Conditional preparation of states containing a definite number of photons

A technique for conditionally creating single-mode or multimode photon-number states is analyzed using Bayesian theory. We consider the heralded $N$-photon states created from the photons produced by an unseeded optical parametric amplifier when the heralding detector is the time-multiplexed photon-number-resolving detector recently demonstrated by Fitch et al. [Phys. Rev. A 68, 043814 (2003)] and simultaneously by Achilles et al. [Opt. Lett. 28, 2387 (2003)]. We find that even with significant loss in the heralding detector, fields with sub-Poissonian photon-number distributions can be created. We also show that heralded multimode fields created using this technique are more robust against detector loss than are single-mode fields.

Figure 1

(Color online) (a) Scheme to produce heralded fields containing a definite number of photons. An unseeded optical parametric amplifier (OPA) produces photon pairs through the process of parametric down conversion. When a time-multiplexed photon-number-resolving detector [shown in (b)] detects $n_i$ photons in the idler field, it heralds the presence of $n_s$ photons in the signal field with probability $P_{\text{sig}}\left(\phantom{|}{n}_s|n_i\right)$.

Figure 2

(Color online) Regions where there is a significant probability of detecting $n$ photons conditioned on $N$ photons being incident on the TMD for three different single-photon detection efficiencies $\left(\eta =1.0,0.66,0.33\right)$. The mean number of photons detected given $N$ photons input into the detection system are plotted using the solid $\left(\eta =1\right)$, dotted $\left(\eta =0.66\right)$, and dashed $\left(\eta =0.33\right)$ lines. The shaded regions are within one standard deviation of the means.

Figure 3

(Color online) Comparison of the photon-number statistics for a typical heralded field with $n_s^{\text{ml}}=5$ and a Poisson-distributed field with a mean photon number of 5. The heralded field, for the case in which $\eta =0.66$ and $g=1$, is conditioned on the detection of four photons in the idler field by the TMD.

Figure 4

(Color online) Uncertainty in the number of photons in a heralded field when $n_i$ photons are detected in the heralding field normalized to the uncertainty in a comparable Poisson-distributed field with mean number of photons $n_s^{\text{ml}}$ for various detection efficiencies $\left(\eta =0.33,0.66,1.0\right)$ and OPA gains $\left(g=0.75,1\right)$.

Figure 5

(Color online) Mandel’s $Q$ parameter versus the gain of the OPA and the single-photon detection efficiency of the TMD for the case where $n_s^{\text{ml}}=5$, i.e., the heralded field is a single-mode field estimated to contain five photons. The regions are labeled with the number of detected idler photons needed to herald the presence of five photons in the signal field. The white contours are lines of constant $P\left(g,\eta ;n_s^{\text{ml}}\right)$, the probability of creating the heralded state. The TMD is assumed to contain five beam splitters $\left(M=32\right)$.

Figure 6

(Color online) Mandel’s $Q$ parameter versus gain of the OPA and the single-photon detection efficiency of the TMD for the case where $n_s^{\text{ml}}=5$ and $\mu =5$, i.e., the signal field is estimated to contain five photons spread between five modes. The regions are labeled with the number of detected idler photons needed to herald the presence of five photons in the signal field. The white contours are lines of constant $P\left(g,\eta ;n_s^{\text{ml}}\right)$, the probability of creating the heralding state.