Conservation laws of optical Bloch equations for the tripod scheme - contribution of double dark-state

Abstract

Results are presented for the conservation laws (CLs) of the optical Bloch equations (OBEs) for the 4-level tripod atomic scheme under the action of three laser fields. CLs are constructed for OBEs that include the spontaneous emission, collisional broadening and time of flight relaxation, and also without these effects, i.e. for the Liouville von Neumann equation (LNE). We used the direct method to construct CLs as the linear combinations of the density matrix elements and their products. These CLs were in turn used to obtain another set of CLs that explicitly depend on time and stand with relaxation due to the time of flight in OBEs. We also constructed CLs for OBEs representing special case of the tripod system that supports the coherent population trapping (CPT), the phenomenon characterized by the ‘dark-state’. Tripod scheme has two dark-states with the same Hamiltonian eigenvalue. As we explain, this peculiarity of tripod scheme contributes to additional CLs of both LNE and OBEs with all relaxation and decoherence effects included, when CPT is present.