Gauge-invariant absorption of light from a coherent superposition of states

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Gauge-invariant absorption of light from a coherent superposition of states

Axel Stenquist, Felipe Zapata, and Jan Marcus Dahlström

Phys. Rev. A 107, 053106 – Published 10 May 2023

Abstract

Absorption and emission of light is studied theoretically for excited atoms in coherent superposition of states subjected to isolated attosecond pulses in the extreme ultraviolet range. A gauge-invariant formulation of transient absorption theory is motivated using the energy operator from Yang's gauge theory. The interaction, which simultaneously couples both bound and continuum states, is simulated by solving the time-dependent Schrödinger equation for hydrogen and neon atoms. A strong dependence on the angular momentum and the relative phase of the states in the superposition is observed. Perturbation theory is used to disentangle the fundamental absorption processes and a rule is established to interpret the complex absorption behavior. It is found that nonresonant transitions are the source of asymmetry in energy and phase, while resonant transitions to the continuum contribute symmetrically to absorption of light from coherent superpositions of states.

Received 13 February 2023
Accepted 19 April 2023

DOI:https://doi.org/10.1103/PhysRevA.107.053106

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Atomic & molecular processes in external fields Coherent control Light-matter interaction Optical transient phenomena Quantum description of light-matter interaction Ultrafast phenomena

Atoms

Configuration interaction Dipole approximation Perturbative methods Schroedinger equation Semiclassical methods

Atomic, Molecular & Optical

Authors & Affiliations

Axel Stenquist , Felipe Zapata ^*, and Jan Marcus Dahlström ^†

Department of Physics, Lund University, 22100 Lund, Sweden

^*felipe.zapata@matfys.lth.se
^†marcus.dahlstrom@matfys.lth.se

Article Text

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Issue

Vol. 107, Iss. 5 — May 2023

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Images

Figure 1
Representation of a hydrogenlike atom prepared in a coherent superposition of states: $2 p + 3 p$ , with all processes that are induced by an attosecond XUV pulse indicated by Roman numerals. (I) Resonant transitions to the continuum (red lines). (II) Resonant transitions to the bound states, which may reside below or above the initial states in the superposition (blue lines). (III) Off-resonant transitions to all states allowed by dipole-selection rules (gray rectangles). The bandwidth of the attosecond pulse is depicted in scale and is larger than the separation of the nondegenerate states in the superposition, which implies that quantum beat phenomena may occur that depend on the phases (or more generally the coherence) of the states.
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Figure 2
Time-resolved energy gain of a hydrogen atom in the coherent superposition states $2 s + 3 s$ (top row) and $2 p + 3 p$ (bottom row) interacting with an attosecond pulse. The left column presents the total gauge-invariant gain in solid lines and the relative energy gain [without the free-electron contribution, see Eq. (21)] in dashed lines for the synchronized relative superposition phase (RSP) $φ = 0$ . The perturbation model results are validated by solving the TDSE numerically (dotted lines). The middle column presents the RSP-resolved total gain, whilst the right column shows the corresponding relative gain. Positive gain implies absorption (red) of energy by the atom, while negative gain implies emission (blue) of energy by the atom.
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Figure 3
Analytical (top row) and numerical (bottom row) energy-resolved absorption of an attosecond pulse by a hydrogen atom in the prepared superpositions: $2 s + 3 s$ and $2 p + 3 p$ . The data are resolved over angular frequency of the field $ω$ and the relative superposition phase (RSP) $φ$ . The left column presents $2 s + 3 s$ and the middle column presents $2 p + 3 p$ ; the dotted black line shows the phase of maximal absorption and the purple line is a contour showing where the absorption is zero. Negative absorption is interpreted as emission of energy to the field by the atom (blue). The right column presents $2 p + 3 p$ with lineouts for the phases $φ = 0$ (solid line), $φ = \frac{3 π}{4}$ (dashed line), and $φ = - \frac{3 π}{4}$ (dash-dotted line).
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Figure 4
Disentangled fundamental processes in energy-resolved absorption, given in arbitrary units, as a function of the relative superposition phase (RSP) $φ$ for a hydrogen atom in superposition: $2 p + 3 p$ . (a) Resonant-continuum contribution [cf. Fig. 1, process (I)]. (b) Resonant-bound contribution, [cf. Fig. 1, process (II)]. (c) and (d) Off-resonant “ $-$ ” and “ $+$ ” frequency contributions, respectively [cf. Fig. 1, process (III)]. Transitions to continuum states (above the ionization threshold) are shown in panels (a), (c), and (d), while in panel (b) pulse parameters with a lower central frequency have been used to show the transitions to the bound states. The $R_{b -}^{j j^{'}}$ contribution has been scaled up by a factor of 10 for clarity of view in panel (b).
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Figure 5
Energy and relative-superposition-phase (RSP)-resolved absorption of a hydrogen atom in the prepared superposition: $2 p + 3 p$ , subject to an attosecond pulse with the intermediate angular momentum restricted to the $s$ wave and the $d$ wave in panels (a) and (b), respectively.
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Figure 6
Energy-resolved absorption by a neon atom subject to an attosecond pulse in the superposition (a) $2 p^{- 1} {(3 s + 4 s)}^{L = 1}$ , (b) $2 p^{- 1} {(3 p + 4 p)}^{L = 2}$ , (c) $2 p^{- 1} {(3 d + 4 d)}^{L = 3}$ , and (d) $2 p^{- 1} {(3 p + 4 p)}^{L = 0}$ , with an estimated scale for the relative superposition phase (RSP) $φ$ . The dotted black line shows the phase of maximal absorption (interpolation is used between discrete simulated values), which can be interpreted using the rule of slope (see main text).
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