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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 

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. 

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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] 

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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 

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