nandi_2022

S. Nandi, E. Olofsson, M. Bertolino, S. Carlström, F. Zapata, D. Busto, C. Callegari, M. Di Fraia, P. Eng-Johnsson, R. Feifel, G. Gallician, M. Gisselbrecht, S. Maclot, L. Neoričić, J. Peschel, O. Plekan, K. C. Prince, R. J. Squibb, S. Zhong, P. V. Demekhin, M. Meyer, C. Miron, L. Badano, M. B. Danailov, L. Giannessi, M. Manfredda, F. Sottocorona, M. Zangrando and J. M. Dahlström, “Observation of Rabi dynamics with a short-wavelength free-electron laser”, Nature 608, 488-493 (2022)

Source record · Catalogue · Review register

DOI: 10.1038/s41586-022-04948-y

Source kind: measurement

Recorded technique: Photoelectron spectroscopy (2 m magnetic-bottle spectrometer, electrons retarded below 1 eV, E/dE about 50) of helium in a pulsed Even-Lavie jet (2 mm cone) driven by the seeded FERMI free-electron laser (fifth harmonic of a 261.08 nm, about 100 fs seed) tuned across the 1s2 1S0 -> 1s4p 1P1 one-photon resonance (‘round trip’ scans 52.50 <-> 51.80 nm, 304,192 of 355,000 shots kept after filtering on the PADRES photon spectrum: bandwidth 20-65 meV FWHM, integrated intensity 0.8-1.6 x 10^5 a.u.; 30 photon-energy bins about 13 meV apart); Richardson-Lucy blind deconvolution with Tikhonov-Miller regularisation at three photon energies. Theory: TDCIS (velocity gauge, Hartree-Fock B-spline orbitals, exterior complex scaling, t-SURFF and iSURF) and an analytical model (Rabi two-level amplitudes in the rotating-wave approximation, flat-top pulse; one-photon ionisation from 1s4p and non-resonant two-photon ionisation from 1s2 by first- and second-order perturbation theory on top), pp. 489-491 and Methods (PDF p. 7).

Materials in structured entries: He

Dataset quantities: transition_dipole_moment

Contents: 2 datasets; 0 formulas.

Curation notes

Identity checked on the first page (PDF p. 1): Nandi et al., Nature 608, 488-493 (2022), 18 August 2022, open access (CC BY 4.0); 15 PDF pages: pp. 488-493 (PDF pp. 1-6), Methods (PDF pp. 7-8), Extended Data Figs. 1-6 (PDF pp. 9-14) and Extended Data Table 1 (PDF p. 15). The Supplementary Information (the analytical model, the lifetimes tau_a, tau_b, the volume averaging) is not in the library. Every number below was read on the PDF’s text layer and, for Extended Data Table 1, on its image.

WHAT IS DRIVEN. A ONE-photon resonance: |a> = 1s2 (1S0) and |b> = 1s4p (1P1), hbar omega_ba = 23.742 eV (quoted from NIST ASD 5.9, p. 489). Photoelectrons come from |b> by one photon (I) and from |a> by two photons (II, non-resonant, through the ‘complement’ states and the continuum); both reach the same continuum near 22.9 eV kinetic energy. No two-photon bound resonance is involved: the ‘two-photon’ of TC-1010’s request (‘He resonant two-photon ionisation’) is the (1+1) resonant ionisation through 1s4p and the non-resonant two-photon ionisation of the ground state.

THE CONVENTION. Omega = e E0 z_ba / hbar with E(t) = E0 cos(omega t), the Hamiltonian (1/2) hbar omega_ba sigma_z + hbar Omega cos(omega t) sigma_x, so in the rotating-wave approximation the off-diagonal element is hbar Omega / 2 and Omega is the full (resonant) Rabi frequency, the Autler-Townes splitting; W = sqrt(Omega^2 + d omega^2) the generalised one (p. 488-489, Eq. 1). This is OEP 8’s Omega_R convention (twice the off-diagonal element), here for one photon. z_ba is the M = 0 (z) matrix element, so f = 2 dE z^2 (dE in hartree) for 1S0 -> 1P1.

THE NUMBERS (pages of the journal; Methods = PDF p. 7): - measured AT splitting hbar Omega = 80 +- 2 meV (p. 489; the error from a fit of two Voigt profiles of equal width to the deconvoluted symmetric doublet); Rabi period 2 pi / Omega ‘about 52 fs’ (recomputed h / 80 meV = 51.7 fs); - the symmetric doublet needs a blue detuning of about 11 meV: photon energy 23.753 eV (Fig. 2a), +-13 meV either side for the asymmetric spectra; the avoided crossing (Fig. 4b) also at about 11 meV blue; TDCIS gives about 14 meV (24.157 eV against its own 24.1432 eV); the analytical model needs d omega = 62 meV (Fig. 3d) for the symmetric doublet. The paper attributes the shift to interference between paths I and II and to the pulse envelope, not to a light shift (pp. 490-491); - pulse: FWHM 56 +- 13 fs (the empirical range tau_seed/n^(1/2) to 7 tau_seed/(6 n^(1/3)) with tau_seed = 100 fs, n = 5: 44.7-68.2 fs recomputed; PERSEO simulation about 66 fs); bandwidth 0.13 nm FWHM at 52.216 nm (59 meV, recomputed 59.1 meV); spot 12 um FWHM, w0 = 10.2 um, Rayleigh length 6.3 mm (recomputed 6.26 mm); 87 uJ at the undulator exit; peak intensity ‘about 1.4 x 10^14 W cm-2’ (Methods). DOUBT: a Gaussian pulse of 87 uJ, 56 fs and w0 = 10.2 um gives about 8.9 x 10^14 W cm-2 at full transmission; the beamline transmission is not stated, so 1.4 x 10^14 presumably includes one (inference, about 16 %); - the effective intensity 2 x 10^13 W cm-2 ‘as obtained from Omega’ (p. 490) and E0 = 0.02388 au (p. 490): recomputed, 80 meV with the CIS dipole 0.124 a0 gives E0 = 0.02371 au and I = 1.97 x 10^13 W cm-2 (I = E0^2 x 3.51 x 10^16, Methods); E0 = 0.02388 au is sqrt(2 x 10^13 / 3.51 x 10^16) = 0.02387, i.e. from the rounded intensity. With the experimental z_ba = 0.1318 a0 the same splitting would give E0 = 0.02231 au and 1.75 x 10^13 W cm-2. The effective intensity is about 7 times below the estimated peak (shot-to-shot and volume averaging, Supplementary Information); - Omega / omega_ba = 0.34 % (p. 490; recomputed 0.337 %): the rotating-wave approximation holds; - pulse area: the Gaussian field envelope exp(-2 ln 2 t^2 / tau^2) of Eq. 2 has area sqrt(pi / (2 ln 2)) tau = 1.505 tau times E0 (the paper’s factor ‘about 1.5’, p. 490), so 56 fs at Omega = 80 meV is 1.63 Rabi periods: the paper’s ‘between 1 and 3/2 Rabi periods’ and its flat-top 3/2-period model; - ionisation is slow: spontaneous emission ‘about 4 ns’ (quoting Zitnik et al. 2003), photoionisation lifetime of the Rabi-cycling atom tau_a+b ‘about 100 ps’ (p. 489, Supplementary Information), so about 6 x 10^-4 of the atoms are ionised in 56 fs; E0 = 0.02388 au ‘implies an ionization-probability ratio of 10^4:1 in favour of the one-photon process from |b>’, while the atomic ‘giant wave’ scaling is ‘about 1:10^4 in favour of the two-photon process’ (p. 490), so the two paths compete; - TDCIS (Methods): omega_ba = 0.887246 au = 24.1432 eV (HF ground state to CIS 1s4p), z_ba = 0.124 a0, Koopmans Ip = 24.9788 eV, Gaussian A(t) of Eq. 2 with tau = 56 fs (intensity FWHM) and 2 x 10^13 W cm-2; the 0.4 eV kinetic-energy offset from experiment is ascribed to correlation missing in CIS (p. 490). The two-photon kinetic energy 23.3076 eV of Fig. 3c,d is the CIS value 2 x 24.1432 - 24.9788 (recomputed), not the experimental 2 x 23.742 - 24.5873 = 22.8967 eV. Ip = 24.5873 eV (p. 491) is NIST’s.

DATASETS. The two transition dipoles of 1s2 1S0 - 1s4p 1P1: - z_ba_chan (0.1318 au, p. 489, reproduced from chan_1991): Chan et al.’s f = 0.0303(7) (chan_1991, f_present, row 4) converted, sqrt(0.0303 / (2 x 23.742 / 27.211386)) = 0.13177 (recomputed); chan_1991’s other rows for n = 4 (Fernley et al. 0.03028, Wiese et al. 0.0302) give 0.1317 and 0.1316. This is the value the experiment is interpreted with; - z_ba_cis (0.124 au, Methods, the paper’s own CIS value, use = check: a model property of the TDCIS calculation, not fitted): with the CIS omega_ba it is f = 2 x 0.887246 x 0.124^2 = 0.0273, 10 % below the measured 0.0303. No uncertainty is stated for either.

NOT TAKEN: - the AT splitting hbar Omega = 80 +- 2 meV, the Rabi period, the effective intensity and E0: no quantity holds a Rabi frequency, and the intensity is derived from the splitting through the CIS dipole (circular); the numbers stand above as a validation case; - the blue shifts (11, 14, 62 meV): interference and pulse-envelope effects, not a light shift of the level, so not light_shift_coefficient; - Extended Data Table 1 (PDF p. 15), ‘Values of the dipole transition elements z_i computed using CIS functions’ from |b> and from the ‘giant wave’ |rho_(!=b)> to the s and d continua: 0.009311 and 0.1056 (s), 0.01298 and -1.300 (d) (checked against the table image). No quantity: bound-free elements whose continuum normalisation, photoelectron energy and units are not stated, and |rho_(!=b)> is a perturbative wave built at the photon energy, not a state; not proposed as vocabulary; - hbar omega_ba = 23.742 eV and Ip = 24.5873 eV: quotations of NIST ASD, which optmat holds directly; the CIS energies 0.887246 au and 24.9788 eV: Hartree-Fock/CIS model values, off by 0.4 eV by the paper’s own account; - the spontaneous-emission limit ‘about 4 ns’ of 1s4p 1P1: a rounded quotation of Zitnik et al. (J. Phys. B 36, 4175 (2003), not in the library), not radiative_lifetime; take it from the original; - the figures (spectra in arbitrary units): no absolute scale.

FOR OEP 8. Nandi et al. test the ONE-photon analogue of OEP 8’s effective-level equations (7), not the two-photon model: a resonantly driven two-level system with Omega_R = d E0 / hbar, populations through about 1.6 Rabi cycles in a 56 fs Gaussian, excited-level photoionisation (Gamma^PI_b) and a direct non-resonant two-photon continuum from the ground level. OEP 8 eliminates exactly such odd levels; here the odd level is resonant, the case its spectral and field-strength conditions exclude. The paper’s central effect, the coherent interference of the one-photon (from |b>) and two-photon (from |a>) ionisation amplitudes into one continuum, is outside OEP 8’s rate treatment of ionisation (incoherent Gamma^PI_f and a separate direct-continuum rate); it shapes the photoelectron spectrum but, with about 6 x 10^-4 ionised, not the bound populations or the polarisation that propagation needs. Usable as a check of a one-photon Bloch integrator: Rabi period 51.7 fs from 80 meV, or from z_ba and E0 (0.1318 x 0.02388 au gives 85.6 meV; the paper’s 80 meV pairs with the CIS 0.124).

Recorded bibliographic identity status: VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets

nandi_2022/z_ba_chan

{
  "id": "z_ba_chan",
  "material": "He",
  "table": "text",
  "quantity": "transition_dipole_moment",
  "unit": "au",
  "abscissa": "label",
  "use": "reproduced",
  "reproduced_from": [
    "chan_1991"
  ],
  "parameters": {
    "states": "1s2 1S0 - 1s4p 1P1",
    "component": "z (M = 0, along the linear polarisation; f = 2 dE z^2)"
  }
}

1 inline points in the source record.

p. 489: ‘the dipole element z_ba = 0.1318 a0’ (ref. 30, Chan, Cooper and Brion 1991): chan_1991’s f = 0.0303(7) for 1 1S -> 4 1P converted with f = 2 dE z^2, dE = 23.742 eV (recomputed 0.13177). The M = 0 component along the linear polarisation. Uncertainty not stated (chan_1991’s 2.3 % on f is 1.2 % on z, plus its 5 % scale).

nandi_2022/z_ba_cis

{
  "id": "z_ba_cis",
  "material": "He",
  "table": "text",
  "quantity": "transition_dipole_moment",
  "unit": "au",
  "abscissa": "label",
  "use": "check",
  "parameters": {
    "states": "1s2 1S0 - 1s4p 1P1",
    "component": "z (M = 0, along the linear polarisation; f = 2 dE z^2)",
    "method": "CIS (configuration-interaction singles on Hartree-Fock B-spline orbitals), the basis of the paper's TDCIS"
  }
}

1 inline points in the source record.

Methods (PDF p. 7): ‘The CIS dipole matrix element between the ground state and the excited state is given by z_ba = 0.124 a0’, between the Hartree-Fock 1s2 and the CIS 1s4p 1P1 at the CIS omega_ba = 0.887246 au (24.1432 eV); with that energy f = 0.0273, 10 % below chan_1991’s 0.0303. The input of the paper’s TDCIS calculation and of its E0 = 0.02388 au; a model value, so check. Uncertainty not stated.

Formulas

No formulas are recorded.

Corrections

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.