Variational approach to atom-membrane dynamics

A time-dependent description of the interaction of a cold atom with a vibrating membrane is formulated using the Dirac-Frenkel variational method. The time-dependent variational state is taken to be a superposition of a free atom with a thermal distribution of phonons and an atom bound to the membrane accompanied by a generalized coherent state of phonons. An effective Lagrangian follows from this variational ansatz. The resulting coupled Euler-Lagrange equations are treated analytically to give a description of the system. Distinct dynamical regimes are identified with the use of zero-pole maps and root-locus diagrams in the complex s-plane.