Computational relativistic quantum dynamics and its application to relativistic tunneling and Kapitza-Dirac scattering
Heiko Bauke, Michael Klaiber, Enderalp Yakaboylu, Karen Z. Hatsagortsyan, Sven Ahrens, Carsten Müller, Christoph H. Keitel
发表信息：
Proc. SPIE 8780, High-Power, High-Energy, and High-Intensity Laser Technology; and Research Using Extreme Light: Entering New Frontiers with Petawatt-Class Lasers, article 87801Q (2013)
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.
Interactions between atoms and light fields are usually described in the electric-dipole approximation. We show that electric-quadrupole terms are important in the Kapitza-Dirac arrangement for light gratings on resonance with a quadrupole atomic transition. We derive the diffraction patterns, which in some cases are experimentally verifiable with the same techniques used with dipole transitions.
We theoretically demonstrate the feasibility of producing electron beam splitter using Kapitza-Dirac diffraction on bichromatic standing waves which are created by the fundamental frequency and the third harmonic. The relativistic electron in Bragg regime absorbs three photons with frequency of w and emits a photon with frequency of 3w, four-photon Kapitza-Dirac effect. In this four-photon Kapitza-Dirac effect distinct spin effects arise in different polarizations of the third harmonic laser beam. It is shown that the shape of Rabi oscillation between initial and scattered states is changed an…
Kapitza-Dirac scattering of nonrelativistic electrons from counterpropagating bichromatic laser waves of linear polarization is studied. The focus lies on the electronic spin dynamics in the Bragg regime when the laser fields possess a frequency ratio of two. To this end, the time-dependent Pauli equation is solved numerically, both in coordinate space and momentum space. Our numerical results are corroborated by analytical derivations. We demonstrate that, for certain incident electron momenta, the scattering crucially relies on the electron spin which undergoes characteristic Rabi-like oscil…
We present the possibility of spin-dependent Kapitza-Dirac scattering based on a two-photon interaction only. The interaction scheme is inspired from a Compton scattering process, for which we explicitly show the mathematical correspondence to the spin-dynamics of an electron diffraction process in a standing light wave. The spin effect has the advantage that it already appears in a Bragg scattering setup with arbitrary low field amplitudes, for which we have estimated the diffraction count rate in a realistic experimental setup at available X-ray free-electron laser facilities.
Spin-dependent Kapitza-Dirac scattering of electron beams from counterpropagating bichromatic laser waves in various polarization geometries is studied. The corresponding scattering probabilities are obtained by analytical and numerical solutions of the time-dependent Dirac equation, assuming a field frequency ratio of 2. When the fundamental field mode is circular-polarized, we show that spin dynamics are generally suppressed at low intensities, but can become distinct at high intensities. Conversely, when a linearly or elliptically polarized fundamental mode is combined with a second harmoni…
We extend the study of Kapitza-Dirac diffraction to the case of two-particle systems. Due to the exchange effects the shape and visibility of the two-particle detection patterns show important differences for identical and distinguishable particles. We also identify a novel quantum statistics effect present in momentum space for some values of the initial particle momenta, which is associated with different numbers of photon absorptions compatible with the final momenta.
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.
We analyze the Bragg regime of the two-particle Kapitza-Dirac arrangement, completing the basic theory of this effect. We provide a detailed evaluation of the detection probabilities for multi-mode states, showing that a complete description must include the interaction time in addition to the usual dimensionless parameter w. The arrangement can be used as a massive two-particle beam splitter. In this respect, we present a comparison with Hong-Ou-Mandel-type experiments in quantum optics. The analysis reveals the presence of dips for massive bosons and a differentiated behavior of distinguisha…
We analyze the diffraction of elementary systems as the electron by light gratings when they are described by charge distributions instead of the usual point-like form. The treatment of the problem is based on the introduction, in analogy with atomic polarizability, of state-dependent non-permanent multi-pole moments for the charge. The diffraction patterns can provide bounds on these moments. With this approach we can experimentally explore some aspects of the interpretation of the wave picture of single charges.
We theoretically investigate the influence of a longitudinal laser polarization component from beam focusing on spin dynamics in Kapitza-Dirac scattering by solving the relativistic Dirac equation with time-dependent perturbation theory. The transverse spacial dependence of the longitudinal beam polarization component is accounted for, by approximating a Gaussian beam with plane-wave components. We find that corrections from a longitudinal laser beam polarization component approximately scale with the second power of the diffraction angle $ε$, from which we conclude that a related influence fr…
In this paper we present an analysis of the spin behavior of electrons propagating through a laser field. We present an experimentally realizable scenario in which spin-dependent effects of the interaction between the laser and the electrons are dominant. The laser interaction strength and incident electron velocity are in the nonrelativistic domain. This analysis may thus lead to novel methods of creating and characterizing spin-polarized nonrelativistic femtosecond electron pulses.
The Kapitza - Dirac effect is the diffraction of a well - collimated particle beam by a standing wave of light. Why is this interesting? Comparing this situation to the introductory physics textbook example of diffraction of a laser beam by a grating, the particle beam plays the role of the incoming wave and the standing light wave the role of the material grating, highlighting particle - wave duality. Apart from representing such a beautiful example of particle - wave duality, the diffracted particle beams are coherent. This allows the construction of matter interferometers and explains why t…
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 with 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 spins of the scattered electrons and of the nonscattered electrons, respectively, precess with a frequency that is of the order of the Bragg-reflection Rabi frequency.…
Harmonic oscillators count among the most fundamental quantum systems with important applications in molecular physics, nanoparticle trapping, and quantum information processing. Their equidistant energy level spacing is often a desired feature, but at the same time a challenge if the goal is to deterministically populate specific eigenstates. Here, we show how interference in the transition amplitudes in a bichromatic laser field can suppress the sequential climbing of harmonic oscillator states (Kapitza-Dirac blockade) and achieve selective excitation of energy eigenstates, Schrödinger cats …
In the Kapitza-Dirac effect, atoms, molecules, or swift electrons are diffracted off a standing wave grating of the light intensity created by two counter-propagating laser fields. In ultrafast electron optics, such a coherent beam splitter offers interesting perspectives for ultrafast beam shaping. Here, we study, both analytically and numerically, the effect of the inclination angle between two laser fields on the diffraction of pulsed, low-energy electron beams. For sufficiently high light intensities, we observe a rich variety of complex diffraction patterns. These do not only reflect inte…
Kapitza-Dirac-Talbot-Lau interferometry (KDTLI) has recently been established for demonstrating the quantum wave nature of large molecules. A phase space treatment permits us to derive closed equations for the near-field interference pattern, as well as for the Moire-type pattern that would arise if the molecules were to be treated as classical particles. The model provides a simple and elegant way to account for the molecular phase shifts related to the optical dipole potential as well as for the incoherent effect of photon absorption at the second grating. We present experimental results for…
The quantum states of the Kapitza pendulum are described within the effective potential obtained by the method of averaging over the fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using approximate model potential. For the lowest states of an inverted pendulum, the spectrum is repeduced by the energies of a harmonic oscillator with perturbation theory corrections. Tunneling effect contribution to the energies of resonance states in the double-well effective potential is estimated. The results of numerical simulations of vibrational and rotational…
Coherent regime of Kapitza-Dirac effect with electrons
Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）；Institute of Physics, Czech Academy of Sciences（捷克科学院物理研究所）；Carl Zeiss Microscopy GmbH（卡尔蔡司显微镜有限公司）
The stationary eigenstates and eigenvalues for the ponderomotive potential of an optical crystal confined in a one-dimensional infinite square well are numerically obtained. The initial states of the incoming particles taken as Gaussian, are expanded in the basis of the stationary eigenstates of the ponderomotive potential, to obtain the subsequent time evolution of the wave function of the particle during the interaction with the optical crystal. From the results of the time evolution of the probability density, it is observed that the particles get localized at equidistant positions in the t…
Research on matter waves is a thriving field of quantum physics and has recently stimulated many investigations with electrons, neutrons, atoms, Bose-condensed ensembles, cold clusters and hot molecules. Coherence experiments with complex objects are of interest for exploring the transition to classical physics, for measuring molecular properties and they have even been proposed for testing new models of space-time. For matter-wave experiments with complex molecules, the strongly dispersive effect of the interaction between the diffracted molecule and the grating wall is a major challenge beca…
We study quantum mechanics problem described by the Schrödinger equation with Kapitza pendulum potential, that is the asymmetric double-well potential on the circle. For the oscillatory states spatially localize around the two stable saddle positions of the potential, we obtain the perturbative eigenvalues and corresponding piecewise wavefunctions. The spectrum is computed by extending the angle coordinate to the complex plane so that the quantization condition is formulated as contour integral along a path extending in the imaginary direction. Quantum tunneling between the wells is computed.
Diffraction of walking drops by a standing Faraday wave
Department of Mathematics, MIT（麻省理工学院数学系）；Department of Mechanical Engineering, Boston University（波士顿大学机械工程系）；Department of Mathematics, UNC, Chapel Hill（北卡罗来纳大学教堂山分校数学系）
It is shown that the Kapitza-Dirac effect with atoms, which has been considered to be evidence of their wavelike character, can be interpreted as a scattering of pointlike objects by the periodic laser field.
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…
We study the tunneling of Dirac fermions in graphene through a double barrier potential allowing the carriers to have an effective mass inside the barrier as generated by a lattice miss-match with the boron nitride substrate. The consequences of this gap opening on the transmission are investigated. The realization of resonant tunneling conditions is also analyzed.
We calculate the tunneling process of a Dirac particle across two square barriers separated a distance $d$, as well as the scattering by a double cusp barrier where the centers of the cusps are separated a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that, the presence of transmission resonances modifies the Lorentizian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range …
We use the Wick-rotated time-dependent supersymmetry to construct models of two-dimensional Dirac fermions in presence of an electrostatic grating. We show that there appears omnidirectional perfect transmission through the grating at specific energy. Additionally to being transparent for incoming fermions, the grating hosts strongly localized states.
Based on the Dirac equation, the behavior of relativistic electrons which tunnel a potential barrier of height V0 for incoming energies between V0 and V0+m is studied by using the wave packet formalism. The choice of this incoming energy zone guarantees that only electrons participate to the tunneling process. In the opaque limit, as shown in a previous analysis, the transit velocity is proportional to the barrier width and for relativistic potential tends to a constant value greater than $c$. In this paper, a new numerical study shows a very surprising result: superluminal transmissions are a…
We propose the quantum simulation of the Dirac equation with potentials, allowing the study of relativistic scaterring and the Klein tunneling. This quantum relativistic effect permits a positive-energy Dirac particle to propagate through a repulsive potential via the population transfer to negative-energy components. We show how to engineer scalar, pseudoscalar, and other potentials in the 1+1 Dirac equation by manipulating two trapped ions. The Dirac spinor is represented by the internal states of one ion, while its position and momentum are described by those of a collective motional mode. …
We discuss the properties of 1D stationary pulses of light in atomic ensemble with electromagnetically induced transparency in the limit of tight spatial confinement. When the size of the wavepacket becomes comparable or smaller than the absorption length of the medium, it must be described by a two-component vector which obeys the one-dimensional two-component Dirac equation with an effective mass $m^*$ and effective speed of light $c^*$. Then a fundamental lower limit to the spatial width in an external potential arises from Klein tunneling and is given by the effective Compton length $λ_C =…
In this article we show that a Dirac Hamiltonian in a curved background spacetime can be interpreted, when discretized, as a tight binding Fermi-Hubbard model with non unitary tunnelings. We find the form of the nonunitary tunneling matrices in terms of the metric tensor. The main motivation behind this exercise is the feasibility of such Hamiltonians by means of laser assisted tunnelings in cold atomic experiments. The mapping thus provide a physical interpretation of such Hamiltonians. We demonstrate the use of the mapping on the example of time dependent metric in 2+1 dimensions. Studying t…
Decoherence can be provided by a dissipative environment as described by the Caldeira-Leggett equation. This equation is foundational to the theory of quantum dissipation. However, no experimental test has been performed that measures for one physical system both the dissipation and the decoherence. Anglin and Zurek predicted that a resistive surface could provide such a dissipative environment for a free electron wave passing close to it. We propose that the electron wave's coherence and energy loss can be measured simultaneously by using Kapitza-Dirac scattering for varying light intensity.
We prove that the classical Dirac equation in the presence of an external (nondynamical) electromagnetic field is a relativistically causal theory. As a corollary, we show that it is impossible to use quantum tunneling to transmit particles or information faster than light. When an electron tunnels through a barrier, it is bound to remain within its future light cone. In conclusion, the relativistic quantum tunneling (if modeled using the Dirac equation) is an entirely subluminal process, and it is not instantaneous.
We examine one-dimensional quantum scattering of a Dirac particle off relativistic potential barriers. With proper considerations of Dirac sea, existence of anomalous tunneling at zero incident-energy is revealed for a particular type of relativistic potential having same magnitudes and opposite signs for scalar and vector components. It is also shown that this leads to an exotic short range limit of the potentials.
In a recent work we have proven the existence of degenerate solutions to the Dirac equation, corresponding to an infinite number of different electromagnetic fields, providing also some examples regarding massless particles. In the present article our results are extended significantly, providing degenerate solutions to the Dirac equation for particles with arbitrary mass, which, under certain conditions, could be interpreted as pairs of particles (or antiparticles) moving in a potential barrier with energy equal to the height of the barrier and spin opposite to each other. We calculate the el…
We study the tunneling time of Dirac fermions in graphene magnetic barrier through an electrostatic potential and a mass term. This latter generates an energy gap in the spectrum and therefore affects the proprieties of tunneling of the system. For clarification, we first start by deriving the eigenspinors solutions of Dirac equation and second connect them to the incident, reflected and transmitted beam waves. This connection allows us to obtain the corresponding phases shifts and consequently compute the group delay time in transmission and reflection. Our numerical results show that the gro…
We study the transmission probability of Dirac fermions in graphene scattered by a triangular double barrier potential in the presence of an external magnetic field. Our system made of two triangular potential barrier regions separated by a well region characterized by an energy gap. Solving our Dirac-like equation and matching the solutions at the boundaries allowed us to express our transmission and reflection coefficients in terms of transfer matrix. We show in particular that the transmission exhibits oscillation resonances that are manifestations of the Klein tunneling effect. The triangu…
This article provides a pedagogical review on Klein tunneling in graphene, i.e. the peculiar tunneling properties of two-dimensional massless Dirac electrons. We consider two simple situations in detail: a massless Dirac electron incident either on a potential step or on a potential barrier and use elementary quantum wave mechanics to obtain the transmission probability. We emphasize the connection to related phenomena in optics, such as the Snell-Descartes law of refraction, total internal reflection, Fabry-Pérot resonances, negative refraction index materials (the so called meta-materials), …
We study the quantum tunneling of non-relativistic electrons for two dimensional condensed matter systems. We employ the Levy-Leblond equation (which is the analogue of the Dirac equation for non-relativistic fermions) and show that the spin of the particle can be incorporated in the 2D tunneling problem. We derive the transmission and reflection coefficients of spin up and down electrons and show that the sum of these coefficients are consistent with the known results for gapless semiconductors.
We obtain the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation regime when the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation originated from the study based on non-relativistic dynamics of tunneling is overcome. The treatment of the problem suggests revealing ins…
We theoretically discuss several aspects of sequential superradiant scattering from atomic Bose-Einstein condensates. Our treatment is based on the semiclassical description of the process in terms of the Maxwell-Schroedinger equations for the coupled matter-wave and optical fields. First, we investigate sequential scattering in the weak-pulse regime and work out the essential mechanisms responsible for bringing about the characteristic fan-shaped side-mode distribution patterns. Second, we discuss the transition between the Kapitza-Dirac and Bragg regimes of sequential scattering in the stron…
Zero-mass lines result in appearance of linear dispersion modes for Dirac fermions. These modes play an important role in various physical systems. However, a Dirac fermion may not precisely follow a single zero-mass line, due to either tunneling between different lines or centrifugal forces. Being shifted from a zero-mass line the Dirac fermion acquires mass which can substantially influence its expected "massless" behavior. In the paper we calculate the energy gap caused by the tunneling between two zero-mass lines and show that its opening leads to the delocalization of linear dispersion mo…
We study the tunneling zone solutions of a one-dimensional electrostatic potential for the relativistic (Dirac to Klein-Gordon) wave equation when the incoming wave packet exhibits the possibility of being almost totally transmitted through the barrier. The transmission probabilities, the phase times and the dwell times for the proposed relativistic dynamics are obtained and the conditions for the occurrence of accelerated tunneling transmission are all quantified. We show that, in some limiting cases, the analytical difficulties that arise when the stationary phase method is employed for obta…
Driven tunneling between graphene Landau levels is theoretically linked to the process of pair creation from vacuum, a prediction of quantum electrodynamics (QED). Landau levels are created by the presence of a strong, constant, quantizing magnetic field perpendicular to a graphene mono-layer. Following the formal analogy between QED and the description of low-energy excitations in graphene, solutions of the fully interacting Dirac equation are used to compute electron-hole pair creation driven by a circularly or linearly polarized field. This is achieved via the coupled channel method, a nume…
