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Physical Review A

Measuring the nonseparability of vector vortex beams

Melanie G. McLaren, Thomas Konrad, Andrew Forbes

Published Aug 18, 2015• 209 citations
Publisher

Abstract

Vector beams have the defining property of nonseparable spatial and polarization degrees of freedom and are now routinely generated in the laboratory and used in a myriad of applications. Here we exploit the nonseparability of such beams, akin to entanglement of quantum states, to apply tools traditionally associated with quantum measurements to these classical fields. We find that the entanglement entropy is a proxy for the average degree of polarization and thus provides a single number for the vector nature of such beams. In addition to providing tools for the analysis of vector beams, we also explore the concept of classical entanglement to explain why these tools are appropriate in the first place.

DOI
10.1103/physreva.92.023833

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Vector beams have the defining property of nonseparable spatial and polarization degrees of freedom and are now routinely generated in the laboratory and used in a myriad of applications. Here we exploit the nonseparability of such beams, akin to entanglement of quantum states, to apply tools traditionally associated with quantum measurements to these classical fields. We find that the entanglement entropy is a proxy for the average degree of polarization and thus provides a single number for the vector nature of such beams. In addition to providing tools for the analysis of vector beams, we also explore the concept of classical entanglement to explain why these tools are appropriate in the first place.
Given the multitude of applications of vector vortex beams there is a need for robust tools to measure them. Here we exploit the non-separability of such beams, akin to entanglement of quantum states, to apply tools traditionally associated with quantum measurements to these classical fields. We apply three measures of non-separability: a Bell inequality, a concurrence, and an entanglement entropy to define the "vectorness" of such beams. In addition to providing novel tools for the analysis of vector beams, we also introduce the concept of classical entanglement to explain why these tools are appropriate in the first place.
Light beams having a vectorial field structure, or polarization, that varies over the transverse profile and a central optical singularity are called vector vortex (VV) beams and may exhibit specific properties such as focusing into ``light needles'' or rotation invariance. VV beams have already found applications in areas ranging from microscopy to metrology, optical trapping, nano-optics, and quantum communication. Individual photons in such beams exhibit a form of single-particle quantum entanglement between different degrees of freedom. On the other hand, the quantum states of two photons can be also entangled with each other. Here, we combine these two concepts and demonstrate the generation of quantum entanglement between two photons that are both in VV states: a form of entanglement between two complex vectorial fields. This result may lead to quantum-enhanced applications of VV beams as well as to quantum information protocols fully exploiting the vectorial features of light.
Vector beams are inhomogeneously polarized optical fields with nonseparable quantumlike correlations between their polarization and spatial components, and they hold tremendous promise for classical and quantum communication across various channels, e.g., the atmosphere, underwater, and in optical fiber. Here, we show that by exploiting their quantumlike features by virtue of the nonseparability of the field, the decay of both the polarization and spatial components can be studied in tandem. In particular, we invoke the principle of channel-state duality to show that the degree of nonseparability of any vector mode is purely determined by that of a maximally nonseparable one, which we confirm using orbital angular momentum (OAM) as an example for topological charges of $\ensuremath{\ell}=1$ and $\ensuremath{\ell}=10$ in a turbulent atmosphere. A consequence is that the well-known cylindrical vector vortex beams are sufficient to predict the behavior of all vector OAM states through the channel. Our method reveals interesting features about the performance of various modes through the channel: the decay in vector quality decreases with an increasing OAM value, even though the spread is higher for larger OAM modes. Our approach offers a fast and easy probe of noisy channels, while at the same time revealing the power of quantum tools applied to classical light.
One of the most prominent features of quantum entanglement is its invariability under local unitary transformations, which implies that the degree of entanglement or nonseparability remains constant during free-space propagation, true for both quantum and classically entangled modes. Here we demonstrate an exception to this rule using a carefully engineered vectorial light field, and we study its nonseparability dynamics upon free-space propagation. We show that the local nonseparability between the spatial and polarization degrees of freedom dramatically decays to zero while preserving the purity of the state and hence the global nonseparability. We show this by numerical simulations and corroborate it experimentally. Our results evince novel properties of classically entangled modes and point to the need for new measures of nonseparability for such vectorial fields, while paving the way for novel applications for customized structured light.
Generation of spatially inhomogeneously polarized (SIP) beams due to polarization spatial mode nonseparability is demonstrated using a polarization Sagnac interferometer. Counter propagating horizontal (H) and vertical (V) polarized Laguerre Gaussian beams of opposite topological charges(LG01)corresponding to the two binary degrees of freedom (DoF) are coherently superposed in the interferometer. Quantum inspired entanglement witness and quantification such as Bell state measurement, violation of CHSH form of Bells inequality and concurrence measurements are experimentally measured to establish nonseparability between the two DoF. As the SIP beams are equivalent to bipartite pure states, point wise Stokes polarimetry measurements are used to calculate the reduced coherency matrix corresponding to polarization DoF. Using these we calculate the linear entropy and maximized Bells measure to quantify the degree of entanglement of the SIP beams.
The nonclassicality of a macroscopic single-mode optical superposition state is potentially convertible into entanglement, when the state is mixed with the vacuum on a beam splitter. Considering light beams with polarization degree of freedom in Euclidean space as coherent product states in a bipartite Hilbert space, we propose a method to convert the two orthogonal polarizations into simultaneous entanglement and classical nonseparability through nonclassicality in the superpositions of coherent and displaced Fock states. Equivalent Bell state emerges from the resulted superpositions and the proportion of mixed entanglement and nonseparablity is determined by the displacement amplitudes along the polarization directions. We characterize the state nonclassicality via features in Wigner distributions and propose an experimental method for generating these states and measuring them via homodyne tomography.