Central-moment description of polarization for quantum states of light

We present a moment expansion for the systematic characterization of the polarization properties of quantum states of light. Specifically, we link the method to the measurements of the Stokes operator in different directions on the Poincaré sphere and provide a scheme for polarization tomography without resorting to full-state tomography. We apply these ideas to the experimental first- and second-order polarization characterization of some two-photon quantum states. In addition, we show that there are classes of states whose polarization characteristics are dominated not by their first-order moments (i.e., the Stokes vector) but by higher-order polarization moments.

Figure 1

Experimental setup. Single-photons A and B, both horizontally polarized, are prepared by spontaneous parametric down-conversion. (P)BS denotes a (polarization) beam splitter. HWP and QWP denote half-wave and quarter-wave plates, respectively. FS denotes a 50:50 fiber splitter and D${}_1$-D${}_4$ denote single-photon avalanche photo diodes.

Figure 2

The second-order central moment $\langle {\widehat{\Delta }}_{\mathbf{n}}^2\rangle$ for the state $|2,0\rangle$. Theoretical plot in (a) and the experimental results in (b).

Figure 3

$\langle {\widehat{\Delta }}_{\mathbf{n}}^2\rangle$ in the plane defined by $\langle {\widehat{\Delta }}_2^2\rangle =0$ for the state $|2,0\rangle$. The theoretically expected result is a solid (blue) line, the experimentally obtained result is a dashed (red) line, and the experimental result, solidly rotated so its eigenvectors coincide with the intended eigenvectors, is a dotted (black) line.

Figure 4

The second-order polarization central moment $\langle {\widehat{\Delta }}_{\mathbf{n}}^2\rangle$ for the state $\left(\right|2,0\rangle \langle 2,0|+|0,2\rangle \langle 0,2\left|\right)/2$. Theoretical plot in (a) and the experimental results in (b).

Figure 5

$\langle {\widehat{\Delta }}_{\mathbf{n}}^2\rangle$ in the plane defined by $\langle {\widehat{\Delta }}_1^2\rangle =0$ for the state $\left(\right|2,0\rangle \langle 2,0|+|0,2\rangle \langle 0,2\left|\right)/2$. The theoretically expected result is a solid (blue) line, the experimentally obtained result is a dashed (red) line, and the experimental result, solidly rotated so its eigenvectors coincide with the intended eigenvectors, is a dotted (black) line.

Figure 6

The absolute value of ${\langle {\widehat{\Delta }}_{\mathbf{n}}^3\rangle }_3$ for the state $\frac{1}{3}|3,0\rangle \langle 3,0|+\frac{1}{2}|1,2\rangle \langle 1,2|+\frac{1}{6}|0,3\rangle \langle 0,3|$.

Figure 7

The absolute value of ${\langle {\widehat{\Delta }}_{\mathbf{n}}^3\rangle }_3$ for the state $\left(\right|0,3\rangle +|3,0\rangle )/\sqrt{2}$.

Figure 8

The absolute value of ${\langle {\widehat{\Delta }}_{\mathbf{n}}^3\rangle }_3$ for the state $\frac{19}{36}|3,0\rangle \langle 3,0|+\frac{15}{36}|1,2\rangle \langle 1,2|+\frac{1}{18}|0,3\rangle \langle 0,3|$.