Observable nonclassicality witnesses for multiplexed detection systems

Abstract

We address the problem of constructing witnesses for nonclassical light that are applicable in state-of-the-art photon-counting devices. The key ingredient for the criteria we derive are generalized and directly measurable counting statistics and matrices of counting moments. Beyond common criteria, we find classes of witnesses that are based on half-integer powers of click moments and counts. Remarkably, this leads to an exponential increase of the number of nonclassicality criteria one can construct and apply. With this finding, special attention is payed to probing even and odd parity states, requiring such distinct witnesses. Our method is applicable to spatial and time-bin multiplexing in optical systems, where each spatial and temporal mode can be measured with both on-off detectors and detectors with partial internal quasi-photon-number resolution. Generalizations to multimode scenarios are provided, allowing for the direct measurement of nonclassical correlations and coincidence counts between an arbitrary number of modes.

Figures (6)

• Figure 1: Nonclassicality witnessing through negativities based on the indicated index sets $\mathcal{I}$ with integers (left column) and half-integers (right column) for even (solid) and odd (dashed) cat states $|\alpha_\pm\rangle$. For a detection loss of $1-\eta=50\%$, criteria based on the photocounts [top row, cf. Eq. \ref{['eq:CountingBasedExample']}] and normally ordered moments [bottom row, cf. Eq. \ref{['eq:MomentBasedExample']}] are shown as a function of $|\alpha|^2$ over several orders of magnitude on a logarithmic scale. Even-parity cat states are sensitive to the half-integer-based test; odd-parity nonclassicality is verified through integer-based criteria. Note that the $C$-based and $M$-based nonclassicality probes behave rather similarly for the states in Eq. \ref{['eq:CatState']}.
• Figure 2: Spatial (top) and time-bin (bottom) multiplexing schemes. Light enters from the left and is distributed via $50/50$ beam splitters (dashed lines). In time-bin multiplexing (bottom), a delay loop separates pulses in time, while spatial multiplexing uniformly distributes the light across different paths. Individual detectors at the end record "light on" and "light off" information as so-called click and no-click events, respectively. The total number $k$ of clicks is recorded. The depicted spatial multiplexing (top) uniformly spreads the light into $N=4$ paths, each measured with one on-off detector. The shown time-bin multiplexing uniformly distributes light into four time bins for the top and bottom path, resulting in up to $N=8$ total click events, using two on-off detectors only. Note that combinations of the two schemes are possible, and that other schemes use uniform illumination of detector arrays, consisting of $N$ diodes BDFL09KASVSH17. Alternative schemes, including deviations from a uniform splitting, have been studied, too LFPR16.
• Figure 3: Nonclassicality witnessing through negative eigenvalues in the click-counting matrix $C$, Eq. \ref{['eq:WitnessClickCounts']}. Each on-off detector has an efficiency of only $50\%$, and we make the unconventional choice of $N=5$ to highlight the applicability of our method even in the case of an odd number of detection bins. (For even $N$s, the approach works analogously.) The index set with whole numbers (top plot) verifies the nonclassicality of the odd coherent state $|\alpha_{-}\rangle$ but is insensitive to the even coherent states $|\alpha_{+}\rangle$. Conversely, the index set with half integers (bottom plot) is sensitive to the opposite parity.
• Figure 4: Nonclassicality witnessing through negative eigenvalues in the matrix $M$ of click moments, Eq. \ref{['eq:MinEigMoments']}. The settings of the detection system are the same as in Fig. \ref{['fig:ClickCountingMatrix']}. Note that, despite the different physical and statistical meaning of the matrix $C$ in Fig. \ref{['fig:ClickCountingMatrix']} and the matrix $M$ here, the plots here are technically identical because $C$ can be mapped to $M$ without altering signs of eigenvalues, using generating functions SVA13.
• Figure 5: Witnessing nonclassicality via a negative minimal eigenvalue of the multinomial click-counting matrix $C$ in Eq. \ref{['eq:MultinomialMatrixCounts']}, using the four possible index sets in Eq. \ref{['eq:MultinomialIndexSets']}. The elements of these index sets $(N_0,N_1,N_2)$ pertain to the number of detectors $N_j$ with no photons detected ($j=0$), one photon detected ($j=1$), and at least two-photons detected ($j=2$), assuming a detection efficiency of $\eta=50\%$. The top-left plot certifies the nonclassicality of the odd cat state using integer-based indices $(N_0,N_1,N_2)\in\mathbb N\times\mathbb N\times\mathbb N$. The top-right plot achieves this too, but with a generally reduced value of negativity, using $(N_0,N_1,N_2)\in\left(\frac{1}{2}\mathbb N\setminus\mathbb N\right)\times \mathbb N\times\left(\frac{1}{2}\mathbb N\setminus\mathbb N\right)$. The bottom-left plot certifies the nonclassicality of the even state via $(N_0,N_1,N_2)\in \mathbb N\times\left(\frac{1}{2}\mathbb N\setminus\mathbb N\right)\times\left(\frac{1}{2}\mathbb N\setminus\mathbb N\right)$. The bottom-right plot does the same, but with generally higher negativities when using $(N_0,N_1,N_2)\in\left(\frac{1}{2}\mathbb N\setminus\mathbb N\right)\times\left(\frac{1}{2}\mathbb N\setminus\mathbb N\right)\times\mathbb N$.