Dynamics of the Quantum Lamb Model

Horace Lamb considered the classical dynamics of a vibrating particle embedded in an elastic medium before the development of quantum theory. Lamb was interested in how the back action of the elastic waves generated can damp the vibrations of the particle. We propose a quantum version of Lamb’s model. We show that this model is exactly solvable using a multimode Bogoliubov transformation. We show that the exact system ground state is a squeezed vacuum state, and we obtain the exact Bogoliubov frequencies by numerically solving a nonlinear integral equation. The damping rate of the particle is calculated, and we find that it agrees with the result obtained by perturbation theory for coupling strengths below a critical value. The model is found to break down for coupling strengths above the critical value where the lowest Bogoliubov frequency vanishes. We show that the addition of an anharmonic elastic term is sufficient to stabilize the system in this strong coupling regime.