1. Theory for Direct Frequency-Comb Spectroscopy Daniel Felinto and Carlos E.E. López 65 th International Symposium on Molecular Spectroscopy June 24, 2010 Departamento de Física, UFPE Recife, Brazil dfelinto@df.ufpe.br

2. Motivation Frequency comb + cw laser High precision spectroscopy rule in frequency space Frequency comb High precision spectroscopy Direct Frequency-Comb Spectroscopy frequency selection complex problem: many modes excites the system simultaneously disadvantage no frequency selection advantagesmultiplexed spectral analysis broader spectral coverage merging with coherent control parallel access to various spectral regions simultaneously frequency combs covers a broader spectral region than cw lasers frequency combs are much more than a rule in frequency space

3. Experimental implementations of DFCS multiplexing Two-photon spectroscopy in Rb (levels 5S,5P,5D,7S) Marian et al, Science 306, 2063 (2004) human breath analysis Thorpe et al, Opt. Express 16, 2387 (2008) broader spectral coverage extreme ultra-violet Witte et al, Science 307, 400 (2005) Gohle et al, Nature 436, 234 (2005) merging with coherent control First attempts with chirped pulses Stowe et al, PRL 96, 153001 (2006) Reinhardt et al, PRA 81, 033427 (2010)

4. Basic problem: interaction of N modes with M-levels system ω E(ω) N frequency modes (frequency comb) M-levels atom / molecule Re{E(ω)} ω arbitrarily shaped frequency comb coherent control PRA 80, 013419 (2010)

5. t Train of pulses TpTp TRTR 1/ T p Frequency comb f R = 1/ T R our approach: solution in time state before n-th pulse Time evolution between pulses n and n+1 pulse-by-pulse iterative solution PRA 80, 013419 (2010)

6. short-pulse approximation t Train of pulses TpTp TRTR T p << T R coherent excitation incoherent redistribution of population decay rate of element ij transition frequency PRA 80, 013419 (2010)

7. spectral resolution under short-pulse approximation T p < 0.1 T R Δf > 10 f R Spectral resolution larger than 10 comb lines Comb contains 10 5 to 10 6 lines Theory model spectral masks with resolutions between 10 and up to 10 6 comb lines (arbitrary shape) PRA 80, 013419 (2010)

8. time evolution for pulse n time evolution for pulse 0 simple diagonal matrix depending only of f 0 and the level struture problem is reduced to calculate one single time-evolution operator, for the first pulse coherent excitation time-evolution operator Interaction potencial PRA 80, 013419 (2010)

9. general representation in time space t 0, t 1,..., t N time grid E i = E(t i ) numerically calculated electric field for each time set of matrices of same dimension as m-th perturbation theory iterate rectangular rule for numerical integration Arbitrarily-shaped frequency combs PRA 80, 013419 (2010)

10. program to calculate time-evolution operator 1) define atomic structure: levels, dipole moments, frequencies 2) define field: shape, center frequency calculation 3) Initialize P matrixes single time integration nested calculation of P matrixes up to m-th order (Porder) PRA 80, 013419 (2010)

11. spectrum calculation after n pulses 1) Calculation of 2) Definition of f R and f 0 3) Iterative time evolution up to t = nT R 4) Storage of the atomic state for the chosen f R and f 0 Resonances in the atomic excitation occur when Transition frequency between states i, j integer PRA 80, 013419 (2010)

12. A. Marian, M.C. Stowe, J.R. Lawall, D. Felinto, and J. Ye, Science 306, 2063 (2004) first comparison with experiment Simplified theory: impulsive approximation pulse area Matrix of dipole moments Rb 87 Pulses so short that the shape can be neglected (up to 4th order in θ a )

13. Pulse-shape effects

14. impulsive T p = 20 fs T p = 150 fs pulse-shape effects: pulse duration Rb 87 PRA 80, 013419 (2010)

15. no chirp positive chirp – lower frequencies first negative chirp – higher frequencies first pulse-shape effects: frequency-chirped pulses type A peak: sequential transition intermediate lower frequency type B peak: two-photon transition Destructive interference between frequency above and below intermediate resonance for transform-limited pulses [Dudovich et al, PRL 86, 47 (2001)] PRA 80, 013419 (2010)

16. sech pulse 0 π pulse pulse-shape effects: 0 π pulses result of propagation of a weak pulse through a dense sample of two-level atoms the 0 π pulse does not transfer population to level 2 Example of spectral mask with high resolution (about 10 comb lines) type A peak: sequential transition type B peak: two-photon transition inhibition of single-photon excitation PRA 80, 013419 (2010)

17. theory to simulate experiments on Direct Frequency Comb Spectroscopy simple approach to model the action of arbitrarily shaped frequency combs application to the excitation of cold ensemble of Rb 87 atoms by pulse trains of sech, frequency chirped and 0 π pulses it provides the complete state of the system evolving in time, typically revealing various aspects that are hard to probe experimentally Conclusions we hope this work may foster future experiments combining frequency combs with coherent control techniques spectral resolution limited to about 10 comb lines