**Abstract**: We report on one- and two-photon resonances in a lambda system excited by a train of femtosecond pulses. Numerical results using Bloch equations reveal the conditions to distinguish between optical pumping, Raman and EIT processes.

The use of an infinite train of short pulses for periodic creation of a coherent superposition of atomic states has been investigated since the first work by Mlynek et al. [1]. In particular, electromagnetically induced transparency (EIT) [2,3] in the ultrashort regime has been observed using a picoseconds laser [4] and a ultrashort pulse train produced by a mode-locked diode laser [5]. More recently, the transient regime to achieve EIT in a degenerated-lambda system was theoretically analyzed [6], and the use of an optical frequency comb to coherently control and entangle atomic qubits was experimentally demonstrated [7].

In this work, we investigate the excitation of one- and two-photon transitions driven by a train of modelocked femtosecond (fs) pulses in a three-level lambda system. First, we analyze the temporal evolution of the atomic populations and coherences as a function of the pulse area, and the number of driving pulses in the excitation train. The interaction of each pulse with the atomic system is described by the optical Bloch equations and the response of the medium is obtained by numerical integration using a standard fourth-order Runge-Kutta method. We consider that the transition 1 - 2 between the two ground states is dipole-forbidden and that the excited state, 3, decay with equal rates, Γ, to both lower states.

Three resonance situations are investigated and the condition at which the atomic system reaches a stationary state of full coherence, for each case, is obtained for two pulse repetition rates: 100 MHz and 1 GHz. Optical pumping is observed when a mode of the frequency comb matches one-photon transition. In this case, no net population is obtained at the excited state and the coherence between the two lower levels, evolves to zero. The Raman resonance is characterized by a pure two-photon transition, and it is achieve whenever the pulse repetition rate or its multiples coincides with the frequency difference between the two ground states [1,4,5]. However, the more interesting situation takes place when both one- and two-photon resonances occur simultaneously, which corresponds to the EIT condition. In this case, the stationary regime of full coherence, is achieve for a number of driving pulses one order of magnitude smaller than that needed for full coherence in Raman resonance.

In order to analyze the efficiency of the excitation process for pulse trains with different repetition rates, we

define the average Rabi frequency

define the average Rabi frequency

where Ωo is the Rabi frequency, Tp is the time pulse duration. This relation implies [8] that to obtain the same value for the coherence, the pulse area times the number of driving pulses, for the 1 GHz pulse train can be one order of magnitude smaller than for the 100 MHz pulse train. This equation also allows to make comparisons with EIT results obtained using a cw lasers [3].

In the stationary regime, the atomic response to the excitation by a train of fs pulses can be studied as a function of the pulse repetition rate. Figure 1 shows (a) the excited state population, and (b) the coherence, as a function of the variation of repetition rate. The results are obtained after a hundred of driving pulses. The maximum value of the excited state population, is modulated by the two-photon resonances of the medium. In Figs. 1(d) and (e) we have a zoom of the two regions inside the Raman resonance. We can clearly see the peaks due to the one-photon resonance condition.

We also investigate the response of a Doppler broadened atomic sample. In this case, we need to integrate the atomic response over the Doppler profile, so the one-photon resonance is burred and only the Raman dip is observed. The population of excitec state in the Doppler profile is shown in Fig. 2 for (a) the optical pumping condition, where the peaks indicate the different one-photon resonances, and (b) when all atoms see the Raman resonance, but only groups of atoms with specific velocities manifest the EIT effect. These results indicate that for a Doppler broadening atomic medium, a velocity selective spectroscopy can be used to probe the EIT effects.

In conclusion, we have present a detailed study of the temporal evolution and repetition rate spectroscopy of the three-level lambda system excited by a train of fs pulses under one- and two-photon resonances. We also mention that it is possible to obtain closed analytical solutions of the Bloch equations, to modeling EIT signal generated by two cw lasers with equal intensities.

This work was supported by CNPq, FACEPE and CAPES (Brazilian Agencies).

[1] J. Mlynek, W. Lange, H. Harde, H. Burggraf, “High-resolution coherence spectroscopy using pulse trains”, Phys. Rev. A 24, 1099-1102 (1981).

[2] K. -J. Boller, A. Imamoglu, and S. E. Harris, “Observation of Electromagnetically Induced Transparency”, Phys. Rev. Lett. 66, 2593-2596 (1991).

[3] M. Fleischhauer, A. Imamoglu, J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media”, Rev. Mod. Phys 77, 633-673 (2005).

[4] L. Arissian, J.-C. Diels, “Repetition rate spectroscopy of the dark line resonance in rubidium”, Opt. Comm. 264, 169–173 (2006).

[5] V. A. Sautenkov, Y. V. Rostovtsev, C. Y. Ye, G. R. Welch, O. Kocharovskaya, and M. O. Scully, “Electromagnetically induced transparency in rubidium vapor prepared by a comb of short optical pulses”, Phys. Rev. A 71 063804 (2005).

[6] A. A. Soares and L. E. E. Araújo, “Coherent accumulation of excitation in the electromagnetically induced transparency of an ultrashort pulse train”, Phys. Rev. A 76, 043818 (2007).

[7] D. Hayes, D.N. Matsukevich, P. Maunz, D. Hucul, Q. Quraishi, S. Olmschenk, W. Campbell, J. Mizrahi, C. Senko, and C. Monroe, “Entanglement of Atomic Qubtis Using an Optical Frequency Comb”, Phys. Rev. Lett. 104, 140501 (2010).

[8] M. Polo, C. A. C. Bosco, L. H. Acioli, D. Felinto and S. S. Vianna, “Coupling between cw lasers and a frequency comb in dense atomic samples”, J. Phys. B: At. Mol. Opt. Phys. 43 055001 (2010).

In conclusion, we have present a detailed study of the temporal evolution and repetition rate spectroscopy of the three-level lambda system excited by a train of fs pulses under one- and two-photon resonances. We also mention that it is possible to obtain closed analytical solutions of the Bloch equations, to modeling EIT signal generated by two cw lasers with equal intensities.

This work was supported by CNPq, FACEPE and CAPES (Brazilian Agencies).

[1] J. Mlynek, W. Lange, H. Harde, H. Burggraf, “High-resolution coherence spectroscopy using pulse trains”, Phys. Rev. A 24, 1099-1102 (1981).

[2] K. -J. Boller, A. Imamoglu, and S. E. Harris, “Observation of Electromagnetically Induced Transparency”, Phys. Rev. Lett. 66, 2593-2596 (1991).

[3] M. Fleischhauer, A. Imamoglu, J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media”, Rev. Mod. Phys 77, 633-673 (2005).

[4] L. Arissian, J.-C. Diels, “Repetition rate spectroscopy of the dark line resonance in rubidium”, Opt. Comm. 264, 169–173 (2006).

[5] V. A. Sautenkov, Y. V. Rostovtsev, C. Y. Ye, G. R. Welch, O. Kocharovskaya, and M. O. Scully, “Electromagnetically induced transparency in rubidium vapor prepared by a comb of short optical pulses”, Phys. Rev. A 71 063804 (2005).

[6] A. A. Soares and L. E. E. Araújo, “Coherent accumulation of excitation in the electromagnetically induced transparency of an ultrashort pulse train”, Phys. Rev. A 76, 043818 (2007).

[7] D. Hayes, D.N. Matsukevich, P. Maunz, D. Hucul, Q. Quraishi, S. Olmschenk, W. Campbell, J. Mizrahi, C. Senko, and C. Monroe, “Entanglement of Atomic Qubtis Using an Optical Frequency Comb”, Phys. Rev. Lett. 104, 140501 (2010).

[8] M. Polo, C. A. C. Bosco, L. H. Acioli, D. Felinto and S. S. Vianna, “Coupling between cw lasers and a frequency comb in dense atomic samples”, J. Phys. B: At. Mol. Opt. Phys. 43 055001 (2010).