Spontaneous and induced emission of soft bosons: Exact non-Markovian solution.

Abstract

In many applications it is assumed that the relaxation process can be described by master equation, $${\dot P_n} = - \sum\limits_k {\left( {{W_{nk}}{P_n} - {W_{kn}}{P_k}} \right).} $$ (1) Here, Pn is the probability that the system is in the state n once Wnk is a probability of transition (per unit time) from state n to state k. This equation is derived in the Markovian approximation (or in the Weisskopf- Wigner approximation for spontaneous radiation1). The necessary condition of this approximation is that the eigenfrequency ωmn has to be much larger than the transition rate Wmn (see, e.g., References 2 and 3). For spontaneous emission of phonons or photons, the transition rate ωmn is typically proportional to |ωmn|3. This means that the following condition $${W_{mn}}/|{\omega _{mn}}| \ll 1$$ (2) can be satisfied at very low frequencies, even for the degenerate levels with ωmn =0. On the other hand, at very low frequencies processes of induced emission and absorption of bosons may prevail over the spontaneous emission. The transition rate Wnm for typical one-boson processes has the form \({W_{nm}} = 2\pi /\bar n\sum\limits_k {|{B_k}{|^2}} \delta \left( {{E_n} - {E_m} \pm \bar n{\omega _k}} \right)\left( {{n_k} + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}} \mp {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} \right)\) where Bk is the interaction parameter and nk is the number of bosons in the kth mode.