Variational approach to atom-membrane dynamics

Using the Dirac–Frenkel variational principle, a time-dependent description of the dynamics of a two-level system coupled to a bosonic bath is formulated. The method is applied to the case of a gas of cold atoms adsorbing to an elastic membrane at a finite temperature via phonon creation. The time-dependence of the system state is analytically calculated using Laplace transform methods, and a closed-form expression for the transition rate is obtained. Atoms in the gas transition to the adsorbed state through a resonance that has contributions from a distribution of vibrational modes of the membrane. The resonance can decay with the creation of a phonon to complete the adsorption process. The adsorption rate at low membrane temperatures agrees with the golden rule estimate to the lowest order in the coupling constant for values greater than a critical coupling strength. Below this critical coupling strength, the adsorption rate is exponentially suppressed by a phonon reduction factor whose exponent diverges with increasing adsorbent size. The rate changes discontinuously with coupling strength for low temperature membranes, and the magnitude of the discontinuity decreases with increasing temperature. These variational results suggest the quantum adsorption model may contain a first-order quantum phase transition.