Single Electron Atoms in Strong Laser Fields

Abstract

In this chapter some modern applications of laser-matter interaction in the field of atomic physics are reviewed. Due to the availability of short and strong laser pulses a range of new and partly counter-intuitive phenomena can be observed. Some of these are:

• Above Threshold Ionization (ATI)

• Multi-Photon Ionization (MPI)

• Localization of Rydberg atoms by Half Cycle Pulses (HCP)

• High-order Harmonic Generation (HHG)

These phenomena can in general not be understood in the framework of perturbation theory. It turns out that the time-dependent wavepacket approach of Chap. 2 is the method of choice to describe and understand a lot of the new atomic physics in strong laser fields. In the present chapter we will almost exclusively deal with the dynamics of a single electron, initially bound in a Coulomb potential. This chapter will therefore begin with a short review of the unperturbed hydrogen atom, together with atomic units. After the discussion of different aspects of field induced ionization, HHG will be reviewed in detail.

Appendix: More on Atomic Units

There is a close connection between atomic units and some expectation values of the hydrogen atom. The Bohr radius, e.g., is related to the expectation value of the position operator in the ground state which is given by

$$\begin{aligned} \langle \hat{r}\rangle=\frac{3}{2}a_0 , \end{aligned}$$

(4.77)

and in addition the maximum of the probability density for finding the particle in the ground state with a radial separation r from the nucleus is at a ₀ [2]. Furthermore, the energy constructed from the atomic base units is twice the absolute value of the ground state energy $E_{1}=-\frac{1}{2}\mbox{ a.u.}\approx-13.6\mbox{ eV}$. One could also measure energy in terms of the ionization potential

$$\begin{aligned} I_{\mathrm{p}}=-E_1 \end{aligned}$$

(4.78)

of hydrogen, however. The corresponding units are so-called Rydberg units (1 a.u.=2 Rydberg), in contrast to the atomic or Hartree units.

The atomic unit for velocity follows most easily by considering the quantum mechanical virial theorem [2], which for the hydrogen atom can be stated as

$$\begin{aligned} \langle \hat{T}_{\mathrm{k}}\rangle=-\frac{1}{2}\langle V_{\mathrm{C}}\rangle . \end{aligned}$$

(4.79)

For the ground state we find with its help that

$$\begin{aligned} \langle \hat{T}_{\mathrm{k}}\rangle=-E_1 \end{aligned}$$

(4.80)

and with the definition

$$\begin{aligned} \langle \hat{T}_{\mathrm{k}}\rangle=:\frac{m_{\mathrm{e}}v_0^2}{2} \end{aligned}$$

(4.81)

the atomic velocity unit $v_{0}=\sqrt{2|E_{1}|/m_{\mathrm{e}}}\approx c\sqrt{27.212 \mbox{ eV}/0.511 \mbox{ MeV}}$ is related to the vacuum speed of light via v ₀=cα=1 a.u. with the fine structure constant α≈1/137. Using v ₀ and a ₀ the atomic unit of time is formed via t ₀=a ₀/v ₀≈24 as. The oscillation period of the electron on the Bohr orbit is therefore given by T _e=2π a.u.

The most important unit for this book is the one for the electric field which is given by ${\mathcal{E}}_{\mathrm{at}}\approx5.1427\cdot10^{11}\mbox{ V}\,\mbox{m}^{-1}$. This is the value of the electric field, due to the proton, experienced by the electron at the Bohr radius. The intensity that corresponds to that field is $I_{\mathrm{at}}=\frac{c\varepsilon_{0}{\mathcal{E}}_{\mathrm {at}}^{2}}{2}\approx 3.5101\cdot10^{16}\mbox{ W}\,\mbox{cm}^{-2}$. An overview of the units mentioned can be found in Table 4.1. As an example, we use this table to convert frequency into wave length. For frequencies given in a.u. (ω=X a.u.), the wave length (in SI units) is

$$\begin{aligned} \lambda= \frac{2\pi c}{\omega} \approx\frac{2\pi 137.036}{\mathrm{X}} \mbox{ a.u.} \approx \frac{45.5636}{\mathrm{X}} \mbox{ nm} . \end{aligned}$$