The detailed calculation of an excitation probability P_{e} of a two-level atom by a travelling light pulse in free space is described in [1–3]. In particular, P_{e}(t) is determined by dynamical coupling strength g(t) which is given by [1]

g(t) = \sqrt{Γ_{p} N} ξ(t).  (1)

Here, N is the average photon number in the coherent state pulse, ξ(t) is the normalized temporal envelope function, and Γ_{p} is the atomic decay rate into the excitation pulse mode. The explicit formula of Γ_{p} can be found in Ref.[1] and expressed in terms of the total spontaneous decay rate Γ as

\begin{aligned}Γ_{p} & = & Γ · (\frac{3Λ}{8π}) , \\ Γ & = & \frac{1}{3π} (\frac{ω_{a}}{c} )^{3} \frac{d^{2}}{h̄ϵ_{0}} , \end{aligned}  (2)

where c is the vacuum speed of light, ϵ_{0} is the permittivity of the vacuum, ω_{a} is the atomic transition frequency, and d is the scalar atomic dipole momentum. The single parameter Λ ∈ [0, \frac{8π}{3} ], which describes the spatial mode overlap, is obtained by weighting the atomic dipole pattern in the solid angle covered by the pulse

Λ = ∑_{λ} ∫ dΩ |u_{k,λ}|^{2} |(ê_{d} · ϵ̂_{k,λ})|^{2},  (3)

where u_{k,λ} are field spatial mode functions, ϵ_{k,λ}(λ = 1, 2) are unit polarization vectors, and e_{d} is the unit atomic dipole vector. Evaluation of the spatial overlap factor with different input mode functions has been done in several recent works [4–6]. Since in our experiment the spatial mode function of the excitation pulse is a strongly focused Gaussian, we will follow results of work [6] to express the overlap integral in terms of the scattering ratio R_{sc}, which further depends on the focusing strength u := w_{L}/f as

Λ = \frac{R_{sc}}{4} \frac{8π}{3} = \frac{π}{2u^{3}} e^{2/u^{2}} [Γ (−\frac{1}{4}, \frac{1}{u^{2}} ) + uΓ (\frac{1}{4}, \frac{1}{u^{2}} )]^{2} ,

(4)

where w_{L} is the input beam waist, f is the focal distance of the coupling lens, and Γ(a, b) = ∫_{b}^{∞} t^{a−1}e^{−t}dt is the incomplete gamma function. For our experimental value u = 0.22, we have Γ_{p} ≈ 0.03Γ and thus Eq. (1) becomes

g(t) = \sqrt{0.03Γ N} ξ(t).  (5)

   1666v2.

[3] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and Application (Wiley Interscience, New York, 1992).

[4] M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, Appl. Phys. B 89, 489 (2007).

[5] G. Zumofen, N. M. Mojarad, V. Sandoghdar, and M. Agio, Phys. Rev. Lett. 101, 180404 (2008).

[6] M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, New Journal of Physics 11, 043011 (2009).

[1] Y. Wang, J. Minar, L. Sheridan, and V. Scarani, Phys.

Rev. A 83, 063842 (2011).

[2] M. Stobińska, G. Alber, and G. Leuchs, Europhys. Lett.

86, 14007 (2009), URL http://arxiv.org/abs/0808.