In the weak excitation limit in dilute gases, when saturation and collision eﬀects are negligible, line broadening occurs due to spontaneous decay (width Γ_{sp}), Doppler eﬀect, and, if atoms interact with tightly focused cw laser beams, also due to limited transit time τ_{tr} of atoms through the laser beam. Consider a two step excitation process, in which Doppler broadening is avoided using counterpropagating laser ﬁelds in the excitation. The conventional knowledge says that the the resultant lineshapes are given by the Lorenz proﬁle [1]
P (Δ) = πΓ^{*}/ (Δ^{2 }+ Γ^{2}); 2Δω =Γ^{*} = Γ_{sp} + 1/τ_{tr}, (1)
where Δ is the one-photon detuning of laser ﬁelds oﬀ from the two-photon resonance, and 2Δω is the FWHM width. If an atom from a supersonic beam at a ﬂow velocity v_{f} crosses a Gaussian laser beam of FWHM L, then it is reasonable to assume τ_{tr} = L/v_{f} .
We consider excitation of the Na(5S1/2) HF sublevel F = 2 with with lifetime τ_{sp} = 76ns by two counter-propagating laser beams, which models an ideal two-level quantum system. Laser in the ﬁrst step is focused to L1 = 30µm using a cylindrical lens and detuned by Δν_{1}=100MHz oﬀ from resonance with the 3S_{1/2}, F'' = 1 → 3p1/2, F' = 2 transition. The second laser is collimated to L_{2} = 1000µm, while its frequency is scanned across the two-photon resonance. The detuning Δν1 is suﬃcient to ensure that the intermediate level is virtual. Both lasers cross the atomic beam with ﬂow velocity of vf = 1200m/s at right angles. The time dependence of the eﬀective Rabi frequencies are given by Ω(t)=Ω_{0} exp(-2t^{2}/τ_{tr}^{2}), which corresponds to a Gaussian eﬀective laser intensity proﬁle I(z) = I_{0} exp(_{−}4z^{2}/L^{2}) along the atomic beam axis z. The spatial distribution of the eﬀective Rabi frequency is given by Ω_{ef f} (z) = Ω_{1}(z)Ω_{2}(z)/Δω_{1}.
We have obtained analytical solutions to this model porlem which show that the lineshape of excitation of the upper state is described by the Voight proﬁle with HWHM which can be approximated (within the accuracy level of 10%) by the expression
Δω_{res} = sqrt(Γ_{sp}/4 + 4.8 _{· }ln(2)/τ_{tran} ^{2}). (2)
Importantly, the value of Δω_{res} exceeds the intuitive width Δω (1) by a factor of three if the broadening occurs predominantly due to limited transit time, i.e. when τ_{tran }< 1/Γ_{sp} = τ_{sp}.