field in the excited guided cavity mode is given by the expression 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 where σ and σ † are the downward and upward operators for the atomic transitions, respec- tively, and 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$_{±}$$_{1}$ = ∓ ( V$_{x}$ ± iV$_{y}$ ) $_{/}$√$_{2 for the spherical components of an arbitrary}$ vector V . We assume that only one spherical component d$_{q}$ = d of the dipole vector d , where q = − 1, 0, or 1, is nonzero. Then, we have d · e ( ω,f,l ) = ( − 1) $^{q}$de ( ω,f,l ) − q . Due to the properties of the mode profile functions [28], we have e ( ω,f,l ) − q ( r, ϕ ) = f 1+ $^{q}$eiq ( π/ 2 − ϕ $^{)}$| e ( ω ) − $_{ql}$( r, ϕ ) | . Here we have introduced the notation | e ( ω ) 0 | = | e ( ω, + , +) z | and | e ( ω ) ± 1 | = ( | e ( ω, + , +) r | ∓ | e ( ω, + , +) ϕ | ) $_{/}$√$_{2.}$ Furthermore, we assume that the probe field and hence the cavity field are counterclockwise circularly polarized, that is, l$_{p}$ = l$_{c}$ = +, and that the atomic transition is σ + polarized, that is, q = 1. Then, we have G = e − i ( m +1) π/ $^{2}$( ω$_{c}$d $^{2}$/ϵ$_{0}$ ¯ hL ) 1 / 2 | e ( ω$_{c}$ ) − 1 | cos ( β$_{c}$z + mπ/ 2). We can remove the phase factor e − i ( m +1) π/ 2 by performing the transformation ae − i ( m +1) π/ 2 = ˜ a for the photon operators or the transformation σe i ( m +1) π/ 2 = ˜ σ for the atomic operators. Therefore, we can take the following expression for the atom-field coupling coefficient: Note that Ω = 2 G is sometimes called the vacuum Rabi frequency. 7 G = √ ω$_{c}$d 2 ϵ$_{0}$ ¯ hL | e ( ω$_{c}$ ) − 1 | cos ( β$_{c}$z + mπ/ 2) . (19) G = 1 2 √ ω$_{c}$ ϵ$_{0}$ ¯ hL d · [ e ( ω$_{c}$,f$_{c}$,l$_{c}$ $^{)}$e if$_{c}$β$_{c}$z + e $^{imπ}$e ( ω$_{c}$, − f$_{c}$,l$_{c}$ $^{)}$e − if$_{c}$β$_{c}$z ] e il$_{c}$ϕ (18) H$_{AF}$ = − i ¯ h ( Gaσ † − H . c . ) , (17) E (+) cav = i 2 √ ¯ hω$_{c}$ ϵ$_{0}$L a [ e ( ω$_{c}$,f$_{c}$,l$_{c}$ $^{)}$e if$_{c}$β$_{c}$z + e $^{imπ}$e ( ω$_{c}$, − f$_{c}$,l$_{c}$ $^{)}$e − if$_{c}$β$_{c}$z ] e il$_{c}$$^{ϕ}$e − iω$_{c}$$^{t}$. (16)