Naman Gupta (intern from Indian Institute of Technology, Kanpur, India)
Employing novel light sources such as
the ELI-Ultra High Field
Facility, for example, that envisage to provide field intensities
in excess of 1020 W/cm2 and field
frequencies in the x-ray domain light-matter interaction in the
relativistic regime may be probed experimentally. Relativistic
quantum mechanics predicts various new phenomena to occur in this
regime, for example, spin effects, multiphoton scattering, radiation
reaction effects, vacuum-polarization effects or even pair creation.
In our research group, we investigate such relativistic quantum
mechanical systems in strong electromagnetic fields from a
theoretical perspective with a strong emphasis on numerical methods.
This also includes the development of numerical algorithms and
high-performance computer codes.
Light-matter interaction in strong laser fields can be modeled within
the frameworks of (relativistic) quantum mechanics and quantum field
theory. The basic equations of nonrelativistic and relativistic
quantum dynamics are the Schrödinger equation and the Dirac equation,
respectively. The Schrödinger equation and the Dirac equation can be
solved analytically for very few systems only. Analytical approaches
fail or involve non-trivial approximations in the case of
light-matter interaction in ultra-strong laser pulses. Thus, the
numerical solution of the time-dependent Schrödinger and Dirac
equations forms the basis for most of our research projects, where
the (relativistic) quantum dynamics is be solved by using
Various different classical models of electrons including their spin degree of freedom are commonly applied to describe the electron dynamics in strong electromagnetic fields. We demonstrate that different models can lead to different or even contradicting predictions how the spin degree of freedom modifies the electron's orbital motion when the electron moves in strong electromagnetic fields. This discrepancy is rooted in the model-specific energy dependency of the spin induced Stern-Gerlach force acting on the electron. The Frenkel model and the classical Foldy-Wouthuysen model are compared exemplarily in the nonrelativistic and the relativistic limits in order to identify parameter regimes where these classical models make different predictions. This allows for experimental tests of these models. In ultra strong laser setups at parameter regimes where effects of the Stern-Gerlach force become relevant also radiation reaction effects are expected to set in. We incorporate radiation reaction classically via the Landau-Lifshitz equation and demonstrate that although radiation reaction effects can have a significant effect on the electron trajectory, the Frenkel model and the classical Foldy-Wouthuysen model remain distinguishable also if radiation reaction effects are taken into account. Our calculations are also suitable to verify the Landau-Lifshitz equation for the radiation reaction of electrons and other spin one-half particles.
Different classical theories ar
e commonly applied in various branches of physics to describe the
relativistic dynamics of electrons by coupled equations for the orbital motion and spin precession.
Exemplarily, we benchmark the Frenkel model and the classical Foldy-Wouthuysen model with spin-
dependent forces (Stern-Gerlach forces) to the quantum dynamics as predicted by the Dirac equation.
Both classical theories can lead to different or even contradicting predictions how the Stern-Gerlach
forces modify the electron’s orbital motion, when the electron moves in strong electromagnetic field
configurations of emerging high-intensity laser facilities. In this way, one may evaluate the validity and
identify the limits of these classical theories via a comparison with possible experiments to provide a
proper description of spin-induced dynamics. Our results indicate that the Foldy-Wouthuysen model is
qualitatively in better agreement with the Dirac theory than the widely used Frenkel model.
The Salecker-Wigner-Peres quantum-clock approach is applied in order to determine the tunneling time of an electron in strong-field tunnel ionization via a time-dependent electric field. Our results show that the ionization of the electron takes a nonvanishing period of time. This tunneling time is of the order of the Keldysh time but strictly larger than the Keldysh time. Comparing the quantum-clock tunneling time to the mean tunneling time as obtained by the virtual-detector approach, one finds that these two complementary methods give very similar results. Due to the asymmetric distribution of the tunneling time, there is a nonnegligible discrepancy between the mean tunneling time and the most probable tunneling time.
Tunneling times in atomic ionization are studied theoretically by a virtual detector approach. A virtual detector is a hypothetical device that allows one to monitor the wave function's density with spatial and temporal resolution during the ionization process. With this theoretical approach, it becomes possible to define unique moments when the electron enters and leaves with highest probability the classically forbidden region from first principles and a tunneling time can be specified unambiguously. It is shown that neither the moment when the electron enters the tunneling barrier nor when it leaves the tunneling barrier coincides with the moment when the external electric field reaches its maximum. Under the tunneling barrier as well as at the exit the electron has a nonzero velocity in the electric field direction. This nonzero exit velocity has to be incorporated when the free motion of the electron is modeled by classical equations of motion.
Ultra strong electromagnetic fields can lead to spontaneous creation of single or multiple electron–positron pairs. A quantum field theoretical treatment of the pair creation process combined with numerical methods provides a description of the fermionic quantum field state, from which all observables of the multiple electron–positron pairs can be inferred. This allows to study the complex multi-particle dynamics of electron–positron pair creation in-depth, including multi-pair statistics as well as momentum distributions and spin. To illustrate the potential benefit of this approach, it is applied to the intermediate regime of pair creation between nonperturbative Schwinger pair creation and perturbative multiphoton pair creation where the creation of multi-pair states becomes nonnegligible but cascades do not yet set in. Furthermore, it is demonstrated how spin and helicity of the created electrons and positrons are affected by the polarization of the counterpropagating laser fields, which induce the creation of electron–positron pairs.
Tunnel ionization belongs to the fundamental processes of atomic physics. The so-called two-step model, which describes the ionization as instantaneous tunneling at the electric field maximum and classical motion afterwards with zero exit momentum, is commonly employed to describe tunnel ionization in adiabatic regimes. In this contribution, we show by solving numerically the time-dependent Schrödinger equation in one dimension and employing a virtual detector at the tunnel exit that there is a nonvanishing positive time delay between the electric field maximum and the instant of ionization. Moreover, we find a nonzero exit momentum in the direction of the electric field. To extract proper tunneling times from asymptotic momentum distributions of ionized electrons, it is essential to incorporate the electron’s initial momentum in the direction of the external electric field.
Interactions between different bound states in bosonic systems can lead to pair creation. We study this process in detail by solving the Klein-Gordon equation on space-time grids in the framework of time-dependent quantum field theory. By choosing specific external field configurations, two bound states can become pseudodegenerate, which is commonly referred to as the Schiff-Snyder-Weinberg effect. These pseudodegenerate bound states, which have complex energy eigenvalues, are related to the pseudo-Hermiticity of the Klein-Gordon Hamiltonian. In this work, the influence of the Schiff-Snyder-Weinberg effect on pair production is studied. A generalized Schiff-Snyder-Weinberg effect, where several pairs of pseudodegenerate states appear, is found in combined electric and magnetic fields. The generalized Schiff-Snyder-Weinberg effect likewise triggers pair creation. The particle number in these situations obeys an exponential growth law in time enhancing the creation of bosons, which cannot be found in fermionic systems.
The Kapitza-Dirac effect, which refers to electron scattering at standing light waves, is studied in the Bragg regime with counterpropagating elliptically polarized electromagnetic waves having the same intensity, wavelength, and degree of polarization for two different setups. In the first setup, where the electric field components of the counterpropagating waves have the same sense of rotation, we find distinct spin effects. The spin of the scattered electrons and of the nonscattered electrons, respectively, precesses with a frequency that is of the order of the Bragg-reflection Rabi frequency. When the electric field components of the counterpropagating waves have opposite sense of rotation, which is the second considered setup, the standing wave has linear polarization and no spin effects can be observed. Our results are based on numerical solutions of the time-dependent Dirac equation and the analytical solution of a relativistic Pauli equation, which accounts for the leading relativistic effects.
We study nonperturbative multiphoton electron-positron pair creation in ultrastrong electromagnetic fields formed by two counterpropagating pulses with elliptic polarization. Our numerical approach allows us to take into account the temporal as well as the spatial variation of the standing electromagnetic field. The spin and momentum resolved pair creation probabilities feature characteristic Rabi oscillations and resonance spectra. Therefore, each laser frequency features a specific momentum distribution of the created particles. We find that, depending on the relative polarization of both pulses, the created electrons may be spin polarized along the direction of field propagation.
Various spin effects are expected to become observable in light-matter interaction at relativistic intensities. Relativistic quantum mechanics equipped with a suitable relativistic spin operator forms the theoretical foundation for describing these effects. Various proposals for relativistic spin operators have been offered by different authors, which are presented in a unified way. As a result of the operators' mathematical properties only the Foldy-Wouthuysen operator and the Pryce operator qualify as possible proper relativistic spin operators. The ground states of highly charged hydrogen-like ions can be utilized to identify a legitimate relativistic spin operator experimentally. Subsequently, the Foldy-Wothuysen spin operator is employed to study electron-spin precession in high-intensity standing light waves with elliptical polarization. For a correct theoretical description of the predicted electron-spin precession relativistic effects due to the spin angular momentum of the electromagnetic wave has to be taken into account even in the limit of low intensities.
The common tunneling picture of electron-positron pair creation in a strong electric field is generalized to pair creation in combined crossed electric and magnetic fields. This enhanced picture, being symmetric for electrons and positrons, is formulated in a gauge-invariant and Lorentz-invariant manner for quasistatic fields. It may be used to infer qualitative features of the pair creation process. In particular, it allows for an intuitive interpretation of how the presence of a magnetic field modifies and, in particular cases, even enhances pair creation. The creation of electrons and positrons from the vacuum may be assisted by an energetic photon, which can also be incorporated into this picture of pair creation.
The Lanczos algorithm is evaluated for solving the time-independent
as well as the time-dependent Dirac equation with arbitrary
electromagnetic fields. We demonstrate that the Lanczos algorithm
can yield very precise eigenenergies and allows very precise time
propagation of relativistic wave packets. The Dirac Hamiltonian's
property of not being bounded does not hinder the applicability of
the Lanczos algorithm. As the Lanczos algorithm requires only
matrix-vector and inner products, which both can be efficiently
parallelized, it is an ideal method for large-scale calculations.
The excellent parallelization capabilities are demonstrated by a
parallel implementation of the Dirac Lanczos propagator utilizing
the Message Passing Interface standard.
We investigate the coupling of the spin angular momentum of light beams with elliptical polarization to the spin degree of freedom of free electrons. It is shown that this coupling, which is of similar origin as the well-known spin-orbit coupling, can lead to spin precession. The spin-precession frequency is proportional to the product of the laser-field's intensity and its spin density. The electron-spin dynamics is analyzed by employing exact numerical methods as well as time-dependent perturbation theory based on the fully relativistic Dirac equation and on the nonrelativistic Pauli equation that is amended by a relativistic correction that accounts for the light's spin density.
Strong rotating magnetic fields may cause a precession of the
electron's spin around the rotation axis of the magnetic field. The
superposition of two counterpropagating laser beams with circular
polarization and opposite helicity features such a rotating magnetic
field component but also carries spin. The laser's spin density, which
can be expressed in terms of the laser's electromagnetic fields and
potentials, couples to the electron's spin via a relativistic
correction to the Pauli equation. We show that the quantum mechanical
interaction of the electron's spin with the laser's rotating magnetic
field and with the laser's spin density counteract each other in such
a way that a net spin rotation remains with a precession frequency
that is much smaller than the frequency one would expect from the
rotating magnetic field alone. In particular, the frequency scales
differently with the laser's electric field strength depending on if
relativistic corrections are taken into account or not. Thus, the
relativistic coupling of the electron's spin to the laser's spin
density changes the dynamics not only quantitatively but also
qualitatively as compared to the nonrelativistic theory. The
electron's spin dynamics is a genuine quantum mechanical relativistic
Meng Wen, Heiko Bauke, Christoph H. Keitel
«Dynamical spin effects in ultra-relativistic laser pulses» arXiv:1406.3659
The dynamics of single laser-driven electrons and many particle systems with spin are investigated on the basis
of a classical theory. We demonstrate that the spin forces can alter the electron dynamics in an ultra-relativistic
laser field due to the coupling of the electron’s spin degree of freedom to its kinematic momentum. High-energy
electrons can acquire significant spin-dependent transverse momenta while passing through a counterpropagating
ultra-relativistic infrared laser pulse. Numerical calculations show that the deflection of the electrons by the laser
pulse is determined by the laser intensity, the pulse duration, and the initial spin orientation of the electron. We
complement our investigation of these dynamical spin effects by performing particle-in-cell simulations and point
out possibilities of an experimental realization of the predicted effect with available laser parameters.
Different operators have been suggested in the literature to describe the electron's spin degree of freedom within the relativistic Dirac theory. We compare concrete predictions of the various proposed relativistic spin operators in different physical situations. In particular, we investigate the so-called Pauli, Foldy-Wouthuysen, Czachor, Frenkel, Chakrabarti, Pryce, and Fradkin-Good spin operators. We demonstrate that when a quantum system interacts with electromagnetic potentials the various spin operators predict different expectation values. This is explicitly illustrated for the scattering dynamics at a potential step and in a standing laser field and also for energy eigenstates of hydrogenic ions. Therefore, one may distinguish between the proposed relativistic spin operators experimentally.
Although the spin is regarded as a fundamental property of the electron, there is no
universally accepted spin operator within the framework of relativistic quantum
mechanics. We investigate the properties of different proposals for a relativistic
spin operator. It is shown that most candidates are lacking essential features
of proper angular momentum operators, leading to spurious zitterbewegung
(quivering motion) or violation of the angular momentum algebra. Only the
Foldy-Wouthuysen operator and the Pryce operator qualify as proper relativistic
spin operators. We demonstrate that ground states of highly charged hydrogen-like
ions can be utilized to identify a legitimate relativistic spin operator
Spin effects in above-threshold ionization of hydrogenlike highly
charged ions in super-strong laser fields are
investigated. Spin-resolved ionization rates in the tunneling regime are
calculated by employing two versions of a relativistic Coulomb-corrected
strong-field approximation (SFA). An intuitive simple-man model is
developed which explains the derived scaling laws for spin-flip and
spin-asymmetry effects. The intuitive model as well as our ab initio
numerical simulations support the analytical results for the spin
effects obtained in the dressed SFA where the impact of the laser field
on the electron spin evolution in the bound state is taken into
account. In contrast, the standard SFA is shown to fail in reproducing
spin effects at ionization even at a qualitative level. The anticipated
spin-effects are expected to be measurable with modern laser techniques
combined with an ion storage facility.
Computational methods are indispensable to study the quantum dynamics of relativistic light-matter interactions in parameter regimes where analytical methods become inapplicable. We present numerical methods for solving the time-dependent Dirac equation and the time-dependent Klein-Gordon equation and their implementation on high performance graphics cards. These methods allow us to study tunneling from hydrogen-like highly charged ions in strong laser fields and Kapitza-Dirac scattering in the relativistic regime.
The electron dynamics in the classically forbidden region during relativistic tunnel ionization is investigated. The classical forbidden region in the relativistic regime is identified by defining a gauge-invariant total-energy operator. Introducing position-dependent energy levels inside the tunneling barrier, we demonstrate that the relativistic tunnel ionization can be well described by a one-dimensional intuitive picture. This picture predicts that, in contrast to the well-known nonrelativistic regime, the ionized electron wave packet arises with a momentum shift along the laser's propagation direction. This is compatible with results from a strong-field approximation calculation where the binding potential is assumed to be zero ranged. Further, the tunneling time delay, stemming from Wigner's definition, is investigated for model configurations of tunneling and compared with results obtained from the exact propagator. By adapting Wigner's time delay definition to the ionization process, the tunneling time is investigated in the deep-tunneling and in the near-threshold-tunneling regimes. It is shown that while in the deep-tunneling regime signatures of the tunneling time delay are not measurable at remote distance, they are detectable, however, in the latter regime.
A relativistic description of the Kapitza-Dirac effect in the so-called
Bragg regime with two and three interacting photons is presented by
investigating both numerical and perturbative solutions of the Dirac
equation in momentum space. We demonstrate that spin flips can be
observed in the two-photon and the three-photon Kapitza-Dirac effects
for certain parameters. During the interaction with the laser field the
electron's spin is rotated, and we give explicit expressions for the
rotation axis and the rotation angle. The off-resonant Kapitza-Dirac
effect, that is, when the Bragg condition is not exactly fulfilled, is
described by a generalized Rabi theory. We also analyze the in-field
quantum dynamics as obtained from the numerical solution of the Dirac
The electron spin degree of freedom can play a significant role in relativistic scattering processes involving intense laser fields. In this contribution we discuss the influence of the electron spin on (i) Kapitza-Dirac scattering in an x-ray laser field of high intensity, (ii) photo-induced electron-positron pair production in a strong laser wave and (iii) multiphoton electron-positron pair production on an atomic nucleus. We show that in all cases under consideration the electron spin can have a characteristic impact on the process properties and their total probabilities. To this end, spin-resolved calculations based on the Dirac equation in the presence of an intense laser field are performed. The predictions from Dirac theory are also compared with the corresponding results from the Klein-Gordon equation.
The tunneling dynamics in relativistic strong-field ionization is investigated with the aim to develop an intuitive picture for the relativistic tunneling regime. We demonstrate that the tunneling picture applies also in the relativistic regime by introducing position dependent energy levels. The quantum dynamics in the classically forbidden region features two time scales, the typical time that characterizes the probability density’s decay of the ionizing electron under the barrier (Keldysh time) and the time interval which the electron spends inside the barrier (Eisenbud-Wigner-Smith tunneling time). In the relativistic regime, an electron momentum shift as well as a spatial shift along the laser propagation direction arise during the under-the-barrier motion which are caused by the laser magnetic field induced Lorentz force. The momentum shift is proportional to the Keldysh time, while the wave-packet’s spatial drift is proportional to the Eisenbud-Wigner-Smith time. The signature of the momentum shift is shown to be present in the ionization spectrum at the detector and, therefore, observable experimentally. In contrast, the signature of the Eisenbud-Wigner-Smith time delay disappears at far distances for pure quasistatic tunneling dynamics.
Electron spin dynamics in Kapitza-Dirac scattering from a standing
laser wave of high frequency and high intensity is studied. We develop
a fully relativistic quantum theory of the electron motion based on
the time-dependent Dirac equation. Distinct spin dynamics, with Rabi
oscillations and complete spin-flip transitions, is demonstrated for
Kapitza-Dirac scattering involving three photons in a parameter regime
accessible to future high-power X-ray laser sources. The Rabi
frequency and, thus, the diffraction pattern is shown to depend
crucially on the spin degree of freedom.
carry out a stability analysis for the real space split operator
method for the propagation of the time-dependent Klein–Gordon
equation that has been proposed in Ruf et al. [M. Ruf, H. Bauke,
C.H. Keitel, A real space split operator method for the
Klein–Gordon equation, Journal of Computational Physics 228
(24) (2009) 9092–9106, doi:10.1016/j.jcp.2009.09.012]. The
region of algebraic stability is determined analytically by means of
a von-Neumann stability analysis for systems with homogeneous scalar
and vector potentials. Algebraic stability implies convergence of the
real space split operator method for smooth absolutely integrable
initial conditions. In the limit of small spatial grid spacings
in each of the
spatial dimensions and small temporal steps
the stability condition becomes
for second order finite differences and
for fourth order finite differences, respectively, with
denoting the speed of light. Furthermore, we demonstrate numerically
that the stability region for systems with inhomogeneous potentials
coincides almost with the region of algebraic stability for
this contribution, we investigate the relativistic ionization
characteristics of highly charged hydrogenlike ions in short intense
laser pulses as a function of the laser pulse parameters by means of
the numerical solution of the time-dependent Dirac equation and the
time-dependent Klein-Gordon equation as well as by the classical
phase-space averaging method. For this purpose, we generalize the
phase-space averaging method such that it is applicable to
relativistically driven particles in arbitrary central potentials. If
the ionization probability is not too small, quantum mechanical and
classical methods give similar results for laser wavelengths in the
range from the near-infrared to soft x-ray radiation. We find that
ionization in few-cycle intense laser pulses depends sensitively on
the pulses' peak intensity but little on the pulse tails and on the
pulse energy. The ionization probability is shown to be strongly
linked to the peak intensity allowing for an estimation of the laser
intensity via ionization yields.
Current generations of graphics processing units have turned into
highly parallel devices with general computing capabilities. Thus,
graphics processing units may be utilized, for example, to solve time
dependent partial differential equations by the Fourier split
operator method. In this contribution, we demonstrate that graphics
processing units are capable to calculate fast Fourier transforms
much more efficiently than traditional central processing
units. Thus, graphics processing units render efficient
implementations of the Fourier split operator method
possible. Performance gains of more than an order of magnitude as
compared to implementations for traditional central processing units
are reached in the solution of the time dependent Schrödinger
equation and the time dependent Dirac equation.
Heiko Bauke and Noya Ruth Itzhak
«Visualizing quantum mechanics in phase space» arXiv:1101.2683
We examine the visualization of quantum mechanics in phase space by
means of the Wigner function and the Wigner function flow as a
complementary approach to illustrating quantum mechanics in
configuration space by wave functions. The Wigner function formalism
resembles the mathematical language of classical mechanics of
non-interacting particles. Thus, it allows a more direct comparison
between classical and quantum dynamical features.
The Klein-Gordon equation is a Lorentz invariant equation of motion
for spinless particles. We propose a real space split operator method
for the solution of the time-dependent Klein-Gordon equation with
arbitrary electromagnetic fields. Split operator methods for the
Schrödinger equation and the Dirac equation typically operate
alternately in real space and momentum space and, therefore, require
the computation of a Fourier transform in each time step. However,
the fact that the kinetic energy operator
in the two-component representation of the Klein-Gordon equation is a
nilpotent operator, that is
, allows us to
implement the split operator method for the Klein-Gordon equation
entirely in real space. Consequently, the split operator method for
the Klein-Gordon equation does not require the computation of a
Fourier transform and may be parallelized efficiently by domain
integration of time-dependent quantum mechanical wave equations
is a fundamental problem in computational physics and computational
chemistry. The wave-function's energy spectrum as well as its
momentum spectrum impose fundamental limits on the performance of
numerical algorithms for the solution of wave equations. We
demonstrate how canonical transforms may be applied to negotiate
these limitations and to increase the performance of numerical
algorithms by up to several orders of magnitude. Our approach
includes the so-called Kramers-Henneberger transform as a special
case and puts forward modifications toward an improved numerical
integration of time dependent quantum mechanical wave equations is a
fundamental problem in computational physics and computational
chemistry. The energy and momentum spectrum of a wave function imposes
fundamental limits on the performance of numerical algorithms for this
problem. We demonstrate how unitary transforms can help to surmount