Two-Photon Transitions Between Discrete States

Abstract

In this note, we discuss different approximation schemes for the evaluation of the two-photon transition rate between discrete states. Non relativistic atomic hydrogen is used as a test of the reliability of the methods. We consider a one particle system described by a Hamiltonian H_o, whose eigenstates and eigenvalues are denoted by |n> and E_n, respectively. In the gauge with divA = 0, the interaction of the particle with the electromagnetic field has the usual form

$${{h}_{j}}=\frac{-e}{mc}A.p\text{+}\frac{1}{2\text{m}}{{\left( \frac{e}{c} \right)}^{2}}{{A}^{2}}$$

(1)

Since we are interested in the two-photon transition rate, we will consider a vector potential

$${{A}_{J}}={{\hat{e}}_{1}}{{A}_{01}}\exp \left[ -i{{\omega }_{1}}t \right]+{{\hat{e}}_{2}}{{A}_{02}}\exp \left[ -i{{\omega }_{2}}t \right]\text{+c}\text{.c}\text{.}$$

(2)

The dipole approximation in (2) is justified in all the cases that we will consider [1].