Dissipation in Quantum Mechanics. Two-Level System. II

Abstract

The theory of a previous article, dealing with two types of two-level systems coupled to a loss mechanism (LM), is extended. The first extension consists of the consideration of the most general type of two-level system (TLS), in which the dipole moment is expanded in terms of the three Pauli spin matrices and unit matrix, the expansion coefficients being vectors (dipole vectors). The second extension consists of the addition to the thermal-reservoir type of LM of a large number of systems identical to the TLS under consideration. The TLS is described in terms of the time development of the Pauli matrices and differential equations are obtained for their expectation value in the presence of arbitrary driving fields. The Bloch equations for a magnetic dipole of spin ½ are exhibited as a special case of these equations, corresponding to a particular combination of the dipole vectors. All other combinations describe electric dipole systems. Equations for two simple special cases of such systems are presented, one treated in the previous article and the other having permanent dipole moment. The frequency of oscillation of a freely decaying TLS is derived and shown to be shifted by an amount that depends on the relationship between the dipole vectors. It is pointed out that the commonly held belief that any TLS can be represented as a magnetic dipole of spin ½ is only approximately correct in the presence of dissipation. The conditions under which the differential equations for the expectation values of the dynamical variables of the TLS can be converted into differential equations for macroscopic variables are discussed.