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IntechOpen eBooks

Separability and Nonseparability of Elastic States in Arrays of One-Dimensional Elastic Waveguides

Pierre Alix Deymier, Jérôme O. Vasseur, Keith Runge, Pierre Lucas

Published Dec 12, 2018• 7 citations
Publisher

Abstract

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.

DOI
10.5772/intechopen.77237

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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 experimentally demonstrate the existence and control of coherent superpositions of elastic states in the direction of propagation of an ultrasonic pseudospin i.e., a φ-bit. The experimental realization of this mechanical pseudospin consists of an elastic aluminum rod serving as a waveguide sandwiched between two heavy steel plates. The Hertzian contact between the rod and the plates leads to restoring forces which couple the directions of propagation (forward and backward). This coupling generates the coherence of the superposition of elastic states. We also demonstrate φ-bit gate operations on the coherent superposition analogous to those used in quantum computing. In the case of a φ-bit, the coherent superposition of states in the direction of propagation are immune to wave function collapse upon measurement as they result from classical waves.
The creation of multilevel quantum states, qudits, has revolutionized concepts for quantum computing. Classical systems that capture behavior analogous to quantum systems have been demonstrated. In this spirit, we consider a three-level classical analogue of the qudit composed of coupled acoustic waveguides. Here, we demonstrate both the experimental realization of a three-level classical analogue of the qudit and the creation and tuning of nonseparable superpositions of two of these analogues, which are classically “entangled.” Measurements of velocities and transmission inform our assignment of these nonseparable states.
We experimentally navigate the Hilbert space of two logical phi-bits supported by an externally driven nonlinear array of coupled acoustic waveguides by parametrically changing the relative phase of the drivers. We observe sharp phase jumps of approximately 180° in the individual phi-bit states as a result of the phase tuning of the drivers. The occurrence of these sharp phase jumps varies from phi-bit to phi-bit. All phi-bit phases also possess a common background dependency on the drivers’ phase. Within the context of multiple time scale perturbation theory, we develop a simple model of the nonlinear array of externally driven coupled acoustic waveguides to shed light on the possible mechanisms for the experimentally observed behavior of the logical phi-bit phase. Finally, we illustrate the ability to experimentally initialize the state of single- and multiple- phi-bit systems by exploiting the drivers’ phase as a tuning parameter. We also show that the nonlinear correlation between phi-bits enables parallelism in the manipulation of two- and multi-phi-bit superpositions of states.
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.
Perelomov coherent states for equally spaced, infinite homogeneous waveguide arrays with Euclidean E(2) symmetry are defined, and new resolutions of the identity are constructed in Cartesian and polar coordinates. The key point to construct these resolutions of the identity is the fact that coherent states satisfy Helmholtz equation (in coherent states labels) an thus a non-local scalar product with a convolution kernel can be introduced which is invariant under the Euclidean group. It is also shown that these coherent states for the Eucliean E(2) group have a simple and natural physical realization in these waveguide arrays.