Exponentially complex nonseparable states in planar arrays of nonlinearly coupled one-dimensional elastic waveguides

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.


Introduction
Superposition of states and entanglement are at the core of today's second quantum revolution [1]. Entangled superpositions of states possess two distinct attributes, namely nonlocality and nonseparability. Nonlocality is a unique feature of quantum mechanics, but nonseparability is not. The notion of 'classical entanglement', i.e., local nonseparable superposition of states has received a significant amount of attention, theoretically and experimentally, in the area of optics [2][3][4]. For instance, degrees of freedom of photon states that span different Hilbert spaces have been locally correlated. Laser beams with spin angular momentum and orbital angular momentum (OAM) can be prepared in a nonseparable state [5][6][7][8][9][10][11][12]. Nonseparability of OAM, polarization and radial degrees of freedom of a beam of light has also been reported [13]. Photonic schemes have also been employed to correlate polarization with propagation direction [14,15]. The notion of classical nonseparability has also been extended beyond the optical field to neutron beams [16]. However, to date, much less attention has been paid to elastic waves; yet, remarkable new behaviors of sound, analogous to quantum physics, such as the notions of elastic pseudospin [17][18][19][20][21][22] and Zak/Berry phase [23][24][25][26][27][28][29], are emerging. Recently, elastic nonseparable states, analogous to 'classically entangled' states were observed experimentally in externally driven systems composed of parallel arrays of one-dimensional elastic waveguides coupled elastically along their length [30]. These classically nonseparable states are Bell states constructed as a superposition of elastic waves, each a product of a plane wave part spanning the length of the waveguides and a spatial part spanning the direction across the array of waveguides. We demonstrated that the amplitude coefficients of the nonseparable superposition of states are complex due to dissipation in the constitutive elastic materials. By tuning these complex amplitudes, we have shown that we can experimentally navigate a sizeable portion of the Bell state's Hilbert space. These states lie in the tensor product Hilbert space of the subspaces associated with the degrees of freedom along and across the waveguide array. The dimension of this product space scales linearly with the number of waveguides as 2N where N is the number of waveguides. In order to achieve the full potential of the second quantum revolution, it is highly desirable to construct classical nonseparable states that lie in an exponentially complex Hilbert space. More specifically, one would like to design a multipartite elastic system composed of Q subsystems or degrees of freedom, each of which able to be in at least two states. The dimension of the Hilbert space of such a system would then take the value 2 Q which scales exponentially with the number Q. We have recently shown such a behavior in a system composed of one-dimensional elastic waveguides which were coupled via quadratic nonlinear forces along their length [31]. The elastic nonlinearity by enabling wavewave interaction was necessary to achieve wave mixing and therefore to allow the formation of waves which frequency and wave number are the sum of the frequencies and wave numbers of parent linear waves. These nonlinear waves were product waves of the parent waves and therefore spanned a 2 2 dimensional Hilbert space which was the product of the spaces supporting the parent waves (each one two-dimensional).
In the present paper, we generalize this approach to planar array of  N 3 nonlinearly coupled waveguides with nonlinear exponent < Q N driven externally by external harmonic forces. We show that to first order in perturbation, if we excite Q of the N possible bands and two plane wave states in each band, the nonlinear elastic system can be visualized as a multipartite two level system which can support superpositions of nonlinear modes which span exponentially complex Hilbert spaces. By partitioning the spatial degrees of freedom across the planar array, i.e., separating the system into two subsystems, we implement an approach based on the density matrix of the subsystems to calculate the entropy of entanglement. This approach is extended to all values of N and Q by the use of singular value decompositions which leads to the possibility of achieving maximally entangled exponentially complex superpositions of nonlinear product states.
The paper is organized as follows. In section 2 we introduce a formalism that enables us to determine the acoustic field of any externally driven planar array of one-dimensional waveguides coupled inelastically along their length. The nonlinear coupling can be any power, Q, of the displacement as long as it is smaller than the number of waveguides. This formalism is based on first order perturbation theory. Section 3 presents three examples with = N 3, 4, 5 and = -Q N 1. We present the nonlinear component of the displacement field as a superposition of product of plane waves. Section 4 introduces the method for the calculation of the entanglement entropy which is then applied to the example systems of section 3. Finally, we draw some conclusions as to the applicability of driven planar arrays of nonlinearly coupled one-dimensional elastic waveguides supporting exponentially complex nonseparable states as quantum analogues in the area of quantum information science.

Planar array of N parallel externally driven waveguides coupled elastically along their length
We consider N one-dimensional elastic waveguides coupled elastically along their length. The system is driven externally at some position = x 0 (figure 1). In the long wavelength limit the wave equation takes the form:  I is the identity matrix. a represents the coupling strength between waveguides with  M being the matrix describing the elastic coupling between the N waveguides.  F represents the driving force. In the case of a planar array of waveguides, the coupling matrix takes the form: represents the displacement in waveguides 1 through N, respectively. In equation (1), w d is the frequency of the driver.
We now define l n and  E n with n=1, 2,K,N, the eigen values and eigen vectors of the matrix  M. The  E n represent the spatial eigen modes across the waveguides with components E , n j , with the jth rod in the nth mode. We can write: Since the  E n form a complete orthonormal basis, we can write the displacement vector as: The N×1 vector,  F , is also expressed in the  E n basis: The F n 's are therefore defined as the dot product   F E . n between the twoŃ 1 vectors. Inserting equations (2)-(4) in equation (1) leads to a set of N equations of the form: The complete displacement field is therefore obtained as with the complex resonant amplitudes given by equation (7). In equations (6) and (9), the summation over the wave numbers, k , n is discrete, thus implying that the waveguides have the same finite length. For a given driving frequency, we now limit the summation over the wave numbers to the two terms which contribute the most to the displacement field. Let k n and ¢ k n be the wave numbers for which w w - We can now approximate equation (9) by: We note that the number of terms in the sum, S , n can be reduced by nullifying some of the complex amplitudes (i.e. choosing some of the F n to be zero).

Planar array of N parallel externally driven waveguides coupled inelastically along their length
We introduce a nonlinear coupling term in equation (1): We assume that the exponent of the nonlinear terms is an integer, ->  N Q 1 2 . We solve equation (11) within perturbation theory. Assuming a small e, we expand the displacement field to first order in perturbation: Inserting equation (13) into equation (11) yields to zeroth order in perturbation: (14) is nothing but the linear equation (1) which solutions, ( )  U , 0 are given by equation (10).  To first order in e we obtain: ⎡ (15), the zeroth order displacement drives the system.
Let us consider the nonlinear terms of the form: In equation (16) we have chosen the driving force  F such that -N Q components, F , n are zero (and therefore the corresponding complex amplitudes are also zero). This choice is highlighted in the summation S n with a ' * '. This enables us to limit the summation over modes to the same number as the exponent of the nonlinear coupling term.
We now seek one particular solution of equation (15) by singling out in the power expansion of equation (16) That is, we define that term as: The driving corresponding to this singled out term: Again since the  E n 's form a complete orthonormal basis, it is possible to expand ( )  g 0 on that basis. For this we write:˜( To first order in perturbation, a particular solution, corresponding to the conditions above, of the nonlinear equation is solution of: And inserting equation (22) into equation (21) and using equation (2), equating the coefficients for each eigenvector,  E , n results in the set of N equations: ⎛ To solve equation (23), we rewrite the product on the right hand side of the equal sign into the sum: where Σʹ sums over all combinations of primed and unprimed k m 's of the allowed Q wave numbers. Here, The complex amplitudes also take two possible values (primed and unprimed) for each of the Q allowable states. There are 2 Q terms in the summation of equation (24). We simplify the notation by reformulating the summation as å j = A . In equation (26), the denominator is expressed in the form: The combination of indices ¼ Q 1, 2, , is that that correspond to the overall index l with K l being the lth term in the sum of the left-hand side of equation (24).
In summary, the first order particular solution for the displacement is with all conventions introduced above:

Examples
The displacement field given by equation (29)  does not involve transfer of energy between the waveguides via the coupling. This trivial case is equivalent to three independent waveguides. The other two eigen modes of the coupling matrix with eigen values l = 1, 2 and l = 3, We now suppose that we are driving the system with = F 0 1 (to eliminate the  E 1 mode) and ¹ ¹ F F 0. 2 3 The vector ( )  g 0 has three components given by equations (19a)-(19c), namely: According to equation (2), we therefore have˜( ) The summation S n in equation (29) has, therefore, only one non-zero term corresponding to n=2. The summation S = l 2 Q 2 has four terms. The four denominators for = l 1, 2, 3, 4, respectively. Here, the four numerators, A , l and plane wave terms, j , l are those of equation (25) and the ( ) A k n n are given by equation (7). The states of the system span a four dimensional Hilbert space product of two two dimensional Hilbert spaces.
For the sake of illustration, we will write explicitly only for that case all the terms contributing to the displacement field: Depending on the value of the complex coefficients of the amplitude of this superposition of states in the product space of plane waves, the superposition may be separable or nonseparable. It is nonseparable if it cannot be written in the form: Here, the subscript amp refers to the amplitude part of equation (33). The degree of separability of the amplitude in equation (33) will be quantified in section 4 by calculating its entropy of 'entanglement. are: We now suppose that we are driving the system with = F 0 1 (to eliminate the  E 1 mode) and The vector ( )  g 0 has four components given by equations (19a)-(19c), namely: According to equation (2), we therefore have for this particular case˜( ) 0. The summation S n in equation (29) has, therefore, only one non-zero term corresponding to n=3. The summation S has eight terms. The eight denominators  Here, the particular first order states span an eight dimensional Hilbert space. This space is the product of three two dimensional Hilbert spaces. are: We now suppose that we are driving the system with = F 0 1 (to eliminate the  E 1 mode) and The vector ( )  g 0 has five components given by equations (19a)-(19c), namely: In that case the particular solution for the displacement field becomes:  Let us now focus on the complex amplitude for one of the waveguides, say for example, the first waveguide, C .
l,1 These amplitudes can be written in the form of a second rank tensor. ⎛ Note that, by construction, the amplitude tensor  A 1 separates the k 2 and ¢ k 2 containing amplitudes by row, top and bottom, and the k 3 and ¢ k 3 containing amplitudes by column, top and bottom. The classical entanglement for this system is seen in the two dimensional Hilbert space of the product states represented by these amplitudes, hence we calculate the entanglement entropy from the amplitudes by forming the product † A A 1 1 which is proportional to the density matrix. The product matrix is diagonalized using the transformation matrix T to give: ⎛ Recalling that the product matrix is proportional to the density matrix, the entropy of entanglement is calculated as the von Neumann entropy with a normalization constraint: The entropy of entanglement can be rewritten in the form: In equation (48),

A A A A
Det Tr For a given finite length of the waveguides (i.e., discrete set of wavenumbers), it might be possible to choose the driving frequency, w , d such that the denominators of the diagonal terms are significantly smaller than those of the off diagonal terms. Let us also assume that the driving frequency is such that the numerators of the diagonal terms are nearly the same. In that case, if we neglect the off diagonal terms compared to the diagonal ones, then ( ( )) ( ) † †

 A A A A Tr
Det 4 1 1 2 1 1 and the ratio  R 1. Under these conditions, = S ln 2, and the system is maximally entangled. This system is equivalent to a maximally entangled Bell state. On the other hand, if the four denominator were the same (in the current situation this is highly improbable unless the wave numbers are the same), i.e., = R 0 and subsequently = S 0. The superposition would be separable. Extending this discussion to the N=4 and Q=3 system, recall the complex coefficients in the superposition of product states: The coefficients can be rearranged in the form of an unfolding tensor The amplitude matrix for this system is 4×2 and is ordered such that the 2×2 matrix on the left has amplitudes including ¢ ¢ k k k k , , 3, 3 as in the previous system and amplitudes that include k , 4 while the 2×2 matrix on the right has amplitudes that include ¢ k , 4 see equation (38). This construction can be seen as a bipartite system which spans an 8 dimensional Hilbert space tensor product of a two-dimensional Hilbert and a 4 dimensional Hilbert space. We employ the singular value decomposition (SVD) to factorize the2 4 complex matrix, A 1 into the product of a2 2complex unitary matrix, X, a2 4 rectangular diagonal matrix, S, with singular real positive values on the diagonal and á 4 4complex unitary matrix, † V , where the dagger refers to the conjugate transpose matrix. The SVD is written as: The rectangular diagonal matrix, ̲ S, then is obtained as: Note that the matrix † A A 1 1 is a 2×2 matrix and the analysis of the entanglement entropy follows the development for the previous system.
The case of the = N 5 and = Q 4, the amplitude matrix is 2×8 and can be constructed by replicating the 2×4 matrix of the previous system i.e., N =4 and Q=3 on left and right, segregating terms in the left and right 2×4 matrices as before while writing amplitudes that depend on k 5 in the right 2×4 matrix and those depending on ¢ k 5 in the left. SVD can be used to generate the entropy of entanglement as in equations (52) and (53) for any planar array of N nonlinearly coupled waveguides with nonlinear exponent < Q N.

Conclusions
We have shown that we can achieve exponentially complex superpositions of product states in externally driven planar arrays of N one-dimensional elastic waveguides coupled nonlinearly along their length. The nonlinear coupling is a power function of the relative displacements between waveguides with exponent Q. Such systems can be realized experimentally by binding near one-dimensional elastic rods with intrinsically nonlinear elastic media to form the parallel arrays of waveguides. Other approach may use granular materials as coupling agent between the waveguides/rods to exploit Hertzian contact as a source of nonlinearity.
To first order in perturbation, if we excite Q of the N possible elastic bands and two plane wave states in each band, the nonlinear elastic system can be visualized as a multipartite two level system which can support superpositions of nonlinear modes which span exponentially complex Hilbert spaces of dimension 2 .
Q By partitioning the spatial degrees of freedom across the planar array, i.e., separating the system into two subsystems, we introduce and implement an approach using singular value decompositions to calculate the entropy of entanglement from the density matrix. We show the possibility of achieving maximally entangled exponentially complex superpositions of nonlinear product states. The exponentially complex Hilbert space can be navigated by varying the driving frequency and the amplitude of the drivers. The nonseparable superpositions of product states can be measured experimentally using contact and/or noncontact probes such as arrays of transducers to determine the relative amplitude and phase of the displacement field in the waveguides to get access to the  E n states. Laser Doppler vibrometry can be used to scan the elastic field along the waveguides. By spatial Fourier transformation of the displacement field along the waveguides measured via noncontact laser vibrometry, we can access the exponentially complex wave number space of K.
This work expands the capability of arrays of nonlinearly coupled elastic waveguides to produce quantum like phenomena. Nonseparable superpositions of elastic product states are not subject to the instability due to decoherence of true quantum systems. Recently, the use of classical light with entangled degrees of freedom have found applications in quantum information [32] and metrology [33,34]. The present study suggests that the same sort of applications can be realized with acoustic systems. Moreover, due to the flexibility of elastic system, the coupling between elastic waveguides can easily be tailored to be linear, periodic, and nonlinear (with different types or degrees of nonlinearity) [35,36], which in turn would widen the Hilbert space accessible to elastic superposition of states. This coupling can be easily manipulated by choices of materials and fabrication. Finally, externally driven arrays of nonlinearly coupled elastic waveguides, with their ability to navigate a large section of an exponentially complex Hilbert space of product of plane wave states, may therefore offer a stable classical alternative to quantum bits for massive information storage or processing. ORCID iDs P A Deymier https:/ /orcid.org/0000-0002-1088-7958 K Runge https:/ /orcid.org/0000-0001-6467-7481 M Arif Hasan https:/ /orcid.org/0000-0001-7452-004X