Abstract
We consider two one-dimensional harmonic chains coupled along their length via linear springs. Casting the elastic wave equation for this system in a Dirac-like form reveals a directional representation. The elastic band structure, in a spectral representation, is constituted of two branches corresponding to symmetric and antisymmetric modes. In the directional representation, the antisymmetric states of the elastic waves possess a plane wave orbital part and a 4x1 spinor part. Two of the components of the spinor part of the wave function relate to the amplitude of the forward component of waves propagating in both chains. The other two components relate to the amplitude of the backward component of waves. The 4x1 spinorial state of the two coupled chains is supported by the tensor product Hilbert space of two identical subsystems composed of a non-interacting chain with linear springs coupled to a rigid substrate. The 4x1 spinor of the coupled system is shown to be in general not separable into the tensor product of the two 2x1 spinors of the uncoupled subsystems in the directional representation.
Highlights
The phenomenon of entanglement of quantum systems is receiving an increasing amount of attention in the light of its importance in the development of quantum information science and computing.[1]
While the states of the bipartite mechanical system are separable in a spectral representation, the possibility of measuring the spinor part of its wave function dictates another partitioning into two identical subsystems, each composed of a single elastic chain coupled to a rigid substrate
We consider a bipartite classical mechanical system composed of two coupled one-dimensional elastic chains whose elastic wave equations can be factored into a Dirac-like equation
Summary
The phenomenon of entanglement of quantum systems is receiving an increasing amount of attention in the light of its importance in the development of quantum information science and computing.[1]. One can establish a one-to-one correspondence between a measurable quantity, namely the transmission coefficient along the chains and the components of the spinor part of the wave function.[10,11,12] While the states of the bipartite mechanical system are separable in a spectral representation, the possibility of measuring the spinor part of its wave function dictates another partitioning into two identical subsystems, each composed of a single elastic chain coupled to a rigid substrate. The non-separability of the states in terms of direction of propagation of the elastic bipartite system relative to single chain subsystems is analogous to the phenomenon of local correlation. These non-separable classical states and their analogy with local quantum states may prove to be useful in quantum information processing
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