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 `__ · :doc:`Catalogue ` · :doc:`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 ~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 :doc:`review register ` for the checks and decisions.