Absorption by laser generated ions the density matrix description

Utilizing the validity of the introduction of a susceptibility, the wave equation (see Eq. 5.10) can be solved analytically if the slowly varying amplitude approximation is employed. The solution is:

e

Epr(x, ω) =Eepr(x0, ω)e2πiω/c

Rx

x₀χ(1)(x′,ω) dx′_{. (5.18)} Expressing the solution in terms of the experimentally relevant intensity Iepr(x, ω) of the

probe field yields

e

Ipr(L, ω) =Iepr(0, ω)e−4πω/c

RL

0 Im[χ(1)(x′,ω)] dx′, (5.19)

where L is the length of the interaction volume. For a homogeneous target medium and the introduction of the single-photon absorption cross section

σ(1)(ω) = 4πω/(c n0)Im

χ(1)(ω), (5.20)

Eq. 5.19 results in the well-known Lambert Beer’s law. At this point it should be men- tioned that the validity of Lambert Beer’s law relies on the condition that in the presence of a coherently populated ensemble of state the probing time (pulse duration of the probe pulse) is substantially shorter than the coherent electronic dynamics which evolve within the electronic structure of the target medium.

5.1.3

Absorption by laser generated ions - the density matrix

description

In view of the availability of short and powerful laser pulses which can be used as the probe pulse, the creation of higher charged states in the atomic ensemble is very likely. In this case neutral atoms as well as the created ions will encode strong characteristic absorption features onto the spectrum of the probe pulse. For electronic systems which undergo an ionization upon the interaction with a strong pump laser field, the description of the polarization response can be described in terms of Slater determinants and the mean-field model [104], [106]. For the general description of the electronic structure problem, the ansatz _|Ψpu(t)i=Piαie−i(Ei−E0)t|ϕii was utilized. Expressing |Ψpu(t)i in terms of Slater

determinants gives: |Ψpu(t)i=α0|Φ0i+ X i X a αa_iei(Ea−Ei)t_|Φa ii (5.21)

where_|Φ0i is the ground state determinant and all particle hole configurations|Φaiiwhich

are obtained by exciting or ionizing an electron from spin orbital i (hole orbital) to an unoccupied spin orbital a (particle). The energies of the spin orbitals are denoted by Ei

and Ea. Substituting ansatz Eq. 5.21 into Eq. 5.7 and following the same steps of the

earlier shown derivation gives an expression for the polarization response which consists of several terms that describe the absorption of Epr by neutral ground state atoms and

strong field created ions. An explicit display of this polarization response would exceed the scope of this work. However, an exemplary description of the polarization response - especially in view of the conducted experiments - of a gas target where the probe pulse only undergoes absorption by singly charged ions will be given. Furthermore it shall be assumed that the probe pulse only triggers hole-hole transitions which means that a core shell electron is excited into the vacancy in the valence shell of the singly charged atomic species A+_.

The NIR generated photoelectrons (particles) can be either slightly bound or in the contin- uum. If they are free photoelectrons they cannot absorb EUV photons. If they still feel the ionic potential, they can absorb EUV photons. However the cross-section for this process is very low and therefore it is a good approximation to assume that the pump field generated photoelectrons do not interact with the probe field. In the particle-hole terminology this would translate into the disregard of EUV initiated particle-particle transitions.

Since the electron excited or ionized by the pump pulse is a spectator of the hole-hole transition of the ion core, it is very convenient to describe the A+ _{ions with a reduced}

density matrix formalism [106]:

ρ(_iiA′+)(t) = e i(Ei−Ei′)tX a αa_iαa_i′∗ =ei(Ei−Ei′)t ρe (A+₎ ii′ . (5.22)

The diagonal elements of the density matrix ρeii are the probabilities of the corresponding

ionic states. The non-zero off-diagonal elements are related to the possibility of observing quantum beats; therefore we will refer to them as coherences.

Considering the temporal regime where pump and probe pulse do not temporally overlap, the matrix ρe(_iiA′+) can be considered as time independent. This results in the following

polarization response: e P(1,A+)(x, ω >0) =n0 X i,i′ e ρ(_iiA′+) X f µf i′ µ_if Ei′ −E_f −iΓ_f/2−ω · e Epr(x, ω+Ei−Ei′). (5.23)

The dipole matrix elementµif describes the filling of a hole in spin orbitaliwith an electron

from spin orbital f and µf i′ depicts the process which fills the hole created in f with an

electron from orbital i′_{. Finally the general linear susceptibility shown in Eq. 5.17 yields}

for singly charged ionic absorbers A+_{, the validity of the short pulse approximation and}

considering only hole-hole transitions in the reduced density matrix formalism:

χ(1,A+)(ω >0, τ) = n0 X i,i′ e ρ(_iiA′+)(τ) X f µf i′ µ_if Ei′ −E_f −iΓ_f/2−ω . (5.24)

It should be emphasized that the wavefunction _|Ψpu(t)i (Eq. 5.21), which describes the

electronic structure problem after interaction with the pump laser pulse, is composed of the wavefunctions of the bound and free electrons, whereas the description of the problem with a reduced density matrix allows to describe a coherent superposition of ionic states without explicitly considering the free electrons. This is in fact the reason for naming it

reduced density matrix formalism.

In the next section an explicit description of the single-photon absorption cross section of krypton singly-charged ions will be given, to describe one of the performed attosecond transient absorption experiments where a few cycle NIR laser pulse was used as the pump pulse and a delayed attosecond XUV pulse probed the strong field created ionic ensemble.

5.2

Real-time observation of valence electron motion

in Kr