Highorder Harmonic and Attosecond Pulse Generation
3.1 Theoretical Tools
wave packet [78, 79, 80].
Paper I presents a method that enabling the complete characterization of the single
photon ionization from the 2p6ground state in neon. Using angleresolved RAB
BIT, the amplitude and phase for different ionization channels are measured as a function of the kinetic energy of the photoelectron. In the following, the key theoretical tools essential for the method are derived. The transition between two continuum states involved in our method is shown to be independent of the atom, i.e. universal, and thus predictable. Finally, the angleresolved photoelectron spec
tra for single and twophoton transitions are analyzed and our method is applied to extract singlephoton amplitudes and phases.
The spherical harmonics are eigenstates of the Laplace operator in spherical co
ordinates. Thus, inserting equation (3.2) into (3.1) and using the simplification fl= rRl, leads to the radial Schrödinger equation:
− ℏ2 2me
d2
dr2 +l(l + 1)ℏ2
2mer2 + V (r)
fl(r) = Efl(r), (3.3) which is effectively a onedimensional version of equation (3.1), when defining the effective potential:
Veff(r) = V (r) + l(l + 1)ℏ2
2mer2 , (3.4)
where the second term is referred to as the centrifugal potential. The solutions of equation (3.3) depend drastically on whether the eigenvalues E are positive or negative. For negative eigenvalues E < 0, the eigenstates are referred to as bound states ϕnlm and are characterized by another discrete quantum number n. Each bound state is associated to a negative energy eigenvalue En∝ −Z2/n2. For positive eigenvalues E > 0, the solutions are referred to as scattering states with the radial part Rkl(r). Here, l remains a discrete quantum number, whereas k is continuous and connected to the energy as k = √
2meE/ℏ. The radial wavefunction can then be written in its asymptotic limit as:
rlim→∞Rkl(r) = r 2
πk 1 r sin
kr +Zln(2kr)
k + ηl(k)−πl 2
. (3.5) The second term in the sine function is a logarithmic divergence characteristic for the Coulomb potential created by the ionic core. The scattering phase ηl(k) consists of two contributions: ηl(k) = ςl(k) + δl(k), where ςl(k) = arg[Γ(l + 1 +iZ/k)] is the Coulomb phase shift and δl(k)is a correction term for the short
range deviation of the potential compared to a pure Coulomb potential. Finally, the last term describes the effect of the centrifugal barrier. It is worth emphasizing that the scattering phase and the centrifugal barrier effect both depend on the quantum number l, which plays an important role in paper I.
The wave function describing an outgoing electron with momentum k can be expanded in what is known as partial waves, as they in practice often combine to a set of eigenfunctions forming a socalled electron wave packet (EWP). Such a superposition of partial waves can be written as:
ϕk(r, θ, φ) = (8π)3/2X
l,m
ileiηl(k)Ylm∗ (ˆk)Rkl(r)Ylm(ˆr), (3.6)
where ˆk = k/kand ˆr = r/r. From this equation, scattering states can be identi
fied as spherical waves with wave number k, modified by the spherical harmonics
-0.5 0 0.5 -0.5
0 0.5
-0.5 0 0.5
-0.5 0 0.5
-0.5 0 0.5
-0.5 0 0.5
-0.5 0 0.5
-0.5 0 0.5
l=0
l=1
l=3 l=2
Figure 3.1: Spherical harmonics Ylm(θ, φ)for l = 0, 1, 2, 3 and m = 0.
Ylm(θ, φ). Figure 3.1 shows the first few orders of the spherical harmonics. The quantum number l, called the angular quantum number or short angular momen
tum, hereby imposes the number of noncentrosymmetric nodes. This dependence is a fundamental element of the method presented in paper I.
3.1.2 TwoPhoton Interferometry
The transition of an electron from the ground state to a scattering state is charac
terized by its transition matrix element. In the following, we focus on laserassisted twophoton transitions, where an XUV photon with angular frequency Ω is ab
sorbed, taking an electron from a bound state|i⟩ to an intermediate scattering state|ν⟩. Subsequently, an infrared photon with angular frequency ω is either ab
sorbed or emitted, in a continuumcontinuum (cc) transition from|ν⟩ to the final scattering state|k⟩. Note that for simplicity the states are written in Dirac nota
tion, where the corresponding wave functions are calculated as: ϕi(r) =⟨r|i⟩ for the initial state with negative energy Eiand ϕkLm=⟨r|k⟩ for the final state with positive energy Ek = k2ℏ2/2me= Ei+ℏΩ ± ℏω. Assuming linear polarization for both the XUV and the infrared field along the axis ˆzand using secondorder perturbation theory, the transition matrix element can be written as [81]:
Mki(±)∝ lim
ϵ→0+
Z
ν
⟨k|z|ν⟩ ⟨ν|z|i⟩
ℏΩ(±)− Eν+ Ei+iϵ. (3.7)
Ekin
(2q − 1)¯hω (2q + 1)¯hω
+¯hω
−¯hω sideband
continuum bound
λ = ℓ ± 1
L = λ ± 1
λ = ℓ ± 1
ℓ
Figure 3.2: Schematic representation of the RABBIT technique. An XUV photon (blue arrow) initiates the transition from the ground state with angular momentum ℓ to the intermediate state with angular momentum λ. An additional infrared photon leads to the final state with angular momentum L. The sideband is reached via two interfering quantum path, as indicated by the solid and dashed lines.
The integral runs over an infinite number of continuum states, while the sum of all discrete states (resonances) are neglected. The (±) refers to two different paths:
the infrared photon is either absorbed (+) with Ω+= (2q− 1)ω or emitted (−) with Ω− = (2q + 1)ω. As shown in figure 3.2, for two consecutive harmonics, the two paths interfere at Ωq = 2qω + Ei/ℏ forming a sideband.
The final energy states can have different angular momenta and magnetic quantum numbers. In addition, the same final state can be reached via a variety of different channels following the selection rules of an electric dipole transition. The angular momentum of the intermediate state λ relates to the ground state angular momen
tum ℓ as λ = ℓ± 1, and the final state angular momentum L to the intermediate state as L = λ± 1.
The transition amplitude is a function of the delay τ between infrared and XUV pulses and can be written as a sum of partial waves:
A(Lm±)(θ, ϕ, τ ) =−ie2
ℏ EXU V(Ω)EIR(ω)e(±)iωτX
λ
MLλℓm(±) YLm(θ, ϕ), (3.8)
where the transition matrix element for each partial wave can be decomposed into its amplitude and its phase:
MLλℓm(±) = CLλmCλℓmσ(Lλℓ±) ei(ϕ(±)Lλ+φ(±)λℓ +Φ2q∓1). (3.9)
Here, m is the magnetic quantum number of the initial state, CLλm, Cλℓmare angu
lar coefficients equal to⟨Lm|Y10|λm⟩ and ⟨λm|Y10|ℓm⟩, and σLλℓ(±)is the radial amplitude. The phase consists of three terms: φ(λℓ±)is the phase associated to one
photon ionization, ϕ(Lλ±)is the continuumcontinuum (cc)phase, and Φ2q∓1 is the phase of the (2q∓ 1)thharmonic field.
In an angleresolved RABBIT experiment, the final states for a given magnetic quantum number m add coherently, whereas states with different m add incoher
ently. The resulting signal of a sideband resolved in angle and delay can be written as [82]:
ISB(θ, τ ) = Z 2π
0
dϕX
m
X
L
A(+)Lm(θ, ϕ, τ ) +A(Lm−)(θ, ϕ, τ ) 2 (3.10)
The angular dependence of the sidebands is therefore given by the contributions of the individual states with different angular momenta as well as the interference between them.
The additional interaction with the second photon leads to an increase in the num
ber of angular channels, as shown in figure 3.5, and modifies the radial amplitude and phase of the outgoing photoionization wavepacket [81, 83, 84]. Hence, the angular structure of the sidebands not only depends on the different partial waves of the angular momentum channels reached via single photon ionization, but is also strongly influenced by the cctransition [83, 85]. In order to disentangle the two contributions, an investigation of the cctransition is presented in the follow
ing. It is found that its universal behavior allows a characterization of the ccphase independently of the atom and that the ccamplitudes can be connected via Fano’s propensity rule.
3.1.3 ContinuumContinuum Transitions
The cctransition has recently attracted much attention [81, 84, 86] and it can be shown that for certain aspects it is independent of the atom, while it mostly depends on the angular momenta involved in the transition. In the following, both the amplitude σ(Lλℓ±) and the phase ϕ(Lλ±) are studied and the universality is demonstrated based on calculations using angularchannelresolved manybody perturbation theory, performed in collaboration with Eva Lindroth and Jimmy Vindbladh [87].
continuum bound
ℓ
λ
(λ + 1) (λ − 1)
αλℓ
5 10 15 20 25
Kinetic energy [eV]
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Figure 3.3: Calculated amplitude ratios for the absorption α+λℓand emission α−λℓprocesses. The left panel sketches how the ratio connects the two cc-transitions exemplified for α+λℓ. The right panels show the calculated curves corresponding to different intermediate states (λ=1,2,3 in blue, red, green, respectively) and different atoms/initial states (square, He, 1s initial state); (cross, Kr, 3d); (triangle, Ne, 2p); (circle, Ar, 3p).
First, the twophoton radial amplitudes σ((λ±)±1)λℓ for a given intermediate state with angular momentum λ and, in accordance with the selection rules, the two final states L = (λ± 1) are compared. Figure 3.3 presents the calculated ratio between the radial amplitudes for increasing and decreasing angular momenta, defined as:
α(±)λℓ = σ((λ±)−1)λℓ σ((λ+1)λℓ±)
. (3.11)
The different curves correspond to the ionization of valence electrons from various atoms and initial states, as indicated in the figure caption. As can be seen, the ratio shows a universal behavior, independent of the atom. Only the orbital angular momentum of the intermediate state, λ, is of importance, as predicted by Fano’s propensity rule for the cctransitions [86]. This result is very promising as it allows us to connect two different ionization channels via the constant αλℓ(±) and hence to reduce the number of unknown parameters in the process.
Further, the phases introduced by the cctransitions are considered. Figure 3.4 shows the result of the calculated ccphase for increasing ϕ((λ+1)λ±) and decreasing
0.1 0.2 0.3 0.4 0.5
(-1)+
5 10 15 20 25
Kinetic energy [eV]
-0.4 -0.3 -0.2 -0.1 0
(-1)-
(-1)-0.1 0.2 0.3 0.4 0.5
(+1)+
5 10 15 20 25
Kinetic energy [eV]
-0.4 -0.3 -0.2 -0.1 0
(+1)-
(+1)-(a) (b)
(c) (d)
Figure 3.4: Calculated continuum-continuum phases ϕ(±)Lλ for the absorption (a,b) and emission (c,d) pro-cesses. The different curves correspond to different intermediate states (λ=1,2,3 in blue, red, green, respectively) and different atoms/initial states (square, He, 1s initial state); (cross, Kr, 3d);
(triangle, Ne, 2p); (circle, Ar, 3p). (a,d) refer to transitions with increasing angular momentum, L = λ + 1, while in (b,c), L = λ− 1 .
angular momenta ϕ((λ±)−1)λ, in case of absorption (+) or emission (). Several col
ors and markers, corresponding to different intermediate angular momenta and atoms respectively, are detailed in the figure caption. For both increasing and de
creasing angular momenta, all curves are in good agreement with each other. Only for certain energies do the ccphases depend slightly on the intermediate angular momentum and not at all on the atomic system. For the absorption, the ccphase decreases as a function of kinetic energy and is positive, while for the emission, it increases and is negative. Note that the phases are not mirror images of one an
other, i.e. ϕ+Lλ ̸= −ϕ−Lλ. To conclude, these results show that the variation of the continuumcontinuum phase can be seen as universal, independent of the atom.