with the experimental data of the metastable
Ne yield. The theoretical result for the SRXS
process (blue curve) shows that the yield rose
sharply just below the inner-shell 2p→1s
resonance at 849 eV toward higher photon
energy and stayed relatively flat until it sharp-
ly declined beginning at the resonance photon
energy of the 1s→3p transition. It is obvious
from the data that both resonances enhanced
the Nestimyield. To see how the theoretical
SRXS data agree with the experimental data,
in Fig. 3B we compare the measured Ne*stim
yield normalized to the Ne*sponyield with the
ratio determined using the data shown in Fig.
3A. We found a good overall agreement of the
ratio as a function of the photon energy. With
the XFEL pulse intensity used in the calcula-
tions, the deviation from the observed ratio was
always smaller than a factor of 2. At photon
energies below 852 eV, the Ne*sponyield de-creased slowly with lower incident photon en-
ergy so that the ratio almost directly reflected
the decreasing Ne*stimyield. On the other
hand, when approaching the 1s→3p reso-
nance, the Ne*sponyield was orders of mag-
nitude larger than the Ne*stimyield, leaving
the ratio close to zero.
We further explored the stimulated x-ray
Raman scattering by investigating the Ne*stim
yield as a function of the intensity of the
XFEL pulses. In Fig. 4A, we show a measure-
ment of the Ne*stimyield at an incident
photon energy of 850 eV as a function of the
XFEL pulse energy. The double logarithmic
plot shows that the data points to a good
approximation followed a quadratic power
lawImwithm= 2 (dashed line). Theoretically,
the Ne*stimyield (Fig. 4B) followed a nearly
quadratic power law at low intensities, and
changed to a power law with a slope ofm=
1.2 at higher XFEL intensities. Obviously, the
inner electron transition is already close to
saturation. We note that at this intensity, the
Rabi oscillation frequency is on the order of
the inverse lifetime of the core excited state.
In Fig. 4C we show the power-law exponent
across the resonance for two intensity regions.
At pulse intensities well below 0.9 × 10^16 W cm–^2 ,
we found an exponent ofm= 2 over a wide
range of incident photon energies, except at
the 1s→3p resonance and toward higher
energies. Off-resonance, we found a quadratic
behavior, as expected for the SXRS process.
Our work establishes PRI as a tool for the
study of nonlinear x-ray physics. In contrast
to direct SXRS measurements, PRI is free
from background noise caused by the pri-
mary radiation and undisturbed by modifica-
tions of the signal due to interactions and
pulse propagation in dense media. In basic
molecular physics, it has been a persistent
objective to implement nonlinear x-ray meth-
ods ( 6 , 7 , 17 ) to investigate and control coherent
wave function evolution and intramolecular1632 25 SEPTEMBER 2020•VOL 369 ISSUE 6511 sciencemag.org SCIENCE
Fig. 3. Comparison of
calculated populations of
excited Ne atoms after
spontaneous and stimu-
lated Raman scattering
with experimental results
as a function of the inci-
dent x-ray photon energy.
(A) The calculated population
of Ne atoms by spontaneous
(black curve) and stimulated
(blue curve) Raman
scattering. The curves have
been multiplied by a single
common factor to match (fit
by eye) the experimental
data for the yield of spontaneously populated Ne atoms (red points). (B) The ratio Nestim/Nesponfrom theoretical data (small red squares) compared with the ratio
Nespon/Ne*stim(blue points) determined from the experimental yields. Error bars denote SD.
Fig. 4. Measured and
calculated dependence of
the Nestimyield on
the XFEL pulse energy.
(A) Nestimyield (counts in
the narrow stripe shown in
Fig. 2C) as a function of the
XFEL pulse energy (inten-
sity) measured at an inci-
dent photon energy of 850
eV. The dashed curve rep-
resents a yield proportional
to the square of the XFEL
pulse energy, also displayed
in (B). The solid black curve
is a weighted fit to the upper
three data points. Error bars
denote SD. (B) Calculated
Nestimyield as a function of
the XFEL intensity using
unchirped pulses at an inci-
dent photon energy of
850 eV. (C)CalculatedNestim
(black squares) yield, dis-
played using unchirped
XFEL pulses (same as
shown in Fig. 3A) together
with the exponentmof
the power lawImcharacter-
izing the intensity depen-
dence of the Ne*stimyield at
intensities below 0.9 × 10^16 W cm–^2 (open blue circles) and above that intensity (red points).
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