Conservation laws for optical Bloch equations for the Λ scheme

Abstract

Results are presented for the conservation laws (CLs) for the optical Bloch equations (OBEs) for the Λ scheme under action of two light fields. The method of multipliers (variational derivative method) is used to obtain CLs which are dependent on the density-matrix elements. Results are classified and discussed by the phenomenological parameters and processes characteristic for OBEs, like the relaxation and decoherence, and the detunings of light fields. CLs for the Liouville von Neumann equation, as a special case of OBEs, without relaxation and decoherence terms, are presented and it is shown that traces, characteristic polynomials and their coefficients, eigen-values and determinants of class of matrices represent CLs. We also presented method, which used CLs obtained by the method of multipliers, to construct another set of CLs, which have the explicit time dependence and are valid for the larger set of parameters, with relaxation due to time of flight and spontaneous emission. Presented results could yield better understanding of the processes in the Λ scheme modeled by the OBEs, and as well have practical applications in various solution methods.