In this chapter, we show the result of the calculation and give explanations of the phenomena.
3.1 Hydrogen
Hydrogen atoms is the simplest system in all atoms. We don’t need to use any approximations about multi-electron effect(only one electron).
We solve the time-dependent SchrÃűdinger equation by TDGPS method.
i∂
∂tϕ(r, t) = [−1
2∇2− 1
r + vext(r, t)]ϕ(r, t) (3.1) The laser fields are polarized along z-axis:
vext(r, t) =−z[εX(t) + εL(t)] (3.2) The SAP field can be defined as follow:
εX(t) = FXexp(−2 ln(2)(t− td)2
τX2 ) cos(ωX(t− td)) (3.3) Here, FX is the peak field strength of the SAP, τX = 140as is its full width at half maximum(FWHM), and ωX = 13.6 eV is its central frequency(here we choose the laser frequency as the ionization energy of 1s orbital because we want to excite
atoms). The SAP peak intensity is 1 × 1010W/cm2. The parameter td represents the time delay between the NIR and SAP; the negative time delay refers to the SAP arriving first. The NIR field has the form:
εL(t) = FLexp(−2 ln(2)t2
τL2 ) cos(ωLt) (3.4)
Here, FLis the peak field of the NIR pulse, τL fs is its FWHM, and ωL is the central frequency of the NIR field(here we choose the laser wavelength as 800 nm [ωL= 1.55 e.V] and 656 nm [ωL = 1.89 e.V]). The NIR laser peak intensity is 1× 1012W/cm2.
Figure 3.1: Illustration of SAP and NIR with time delay -5 fs.
After the time-propagation procedure, the dipole moment and the dipole accel-eration can be expressed as follow:
d(t) =hϕ(r, t)|r|ϕ(r, t)i (3.5)
a(t) =hϕ(r, t)|∇(1
r − vext(r, t))|ϕ(r, t)i (3.6) The spectral density of the radiation energy is given by the following expression:
S(ω) = 2 3πc3|
Z ∞
−∞
a(t) exp(−iωt)dt|2 (3.7)
Here ω is the frequency of radiation, c is the velocity of light. S(ω) has the meaning of the energy emitted per unit frequency.
In the calculation, we use 128 radial and 32 angular grid points and the time step ω 2π
L1024 (nearly 0.1 a.u.). The maximum radius is 60 a.u. and we place absorber between 40 a.u. and 60 a.u. describe the ionization process. The time delay was varied in steps of 4td = 20as within the range of −20fs 6 td 6 20fs (2048 steps in total).
Figure 3.2: Photon emission energy spectrum of the exicted states (2p[3s− 5s] and 2p[3d − 4d] as a function of the time delay between the NIR pulse and SAP. The yellow color indicates the highest energy emitted. The color bars are represented by the log10S(ω) of the spectral density in Eq. 3.7
In Fig. 3.2 we show the 3D plot of the photon emission spectrum as a function of td for the excited states 1snp(n ≤ 5). The higher excited states (1s4p and 1s5p) are shifted by the pondermotive potential Up of the NIR field, where Up = ε2L
(2ωL)2; for the field strength and frequency used, Up = 0.17 eV.
Figure 3.3: Photon emission energy spectrum of the 1s2p excited state as a function of the time delay between the NIR pulse and SAP.
Figure 3.4: Photon emission energy spectrum of the 1s3p excited state as a function of the time delay between the NIR pulse and SAP.
Figure 3.5: Photon emission energy spectrum of the 1s4p excited state as a function of the time delay between the NIR pulse and SAP.
The density plots of the photon emission spectrum in Fig. 3.3-3.5 depict the transition from 1s2p, 1s3p and 1s4p as the function of td. We can observe the oscillation structure in the region where the NIR and SAP overlap. The period of the oscillation is 1.1 fs, which is half of the NIR laser optical cycle. This phenomenon was also observed in theoretical calculation where they use absorption spectra[12].
In Fig. 3.3 and Fig. 3.4 we observe the splitting of the lines near td∼ 10fs. The electron absorbs one XUV photon to np states and then absorbs more NIR photons to forbidden states, ns or nd. If we try the NIR with wavelength of 656nm, the transition is more obvious. The splitting has been known as result of Autler-Townes effect[40]. We can identify this splitting by the Hamiltonian without some of the excited states. For 1s2p transition, we choose the td = 10f s and remove 3s and 3d states in the Hamiltonian and for 1s3p transition we choose the td= 10f s and remove 2s states in the Hamiltonian in Fig. 3.2. The splitting disappears in both of them and make sure this splitting can be explained in terms of two-photon absorption and emission process. The SAP excites the ground state to 1snp states; then the NIR
0
Spectral density of radiation energy
Photon energy (eV)
Spectral density of radiation energy
Photon energy (eV)
full calculation 1s2s removed
Figure 3.6: (left) Energy emitted near 1s2p transition (right) Energy emitted near 1s3p transition
couples these states to forbidden states 1sns and 1snd which causes splitting and shift.
Figure 3.7: Illustration of Autler-Townes effect in hydrogen. When SAP comes first, the electron can be exicted to np orbitals. The NIR is not weak for excited states, so it can make 2p3s and 2p3d transition happen.
0 5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06
-10 -5 0 5 10 15 20 25 30
P
Time Delay (fs)
1s3s 1s2p 1s3p 1s4p 1s3d
Figure 3.8: Population of several excited states as a function of the time delay between the NIR pulse and SAP. The center frequency of NIR is 800 nm.
Here we calculate the populations of the exited states in Fig. 3.8. We can see the resonances between the 1s2p and 1s3s and 1s3d states, when 1s2p population goes down, and the 1s3s and 1s3d go up in the region where SAP and NIR overlap(−8 ≤ td≤ 8). The 1s2p state is substantially ionized or excited to forbidden states in the two photon absorption, SAP and NIR photon.
Figure 3.9: Illustration of two photon absorption.
And we try use the 656nm(=1.89 eV, energy different the n=2 and n=3) as the NIR frequency. The excited states 1s3s and 1s3d are more obvious. That’s because of Autler-Townes effect and it’s also shown in [12].
0 5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06
-10 -5 0 5 10 15 20 25 30
P
Time Delay(fs)
1s3s 1s2p 1s3p 1s4p 1s3d
Figure 3.10: Population of several excited states as a function of the time delay between the NIR pulse and SAP.The center frequency of NIR is 800 nm.
3.2 Helium
Next atom is Helium, which has two electrons on 1s orbital. To obtain the time-dependent electron density and calculate the induced dipole moments, one has to solve a set of the time-dependent Kohn-Sham equations for the spin-orbitals ϕiσ(r, t):
i∂
∂tϕiσ(r, t) = [−1
2∇2− Z
r + vext(r, t) + VσKLI(r, t)]ϕiσ(r, t), i = 1, . . . , Nσ (3.8) Here the Z = 2 is the nucleus charge and vext is the interaction of the electron with the external field. Compared with Eq. 3.1, we add the VσKLI(r, t) from the TD-KLI-SIC procedure to describe electron correlation effect.
Here we use the same laser pulse and parameters as we do in hydrogen, but we change the central frequency of the NIR to 750 nm and increase the NIR peak intensity to 3× 1012W/cm2.
Figure 3.11: Photon emission energy spectrum of the exicted states (1s[2p− 5p] as a function of the time delay between the NIR pulse and SAP. The yellow color indicates the highest energy emitted. The color bars are represented by the log10S(ω) of the spectral density in Eq. 3.7
In Fig. 3.11 we show the 3D plot of the photon emission spectrum as a function of td for the excited states 1snp(n ≤ 5). The higher excited states (1s4p and 1s5p) are shifted by the pondermotive potential Up of the NIR field, where Up = ε2L
(2ωL)2; for the field strength and frequency used, Up = 0.15 eV.
Figure 3.12: Photon emission energy spectrum of the 1s2p excited state as a function of the time delay between the NIR pulse and SAP.
Figure 3.13: Photon emission energy spectrum of the 1s3p excited state as a function of the time delay between the NIR pulse and SAP.
Figure 3.14: Photon emission energy spectrum of the 1s4p excited state as a function of the time delay between the NIR pulse and SAP.
We show the more clearer 3D plot of photon emission spectrum transition from 1s2p, 1s3p and 1s4p in Figs. 3.12-3.14. In the region where the NIR pulse and SAP overlap (−8 ≤ td ≤ 8 fs), the photon emission lines have oscillations with a period of 1.3 fs, which is half of the NIR laser optical cycle. This is a instantaneous shift of the electronic energy levels in the NIR laser field(or instantaneous Stark shift[41, 42, 43, 44]). This phenomena was also observed in recent experimental works[16, 15] where the transient absorption technique was used.
In Fig. 3.12 and 3.13 we see the splitting of the lines in the photon emission spectrum.
It’s understood as Autler-Townes effect[40]. We can remove some of excited states in Hamiltonian like we’ve done in hydrogen atoms. But for 1s3p in helium atoms, we found the splitting is not from the 2s state but the other nd states.
0
Spectral density of radiation energy
Photon energy (eV)
Spectral density of radiation energy
Photon energy (eV)
full calculation 1s(n<8)d removed
Figure 3.15: (left) Energy emitted near 1s2p transition (right) Energy emitted near 1s3p transition
Figure 3.16: Population of several excited states as a function of the time delay between the NIR pulse and SAP.
Here we show the population of the several excited states as a function of the time delay. Before the overlapping of two pulses(td ≤ −8fs), the population of 1s3s and 1s3d are zero because of selection rule. The population of 1s2p , 1s3p and
1s4p are straight line when NIR pulse comes first because NIR pulse is too weak to change the population of excited state. But for the region where the NIR and SAP (−8 ≤ td≤ 8fs), the NIR pulse is not weak for excited states. The population of 1s3s and 1s3d are increasing and oscillation with period of subcycle as the decreasing of 1s2p. Even for td≥ 8fs, we can still observe the oscillation.
3.3 Neon
Neon has ten electrons with configuration 1s22s22p6. Because intensity of the laser field is weak to ionize the inner shell electrons 1s2, we don’t need to propagate the 1s orbital. And for p orbitals, we need to identify the 2p2m=0 on z-axis and 2p4m=1 on xy-plane because of laser field. To obtain the time-dependent electron density and calculate the induced dipole moments, one has to solve a set of the time-dependent Kohn-Sham equations for the spin-orbitals ϕiσ(r, t):
i∂
∂tϕiσ(r, t) = [−1
2∇2− Z
r + vext(r, t) + VσKLI(r, t)]ϕiσ(r, t), i = 1, . . . , Nσ (3.9) Here the Z = 10 is the nucleus charge and vext is the interaction of the electron with the external field. Compared with Helium, we have 4 orbitals and it’s a little complicated. But we use the SAP with center frequency 0.808 eV(the ionization energy of 2p) and the laser field is on z-axis, so we focus on the transition of 2p0.
In the calculation, we use 256 radial and 32 angular grid points and the time step 1024ω2π
L (nearly 0.1 a.u.). The maximum radius is 100 a.u. and we place absorber between 60 a.u. and 100 a.u. describing the ionization process. The time delay was varied in steps of 4td = 20as within the range of −20fs 6 td 6 20fs (2048 steps in total).
Figure 3.17: Photon emission energy spectrum of the excited states (2p[3s− 5s] and 2p[3d − 4d] as a function of the time delay between the NIR pulse and SAP. The yellow color indicates the highest energy emitted. The color bars are represented by the log10S(ω) of the spectral density in Eq. 3.7
In Fig. 3.17 we show the 3D plot of the photon emission spectrum as a function of td for the excited states 2p[3s− 5s] and 2p[3d − 4d]. The higher excited states (1s4p and 1s5s) are shifted by the pondermotive potential Up of the NIR field, where Up = ε2L
(2ωL)2; for the field strength and frequency used, Up = 0.15 eV. The other transition can be seen only if we zoom in the plot on next page.
Figure 3.18: Photon emission energy spectrum of the 2p3s excited state as a function of the time delay between the NIR pulse and SAP.
Figure 3.19: Photon emission energy spectrum of the 2p4s excited state as a function of the time delay between the NIR pulse and SAP.
Figure 3.20: Photon emission energy spectrum of the 2p3d excited state as a function of the time delay between the NIR pulse and SAP.
Figure 3.21: Photon emission energy spectrum of the 2p5s , 2p4d and 2p6s(down to up) excited states as a function of the time delay between the NIR pulse and SAP.
We show the more clearer 3D plot of photon emission spectrum transition from 2p3s, 2p4s, 2p5s ,2p6s,2p3d and 2p4d in Figs. 3.18-3.21. In the region where the NIR pulse and SAP overlap (−8 ≤ td≤ 8 fs), the photon emission lines have oscilla-tions with a period of 1.3 fs, which is half of the NIR laser optical cycle. This is a instantaneous shift of the electronic energy levels in the NIR laser field(or instanta-neous Stark shift). This phenomena was also observed in recent experimental works where the transient absorption technique was used[20]. Compared with Helium, the transition from 2p is more complicated because p orbitals can go to s and d orbitals.
Moreover, their energies are sometimes close to one other in Fig. 3.21 ,so it’s hard to identify each excited states. We don’t see the Aulter-Townes effect in the photon emission spectra, but we can observe the population of np orbitals and it’s too weak to show.
0 5e-08 1e-07 1.5e-07 2e-07 2.5e-07 3e-07
-10 -5 0 5 10 15 20
P
Delay (fs)
2p3s 2p4s 2p5s 2p3d 2p4d
Figure 3.22: Population of several excited states as a function of the time delay between the NIR pulse and SAP.
Here we show the population of excited state as the function of the time delay.
In the region where the NIR and SAP overlap we can see the 2p3s decreasing and 2p4s and 2p3d populating. These are also shown on the photon emission spectra.
We also see the populations of excited states are changed with period of half optical cycle.