Einstein A coefficient
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Definition
Spontaneous emission rate, postulated by Einstein in a rate-equation analysis of the energy exchange between light and atoms in thermal equilibrium. The spontaneous emission rate is often denoted by the symbol γ, and is given by (in free-space)
where ε0 is the permitivitty of free-space, ω is the resonant frequency, and d12 is the dipole matrix element between the two atomic levels.
Derivation
The derivation of the above expression can be found in most quantum optics textbooks, see the references below. A general (qualitative) sketch goes as follows:
1) Assume a two level atom with states |1> and |2>, with energies E1 and E2. We want to consider an atom that is initially in upper state |2> and which undergoes a transition to state |1> due to its interaction with a radiation field. Initially, we can consider the transition rate due to the interaction with a single radiation mode.
2) Calculate this transition rate using Fermi's golden rule and the standard dipole interaction Hamiltonian. The states that are coupled are labeled |2,n> (upper state with n photons) and |1,n+1> (lower state with n+1 photons).
3) The calculated rate consists of two parts, one that is an induced rate (stimulated emission) and is proportional to n, and another that is independent of photon number and is the spontaneous rate.
4) All that is left to do is to sum over all radiation modes; this is done by changing the summation over modes to an integral using the density of states in free space.
References
Amnon Yariv, Quantum Electronics, 1989. In particular, Chapter 8, pp. 164-166
Howard Carmichael, Statistical Methods in Quantum Optics I: Master Equations and Fokker-Planck Equations, 1999. In particular, Chapter 2, pp. 33-34 and Chapter 7, pp. 256-258 Albert Einstein Quotes
