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.

Atoms interacting with intense laser fields can emit electrons and photons of very high energies. An intuitive and quantitative explanation of these highly nonlinear processes can be found in terms of a generalization of classical Newtonian particle trajectories, the so-called quantum orbits. Very few quantum orbits are necessary to reproduce the experimental results. These orbits are clearly identified, thus opening the way for an efficient control as well as previously unknown applications of these processes.

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...

An efficient method is investigated for the generation of circularly polarized high-order harmonics by a bichromatic laser field whose two components with frequencies and 2 are circularly polarized in the same plane, but rotate in opposite directions. The generation of intense harmonics by such a driving-field configuration was already confirmed by a previous experiment. With the help of both a semiclassical three-step model as well as a saddle-point analysis, the mechanism of harmonic generation in this case is elucidated and the plateau structure of the harmonic response and their cutoffs are established. The sensitivity of the harmonic yield and the polarization of the harmonics to imperfect circular polarization of the driving fields are investigated. Optimization of both the cutoff frequency and the harmonic efficiency with respect to the intensity ratio of the two components of the driving field is discussed. The electron trajectories responsible for the emission of particular harmonics are identified. Unlike the case of a linearly polarized driving field, they have a nonzero start velocity. By comparison with the driving-field configuration where the two components rotate in the same direction, the mechanism of the intense harmonic emission is further clarified. Depending on the ͑unknown͒ saturation intensity for the bichromatic field with counter-rotating polarizations, this scheme might be of practical interest not only because of the circular polarization of the produced harmonics, but also because of their production efficiency.

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The strong-field approximation can be and has been applied in both length gauge and velocity gauge with quantitatively conflicting answers. For ionization of negative ions with a ground state of odd parity, the predictions of the two gauges differ qualitatively: in the envelope of the angularresolved energy spectrum, dips in one gauge correspond to humps in the other. We show that the length-gauge SFA matches the exact numerical solution of the time-dependent Schrödinger equation.PACS numbers: 42.50. Hz, 32.80.Rm, 32.80.Gc Quantum mechanics is gauge invariant: it is easily proven that a given physical quantity can be evaluated in any gauge with the same result [1]. In nonrelativistic quantum mechanics, when the dipole approximation is adopted, the interaction of an atom with a timedependent field such as a laser field is usually described in either one of two gauges: the length gauge (L gauge) or the velocity gauge (V gauge). In numerical solutions of the time-dependent Schrödinger equation (TDSE), gauge invariance has been confirmed many times. In analytical work, however, some approximations almost always have to be adopted. There is no formal reason of why after such approximations the resulting theory should still be gauge invariant. Indeed, the lack of gauge invariance after what seems like very well justified approximations has driven many a researcher to despair [2].In this paper, we will address one of the most glaring manifestations of this "gauge problem": the lack of gauge invariance of the strong-field approximation (SFA) in intense-laser-atom physics [3]. The SFA underlies almost any analytical approach to total ionization rates, above-threshold ionization, high-order harmonic generation, and nonsequential double ionization, both of atoms and of molecules. Briefly, it assumes that the initial bound state of the atom or molecule is unaffected by the laser field while the final state, which is in the continuum, does not feel the presence of the binding potential. The lack of gauge invariance of the SFA has been noted many times; see, e.g., Ref. [4]. Comparisons that have been carried out indeed have exhibited significant disagreements between the results obtained from L gauge and V gauge [5]. Different authors have preferred different gauges. The question of which gauge is superior for which problem has often been raised, but never led to any consensus about its answer. Below, we will give an answer for the case of a short-range binding potential, where the SFA is expected to be most accurate [6], by comparing the SFA in L gauge and V gauge with the numerical solution of the TDSE.For a fixed nucleus and in the single-active-electron approximation, where the effects of all electrons but one are absorbed into an effective binding potential, the complete Hamiltonian in the presence of an external electromagnetic field can be decomposed aswhere the subscript x specifies the gauge (x = L, V) andThis operator contains the binding potential V (r) and is independent of the gauge. With the dipole approximat...

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