Obtaining a numerical solution of the time-dependent Schrödinger equation requires an initial state for the time evolution. If the system Hamiltonian can be split into a time-independent part and a time-dependent perturbation, the initial state is typically chosen as an eigenstate of the former. For propagation using approximate methods such as operator splitting, we show that both imaginary-time evolution and diagonalization of the time-independent Hamiltonian produce states that are not exactly stationary in absence of the perturbation. In order to avoid artifacts from these non-stationary initial states, we propose an iterative method for calculating eigenstates of the real-time propagator. We compare the performance of different initial states by simulating ionization of a model atom in a short laser pulse and we demonstrate that much lower noise levels can be achieved with the real-time propagator eigenstates.

Compton scattering is one of the fundamental interaction processes of light with matter. When discovered(1), it was described as a billiard-type collision of a photon 'kicking' a quasi-free electron. With decreasing photon energy, the maximum possible momentum transfer becomes so small that the corresponding energy falls below the binding energy of the electron. In this regime, ionization by Compton scattering becomes an intriguing quantum phenomenon. Here, we report on a kinematically complete experiment studying Compton scattering off helium atoms in that regime. We determine the momentum correlations of the electron, the recoiling ion and the scattered photon in a coincidence experiment based on cold target recoil ion momentum spectroscopy, finding that electrons are not only emitted in the direction of the momentum transfer, but that there is a second peak of ejection to the backward direction. This finding links Compton scattering to processes such as ionization by ultrashort optical pulses(2), electron impact ionization(3,4), ion impact ionization(5,6) and neutron scattering(7), where similar momentum patterns occur. Compton scattering experiments off helium atoms for photon energies close to the ionization threshold reveal that electrons are not only emitted in the direction of the momentum transfer but also backwards.

Streaking with a weak probe field is applied to ionization in a two-dimensional strong field tailored to mimic linear polarization, but without disturbance by recollision or intracycle interference. This facilitates the observation of electron-momentum-resolved times of ionization with few-attosecond precision, as demonstrated by simulations for a model helium atom. Aligning the probe field along the ionizing field provides meaningful ionization times in agreement with the attoclock concept that ionization at maximum field corresponds to the peak of the momentum distribution, which is shifted due to the Coulomb force on the outgoing electron. In contrast, this attoclock shift is invisible in orthogonal streaking. Even without a probe field, streaking happens naturally along the laser propagation direction due to the laser magnetic field. As with an orthogonal probe field, the attoclock shift is not accessible by the magnetic-field scheme. For a polar molecule, the attoclock shift depends on orientation, but this does not imply an orientation dependence in ionization time.

Ionization of atoms by strong laser fields produces photoelectron momentum distributions that exhibit modulations due to the interference of outgoing electron trajectories. For a faithful modeling, it is essential to include previously overlooked phase shifts occurring when trajectories pass through focal points. Such phase shifts are known as Gouy's phase anomaly in optics or as Maslov phases in semiclassical theory. Because of Coulomb focusing in three dimensions, one out of two trajectories in photoelectron holography goes through a focal point as it crosses the symmetry axis in momentum space. In addition, there exist observable Maslov phases already in two dimensions. Clustering algorithms enable us to implement a semiclassical model with the correct preexponential factor that affects both the weight and the phase of each trajectory. We also derive a simple rule to relate two-dimensional and three-dimensional models for linear polarization. It explains the shifted interference fringes and weaker high-energy yield in three dimensions. The results are in excellent agreement with solutions of the time-dependent Schrödinger equation.

The attoclock technique which maps the emission time of a photoelectron to its detection angle is an important tool in strong-field physics. Previously, it was implemented only with circularly or elliptically polarized laser fields. Here, we show how counter-rotating bicircular laser fields can be used as an attoclock to investigate the ionization dynamics in quasilinear polarization. This is achieved by choosing the ratio of the two field strengths in a way such that the vector potential has aspects of the attoclock and time is mapped directly to the photoelectron momentum, but the shape of the electric field corresponds to approximately linear polarization during three intervals per optical cycle. We report momentum distributions calculated by solving the time-dependent Schr\"odinger equation for a model helium atom and obtain the mapping from photoelectron momentum to ionization time using a trajectory-free method. Unlike circular polarization where the time of maximal ionization rate typically deviates less than 5 attoseconds from the maximium of the electric field, we find positive ionization times of more than 10 attoseconds in the quasilinear case.

Streaking with a weak probe field is applied to ionization in a two-dimensional strong field tailored to mimic linear polarization but without disturbance by recollision or intracycle interference. This facilitates the observation of electron-momentum-resolved times of ionization with few-attosecond precision as demonstrated by simulations for a model helium atom. Aligning the probe field along the ionizing field provides meaningful ionization times in agreement with the attoclock concept that ionization at maximum field corresponds to the Coulomb-shifted peak of the momentum distribution. In contrast, this attoclock shift is invisible in orthogonal streaking. Even without a probe field, streaking happens naturally along the laser propagation direction due to the laser magnetic field. As with an orthogonal probe field, the attoclock shift is not accessible by this scheme. For a polar molecule, the attoclock shift depends on orientation, but this does not imply an orientation dependence in ionization time.

Ionization of atoms by linearly polarized strong laser fields produces cylindrically symmetric photoelectron momentum distributions that exhibit modulations due to the interference of outgoing electron trajectories. For a faithful modeling, it is essential to include previously overlooked phase jumps occurring when trajectories pass through focal points. Such phase jumps are known as Gouy’s phase anomaly in optics or as Maslov phases in semiclassical theory. Most importantly, because of Coulomb focusing in three dimensions, one out of two trajectories in photoelectron holography goes through a focal point as it crosses the symmetry axis in momentum space. In addition, there exist observable Maslov phases already in two dimensions. Clustering algorithms enable us to implement a semiclassical model with the correct preexponential factor that affects both the weight and the phase of each trajectory. We also derive a simple rule to relate two-dimensional and three-dimensional models. It explains the shifted interference fringes and weaker high-energy yield in three dimensions. The results are in excellent agreement with solutions of the time-dependent Schrödinger equation.

Wave-particle duality is an inherent peculiarity of the quantum world. The double-slit experiment has been frequently used for understanding different aspects of this fundamental concept. The occurrence of interference rests on the lack of which-way information and on the absence of decoherence mechanisms, which could scramble the wave fronts. Here, we report on the observation of two-center interference in the molecular-frame photoelectron momentum distribution upon ionization of the neon dimer by a strong laser field. Post-selection of ions, which are measured in coincidence with electrons, allows choosing the symmetry of the residual ion, leading to observation of both, gerade and ungerade, types of interference.

We investigate tunneling ionization of a model helium atom in a strong circularly polarized short laser pulse using the classical backpropagation method and compare ten different tunneling criteria on the same footing, aiming for a consistent classical picture of the tunneling dynamics. These tunneling criteria are categorized into velocity-based, position-based, and energy-based criteria according to different notions of a tunnel exit. We find that velocity-based criteria give consistent tunneling exit characteristics with nonadiabatic effects fully included. Other criteria are either inconsistent or only able to include nonadiabatic effects partially. Furthermore, we construct a simple tunneling rate formula, identify a term in the rate responsible for the nonadiabatic effects, and demonstrate the importance of this term.

We report on the nonadiabatic offset of the initial electron momentum distribution in the plane of polarization upon single ionization of argon by strong field tunneling and show how to experimentally control the degree of nonadiabaticity. Two-color counter- and corotating fields (390 and 780 nm) are compared to show that the nonadiabatic offset strongly depends on the temporal evolution of the laser electric field. We introduce a simple method for the direct access to the nonadiabatic offset using two-color counter- and corotating fields. Further, for a single-color circularly polarized field at 780 nm, we show that the radius of the experimentally observed donutlike distribution increases for increasing momentum in the light propagation direction. Our observed initial momentum offsets are well reproduced by the strong-field approximation. A mechanistic picture is introduced that links the measured nonadiabatic offset to the magnetic quantum number of virtually populated intermediate states.

Using a purely quantum mechanical approach without trajectories, we are able to compute momentum-resolved ionization times of electrons released by the strong-field ionization of atoms. For the attoclock setting, we show that the dominant emission angle corresponds well to instantaneous ionization at the maximum field if geometrical factors are taken into account appropriately and if the field intensity is small enough to avoid depletion of the bound state.

In strong-field ionization processes, two-color laser fields are frequently used for controlling sub-cycle electron dynamics via the relative phase of the laser fields. Here we apply this technique to velocity map imaging spectroscopy using an unconventional orientation with the polarization of the ionizing laser field perpendicular to the detector surface and the steering field parallel to it. This geometry allows not only to image the phase-dependent photoelectron momentum distribution (PMD) of low-energy electrons that interact only weakly with the ion (direct electrons), but also to investigate the low yield of higher-energy rescattered electrons. Phase-dependent measurements of the PMD of neon and xenon demonstrate control over direct and rescattered electrons. The results are compared with semi-classical calculations in three dimensions including elastic scattering at different orders of return and with solutions of the three-dimensional time-dependent Schrödinger equation.

Two-color ionization with a strong 800nm field and a weak orthogonal 400nm field allows for the retrieval of ionization times and relative amplitudes of short and long trajectories directly from the photoelectron momentum distribution by observing the signal as a function of the relative phase between the two fields. By numerical solution of the time-dependent Schrodinger equation in three dimensions we show that the amplitudes are strongly affected by Coulomb focusing. We determine these amplitudes as a function of the lateral momentum, revealing holographic structures in the delay scan.

High harmonic interferometry of the Lorentz force in strong midinfrared laser fields Emilio Pisanty, Daniel D. Hickstein, Benjamin R. Galloway, Charles G. Durfee, Henry C. Kapteyn, Margaret M. Murnane, and Misha Ivanov 1 Blackett Laboratory, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom 2 Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max Born Strasse 2a, 12489 Berlin, Germany 3 JILA – Department of Physics, University of Colorado and NIST, Boulder, Colorado 80309, USA 4 Colorado School of Mines, Department of Physics, Golden, Colorado 80401 USA 5 Department of Physics, Humboldt University, Newtonstrasse 15, 12489 Berlin, Germany High harmonic radiation is a useful source of short pulses of high-frequency XUV radiation, and the cutoff frequency should increase when the driving field is stronger and has a longer wavelength. Unfortunately, in those regimes the magnetic field of the driver acts to deflect the electron excursion and prevent its recollision with the core, thus inhibiting harmonic emission, as a result of a breakdown of the dipole approximation in the long-wavelength regime. We show that the combination of two non-collinear counter-rotating circularly polarized beams produces a small forwards ellipticity, and that this can be used to probe, measure, control, and cancel the effect of this magnetic field. Thus we obtain a flexible scheme to re-enable harmonic emission deep in the long-wavelength and strong field regimes. We show, moreover, that the beam configuration can be used with currently-available sources to demonstrate clear traces, in the form of even harmonics, of the breakdown of the dipole approximation. [1] E. Pisanty, D.D. Hickstein, B.R. Galloway et al. High harmonic interferometry of the Lorentz force in strong mid-infrared laser fields. arXiv:1606.01931. The circular w+2w scheme: symmetry breaking and ways to control the ellipticity of attosecond bursts Álvaro Jiménez-Galán, Marcel Schloz, David Ayuso, Felipe Morales, Olga Smirnova, Mikalai Zhavarankau and Misha Ivanov 1 Max Born Institute, Max Born Strasse 2A, D-12489 Berlin, Germany 2 Department of Physics, Humboldt University, Newtonstrasse 15, D-12489 Berlin, Germany 3 Department of Physics, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom Generation of coherent light sources with attosecond duration and controllable ellipticity will enable studies on chiral-sensitive systems at ultrafast time-scales [1,2]. An elegant approach to the generation of such pulses consists of combining a circularly polarized fundamental field with a counter-rotating second harmonic [3]. This scheme leads to harmonic peaks at (3N+1) and (3N+2) lines, with the helicity of the fundamental field and the second harmonic, respectively, while 3N harmonics are absent due to symmetry. Suppression of one of the lines will thus yield highly elliptical pulses. Several studies [4,5] have investigated this possibility, but control over such ellipticity is still to be achieved. Moreover, recent experiments have reported the appearance of forbidden 3N harmonics even for perfectly three-foil-symmetric pulses, implying a breaking of the symmetry that is yet to be explained. In this work, we outline the physical mechanism responsible for the different strengths of the lines observed in the harmonic spectrum and the reasons behind the breaking of the symmetry in the system. We do so by solving the time-dependent Schrödinger equation in the single-active electron approximation for both helium and neon, and compare our results to experiments. Additionally, we derive ionization and recombination propensity rules in the harmonic process, and show how they can be used to control the ellipticity of the generated attosecond bursts. [1] C. Lux et al., Angew. Chem. Int. Ed., 51: 5001-5005 (2012) [2] R. Cireasa et al., Nat. Phys. 11 654-658 (2015) [3] D.B. Milosevic et al., Phys. Rev. A 61 063403 (2000) [4] O. Kfir et al., Nat. Phot. 9 99 (2014) [5] L. Medisauskas et al., Phys. Rev. Lett. 115 153001 (2015) Illuminating Molecular Symmetries with Bicircular High-Order Harmonic Generation Daniel M. Reich and Lars Bojer Madsen 1 Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark 2 Theoretische Physik, Universität Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany I will present a general theory of bicircular high-order-harmonic generation from N-fold rotationally symmetric molecules that has recently been formulated [1]. Using a rotating frame of reference it is possible to predict the complete structure of the high-order-harmonic spectra for arbitrary driving frequency ratios and molecular symmetries can be directly identified from the high-harmonic signal. These findings reveal that a characteristic fingerprint of rotational molecular symmetries can be universally observed in the ultrafast response of molecules to strong bicircular fields. [1] Daniel M. Reich and Lars Bojer Madsen, Phys. Rev. Lett. 117, 133902 Recollision with spin-polarized electrons in tailored laser fields David Ayuso, Álvaro Jiménez-Galán, Felipe Morales, Misha Ivanov and Olga Smirnova Max-Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-BornStrasse 2A, D-12489 Berlin, Germany Department of Physics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK Institute für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, D-12489 Berlin, Germany Technische Universität Berlin, Ernst-Ruska-Gebäude, Hardenbergstraße 36A, 10623 Berlin, Germany Ionization of noble gases by strong infrared circularly-polarized laser pulses can produce electron currents with a controllable degree of spin polarization [1-4]. Spin polarization arises as a result of (1) entanglement between the emitted electron and the parent ion, and (2) sensitivity of ionization to the sense of electron rotation in the initial state. The use of two-color counter-rotating bi-circular fields [5] opens new opportunities for introducing the spin degree of freedom into attosecond science [6], since the liberated electron can be driven back towards the parent ion within one optical cycle. We show that electrons recolliding with the ionic core upon tunnel ionization of xenon atoms driven by strong bi-circular fields are spin polarized and that their degree of polarization depends strongly on the recollision time (energy). We have found that the level of polarization can be modified by tailoring the driving fields, opening the door for attosecond control of spin-resolved dynamics. [1] I. Barth and O. Smirnova, Phys. Rev. A 84, 063415 (2011) [2] I. Barth and O. Smirnova, Phys. Rev. A 87, 013433 (2013) [3] I. Barth and O. Smirnova, Phys. Rev. A 88, 013401 (2013) [4] A. Hartung et al., Nat. Phot. 10, 526 (2016) [5] D. B. Milošević and W. Becker, Phys. Rev. A 62 011403 (2000) [6] D. B. Milošević, Phys. Rev. A 93, 051402(R) (2016) [7] O. Smirnova and M. Ivanov, Multielectron High Harmonic Generation: simple man on a complex plane, chapter 7 in Attosecond and XUV physics, edited by T. Schultz and M. Vrakking, Wiley (2013) Optimization of strong laser field-free alignment using tailored light fields Evangelos T. Karamatskos, A. Trabattoni, T. Mullins, K. Dlugolecki, S. Trippel and J. Küpper, S. Raabe, M.J.J. Vrakking, A. Rouzée, R.R. Johansen and H. Stapelfeldt 1 Center for Free-Electron Laser Science, DESY, Hamburg, Germany 2 Department of Physics, University of Hamburg, Germany 3 The Hamburg Center for Ultrafast Imaging, University of Hamburg, Germany 4 Max Born Institute, Berlin, Germany 5 FOM Institute AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands 6 Department of Chemistry, Aarhus University, Denmark 7 Interdisciplinary Nanoscience Center (iNANO), Aarhus University, Denmark Alignment of molecules with respect to the laboratory fixed frame enables the realization of a large variety of experiments such as the determination of molecular frame photoangular distribution (MFPAD's) or laser induced electron diffraction (LIED) where typically a strong degree of alignment is needed. We present a combined theoretical and experimental effort to optimise the degree of laser field-free alignment of molecules in the gas phase. We start by solving the time-dependent rotational Schrödinger equation coupled to a non-resonant laser field and a static electric field and use an iterative learning-loop algorithm to determine the ideal pulse shape that optimises the degree of alignment. These calculations serve as a guide to complement the experiments where the alignment laser pulse form is optimally tailored. We discuss the simulation results and the experimental realization of two-pulse impulsive alignment on the example of the linear molecule carbonyl sulfide (OCS) and give an outlook for the use of pulse shaping techniques to achieve strongly aligned asymmetric top molecules. Interaction of Optical Vortices with Atoms and Molecules Carlos M. Granados Castro, J. Berakdar Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, Karl-Freiherr-vonFritsch Straße 3, 06120 Halle (Saale), Germany Over the last decades, the study of photoionization of atoms and molecules has been an active field in physics, and quite a few theoretical methods have been proposed and applied successfully [1,2]. Most of these studies were carried considering interaction with photons —specially described by plane waves or short laser pulses— within the dipolar approximation only (transferring only one unit of angular momenta). However, in recent years, light beams carrying orbital angular momentum (OAM), such as Laguerre– Gaussian beams, become available, allowing one to go beyond the dipolar approximation. Then, some extensions to the current theoretical methods are needed, in order to explore this new framework in nonlinear and quantum optics. In this contribution we explore the implementation of the Sturmian approach [2,3], based o