Fundamental limits on determination of photon number statistics from measurements with multiplexed on/off detectors

Abstract

We investigate fundamental bounds on the ability to determine photon number distribution and other related quantities from tomographically incomplete measurements with an array of M detectors that can only distinguish the absence or presence of photons. We show that the lower and upper bounds on photon number probabilities can be determined by solving a linear program. We present and discuss numerical results for various input states including thermal states, coherent states, squeezed states, and highly non-classical single-photon subtracted squeezed vacuum states. Besides photon number probabilities we also investigate bounds on the parity of photon number distribution that determines the value of Wigner function of the state at the origin of phase space. Moreover, we also discuss estimation of mean photon number as an example of a quantity described by an unbounded operator. Our approach and results can provide quantitative guidance on the number of detection channels required to determine the photon number distribution with a given precision.

Figures (12)

• Figure 1: Multiplexed detector of photons. The input signal is evenly split among $M$ output modes, e.g., by an array of $M-1$ unbalanced beam splitters BS$_k$ with suitably chosen transmittances $T_k=(M-k)/(M-k+1)$. Each output mode is measured with a binary detector $D$ that can distinguish the presence and absence of photons. The number of detector clicks $m$ represents the measured signal.
• Figure 2: Four different photon number distributions $p_n$ are displayed that all yield the same click statistics when measured with balanced multiplexed detector with $M=10$ channels and $\eta=1$. The click statistics corresponds to statistics generated by thermal state with mean photon number $\bar{n}=3$. For clarity, only $p_n$ for $n \leq 14$ are shown in the figure.
• Figure 3: Lower bounds (green bars) and upper bounds (pink bars) on the photon number probabilities $p_{n}$ are plotted for four different states with $\bar{n}=2$: thermal state (a), coherent state (b), squeezed vacuum state (c), and single-photon subtracted squeezed vacuum state (d). Blue diamonds indicate the true photon number distributions. Parameters of the multiplex detector read $M=10$ and $\eta=1$, and the photon number cutoff was set to $N=80$ in the numerical calculations.
• Figure 4: Lower and upper bounds on photon number probability $p_5$ are plotted in dependence on the mean photon number $\bar{n}$ for four different states: thermal state (a), coherent state (b), squeezed vacuum state (c), and single-photon subtracted squeezed vacuum state (d). Blue lines indicate the true values of $p_5$. Parameters of the multiplexed detector read $M=10$ and $\eta=1$.
• Figure 5: The photon number uncertainties $\Delta p_n$ are plotted as functions of the number of detector channels $M$ for thermal state with $\bar{n}=2$. Photon number cutoff is set to $N=80$ and $\eta=1$ is assumed in the numerical calculations.