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Optics Letters

Twist phase and classical entanglement of partially coherent light

Sergey A. Ponomarenko

Published Nov 8, 2021• 27 citations
Publisher

Abstract

We demonstrate that the presence of a twist phase in a random light beam leads to classical entanglement between phase space degrees of freedom of the beam. We find analytically the bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) source and quantify its entanglement by evaluating the Schmidt number of the decomposition. We show that (i) classical entanglement of a TGSM source vanishes concurrently with the twist in the fully coherent limit and (ii) entanglement dramatically increases as the source coherence level decreases. We also show that the discovered type of classical entanglement of a Gaussian Wigner function does not degrade on beam propagation in free space.

DOI
10.1364/ol.445258

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We demonstrate that the presence of a twist phase in a random light beam leads to classical entanglement between phase space degrees of freedom of the beam. We find analytically the bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) source and quantify its entanglement by evaluating the Schmidt number of the decomposition. We show that (i) classical entanglement of a TGSM source vanishes concurrently with the twist in the fully coherent limit and (ii) entanglement dramatically increases as the source coherence level decreases. We also show that the discovered type of classical entanglement of a Gaussian Wigner function does not degrade on beam propagation in free space.
We examine the impact of the atmospheric turbulence on a recently discovered type of classical entanglement of partially coherent beams endowed with a twist phase. We derive a compact analytical expression for the Schmidt number of a bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) beam propagating through the turbulent atmosphere. We elucidate conditions for a TGSM source to generate a strongly classically entangled paraxial field over a desired propagation distance in the turbulent atmosphere. Our results will find applications to free-space optical communications and motivate further research into classical entanglement with random light.
Recently, a new form of classical entanglement between phase space degrees of freedom in twisted partially coherent beams has been theoretically predicted [Opt. Lett.46, 5958 (2021)10.1364/OL.445258]. Here, we report the first, to our knowledge, experimental observation of such phase-space entanglement (nonseparability) in a twisted Gaussian Schell-model (TGSM) beam via four-dimensional Wigner distribution function measurements, realized through the scanned windowed Fourier transform method. The Wigner distribution measurement enables direct visualization of phase-space nonseparability in the beam. Moreover, we demonstrate that the twist factor of the TGSM beam can be precisely determined from phase-space measurements. Furthermore, the Wigner distribution reveals the nonuniform spatial correlation properties of the TGSM beam and the rotational symmetry of its spatial coherence structure. Our method provides a versatile platform for probing phase-space nonseparability in random light fields.
We introduce the Twisted Gaussian Schell (TGS) state, a two-mode mixed Gaussian state defined as the quantum-optical analog of the Twisted Gaussian Schell-model beam of classical paraxial optics, characterized by the so-called twist phase. In the TGS state, the twist parameter arises when an asymmetric two-mode thermal state is subject to local squeezing after the action of phase shifters and a beam splitter. Its defining quantum feature is nonclassicality: although the state is separable in its natural bipartition, when the twist parameter is nonzero there are global quadratures that can be squeezed below the shot-noise limit. The nonclassicality has also a direct signature in the joint photon-number distribution, which we obtain in closed form. Moreover, coupling each mode to an ancillary vacuum at a balanced beam splitter yields a four-mode state with entanglement in select 2×22\times22×2 bipartitions, with local description given by two TGS states, and all 1×31\times31×3 bipartitions. For fixed input squeezing, increasing the twist parameter activates entanglement where the state is otherwise separable and deepens it where already present. The classical physicality bound on the twist parameter coincides with the quantum physicality condition. These results advance the two-way bridge between classical beam engineering and quantum information.
Abstract Entanglement of states and Complete Positivity of maps are concepts that have achieved physical importance with the recent growth of quantum information science. They are however mathematically relevant whenever tensor products of complex linear (Hilbert) spaces are involved. We present such situations in classical scalar paraxial wave optics where these concepts play a role: propagation characteristics of coherent and partially coherent Gaussian beams; and the definition and separability of the family of Twisted Gaussian Schell Model (TGSM) beams. In the former, the evolution of the width of a projected one‐dimensional beam is shown to be a signature of entanglement in a two‐dimensional amplitude. In the latter, the partial transpose operation is seen to explain key properties of TGSM beams.
Partially coherent optical beams present phenomena that may not appear in perfectly coherent beams. In this regard, spontaneous parametric downconversion is an intriguing physical process, since the optical properties of the pump laser beam are transferred to the quantum properties of the down-converted photon pairs. Here, the authors study the transfer of twist phase, a novel property of the so-called twisted Gaussian Schell model beams, from a pump beam to down-converted photons. The authors show that a non-zero twist phase can lead to highly entangled yet highly mixed states. The correlations that appear in near/far-field intensity measurements, in the two-photon covariance matrix as well as in orbital angular momentum modes, are explored. The authors derive expressions for the purity of the two-photon state and relate it to the transverse spatial coherence of the pump beam. Their main results utilize a Wigner function formalism that can be applied to a number of scenarios involving partially coherent downconversion sources and contribute to the recent interest in spatially correlated photon pairs produced in mixed states, with possible applications in quantum communication and imaging.