Time-Resolved Holography with Photoelectrons

Experiments on atoms in intense laser pulses and the corresponding exact ab initio solutions of the timedependent Schrödinger equation (TDSE) yield photoelectron spectra with low-energy features that are not reproduced by the otherwise successful work horse of strong field laser physics: the "strong field approximation" (SFA). In the semi-classical limit, the SFA possesses an appealing interpretation in terms of interfering quantum trajectories. It is shown that a conceptually simple extension towards the inclusion of Coulomb effects yields very good agreement with exact TDSE results. Moreover, the Coulomb quantum orbits allow for a physically intuitive interpretation and detailed analysis of all low-energy features in the semi-classical regime, in particular the recently discovered "low-energy structure" [C.I. Blaga et al., Nature Physics 5, 335 (2009) The development of analytical and numerical methods capable of treating strongly-driven quantum systems is of great interest in many areas of physics. By "strongly-driven" we understand that conventional time-dependent perturbation theory is not applicable. A prime example for such a system is an atom in an intense laser field. The force on valence electrons due to the electric field of the electromagnetic wave delivered by present-day intense lasers can easily compete with the binding force. As a consequence, the photoelectron spectra may show strong nonperturbative features such as plateaus and cut-offs [1], instead of a simple exponential decrease with the number of absorbed photons, as expected from perturbation theory. Recently, an "ionization surprise" [2] at wavelength λ = 2 µm and intensity I = 80-150 TW/cm 2 , the socalled "low-energy structure" (LES) [3,4], has been reported. The LES is a strong but narrow enhancement of the differential ionization probability along the polarization direction of the laser at low energies. This result was so astonishing not only because it is unpredicted by the "strong field approximation" (SFA) [5] but also because it is observed in a regime where matters were actually expected to simplify. In fact, if the number of photons N of energyhω required to overcome the ionization potential I p is large, N = I p /hω ≫ 1, and the time the electron needs to tunnel through the Coulombbarrier is small compared to a laser period, i.e., the Keldysh parameter γ = I p /2U p with U p the ponderomotive potential, is small, a quasi-static tunneling theory appears to apply [6]. As the tunneling ionization rate in a static electric field is a smooth, featureless function of the final momenta p and p ⊥ parallel and perpendicular to the electric field, respectively, no LES has been expected. In the present Letter we reveal the origin of the LES using our trajectory-based Coulomb-SFA (TC-SFA). The fact that the TC-SFA allows recourse to trajectories provides an unprecedented insight into the origin of any spectral feature of interest, as constructive or destructive interference of trajectories or the Coulomb-focusing of them [7] can be analyz...

The history and current status of the Keldysh theory of strong field ionization are reviewed. The focus is on the fundamentals of the theory, its most important applications and those aspects which still raise difficulties and remain under discussion. The Keldysh theory is compared with other nonperturbative analytic methods of strong field atomic physics and its important generalizations are discussed. Among the difficulties, the gauge invariance problem, the tunneling time concept, the conditions of applicability and the application of the theory to ionization of systems more complex than atoms, including molecules and dielectrics, are considered. Possible prospects for the future development of the theory are also discussed.

A theory describing above-threshold ionization of atoms and ions in a strong electromagnetic field is presented. It is based on the widely known strong field approximation and incorporates the Coulomb interaction between the photoelectron and the nucleus using the method of complex classical trajectories. A central result of the theory is the Coulomb-corrected ionization amplitude whose evaluation requires little extra numerical effort. By comparing our predictions with the results of ab initio numerical solutions for two examples we show that the new theory provides a significant improvement of the Coulomb-free strong field approximation. For the case of above-threshold ionization in elliptically polarized fields a com-