Numerical reconstruction of photon-number statistics from photocounting statistics: Regularization of an ill-posed problem

We demonstrate a practical possibility of loss compensation in measured photocounting statistics in the presence of dark counts and background radiation noise. It is shown that satisfactory results are obtained even in the case of low detection efficiency and large experimental errors.

Figure 1

(Color online) The photon-number, $p_n$, and photocounting, ${\mathcal{P}}_n$, distributions of the thermal state, ${\overline{n}}_{\text{th}}=30$. The circles and triangles are the initially chosen and reconstructed ($\tilde{\eta }=0.35$, ${\tilde{N}}_{\text{nc}}=0.29$) photon-number distributions, respectively. The asterisks show the simulated photocounting distribution ($\nu =5\times {10}^4$, $\eta =0.34$, $N_{\text{nc}}=0.30$).

Figure 2

(Color online) The photon-number, $p_n$, and photocounting, ${\mathcal{P}}_n$, distributions of the SPATS, ${\overline{n}}_{\text{th}}=10$. The circles and triangles are the initially chosen and reconstructed ($\tilde{\eta }=0.77$, ${\tilde{N}}_{\text{nc}}=0.75$) photon-number distributions, respectively. The asterisks show the simulated photocounting distribution ($\nu =5\times {10}^3$, $\eta =0.7764$, $N_{\text{nc}}=0.748$).

Figure 3

(Color online) The photon-number distribution, $p_n$, of the SPATS, ${\overline{n}}_{\text{th}}=10$. The circles and triangles are the initially chosen and reconstructed ($\nu =5\times {10}^5$, $\eta =0.7764$, $N_{\text{nc}}=0.748$) distributions in Eq. (6), respectively. The result of such a reconstruction demonstrates much stronger noise effect in comparison with the Landweber algorithm (see Fig. 2).

Figure 4

(Color online) The photon-number, $p_n$, and photocounting, ${\mathcal{P}}_n$, distributions of the superposition of the coherent states ${\left|\alpha \right|}^2=23.9$. The circles and triangles are the initially chosen and reconstructed ($\tilde{\eta }=0.59$, ${\tilde{N}}_{\text{nc}}=1.77$) photon-number distributions, respectively. The asterisks show the simulated photocounting distribution ($\nu =5\times {10}^3$, $\eta =0.613749$, $N_{\text{nc}}=1.763442$).