a, The strong electric field of a near-infrared (NIR) laser pulse (in red) with a duration of τ_L < 4 fs liberates electrons from the 4p valence subshell of krypton atoms to generate singly charged 4p⁻¹, doubly charged 4p⁻² or triply charged 4p⁻³ ions in the 4p⁻¹, 4p⁻² and 4p⁻³ manifolds of quantum states, respectively, by means of optical field ionization (indicated by red arrows). A subfemtosecond EUV pulse (in violet) with a carrier photon energy of ~80 eV is passed through the ions and promotes them to core-hole excited-state manifolds 3d⁻¹, 3d⁻¹4p⁻¹ and 3d⁻¹4p⁻² (as indicated by the violet arrows). Transient EUV absorption spectra are acquired by recording the attosecond EUV pulse spectrum transmitted through the ionized gas target as a function of pump–probe delay with an EUV spectrometer. b, c, Spectral intensity distribution of the relevant part of the broadband attosecond probe pulse (b) and schematic of the experimental set-up (c). The pulse is transmitted through an ensemble of krypton atoms (~10¹⁸ cm⁻³; interaction length, ~1 mm) before their ionization (blue curve) and after their ionization (red curve) by the laser pulse with a peak intensity of ~7 × 10¹⁴ W cm⁻², as well as by several optical elements shown in c and discussed in detail in Methods. arb.u., arbitrary units.

Absorbance is defined as A(E, τ) = ln(I_trans(E, τ)/I₀(E)), where I₀(E) is the spectral density recorded at a negative delay of −10 fs, that is, the attosecond probe precedes the ionizing laser pulse by 10 fs in the atomic sample, and I_trans(E, τ) is the spectral density recorded at a pump–probe delay τ. The delay is varied in steps of 200 as. Error bars indicate the standard error of the mean values acquired from several spectra recorded at the same delay. The EUV probe pulse shows the formation of charge states up to Kr³⁺ as indicated in the spectrum shown in the background, which is recorded at τ ≈ 10 fs. Disregarding a forerunner in the main Kr⁺ line, the origin of which is unclear, the Kr²⁺ lines appear with a significant, well-resolved delay of about one-half the laser period (T_L/2 ≈ 1.25 fs) after the Kr⁺ lines, and the Kr³⁺ lines appear with approximately the same delay after the Kr²⁺ lines. The decrease in the neutral krypton population in the atomic sample manifests itself as a reduction of the absorption in the range 91–93 eV, where neutral krypton atoms absorb resonantly. Relative occupations between the ionic states Kr⁺, Kr²⁺ and Kr³⁺ are estimated as = 1:0.875:0.25.

a, Temporal evolution of the diagonal elements of the reduced density matrix of Kr⁺ in the presence of a 750-nm, 3.8-fs laser pulse with a peak intensity of ~3 × 10¹⁴ W cm⁻² and a sinusoidal waveform. The notation of the matrix elements is explained in the text. The populations have been normalized such that the trace of the reduced density matrix at τ = 10 fs equals one. The dashed line shows the electric field of the laser pulse (in atomic units (a.u.)) used in the simulations. The hole density distributions of the corresponding orbitals of the 4p subshell are also depicted. b, The degree of coherence, g(t) (see equation (1)), averaged over time intervals of the laser half-cycle (~1.25 fs), is shown with red dots. Squares show the half-cycle-averaged degree of coherence for an ionizing laser pulse with a duration of 7.6 fs at the same peak intensity. The degree of coherence is defined such that, after the NIR pump pulse, it equals one for a perfectly coherent hole wave packet. The detailed behaviour of the degree of coherence during ion formation is not yet understood.

a, Energy-level diagram showing the spin–orbit splitting of the 4p and 3d subshells in Kr⁺. A sub-4-fs NIR laser pulse (red wave) liberates an electron from the 4p subshell and leaves the ensemble of ions in a coherent superposition of and states. Single-photon EUV absorption promotes the ions from these states to the core-excited state. b, Simulated transient EUV absorption spectra reveal characteristic modulations present in the    and    transitions as functions of pump–probe delay. The modulation depth is highly sensitive to the degree of coherence, g(t). Mb, megabarn. c, False-colour plot of an attosecond absorption spectrogram comprising 40 transient absorption spectra recorded at delays increased in steps of 1 fs with a sub-4-fs, ~750-nm laser pump and a sub-150-as, ~80-eV EUV probe. The reference spectrum was recorded at −6 fs. The absorption spectrum plotted in the background is taken at a delay of 30 fs. The linewidths are determined by the ~0.45-eV resolution of our EUV spectrometer. The lower modulation depth of the    transition relative to the calculations shown in b is a result of the spectral resolution being limited in comparison with the ~88-meV natural linewidth of the studied transitions. The zero of the delay scale is set to coincide with the instant when the main    absorption line reaches 95% of its quasistationary value.

a, Absorbance (dots) averaged over the photon energy range 81.20–81.45 eV corresponding to the    transition (see Fig. 4c), as a function of time elapsed since zero as defined in the text. The full line shows the result of our modelling for the values of the fit parameters given in the text. The modulation occurs with a period of 6.3 ±0.1 fs. Error bars depict the standard error of the values extracted from several data sets recorded under identical experimental conditions. b, Quantum phase of the 4p superposition state φ(t) (see equation (2)), as retrieved from the measured attosecond absorption spectrogram shown in Fig. 4c. Uncertainty in the values, resulting from our measurement and modelling, indicates accuracy of reconstruction of the superposition of ~π/5. The lower diagram shows ensemble-averaged hole density distributions in the 4p subshell of Kr⁺ reconstructed from the measured φ(t) and the measured components of the density matrix, at instants separated by 1 fs, within an interval of 17–25 fs following ionization.