Non-equilibrium Lifshitz theory as a steady state of a full dynamical quantum system

TL;DR

This paper demonstrates that Lifshitz's theory for Casimir pressure in non-equilibrium scenarios emerges as the steady state of a fully quantized dynamical system, clarifying the role of initial conditions and baths.

Contribution

It provides a full quantum dynamical derivation of Lifshitz theory, including transient effects and the conditions for steady state contributions from initial conditions.

Findings

Lifshitz's framework corresponds to the long-time limit of a quantum dynamical system.

In the steady state, only the thermal baths contribute to the Casimir pressure.

Initial conditions can have a non-vanishing contribution in finite-width slab scenarios.

Abstract

In this work we analyze the validity of Lifshitz's theory for the case of non-equilibrium scenarios from a full quantum dynamical approach. We show that Lifshitz's framework for the study of the Casimir pressure is the result of considering the long-time regime (or steady state) of a well-defined fully quantized problem, subjected to initial conditions for the electromagnetic field interacting with real materials. For this, we implement the closed time path formalism developed in previous works to study the case of two half spaces (modeled as composite environments, consisting in quantum degrees of freedom plus thermal baths) interacting with the electromagnetic field. Starting from initial uncorrelated free subsystems, we solve the full time evolution, obtaining general expressions for the different contributions to the pressure that take part on the transient stage. Using the analytic properties of the retarded Green functions, we obtain the long-time limit of these contributions to the total Casimir pressure. We show that, in the steady state, only the baths' contribute, in agreement with the results of previous works, where this was assumed without justification. We also study in detail the physics of the initial conditions' contribution and the concept of modified vacuum modes, giving insights about in which situations one would expect a non vanishing contribution at the steady state of a non-equilibrium scenario. This would be the case when considering finite width slabs instead of half-spaces.