Non-resonant two-photon x-ray absorption in Cu
J. J. Kas and J. J. Rehr
∗
Dept. of Physics, Univ. of Washington Seattle, WA 98195
J. St¨ohr
SLAC National Accelerator Laboratory, Menlo Park, CA 94025
J. Vinson
Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland, 20899
(Dated: July 11, 2025)
We present a real-space Green’s function theory and calculations of two-photon x-ray absorption
(TPA). Our focus is on non-resonant K-shell TPA in metallic Cu, which has been observed experi-
mentally at intense x-ray free electron laser (XFEL) sources. The theory is based on an independent-
particle Green’s function treatment of the Kramers-Heisenberg equation and an approximation for
the sum over non-resonant intermediate states in terms of a static quadrupole transition operator.
XFEL effects are modeled by a partially depleted d-band. This approach is shown to give results for
K-shell TPA in quantitative agreement with XFEL experiment and with a Bethe-Salpeter Equation
approach. We also briefly discuss many-body corrections and TPA sum-rules.
Keywords: Green’s function, Two-photon absorption, XAS, XFEL
I. INTRODUCTION
Two-photon absorption (TPA) and emission (TPE)
processes were originally predicted theoretically by Maria
Goeppert-Mayer in her doctoral dissertation.
1,2
However,
TPA was not observed until lasers became available and
then only for optical frequencies.
3
More recently, TPA
of hard x-rays has been observed for metallic Cu using
intense x-ray free electron laser (XFEL) sources.
4
For-
mally, the theory of TPA is based on a sum of amplitudes
for two successive dipole transitions over all possible in-
termediate states. This sum is given by the Kramers-
Heisenberg (KH) equation. Energy is conserved only for
the net transition, with the transition energy equal to the
sum of the two photon energies.
1,5–7
This process is illus-
trated by the Feynman diagrams
8
in Fig. 1. The left di-
agram depicts a process in which the first photon excites
an occupied p−state to the final s− or d−photoelectron
state and the second photon excites the 1s electron to the
now empty p−state. The right diagram depicts the other
FIG. 1. Feynman diagrams
8
for the TPA amplitude: Incident
photons are represented by wavy lines (blue), the single par-
ticle state |i⟩ by the black line, the the photoelectron |f⟩ by
the green line, and intermediate states |n⟩ by the orange line.
The left diagram indicates occupied intermediate states, while
the right diagram indicates unoccupied intermediate states.
possible process, in which the first photon excites the 1s
electron to an unoccupied p-photoelectron state, and the
second photon scatters this photoelectron to the final
s− or d−photoelectron state. While the KH approach
is tractable for atomic systems,
5,6
and non-linear ap-
proaches have been developed for optical spectra,
9
quan-
titative TPA calculations are computationally challeng-
ing for condensed matter. However, for K-shell TPA in
Cu with ≈ 4500 eV photons,
4
only non-resonant interme-
diate states are possible, greatly simplifying the theory.
For this case an approximation for K-shell TPA based on
the Bethe-Salpeter Equation (BSE) has been proposed.
10
Our goal here is to develop a real-space Green’s func-
tion (RSGF) approach for deep core TPA in condensed
matter that only includes non-resonant contributions and
is applicable for simulations of XFEL spectra. We show
that this method can be expressed in a form analogous to
one-photon (OPA) x-ray absorption spectra (XAS), but
with an effective static quadrupole transition operator.
TPA calculations are presented based on an extension of
the RSGF XAS code FEFF10.
11
XFEL effects on the
near-edge are modeled by a partially depleted d-band.
This theory yields K-shell TPA spectra for Cu in good
agreement with XFEL experiment
4
and with the BSE
approach.
6
II. TPA THEORY
Within 2nd-order perturbation theory in the electron-
photon interaction, the TPA cross-section σ
2P
XAS
is given
