Theory of decoherence of N-state quantum systems in the Born–Markov approximation

Robert Joynt, Bich Ha Nguyen and Van Hieu Nguyen

Abstract

We present a systematic formalism for the computation of the density matrix of an N-state quantum system in the presence of classical noise or a coupling to the environment. In this formalism, the density matrix of the system is given as an expansion in the generators of the SU(N) group with real coefficients. This leads to a system of master equations. The parameters in these equations may be approximately expressed in terms of the components of the Redfield tensor, when the Born and Markov approximations are valid. The general form of the solution of the system of master equations is established. All relaxation and dephasing rates are then very simply expressed as eigenvalues of a certain matrix. This gives the formulation its simplicity and makes it uniquely suitable for numerical computation. The spectral representation of the components of the Redfield tensor is derived in the case when the environment is a harmonic oscillator bath in thermal equilibrium. Beyond the Born approximation, the decoherence of the system is determined by the Lindblad formula for the Liouvillian superoperator. The Lindblad formulae of some models of multi-state quantum systems are also presented