Extracting Phase and Amplitude Modifications of Laser-Coupled Fano Resonances

Andreas Kaldun, Christian Ott, Alexander Blättermann, Martin Laux, Kristina Meyer, Thomas Ding, Andreas Fischer, and Thomas Pfeifer

Phys. Rev. Lett. 112, 103001 – Published 10 March 2014

Abstract

Fano line shapes observed in absorption spectra encode information on the amplitude and phase of the optical dipole response. A change in the Fano line shape, e.g., by interaction with short-pulsed laser fields, allows us to extract dynamical modifications of the amplitude and phase of the coupled excited quantum states. We introduce and apply this physical mechanism to near-resonantly coupled doubly excited states in helium. This general approach provides a physical understanding of the laser-induced spectral shift of absorption-line maxima on a sub-laser-cycle time scale as they are ubiquitously observed in attosecond transient-absorption measurements.

Received 10 December 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.103001

Authors & Affiliations

Andreas Kaldun^*, Christian Ott, Alexander Blättermann, Martin Laux, Kristina Meyer, Thomas Ding, Andreas Fischer, and Thomas Pfeifer^†

Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

^*andreas.kaldun@mpi-hd.mpg.de
^†tpfeifer@mpi-hd.mpg.de

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Vol. 112, Iss. 10 — 14 March 2014

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Images

Figure 1
(a) Doubly excited states of helium and the embedded three-level laser-coupled system. The XUV pulse coherently excites the dipole-allowed $2 s 2 p$ and ${s p}_{2, 3 +}$ states (straight arrows), while the energetically intermediate $2 p^{2}$ state cannot be accessed from the ground state $1 s^{2}$ . Both the $2 s 2 p$ and ${s p}_{2, 3 +}$ states are resonantly coupled by two photons of a VIS pulse, via the $2 p^{2}$ state. (b) The time-dependent atomic dipole moment $d (t)$ , after $δ$ -like XUV excitation at time $t = 0$ followed by natural decay. The observed spectral response [optical density (OD)], see inset, depends on the intensity and time delay of a VIS pulse (red shading), resonantly coupling the autoionizing states.
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Figure 2
(a) Measured (dots) Fano resonance absorption spectrum [optical density (OD)] near the He ${1 s}^{2} \to {s p}_{2, 3 +}$ transition. The resonant line shape changes with time delay $τ$ from 8.6 to 9.8 fs of the VIS pulse following the XUV excitation. The VIS intensity was set to $0.5 \times 10^{12} W / {cm}^{2}$ . Analytical formula [Eqs. (6) and (7)] (solid lines) for a fixed resonance-energy position. (b) Measured absorption spectra for varying time delay. (c) Analytically calculated absorption spectra [Eqs. (6) and (7)] after convolution with the spectrometer resolution (20 meV). The maxima of each of the spectra are also highlighted by the solid oscillatory trace. The oscillation of the maxima is due to a line-shape change while the resonance energy remains constant.
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Figure 3
Graphical representation of the complex amplitude of an excited state, resonantly coupled by $δ$ -like pulsed laser interaction to two other levels: a (dark) initially nonpopulated intermediate state and a coherently excited (bright) state. In second-order perturbation theory, the amplitude contribution for transition in and out of the dark state $A_{self}$ is phase shifted by $π$ with respect to the original excitation amplitude $A_{1}$ , resulting in an amplitude reduction due to population transfer. The contribution of the coherently excited bright state $A_{inter}$ describes a circle in the complex plane as a function of time delay due to the energy difference between the states and their corresponding field-free time-dependent phase evolution.
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Figure 4
(a) Spectral dipole amplitudes and phases as a function of the VIS intensity and time delay. Each layer represents a complex plane in which the time-delay evolution of the dipole moment is shown (clockwise for increasing time delay). The dots show the experimental data, the solid lines are ellipsoidal fits. One half cycle of the VIS pulse corresponds to seven data points. The time delay $τ$ ranges from 9.8 to 11.0 fs. By extracting the circle radius and center as a function of VIS intensity, the different resonant coupling pathways can be separated [as shown in (b) and (c)]. With increasing intensity and correspondingly larger radius approaching the size of the initial amplitude, the circles become more and more elliptical. (b) The absolute value of the complex-valued center point of each ellipse (upper curve and left $y$ axis) and the corresponding phase (lower curve and right $y$ axis) for each measured VIS intensity. The center point of the ellipse changes its position due to population transfer with increasing VIS intensity. The phase of the center point remains approximately constant for all VIS intensities. (c) Length of the major and minor axes of each ellipse, the corresponding mean value, and, relative to the center point, the phase of the point at fixed time delay 10.6 fs for each measured VIS intensity.
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