The detailed calculation of an excitation probability P_{e} of a two-level atom by a travelling light pulse is desctibed in details 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 number of photons in a pulse with a coherent statistics, ξ(t) is the normailzed temporal envelope mode function and Γ_{p} is the decay rate of an atom into a spatial mode subtended by an excitation pulse. Since in the theoretical description of experiments described below, Γ_{p} will be the only parameter, we briefly outline its evaluation for experimental settings.

We start with the expression of the electric field operator

Ê^{+}(r, t) = i ∑_{λ} ∫ d^{3}k \sqrt{\frac{h̄ω_{k}}{(2π)^{3} 2ϵ_{0}}} â_{k,λ} ϵ_{k,λ} u_{k,λ}(r, t) e^{−iω_{k}t},

(2)

where ω_{k} = c|k|, c is the vacuum speed of light, ϵ_{0} is the permittivity of the vacuum, u_{k,λ} are spatial mode functions, and ϵ_{k,λ} are unit polarization vectors λ = 1, 2. A two-level atom, represented by a dipole operator d̂ = |d|σ̂_{x,y,z}e_{d} interacts with a field at its location r_{a} with a coupling strength (dipole matrix element, Rabi frequency, interaction Hamiltonian, etc...) g(r_{a}) = (d̂· Ê^{+})

g(r_{a}) = |d|\sqrt{\frac{ω_{k}}{(2π)^{3} 2h̄ϵ_{0}}} u_{k,λ}(e_{d} · ϵ_{k,λ}),  (3)

where e_{d} is the unit dipole vector defining atomic transition symmetry. Using Wigner-Weisskopf approximation, one arrives to spontaneous decay rate Γ_{p} of an atom into pulse mode [4]

Γ_{p} = 2π ∑_{λ} ∫ d^{3}k |g_{k,λ}(r_{a})|^{2} δ(ω_{k} − ω_{a}). (4)

Substitution of Eq. 3 into Eq. 4 and employing spherical coordinates yields

\begin{aligned}Γ_{p} & = & Γ · (\frac{3Λ}{8π}) \\ Γ & = & \frac{1}{3π} (\frac{ω_{a}}{c} )3 \frac{|d|^{2}}{h̄ϵ_{0}} \\ Λ & = & ∫ dΩ |u_{k⃗,λ}|^{2} |(ê_{d} · ϵ̂_{k⃗,λ})|^{2} \end{aligned}  (5)

Here Γ is the overall spontaneous decay rate [3]. The overlap integral Λ ∈ [0, \frac{8π}{3} ] shows the overlap between the spatial mode subtended by a pulse and atomic dipolar emission modes. Evaluation of this integral for different input mode functions was done in several recent works [5– 7]. Since in our experiment the spatial mode function of the excitation pulse is a strongly focused Gaussian, we will follow results of work [7] to express the overlap integral in terms of the focusing strength u := w_{L}/f defined through input beam waist w_{L} and focal distance of the coupling lens f

Λ = \frac{2π R_{sc}}{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}

(6)

For our experimental value u = 0.22, Γ_{p} ≈ 0.03Γ and coupling strength of equation 1 becomes

g(t) = \sqrt{0.03Γ N} ξ(t).  (7)

[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.
                                                           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. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, UK, 1997).

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

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

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