Theory Framework of Multiplexed Photon-Number-Resolving Detectors

TL;DR

This work addresses the challenge of achieving true photon-number resolution by proposing a multiplexed array of ON-OFF detectors. It develops a unified theoretical model for two architectures and proves that higher-order photon-number moments converge to the true moments with error scaling as $O(1/n)$. Applied to squeezed-vacuum statistics and cat-state breeding, the results show that realistic on-chip MPNR detectors with around 20–100 detectors and efficiencies near 95% can deliver high-fidelity non-Gaussian states at high rates, exemplified by a fidelity of ~0.88 and a 3.8% success probability for 7 dB squeezing with 20 detectors. This work provides a pathway toward practical, scalable PNR detection for quantum information tasks and outlines directions to incorporate device imperfections.

Abstract

Photon counting is a fundamental component in quantum optics and quantum information. However, implementing ideal photon-number-resolving (PNR) detectors remains experimentally challenging. Multiplexed PNR detection offers a scalable and practical alternative by distributing photons across multiple modes and detecting their presence using simple ON-OFF detectors, thereby enabling approximate photon-number resolution. In this work, we establish a theoretical model for such detectors and prove that the estimation error in terms of photon number moments decreases inverse proportionally to the number of detectors. Thanks to the enhanced PNR capability, multiplexed PNR detector provides an advantage in cat-state breeding protocols. Assuming a two-photon subtraction case, $7$dB of squeezing, and an array of 20 detectors of efficiency $95\%$, our calculation predicts fidelity $\sim0.88$ with a success probability $\sim 3.8\%$, representing orders-of-magnitude improvement over previous works. Similar enhancement also extends to cat-state generation with the generalized photon number subtraction. With experimentally feasible parameters, our results suggest that megahertz-rate cat-state generation is achievable using an on-chip array of \emph{tens} of ON-OFF detectors.

Figures (6)

• Figure 1: Main result of the paper. We develop a theoretical model that captures two realistic detection architectures, namely (a) a nanowire sequential detector that decodes pixel information from time-resolved electrical signals and (c) a parallel detector that decodes pixel information independently from separate electrical signals. As shown in (b), the theory model has an overall loss parameter and dark count parameter.
• Figure 2: Ratio of estimation error to the true value for higher moments of photon number distribution. The $x$-axis represents the number of ON-OFF detectors, and the $y$-axis shows the ratio of the estimation error, $\left|N^2_{\rm mpnr}-N_{\rm true}^2\right|$, to the true value $N^2$. Numerical results are presented for a coherent state $|\underline{\sqrt 2/2}\rangle$ (solid red line, with a scaling trend of $1/n$ shown as a red dashed line) and a cat state $\propto |\underline{\sqrt 2/2}\rangle+|\underline{-\sqrt 2/2}\rangle$ (solid blue line, with a scaling trend of $1/n$ shown as a blue dashed line). Both axes are plotted on a logarithmic scale.
• Figure 3: Observed photon distributions for squeezed vacuum states. (a), (b), and (c) show the observed photon number distributions $\{p_k\}$ for ON-OFF detector numbers $n=10,20$, and $50$, respectively, for a single-mode squeezed vacuum state with a squeezing level of $7$dB. (d) Displays the true photon number distribution. (e) Depicts the odd-photon error probability $p_{\rm odd-err}$ where the solid blue line represents calculated values, the solid light blue line denotes the error probability $p_{\rm odd-err}$ with detector efficiency $95\%$, the dashed blue line indicates a fitting function $\sim\mathcal{O}(n^{-0.5})$, the dotted blue line corresponds to $\sim\mathcal{O}(n^{-1})$.
• Figure 4: Schematic of the cat state generation protocol with MPNR detection. (a) Photon subtraction (b) Generalized photon subtraction. The beamsplitter splits $\eta$ portion of light towards the cat state output. The MPNR detector figure is adopted from Ref. divochiy2008superconducting.
• Figure 5: Impact of detector number $n$ and efficiency $\kappa$ on cat-state breeding. (a,c) Maximum success probability versus fidelity, obtained by tuning the transmissivity $\eta$ (in the range $0 \le \eta \le 1$ for (a) and $0.4 \le \eta \le 1$ for (c)). (b,d) transmissivity values $\eta$ corresponding to a given fidelity. Panels (a) and (b) present results for photon subtraction, whereas panels (c) and (d) illustrate generalized subtraction. We use a smooth color gradient from light blue to blue to represent increasing numbers of ON–OFF detectors, with $n=2$ shown in light blue and $n=\infty$ in blue. Solid, dashed, and dotted lines indicate detector efficiencies $\kappa = 1$, $0.95$, and $0.7$, respectively. All data are produced by setting input states with 7 dB squeezing and an MPNR detector registering $k=2$ clicks.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2
• Proposition 3: Measuring squeezed states
• Theorem 4
• Definition 5: MPNR
• Definition 6: Measurement after beamsplitting
• Lemma 7: Higher-order moments