Fermi\u2019s Golden Rule and the Second Law of Thermodynamics

D Braak,J Mannhart

doi:10.1007/s10701-020-00380-2

Abstract

We present a Gedankenexperiment that leads to a violation of detailed balance if quantum mechanical transition probabilities are treated in the usual way by applying Fermi’s “golden rule”. This Gedankenexperiment introduces a collection of two-level systems that absorb and emit radiation randomly through non-reciprocal coupling to a waveguide, as realized in specific chiral quantum optical systems. The non-reciprocal coupling is modeled by a hermitean Hamiltonian and is compatible with the time-reversal invariance of unitary quantum dynamics. Surprisingly, the combination of non-reciprocity with probabilistic radiation processes entails negative entropy production. Although the considered system appears to fulfill all conditions for Markovian stochastic dynamics, such a dynamics violates the Clausius inequality, a formulation of the second law of thermodynamics. Several implications concerning the interpretation of the quantum mechanical formalism are discussed.

Highlights

The probabilistic nature of quantum physics is related to a process whose correct interpretation and mathematically sound formulation is still under debate, the socalled collapse of the wave function [1]
The rate equations derived from the quantum mechanical interaction Hamiltonian (1) together with the golden rule entail thermal equilibration according to Clausius’ formulation of the second law of thermodynamics: the two gases exchange heat which flows from the hotter to the colder subsystem until a uniform temperature and maximum entropy of the compound system is reached [15, 19]
There are feasible experimental setups violating the detailed balance condition while satisfying all other prerequisites for the application of the golden rule. The consequence of this violation is a macroscopic disagreement with the second law of thermodynamics

Summary

Introduction

The probabilistic nature of quantum physics is related to a process whose correct interpretation and mathematically sound formulation is still under debate, the socalled collapse of the wave function [1]. The rate constants , ′ depend on the coupling g and the density of states of the radiation continuum around Ω (see below) These equations describe the irreversible change of average quantities and use the ensemble picture of statistical mechanics [15]. The coupling to a continuum of modes leads to real and irreversible microscopic processes, the emission or absorption of light quanta, no macroscopic measurement apparatus is involved Such a microscopic collapse process is tacitly assumed whenever the golden rule is employed. The rate equations derived from the quantum mechanical interaction Hamiltonian (1) together with the golden rule entail thermal equilibration according to Clausius’ formulation of the second law of thermodynamics: the two gases exchange heat which flows from the hotter to the colder subsystem until a uniform temperature and maximum entropy of the compound system is reached [15, 19]. The consequence of this violation is a macroscopic disagreement with the second law of thermodynamics

The Gedanken Experiment

Conflict with the Second Law of Thermodynamics

Discussion and Conclusions