Open access peer-reviewed chapter

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

By Pierre Alix Deymier, Jerome Olivier Vasseur, Keith Runge and Pierre Lucas

Submitted: February 20th 2018Reviewed: April 13th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.77237

Downloaded: 206

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 φ-bit) is isomorphic to the spin of a quantum particle. The pseudospin states of elastic waves in these systems can be described via a Dirac-like equation and possess 2 ×1 spinor amplitudes. Unlike the quantum systems, these amplitudes are, however, measurable through the measurement of transmission coefficients. The notion of measurement is an important one as it has been realized that separability is relative to the choice of the partitioning of a multipartite system. Indeed, it is known that given a multipartite physical system, whether quantum or classical, the way to subdivide it into subsystems is not unique [22, 23]. For instance, the states of a quantum system may not appear entangled relative to some decomposition but may appear entangled relative to another partitioning. The criterion for that choice may be the ability to perform observations and measurements of some degrees of freedom of the subsystems [23].

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 Ncoupled waveguides span an N-dimensional subspace. Subsequently, the Hilbert space spanned by the elastic modes is a 2N-dimensional space, comprised of the tensor product of the directional and waveguides subspaces. This representation is isomorphic to the degrees of freedom of photon states in a beam of light. While beams of light cannot be decomposed into subsystems, an elastic system composed of coupled 1D waveguides can. Indeed, the elastic system considered here forms a multipartite system composed of N1D waveguide subsystems. We show that, since each waveguide possesses two directional degrees of freedom, one can represent the elastic states of the N-waveguide system in the 2Ndimensional tensor product Hilbert space of N2D spinor subspaces associated with individual waveguides. The elastic modes in this representation obey a 2Ndimensional nonlinear Dirac-like equation. These modes span the same space as that of uncoupled waveguides grounded to a rigid substrate, i.e., Nφ-bits. However, the modes’ solutions of the nonlinear Dirac equation cannot be expressed as tensor products of the states of Nuncoupled grounded waveguides, i.e., φ-bit states.

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 2Nand leads to a description of the elastic system with exponential complexity. The linear representation enables us to operate easily on the states in the exponential space.

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:

H.IN×N+α2MN×NuN×1=0E1

Here, the propagation of elastic waves in the direction xalong the waveguides is modeled by the dynamical differential operator, H=2t2β22x2. The parameter βis proportional to the speed of sound in the medium constituting the waveguides and the parameter α2characterizes the strength of the elastic coupling between them (here, we consider that the strength is the same for all coupled waveguides). uN×1is a vector with components, ui,i=1N, representing the displacement of the ith waveguide. The coupling matrix operator MN×Ndescribes the elastic coupling between waveguides which, in the case of N = 3 parallel waveguides in a closed ring arrangement with first neighbor coupling, takes the form:

MN=3×N=3=211121112E2

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 H.IN×N+α2MN×N. In this factorization, the dynamics of the system are represented in terms of first derivatives with respect to time, t, and position along the waveguides, x. There are two possible Dirac equations:

UN×Nσxt+βUN×Niσyx±U2N×2NMN×NσxΨ2N×1=0E3

In Eq. (3), UN×Nand U2N×2Nare antidiagonal matrices with unit elements. σx=0110and σy=0ii0are two of the Pauli matrices.Ψ2N×1is a 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 MN×Nis the square root of the coupling matrix. The square root of a matrix is not unique but we will show later that we can pick any form without loss of generality.

We choose components of the Ψ2N×1vector in the form of plane waves ψI=aIeikxeiωtwith I=1,,2Nand kand ωbeing the wave number and angular frequency, respectively, Eq. (3) becomes:

ωA2N×2N+βkB2N×2N±αC2N×2Na2N×1=0E4

where

A2N×2N=IN×NI2×2E5a
B2N×2N=IN×NσzE5b
C2N×2N=MN×NσxE5c

In Eqs. (4) and (5), σz=1001is the third Pauli matrix, IN×Nis the identity matrix of order N and a2Nx1 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 U2N×2N.

Writing Eq. (4) as a linear combination of tensor products of N×Nand 2×2matrix operators:

IN×NωI2×2βkσz±αMN×Nσxa2N×1=0E6
we seek solutions in the form of tensor products:
a2N×1=EN×1s2×1.E7

While the degrees of freedom associated with EN×1span an N dimensional Hilbert subspace, the degrees of freedom associated with s2×1span a 2D space.

Replacing a2N×1from Eq. (7) in Eq. (6) yields:

IN×NEN×1ωI2×2βkσzs2×1±αMN×NEN×1σxs2×1=0.E8

Choosing EN×1to be an eigenvector, en, of the matrix MN×Nwith eigen value λnEq. (8) reduces to:

enωI2×2βkσz±αλnσxs2×1=0E9

For nontrivial eigenvectors en, the problem in the space of the directions of propagation reduces to finding solutions of

ωI2×2βkσz±αλnσxs2×1=0E10

In obtaining Eq. (9), we have also used the fact that enis an eigen vector of IN×Nwith eigen value 1 and we note that Eq. (9) is the 1D Dirac equation for an elastic system which solutions, s2×1, have the properties of Dirac spinors [18, 19, 20, 21]. The components of the spinor represent the amplitude of the elastic waves in the positive and negative directions along the waveguides, respectively.

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

ωnβk±αλn±αλnωn+βks1s2=0E11

This eigen equation gives the dispersion relation ωn2=βk2+αλn2(vide infra) and the following eigen vectors projected into the space of directions of propagation:

s2×1=s0ωn+βk±ωnβkE12

To determine the eigen vectors of MN×N, we note that they are identical to the eigen vectors of the coupling matrix MN×Nand the eigen values of MN×Nare also λn2. These properties indicate that we do not have to determine the square root of the coupling matrix to find the solutions a2N×1. All that is required is to calculate the eigen vectors and the eigen values of the coupling matrix. Hence, the nonuniqueness of MN×Ndoes not introduce difficulties in determining the elastic modes of the coupled system in the Dirac representation.

In the case of the coupling matrix, M3×3, presented in Eq. (2), the eigen values and real eigen vectors are obtained as λ02=0,λ12=λ22=3,and

e0=13111,e1=2311212ande2=2312112E13

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

Ψ2N×1nk=enNs2×1keikxeiωnktE14

In Eq. (14), we have expressed the dependencies on the wave number kand the number of waveguides N. The eigen vectors enNdepend on the connectivity of the 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 2N

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 φ-bit. The waveguides are not coupled to each other. The dynamics of the system can be described by a single equation which is constructed as follows:

σxσxσxt+σyσxσxx1+σxσyσxx2+σxσxσyx3±I2×2σxσx±σxI2×2σx±σxσxI2×2Ψ8×1=0E15

In Eq. (15), we are now defining a positional variable for each waveguide, namely, x1,x2,x3. The quantity αis a measure of the strength of the elastic coupling to the rigid substrate. The solutions are the 8 × 1 vectors Ψ8×1=Ψ1Ψ2Ψ3Ψ4Ψ5Ψ6Ψ7Ψ8. When seeking solutions in the form of tensor products of spinor solutions for the three waveguides (as indicated by the upper scripts)

Ψ8×1=ψ1ψ2ψ3=ψ11ψ21ψ12ψ22ψ13ψ23=ψ11ψ12ψ13ψ11ψ12ψ23ψ11ψ22ψ13ψ11ψ22ψ23ψ21ψ12ψ13ψ21ψ12ψ23ψ21ψ22ψ13ψ21ψ22ψ23E16
it is straightforward to show that one recovers from Eq. (15), the six Dirac equations of Eq. (3) with MN=3×N=3=I3×3. The solutions of Eq. (16) are obtained from the spinor solution for individual waveguides (j):
ψj=s0ω+βk±ωβkeikxeiωtE17

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 φ-bits span a space when dimension is 2N.

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:

σxσxσxt+σyσxσxx1+σxσyσxx2+σxσxσyx3±ε8×8Ψ8×1=0E18

The matrix αε8×8represents the coupling between the waveguides in the 2N=3space. We are still seeking solutions in the form of tensor products (Eq. (16)). After a lengthy algebraic manipulation, we find that we can reproduce Eq. (3) with the coupling matrix of Eq. (2) if one chooses εij=0excepting

ε14=ε41=ε23=ε31=2ψ11ψ12ψ13ψ11;ε16=ε61=ε25=ε52=2ψ12ψ11ψ13ψ12;ε17=ε71=ε35=ε53=2ψ13ψ11ψ12ψ13;ε28=ε82=ε46=ε64=2ψ23ψ21ψ22ψ23;ε38=ε83=ε47=ε74=2ψ22ψ21ψ23ψ22;ε58=ε85=ε67=ε76=2ψ21ψ22ψ23ψ21E19

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:

σxNt+σyσxN1x1+σxσyσxN2x2++σxN1σyxN±iαε2N×2NΨ2N×1=0E20

where nonzero components of ε2N×2Ndepend on the ψij,i=1,2;j=1,Nthat appear in the solution Ψ2N×1=ψ1ψ2ψN. The solutions of the nonlinear Dirac equation for the coupled waveguides span the same space as that of the system of φ-bits, i.e., uncoupled waveguides connected to rigid substrates. The next subsection addresses the question of separability of the coupled waveguide system into a system of uncoupled φ-bits.

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 22space can be constructed in the form of the following linear combination of tensor products:

Ψ4×1=s02ω+βk±ωβkω+βk±ωβkei2kxei2ωts02ωβk±ω+βkωβk±ω+βkei2kxei2ωtE21

Choosing s0=s0and writing Eq. (21) at the location x=0, one gets:

Ψ4×1x=0=ω+βkω+βk±ω+βkωβk±ωβkω+βkωβkωβkωβkωβk±ωβkω+βk±ω+βkωβkω+βkω+βkei2ωtE22

The bracket takes the form:

ω+βkω+βkωβkωβk1001E23

The vector 1001is not separable into a tensor product of two 2×1vectors. Considering on the basis 0=10and 1=01, one can write the state given in Eq. (22) in the form of the nonseparable Bell state:

Ψ4×1x=0=ω+βkω+βkωβkωβk0011ei2ωtE24

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):

Ψ2N×1nnkk=χnenNs2×1keikxeiωnkt+χnenNs2×1keikxeiωnktE25

The nand ncorrespond to two different nonzero eigen values, λnand λn, i.e., they correspond to two different dispersion relations ωnkand ωnk. We also choose the wave number ksuch that ωnk=ωnk=ω0. These modes are illustrated in Figure 1 in the case of an N = 9 waveguide system. χnand χnare the coefficients of the linear combination.

Figure 1.

Schematic illustration of the band structure (circular frequency in rad s−1 versus the wave number in m−1) for an array of nine elastically coupled waveguides arranged in a ring pattern. The four upper bands are doubly degenerate. We have taken β = 1 and α = 1 . The two modes with wave number k and k ′ ( n = 3 and n ′ = 2 ) have the same frequency ω 0 .

With enN=A1A2ANand enN=A1A2ANwhere the specific values of the components AIand AIare determined by the connectivity and coupling of the waveguides, the state of Eq. (25) can be rewritten as:

Ψ2N×1nnkk=χnA1ω0+βkeikx+χnA1ω0+βkeikxχnA1ω0βkeikx+χnA1ω0βkeikxχnANω0+βkeikx+χnANω0+βkeikxχnANω0βkeikx+χnANω0βkeikxeiω0t=φ11φ21φ1Nφ2NE26

Here, we have chosen, for the sake of simplicity, the +of the ±in the s2×1terms.

The first two terms in Eq. (26) form a 2×1spinor, φ1=φ11φ21, which corresponds to the first waveguide, the next two terms form a spinor φ2for the second waveguide, etc. We can then construct a solution of the nonlinear Dirac Eq. (20) in the exponential space as the tensor product:

Φ2N×1=φ1φ2φNE27

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 φjare expressed on the basis of 2×1vectors. 0=10and 1=01. If one desires to express Φ2N×1as a nonseparable state, one has to define a new basis in which this wave function cannot be expressed as a tensor product. This is done in Section 2.5. However, prior to demonstrating this, we illustrate in the next subsection how one can manipulate states of the form Φ2N×1in the exponential space.

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:

MN=3×N=3=311131113

This matrix has three nonzero eigen values λ02=1,and λ12=λ22=4corresponding to two dispersion relations ωn2=βk2+αλn2with cutoff frequencies. The second band is doubly degenerate. The eigen vectors are also given in Eq. (13). We now consider an elastic mode in the linear space that is a linear combination of these eigen modes (see Eq. (26)):

Ψ6×1nnkk=φ11φ21φ12φ22φ13φ23=χnA1ω0+βkeikx+χnA1ω0+βkeikxχnA1ω0βkeikx+χnA1ω0βkeikxχnA2ω0+βkeikx+χnA2ω0+βkeikxχnA2ω0βkeikx+χnA2ω0βkeikxχnA3ω0+βkeikx+χnA3ω0+βkeikxχnA3ω0βkeikx+χnA3ω0βkeikxeiω0tE28

In Eq. (28), the AI’s can be the components of the eigen vector e0and the AI’s can be linear combinations of the components of the eigen vectors e1and e2. We can calculate the tensor product of the spinor components in the form of Eq. (27)

Φ23×1=φ1φ2φ3E29

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

Φ23×1={ζ1ξ1ζ2ξ2ζ3ξ3+ζ1ξ1ζ2ξ2ζ3ξ3+ζ1ξ1ζ2ξ2ζ3ξ3+ζ1ξ1ζ2ξ2ζ3ξ3+ζ1ξ1ζ2ξ2ζ3ξ3+ζ1ξ1ζ2ξ2ζ3ξ3+ζ1ξ1ζ2ξ2ζ3ξ3+ζ1ξ1ζ2ξ2ζ3ξ3}ei3ω0tE30

In Eq. (30), we have defined

ζIξI=χnAIeikxω0+βkω0βk=χnAIeikxs2×1E31a
ζIξI=χnA1eikxω0+βkω0βk=χnA1eikxs2×1E31b

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

Φ23×1={χn3A1A2A3ei3kxs2×1s2×1s2×1+χn2χnA1A2A3ei2kxeikxs2×1s2×1s2×1+χn2χnA1A2A3ei2kxeikxs2×1s2×1s2×1+χn2χnA1A2A3ei2kxeikxs2×1s2×1s2×1+χn2χnA1A2A3eikxei2kxs2×1s2×1s2×1+χn2χnA1A2A3eikxei2kxs2×1s2×1s2×1+χn2χnA1A2A3eikxei2kxs2×1s2×1s2×1+χn3A1A2A3ei3kxs2×1s2×1s2×1}ei3ω0tE32

The spinors s2×1and s2×1can be expressed on the basis 0=10and 1=01:

s2×1=s10+s21E33a
s2×1=s10+s21E33b

With s1=ω0+βk, s2=ω0βk, s1=ω0+βkand s2=ω0βk. Inserting Eqs. (33a) and (33b) into Eq. (32), we can express the tensor product Φ23×1on the basis 000001010100011101110111. In defining the basis vectors for the exponential space, we have omitted the symbols . It is also implicit that the left, middle, and right elements in the tensor product abccorrespond to the first, second, and third waveguides, respectively.

We find

Φ23×1=T1000+T2001++T8111ei3ω0tE34
with
T1=Q1s1s1s1+Q2s1s1s1+Q3s1s1s1+Q4s1s1s1+Q5s1s1s1+Q6s1s1s1+Q7s1s1s1+Q8s1s1s1E35a
T2=Q1s1s1s2+Q2s1s1s2+Q3s1s1s2+Q4s1s1s2+Q5s1s1s2+Q6s1s1s2+Q7s1s1s2+Q8s1s1s2E35b
T3=Q1s1s2s1+Q2s1s2s1+Q3s1s2s1+Q4s1s2s1+Q5s1s2s1+Q6s1s2s1+Q7s1s2s1+Q8s1s2s1E35c
T4=Q1s2s1s1+Q2s2s1s1+Q3s2s1s1+Q4s2s1s1+Q5s2s1s1+Q6s2s1s1+Q7s2s1s1+Q8s2s1s1E35d
T5=Q1s1s2s2+Q2s1s2s2+Q3s1s2s2+Q4s1s2s2+Q5s1s2s2+Q6s1s2s2+Q7s1s2s2+Q8s1s2s2E35e
T6=Q1s2s1s2+Q2s2s1s2+Q3s2s1s2+Q4s2s1s2+Q5s2s1s2+Q6s2s1s2+Q7s2s1s2+Q8s2s1s2E35f
T7=Q1s2s2s1+Q2s2s2s1+Q3s2s2s1+Q4s2s2s1+Q5s2s2s1+Q6s2s2s1+Q7s2s2s1+Q8s2s2s1E35g
T8=Q1s2s2s2+Q2s2s2s2+Q3s2s2s2+Q4s2s2s2+Q5s2s2s2+Q6s2s2s2+Q7s2s2s2+Q8s2s2s2E35h
with

Q1=χn3A1A2A3ei3kx; Q2=χn2χnei2kxeikxA1A2A3; Q3=χn2χnei2kxeikxA1A2A3; Q4=χn2χnei2kxeikxA1A2A3; Q5=χn2χneikxei2kxA1A2A3; Q6=χn2χneikxei2kxA1A2A3; Q7=χn2χneikxei2kxA1A2A3; Q8=χn3A1A2A3ei3kx.

In a true quantum system composed of three spins for instance, states can be created in the form of linear combinations like m1000+m2001++m8111. For the quantum system, the linear coefficients m1, m2,…, m8are independent. The classical elastic analogue, introduced here, states which are given in Eq. (34) possesses linear coefficients T1,…, T8are interdependent. While somewhat restrictive compared to true quantum systems, the coefficients TIdepend on an extraordinary number of degrees of freedom which allows exploration of a large volume of the exponential tensor product space. In the case of the three waveguides, these degrees of freedom include (a) the components (or linear combinations) of the eigen vectors of the coupling matrix through the choice of the eigen modes or the application of a rotational operation that creates cyclic permutations of the eigen vector components, (b) the linear coefficients χnand χnused to form the multiband linear superposition of states in the linear space, (c) the frequency and therefore wave number which affect the spinor states and the phase factors eikxand eikx, and (d) a phase added to the terms eikxand eikx.

In the case of N>3,Eq. (21) can be extended to linear combinations of more than two modes with the same frequency, leading to additional freedom in the control of the TI. Furthermore, the elastic coefficients βof the waveguides and the coupling elastic coefficient αcould also be modified by using constitutive materials with tunable elastic properties via, for instance, the piezoelectric, magneto-elastic or photoelastic effects [24, 25, 26]. Also note that in all the examples we considered, the coupling of the waveguides had the same strength. Tunability of the coupling elastic medium would lead to the ability to modify the connectivity of the waveguides and therefore the coupling matrix. Exploration of the elastic modes given in Eq. (34) can be realized by varying any number of these variables. We illustrate in Figure 2 an example of operation in a very simple case. Figure 2b shows that by varying a single parameter one may achieve a wide variety of states. For instance, one can obtain states with T1>0and T2>0or T1>0and T2=0or T1>0and T2<0or T1=0and T2=0. Another interesting example occurs at χn˜0.6, there only T5and T8are different from zero. Then, Φ23×1=T50.6011+T80.6111ei3ω0tcan be written as the tensor product state T50.60+T80.6111ei3ω0t. A similar state is also obtained for χn0.25. This is the state T10.250+T40.25100ei3ω0t. Varying χncan be visualized as a matrix operator. For example, in this latter case, one can define the operation:

q1100000000000000000000000000q44000000000000000000000000000000000000T1χnT2χnT3χnT4χnT5χnT6χnT7χnT8χn=T10.25T20.25T30.25T40.25T50.25T60.25T70.25T80.25=T10.2500T40.250000E36

Figure 2.

(a) Schematic illustration of the band structure for an array of three elastically coupled waveguides arranged in a ring pattern. Each of the waveguides is also grounded elastically to a rigid substrate. The upper band is doubly degenerate. We have taken β = 1 and α = 1 . We highlight the frequency ω 0 = 2.5 corresponding to the wave numbers k = 2.292 and k ′ = 1.5 . (b) Calculated values of T 1 (open circles), T 2 and T 3 (closed circles), T 4 (open triangles), T 5 (closed triangles), T 6 and T 7 (open squares) and T 8 (closed squares) as functions of χ n ′ for χ n = 0.4 . We have fixed x = 0 .

with q11=T10.25T1χnand q44=T40.25T4χn. Another interesting state occurs at χn=0.5. Here, we have T1=T5, T2=T3, T4=T8and T6=T7. We also have T3=T5and T4=T6This state can be written as the tensor product T10.50T60.510101.

This simple example indicates the large variability in TI’s (i.e., of states) that we can achieve with a single variable. The large number of available variables will lead to even more flexibility in defining states and operators in the exponential tensor product space.

2.5. Nonseparability of states in exponentially complex space

States given in Eq. (27) are tensor products on the basis 000001010111. They are therefore always separable in that basis. Consequently these states cannot be written as nonseparable Bell states. However, we might be able to identify a basis in which Eq. (27) is not separable.

Since the Dirac equation (Eq. (15)) for the uncoupled waveguides is linear, its solutions can be a tensor product of a linear combination of Ndifferent individual spinors. For instance, we can write:

Ψ2N×1=ρ1ω1+βk1eik1x+μ1ω1βk1eik1xρ1ω1βk1eik1x+μ1ω1+βk1eik1xeiω1tρNωN+βkNeikNx+μNωNβkNeikNxρNωNβkNeikNx+μNωN+βkNeikNxeiωNtE37

ρIand μI, I=1,,Nare linear coefficients.

The state of the coupled waveguide system will be separable in the exponential space into the state of φ-bits if

Φ2N×1=Ψ2N×1E38

A necessary condition for satisfying Eq. (38) is that Nω0=ω1++ωN.

Furthermore, the first two terms in the column vector Φ2N×1of the coupled system are φ11φ12φ1Nand φ11φ12φ2N. Their ratio is simply equal to rNc=φ1Nφ2N. Similarly, the ratio of the first two terms in the vector Ψ2N×1of the φ-bit system is given by rNu=ρNωN+βkNeikNx+μNωNβkNeikNxρNωNβkNeikNx+μNωN+βkNeikNx. Anecessary condition for Eq. (38) to be satisfied is that

rNC=rNuE39

This leads to

χnANω0+βkeikx+χnANω0+βkeikxχnANω0βkeikx+χnANω0βkeikx=ρNωN+βkNeikNx+μNωNβkNeikNxρNωNβkNeikNx+μNωN+βkNeikNx
which can be rewritten as:
ueikx+ueikxveikx+veikx=γeikNx+γeikNxδeikNx+δeikNx

This condition takes the more compact form:

Peik+kNx+QeikkNx+Reik+kNx+SeikkNx=0E40

P,Q,R,Sare real. Eq. (40) is true for all values of position x. At x=0, we obtain the relation P+Q+R+S=0. Inserting that relation into Eq. (27) and eliminating Q yields:

{(R+S)coskNx[cos(kk)x1]+(RS)sin(kk)sinkNx}+i{(R+S)sin(kk)x+sinkNx[(RS)cos(kk)x+(R+S)]}

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 x’s by setting k=k. In this case, equating the imaginary parts leads to R=P. However, when kk, Eq. (39) and therefore Eq. (38) are not satisfied. When kkwhich corresponds to considering a linear combination of multiband states, Φ2N×1is not separable into the tensor product of individually uncoupled φ-bit waveguides. Therefore, we conclude that there are a large number of solutions of the nonlinear Dirac equation (Eq. (20)) representing states of arrangements of elastically coupled 1-D waveguides that are not separable in the 2Ndimensional tensor product Hilbert space of individual φ-bits.

We illustrate the notion of nonseparability of exponentially complex states of a coupled system composed of N=2waveguides on a basis in the exponential Hilbert space of two individual φ-bits. The waveguides are coupled to each other but also to a rigid substrate such that the coupling matrix, MN×N, takes the form:

M2×2=2112

The eigen values and real eigen vectors of this coupling matrix are λ02=1,and λ12=3and

e0=A1A2=1211,e1=A1A2=1211E41

Following the procedure of Section 2.4, we construct a tensor product state in the 22exponential space:

Φ22×1={χn2A1A2ei2kxs2×1s2×1+χnχnA1A2eikxeikxs2×1s2×1+χnχnA1A2eikxeikxs2×1s2×1+χn2A1A2ei2kxs2×1s2×1}ei2ω0tE42

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

On the basis, η1=ei2ω0tei2kxs2×1s2×1, η2=ei2ω0teikxeikxs2×1s2×1, η3=ei2ω0teikxeikxs2×1s2×1, and η4=ei2ω0tei2kxs2×1s2×1, Eq. (42) can be rewritten as:

Φ22×1=a11η1+a12η2+a21η3+a22η4E43
with a11=χn2A1A2=12χn2, a12=χnχnA1A2=12χnχn, a21=χnχnA1A2=12χnχn, and a22=χn2A1A2=12χn2. It is then easy to demonstrate that deta11a12a21a22=12χn212χnχn12χnχn12χn2=0, which indicates that the state Φ22×1is separable on the basis η1η2η3η4. At this stage, there is nothing surprising as the state Φ22×1was constructed as a tensor product. We now try to express the state given in Eq. (42) on a basis of two individually uncoupled φ-bits. Considering the Hilbert space of the first φ-bit, H1, we use the spinor solutions for uncoupled waveguides given in Eq. (17) to construct the orthonormal basis
ψ11=12ω1ω1+β1k1ω1β1k1eik1xeiω1t=s11k1eik1xeiω1tE44a
ψ21=12ω1ω1β1k1ω1+β1k1eik1xeiω1t=s21k1eik1xeiω1tE44b

Similarly, we define the orthonormal basis in the Hilbert space, H2,of the second φ-bit,

ψ12=12ω2ω2+β2k2ω2β2k2eik2xeiω2t=s12k2eik2xeiω2tE45a
ψ22=12ω2ω2β2k2ω2+β2k2eik2xeiω2t=s22k2eik2xeiω2tE45b

In these equations, we have used s11k1, s21k1, s12k2, and s22k2as short-hands for the spinor parts of the basis functions.

The basis in the tensor product space H1H2is given by the four functions:

τ1=ψ11ψ12,τ2=ψ11ψ22,τ3=ψ21ψ12,τ4=ψ21ψ22E46

We have

τ1=s11k1s12k2eik1+k2xeiω1+ω2tE47a
τ2=s11k1s22k2eik1k2xeiω1+ω2tE47b
τ3=s21k1s12k2eik1+k2xeiω1+ω2tE47c
τ4=s21k1s22k2eik1k2xeiω1+ω2tE47d

It is straightforward to show that τ1τ2τ3τ4form an orthogonal basis. That is, τiτj=0if ij, where τiis the Hermitian conjugate of τi.

We want now to express the state Φ22×1in the τbasis:

Φ22×1=b11τ1+b12τ2+b21τ3+b22τ4E48

For this, we now need to expand the basis vectors η1η2η3η4on the basis τ1τ2τ3τ4

We define the expansions:

η1=c11τ1+c12τ2+c13τ3+c14τ4E49a
η2=c21τ1+c22τ2+c23τ3+c24τ4E49b
η3=c31τ1+c32τ2+c33τ3+c34τ4E49c
η4=c41τ1+c42τ2+c43τ3+c44τ4E49d

Note that the cij’s are functions of k1,k2, x, and t.

We can find the coefficients cijby exploiting the orthogonality of the τis. For instance, we can multiply Eq. (49a) to the left by the Hermitian conjugate τ1, leading to

τ1η1=c11τ1τ1+c12τ1τ2+c13τ1τ3+c14τ1τ4=c11τ1τ1E50
or
c11k1k2xt=s11k1s12k2s2×1s2×1eik1+k2xei2kxeiω1+ω2tei2ω0t/s11k1s12k2s11k1s12k2E51

We can obtain all other cij’s in a similar fashion. Eqs. (49a)–(49d) can be rewritten in the form:

η1η2η3η1=η4×1=c11c12c13c14c21c22c23c24c31c32c33c34c41c42c43c44τ1τ2τ3τ4=C4×4τ4×1E52

The matrix C4×4can be diagonalized. Let d1, d2, d3, and d4be the four eigen values of C4×4with their associated eigen vectors v11v12v13v14, v21v22v23v24, v31v32v33v34and v41v42v43v44. We can construct the following matrix out of the four eigen vectors:

V4×4=v11v21v31v41v12v22v32v42v13v23v33v43v14v24v34v44

On the new basis τ˜1τ˜2τ˜3τ˜4constructed by using the relation τ˜4×1=V4×41τ4×1V4×4, the matrix that couples the ηbasis and the τbasis takes the form: C˜4×4=d10000d20000d30000d4so η1=d1τ˜1, η2=d2τ˜2, η3=d3τ˜3and η4=d4τ˜4. On the τ˜basis, Eq. (43) can be rewritten as

Φ22×1=a11d1τ˜1+a12d2τ˜2+a21d3τ˜3+a22d4τ˜4E53

Then on the basis τ˜, we can investigate the separability or nonseparability of Φ22×1by calculating the determinant of the linear coefficients in Eq. (53), that is

deta11d1a12d2a21d3a22d4=12χn2d112χnχnd212χnχnd312χn2d4=14χn2χn2d1d4d2d3E54

Only in the unlikely event of degenerate eigen values, d1, d2, d3, and d4, would this determinant be equal to zero. A nonzero determinant given in Eq. (54) indicates that the state Φ22×1is nonseparable on the basis τ˜1τ˜2τ˜3τ˜4.

The existence of nonseparable solutions to the nonlinear Dirac equation raises the possibility of exploiting these solutions for storing and manipulating data within the 2N 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.

Figure 3.

Schematic illustration of a five waveguide system. The waveguides are composed of an elastic medium 1 in which mass density and elastic stiffness determine the physical parameter β . The waveguides are embedded in an elastic medium 2 in which mass density and stiffness relate to the parameter α . The waveguides are actuated via transducers (see the text for details).

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 enand en. These modes can be excited by applying a superposition of signals on the transducers with the appropriate phase, amplitude and frequency relations. The frequencies ωnkand ωnkare used to control the spinor parts of the wave functions s2×1kand s2×1k. The spinor components which represent a quasistanding wave can be quantified by measuring the transmission coefficient (normalized transmitted amplitude) along any one of the waveguides. It is then possible to operate on the eigen vectors enand enwithout affecting the spinor states or vice versa. For instance, one could apply a rotation that permutes cyclically the components of enby changing the phase of the signal generators. Such an operation could be quantified by measuring the phase of the transmission amplitude at the output end of the waveguides.

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 2Ndimensional 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 (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 2Ndimensional and the 2Ndimensional representations of the elastic system leads to the capacity for exploring an exponentially scaling space by handling a linearly growing number of waveguides (i.e., preparation, manipulation, and measurement of these states). The scalability of the elastic system, the coherence of elastic waves at room temperature, and the ability to measure classical superpositions of states may offer an attractive way for addressing exponentially complex problem through the analogy with quantum systems.

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.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Pierre Alix Deymier, Jerome Olivier Vasseur, Keith Runge and Pierre Lucas (November 5th 2018). Separability and Nonseparability of Elastic States in Arrays of One-Dimensional Elastic Waveguides, Phonons in Low Dimensional Structures, Vasilios N. Stavrou, IntechOpen, DOI: 10.5772/intechopen.77237. Available from:

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