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. The quantum theory of this model is relevant to a variety of mesoscopic and nanoscopic mechanical systems currently under experimental study. We propose the quantum version of the Lamb model, and we find that this model is exactly solvable. We find that the exact system ground state is a squeezed vacuum state, and we obtain the exact eigenenergies numerically from a nonlinear integral equation. We then calculate the exact damping rate of the particle, and we find that it agrees with the result obtained by perturbation theory for coupling strengths below a critical value; the model, however, breaks down for coupling strengths above this critical value, as the spectrum of the quantum Lamb Hamiltonian is unbounded below. We find that by including an anharmonic elastic term in the Hamiltonian, the model's stability can be restored.