content_image/371074.jpg content_image/371073.jpg
nanofiber has been realized experimentally [27]. With the above assumption, we can neglect the kinetic energy and the potential in the Hamiltonian (23). Then, we obtain the following effective Hamiltonian: 

<img src='content_image/371074.jpg'>

We emphasize that the effect of the van der Waals potential on the atomic transition fre- quency ω$_{a}$ and the detuning Δ$_{c}$ is kept in the above Hamiltonian. It is clear that the effect of the van der Waals potential on the atomic transition frequency is negligible in the region of large atom-to-surface distances r − a but significant in the region of small distances r − a . In our numerical calculations presented in Sec. IV, we assume for simplicity that the van der Waals potential in the case of the fiber is the same as that in the case of a flat surface, that is, V$_{α}$ = − C$_{3}$$_{α}$/ ( r − a ) $^{3}$, where α = g, e . 

## III. DENSITY-MATRIX EQUATIONS 

Let ρ be the density operator for the combined atom-field system. In the presence of the atomic decay and the cavity damping, the time evolution of ρ is governed by the master equation 

<img src='content_image/371073.jpg'>

Here γ = γ$_{gyd}$ + γ$_{rad}$ is the decay rate of the atom in the presence of the nanofiber and in the absence of the FBG cavity. 

We use the basis | α, n 〉 formed from the internal states | α 〉 = | e 〉 , | g 〉 of the atom and the number states | n 〉 of the guided cavity field. For the matrix elements ρ$_{α,n}$$_{;}$$_{α}$ ′ ,n ′ = 〈 α, n | ρ | α $^{′}$, n$^{′}$〉 

