Lamb Mode Tunneling

• Lamb mode tunneling is a phenomenon enabling energy transfer across classically forbidden regions via evanescent, quantized modes in elastic plates and quantum circuits.
• It leverages local changes in geometry or material properties to create turning points, producing an exponential transmission described by WKB approximations and Airy-function matching.
• Experimental studies in precision-machined elastic waveguides and superconducting circuits validate the theory through controlled dispersion relations and photon-number-dependent loss mechanisms.

Lamb mode tunneling denotes the phenomenon where energy is transferred across classically forbidden (evanescent) regions in structured wave systems through the mediation of specific quantized modes—in particular Lamb modes in elastic plates, and resonant photon modes in engineered quantum circuits. This effect arises whenever a propagative mode encounters a spatial domain where it becomes evanescent due to local changes in medium geometry or material parameters, yet finite transmission persists via quantum or semiclassical tunneling mechanisms. Lamb mode tunneling has been observed in both elastic waveguides with spatially varying thickness and in superconducting microwave circuits exhibiting engineered, Fock-state-dependent Lamb shifts, unifying disparate wave-propagation scenarios under the concept of mode-mediated tunneling.

1. Fundamental Principles and Physical Mechanisms

In elastic plates, Lamb waves comprise symmetric (S₀, S₁, S₂, …) and antisymmetric (A₀, A₁, …) mode families, each with characteristic cutoff frequencies below which propagation is forbidden in a uniform-thickness medium. In a plate with linearly varying thickness,

$h(x) = h_0 + \alpha x,$

the local cutoff frequency for a given mode becomes position-dependent. If a driving frequency $\omega$ is fixed so that in a thick region $k^2(x) > 0$ (real wavenumber) but $k^2(x) < 0$ in a thinner segment (evanescent), a spatial interval of forbidden propagation emerges. In this region, exponential decay of mode amplitude is observed, yet a nonzero transmission persists and is identified as tunneling of elastic energy via evanescent Lamb modes. The physical picture is analogous to quantum tunneling across a potential barrier and is quantitatively described using WKB-type approximations for wave equations with slowly varying parameters (Charau et al., 11 Jan 2026).

In high-impedance microwave resonators coupled to superconducting tunnel junctions, similar tunneling phenomena appear via engineered system-bath interactions. The Hamiltonian

$$\hat{H}_T = \hat{B}\,e^{\,i\lambda(\hat a + \hat a^\dagger)} + \text{h.c.}$$

enables quasiparticle tunneling events to selectively absorb multiple cavity photons, with selection rules and loss rates determined by the coupling parameter $\lambda$ and the occupation number basis. These processes induce Fock-state-dependent dissipations and energy shifts (“Lamb shifts”) interpreted as Lamb mode tunneling in the circuit-QED context (Aiello et al., 2022).

2. Dispersion Relations, Turning Points, and Barrier Formation

The propagation of Lamb waves is governed by the local symmetric-mode dispersion relation:

$$\tan(qh) = \frac{4 k^2 p q}{(q^2 - k^2)^2},\quad p^2 = \frac{\omega^2}{c_L^2} - k^2, \quad q^2 = \frac{\omega^2}{c_T^2} - k^2,$$

where $c_L$ and $c_T$ are the bulk longitudinal and shear wave speeds, respectively. For a non-uniformly thick plate, $h$ is replaced by $h(x)$, and one solves for $k(x)$ at fixed $\omega$. The condition $k^2(x) = 0$ defines classical turning points $x_1$ and $x_2$, demarcating the boundaries of the evanescent barrier. The width of the forbidden region is

$$L_{\rm barrier} = x_2 - x_1 = \frac{2(h_c - h_0)}{\alpha}$$

for a symmetric crossing, where $h_c$ is the cutoff thickness at the given frequency (Charau et al., 11 Jan 2026). Within this region, the transmission amplitude decays exponentially as a function of barrier width, but tunneling persists for arbitrarily wide but finite barriers.

3. Quantitative Description: WKB Transmission and Bath Engineering

For the evanescent interval $x_1 < x < x_2$, the local wavenumber is $k(x) = i \kappa(x)$ with $\kappa(x) > 0$. The transmission coefficient is given by the WKB formalism:

$$T \approx \exp\left( -2 \int_{x_1}^{x_2} \kappa(x)\,dx \right)$$

with

$\kappa(x) = \sqrt{-k^2(x)}$

obtained from the local dispersion relation. Near turning points, Airy-function matching yields the same exponential scaling. This approach accurately describes experimental measurements of tunneling transmission as a function of barrier width in micron-precision machined metallic plates (Charau et al., 11 Jan 2026).

In superconducting quantum circuits, the photon-number-dependent loss rate associated with $l$-photon transitions is

$$\alpha_{n, l} = \frac{\lambda^{2l} e^{-\lambda^2} n!}{(n+l)!}[L_n^{(l)}(\lambda^2)]^2,$$

where $L_n^{(l)}$ is the generalized Laguerre polynomial. The corresponding Lamb shift for Fock state $|n\rangle$ is

$$\delta E_n = \hbar\,\delta\omega_n = \hbar \mathcal{P} \int_0^\infty \frac{d\omega}{\pi} \frac{\Gamma_n(\omega)}{\omega_0 - \omega},$$

with the bath spectral density $\Gamma_n(\omega)$ determined by the junction I–V characteristic and multi-photon matrix elements (Aiello et al., 2022). The selection of dissipation channels is tunable via the junction bias $V$, enabling regimes in which only $l$-photon loss is allowed, corresponding to engineered Lamb mode tunneling processes in the Fock basis.

4. Material and Device Parameter Dependencies

The precise cutoff and tunneling characteristics in elastic Lamb mode systems are highly sensitive to material parameters, notably Poisson’s ratio $\nu$. The bulk speeds

$$c_L = \sqrt{\frac{E(1-\nu)}{\rho(1+\nu)(1-2\nu)}}, \quad c_T = \sqrt{\frac{E}{2\rho(1+\nu)}}$$

depend on Young’s modulus $E$, density $\rho$, and $\nu$. Numerically, increasing $\nu$ increases $h_c$ and thus the barrier width for fixed frequency, broadening and modifying the tunneling region. For metallic plates ($\nu \approx 0.3$), the barrier width varies by several tens of percent as $\nu$ traverses the experimentally accessible range (Charau et al., 11 Jan 2026).

In high-impedance circuit-QED systems, the dimensionless coupling $\lambda = \sqrt{\pi Z_c/R_K}$ (with $Z_c$ the mode impedance and $R_K = h/e^2$ the quantum of resistance) sets the strength of displacement and thus the participation of multi-photon tunneling terms. The voltage thresholds for $l$-photon loss are governed by the superconducting gap $\Delta$ and mode frequency $\omega_0$:

$V \geq \frac{2\Delta - l \hbar\omega_0}{e}.$

This enables bias-controlled selection of the active tunneling channels and the observation of shifts in transition frequencies (δω₀₁, δω₁₂, etc.) consistent with theory.

5. Experimental Verification and Observables

In elastic waveguides, experimental configurations employ brass or steel plates with precision-machined linear tapers, excited by piezoelectric transducers in the $S_2$ mode, and detected using laser vibrometry. As the thickness sweeps below the mode cutoff, a marked drop in transmission is observed, followed by an exponentially decaying but nonzero signal indicative of tunneling. The measured transmission curves for varying taper length match WKB and finite element predictions, with the barrier width and transmission rates tunable via both geometry and material (Charau et al., 11 Jan 2026).

For quantum circuits, measurements of the mode frequency as a function of bias voltage reveal Fock-state-dependent Lamb shifts and dissipative kinks at the onset of each multi-photon channel. Experimental values for the shift of the $|0\rangle\to|1\rangle$ and $|1\rangle\to|2\rangle$ transitions (e.g., at $V=383\,\mu\text{V}$, $\delta\omega_{01}/2\pi \approx -30\,\text{MHz}$, $\delta\omega_{12}/2\pi \approx -72\,\text{MHz}$) quantitatively confirm ab initio theory (Aiello et al., 2022).

6. Dirac Cones, Mode Hybridization, and Applications

Near the “Dirac cone” regime, where two Lamb branches (such as $S_2$ and $A_3$) exhibit linear crossing and degeneracy, the barrier width between turning points vanishes and the tunneling probability reaches unity, i.e., perfect transmission. Engineering such Dirac-like geometries via corrugation or thickness patterning unlocks possibilities including unidirectional elastic waveguides, topologically protected energy localization, and robust mode conversion. These phenomena are directly relevant to nondestructive evaluation (where tunneling delays signal defects), acoustic lensing, and the realization of phononic circuitry such as elastic-wave diodes (Charau et al., 11 Jan 2026).

In quantum bath-engineered circuits, Lamb mode tunneling underpins protocols for rapid reservoir stabilization, two-photon cat-state engineering, strong Kerr nonlinearity in nominally linear resonators, and the exploration of quantum Zeno dynamics by inducing state-selective dissipation and energy shifts (Aiello et al., 2022).

7. Conceptual Significance and Broader Context

Lamb mode tunneling unifies a class of phenomena in which mode structure, local dispersion anomalies, and boundary/geometry interplay enable controlled transmission across forbidden regions. Both in mechanical and electromagnetic settings, the effect is governed by a combination of local dispersion relations, tunneling via evanescent or virtual states, and causality-enforced energy shifts encapsulated by generalized Kramers–Kronig relations. This conceptual framework extends beyond traditional quantum mechanical tunneling, highlighting the universality of mode-based, barrier-crossing transport in structured media.