The theoretical description and the experimental methods and results for above-threshold ionization (ATI) by few-cycle pulses are reviewed. A pulse is referred to as a few-cycle pulse if its detailed shape, parametrized by its carrier-envelope phase, affects its interaction with matter. Angular-resolved ATI spectra are analysed with the customary strong-field approximation (SFA) as well as the numerical solution of the time-dependent Schrödinger equation (TDSE). After a general discussion of the characteristics and the description of few-cycle pulses, the behaviour of the ATI spectrum under spatial inversion is related to the shape of the laser field. The ATI spectrum both for the direct and for the rescattered electrons in the context of the SFA is evaluated by numerical integration and by the method of steepest descent (saddle-point integration), and the results are compared. The saddle-point method is modified to avoid the singularity of the dipole transition matrix element at the steepest-descent times. With the help of the saddle-point method and its classical limit, namely the simple-man model, the various features of the ATI spectrum, their behaviour under inversion, the cut-offs and the presence or absence of ATI peaks are analysed as a function of the carrier-envelope phase of the few-cycle laser field. All features observed in the spectra can be explained in terms of a few quantum orbits and their superposition. The validity of the SFA and the concept of quantum orbits are established by comparing the ATI spectra with those obtained numerically from the ab initio solution of the TDSE.

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...

A new scheme for a double-slit experiment in the time domain is presented. Phase-stabilized few-cycle laser pulses open one to two windows ("slits") of attosecond duration for photoionization. Fringes in the angle-resolved energy spectrum of varying visibility depending on the degree of whichway information are observed. A situation in which one and the same electron encounters a single and a double slit at the same time is discussed. The investigation of the fringes makes possible interferometry on the attosecond time scale. The number of visible fringes, for example, indicates that the slits are extended over about 500 as.The conceptually most important interference experiment is the double-slit scheme, which has played a pivotal role in the development of optics and quantum mechanics. In optics its history goes back to Young's double-slit experiment. Its scope was greatly expanded by Zernike's work and continues to deliver new insights into coherence to the present day [1]. One of the key postulates of quantum theory is interference of matter waves, experimentally confirmed by electron diffraction [2,3]. More than 30 years later, Jönsson was the first to perform a double-slit experiment with electrons [4]. Of particular importance for interpreting quantum mechanics have been experiments with a single particle at any given time in the apparatus [5,6]. More recent work has illuminated the fundamental importance of complementarity in which-way experiments [7] and of quantum information in quantum-eraser schemes [8].In this letter a novel realization of the double-slit experiment is described. It is distinguished from conventional schemes by a combination of characteristics: (i) The double slit is realized not in position-momentum but in time-energy domain.(ii) The role of the slits is played by windows in time of attosecond duration. (iii) These "slits" can be opened or closed by changing the temporal evolution of the field of a few-cycle laser pulse. (iv) At any given time there is only a single electron in the double-slit arrangement. (v) The presence and absence of interference are observed for the same electron at the same time.Interference experiments in the time-energy domain are not entirely new. Interfering electron wave packets were created by femtosecond laser pulses [9]. Accordingly, the windows in time (or temporal slits) during Temporal variation of the electric field E (t) = E0(t) cos(ωt + ϕ) of few-cycle laser pulses with phase ϕ = 0 ("cosine-like") and ϕ = −π/2 ("sine-like"). In addition, the field ionization probability R(t), calculated at the experimental parameters, is indicated. Note that an electron ionized at t = t0 will not necessarily be detected in the opposite direction of the field E at time t0 due to deflection in the oscillating field.which these wave packets are launched were comparable to the pulse duration. In the present experiment, in contrast, the slits are open during a small fraction of an optical cycle, which gives the attosecond width. A number of experiments, in particular...

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