Analysis of counting measurements on narrowband frequency up-converted single photons and the influence of heralding detector dead time

We present a theoretical model of photon counting measurements on conditionally generated narrowband single-photon states that are produced via cavity-enhanced spontaneous parametric down-conversion, then frequency up-converted from the telecom wavelength of 1550 nm to the visible wavelength of 532 nm. The highly nonclassical character of the up-converted states is certified by a quantum non-Gaussianity witness that is determined from coincidence measurements with single-photon detectors in a Hanbury-Brown–Twiss configuration. We find our model in good agreement with the experimental data, and we investigate a useful effect caused by the dead time of the trigger detector, whose clicks herald conditional preparation of the single-photon state. Due to the dead time, a click of the trigger detector excludes the possibility of a trigger event at a certain preceding time interval, during which the measured idler beam is thus projected onto a vacuum state. Due to quantum correlations between signal and idler beams, this reduces the multiphoton contributions in the conditionally generated state of the signal beam and accordingly increases the value of the quantum non-Gaussianity witness. We also show that spurious heralding detections due to after-pulsing can be suppressed by accepting a click of the trigger detector only if its distance from a previous click of this detector exceeds a certain suitably chosen threshold.

Figure 1

Experimental setup [31]. See text for details.

Figure 2

The quantum non-Gaussianity witness $W$ is plotted as a function of squeezing constant $r$ for ${\eta }_S=0.20$ and ${\eta }_I=0.11$.

Figure 3

Discretization of the coincidence window into $2N_S+1$ rectangular modes of width ${\tau }_0$. The trigger detector clicks at time $t_0=0$, which is modeled by assuming that APD-T detects a photon belonging to the rectangular temporal mode highlighted by gray color. The figure also shows the rectangular temporal modes of the idler beam that are introduced to account for the dead time of the trigger detector APD-T $t_D=N_I{\tau }_0$.

Figure 4

Dependence of the non-Gaussianity witness $W$ on the coincidence window size $\Delta t$. Blue dots represent experimental results for three different pump powers $P$, and solid lines represent predictions of the theoretical model for (a) $\gamma =\pi \times 31\mathrm{MHz},{\eta }_I=0.11,t_D=29\mathrm{ns},\kappa =1.4\gamma ,{\eta }_S=0.235,\epsilon =0.070\gamma$, (b) ${\eta }_S=0.252,\epsilon =0.109\gamma$, and (c) ${\eta }_S=0.260,\epsilon =0.194\gamma$. The error bars were estimated assuming Poissonian statistics of the count rates and they represent one standard deviation.

Figure 5

Dependence of the quantum non-Gaussianity witness $W$ on ${\eta }_I$. The other parameters read $\gamma =\pi \times 31\mathrm{MHz},{\eta }_S=0.252,t_D=29\mathrm{ns},\kappa =1.4\gamma ,\epsilon =0.109\gamma$, and $\Delta t=50\mathrm{ns}$. For comparison, the dashed line indicates results for a detector with $t_D=0$.

Figure 6

Dependence of probabilities $P_S$ and $P_C$ on the dead time of the trigger detector $t_D$. The other parameters read $\gamma =\pi \times 31\mathrm{MHz},{\eta }_S=0.25,{\eta }_I=0.4,\kappa =1.4\gamma ,\Delta t=50\mathrm{ns},\epsilon =0.1\gamma$ (solid line) and $\epsilon =0.2\gamma$ (dashed line).

Figure 7

Dependence of the quantum non-Gaussianity witness $W$ on ${\eta }_S$. The other parameters read $\gamma =\pi \times 31\mathrm{MHz},{\eta }_I=0.11,t_D=29\mathrm{ns},\kappa =1.4\gamma$, and $\epsilon =0.109\gamma$.

Figure 8

Experimentally determined dependence of $p_0$ (solid line), $p_1$ (dashed line), and $W$ on $t_{\min }$. The results are shown for ${\epsilon }_{\mathrm{nom}}=0.16\gamma$.