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1 Femtosecond two-photon Rabi oscillations in excited He driven by ultrashort intense laser fields M. Fushitani, 1,2 C.-N. Liu, 3 A. Matsuda, 1 T. Endo, 1 Y. Toida, 1 M. Nagasono, 2 T. Togashi, 4 M. Yabashi, 2,4 T. Ishikawa, 2 Y. Hikosaka, 2,5 T. Morishita, 6 A. Hishikawa 1,2, * 1 Department of Chemistry, Graduate School of Science, Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi , Japan. 2 RIKEN, SPring-8 Center, Sayo, Hyogo , Japan. 3 Department of Physics, Fu-Jen Catholic University, Taipei 24205, Taiwan. 4 Japan Synchrotron Radiation Research Institute, Sayo, Hyogo , Japan. 5 Graduate School of Medicine and Pharmaceutical Sciences, University of Toyama, Toyama , Japan 6 Department of Engineering Science and Institute for Advanced Science, The University of Electro-communications, Chofu-ga-oka, Chofu-shi, Tokyo , Japan * hishi@chem.nagoya-u.ac.jp NATURE PHOTONICS 1

2 1. Dependence of two-photon transition moments on the initial magnetic sublevels When the initial (i) and the final (f) levels are coupled by two-photon transition via a virtual intermediate state (a), the two-photon Rabi frequency is expressed 1 as Ω (2) 0 = Ω fa Ω ai / 2Δ, where Ω fa =µ fa F and Ω ia =µ ia F with µ fa and µ ia being the corresponding transition moments, and Δ represents the detuning of the intermediate state. The transition moment µ kj may be expressed as µ kj = R kj A (J k J j ;M) with the radial R kj and the angular A(J k J j ;M) = <J k M cosθ J j M> component. If J a = 2 is adopted for a virtual intermediate state (1s3d 1 D 2 ) for the two-photon transition (see Fig.1b), a ratio of A (J f J a ;0) A (J a J i ;0) / Α (J f J a ;±1) Α (J a J i ;±1) = (3/2) 1/2 is obtained, supporting the faster Rabi oscillations for the parallel configuration (M = 0). 2 NATURE PHOTONICS

3 2. Contributions from np states Figure S1 shows that the population transfer is dominated by the nf (n = 5-7) Rydberg states. The population of the np state remains small (at most 1.5 %) in the present NIR intensity range. The difference between the 1snp or 1snf states can be understood in terms of the magnitude of transition dipole moments from the virtual intermediate 1s3d state for the two-photon transition. The transition dipole moments for the 1snp are calculated from the literature values 2 to be µ(1s3d-1s5p) = au and µ(1s3d-1s6p) = au, while substantially larger values µ(1s3d-1s5f) = au and µ(1s3d-1s6f) = au are obtained for the f states, which explains dominance of the nf states over the np states in the two-photon Rabi oscillation. Fig. S1: Population of the 1snp (n = 2-7) states (left) and 1snf (n = 4-8) states (right) after the interaction with the laser pulse, obtained by the TDSE calculation for M = ±1. The relative polarization between FEL and NIR laser pulses is perpendicular. NATURE PHOTONICS 3

4 3. Effect of initial magnetic sublevels on population transfer Figure S2a shows that the two-photon Rabi oscillations in the 6f state have different periods depending on the initial magnetic sublevels (M = 0, ±1). As a result, anti-phase-like oscillations are obtained, as observed in the photoelectron yields in Fig.3. A similar behavior is observed for the 5f state (Fig.S2b), but the population transfer is more significant for M = 0 than for M = ±1. This is attributed to the contribution from the additional ac-stark shift by the 1s3s state (located 1.7 ev above the 2 1 P state), which is absent in the M = ±1 manifold. Fig. S2: a Laser-field intensity dependence of the 6f population obtained by the full TDSE calculation for M=±1 states (solid line) and M = 0 states (dashed line). b Same as a, but for the 5f state. 4 NATURE PHOTONICS

5 4. Results of 4-level model calculations Figure S3 shows the results of 4-level model calculations consisting of the 2p, 3d, 5f, 6f singlet states of helium. The overall features are in good agreement with those obtained by the full TDSE calculations in Fig.3c. Fig. S3: 4-level model calculation on the 1s2p, 1s3d, 1s5f and 1s6f state populations. NATURE PHOTONICS 5

6 5. Dressed state analysis NIR laser field intensity dependence of the dressed energy levels is shown in Fig.S4. The (avoided) crossing between the 2p> (yellow) and 6f> (red) ( 5f>, blue) states drives a high-contrast two-photon Rabi oscillation for the 6f (5f) state. Fig.S4: (a) Energy levels of the 1s2p, 1s3d, 1s5f and 1s6f states used for the four- or five-level system simulation based on the dressed state approach. (b) Energy levels in dressed-state representation plotting 2p, N+1>, 3d, N>, 5f, N-1>, 6f, N-1> in the M = ±1 manifold in the four-level system (1s2p, 1s3d, 1s5f and 1s6f). (c) Same as (b), but for the M = 0 manifold including the 1s3s state ( 3s, N>). (d) Same as (b), but the 3d state shifted to have a negative detuning of Δ = -0.2 ev. References 1 Gentile, T. R., Hughey, B. J., Kleppner, D. & Ducas, T. W. Experimental Study of One- and Two-Photon Rabi Oscillations. Phys. Rev. A 40, (1989). 2 Wiese, W. L. & Fuhr, J. R. Accurate Atomic Transition Probabilities for Hydrogen, Helium, and Lithium. J. Phys. Chem. Ref. Data 38, (2009). 6 NATURE PHOTONICS