Resonant excitation of a two-level system with relaxations

5 Resonant excitation of a two-level system with relaxations 

A real two-level particle is inevitably subject to the decay of its excited state, owing to at least spontaneous emission (Fig. 2.4(b)). Giving consideration to the decay of the excited state immediately makes the absorption spectral line of the transition 1 2 have a finite width even in the absence of an external field. In actual fact, the interaction of the particle with its surroundings, which can be considered as a thermal bath, can additionally shorten the lifetime of the excited state. In the general case, the relaxation of the population of the excited level to its equilibrium state is called longitudinal relaxation and is characterized by a longitudinal relaxation time Ti (Bloch 1946). 

The interaction of the particle with its surroundings causes a random shift of the phase of the particle s wave function in each of its steady states, which is not necessarily accompanied by the decay of the particle to the lower level. The mean time of such a phase relaxation is denoted by T2, and the relaxation itself is frequently referred to as transverse relaxation (Bloch 1946). The phase relaxation has no effect on the relaxation of the population of levels, but it broadens the spectral line of the 1 — 2 transition. The homogeneous half-width F of the Lorentzian in eqn (2.47) is related to the time T2 by a simple relation at Ti T2  

The excitation of a quantum system under conditions of population and phase relaxation cannot be described by the probability amplitudes a (t). Instead, the time evolution of the particle is described by the combinations a (t)aj(t) averaged over the ensemble. This is a standard procedure in quantum mechanics, according to which the quantity a t)aj(t)) is an element Pji t) of a density matrix (see, for example, Sargent et al. 1974)  

The population probability of state i introduced above is described by a diagonal 

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