2 research outputs found
Quantumâmechanical derivation of the Bloch equations: Beyond the weakâcoupling limit
This is the publisher's version, also available electronically from http://scitation.aip.org/content/aip/journal/jcp/94/6/10.1063/1.460626Two nondegenerate quantum levels coupled offâdiagonally and linearly to a bath of quantumâmechanical harmonic oscillators are considered. In the weakâcoupling limit one finds that the equations of motion for the reduced densityâmatrix elements separate naturally into two uncoupled pairs of linear equations for the diagonal and offâdiagonal elements, which are known as the Bloch equations. The equations for the populations form the simplest twoâcomponent master equation, and the rate constant for the relaxation of nonequilibrium population distributions is 1/T 1, defined as the sum of the ââupââ and ââdownââ rate constants in the master equation. Detailed balance is satisfied for this master equation in that the ratio of these rate constants is equal to the ratio of the equilibrium populations. The relaxation rate constant for the offâdiagonal densityâmatrix elements is known as 1/T 2. One finds that this satisfies the wellâknown relation 1/T 2=1/2T 1. In this paper the weakâcoupling limit is transcended by deriving the Bloch equations to fourth order in the coupling. The equations have the same form as in the weakâcoupling limit, but the rate constants are calculated to fourth order. For the populationârelaxation rate constants this results in an extension to fourth order of Fermiâs golden rule. We find that these higherâorder rate constants do indeed satisfy detailed balance. Comparing the dephasing and populationârelaxation rate constants, we find that in fourth order 1/T 2â 1/2T 1