Wigner Distribution of Twisted Photons

We present the first experimental characterization of the azimuthal Wigner distribution of a photon. Our protocol fully characterizes the transverse structure of a photon in conjugate bases of orbital angular momentum (OAM) and azimuthal angle. We provide a test of our protocol by characterizing pure superpositions and incoherent mixtures of OAM modes in a seven-dimensional space. The time required for performing measurements in our scheme scales only linearly with the dimension size of the state under investigation. This time scaling makes our technique suitable for quantum information applications involving a large number of OAM states.

Figure 1

Left: Schematic diagram for experimental characterization of a structured laser beam. Middle and right: Experimental results for characterization of an OAM mode with $\ell =-1$. The plots in the middle column show the density matrix in the ANG basis, and the plots in the right column present the azimuthal Wigner distribution along with the corresponding marginal distributions in the ANG and OAM bases. The real and imaginary parts of the density matrix are plotted with an identical range of values for the vertical axes.

Figure 2

(a) The intensity pattern of a pure superposition (top) and (bottom) an incoherent mixture of $\ell =1$ and $\ell =-1$ OAM modes with equal weights. (b) The azimuthal Wigner distribution from the experiment. (c) The marginal distributions in the OAM and ANG bases. (d) The real part of the OAM density matrices.

Figure 3

Left: Single photons from nondegenerate parametric down-conversion are separated by a dichroic mirror. The idler photons (830 nm) are detected by an APD, which heralds the detection of signal photons (790 nm) with an ICCD. A $q$ plate ($q=1/2$) is placed between two crossed polarizers to prepare an equal superposition of $\ell =1$ and $\ell =-1$ OAM modes. Inset: The transverse structure of single photons captured with an accumulation of 5-ns-coincidence events over a 1200 sec exposure time. Right: The Wigner distribution, the OAM and ANG marginals, and the real and imaginary parts of the OAM density matrix from experiment.