IV. Coherent ionization dynamics in crystals 41
IV.2.1. Schrödinger equation for non-interacting electrons
The equations of motion for the spin-independent 1RDM %nm(k, k0) := hc†nkσcmk0σi can be derived from the Heisenberg equations for the creation and annihilation operators:
idtd%nm(k, k0) = (mk0−nk)%nm(k, k0) +X
j
Wmj(k)%nj(k, k0)−Wnj∗ (k)%jm(k, k0) , (IV.4) where the matrix elements are given byWnj(k) :=αA(t)hψnkσ|pˆ|ψjkσi. Due to the simple form of the HamiltonianH, the equations of motion automatically decouple for every pairk, k0. As the analyses in this chapter concentrate on occupation probabilities of bands, it is sufficient to restrict Eq. (IV.4) to thek-diagonal part of the1RDM%nm(k) :=%nm(k, k). Alternatively, one can argue that thek-off-diagonal parts remain zero for all times according to Eq. (IV.4) if the initial state is diagonal. In both cases, one obtains the following closed set of equations of motion for%nm(k):
Time-dependent Schrödinger equation(TDSE) idtd%nm(k) = (mk−nk)%nm(k) +X
j
Wmj(k)%nj(k)−Wnj∗(k)%jm(k)
. (IV.5)
Being a direct consequence of thetime-dependent Schrödinger equation(TDSE) with respect to the Hamiltonian in Eq. (IV.3), these equations of motion are referred to asTDSEor Schrödinger theory below.
45
IV.2. Equations of motion
The physical quantities of interest are the average occupation probabilities of all bands. For this purpose, Eq. (IV.5) has to be solved for an appropriate discretization of thekdomain, the firstBZ. To be more precise, the periodic boundary conditions suggest that thekdomain should be divided into a uniform grid [261]. For each Bloch wave vectork, there is a corresponding equation of the form Eq. (IV.5) that is decoupled from the values of the1RDM at a distinct k0 6=k. The average occupation probabilities of a bandnare then obtained as thekaverage over
%nn(k). For the sake of brevity, the parametric dependence onkis omitted in spite of actually being present in Eq. (IV.5) and equations below.
IV.2.2. Resonant approximation
The derivation of the rate description in the present and the ensuing subsection follows the procedure fromRossi and Kuhn (2002)[300]. This simplified perspective, where ultimately only the diagonal elements of the1RDM%nn(k)appear in the dynamical equations, is known to be applicable if both non-resonant contributions and the memory effects can be safely disregarded.
To distinguish the implications of these two distinct approximations, an approach will be introduced which solely requires the assumption of the dominance of resonant terms in the equations of motions. This intermediate level of theory between theTDSEand the rate-equation approach is straightforwardly achieved by using an intermediate result of the aforementioned derivation.
For this purpose, the strategy consists in the first instance of a reformulation of Eq. (IV.5) emphasizing on the occupation probabilities:
In this step, the generation rategjn(t)for the occupation of statenstemming from statej was introduced. Eqs.IV.6and IV.7obviously do not constitute a closed set of equations because the generation rate depends on the off-diagonal elements of %nm. To resolve this issue, the off-diagonal elements are replaced in Eq. (IV.7) by the formal solution of theTDSEwith respect to the initial condition lim with the transition frequencyωnm:=n−m. For a compact notation, one defines the auxiliary quantitiesW˜nj(t) :=eiωnjtWnj(t)and%enj(t) := e−iωnjt%nj(t). Then, one obtains the generation
46 IV. Coherent ionization dynamics in crystals
So far, no further approximation compared to theTDSEhas been made. Now, it will be shown that under appropriate conditions the main contribution to the integral in Eq. (IV.9) stems from diagonal elements. This procedure allows to neglect the off-diagonal terms and thus yields a theory that is closer to the rate description. To be able to continue with the derivation, the form of the laser pulse, which is characterized by a time-dependent vector potentialA(t), is further constrained by assuming that
A(t) =A0(t) cos(ω0t), (IV.10) whereω0 is interpreted as the photon energy andA0is the slowly varying amplitude (compared to the fast time scale of1/ω0). The premise of the approximative treatment is that oscillations on the fast time scale1/ω0 and1/ωnmforn 6=maverage out in the integral in Eq. (IV.9). Note that the time dependence of%nmis, based on Eq. (IV.8), roughly given by%nm(t)∝eiωnmt, i.e., e
%nm(t)has a component on the slow time scale for alln, m. In contrast,W˜nm(t)contributes if and only if|ωnm| ≈ω0. It is assumed that for eachkthere is only one pair of indicesn, msuch that this condition is fulfilled. Now, two cases are distinguished.
(i) Letn, jbe the pair of indices such that|ωnj| ≈ω0. Then, the sum in Eq. (IV.9) is reduced to l =nin the first term andl =j in the second term. This procedure yields
gjn(t)≈gjnres(t) := 2<
W˜nj(t) Zt
−∞
dt0W˜jn(t0) %jj(t0)−%nn(t0)
. (IV.11)
(ii) Letn, jbe a pair of indices such that|ωnj| ≈ω0 is not fulfilled. This implies that the factor in front of the integral causes the generation rate itself to vary on the fast time scale and thus even the slowly varying contributions to the integral do not affect the occupation numbers on the time scale of the ionization process. Correspondingly, one can assume that Eq. (IV.11) holds approximately in this case as well. It should be mentioned that this choice is the reason that the fast time scale is still present in the generation rate of the preliminary approach in this subsection.
An alternative derivation of the generation rate Eq. (IV.11) relies on treating the laser pulse as a perturbation. However, the rather complicated consideration presented here based on different time scales has the advantage that it is not limited to weak pulses, so it is able to explain the results close to those obtained with the TDSEfor relatively large intensities (see Sec. IV.3).
Of course, this is only valid to a certain extent because increasing the intensity leads to faster ionization dynamics and thus a more rapid time scale of occupation probabilities. Therefore, the separation of time scales breaks down for sufficiently high intensities.
Lastly, arotating wave approximation(RWA) is introduced by replacing
W˜nj(t)→αA0(t)eiδωnjthψnkσ|pˆ|ψjkσi, (IV.12) where δωnj :=
(ωnj −ω0 forωnj >0, ωnj +ω0 forωnj <0.
The resonant approximationis now defined as the equation of motion in Eq. (IV.6) with the approximative generation rate in Eq. (IV.11) including theRWA. It can be equivalently formulated as
47
IV.2. Equations of motion
Resonant approximation
idtd%nn =X
j
W˜mj%nj−W˜nj∗ %jm
,
idtd%nm= ˜Wmn(%nn−%mm), forn6=m.
(IV.13)
Evidently, oscillations on the fast time scale are still present in the resonant approximation;
despite the assumption that the fast time scale does not contribute to the occupation probabilities, not all of the corresponding terms have been dropped.
IV.2.3. Markovian theory
The final step for achieving the rate-equation limit within the procedure byRossi and Kuhn (2002) [300] is to approximate the integral in the resonant generation rate, Eq. (IV.11), by assuming short-lived memory effects and applying a Markovian approximation (cf. Sec.III.2.2).
Provided that the amplitude of the vector potential is described by a Gaussian function, i.e., A0(t) = ¯A0exp
− t2 2τ02
, (IV.14)
one obtains the followingMarkovian generation rate: Markovian generation rate
gjnMarkov(t) = %jj(t)−%nn(t)
2πWnj(0)(t)2S(δωnj). (IV.15) Here,Wnj(0)(t)is defined asWnj(0)(t) := αA0(t)hψnkσ|pˆ|ψjkσiand contains no frequency compo-nent in the range ofω0 andωnmforn6=m. Further, spectral properties of the transform-limited pulse enter the Markovian generation rate throughS(ω) := √τ2π0 exp(−τ0ω2). Note that Fermi’s golden rule can be recovered from Eq. (IV.15) in the limitτ0 → ∞ as S(ω) → δ(ω). The equations of motion defined by the Markovian generation rate in conjunction with Eq. (IV.6) will be referred to asMarkovian theoryin the following. Originally, the Markovian generation rate was termedsemiclassical generation ratebyRossi and Kuhn (2002)[300] but renamed here in order to avoid confusion with the more common definition ofsemiclassicalon page20.
It should be mentioned that relaxation processes occur in real systems characterized by an energy and a phase relaxation time, τE and τph. There is a relationship between τE and τph
that follows directly from the Cauchy-Schwarz inequality [220], namely τph ≤ 2τE. This means that the decay time of off-diagonal elementsτphof the1RDMis bounded by the decay time of diagonal elementsτEso that the property of the1RDMof being positive semidefinite is not violated. Contrariwise, there is no lower bound on τph, i.e., the case τph τE is conceivable and is known in various physical systems to result from elastic collisions [296]. In the cases considered here, the energy relaxation timeτEis typically fast due to Auger decay or fluorescence (cf. ChaptersIandII). To the author’s knowledge, there have been no conclusive
48 IV. Coherent ionization dynamics in crystals
results measuring the phase relaxation timesτphatXFELbased experiments so far; by contrast, the energy relaxation timesτEare widely known [103,301]. For instance, theL-hole lifetime in aluminum13Al around19 fs[131], i.e., for hole states corresponding to electron binding energies around100 eV. There are two things to learn from that. Firstly, the models discussed so far break down ifτEis on the order of the pulse durationτ or lower (cf. ChapterII). Secondly, if off-diagonal elements decay rapidly, i.e., τph τE, the Markovian theory may even provide a more accurate description than the TDSE, Eq. (IV.5). Hence, the Markovian theory can be not only regarded as an approximate to theTDSEbut can also be interpreted as the limit τph τ0 τEsince the coherences exhibit a fast decay so that memory effects in Eq. (IV.8) are naturally avoided.