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field in the excited guided cavity mode is given by the expression 

<img src='content_image/343181.jpg'>

Hence we find that the interaction between the atom and the quantum guided cavity field in the dipole and rotating-wave approximations is described by the Hamiltonian 

<img src='content_image/343180.jpg'>

where σ and σ † are the downward and upward operators for the atomic transitions, respec- tively, and 

<img src='content_image/343179.jpg'>

is the coupling coefficient. Here r , ϕ , and z are the cylindrical coordinates of the position of the atom. 

In general, the atomic dipole vector d of a realistic atom is a complex vector. We use the notation V$_{0}$ = V$_{z}$ and V$_{±}$$_{1}$ = ∓ ( V$_{x}$ ± iV$_{y}$ ) $_{/}$√$_{2 for the spherical components of an arbitrary}$ vector V . We assume that only one spherical component d$_{q}$ = d of the dipole vector d , where q = − 1, 0, or 1, is nonzero. Then, we have d · e ( ω,f,l ) = ( − 1) $^{q}$de ( ω,f,l ) − q . Due to the properties of the mode profile functions [28], we have e ( ω,f,l ) − q ( r, ϕ ) = f 1+ $^{q}$eiq ( π/ 2 − ϕ $^{)}$| e ( ω ) − $_{ql}$( r, ϕ ) | . Here we have introduced the notation | e ( ω ) 0 | = | e ( ω, + , +) z | and | e ( ω ) ± 1 | = ( | e ( ω, + , +) r | ∓ | e ( ω, + , +) ϕ | ) $_{/}$√$_{2.}$ Furthermore, we assume that the probe field and hence the cavity field are counterclockwise circularly polarized, that is, l$_{p}$ = l$_{c}$ = +, and that the atomic transition is σ + polarized, that is, q = 1. Then, we have G = e − i ( m +1) π/ $^{2}$( ω$_{c}$d $^{2}$/ϵ$_{0}$ ¯ hL ) 1 / 2 | e ( ω$_{c}$ ) − 1 | cos ( β$_{c}$z + mπ/ 2). We can remove the phase factor e − i ( m +1) π/ 2 by performing the transformation ae − i ( m +1) π/ 2 = ˜ a for the photon operators or the transformation σe i ( m +1) π/ 2 = ˜ σ for the atomic operators. Therefore, we can take the following expression for the atom-field coupling coefficient: 

Note that Ω = 2 G is sometimes called the vacuum Rabi frequency. 

<img src='content_image/343178.jpg'>

