Coherence measures for heralded single-photon sources

Single-photon sources (SPSs) are mainly characterized by the minimum value of their second-order coherence function, viz. their $g^{\left(2\right)}$ function. A precise measurement of $g^{\left(2\right)}$ may, however, require high time-resolution devices, in whose absence, only time-averaged measurements are accessible. These time-averaged measures, standing alone, do not carry sufficient information for proper characterization of SPSs. Here, we develop a theory, corroborated by an experiment, which allows us to scrutinize the coherence properties of heralded SPSs that rely on continuous-wave parametric down conversion. Our proposed measures and analysis enable proper standardization of such SPSs.

Figure 1

(Color online) Experimental setup for our heralded single-photon source. A blue laser is cleaned from infrared fluorescence by a color filter, is focused by cylindrical lenses onto a PPKTP crystal (cut and poled for type-II), and is then removed from the down-converted beam by a dispersive prism. A PBS splits the down-converted beam into the signal and idler arms. The idler beam $i$ is used as a trigger and, in an HBT interferometer, the signal beam is split by a 50:50 beam splitter into $s1$ and $s2$ for the coherence function measurement. Bandpass filters block background light from entering the multimode fibers that couple the light to the single-photon detectors.

Figure 2

(Color online) Top: Measurements (symbols) and theory predictions (lines) of the time-averaged coherence function ${\overline{g}}_{si}^{\left(2\right)}\left(\tau \right)$ for the signal and idler photons. The low-gain regime theory curves are in striking agreement with the data using the following parameter values $R_{\text{SPDC}}=43\text{MHz}$, ${\tau }_d=350\text{ps}$, and $1/\Delta t=3\text{THz}$, where the last one was measured separately by spectroscopy and the other two were adapted to provide the subjective best visual fit simultaneously to all three sets of data. Bottom: Same data as top but with magnified ordinate in order to reveal structure that is caused by double reflection of photons from the fiber input end face and the surface of another optical element in the experiment. While this effect causes only very small deviations (0.2% of the central peak) from the expected flat line, these deviations cause noticeable ringing of the measured ${\overline{g}}_c^{\left(2\right)}$ (see Fig. 3).

Figure 3

(Color online) Measured (symbols) and calculated (lines) time-averaged conditional coherence functions ${\overline{g}}_c^{\left(2\right)}\left(\tau \right)$. The theory lines were calculated using the same parameter values as in Fig. 2. The purely statistical errors of our data are on the order of the symbol size in the figure and therefore not shown. As explained in the caption of Fig. 2 photons that are reflected twice cause the apparent ringing.

Figure 4

Experimental (symbols) and theoretical (line) results for the minimum of the time-averaged conditional coherence function, ${\overline{g}}_c^{\left(2\right)}\left(0\right)$, as a function of the coincidence window $2{\tau }_{\text{coin}}$ using the same set of parameters as in Fig. 2. The dashed line is for ideal photodetectors $\left({\tau }_d=0\right)$.