arXiv:1012.4930v1 [physics.atom-ph] 22 Dec 2010
Novel light from high-order harmonic generation manipulated by xuv light
Christian Buth,1, ∗Markus C. Kohler,1Joachim Ullrich,1,2and Christoph H. Keitel1
1Max-Planck-Institut f¨ ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
2Max-Planck Advanced Study Group at CFEL, 22607 Hamburg, Germany
(Dated: December 23, 2010)
The combination of high harmonic generation (HHG) with resonant xuv excitation of a core
electron into the transient valence vacancy that is created in the course of the HHG process is inves-
tigated theoretically. In this setup, the first electron performs a HHG three-step process whereas,
the second electron Rabi flops between the core and the valence vacancy. The modified HHG spec-
trum due to recombination with the valence and the core is determined and analyzed for krypton on
the 3d −→ 4p resonance in the ion. We assume an 800nm laser with an intensity of about 1014
and xuv radiation from the Free Electron Laser in Hamburg (FLASH) with an intensity in the
cm2. Our prediction offers novel prospects for nonlinear xuv physics, attosecond
x rays, and tomographic imaging of core orbitals.
PACS numbers: 42.65.Ky, 32.80.Aa, 32.30.Rj, 41.60.Cr
The light from high harmonic generation (HHG)  is
studied where HHG is manipulated by intense xuv light
from the newly constructed free electron lasers (FEL)—
e.g., the Free Electron Laser in Hamburg (FLASH) —
which induces Rabi flopping of a second electron be-
tween the core and the valence vacancy that is created
when a valence electron tunnel ionizes.
proach makes single attosecond x-ray pulses feasible us-
ing the same methods that are used conventionally .
The novel scheme also may represent an optical gating
to perform frequency resolved optical gating (FROG) 
with xuv pulses thus offering the long-sought pulse char-
acterization for chaotic self-amplification of spontaneous
emission (SASE) xuv light . Above all tomographic
imaging of core orbitals with HHG comes into reach —
which has only been possible for valence orbitals so far.
Such a proposed combination of optical and xuv light—
a so-called two-color problem—has already offered excit-
ing prospects [6, 7] for studying atoms and molecules
and controlling the interaction of xuv radiation with
Our new ap-
A few methods exist to increase the maximum pho-
ton energy (cutoff) of HHG for a given laser intensity.
First, a two-electron scheme was considered that uses se-
quential double ionization by a laser with a subsequent
nonsequential double recombination; in helium it leads
to a second plateau with about 12 orders of magnitude
lower yield than the primary HHG plateau . Second,
two-color HHG (optical plus UV) has been studied in a
one-electron model; the UV assists thereby in the ioniza-
tion process leading to an overall increased yield  and
the emergence of a new plateau , the latter, however,
at a much lower yield. Third, by optimizing the quan-
tum path of the continuum electron, the HHG cutoff is
increased by a factor of 2.5 .
Our principal idea of an efficient HHG process in the
xuv regime is sketched in Fig. 1.
the three-step model [13, 14], HHG proceeds as follows:
In the parlance of
FIG. 1: (Color online) Schematic of the three-step model for
the HHG process augmented by xuv excitation of a core elec-
(a) the atomic valence is tunnel ionized; (b) the liberated
electron propagates freely in the electric field of the op-
tical laser; (c) the direction of the laser field is reversed
and the electron is driven back to the ion and eventually
recombines with it emitting HHG radiation. The excur-
sion time of the electron from the ion is approximately 1fs
for typical 800nm laser light. During this time, one can
manipulate the ion such that the returning electron sees
the altered ion as depicted in Fig. 1. Then, the emit-
ted HHG radiation bears the signature of the change.
Perfectly suited for this modification during the propa-
gation step is xuv excitation of an inner-shell electron
to the valence shell. The recombination of the return-
ing electron with the core hole leads to a large increase
of the energy of the emitted HHG light as the energy of
the xuv photons ωXis added shifting the HHG spectrum
towards higher energies. A prerequisite for this to work
certainly is that the core hole is not too short lived, i.e.,
it should not decay before the electron returns. We apply
our theory to study HHG in krypton atoms where tunnel
ionization leads to 4p vacancies. The xuv light is tuned
to the 3d −→ 4p resonance in the cation. If the xuv light
is very intense, even Rabi flopping is possible. Such Rabi
oscillations were predicted in neon atoms (without opti-
cal laser) . Our equations are formulated in atomic
The spatial one-electron states of relevance to the prob-
lem are the valence state |a? and the core state |c? of the
closed-shell atom. In strong-field approximation, contin-
uum electrons are described by free-electron states |?k?
for all?k ∈ R3. The associated level energies are Ea,
2. We need to consider three different classes
of two-electron basis states to describe the two-electron
dynamics: first, the ground state of the two-electron sys-
tem is given by the Hartree product |a?⊗|c?; second, the
valence-ionized states with one electron in the continuum
and one electron in the core state are |?k? ⊗ |c?; third,
the core-ionized states with one electron in the contin-
uum and one electron in the valence state are |?k? ⊗|a?.
We apply the three assumptions of Lewenstein et al. 
in a somewhat modified way by considering also phe-
nomenological decay widths of the above three state: Γ0
and Γa to account for direct valence ionization by the
xuv light by |a? ⊗ |c? and |?k? ⊗ |c?, respectively, and
Γcto represent direct valence ionization by the xuv and
Auger decay of core holes for |?k?⊗|a? with radiative de-
cay of the core hole being safely neglected. Further, the
effect of the xuv light is confined to the two-level system
of |?k? ⊗ |a? and |?k? ⊗ |c?.
The two-electron Hamiltonian of the atom in two-color
light (laser and xuv) reads
ˆH =ˆHA+ˆHL+ˆHX; (1)
it consist of three parts: the atomic electronic struc-
tureˆHA, the interaction with the laserˆHL, and the inter-
action with the xuv lightˆHX. We constructˆH from ten-
sorial products of the corresponding one-particle Hamil-
toniansˆhA,ˆhL, andˆhX. The laser and xuv interaction
is treated in dipole approximation in length form .
We make the following ansatz for the two-electron
|Ψ,t? = a(t)e−i
2(Ea+Ec−ωX)t+iIPt|a? ⊗ |c?
2(Ea+Ec−ωX)t+iIPt|?k? ⊗ |c?
2(Ea+Ec+ωX)t+iIPt|?k? ⊗ |a??d3k ,
2(Ea+ Ec+ ωx) = −Ea+δ
on the amplitudes ba(?k,t) and bc(?k,t) indicates which or-
bital contains the hole. The detuning of the xuv photon
energy from the energy difference of the two ionic levels
is δ = Ea− Ec− ωX.
We insert |Ψ,t? into the time-dependent Schr¨ odinger
equation and project onto the three classes of basis states
which yields equations of motion (EOMs) for the in-
volved coefficients. We obtain the following EOM for
the ground-state population
where IP= −1
2. The index
dta(t) = −Γ0
2a(t) − i
ba(?k,t) ?a|ˆhL|?k? d3k . (3)
The other two EOMs are written as a vector equation
defining the amplitudes?b(?k,t) =?ba(?k,t),bc(?k,t)?T, the
Rabi frequency R0X  for continuous-wave xuv light
and the Rabi matrix
?−δ − iΓa
δ − iΓc
This yields for a continuous-wave laser field EL(t):
?b(?k,t) = −i
?R + (?k2+ 2IP)
???b(?k,t) + EL(t)(5)
?b(?k,t) − ia(t)EL(t) ??k|ˆhL|a?
We change to the basis of xuv-dressed states using the
eigenvectors U and eigenvalues λ+, λ−of R.
To determine the HHG spectrum, we solve Eq. (3) for
a low tunnel ionization rate—by neglecting the second
term on the right-hand side as in Ref. —and a con-
stant xuv flux starting at t = 0 and ending at t = TP.
The HHG spectrum is given by the Fourier transform
of ?Ψ0,t|ˆD|Ψc,t?, whereˆD is the two-electron electric
dipole operator , |Ψ0,t? is the ground-state part of
the wavepacket (2) and |Ψc,t? is the continuum part,
˜D(Ω) = −i
bM−N,i(τ)hM−N,0,i(Ω,τ) dτ .
Definitions of the constants and functions here little rel-
evant can be found in Ref. . Further,
2τ(1 − e−(Γ0+i˜ΩM,N,i−Ω)TP)
Γ0+ i(˜ΩM,N,i− Ω)
Neglecting the factor e−Γ0
the hM,N,i(Ω,τ) peak at Ω =˜ΩM,N,i = (2(M + N) +
δi,a)ωL+δi,cωX. In other words, for i = a the peaks are
at the positions of the harmonics without ground-state
depletion; for i = c, the harmonics are shifted by ωXwith
respect to the harmonics for i = a such that, in general,
they do not coincide with higher orders of radiation from
valence recombination. Above all, the hM,N,i(Ω,τ) have
a finite width; their real part is in the limit TP → ∞
proportional to a Lorentzian with a width of 4Γ0. The
harmonic photon number spectrum (HPNS) for a sin-
gle atom—the probability to find a photon with specified
energy—along the x axis is given by
for now, we see that
= 4π Ω̺(Ω) |˜D(Ω)|2,
with the density of free-photon states ̺(Ω)  and the
solid angle ΩS. The cases of continuous-wave xuv light
0.00.51.0 1.5 2.02.5 3.0
FIG. 2: (Color online) time evolution of the probabilities to
find the electron in the valence [solid black line] or the core
state [dashed red line] for xuv intensities of (a) 1013
without ground-state depletion and arbitrary xuv pulses
will be treated in Ref. .
We apply our theory to krypton atoms. The energy
levels are Ea = −13.0eV for Kr4p and Ec = −96.6eV
for Kr3d with a radial dipole transition matrix element
of 0.206Bohr. The xuv light has a central frequency
which is resonant with the 3d −→ 4p transition, i.e., the
photon energy is ωX = Ea− Ec. The laser intensity is
set to I0L = 5 × 1014 W
The experimental value for the decay width of Kr3d va-
cancies is Γexpt = 88meV which corresponds to a life-
time of 7.5fs . This lifetime is much longer than the
typical excursion time of electrons in the HHG process
of ≈ 1fs for a laser with 800nm wavelength and is thus
very well suited for our purposes here. The decay width
due to direct valence ionization of the atom and the ion
are obtained from the responsible photoionization cross
sections; we find for an xuv-intensity of I0X= 1013 W
Γ0= 0.17meV, Γa= 0.17meV, and Γc= 88meV, and
for an xuv-intensity of I0X= 1016 W
Γa= 170meV, and Γc= 300meV.
Before we investigate HHG, we would like to examine
the isolated case of Rabi flopping of an electron in the
two-level system of the core state and the valence hole
which occurs after tunnel ionization of the atom. We
notice that Eq. (5) becomes the well-known Rabi equa-
tion  by omitting all terms related to HHG. For this
situation, we display in Fig. 2 the time evolution of the
population of the probabilities to find the electron in the
valence and the core state where t = 0fs is the instant of
tunnel ionization of the valence electron. For moderate
xuv intensity [Fig. 2a] the population of the 4p level is
small on the time scale of the laser period. Applying a
higher intensity in Fig. 2b, the peak of the first Rabi os-
cillation is located slightly above 1fs. As the excursion
cm2 at a wavelength of 800nm.
cm2: Γ0= 170meV,
Harmonic order h
FIG. 3: (Color online) Photon number of hth harmonic or-
der for xuv intensities of (a) 1013
black solid lines show the contribution from recombination
with a valence hole whereas the red dashed lines correspond
to recombination with a core hole. The lines represent har-
monic strengths obtained by integrating over the finite peaks
in Eq. (8).The thin lines are spectra where ground-state
depletion due to direct valence ionization by the xuv light
is neglected; the resulting HHG rate  is multiplied by the
xuv pulse duration TPfrom Fig. 4 of three optical laser cycles.
cm2 and (b) 1016
time of the tunneled electron is on this time scale, the
returning electron encounters with a high probability a
In Fig. 3 we show the corresponding single-atom
HHG spectra to Fig. 2 modified by xuv light where we
distinguish parts due to valence- and core-hole recom-
bination with their plateaus slightly overlapping. The
width of the overlap can be tuned by changing the laser
intensity and interferences between both terms occur
when the lines from both contributions overlap. Given
the reasoning from the previous paragraph, it is at first
sight striking that also in the case of a moderate xuv in-
tensity [Fig. 3a] the emission rate of HHG is substantial.
However, a closer inspection reveals that the strength
of the HHG emission due to core-hole recombination is
roughly proportional to the population of the upper state
around 1fs in Fig. 2. Comparing the emission from core
recombination with the emission from valence recombina-
tion, we find an appreciably lower yield of the former with
respect to the latter. This is found also for an xuv in-
tensity of 1016 W
cm2although the probability of finding the
Rabi flopping electron in either of the bound states is on
average similar. The reason for the observed differences
in the HHG yield lies in the different dipole matrix el-
ements for valence- and core-hole recombination where
the former is somewhat larger than the latter. Ignoring
the impact of depletion due to direct valence ionization
by the xuv radiation of the neutral atom and the cation
leads to the thin-line HHG spectra in Fig. 3. We see
that even for strong xuv intensity the consequences of
Envelope ?Eh??e a0??
20 40 6080 100 120140
Harmonic order h
FIG. 4: (Color online) (a) a sample SASE pulse and (b) the resulting harmonic photon number spectrum [Eq. (8)]. In (a) the
amplitude is given by the solid black line and the phase by the dashed red line. Line styles in (b) as in Fig. 3.
depletion are low.
So far we have assumed entirely coherent xuv light
with constant amplitude. This is not the case for radia-
tion from present-day FELs which is generated with the
SASE principle . Such light is transversally coherent
but exhibits only limited longitudinal coherence. In our
case, the coherence time is about 5fs  which is much
longer than the laser cycle and thus the HHG process can
occur within an interval where the laser and the xuv ra-
diation are coherent. To explore the implications of par-
tially coherent radiation with fluctuating amplitudes, we
construct model pulses following Ref. [21, 22]. We employ
a Gaussian spectrum centered on the 3d −→ 4p transi-
tion energy with a full width at half maximum of 0.7%
of the transition energy . Further, the pulses are con-
structed to have a cosine-square temporal shape on av-
erage and a total duration of three cycles of the optical
laser with a peak intensity of 1016 W
envelope and phase of a sample pulse in Fig. 4a. The
HHG spectrum that is produced by the pulse [Fig. 4b] is
determined in zero-order approximation of the equations
for arbitrary xuv pulses . We find that the spectra
deviate only moderately for different pulse shapes and
the fluctuating phase does not destroy the spectra.
The manipulation of HHG with xuv radiation gener-
ates HHG light from core orbitals with an efficiency which
is very close to HHG from valence recombination (about
an order of magnitude lower efficiency in Fig. 3b). The
extension of the HHG cutoff is thus not bought dearly
by a diminishing yield in contrast to the conventional
route [9, 11, 23]. Using a future hard x-ray FEL tuned to
the 1s −→ 4p resonance in the krypton ion (14.3keV ),
instead of xuv radiation, leads to an enormous upshift
of the HHG cutoff.
In conclusion, we predict novel light from resonant ex-
citation of transient ions in HHG that allows insights into
the physics of core electrons and has various applications:
it allows one to generate isolated attosecond x-ray pulses
by ionizing atoms near the crests of a single-cycle optical
laser pulse . The HHG spectra depend sensitively on
the xuv pulse shape; a reconstruction with the FROG
method  may be possible but requires further theo-
retical investigation. Finally, the emitted upshifted light
cm2. We show the
due to core recombination bears the signature of the core
orbital; thus it can be used for its tomographic recon-
struction  which is not feasible so far. This allows one
to extend tomographic imaging to all orbitals that cou-
ple to the transient valance vacancy by suitably tuned
C.B. and M.C.K. were supported by a Marie Curie In-
ternational Reintegration Grant within the 7thEuropean
Community Framework Program (call identifier: FP7-
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