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Even though our simulation for Kr II handles 16+2 levels (including full degeneracy) and could in principle handle many more, we must always bear in mind the levels and continua not captured by this. For the simulation to make sense we must have that the electric dipole interaction is dominant and that other levels and continua are energetically far enough away. A very strong probe electric field would of course violate this by causing ionization but if the field is not too strong then ionization should be exponentially suppressed.

When we assume we are in a safe regime for the probe strength we can analyze how the density matrix depends polynomially on the probe electric field strength, in other words a

0.5 1 1.5 2 Time [fs] -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 Energy / E 4

Fourth Order Hamiltonian Expectation

Figure 5.13: Fourth order perturbative contribution to the Hamiltonian expectation for a Krypton cation, resonantly driven by a 80 eV, 150 as, 0.2 au peak amplitude probe pulse polarized in the z-direction. Note how the fourth order effect is opposite in sign from the second order effect and is rather small in size for reasonable probe electric field strength. Energies are scaled with the fourth power of the peak electric field amplitude of the probe pulse.

perturbative expansion. In principle it is possible to compute this expansion analytically with perturbative density matrix theory but the formulae become cumbersome and unwieldy very quickly for higher orders. It is much easier to simulate one probe pulse multiplied by a set of scale factors. The set of results can be used to back-out the perturbative contributions. We have found that perturbative contributions up to order 8 can be recovered with good numerical accuracy. This does depend on the size of the contribution and the numerical quality of the simulation. Note that contributions to the diagonal of the density matrix are even in the probe electric field, as are the contributions to the expected Hamiltonian. The induced polarization is odd in the electric field.

In Figures (5.10) and (5.11) we show the second and fourth order terms for the populations a.k.a. probabilities a.k.a. density matrix diagonal elements for Kr II 4p and 3d. Note how

probability is transferred from the 4p states to the 3d states. Note how the fourth order effect acts in the opposite direction from the second order effect. For a rather strong electric field of 0.2 au we see that the fourth order term provides a small correction of around -0.4%. Notice also that while the second order contribution at the time of the center of the probe pulse is about 1/4 of the value right after the pulse this ratio is actually close to 1/16 for the fourth order contribution. These ratios, 2−2 _{and 2}−4 _{are consistent with quadratic and} fourth power resonant growth during the body of the probe pulse.

0.5 1 1.5 2 Time [fs] -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 Induced Polarization / E

First Order Induced Polarization

Figure 5.14: First order perturbative contribution to the induced polarization for a Krypton cation, reso- nantly driven by a 80 eV, 150 as, 0.2 au peak amplitude probe pulse polarized in the z-direction. Polarization is scaled with the peak electric field amplitude of the probe pulse.

The second and fourth order contributions to the expected Hamiltonian are shown in Fig- ures (5.12) and (5.13) with, as expected, the same patterns as we saw for the probabilities. The first and third order contributions to the induced polarization are shown in Figures (5.14)

0.5 1 1.5 2 Time [fs] -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 Induced Polarization / E 3

Third Order Induced Polarization

Figure 5.15: Third order perturbative contribution to the induced polarization for a Krypton cation, resonantly driven by a 80 eV, 150 as, 0.2 au peak amplitude probe pulse polarized in the z-direction. Note how the third order contribution has opposite sign from the first order contribution. Polarization is scaled with the third power of the peak electric field amplitude of the probe pulse.

and (5.15). We see that the third order effect acts in the opposite direction from the first order effect and is of relative size of about -0.7% for a probe peak electric field of 0.2 au. Looking again at the ratios for midway versus the end of the probe pulse we find about 1/2 for the first order and 1/8 for the third order. These values, 2−1 _{and 2}−3_{, are consistent with} linear and cubic resonant growth during the body of the probe pulse.

It turns out that these higher order features of our multi-level system can be understood qualitatively and to some extent quantitatively in the context of the toy two-level model from Equation (5.8). When solving the toy-model non-perturbatively but with the simplifying rotating wave approximation and using an exactly resonant constant amplitude external force

f (t) = A sin(t) we find the familiar Rabi oscillation

ρ11 = sin2(At/2) ρ01 = 1₂eitsin(At)

hDi = cos(t) sin(At) (5.14)

with perturbative expansion in the external field strength A

ρ11 = 1₄A2t2− ₄₈1A4t4+ . . . hDi = cos(t) · At − 1

6A

3_t3_{+ . . .
(5.15)}

Comparing with the full Kr II simulation we observe the same oddness of the induced polar- ization hDi, the evenness of excited population ρ11, terms of order An grow resonantly as tn and even the sign of the higher order terms versus the lower order ones is correct.

5.3

Conclusion

In this chapter we have simulated the effects of NIR pump and XUV probe pulses. We have used our source-cation-sink method to model the coherent multichannel ionization of noble gases and analyzed the dependence on the strength and length of the pump pulse. Our method, based on Lindblad terms, is very much simpler than full TDSE simulations of the ionization process and could be of practical use by itself or in the context of larger simulations. As a future direction of research it would be interesting to see how this method compares with more elaborate multi-channel ionization approaches. The implied density for 4p3/2 and 4p1/2 levels in the m = ±1/2 channels in the krypton experiment by Goulielmakis et al.[2] fit well both with our results and those from full TDSE treatments [35][36]. However, the implied diagonal density matrix elements by Goulielmakis et al. for 4p3/2 in the m = ±3/2 channels are very far (factor 7) off the mark compared to TDSE [35][36]. So more work needs

0 0.5 1 1.5 2 2.5 Time [fs] 0 2e-8 4e-8 6e-8 8e-8 1e-7 1.2e-7 1.4e-7 Probability

Kr Probabilities 3d_5/2 and 3d_3/2, 80 eV, 150 as probe pulse, HHG

3d 5/2,1/2 3d 5/2,3/2 3d_3/2,1/2 3d 3/2,3/2

Figure 5.16: Population probabilities of the excited Krypton cation 3d5_/2and 3d3_/2states, resonantly driven

by a 80 eV, 150 as, 0.001688 au peak amplitude probe pulse polarized in the z-direction, corresponding to 1011

W/cm2

, an intensity typical of HHG.

to be done here. Note that our method has no opinion on these relative populations but can accommodate the data by adjusting the πi parameters from equation (4.63).

We simulated how probe pulses can resonantly populate the excited states of Kr II and studied in detail how populations, coherences and induced dipole moments evolved in time, including the effects of Auger decay. The onset of nonlinear effects can be estimated from the perturbative contributions we derived in the previous section. The probe peak electric field strength for which the second order and fourth order contributions to the excited populations are of equal strength is E = 3.5 au (corresponding to 4 × 1017 _W/cm2_{), while the electric} field strength for which the first and third order contributions to the induced dipole are of

0 0.5 1 1.5 2 2.5 Time [fs] 0 0.02 0.04 0.06 0.08 0.1 0.12 Probability

Kr Probabilities 3d_5/2 and 3d_3/2, 80 eV, 150 as probe pulse, FEL

3d_5/2,1/2

3d_5/2,3/2

3d_3/2,1/2

3d

3/2,3/2

Figure 5.17: Population probabilities of the excited Krypton cation 3d5_/2 and 3d3_/2 states, resonantly

driven by a 80 eV, 150 as, 1.688 au peak amplitude probe pulse polarized in the z-direction, corresponding to 1017

W/cm2

, an intensity typical of the free electron laser (FEL).

equal strength is E = 2.4 au (corresponding to 2 × 1017 _W/cm2_{). So nonlinear effects are} not relevant for the typical HHG regime of 1012 _W/cm2_{, while they do become relevant for} free electron laser (FEL) intensities which are on the order of 1017 _W/cm2_{, see Figure (5.17).} We must be careful to note that some physical processes, such as non-Auger ionization into Kr III and beyond, go unmodeled here and that, therefore, the value 1017 _W/cm2 _{should be} viewed rather as an upper limit for nonlinear onset. One reason why the modelling approach used here can work well quantitatively for a large range of intensities is that we focus on a small part of the spectrum (79 to 82 eV) and that we drive transitions in this spectral range resonantly. The working assumption is that unmodeled processes have no resonances in this

tight spectral range and would only manifest themselves as modest backgrounds. Of course when the probe field is too strong severe depletion and saturation effects come into play. To understand this in detail would require a full treatment of the multi-electron dynamics involving very extensive TDSE simulations with large numbers of levels. The development of free electron lasers has made it possible to start probing the regime of 1017 _W/cm2 _so experimental answers to these questions are becoming possible. The most straightforward measure of nonlinearity would be to experimentally determine beyond which field strength the probe absorbance develops an interesting dependence on the probe intensity.

Chapter 6

Propagation Revisited

6.1

The Propagation Equation

In this chapter we derive the propagation equation for a laser pulse from the Maxwell equa- tions, the induced polarization from low order density matrix perturbation theory and the Beer-Lambert absorption law for linear susceptibility. The derivation for the induced po- larization is given in some detail and forms the basis for our results in Section 6.3 on the breaking of the Beer-Lambert law due to coherence.