Editing Ionization (section)

==Kramers–Henneberger frame==
Sources:<ref>{{Cite book |last=Kramers |first=H.A. |title=Collected Papers |date=1956 |publisher=North Holland}}</ref><ref>{{Cite journal |last=Henneberger |first=W.C. |date=1968 |title=Perturbation Method for Atoms in Intense Light Beams |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.21.838 |journal=Physical Review Letters |volume=21 |pages=838 |doi=10.1103/PhysRevLett.21.838}}</ref>

The Kramers–Henneberger(KF) frame is the non-inertial frame moving with the free electron under the influence of the harmonic laser pulse, obtained by applying a translation to the laboratory frame equal to the quiver motion of a classical electron in the laboratory frame. In other words, in the Kramers–Henneberger frame the classical electron is at rest.<ref>{{Cite journal |last=Gavrila |first=Mihai |date=2002-09-28 |title=Atomic stabilization in superintense laser fields |url=https://iopscience.iop.org/article/10.1088/0953-4075/35/18/201 |journal=Journal of Physics B: Atomic, Molecular and Optical Physics |volume=35 |issue=18 |pages=R147–R193 |doi=10.1088/0953-4075/35/18/201 |issn=0953-4075}}</ref> Starting in the lab frame (velocity gauge), we may describe the electron with the Hamiltonian:
:<math> H_{lab}=\frac{1}{2}(\mathbf{P}+\frac{1}{c}\mathbf{A}(t))^2 + V(r)</math>
In the dipole approximation, the quiver motion of a classical electron in the laboratory frame for an arbitrary field can be obtained from the vector potential of the electromagnetic field:
:<math> \mathbf{\alpha}(t) \equiv \frac{1}{c} \int^{t}_{0}\mathbf{A}(t')dt' =  (\alpha_0/E_0)\mathbf{E}(t)</math>
where <math>\alpha_0 \equiv E_0\omega^{-2} </math> for a monochromatic plane wave.

By applying a transformation to the laboratory frame equal to the quiver motion <math>\mathbf{\alpha}(t)</math> one moves to the ‘oscillating’ or ‘Kramers–Henneberger’ frame, in which the classical electron is at rest. By a phase factor transformation for convenience one obtains the ‘space-translated’ Hamiltonian, which is unitarily equivalent to the lab-frame Hamiltonian, which contains the original potential centered on the oscillating point <math>-\mathbf{\alpha}(t)</math>:
:<math>H_{KH}=\frac{1}{2}\mathbf{P}^2 + V(\mathbf{r} + \mathbf{\alpha}(t)) </math>

The utility of the KH frame lies in the fact that in this frame the laser-atom interaction can be reduced to the form of an oscillating potential energy, where the natural parameters describing the electron dynamics are <math>\omega </math> and <math> \alpha_0 </math>(sometimes called the “excursion amplitude’, obtained from <math>\mathbf{\alpha}(t)</math>).

From here one can apply Floquet theory to calculate quasi-stationary solutions of the TDSE. In high frequency Floquet theory, to lowest order in <math>\omega^{-1}</math> the system reduces to the so-called ‘structure equation’, which has the form of a typical energy-eigenvalue Schrödinger equation containing the ‘dressed potential’ <math> V_0(\alpha_0,\mathbf{r}) </math> (the cycle-average of the oscillating potential). The interpretation of the presence of <math> V_0 </math> is as follows: in the oscillating frame, the nucleus has an oscillatory motion of trajectory <math>-\mathbf{\alpha}(t)</math> and <math>V_0</math> can be seen as the potential of the smeared out nuclear charge along its trajectory.

	The KH frame is thus employed in theoretical studies of strong-field ionization and atomic stabilization (a predicted phenomenon in which the ionization probability of an atom in a high-intensity, high-frequency field actually decreases for intensities above a certain threshold) in conjunction with high-frequency Floquet theory.<ref>Gavrila, Mihai. "Atomic structure and decay in high-frequency fields." ''Atoms in Intense Laser Fields,'' edited by Mihai Gavrila, Academic Press, Inc, 1992, pp. 435-508.</ref>

	The KF frame was successfully applied for different problems as well e.g. for [[High harmonic generation|higher-hamonic generation]] from a metal surface in a powerful laser field<ref>{{Cite journal |last=Varró |first=S. |last2=Ehlotzky |first2=F. |date=1994 |title=Higher-harmonic generation from a metal surface in a powerful laser field |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.49.3106 |journal=Physical Review A |volume=49 |issue=3 |pages=3106 |doi=10.1103/PhysRevA.49.3106}}</ref>