In this section, we devote to explain the significant change of the behavior of the ionization probability with laser intensity below γ = 2.5 that we have mentioned in the section 3.2 (Fig. 3.11). Before proceeding, we discuss atomic stabilization first [78, 79, 80, 81, 82].
The atomic stabilization is easily understood in the Kramers-Henneberger (KH) frame which is the frame of the moving electron in the laser field. The Hamiltonian of an atom interacting with laser field in the length gauge is
H = p2
2 + V (r) + E(t)· r (3.8)
E(t) = E0cos(ωt)ˆz (3.9)
where ω is the laser frequency and E0is the field strength. By doing the following unitary transformation [81]: (
where α = E0/ω2 is the amplitude of free-electron oscillation in a laser field.
The oscillating potential in Eq. (3.13) can be expanded in a Fourier series to take the form of a sum of harmonics of the frequency ω. The zero-order harmonics, which is the time-averaged part of the sum, represents a stationary potential. When the laser frequency is much higher than the frequency of a atomic state, ω ≫ |En|/¯h, except the zero-order harmonics, all other higher order terms can be neglected. The KH Hamiltonian is thus independent of time, then the electron is stabilized in that state which is known as KH stabilization.
Rydberg state is not a clearly-defined physical noun. It is usually used to indicate a highly excited state without defining to what extent it should be.
Back to our system, the time scale of the Rydberg states is in microwave range, whereas the 785nm infrared laser is at a much higher frequency than those of microwaves.
Then a electron in Rydberg states almost doesn’t respond to the laser field and thus is sta-bilized in those states. This is what we mean ”Rydberg stabilization”.
It is interesting that stabilization of Rydberg states are accompanied by the transition from direct 4-photon ionization to direct 5-photon ionization. In Fig. 3.18, we show a
Figure 3.18: Schematic plot of the process of transition from direct 4-photon ionization to direct 5-photon ionization. During this process, it is strongly coupling to Rydberg states.
|0⟩ is the ground state, |b⟩ is any intermediate bound state, IP is ionization potential, I0is laser intensity, Up is ponderomotive shift.
schematic plot for the transition process. At smaller intensity, electron can be ionized by absorbing 4 photons. However, with increasing intensity, the ionization threshold (ionization potential plus the ponderomotive shift) rises gradually. Finally, the energy of absorbing 4 photons are lower than the ionization threshold, no longer afford to ionization, but coupling to the Rydberg states instead. From Fig. 3.17, we know that transition from direct 4-photon ionization to direct 5-photon ionization occur at about 90× 1011W/cm2 (γ∼ 2.3). We now proceed to check is there anything happening at (γ ∼ 2.3).
In Fig. 3.19, we show the ionization probability and the sum of the occupation prob-ability of the low-lying bound states with principal quantum number n≤ 4 and those of Rydberg states with n ≥ 5 for 30fs, 10fs, and 6fs pulses. We find that there is a dra-matic occupation to Rydberg states from (γ ∼ 2.3) for all three cases. We also find that the trend of the ionized probability is closely related to that in the Rydberg states. This manifest itself the Rydberg stabilization indeed dominate the features of the ionization probability below γ = 2.3. The recover and oscillation of ionization probabilities for 10fs and 6fs pulses are due to the broader bandwidth of these shorter pulses. In 30fs case, the narrower bandwidth can concentrate on the Rydberg states and lasts for even higher intensities (smaller γ). In 10fs case, the energy bandwidth can also concentrate on the
Figure 3.19: Ionization probability and the sum of the occupation probability of the low-lying bound states with principal quantum number n ≤ 4 and those of Rydberg states with n≥ 5 for 30fs, 10fs, and 6fs pulses.
Rydberg states at γ ∼ 2.3. However, when γ reach to about 1.5. The energy bandwidth start to cover lower levels where electron is no longer stabilized under the laser field and thus ionized [83]. In 10fs case, the broader bandwidth can’t concentrate on the Rydberg states only, it always covers the low-lying states. So the phenomena of ionization trapping is not so dramatic as 30fs and 10fs pulses, it only exhibits a reduction of the ionization rate. In addition to Rydberg stabilization, this reduced ionization rate is partially due to the the direct 5-photon ionization dominant in this regime which is the next order to the direct 4-photon ionization of the perturbation series.
Figure 3.20: Ratio of the sum of the occupation probability of the Rydberg states (n≥ 5) to that of total bound states.
In Fig. 3.20, we show the ratio of the sum of the occupation probability of the Rydberg states (n ≥ 5) to that of total bound states. We find that, at about γ = 2.3 (the critical value for transition from direct 4-photon ionization to direct 5-photon ionization), the bound electron dramatically occupy Rydberg states. For 30fs pulse, over 90% bound electron occupy the Rydberg state, over 80% for 10fs pulse, and over 60% for 10fs pulse.
This also confirms the Rydberg stabilization. The drop of the ratio of the Rydberg states to the bound states at γ = 1 for 30fs pulse is simply because the 4-photon arrow starts to couple to low-lying states with keeping increasing laser intensity.