## 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 2N 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.

### Keywords

- one-dimensional elastic waveguides
- nonseparability
- elastic waves
- elastic pseudospin
- coupled waveguides

## 1. Introduction

Quantum bit-based computing platforms can capitalize on exponentially complex entangled states which allow a quantum computer to simultaneously process calculations well beyond what is achievable with serially interconnected transistor-based processors. Ironically, a pair of classical transistors can emulate some of the functions of a qubit. While current manufacturing can fabricate billions of transistors on a chip, it is inconceivable to connect them in the exponentially complex way that would be required to achieve nonseparable quantum superposition analogues. In contrast, quantum systems possess such complexity through the nature of the quantum world. Outside the quantum world, the notion of classical nonseparability [1, 2, 3] has been receiving a lot of attention from the theoretical and experimental point of views in the field of optics. Degrees of freedom of photon states that span different Hilbert spaces can be made to interact in a way that leads to local correlations. Correlation has been achieved between degrees of freedom that include spin angular momentum and orbital angular momentum (OAM) [4, 5, 6, 7, 8, 9], OAM, polarization and radial degrees of freedom of a beam of light [10] as well as propagation direction [11, 12]. Recently, we have extended this notion to correlation between directional and OAM degrees of freedom in elastic systems composed of arrays of elastic waveguides [13]. This classical nonseparability lies only in the tensor product Hilbert space of the subspaces associated with these degrees of freedom. This Hilbert space does not possess the exponential complexity of a multiqubit Hilbert space, for instance. It has been suggested theoretically and experimentally that classical systems coupled via nonlinear interactions may have computational capabilities approaching that of quantum computers [14, 15, 16].

We demonstrated in Ref. [17] that nonlinear elastic media can be used to produce phonons that can be correlated simultaneously in time and frequency. We have also shown an analogy between the propagation of elastic waves on elastically coupled one-dimensional (1D) wave guides and quantum phenomena [18, 19, 20, 21]. More specifically, the projection on the direction of propagation of elastic waves in an elastic system composed of a 1D waveguide grounded to a rigid substrate (denoted

The objective of this paper is to investigate the notion of separability and nonseparability of multipartite classical mechanical systems supporting elastic waves. These systems are composed of 1D elastic waveguides that are elastically coupled along their length to each other and/or to some rigid substrate. The 1D waveguides support spinor-like amplitudes in the two-dimensional (2D) subspace of directional degrees of freedom. The amplitudes of

In Section 2 of this chapter, we introduce the mathematical formalism that is needed to demonstrate the nonseparability of elastic states of coupled elastic waveguides in an exponentially complex space. Throughout this section, we use illustrations of the concepts in the case of systems composed of small numbers of waveguides. However, the approach is fully scalable and can be generalized to any large number of coupled waveguides. In Section 3, we draw conclusions concerning the applicability of this approach to solve complex problems.

## 2. Models and methods

We have previously considered systems constituted of *N* one-dimensional (1D) waveguides coupled elastically along their length [13]. In this section, we summarize the results of these previous investigations to develop a formalism to address our current considerations. The parallelly coupled waveguides can be arranged in any desired way. The propagation of elastic modes is limited to longitudinal modes along the waveguides in the long wavelength limit, i.e., the continuum limit. We consider the representations of the modes of the coupled waveguide systems in two spaces. The first space scales linearly with *N*. The second space scales as

### 2.1. Representation of elastic states in a space scaling linearly with *N*

A compact form for the equations of motion of the *N* coupled waveguides is:

Here, the propagation of elastic waves in the direction *i*th waveguide. The coupling matrix operator *N* = 3 parallel waveguides in a closed ring arrangement with first neighbor coupling, takes the form:

Eq. (1) takes the form of a generalized Klein-Gordon (KG) equation and its Dirac factorization introduces the notion of the square root of the operator

In Eq. (3), *2 N* dimensional vector which represents the modes of vibration of the *N* waveguides projected in the two possible directions of propagation (forward and backward) and

We choose components of the

where

In Eqs. (4) and (5), *a*_{2Nx1} is a *2 N* dimensional vector whose components are the amplitudes *aI*. In obtaining Eq. (4), we have multiplied all terms in Eq. (3) on the left by

Writing Eq. (4) as a linear combination of tensor products of

While the degrees of freedom associated with *N* dimensional Hilbert subspace, the degrees of freedom associated with

Replacing

Choosing

For nontrivial eigenvectors

In obtaining Eq. (9), we have also used the fact that

Eq. (10), now written in the matrix form, can now be solved for a given

This eigen equation gives the dispersion relation *vide infra*) and the following eigen vectors projected into the space of directions of propagation:

To determine the eigen vectors of

In the case of the coupling matrix,

Eq. (3) being linear, its solutions can be written as linear combinations of elastic wave functions in the form:

In Eq. (14), we have expressed the dependencies on the wave number *N*. The eigen vectors *N* waveguides. The space spanned by these solutions scales linearly with the number of waveguides, i.e., as *2N*.

### 2.2. Representation of elastic states in a space scaling as 2 N

We first illustrate the notion of exponential space in the case of three waveguides. Each guide is connected to a rigid substrate and therefore constitutes a *single* equation which is constructed as follows:

In Eq. (15), we are now defining a positional variable for each waveguide, namely,

*j*):

The Hilbert space spanned by the solutions of Eq. (15) is the product space of the three 2D subspaces associated with each waveguide. The states of a system composed of *N*

The question that arises then concerns the possibility of writing an equation in the exponential Hilbert space for *N* waveguides coupled to each other. For instance, we wish to obtain the states of the system composed of three waveguides coupled in a ring arrangement from an equation of the form:

The matrix

The Dirac equation of the three coupled waveguides in the exponential space is therefore nonlinear. Generalization to *N* coupled chains will result in the following nonlinear equation:

where nonzero components of

### 2.3. Elastic states in the exponential space

For a system of waveguides that are not coupled, the elastic states, solutions of linear equations of the form of Eq. (15), are tensor products but also linear combinations of tensor products of spinor solution for individual waveguides (see Eq. (17)). It is therefore possible to construct nonseparable states in the exponential space for systems of uncoupled waveguides. For example, if we consider a system of two uncoupled waveguides, a possible state of the system in the

Choosing

The bracket takes the form:

The vector

Since the waveguides are not coupled, it is, however, not possible to manipulate the state of one of the waveguides by manipulating the state of the other one. Simultaneous manipulation of the state of waveguides in the exponential space requires coupling. We now address elastic states in the coupled waveguides system.

For a system of *N* coupled waveguides, we construct a solution of Eq. (3) that takes the form of a linear combination of solutions given in Eq. (14):

The *N* = 9 waveguide system.

With

Here, we have chosen, for the sake of simplicity, the

The first two terms in Eq. (26) form a

Since Eq. (20) is nonlinear, linear combinations of tensor product solutions of the form above are not solutions. Solutions of the nonlinear Dirac equation always take the form of a tensor product when the spinor wave functions

### 2.4. Operating on exponentially-complex tensor product elastic states

In this subsection, we expand tensor product states of the form given in Eq. (27) in linear combinations of tensor products of pure states in the exponential space. We illustrate this expansion in the case of three parallel waveguides elastically coupled to each other. Each waveguide is also coupled elastically to a rigid substrate. We treat the case where the strength of all the couplings is the same. In that case, the coupling matrix is:

This matrix has three nonzero eigen values

In Eq. (28), the

Eq. (29) can be rewritten after some algebraic manipulations in the form of the linear combination:

In Eq. (30), we have defined

The tensor product of Eq. (30) then reduces to

The spinors

With

We find

In a true quantum system composed of three spins for instance, states can be created in the form of linear combinations like

In the case of

with

This simple example indicates the large variability in

### 2.5. Nonseparability of states in exponentially complex space

States given in Eq. (27) are tensor products on the basis

Since the Dirac equation (Eq. (15)) for the uncoupled waveguides is linear, its solutions can be a tensor product of a linear combination of

The state of the coupled waveguide system will be separable in the exponential space into the state of

A necessary condition for satisfying Eq. (38) is that

Furthermore, the first two terms in the column vector

This leads to

This condition takes the more compact form:

For this condition to be satisfied, one needs the real part of the right-hand side of the equation to be equal to zero. This can be achieved for all

We illustrate the notion of nonseparability of exponentially complex states of a coupled system composed of

The eigen values and real eigen vectors of this coupling matrix are

Following the procedure of Section 2.4, we construct a tensor product state in the

Eq. (42) is equivalent to Eq. (32) but for two coupled waveguides.

On the basis,

Similarly, we define the orthonormal basis in the Hilbert space,

In these equations, we have used

The basis in the tensor product space

We have

It is straightforward to show that

We want now to express the state

For this, we now need to expand the basis vectors

We define the expansions:

Note that the *x,* and *t*.

We can find the coefficients

We can obtain all other

The matrix

On the new basis

Then on the basis

Only in the unlikely event of degenerate eigen values,

The existence of nonseparable solutions to the nonlinear Dirac equation raises the possibility of exploiting these solutions for storing and manipulating data within the 2*N* dimensional tensor product Hilbert space. The exploration of algorithms for exploiting these solutions is beyond the scope of this chapter; however, we note that these solutions may well be observed in physical systems including elastic waveguides which are embedded in a coupling matrix. The manipulation of the system could be achieved either by externally altering the parameters of the system, i.e., the elastic properties of the waveguides, or by changing the frequency and wavenumber of input waves. These possibilities are illustrated for a five-waveguide system driven by transducers in Section 2.6.

### 2.6. Physical realization and actuation

Figure 3 illustrates a possible realization of a five waveguide system. The parallel elastic waveguides are embedded in an elastic medium which couples them elastically. The waveguides are arranged in a ring pattern.

Modes of the form given in Eq. (21) can be excited with *N* transducers attached to the input ends of the *N* waveguides and connected to *N* phase-locked signal generators to excite the appropriate eigen vectors

## 3. Conclusions

We have shown that the directional projection of elastic waves supported by a parallel array of *N* elastically coupled waveguides can be described by a nonlinear Dirac-like equation in a *N* uncoupled waveguides grounded elastically to a rigid substrate (which we called *φ*-bits). We demonstrate that we can construct tensor product states of the elastically coupled system that are nonseparable on the basis of tensor product states of *N* uncoupled *φ*-bits. A *φ*-bit exhibits superpositions of directional states that are analogous to those of a quantum spin, hence it acts as a pseudospin. Since parallel arrays of coupled waveguides span the same exponentially complex space as that of uncoupled pseudospins, the type of elastic systems described here may serve as a simulator of interacting spin networks. The possibility of tuning the elastic coefficients and the elastic coupling constants of the waveguides would allow us to explore the properties of spin networks with variable connectivity and coupling strength. The mapping between the

## Acknowledgments

We acknowledge the financial support of the W.M. Keck Foundation. We thank Saikat Guha and Zheshen Zhang for useful discussions.

## Conflict of interest

The authors declare that they have no affiliations with or involvement in any organization or entity with any financial interest or nonfinancial interest in the subject matter or materials discussed in this manuscript.