Reconstruction of photon-number distribution using low-performance photon counters

The output of a photodetector consists of a current pulse whose charge has the statistical distribution of the actual photon numbers convolved with a Bernoulli distribution. Photodetectors are characterized by a nonunit quantum efficiency, i.e., not all the photons lead to a charge, and by a finite resolution, i.e., a different number of detected photons leads to a discriminable values of the charge only up to a maximum value. We present a detailed comparison, based on Monte Carlo simulated experiments and real data, among the performances of detectors with different upper limits of counting capability. In our scheme the inversion of Bernoulli convolution is performed by maximum-likelihood methods assisted by measurements taken at different quantum efficiencies. We show that detectors that are only able to discriminate between zero, one and more than one detected photons are generally enough to provide a reliable reconstruction of the photon number distribution for single-peaked distributions, while detectors with higher resolution limits do not lead to further improvements. In addition, we demonstrate that, for semiclassical states, even on/off detectors are enough to provide a good reconstruction. Finally, we show that a reliable reconstruction of multipeaked distributions requires either higher quantum efficiency or higher resolution.

Figure 1

(Color online) Maximum-likelihood reconstruction of photon number distribution using photodetectors with different counting capability (different values of $M$) and maximum quantum efficiency ${\eta }_{\max }=0.2$. First line, photon distribution of a coherent state with $⟨a^{†}a⟩=3$; panel (a) the reconstructed ${\rho }_n$; (b) total absolute error ${\epsilon }^{\left(i\right)}$ according to Eq. (8); (c) fidelity $G^{\left(i\right)}$ according to Eq. (9). Results are reported for $M=1$ (on/off detector), $M=2$, $M=3$, and $M=6$. We use $K=30$ different quantum efficiency values $\eta ={\eta }_{\nu }$, uniformly distributed in $[{\eta }_{\max }∕K,{\eta }_{\max }]$. The Hilbert space have been truncated at $N=30$ and $n_{\nu }={10}^6$ runs have been performed for each value ${\eta }_{\nu }$. The last iteration is $I=n_{\nu }$ except for the Fock state, $M=1$, for which $I={10}^7$. Other lines, same as first line for thermal state with $⟨a^{†}a⟩=3$, Fock state with $n=4$; superposition of two Fock states $\frac{1}{\sqrt{2}}(\mid 3⟩+\mid 5⟩)$. The inset of panel (c) in the third line shows the last iterations $i$ in an expanded scale.

Figure 2

(Color online) Fidelity parameter at the last iteration, $G^{\left(I\right)}$, as a function of $⟨a^{†}a⟩$ for four different statistics of $\left\{{\varrho }_n\right\}$: coherent, panels labeled (a); thermal, label (b); Fock state, label (c); superposition of two Fock states, $\frac{1}{\sqrt{2}}(\mid ⟨a^{†}a⟩-1⟩+\mid ⟨a^{†}a⟩+1⟩)$, label (d). The reconstructions have been performed using ${\eta }_{\max }=0.2$, 0.5, and 0.8, panels labeled with (i), (ii) and (iii), respectively. Number of runs, $n_{\nu }={10}^4$. According to the rule given in the text, the last iteration performed is $I=n_{\nu }$ except for the coherent state with $⟨a^{†}a⟩=1$ to 4 for which $I={10}^5$ (only for $M=1$), and for the superpositions state. Inset of (di) show $G^{\left(I\right)}$ for the superposition $\frac{1}{\sqrt{2}}(\mid 5⟩+\mid 7⟩)$, $M=1$, quantum efficiencies $\eta =0.2$ and $\eta =0.8$ as a function of $n_{\nu }$. Each point of the curves is the mean of 40 simulations, the error bars represent the standard deviations. The other parameters are the same as in Fig. 1.

Figure 3

(Color online) Experimental setup for reconstruction the photon number distribution of a coherent state. The laser and the detector (named hybrid) are described in the text. VA is a variable attenuator; SGI is a synchronous gated integrator; MCA is a multichannel analyzer.

Figure 4

(Color online) Experimental reconstruction of the photon number distribution of a coherent state. (a) Reconstructed ${\rho }_n$ and coherent best fit, $⟨a^{†}a⟩=2.98$; (b) total absolute error ${\epsilon }^{\left(i\right)}$; (c) fidelity $G^{\left(i\right)}$ calculated a posteriori. The reconstruction is performed with a hybrid photodetector operated by taking $M=1$ (on/off detector), $M=2$ and $M=3$; $K=100$ different quantum efficiencies $\eta ={\eta }_{\nu }$ distributed in $[0,{\eta }_{\max }]$, ${\eta }_{\max }=0.4$; the Hilbert space is truncated at $N=30$; $n_{\nu }={10}^4$ number of runs have been performed for each $\eta$; the algorithm is stopped at iteration $I=n_{\nu }$.