Real-time observation of valence electron motion

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The superposition of quantum states drives motion on the atomic and subatomic scales, with the energy spacing of the states dictating the speed of the motion. In the case of electrons residing in the outer (valence) shells of atoms and molecules which are separated by electronvolt energies, this means that valence electron motion occurs on a subfemtosecond to few-femtosecond timescale (1fs = 10−15s). In the absence of complete measurements, the motion can be characterized in terms of a complex quantity, the density matrix. Here we report an attosecond pump–probe measurement of the density matrix of valence electrons in atomic krypton ions1. We generate the ions with a controlled few-cycle laser field2 and then probe them through the spectrally resolved absorption of an attosecond extreme-ultraviolet pulse3, which allows us to observe in real time the subfemtosecond motion of valence electrons over a multifemtosecond time span. We are able to completely characterize the quantum mechanical electron motion and determine its degree of coherence in the specimen of the ensemble. Although the present study uses a simple, prototypical open system, attosecond transient absorption spectroscopy should be applicable to molecules and solid-state materials to reveal the elementary electron motions that control physical, chemical and biological properties and processes.

At a glance


  1. Probing intra-atomic electron motion by attosecond absorption spectroscopy.
    Figure 1: Probing intra-atomic electron motion by attosecond absorption spectroscopy.

    a, The strong electric field of a near-infrared (NIR) laser pulse (in red) with a duration of τL<4fs 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 ~80eV 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 (~1018cm−3; interaction length, ~1mm) before their ionization (blue curve) and after their ionization (red curve) by the laser pulse with a peak intensity of ~7×1014Wcm−2, as well as by several optical elements shown in c and discussed in detail in Methods. arb.u., arbitrary units.

  2. Transient absorption spectra of krypton ions.
    Figure 2: Transient absorption spectra of krypton ions.

    Absorbance is defined as A(E, τ) = ln(Itrans(E, τ)/I0(E)), where I0(E) is the spectral density recorded at a negative delay of −10fs, that is, the attosecond probe precedes the ionizing laser pulse by 10fs in the atomic sample, and Itrans(E, τ) is the spectral density recorded at a pump–probe delay τ. The delay is varied in steps of 200as. 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 τ10fs. 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 (TL/21.25fs) 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 91–93eV, where neutral krypton atoms absorb resonantly. Relative occupations between the ionic states Kr+, Kr2+ and Kr3+ are estimated as = 1:0.875:0.25.

  3. Build-up of electronic coherence in Kr+ produced by optical field ionization (theory).
    Figure 3: Build-up of electronic coherence in Kr+ produced by optical field ionization (theory).

    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 ~3×1014Wcm−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 = 10fs 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 (~1.25fs), is shown with red dots. Squares show the half-cycle-averaged degree of coherence for an ionizing laser pulse with a duration of 7.6fs 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.

  4. Attosecond absorption spectroscopy reveals intra-atomic electron wave-packet motion in Kr+.
    Figure 4: Attosecond absorption spectroscopy reveals intra-atomic electron wave-packet motion in Kr+.

    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 right arrow and right arrow 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 1fs with a sub-4-fs, ~750-nm laser pump and a sub-150-as, ~80-eV EUV probe. The reference spectrum was recorded at −6fs. The absorption spectrum plotted in the background is taken at a delay of 30fs. The linewidths are determined by the ~0.45-eV resolution of our EUV spectrometer. The lower modulation depth of the right arrow 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 right arrow absorption line reaches 95% of its quasistationary value.

  5. Reconstruction of valence-shell electron wave-packet motion.
    Figure 5: Reconstruction of valence-shell electron wave-packet motion.

    a, Absorbance (dots) averaged over the photon energy range 81.20–81.45eV corresponding to the right arrow 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±0.1fs. 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 1fs, within an interval of 17–25fs following ionization.


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Author information

  1. These authors contributed equally to this work.

    • Eleftherios Goulielmakis &
    • Zhi-Heng Loh


  1. Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

    • Eleftherios Goulielmakis,
    • Adrian Wirth,
    • Vladislav S. Yakovlev,
    • Sergey Zherebtsov,
    • Matthias F. Kling &
    • Ferenc Krausz
  2. Departments of Chemistry and Physics, University of California, Berkeley, California 94720, USA

    • Zhi-Heng Loh,
    • Thomas Pfeifer &
    • Stephen R. Leone
  3. Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

    • Zhi-Heng Loh,
    • Thomas Pfeifer &
    • Stephen R. Leone
  4. Argonne National Laboratory, Argonne, Illinois 60439, USA

    • Robin Santra
  5. Department of Physics, University of Chicago, Chicago, Illinois 60637, USA

    • Robin Santra
  6. Lawrence Livermore National Laboratory, Livermore, California 94551, USA

    • Nina Rohringer
  7. Department für Physik, Ludwig-Maximilians-Universität München, Am Coulombwall 1, D-85748 Garching, Germany

    • Vladislav S. Yakovlev &
    • Ferenc Krausz
  8. Physics and Astronomy Department, King Saud University, Riyadh, 11451, Kingdom of Saudi Arabia

    • Abdallah M. Azzeer
  9. Present address: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany.

    • Thomas Pfeifer


E.G., Z.-H.L. and A.W. conceived and designed the experiments; E.G., A.W. and Z.-H.L. performed the measurements; A.W., Z.-H.L., E.G., T.P., S.Z., A.M.A., M.F.K., S.R.L and F.K. evaluated, analysed and interpreted the experimental data; and R.S., N.R. and V.S.Y. performed the theoretical modelling. All authors discussed the results and contributed to the final manuscript.

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