self-correlation even when their individual cavity modes have different lifetimes. Figure 6c confirms this prediction in the time domain with a direct comparison (2) between g(2) ss (τ) and gii (τ). The identical self-correlation agrees with the prediction of biphoton spectra in the frequency domain (eq 1), where the energy conservation of the FWM

process restricts signal and idler photons to exhibit identical emission spectra profiles. As illustrated in Figure 7, although the signal and idler cavity modes of the passive cavity have different cavity Qs and thus different linewidths, the emitted photons have the same spectra profile located around the signal (or idler) frequency because of the energy conservation. Of particular interest is that eq 3 reveals that the selfcorrelation function does not show a pure exponential function, which is contrary to the common adoption of exponential F


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fittings for the self-correlation functions.14,38,52,55−57 Eq 3 also suggests that the self-correlation function has a much broader linewidth than that of the cross-correlation function. The nonexponential and linewidth-broadening features can be seen more clearly when the signal and idler photons share the same lifetime (Γts = Γti = Γt). In this case, eq 3 reduces to g(2) jj (τ) = (1 + Γt |τ|/2)2e−Γt|τ|+1. These features are observed experimentally by comparing Figure 6c with Figure 5c. Clearly, g(2) jj (τ) shows a waveform with a full-wave half-maximum (fwhm) time around 0.6 ns (Figure 6c), which is two times broader than the fwhm time of g(2) si (τ) (Figure 5c). Although eq 3 is obtained here for a SFWM process, it is universal to a cavity-enhanced nonlinear process regardless of SFWM or SPDC.59 Heralded Single Photons. The high efficiency and purity of the photons readily imply the application for heralding highpurity single photon Fock state. Heralding efficiency is an important figure of merit in heralded single photon sources.9−14 A critical parameter distinct in the cavity quantum electrodynamic system is photon extraction efficiency ηE. After a biphoton is created inside the cavity, individual photons can either be extracted out of the cavity into the coupling waveguide with an external coupling rate of Γe or be lost inside the cavity at an intrinsic photon decay rate of Γ0 due to intrinsic material absorption or device scattering (Figure 8a). Because only the extracted photons can be detected and heralded, the photon extraction efficiency of the heralded photons in the coupling waveguide is given by ηE = Γe/Γt. This efficiency can be quantified by the transmissivity at the center of the resonance Tc, given by ηE = (1 ± Tc )/2 , where the plus and minus signs correspond to the over- and undercoupled conditions, respectively. In practice, as the signal mode at 1534.4 nm has an external coupling rate higher than that of the idler mode at 1497.1 nm (see Figure 3b), we use the detection of an idler photon to herald the arrival of a signal photon. The flexible external coupling in our system results in widely tunable photon extraction efficiency, as shown in Figure 8b, where a maximum ηE of 76% is achieved by overcoupling the device. To characterize the performance of photon sources, we defined preparation efficiency13 as the heralding efficiency in the singlemode fiber with the effects of optical components and detectors excluded. At the above coupling condition, the preparation efficiency is 51%. The corresponding Klyshko efficiency60 (i.e., raw heralding efficiency) is 3.4%, which is also the ratio of the measured photon flux and the generated photon flux out of the device. This efficiency is primarily due to the detection efficiency of the InGaAs SPDs (15%), the photon extraction efficiencies (Figure 8b), and the loss of optical components (−3.55 dB). In Figure 5, the Klyshko efficiency is approximately 50% lower due to the low detection efficiency of NbN SNSPD (7%). For practical quantum applications, the Klyshko efficiency needs to be further improved by using detectors with higher efficiencies, optimizing the fiber device coupling conditions, and using optical components with smaller optical loss. The quality of heralded single photons can be characterized by the conditional self-correlation gc(2)(0) ≡

⟨Es†(t )Es†(t )Es(t )Es(t )⟩i ⟨Es†(t )Es(t )⟩i2

feature. To characterize the photon antibunching quality of the heralded single photons, we perform a Hanbury−Brown− Twiss experiment to measure the conditional self-correlation (Figure 2d). The signal photons are separated into two channels through a 50:50 beam splitter, and the triple coincidence is detected among the idler channel and the two split signal channels. The conditional self-correlation of the heralded single photon is thus given by61,62 gc(2)(0) =

Nis1s2Ni Nis1Nis2

(4)

where Nis1s2 is the triple coincidence rate, Ni is the counting rate of the idler photons, and Nis1/Nis2 are the double coincidence rates between the idler photon and the single photon in the two split channels, respectively. Figure 8c shows the recorded g(2) c (0) versus biphoton flux in three different coupling conditions, whose photon extraction efficiencies are shown in Figure 8b. The solid data points are measured following eq 4. The empty data points are measured with detector dark counts subtracted, as given by

Nis1s2(Ni − ND) Nis1Nis2

,

where the dark counts are only considered in the singlet counts, as the contribution of dark counts to the high order correlations are negligible. g(2) c (0) remains below 0.08 over the entire recorded range of photon flux up to 3.9 × 106 pairs/s with a lowest value of 0.0059 ± 0.0014 (0.0035 ± 0.0008) recorded at a pair flux of 1.95 × 105 pairs/s without (with) subtracting detector dark counts. Moreover, for a certain photon pair flux, the conditional self-correlation decreases as photon extraction efficiency increases. At the same photon-pair flux, a higher photon extraction efficiency results in a reduced photon number inside the cavity, which in turn reduces the probability of multiphoton generation. At a photon extraction efficiency of 76% (blue dots in Figure 8c), g(2) c (0) becomes 0.026 ± 0.001 at a pair flux of 2.93 × 106 pairs/s. To the best of our knowledge, these values are among the lowest g(2) c (0) at corresponding biphoton fluxes reported to date,12,13 which demonstrate the superior quality of the heralded single photon Fock state produced in our device.

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DISCUSSION In summary, we demonstrate the nonclassical nature of biphotons produced by cavity-enhanced FWM on a silicon 4 chip with a cross-correlation of g(2) si (0) = (2.58 ± 0.16) × 10 , the highest value reported to date.9−41,45−57 The generated photons are essentially within a single whispering gallery mode, which is confirmed with self-correlations of g(2) ss (0) = 1.90 ± 0.05 and g(2) ii (0) = 1.87 ± 0.06, respectively. The demonstrated correlations violate the classical Schwartz inequality, g(2) si (0) ≤ 1/2 (2) [g(2) (0) g (0)] , by 4 orders of magnitude. The device has a ss ii spectral brightness of 1.06 × 109 pairs/s/mW2/GHz, which is the highest in current FWM-based devices.9−11,13−16,18−22,24−28,30,33−40,43 In particular, we observe the waveform asymmetry of the biphoton cross-correlation and reveal the identity and nonexponential nature of the selfcorrelation functions of individual signal and idler photons. The strong photon correlation and high efficiency of biphoton production enable us to herald single photons with conditional self-correlation g(2) c (0) as low as 0.0059 ± 0.0014 at a pair flux of 1.95 × 105 pairs/s, which remains below 0.026 ± 0.001 for biphoton flux up to 2.93 × 106 pairs/s with a photon

,

where the subscript on ⟨ ⟩i denotes the condition of detecting an idler photon.61 The value of g(2) c (0) describes the probability of heralding a multiphoton state. For an ideal heralded single photon Fock state, g(2) c (0) = 0 indicates a perfect antibunching G


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Figure 9. Schematic of biphoton generation in a silicon microdisk resonator.

preparation efficiency in the single-mode fiber up to 51%. The 12,13 recorded g(2) c (0) values are among the best values reported. The high quality and high purity biphotons and heralded single photons produced directly in the telecom band, together with the CMOS compatibility of the device platform, render our device a promising source for a variety of integrated quantum photonic applications, including quantum key distribution, hybrid macroscopic-microscopic entanglement, and intra/inter-chip quantum state teleportation. The exploration of the detailed waveform structure of cross- and selfphoton correlations advance our understanding of biphoton statistics of cavity-enhanced nonlinear optical processes, which may have profound impact on quantum functionalities such as quantum frequency conversion and light-matter interfacing that require detailed information on photon wavepackets. There has been significant development in the past few years on various chip-scale platforms for photon generation.9−41 Our work unambiguously demonstrates that silicon photonic chips are superior material and device platforms for integrated quantum photonics. Further development of quantum photonic functionalities based upon our devices would form a fundamental building block toward an integrated quantum photonic interconnect.

where the intracavity field operator aj (j = p,s,i) is normalized such that a†j aj represents the photon number operator. Γ0j and Γtj = Γ0j + Γej (j = p,s,i) are the photon decay rates of the intrinsic and loaded cavity, respectively. bj is the field operator of the incoming wave at carrier frequency ωj inside the coupling waveguide, which is normalized such that b†j bj represents the input photon flux. bj satisfies the commutation relation of [bj(t), b†j (t′) ]= δ(t − t′). uj is the noise operator associated with the intrinsic cavity loss, which satisfies the commutation relation of [uj(t), uj†(t′) ]= δ(t − t′). g = cηn2ℏωp ωsωi /(nsn i V) ̅ describes the vacuum coupling rate of the FWM process, where η, n2, and V are the spatial overlap among the interacting modes, the Kerr nonlinear coefficient, and the effective mode volume, respectively. In the theoretical prediction, the mode overlap (η) is unity because the three optical modes share the same spatial mode profile. The third-order nonlinearity is n2 = 2.23 × 10−5 cm2/GW. The effective volume of the employed optical modes is V = 9.01 μm3, simulated by the finite-element method. The pump mode is launched with a continuous-wave laser and can be treated as a classical field. For the spontaneous FWM process, the signal and idler modes stay at single photon regime, and the pump mode can be assumed to be nondepleted. Making a Fourier transform as Fj(ω) = i(ω−ωj) t ∫ +∞ Fj(t) dt, where Fj stands for operator aj, bj, or uj −∞e (j = p,s,i), we can solve eqs 5−7 analytically in the frequency domain to obtain

■

METHODS In this section, we provide a theoretical description of biphoton generation through spontaneous degenerate FWM in the triply resonant cavity. We only consider Kerr nonlinear optical interactions because two-photon absorption and free-carrier effects are negligible for the optical power employed in our system. As shown in Figure 9, a pump wave is launched into the resonator to generate signal and idler photon pairs, which are coupled out of the cavity into the delivery waveguide. The dynamics among the three cavity modes at frequencies ω0j (j = p,s,i) can be described by the following equations of motion,37 da p dt

(5)

(9)

a i(ω) = A(ω)n i(ω) + B(ω)ns†( −ω)

(10)

ns(ω) ≡ i Γes bs(ω) + i Γ0s us(ω)

(11)

n i(ω) ≡ i Γei bi(ω) + i Γ0i u i(ω)

(12)

A(ω) = (6)

B(ω) =

da i = ( −iω0i − Γti/2)ai + 2iga p†a pai + igas†a p2 dt + i Γei bi(t ) + i Γ0i u i(t )

as(ω) = A(ω)ns(ω) + B(ω)n i†( −ω)

In eqs 9 and 10, A(ω) and B(ω) have the expressions

das = ( −iω0s − Γts/2)as + 2iga p†a pas + iga i†a p2 dt + i Γes bs(t ) + i Γ0s us(t )

(8)

where the noise operators ns and ni are given by

= ( −iω0p − Γtp/2)a p + iga p†a p2 + i Γep bp(t ) + i Γ0p u p(t )

a p(ω) = i Γep bp(ω)/(Γtp/2 − iω)

Γti/2 − iω (Γts/2 − iω)(Γti/2 − iω) − (gNp)2

(13)

−iga p2 (Γts/2 − iω)(Γti/2 − iω) − (gNp)2

(14)

where Np = ⟨a†pap⟩ is the average photon number of the pump wave inside the cavity. Eqs 13 and 14 include the multiphoton

(7) H


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Figure 10. Theoretical cross/self-correlations. (a) Normalized cross-correlations without broadening from the instrument response function (IRF) show asymmetric exponential decay in solid red and blue lines. The green arrow specifies a full-wave at half-maximum (fwhm) time of 0.18 ns. (b) Self-correlations without IRF show the identical and nonexponential spectral profiles for both signal and idler photons with a solid black line. The green arrow specifies a fwhm time of 0.55 ns, which is much broader than the fwhm time in cross-correlations. The dashed lines in a and b correspond to the predictions of cross- and self-correlations with IRF in Figures 5c and 6c, respectively.

generation induced by the stimulated FWM. In the singlephoton regime, where gNp ≪ (Γts and Γti), these equations reduce to 1 A(ω) ≈ Γts/2 − iω

B(ω) ≈

In particular, the cross-correlation between the signal and idler photons is found to be gsi(2)(τ ) ≡

(15)

⟨fs† (t )fs (t )⟩⟨fi† (t )fi (t )⟩ +∞

−iga p2 (Γts/2 − iω)(Γti/2 − iω)

=

(16)

fj = bj + i Γej a j

Eqs 8−17 can be applied to find the photon statistics. For example, the spectra of the signal and idler photons are given by (18)

Si(ω) ≡ ⟨fi† (ω)fi (ω)⟩ = ΓeiΓts|B(ω)|2

(19)

+∞

2

2

+1

⎧ ΓtsΓti −Γ τ e ts + 1 (τ ≥ 0) ⎪ 2 ⎪ 4(gNp) =⎨ ⎪ ΓtsΓti Γtiτ ⎪ 4(gN )2 e + 1 (τ < 0) p ⎩

(17)

Ss(ω) ≡ ⟨fs† (ω)fs (ω)⟩ = ΓesΓti|B(ω)|2

∫−∞ B(−ω)[ΓtsA(ω) − 1]e−iωτ dω ΓtsΓti ∫ |B(ω)|2 dω −∞

As the transmitted field for each photon mode is given by63

(23)

which is eq 2 in the main text. Therefore, the emission flux of correlated biphotons is given by

which shows that the signal and idler exhibit identical spectral profiles, a result of energy conservation for the FWM process, Sj(ω) ∝ |B(ω)|2 =

⟨fi† (t )fs† (t + τ )fs (t + τ )fi (t )⟩

R c = ⟨fs† (t )fs (t )⟩⟨fi† (t )fi (t )⟩

g 2Np2

=

[ω 2 + (Γts/2)2 ][ω 2 + (Γti/2)2 ]

+∞

∫−∞

[gsi(2)(τ ) − 1] dτ

2 ΓesΓei 2(gNp) ΓtsΓti Γ̅

(20)

(24)

which is eq 1 in the main text. Interestingly, eq 20 also shows that the emission spectrum of signal/idler photons exhibits a spectral width narrower than that of the passive cavity mode. This spectral narrowing arises from the energy conservation among interacting cavity modes regardless of SFWM or SPDC processes.59,64 At the same time, the emitted photon fluxes of individual signal and idler are given by

On the other hand, the self-correlations of signal or idler photons are found to be

⟨fs† (t )fs (t )⟩ =

ΓesΓti 2π

+∞

∫−∞

|B(ω)|2 dω ≈

g jj(2)(τ )

+∞

∫−∞

|B(ω)|2 dω ≈

=

ΓtsΓ̅

∫−∞ |B(ω)|2 e−iωτ dω +∞

2

∫−∞ |B(ω)| dω

2

2

+1

⎡ Γ e−Γti |τ| /2 − Γ e−Γts|τ| /2 ⎤2 ti ⎥ +1 = ⎢ ts ⎢⎣ ⎥⎦ Γts − Γti

2Γei(gNp)2 ΓtiΓ̅

⟨f j† (t )fj (t )⟩2 +∞

2Γes(gNp)2 (21)

ΓΓ ⟨fi† (t )fi (t )⟩ = ei ts 2π

≡

⟨f j† (t )f j† (t + τ )fj (t + τ )fj (t )⟩

(22) I

(25)


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Osellame, R.; Galvão, E. F.; Sciarrino, F. Experimental validation of photonic boson sampling. Nat. Photonics 2014, 8, 615−620. (8) Kimble, H. J. The quantum internet. Nature 2008, 453, 1023− 1030. (9) Davanço, M.; Ong, J. R.; Shehata, A. B.; Tosi, A.; Agha, I.; Assefa, S.; Xia, F.; Green, W. M. J.; Mookherjea, S.; Srinivasan, K. Telecommunications-band heralded single photons from a silicon nanophotonic chip. Appl. Phys. Lett. 2012, 100, 261104. (10) Clark, A. S.; Husko, C.; Collins, M. J.; Lehoucq, G.; Xavier, S.; De Rossi, A.; Combrié, S.; Xiong, C.; Eggleton, B. J. Heralded singlephoton source in a III-V photonic crystal. Opt. Lett. 2013, 38, 649− 651. (11) Collins, M. J.; Xiong, C.; Rey, I. H.; Vo, T. D.; He, J.; Shahnia, S.; Reardon, C.; Krauss, T. F.; Steel, M. J.; Clark, A. S.; Eggleton, B. J. Integrated spatial multiplexing of heralded single-photon sources. Nat. Commun. 2013, 4, 2582. (12) Krapick, S.; Herrmann, H.; Brecht, B.; Quiring, V.; Suche, H.; Silberhorn, C. An efficient integrated two-color source for heralded single photons. New J. Phys. 2013, 15, 033010. (13) Spring, J. B.; Salter, P. S.; Metcalf, B. J.; Humphreys, P. C.; Moore, M.; Thomas-Peter, N.; Barbieri, M.; Jin, X.-M.; Langford, N. K.; Kolthammer, W. S.; Booth, M. J.; Walmsley, I. A.; Peter, C.; Steven, W. On-chip low loss heralded source of pure single photons. Opt. Express 2013, 21, 13522−13532. (14) Reimer, C.; Caspani, L.; Clerici, M.; Ferrera, M.; Kues, M.; Peccianti, M.; Pasquazi, A.; Razzari, L.; Little, B. E.; Chu, S. T.; Moss, D. J.; Morandotti, R. Integrated frequency comb source of heralded single photons. Opt. Express 2014, 22, 6535−6546. (15) Sharping, J. E.; Lee, K. F.; Foster, M. A.; Turner, A. C.; Schmidt, B. S.; Lipson, M.; Gaeta, A. L.; Kumar, P. Generation of correlated photons in nanoscale silicon waveguides. Opt. Express 2006, 14, 12388−12393. (16) Takesue, H.; Fukuda, H.; Tsuchizawa, T.; Watanabe, T.; Yamada, K.; Tokura, Y.; Itabashi, S. Entanglement generation using silicon wire waveguide. Appl. Phys. Lett. 2007, 91, 201108. (17) Zhang, Q.; Xie, X.; Takesue, H.; Nam, S. W.; Langrock, C.; Fejer, M. M.; Yamamoto, Y. Correlated photon-pair generation in reverse-proton-exchange PPLN waveguides with integrated mode demultiplexer at 10 GHz clock. Opt. Express 2007, 15, 10288−10293. (18) Clemmen, S.; Huy, K. P.; Bogaerts, W.; Baets, R. G.; Emplit, P.; Massar, S.; Phan Huy, K.; Bogaerts, W.; Baets, R. G.; Emplit, P.; Massar, S. Continuous wave photon pair generation in silicon-oninsulator waveguides and ring resonators. Opt. Express 2009, 17, 16558−16570. (19) Harada, K.; Takesue, H.; Fukuda, H.; Tsuchizawa, T.; Watanabe, T.; Yamada, K.; Tokura, Y.; Itabashi, S. Frequency and polarization characteristics of correlated photon-pair generation using a silicon wire waveguide. IEEE J. Sel. Top. Quantum Electron. 2010, 16, 325−331. (20) Xiong, C.; Monat, C.; Clark, A. S.; Grillet, C.; Marshall, G. D.; Steel, M. J.; Li, J.; O’Faolain, L.; Krauss, T. F.; Rarity, J. G.; Eggleton, B. J. Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide. Opt. Lett. 2011, 36, 3413−3415. (21) Azzini, S.; Grassani, D.; Strain, M. J.; Sorel, M.; Helt, L. G.; Sipe, J. E.; Liscidini, M.; Galli, M.; Bajoni, D. Ultra-low power generation of twin photons in a compact silicon ring resonator. Opt. Express 2012, 20, 23100−23107. (22) Collins, M. J.; Clark, A. S.; He, J.; Choi, D.-Y.; Williams, R. J.; Judge, A. C.; Madden, S. J.; Withford, M. J.; Steel, M. J.; LutherDavies, B.; Xiong, C.; Eggleton, B. J. Low Raman-noise correlated photon-pair generation in a dispersion-engineered chalcogenide As2S3 planar waveguide. Opt. Lett. 2012, 37, 3393−3395. (23) Horn, R.; Abolghasem, P.; Bijlani, B. J.; Kang, D.; Helmy, A. S.; Weihs, G. Monolithic source of photon pairs. Phys. Rev. Lett. 2012, 108, 153605. (24) Matsuda, N.; Le Jeannic, H.; Fukuda, H.; Tsuchizawa, T.; Munro, W. J.; Shimizu, K.; Yamada, K.; Tokura, Y.; Takesue, H. A monolithically integrated polarization entangled photon pair source on a silicon chip. Sci. Rep. 2012, 2, 817.

which is eq 3 in the main text. The signal and idler photons exhibit identical self-correlation profiles regardless of the potential Q difference of their passive cavity modes. When the signal and idler modes exhibit the same photon decay rate, Γts = Γti ≡ Γt, eq 25 reduces to a simple form of g(2) jj (τ) = (1 + Γt |τ|/2)2e−Γt|τ| + 1, which clearly shows the nonexponential nature of the self-correlation. In practice, the coincidence counting system generally exhibits a finite instrument response primarily due to the timing jitters of the single photon detectors (gray dots in Figure 5). As a result, the experimentally recorded photon correlations are the convolution between eq 23 (or eq 25) and the instrument response function (IRF) of the coincidence counting system. Figure 10 shows this effect, where the dashed lines include the instrument response, compared with the ideal case of eqs 23 and 25 (solid lines). The full-width at halfmaximum (fwhm) of the IRF of our detection system is approximately 106 ps. It broadens the cross-correlation function by a certain extent (Figure 10a), but its impact on the self-correlation is small, only decreasing its peak value by a small amount from 2 to 1.98. In particular, Figure 10a verifies the waveform asymmetry of the cross-correlation. Figure 10b shows the nonexponential profile for self-correlation, whose fwhm time (0.55 ns) is much broader than the fwhm time of cross-correlation (0.18 ns). As discussed above, the selfcorrelations are identical for signal and idler photons (Figure 10b), and they share the same emission spectra profile (eqs 18−20).

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Stefan Preble for the loan of superconducting nanowire single-photon detectors. We thank Oskar Painter for helpful discussions. This work is supported by National Science Foundation (NSF) under Grant ECCS-1351697. The device fabrication was performed in part at the Cornell NanoScale Science & Technology Facility (CNF), a member of the National Nanotechnology Infrastructure Network.

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REFERENCES

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Biphoton Statistics of Quantum Light Generated on a Silicon Chip

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