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Journal of Physics B Atomic Molecular and Optical Physics

Elastic waves with correlated directional and orbital angular momentum degrees of freedom

Pierre Alix Deymier, Keith Runge, Jérôme O. Vasseur, Anne-Christine Hladky, Pierre Lucas

Published May 14, 2018• 14 citations
Publisher

Abstract

Abstract We introduce a formalism that enables the calculation of elastic wave functions supported by parallel arrays of coupled one-dimensional elastic waveguides. These wave functions are expressed as tensor products of a spinor part associated with directional degrees of freedom and an orbital angular momentum (OAM) part associated with the phase of the coupled waveguides. We demonstrate that one can construct wave functions as a superposition of these elastic waves, which cannot be written as a tensor product of a spinor part and an OAM part. These elastic wave functions are not separable in the tensor product Hilbert space of directional and OAM subspaces. We show that we can construct maximally nonseparable states that are similar to Bell states.

DOI
10.1088/1361-6455/aac446

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Abstract We introduce a formalism that enables the calculation of elastic wave functions supported by parallel arrays of coupled one-dimensional elastic waveguides. These wave functions are expressed as tensor products of a spinor part associated with directional degrees of freedom and an orbital angular momentum (OAM) part associated with the phase of the coupled waveguides. We demonstrate that one can construct wave functions as a superposition of these elastic waves, which cannot be written as a tensor product of a spinor part and an OAM part. These elastic wave functions are not separable in the tensor product Hilbert space of directional and OAM subspaces. We show that we can construct maximally nonseparable states that are similar to Bell states.
We show that the directional projection of longitudinal waves propagating in a parallel array of N elastically coupled waveguides can be described by a nonlinear Dirac-like equation in a 2 N dimensional exponential space.This space spans the tensor product Hilbert space of the two-dimensional subspaces of N uncoupled waveguides grounded elastically to a rigid substrate (called φ-bits).The superposition of directional states of a φ-bit is analogous to that of a quantum spin.We can construct tensor product states of the elastically coupled system that are nonseparable on the basis of tensor product states of N φ-bits.We propose a system of coupled waveguides in a ring configuration that supports these nonseparable states.
Externally driven parallel arrays of elastic waveguides inelastically coupled along their length are shown to support inelastic modes that span exponentially complex Hilbert spaces. The nonlinear coupling takes the form of power functions of the relative displacement between waveguides. First order perturbation theory is employed to obtained closed form solutions for the contribution of nonlinearity to the displacement field in the form of nonseparable superpositions of product states of plane waves. When the system is dissipative, the coefficients of the superposition of product states are complex and can be varied by controlling the characteristics of the way the array is driven externally. The entropy of 'entanglement' of these nonseparable superpositions of product states is calculated from the complex amplitudes through the density matrix. Navigation of the exponentially complex Hilbert space of product states enables manipulation of the degree of nonseparability of the superpositions.
We show that when an electron or photon propagates in a cylindrically symmetric waveguide, its spin angular momentum (SAM) and its orbital angular momentum (OAM) interact. Remarkably, we find that the dynamics resulting from this spin-orbit interaction are quantitatively described by a single expression applying to both electrons and photons. This leads to the prediction of several rotational effects: the spatial or time evolution of either particle's spin-polarization vector is controlled by its OAM quantum number or, conversely, its spatial wave function is controlled by its SAM. We show that the common origin of these effects in electrons and photons is a universal geometric phase. We demonstrate how these phenomena can be used to reversibly transfer entanglement between the SAM and OAM degrees of freedom of two-particle states.
Introduction In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics. Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes , which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11]. There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m , which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.
A continuum of coupled oscillators is considered, described by a continuous tensor product of Hilbert spaces. The mode position Ux and the mode momentum Up are operators which act collectively on all oscillators. They obey equations of motion which are very similar to those of a harmonic oscillator. Q‐functionals are used to introduce entropic quantities that describe correlations among the oscillators. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement; and the speed with which the entanglement propagates.