General and complete description of temporal photon correlations in cavity-enhanced spontaneous parametric down-conversion

Heralded single-photon sources are the most commonly used sources for optical quantum technology applications. There is strong demand for accurate prediction of their spectral features and temporal correlations with ever-increasing precision. This is particularly important in connection with the intrinsically stochastic photon-pair generation process in heralded sources. Here we present a complete theoretical description of the temporal correlation of a signal-idler, signal-signal, and signal-signal-idler coincidences of photons generated by continuous-wave pumped cavity-enhanced spontaneous parametric down-conversion. The theory excellently predicts the measurements, which has been experimentally confirmed in our setup utilizing single-photon detectors with high temporal resolution. This enables us to resolve and analyze the multiphoton correlation functions in great detail.

Figure 1

Setup for measuring the signal-idler correlation. (a) The SPDC source consists of a 2-cm PPKTP crystal, which is placed in a low-finesse cavity to generate photons with 100-MHz bandwidth [19]. Residual pump light and non-phased-matched background are suppressed by a long-pass filter (LP) and a 1-nm bandpass filter for 894 nm (BP) after the cavity. Since the crystal is phase matched for type-II down-conversion, the signal and idler photons are split by a polarizing beam splitter (PBS) and collected into two separate single-mode fibers. SSPDs (Scontel) with a temporal resolution of 40 ps for each detector are used to detect the signal and idler photons. (b) Calculated joint spectral intensity of the SPDC source with a central emission wavelength of 894.3 nm. The joint spectral intensity shows separated dots with a decreasing intensity for increasing wavelength difference $\mathrm{\Delta }\lambda$. This dotted pattern is caused by the cavity.

Figure 2

Measured and theoretical second-order signal-idler correlations. (a) Measured data (dots) and theoretical curve (dashed line) calculated with Eq. (8), using the phase-matching function of our source [36] convoluted with the overall temporal resolution of the setup. The distance between the peaks corresponds to the round-trip time of photons in the cavity. The exponential slope of the individual fringes is determined by the reflectivity of the outcoupling mirror. Negative correlation times correspond to the case where the signal photon left the cavity and was detected before the idler photon. (b) A close-up and a theoretical curve with a much higher temporal resolution (solid line). The substructures, which cannot be resolved in the experiment, originate from the birefringence of the nonlinear crystal [32].

Figure 3

Setup for measuring the signal-signal correlation. The source is the same as for the signal-idler correlation, where signal and idler photons are orthogonally polarized. The emitted spectrum is tuned so that signal and idler photons have different frequencies. Therefore, the idler photons can be filtered out with a polarizer and a bandpass filter. The signal photons are send on a half waveplate (HWP) and PBS combination to split them equally to two SSPDs (Scontel) with a temporal resolution of 40 ps.

Figure 4

Measured (dots) and theoretical signal-signal correlations with a convolution of the overall temporal resolution of the setup (dashed line). Some of the idler photons passed the filter system and were detected. Hence, the theoretical line is a combination of 63% signal-signal correlation with a residual contribution of 37% signal-idler correlations as shown in Eq. (15). (a) The correlation has peaks as for the signal-idler correlation, where the distance corresponding to the round-trip time. (b) A close-up and a much higher temporal resolution assumed in the theoretical curve (solid line) reveals a substructure again, which arises by the combination of the idler and signal probability to stimulate the second pair. The substructure has positive and negative contributions since both detectors have an equal probability to detect the first photon.

Figure 5

Second-order signal-signal-idler correlation of two simultaneously generated photon pairs. Calculated with Eq. (20) for a temporal resolution of 1 ps (the left picture is binned to 30 ps afterward for improved illustration). The highest triple coincidence probability is normalized to one. Here $\mathrm{\Delta }{\mathrm{t}}_1$ ($\mathrm{\Delta }{\mathrm{t}}_2$) is the time difference between a detected signal photon 1 (signal photon 2) and the detection of the idler photon. The highest probability of a triple coincidence is when all involved photons leave the cavity together. This probability decreases with each additional round-trip of one of the photons. Zooming in shows a substructure of the correlation. This is caused by birefringence in the crystal for the signal and idler photons, which are polarized orthogonal to each other. The substructure is similar to the substructure in the signal-idler correlation (see Fig. 2).

Figure 6

Setup for measuring the second-order signal-signal-idler correlation. The source is identical to the one used to measure the signal-idler correlation (see Fig. 1). The signal and idler photons are split on a PBS. The idler photons are detected by a fiber-coupled avalanche photodiode (APD) with a timing resolution of 60 ps. The signal photons are collected with a polarization maintaining fiber and sent to a polarizer to suppress remaining idler photons. The transmitted signal photons pass a HWP and a PBS. The HWP rotates the polarization of the signal photons to diagonal so that the following PBS acts like a 50:50 beam splitter. The signal photons are detected individually with the two SSPDs with a temporal resolution better than 40 ps (Scontel).

Figure 7

Theoretical second-order signal-signal-idler correlation taking uncorrelated photon pairs into account, calculated with a resolution of 1 ps (the left picture is binned to 30 ps for improved illustration). The coincidence maximum is normalized to one. The time difference between a detected photon at the heralding APD and SSPD 1 (SSPD 2) is described by $\mathrm{\Delta }{\mathrm{t}}_1$ ($\mathrm{\Delta }{\mathrm{t}}_2$). The peak positions of the coincidences are equal to the case of simultaneously generated photons. Additionally, stripes can be seen which are caused by uncorrelated signal photons. At least one of the two signal photons has a time correlation to the detected idler photon, allowing the detection of a triple coincidence only at certain times. The uncorrelated signal photons are generated at a different time than the detected idler photon, resulting in the constant stripes.

Figure 8

Measured second-order signal-signal-idler correlation with a binning resolution of 50 ps. The correlation resembles a cross with the highest coincidence rates in the middle. The triple coincidence probability decreases rapidly with increasing time difference as expected from the theory (see Fig. 7). The stripes are caused by signal photons, which were not generated at the same time as the detected idler photons. The stripes are constant over the entire measurement as predicted by the theory.

Figure 9

Comparison of measured and theoretically predicted second-order signal-signal-idler correlation with a binning of 50 ps. (a) Measured correlation function. The measurement shows separated peaks with high triple coincidence probability and stripes from uncorrelated photons. (b) Theoretical correlation function which is identical to Fig. 7 but binned to a resolution of 50 ps to match the measurement and convoluted with an overall detector resolution of 125 ps. (c) Comparison of the measured (dots) and theoretical models (solid line) at $\mathrm{\Delta }{t}_2=-2.25$ ns. To compensate for the noise in the measurement, the average value over five bins is used for the direct comparison. (d) Same as (c) but for $\mathrm{\Delta }{t}_2=0$ ns.