The operators a$_{µ}$ and a † µ satisfy the continuous-mode bosonic commutation rules [ a$_{µ}$, a † µ ′ ] = δ ( ω − ω $^{′}$) δ$_{ff}$ ′ δ$_{ll}$ ′ . The explicit expression for the mode function e ( µ ) is given in Refs. [28, 30]. According to Ref. [30], the rate of spontaneous emission into guided modes is given by γ$_{gyd}$ = ω$_{0}$ 2 ϵ$_{0}$ ¯ hv$_{g}$ ∑ fl ∣ d · e ( ω 0 ,f,l $^{)}$∣ ∣$^{2}$ . (2) Here d is the matrix element of the electric dipole moment of the atom and v$_{g}$ = 1 /β $^{′}$( ω$_{0}$ ) is the group velocity of the guided field. atom FBG FBG nanofiber input guided probe light  FIG. 1: (Color online) An atom in the vicinity of a nanofiber with two fiber-Bragg-grating mirrors driven by a weak guided probe light field. Regarding the radiation modes, the longitudinal propagation constant β for each fre- quency ω can vary continuously, from − k to k , with k = ω/c being the wave number. We label each radiation mode by an index ν = ( ω, β, m, l ), where m = 0 , ± 1 , ± 2 , . . . is the mode order and l = + , − is the mode polarization. The quantum expression for the electric positive-frequency component E (+) rad of the field in the radiation modes is [30] E (+) rad = i ∑ ν √ ¯ hω 4 πϵ$_{0}$ a$_{ν}$ e ( ν $^{)}$e i ( βz + mϕ ) − $^{iωt}$. (3) Here e ( ν ) = e ( ν $^{)}$( r, ϕ ) is the profile function of the radiation mode ν in the classical problem, a$_{ν}$ is the corresponding photon annihilation operator, and ∑ ν = ∑ ml ∫ ∞ 0 dω ∫ k − k dβ is the summation over the radiation modes. The operators a$_{ν}$ and a † ν satisfy the continuous-mode bosonic commutation rules [ a$_{ν}$, a † ν ′ ] = δ ( ω − ω $^{′}$) δ ( β − β $^{′}$) δ$_{mm}$ ′ δ$_{ll}$ ′ . The explicit expression for the mode function e ( ν ) is given in Refs. [28, 30]. According to Ref. [30], the rate of spontaneous emission into radiation modes is given by γ$_{rad}$ = ω$_{0}$ 2 ϵ$_{0}$ ¯ h ∑ ml ∫ k 0 − k 0 dβ ∣ d · e ( ω 0 ,β,m,l $^{)}$( r, ϕ ) ∣ ∣$^{2}$ . (4) 4