Variational approach to atom-membrane dynamics

TL;DR

This paper develops a variational method to describe the quantum dynamics of atoms adsorbing onto membranes, revealing a potential first-order quantum phase transition influenced by temperature and coupling strength.

Contribution

It introduces a variational approach based on the Dirac-Frenkel principle to analytically study atom-membrane interactions and adsorption rates at finite temperatures.

Findings

Adsorption rate matches golden rule estimates at low temperatures and strong coupling.

Below a critical coupling, the adsorption rate is exponentially suppressed.

Discontinuous changes in adsorption rate suggest a quantum phase transition.

Abstract

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 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 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.