Elastic Hamiltonians for quantum analog applications

Elastic waves are complex mixtures of transverse and longitudinal oscillations even in isotropic and homogeneous media, in contrast to the quantum, electromagnetic, or acoustic waves which could share the same formalism of Hamiltonian and application techniques. Here, we reformulate the elastic wave equation into a set of polarization-dependent decoupled Hamiltonians, to enable the quantum analogous techniques for higher functionalities. As an application example, we adopt the supersymmetric transformation from particle physics and apply it to elastic Hamiltonians, for the demonstration of spatial- and polarization-selective separation of guided elastic waves. Enabling the application of quantum-analogous techniques under the established elastic Hamiltonian formulation, our approach provides a pathway for controlling elastic waves, not limited to the control of an individual guided mode for arbitrary elastic waves, demonstrated here with supersymmetric technique.

Figure 1

Quantum-analogous potential description of elastic materials and application of the SUSY transformations. (a) Inhomogeneous elastic media defined by three wave parameters (ρ, λ, and μ) and their elastic potential landscape representation ($V^{\mathrm{L}}, V^{\mathrm{S}}$) each for L and S polarizations. (b) Illustration of SUSY-transformed elastic potentials and their successive ground-state annihilations for each L- and S-polarization axis. These SUSY transformations on $V^{\mathrm{L}}$ and $V^{\mathrm{S}}$ are independent of each other.

Figure 2

Polarization-selective SUSY ladders for elastic waves. (a) Longitudinal- and (b) shear-potential landscapes. (a, b) Black lines for SUSY ladder potentials $V^{\mathrm{L}\left(\mathrm{\Sigma }\right)}$ and $V^{\mathrm{S}\left(\mathrm{\Sigma }\right)}$. Regions in color are individual SUSY partner potentials $V^{\mathrm{L}\left(p\right)}$ and $V^{\mathrm{S}\left(q\right)}$ composing SUSY ladders. Here, potentials are normalized to the negative square of propagation constants of a plane wave. Eigenstates of each $V^{\mathrm{L}\left(p\right)}$ and $V^{\mathrm{S}\left(q\right)}$ are overlaid for lowest, first higher, and second higher modes, in red, green, and blue colors, respectively. (c) Spatial profiles of elastic wave parameters (ρ, λ, and μ) calculated from elastic SUSY ladder potentials $V^{\mathrm{L},\mathrm{S}\left(\mathrm{\Sigma }\right)}$ designed in (a) and (b). To reduce the design freedom, ρ($x$) was set to a constant ${\rho }_0$. (d–i) Wave propagations through elastic SUSY ladders for the input of different eigenmodes ${{\psi }_i}^{\mathrm{L},\mathrm{S}\left(0\right)}$ to the input waveguide of $V^{\mathrm{L},\mathrm{S}\left(0\right)}$ (position centered at $x=-0.2$). Full coupling is achieved between phase-matched eigenstates: (d, g) the ground state ${{\psi }_0}^{\mathrm{L},\mathrm{S}\left(0\right)}$ trapped in the input waveguide of $V^{\mathrm{L},\mathrm{S}\left(0\right)}$, (e, h) the first excited state ${{\psi }_1}^{\mathrm{L},\mathrm{S}\left(0\right)}$ of $V^{\mathrm{L},\mathrm{S}\left(0\right)}$ penetrating into the adjacent waveguide of first SUSY-partner potential $V^{\mathrm{L},\mathrm{S}\left(1\right)}$ of ${{\psi }_0}^{\mathrm{L},\mathrm{S}\left(1\right)}$, and (f, i) the second excited state ${{\psi }_2}^{\mathrm{L},\mathrm{S}\left(0\right)}$ penetrating into waveguides of first and second SUSY-partner potentials $V^{\mathrm{L},\mathrm{S}\left(1\right)}$ and $V^{\mathrm{L},\mathrm{S}\left(2\right)}$. It is noted that SUSY potentials and eigenmodes here are obtained by using finite difference method with MATLAB, and (ρ, λ, and μ) values in (c) have been normalized to that of aluminum with a plane strain approximation [35] at the target frequency of 200 kHz.

Figure 3

Elastic metamaterial template for elastic potentials. (a) An illustration of the employed elastic metamaterial, consisting of a square lattice unit cell with a rectangular hole. The lattice constant $a=1$ mm satisfies the subwavelength conditions of $\sim {\mathrm{\Lambda }}_{\mathrm{L}}/30$ and $\sim {\mathrm{\Lambda }}_{\mathrm{S}}/15$, where ${\mathrm{\Lambda }}_{\mathrm{L}}$ and ${\mathrm{\Lambda }}_{\mathrm{S}}$ denote the wavelength of the L and S polarization, respectively, in the Al plate at a 200-kHz frequency. (b–e) FEM (COMSOL) obtained elastic parameters ${\rho }_{\mathrm{L}}, {\rho }_{\mathrm{S}}$, λ, and μ for different hole widths $d_1$ and $d_2$. (f, g) Elastic potentials $V^{\mathrm{L}}$ and $V^{\mathrm{S}}$ corresponding to ${\rho }_{\mathrm{L}}, {\rho }_{\mathrm{S}}$, λ, and μ in (b)–(e).

Figure 4

Design and operation of the polarization-diversified SUSY ladder. (a) The landscapes of elastic potentials for the polarization-diversified SUSY ladders. (b) The structural parameters of elastic metamaterials corresponding to the potentials in (a). (c) Metamaterial structures in the vicinity of each waveguide center. (d) Snapshots of mode splitting for a randomly mixed input of spatial and polarization modes at $t=0.5$, 2.0, and 3.5 ms. The input of randomly mixed spatial and polarization modes are fed into the input waveguide around $x=0$, at $t=0\mathrm{ms}$. For the time domain simulation, the eigenmode expansion method has been used based on FEM (COMSOL Multiphysics) solid mechanics simulation results. (e) The steady-state finite difference method (MATLAB) solution obtained from the analytical scalar elastic Hamiltonians, using the potentials shown in (a).