Down-converted photon pairs in a high-Q silicon nitride microresonator

Effective χ
⁽²⁾ creation through SHG

To induce an effective χ⁽²⁾ and simultaneously quasi-phase-match the down-conversion process, a space-charge grating was created using the experimental set-up shown in Extended Data Fig. 1a. The temperature of the microresonator was carefully controlled to frequency-match the visible mode and the near-infrared cavity mode. These frequencies were monitored using a near-infrared band tunable laser (1,560 nm) that was frequency-doubled using a periodically poled lithium niobate crystal. Once this was done, the frequency of the near-infrared laser was tuned to the cavity resonance. A periodic space-charge field then built up and generated second-harmonic power, as described elsewhere¹⁶. A steady state was achieved within a few seconds, and the grating persisted for a long time without the external excitation. The second-harmonic power conversion efficiency and χ⁽²⁾ strength were then characterized using a lower-power near-infrared pump power, as shown in Extended Data Fig. 1b. We observed an 11-mW on-chip 780 nm second-harmonic signal when 41 mW of 1,560 nm laser power was launched into the waveguide. This corresponds to a second-harmonic efficiency (η) of 651% per watt. We believe this value is limited by saturation of the second-harmonic process and the presence of cascaded sum-frequency generation to 520 nm (ref. ⁴²).

For comparison, the second-harmonic efficiency can also be estimated from the measured photon-pair generation rate R and visible (780 nm) pump power P_vis in the SPDC process¹⁰:

where κ_IR,L (κ_IR,e) is the loaded (external) cavity loss rate of the near-infrared mode. At 1.5-mW on-chip pump power, the photon-pair generation rate was measured as 800 kHz. Using the above expression and optical loss rates from the main text, the calculated SHG efficiency η = 624% per watt, in close agreement with the above measured value. As an aside, we expect that this calculation underestimated the peak SHG efficiency, because, as noted in the main text, the charge distribution faded away in the presence of the 780 nm pump (see Methods for more details). In particular, the 800 kHz rate was recorded a few minutes after the 780 nm pump had been coupled into the resonator, and the charge distribution would, therefore, not be as strong as the initial state. Overall, the agreement of these two inferred efficiency values might be fortuitous in view of the experimental uncertainties.

SPDC spectral measurements

We used a a low-noise liquid-nitrogen-cooled spectrometer for the spectral measurements (PyLoN IR 1024-1.7). It had a quantum efficiency of more than 75%. The spectrometer had a 300 lines per millimetre grating (1.2 μm blaze) and an efficiency of more than 50% for wavelengths below 1,600 nm.

Temperature dependence of the photon-pair wavelengths

The wavelengths of the generated photon pairs are determined by the frequency-matching condition between the visible pump mode and the near-infrared-band mode family. To determine this matching condition, the dispersion of the near-infrared mode family was first measured using a calibrated Mach–Zehnder interferometer in combination with a wavelength-tunable laser⁴³. The resonant frequency ω_m of mode m can be approximated as a Taylor series relative to a mode at ω₀ defined to have a relative mode number m = 0:

where D_j is the jth-order dispersion coefficient. Specifically, D₁/2π equals the FSR, and D₂ is the second-order dispersion coefficient at mode m = 0. As shown in Extended Data Fig. 2a, a second-order expansion agrees well with the measurements over a range of mode numbers within a few hundreds of m = 0. The parabolic fitting gives D₂/2π = −863.7 kHz.

The resonant frequency tuning coefficients δω/δT were directly measured by varying the temperature of the microresonator chip using a thermoelectric cooler. The results are shown in Extended Data Fig. 2b. Using these measurements, the relative frequency detuning rate (δω_vis/δT − 2δω_IR/δT) was determined to be −814.8 MHz K⁻¹, where ω_vis and ω_IR represent the resonance frequency of the near-visible and near-IR modes respectively. Combined with the negative D₂ measured above, the temperature detuning coefficient changed the SPDC process from degenerate to non-degenerate on changing the resonator temperature. Accordingly, suppose that the degenerate SPDC process is frequency-matched to near-infrared mode number m = 0 at temperature T₀ (ω_vis(T₀) = 2ω₀(T₀)). Then, using the dispersion expansion (equation (2)), the non-degenerate frequency-matching condition at temperature T can be written as,

where mode numbers  +m and  −m are the relative mode numbers of the near-infrared modes involved in the SPDC process. From equation (4) and using the measured value for δω_vis/δT − 2δω_IR/δT, the quadratic coefficient of the chip temperature as a function of the photon-pair wavelength was calculated to be 0.0131 K nm⁻², which was used to plot the quadratic function in Fig. 2d.

Bandwidth of the down-converted photon pairs

In the spontaneous down-conversion process, a few near-infrared mode pairs can frequency-match with the visible pump mode because a slight frequency mismatch caused by dispersion can be smaller than the resonator linewidth. The frequency-matching condition can be written as:

where  ±m_max (±m_min) is the largest (smallest) relative mode number of the frequency-matched mode pairs, and Δω = ω/Q is the full-width at half-maximum of the near-infrared resonances. In the degenerate case, m_min = 0. m_max is then given by,

Using current resonator values, m_Deg,max was calculated to be 5.7, which is consistent with the dark blue trace in Fig. 2c.

A condition for m_Deg,max = 1 is given by D₂ > 2Δω = 2ω₀/Q where Q is the loaded cavity Q factor of the near-infrared modes. By rewriting D₂ in terms of the FSR Δν_FSR (hertz) and the waveguide group velocity dispersion parameter β₂ (ref. ⁴³), the following design condition for single-mode SPDC emerges:

where λ₀ and n are the wavelength and effective index of the mode m = 0, respectively. β₂ is related to D₂ as \(\beta _2=-nD_2/cD_1^2\), and was calculated to be 540 ps² km⁻¹, using the parameters from the previous section. c stands for the speed of light in the vacuum. For other values typical of the ultra-low-loss Si₃N₄ system, this gives QΔν_FSR² > 5.49 × 10¹¹ GHz². Assuming Q = 100 million, which is readily attainable by this system, Δν_FSR > 74 GHz is sufficient to guarantee single-mode degenerate SPDC.

In the non-degenerate case, the equation can be modified as:

where \(\overlinem=(m_\textmax+m_\min )/2\) and \(\Delta m=(m_\textmax-m_\min )/2\). At 23.7 °C, \(\overlinem=16\) and Δm was calculated to be 1. This is again consistent with observations, as for both long and short wavelengths of the spectrum, photon fluxes were observed in only one neighbouring mode on each side of the strongest mode. At even higher temperatures, no side peaks were observed in the spectrum.

g
⁽²⁾ measurements

The g⁽²⁾(t) and pair generation rates were measured with a pair of 1.55-μm SNSPDs with a quantum efficiency of 85% provided by ID Quantique. The signals from the SNSPDs were recorded by a ID900 time-to-digital converter with a temporal resolution of 2 ns.

Temporal degradation of the effective χ
⁽²⁾

As mentioned in the main text, the SPDC rate was observed to decay in time. We believe that this results from multiphoton excitation of conduction band electrons by the 780 nm pump. These electrons would gradually neutralize the space charge, thereby diminishing the effective χ⁽²⁾. This section investigates the speed of this effect. It is experimentally challenging to reconstruct the decay directly from the SPDC rate, because the SPDC rate is sensitive to temperature and laser detuning. Alternatively, we chose to monitor the charge through the SHG signal when scanning the 1,560 nm laser across the resonance. We recorded the peak value of the second-harmonic signal, as shown in Extended Data Fig. 1b. This measurement samples a range of different detunings and should be less affected by the temperature change.

The experimental set-up is the same as in Extended Data Fig. 1a. The space-charge grating was first introduced through SHG as described in the section ‘Effective χ⁽²⁾ creation through SHG’. Then, the 1,560 nm laser was turned off and the 780 nm laser was tuned to resonance to simulate the condition for SPDC. After each 15-s interval, the 780 nm signal was turned off and the 1,560 nm laser was scanned across the resonance to collect the second-harmonic peak value. Two on-chip 780 nm pump powers (0.5 and 1.5 mW) were used. The result is shown in Extended Data Fig. 3. An exponential decay fitted to the data gives lifetimes of 68 and 40 s for pumping powers 0.5 and 1.5 mW, respectively. As expected, the decay was faster with the higher pump power. Also note that as time increased, the data significantly deviated from the exponential fit and the decay rate decreased. Possible mechanisms that could impact the SHG efficiency include changes in the pump laser phase or polarization. As the SPDC measurement in the main text requires a wait time of a few minutes for the chip temperature to achieve a steady state, the observed decay times are consistent with our observation that the SPDC rate can be observed over several minutes.