Atomic Coherent States in Quantum Optics

Abstract

The central problem of Quantum Optics (laser theory, super-radiance, resonant propagation, etc.) is the description of the interaction between N atoms and an electromagnetic field confined in a cavity of finite volume. A suitable model Hamiltonian for this problem is the following one (ℏ = 1)

$$\begin{array}{*{20}{l}} {{\rm{H = }}\sum\limits_{\rm{k}} {{{\rm{\omega }}_{\rm{k}}}} {\rm{a}}_{\rm{k}}^{\rm{ + }}{{\rm{a}}_{\rm{k}}}{\rm{ + }}\frac{{{{\rm{\omega }}_{\rm{o}}}}}{{\rm{2}}}\sum\limits_{{\rm{i = 1}}}^{\rm{N}} {{{\rm{S}}_{{\rm{3}}\left( {\rm{i}} \right)}}} }\\ {{\rm{ + }}\sum\limits_{{\rm{k,i}}} {{{\rm{g}}_{\rm{k}}}\left( {{{\rm{a}}_{\rm{k}}}{\rm{S}}_{\rm{i}}^{\rm{ + }}{{\rm{e}}^{{\rm{i \bullet }}{{{\rm{}}}_{\rm{i}}}}}{\rm{ + a}}_{\rm{k}}^{\rm{ + }}{\rm{S}}_{\rm{i}}^{\rm{ - }}{{\rm{e}}^{{\rm{ - i \bullet }}{{{\rm{}}}_{\rm{i}}}}}} \right)} {\rm{,}}} \end{array}$$

where a_k, a ⁺_k are Bose operators describing the k-th field mode and S ^±_i ,S_3i are Pauli operators describing the atom located at position x _i as a two-level system.