Spontaneous and Induced Emission of Soft Bosons: Exact Non-Markovian Solutions View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1989

AUTHORS

Benjamin Fain

ABSTRACT

In many applications it is assumed that the relaxation process can be described by master equation, (1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot P_n} = - \sum\limits_k {\left( {{W_{nk}}{P_n} - {W_{kn}}{P_k}} \right).} $$\end{document}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 (2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W_{mn}}/|{\omega _{mn}}| \ll 1$$\end{document}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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W_{nm}} = 2\pi /\bar n\sum\limits_k {|{B_k}{|^2}} \delta \left( {{E_n} - {E_m} \pm \bar n{\omega _k}} \right)\left( {{n_k} + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}} \mp {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} \right)$$\end{document} where Bk is the interaction parameter and nk is the number of bosons in the kth mode. More... »

PAGES

281-285

Book

TITLE

Coherence and Quantum Optics VI

ISBN

978-1-4612-8112-2
978-1-4613-0847-8

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-1-4613-0847-8_52

DOI

http://dx.doi.org/10.1007/978-1-4613-0847-8_52

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1036570865


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