Continuous-variable entangled states of light carrying orbital angular momentum

The orbital angular momentum of light, unlike spin, is an infinite-dimensional discrete variable and may hence offer enhanced performances for encoding, transmitting, and processing quantum information. Hitherto, this degree of freedom of light has been studied mainly in the context of quantum states with definite number of photons. On the other hand, field-quadrature continuous-variable quantum states of light allow implementing many important quantum protocols not accessible with photon-number states. Here, we realize a scheme based on a $q$-plate device for endowing a bipartite continuous-variable Gaussian entangled state with nonzero orbital angular momentum. We then apply a reconfigurable homodyne detector working directly with such nonzero orbital angular momentum modes in order to retrieve experimentally their entire quantum-state covariance matrix, thus providing a full characterization of their quantum fluctuation properties. Our work is a step towards generating multipartite continuous-variable entanglement in a single optical beam.

Figure 1

Schematic of the experimental setup. An internally frequency-doubled Nd:YAG pumps an OPO based on an $\alpha$-cut PPKTP type-II crystal that generates a pair of frequency-degenerate orthogonally polarized modes. The OPO [23] oscillation threshold is $\approx 70$ mW and it is operated at $\approx 70\%$ of the threshold value. The yellow shadowed area (bottom-left inset) including two quarter-wave plates (QWP1 and QWP2), a $q$-plate (qP), and a half-wave plate (HWP) represents the optical setup for manipulating the SAM and OAM d.o.f. of the entangled beam. A similar set of optical components is placed along the LO path for effective OAM homodyning (dark orange shadow). The bottom-right inset shows an enlarged view of the homodyne detector.

Figure 2

Histogram of the values obtained for the kurtosis relative to all the 700 quadrature marginal distributions used in the data analysis. Each data set, corresponding to a single detected mode, is partitioned into 100 phase bins each containing a marginal distribution of $10000$ points. The mean of the kurtosis distribution is 3.01 with a standard deviation of 0.05, thus confirming that the quadrature marginal distributions are Gaussian and so are the corresponding field states.

Figure 3

Graphic representation of the covariance matrix given in Eq. (7). The nonzero elements outside the main diagonal are a signature of quantum correlation between pairs of quadratures of modes carrying different OAM.

Figure 4

Histogram of the purity $\mu$ [see Eq. (13)] for the reconstructed pure states obtained from the Monte Carlo statistical analysis. The average of the distribution is 0.99 with a standard deviation of 0.03. The distribution median is at 0.966.

Figure 5

Joint photon number probability distribution $p(n;m)$ of the pure twin-beam state generated inside the crystal. This state propagates all the way down to the homodyne detector and, after collection and detection losses, is represented by the CM given by Eq. (7).