Path-Optimized Fast Quasi-Adiabatic Driving in Coupled Elastic Waveguides

TL;DR

The paper extends shortcuts to adiabaticity by incorporating path optimization in a two-parameter space $(\kappa,\delta)$ alongside velocity scheduling to realize FAQUAD in elastic-waveguide systems. By extracting a $2\times2$ effective Hamiltonian from transfer-matrix data, it maps the band structure and identifies a method to minimize the adiabaticity parameter $A$ along both the path and velocity profile. Through analytic modeling, full-wave simulations, and experimental demonstrations using scanning laser Doppler vibrometry, the authors show that optimized paths enable complete energy transfer between coupled waveguides within compact device lengths, staying on the target upper band $k_2$ with $\langle B\rangle\approx+1$. The elastic-wave platform thus provides a direct, visualizable classical analogue for multidimensional STA control and paves the way for compact metamaterial devices implementing STA protocols.

Abstract

Fast quasi-adiabatic driving (FAQUAD) is a central technique in shortcuts to adiabaticity (STA), enabling accelerated adiabatic evolution by optimizing the rate of change of a single control parameter. However, many realistic systems are governed by multiple coupled parameters, where the adiabatic condition depends not only on the local rate of change but also on the path through parameter space. Here, we introduce an enhanced FAQUAD framework that incorporates path optimization in addition to conventional velocity optimization, extending STA control to two-dimensional parameter spaces. We implement this concept in a coupled elastic-waveguide system, where the synthetic parameters-detuning and coupling-are controlled by the thicknesses of the waveguides and connecting bridges. Using scanning laser Doppler vibrometry, we directly map the flexural-wave field and observe adiabatic energy transfer along the optimized path in parameter space. This elastic-wave platform provides a versatile classical analogue for exploring multidimensional adiabatic control, demonstrating efficient and compact implementation of shortcut-to-adiabaticity protocols.

Figures (4)

• Figure 1: Two-waveguide elastic system and band structure. (a) Schematic of the two-waveguide system connected by elastic bridges. (b) Unit cell of the elastic system, where $h_1$ and $h_2$ are used to adjust the difference in the propagation constants between the two waveguides. (c) Procedure for extracting the transfer matrix $T$, with fields evaluated on both sides of the central unit (indicated by the red planes). (d) Components of the effective $2\times2$ Hamiltonian $H_{ij}$ as functions of the coupling parameter $\kappa$ for several detuning values $\delta$, demonstrating the dependence of mode coupling on geometric and material variations. (e) Band structures in the two-dimensional parameter space $(\kappa,\delta)$, with orange dots denoting experimentally verified bands. Modal parity parameter $S$ is plotted as color map on the surfaces. In panels (b) and (d), $\kappa$ and $\delta$ refer to geometric tuning parameters (bridge width and thickness difference), whereas in (e) they represent the effective coupling and detuning extracted from the transfer matrix. The elastic parameters used in the COMSOL simulations are $f=3\,\mathrm{kHz}$, $\omega=2\pi f$, $a=b=w=10\,\mathrm{mm}$, cross-sectional area $A=bh$, second moment-of-area $I=bh^3/12$, Young's modulus $E=3.4\,\mathrm{GPa}$, density $\rho=1190\,\mathrm{kg/m^3}$, shear modulus $\mu=0.35$, and longitudinal sound-speed $c=\sqrt{E/(12\rho)}$. Quantities with subscript zero correspond to the background double-waveguide (without bridges), where $h_0=4\,\mathrm{mm}$ and all other parameters remain unchanged.
• Figure 2: Optimized paths in the parameter space. (a,b) Paths in the $(\kappa,\delta)$ parameter space for the two designed paths. The black curves denote paths obtained from the analytic model for paths 1 and 2, respectively. The green curves correspond to the same paths executed over a much longer waveguide length (601 instead of 41 unit cells), representing the quasi-adiabatic limit. The colored surfaces show the band structure in $(\kappa,\delta)$, where the propagation constant $k$ acts as the eigenvalue analogue of energy in a time-dependent system. The color indicates the modal parity parameter $S$, quantifying the composition of the two hybridized modes along the path.
• Figure 3: Comparison between unoptimized and optimized paths. (a,c) Vector field of $\vec{E}$ in the $(\kappa,\delta)$ parameter space, where color denotes the field intensity $|\vec{E}|$ and orange arrows indicate its local direction. The black dots represent the discrete path along which the system evolves: (a) a straight, constant-speed path (path 1) and (c) an optimized path with spatially varying speed (path 2).(b,d) Corresponding adiabaticity parameters in discrete form for the two paths, showing the local degree of adiabatic following between adjacent unit cells. The discrete adiabaticity parameter is obtained from Eq. \ref{['eq:adiabaticity']} by replacing $\dot{\ket{\psi_1}}$ with $(\ket{\psi_1(i+1)}-\ket{\psi_1(i)})/a$, where $a$ is the unit-cell length.
• Figure 4: Experimental validation of the optimized and unoptimized paths. (a) Photograph of the experimental setup. The upper (lower) beam corresponds to WG 1 (2). Blue tack is attached at both ends to serve as a perfect matching layer for suppressing reflections. A piezoelectric transducer is coupled to WG 1 to generate the prescribed input excitation. The structure contains 41 unit cells, with unit 1 located on the left and unit 41 on the right. The inset provides a side view of the fabricated sample. (b,c) Field distributions obtained from simulations and experiments for the paths shown in Fig. 2(a) and Fig. 2(b), respectively. The color maps represent the measured displacement fields, while the curves show the normalized real parts of the wave amplitudes $W_1$ and $W_2$ for the two coupled waveguides, comparing simulation (black and red) and experiment (blue and orange). (d,e) Corresponding band indices $\langle B \rangle$ extracted from simulations (black dots) and experiments (red dots) for the two paths, showing consistent evolution of modal character along the propagation direction.