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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

range 1013–1016

cm2. Our prediction offers novel prospects for nonlinear xuv physics, attosecond

x rays, and tomographic imaging of core orbitals.

W

cm2

W

PACS numbers: 42.65.Ky, 32.80.Aa, 32.30.Rj, 41.60.Cr

The light from high harmonic generation (HHG) [1] 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) [2]—

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 [3].

The novel scheme also may represent an optical gating

to perform frequency resolved optical gating (FROG) [4]

with xuv pulses thus offering the long-sought pulse char-

acterization for chaotic self-amplification of spontaneous

emission (SASE) xuv light [2]. Above all tomographic

imaging of core orbitals with HHG comes into reach [5]—

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

atoms [8].

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 [9]. 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 [10] and

the emergence of a new plateau [11], 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 [12].

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-

tron.

(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) [15]. Our equations are formulated in atomic

units.

The spatial one-electron states of relevance to the prob-

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2

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,

Ec, and

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. [16]

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

?k2

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 [17].

We make the following ansatz for the two-electron

wavepacket

|Ψ,t? = a(t)e−i

2(Ea+Ec−ωX)t+iIPt|a? ⊗ |c?

?ba(?k,t)e−i

+ bc(?k,t)e−i

(2)

+?

R3

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

d

dta(t) = −Γ0

2a(t) − i

?

R3

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 [18] for continuous-wave xuv light

and the Rabi matrix

R =

?−δ − iΓa

R0X

R0X

δ − iΓc

?

.

(4)

This yields for a continuous-wave laser field EL(t):

∂

∂t

?b(?k,t) = −i

2

?R + (?k2+ 2IP)

∂

∂kz

???b(?k,t) + EL(t)(5)

×

?b(?k,t) − ia(t)EL(t) ??k|ˆhL|a?

?1

0

?

.

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. [16]—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 [17], |Ψ0,t? is the ground-state part of

the wavepacket (2) and |Ψc,t? is the continuum part,

i.e.,

˜D(Ω) = −i

?

i∈{a,c}

j∈{+,−}

Uijwj

∞ ?

0

?

(−2πi)3

τ3

e−iF0,j(τ)

×

∞

?

∞

?

N=−∞

iNJN

?UP

ωLC(τ)?eiN ωLτ

(6)

×

M=−∞

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. [17]. Further,

hM,N,i(Ω,τ) =e−Γ0

2τ(1 − e−(Γ0+i˜ΩM,N,i−Ω)TP)

Γ0+ i(˜ΩM,N,i− Ω)

.

(7)

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

2τ

for now, we see that

d2P(Ω)

dΩ dΩS

= 4π Ω̺(Ω) |˜D(Ω)|2,

(8)

with the density of free-photon states ̺(Ω) [18] and the

solid angle ΩS. The cases of continuous-wave xuv light

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?a?

10?6

100

10?4

10?2

100

Probability

?b?

0.00.51.0 1.5 2.02.5 3.0

10?6

10?4

10?2

Time ?fs?

Probability

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

(b) 1016

W

cm2and

W

cm2.

without ground-state depletion and arbitrary xuv pulses

will be treated in Ref. [17].

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 [19]. 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 [18] 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:

cm2: Γ0= 170meV,

?a?

10?13

10?10

10?7

dPh?d?S

?b?

20406080100120

10?13

10?10

10?7

Harmonic order h

dPh?d?S

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 [17] is multiplied by the

xuv pulse duration TPfrom Fig. 4 of three optical laser cycles.

W

cm2 and (b) 1016

W

cm2. The

time of the tunneled electron is on this time scale, the

returning electron encounters with a high probability a

core-excited ion.

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

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4

?a?

01234567

0.0

0.2

0.4

0.6

0.8

1.0

1.2

?Π

0

Π

Time ?fs?

Envelope ?Eh??e a0??

Phase

?b?

20 40 6080 100 120140

10?14

10?11

10?8

10?5

Harmonic order h

d2P?????d? d?S?

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 [2]. Such light is transversally coherent

but exhibits only limited longitudinal coherence. In our

case, the coherence time is about 5fs [20] 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 [20]. 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 [17]. 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 [7]),

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 [3]. The HHG spectra depend sensitively on

the xuv pulse shape; a reconstruction with the FROG

method [4] 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 [5] 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

xuv light.

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-

PEOPLE-2010-RG, proposal No. 266551).

∗Corresponding

christian.buth@web.de

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