Operational definition of quantum correlations of light

Quantum features of correlated optical modes define a major aspect of the nonclassicality in quantized radiation fields. However, the phase-sensitive detection of a two-mode light field is restricted to interferometric setups and local intensity measurements. Even the full reconstruction of the quantum state of a single radiation mode relies on such detection layouts and the preparation of a well-defined reference light field. In this work, we establish the notion of the essential quantum correlations of two-mode light fields. It refers to those quantum correlations which are measurable by a given device, i.e., the accessible part of a nonclassical Glauber-Sudarshan phase-space distribution, which does not depend on a global phase. Assuming a simple four-port interferometer and photon-number-resolving detectors, we derive the reconstruction method and nonclassicality criteria based on the Laplace-transformed moment-generating function of the essential quasiprobability. With this technique, we demonstrate that the essential quantum correlations of a polarization tomography scheme are observable even if the detectors are imperfect and cannot truly resolve the photon statistics.

Figure 1

The plot shows $M_{\mathrm{ess}}(\mathit{e};0)\mathit{e}$ [cf. the map in Eq. (43) for $t=1$ and $\tau =0]$, that is, the essential MGF evaluated on a unit sphere, $\mathit{e}$, and plotted in the direction of $\mathit{e}$. For the $z$ direction, $\mathit{e}={(0,0,\pm 1)}^{\mathrm{T}}$, we have $M_{\mathrm{ess}}(\mathit{e};0)=0$.

Figure 2

The determinant in Eq. (25) is plotted for $\tau ={\tau }^{\prime }=0={\mathit{t}}^{\prime }$ as a function of ${\left|T\right|}^2$. The other vector is $\mathit{t}=t\mathit{e}$, with $\mathit{e}$ as given in Eq. (6). A negative value of the determinant in the plot certifies essential nonclassical correlations. Note there is no dependency on the phases of $T$ or $R$; cf. Eq. (40). The parameter $t$ is chosen as $t=\sqrt{2}$ (dotted line), $t=\sqrt{3}$ (solid line), and $t=\sqrt{4}$ (dashed line). For ${\left|T\right|}^2={\left|R\right|}^2=1/2$, we have a balanced beam splitter. The values ${\left|T\right|}^2=1$ or ${\left|R\right|}^2=1$ describe a perfect transmission or reflection, respectively.

Figure 3

Schematics of the setup for measuring the polarization of the signal (SI). The SU(2) transformation can be realized with half-wave and quarter-wave plates. PBS is a polarizing beam splitter, ND is a neutral density filter, and CC denotes a click-counting detector. Bottom pattern: Two possible implementations of a click-counter; see Ref. [50] for a recent review. The upper version shows the time bin [51, 52] and the lower version the spatial [53, 54] multiplexing configuration. In each step, the signal is split on a $50:50$ beam splitter. Avalanche photodiodes in the Geiger mode (APDs) serve as on-off detectors.

Figure 4

Left: The measurable MGF, $M_{\mathrm{ess}}(t\mathit{e};\tau )$, is shown depending on $\tau$ and $-\tau \le t\le \tau$, with $\mathit{e}={(0,0,1)}^{\mathrm{T}}$ and $tanh\left(\xi \right)=1/2$. The transparent gray area shows the same for a vacuum state for comparison. Center: $M_{\mathrm{ess}}(t\mathit{e};\tau )$ in the direction of $\mathit{e}$ is depicted as a function of $\mathit{e}={[sin\left(\vartheta \right)cos\left(\phi \right),sin\left(\vartheta \right)sin\left(\phi \right),cos\left(\vartheta \right)]}^{\mathrm{T}}$, with $\tau =t=1$ and $tanh\left(\xi \right)=3/4$. The transparent surface shows the result for a vacuum state. Right: The verification of nonclassicality (negative values) by the determinant in Eq. (25) is given as a function of $tanh\left(\xi \right)$ and $\tau$ ($0\le 2\tau \le 1$), with $\mathit{e}={\mathit{e}}^{\prime }={(0,0,1)}^{\mathrm{T}}$ and ${\tau }^{\prime }=t^{\prime }=\tau =-t$.