Nonclassical photocounting statistics with a single on-off detector

TL;DR

The paper tackles the challenge that a single on-off photocounter cannot reveal nonclassical light because its statistics can be mimicked by coherent states. It proposes a simple experimental modification: inserting a tunable attenuator to create a discrete set of detection efficiencies and testing the resulting no-click probabilities against the convex hull of classical (coherent-state) responses. By employing tight, geometry-based inequalities (linear and nonlinear) and, for uniformly distributed efficiencies, the Hausdorff moment approach, the authors derive a complete set of criteria that certify nonclassicality when the measured statistics lie outside the classical set. They demonstrate the method on phase-squeezed coherent states, showing that nonclassicality can be witnessed with as few as three efficiency settings, even in regimes where standard phase-insensitive tests fail, and discuss robustness to imperfect knowledge of efficiencies. The approach is computationally tractable, experimentally feasible, and broadly applicable to practical photodetection-based nonclassicality tests.

Abstract

Any single on-off photocounter, which can only detect the presence or absence of photons without discriminating their number, is not capable of identifying nonclassical nature of light. This limitation arises because any photocounting statistics obtained with such a detector can be easily reproduced with coherent states of a light mode. We show that a simple modification of an on-off detector -- introducing controlled attenuation as a tunable setting -- enables such detectors to reveal nonclassical properties of radiation fields.

Figures (7)

• Figure 1: The scheme of experiment for detecting nonclassicality of photocounting statistics with an on-off detector. The detector is preceded by an amplitude modulator, whose transmittance may take the values $\eta_i$, $i=1\ldots N$.
• Figure 2: Classical region for $N=2$ settings (shaded), its boundary (solid line), and points related to the Fock states $\left|n\right\rangle$ attenuated with the efficiency $\eta_{\mathrm{c}}$. In this example, $\eta_1=1/2$ and $\eta_2=1$.
• Figure 3: The curve $\mathcal{C}$ and its convex hull $\mathcal{H}$ for the case of $N=3$. Also the vectors $\boldsymbol{\lambda}(t_1;\tau)$ and the vectors $\Delta \boldsymbol{\Pi}(t_1,\tau)$ and $\dot{\boldsymbol{\Pi}}(t_1)$ defining supporting planes of $\mathcal{H}$ are shown for $\tau=0$.
• Figure 4: The function $\boldsymbol{\lambda}(\boldsymbol{t}_m; \tau)\cdot\boldsymbol{\Pi}(t)$ for $\eta_i=i/N$, $N=5$ (i.e., $m=2$), and $\tau=0$. The points $t = t_1$, $t = t_2$, and $t = \tau$ correspond to the global maximum. Depending on whether $t_1 = t_2$ or $t_2 = \tau$, the function exhibits $2^2 = 4$ different variants [see Eq. (\ref{['Eq:NumbMax-Odd']})], each with a different number of maxima. Here, $\mathcal{C} = \inf_{t \in [0, 1]} \boldsymbol{\lambda}(t_m;\tau) \cdot \boldsymbol{\Pi}(t)$ is a scaling factor introduced to make all plots comparable in scale.
• Figure 5: The functions (a) $\boldsymbol{\lambda}(\boldsymbol{t}_m)\cdot\boldsymbol{\Pi}(t)$ and (b) $\boldsymbol{\lambda}(\boldsymbol{t}_{m-1};0,1)\cdot\boldsymbol{\Pi}(t)$ are shown for $\eta_i = i/N$ with $N=4$ (i.e., $m=2$). For case (a), the points $t = t_1$ and $t = t_2$ correspond to the global maximum. For case (b), the points $t=0$, $t = t_1$ and $t=1$ correspond to the global maximum. Depending on whether $t_1 = t_2$, the function exhibits two distinct variants in case (a), and $2^3-1=7$ variants in case (b), where the cases $t_i=0$ and $t_i=1$ are also included. Here, $\mathcal{M} = \sup_{t \in [0, 1]} \boldsymbol{\lambda}(t_m) \cdot \Pi(t)$ and $\mathcal{C} = \inf_{t \in [0, 1]} \boldsymbol{\lambda}(t_{m-1};0,1) \cdot \Pi(t)$ are scaling factors introduced to make all plots comparable in scale.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2: Karlin