Elsevier

Wave Motion

Volume 113, August 2022, 102966
Wave Motion

Navigating the Hilbert space of elastic bell states in driven coupled waveguides

https://doi.org/10.1016/j.wavemoti.2022.102966Get rights and content

Highlights

  • Driven elastic waveguides capture the characteristic of classical “entanglement”.

  • The ability to tune the degree of nonseparability has been demonstrated experimentally.

  • Demonstrates critical parameters for exploring elastic Bell states’ Hilbert space.

  • Hilbert space navigation enables transformation of states analogous to quantum gates.

Abstract

Externally driven arrays of coupled elastic waveguides have been shown to support nonseparable elastic superpositions of states that are analogous to entangled Bell states in a multipartite quantum system. Here, the “subsystems” correspond to spatial eigen modes characterized by the amplitude and phase difference between the waveguides. We show experimentally that the driving frequency, the relative amplitudes, and phases of the drivers applied to the waveguides, are critical parameters for exploring the elastic Bell states’ Hilbert space. We also demonstrate experimentally the capability of tuning the degree of nonseparability of the superpositions of elastic states. The degree of nonseparability is quantified by calculating the entropy of entanglement. Finally, in support of the experimental observations, we show theoretically that nonlinearity in the elastic behavior of the coupling medium (epoxy) and heterogeneities in the coupling along the waveguides can serve as design parameters in extending the range of the elastic Bell states’ Hilbert space that can be explored.

Introduction

Quantum entanglement, the nonclassical correlation between quantum systems, is an essential ingredient for applications in quantum information science [1]. One of the properties of quantum entangled states is nonseparability, which is however not limited to quantum systems. Nonseparable states of classical waves [2] are sometimes referred to as classically or nonquantum entangled [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. While such classical nonseparable waves do not exhibit the uniquely quantum property of nonlocality, they manifest all other properties of locally entangled states [15], such as nonseparable linear combinations of tensor product states between different degrees of freedom of the same physical manifestation similar to the degree of freedom of a single quantum particle [16]. They have found applications in quantum information science [4], [17], [18] and metrology [4], [19]. Another rather important application is in quantum computing since quantum computing harnesses the nonseparability of entangled states [20]. To date, the study of local nonseparable superpositions of states has essentially focused on the area of optics [7], [19], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], and much less attention has been paid to other classical waves such as elastic waves; yet, remarkable quantum analogous behaviors of sound are emerging, such as the notions of elastic pseudospin [31], [32], [33] and Zak/Berry phase [34], [35], [36]. Recently, we have shown that parallel arrays of one-dimensional (1D) elastic waveguides composed of aluminum rods that are coupled along their length with epoxy and are driven externally, can capture the characteristic of nonseparability (classical entanglement) between different degrees of freedom of the same physical manifestation [12], [13], [14]. These nonseparable superpositions of elastic states, analogous to Bell states, are constructed as a superposition of elastic waves, each a product of a spatial eigen mode part and a plane wave. The plane wave part describes the elastic wave along the length of the waveguides and the spatial eigen modes characterize the amplitude and phase difference between waveguides i.e., across the array of waveguides. These states lie in the tensor product Hilbert space of the two-dimensional subspaces associated with the degrees of freedom along and across the waveguide array. Experimentally we demonstrated that the amplitude coefficients of these nonseparable states are complex due to the dissipative nature of the coupling medium. Navigating portions of the elastic Bell states’ Hilbert space necessitated means of tuning these complex amplitudes [12], [13], [14]. In the area of optics, the generation of Bell states and the tuning of their amplitudes is well established [37], [38], [39], [40], [41]. In optics, the phase of the entangled states are commonly used to create a wide range of quantum states with controllable degrees of entanglement [40], [42]. The phase of the entangled states are selected by changing the half-wave plate and quarter-wave plate orientation [40], [42]. However, to the best of our knowledge, no similar work has been done in elastic systems. It is the objective of the present study to experimentally demonstrate the possibility of not only generating elastic Bell states but also tuning their amplitude over a broad region of the Bell states’ Hilbert space. More specifically, we show experimentally that the frequency, relative amplitudes and phases of the external drivers applied to the waveguides, are critical parameters for navigating the elastic Bell states’ Hilbert space. To explain the experimental observations, we have developed a theoretical model, which in addition to dissipation, accounts for the weak elastic nonlinearity of the medium (epoxy) coupling the essentially linear waveguides, as well as for heterogeneities in the coupling medium along the waveguides. Nonlinear elasticity and heterogeneity of the coupling medium are shown to be potential design parameters in extending the range of the elastic Bell states’ Hilbert space that can be explored via the external drivers.

Section snippets

Background

We briefly review the theory behind the behavior of nonseparable elastic states in parallel arrays of coupled 1D elastic waveguides. The experimental realization of nonseparable superposition of elastic states requires a mechanical system in which elastic wave behavior is effectively described by: H.IN×N+kc2MN×NUN=0,where H=2t2β22x2 is the dynamical differential operator that models the propagation of elastic waves along the waveguides (in xdirection) and β is proportional to sound speed

Experimental results

The experimental set up is the same as that reported in Refs. [12], [13], [14]. Some details are provided in the Supplemental Material. To experimentally realize a nonseparable superpositions of E2 and E3 spatial modes, as shown in Eq. (4), we need to drive the coupled waveguides at isofrequency state ωI and need to tune that state between the E2 and E3 pure states. We identified two isofrequency states, ωI=48.8kHz and ωI=60kHz that enable us to overlap E2 and E3 modes (see [12], [13] and the

Numerical modeling: Mass-spring waveguides

To numerically model the coupled waveguides, we assume that each rod constituting the experimental waveguide can be represented by a 1D crystal with harmonic approximation. Each rod consists of finite number of masses and springs with no pre-compression. The masses are constrained to move horizontal direction only. Therefore, the experimental coupled waveguides can be represented as a set of three 1D crystals with longitudinal harmonic springs coupled them along their length.

The experimental

Conclusion and discussion

Externally driven classical elastic waveguide systems, which are composed of parallel arrays of 1D aluminum rods coupled along their length with epoxy, are able to capture the characteristic of classical “entanglement” i.e., they can support local nonseparable superpositions of product states. These nonseparable states, analogous to the Bell states, are the superposition of elastic waves, each a product of a plane wave part and spatial eigen modes. We have experimentally explored the different

CRediT authorship contribution statement

M. Arif Hasan: Conceived the idea of the research, Performed the theoretical studies, Fabricated the samples and built the experimental setups, Conducted the measurements, Analyzed the data, Scientific discussion and to writing the manuscript. Trevor Lata: Fabricated the samples and built the experimental setups, Scientific discussion and to writing the manuscript. Pierre Lucas: Scientific discussion and to writing the manuscript. Keith Runge: Conceived the idea of the research, Performed the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We acknowledge financial support from the W.M. Keck Foundation. M.A.H. thanks Wayne State University Startup funds for support.

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