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 $λ_C = \hbar/(m^* c^*)$. Since $c^*$ and $m^*$ can be externally controlled and can be made small it is possible to observe effects of the relativistic dispersion for rather low energies or correspondingly on macroscopic length scales.
We propose an experimental scheme to simulate and observe the Klein tunneling of relativistic Maxwell particles with trapped ions. We explore the scattering dynamics of the pseudospin-one Maxwell particles and demonstrate that the scattered state should be a superposition of a reflection state, a localization state, and a transmission state. The probabilities of these states can be analytically obtained by the approach of Landau-Zener transition. We further show that the Maxwell Hamiltonian and the associated scattering dynamics can be mimicked with two trapped ions. The Maxwell spinors are en…
We study a transmission coefficient of graphene nanoribbons with a top gate which acts as an oblique barrier. Using a Green function method based on the Dirac-like equation, scattering among transverse modes due to the oblique barrier is taken into account numerically. In contrast to the 2-dimensional graphene sheet, we find that the pattern of transmission in graphene ribbons depends strongly on the electronic structure in the region of the barrier. Consequently, irregular structures in the transmission coefficient are predicted while perfect transmission is still calculated in the case of me…
We calculate the transmission coefficient for a particle crossing a potential barrier in monolayer graphene with Rashba spin-orbit coupling and in bilayer graphene. We show that in both the cases one can go from Klein tunneling regime, characterized by perfect normal transmission, to anti-Klein tunneling regime, with perfect normal reflection, by tuning the Rashba spin-orbit coupling for a monolayer or the interplane coupling for a bilayer graphene. We show that the intermediate regime is characterized by a non-monotonic behavior with oscillations and resonances in the normal transmission ampl…
We propose an experimental scheme to detect macroscopic Klein tunneling with spin-orbit coupled Bose-Einstein condensates (BECs). We show that a nonlinear Dirac equation with tunable parameters can be realized with such BECs. %in a simple configuration to generate the artificial spin-orbit coupling. Through numerical calculations, we demonstrate that macroscopic Klein tunneling can be clearly detected under realistic conditions. Macroscopic quantum coherence in such relativistic tunneling is clarified and a BEC with a negative energy is shown to be able to transmit transparently through a wide…
Klein tunneling and conductance for Dirac fermions in ABC-stacked trilayer graphene (ABC-TLG) through symmetric and asymmetric double potential barriers are investigated using the two and six-band continuum model. Numerical results for our system show that the transport is sensitive to the height, the width and the distance between the two barriers. Klein paradox at normal incidence and resonant features at non-normal incidence in the transmission result from resonant electron states in the wells or hole states in the barriers and strongly influence the ballistic conductance of the structures.
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), …
Statistical complexity and Fisher-Shannon information are calculated in a problem of quantum scattering, namely the Klein tunneling across a potential barrier in graphene. The treatment of electron wave functions as masless Dirac fermions allows us to compute these statistical measures. The comparison of these magnitudes with the transmission coefficient through the barrier is performed. We show that these statistical measures take their minimum values in the situations of total transparency through the barrier, a phenomenon highly anisotropic for the Klein tunneling in graphene.
Optical lattices have proven to be powerful systems for quantum simulations of solid state physics effects. Here we report a proof-of-principle experiment simulating effects predicted by relativistic wave equations with ultracold atoms in a bichromatic optical lattice that allows for a tailoring of the dispersion relation. We observe the analog of Klein-tunneling, the penetration of relativistic particles through a potential barrier without the exponential damping that is characteristic for nonrelativistic quantum tunneling. Both linear (relativistic) and quadratic (nonrelativistic) dispersion…
We study particle interaction with a Dirac step potential. In the standard Klein energy zone, the hypothesis of Klein pair production predicts the existence of free/oscillatory antiparticles. In this paper, we discuss the tunneling energy zone characterized by evanescent wave functions in the classically forbidden region. We ask the question of the nature, particle or antiparticle, of the densities within the classically forbidden region. The answer to this question is relevant to the correct form of the reflection coefficient.
General physical conditions for the occurrence of photonic Klein tunneling are studied, where (controlled) spontaneous emission from the devices considered plays a key role. The specific example of a simple dielectric slab bounded by two dielectric half spaces with arbitrary refractive indices is worked out quite in detail, the measured reflection and transmission probabilities being calculated analytically. It is found that, in given cases, the measured reflection probability may be arbitrarily large (for large incident wavelengths) irrespective of the fact that the transmission probability i…
Reflection of two strongly interacting bosons with long-rage interaction hopping on a one-dimensional lattice scattered off by a potential step is theoretically investigated in the framework of the extended Hubbard model. The analysis shows that, in the presence of unbalanced on-site and nearest-neighbor site interaction, two strongly correlated bosons forming a bound particle state can penetrate a high barrier, despite the single particle can not. Such a phenomenon is analogous to one-dimensional Klein tunneling of a relativistic massive Dirac particle across a potential step.
We explore the dynamics of relativistic quantum waves in a potential step by using an exact solution to the Klein-Gordon equation with a point source initial condition. We show that in both the propagation, and Klein-tunneling regimes, the Zitterbewegung effect manifests itself as a series of quantum beats of the particle density in the long-time limit. We demonstrate that the beating phenomenon is characterized by the Zitterbewegung frequency, and that the amplitude of these oscillations decays as $t^{-3/2}$. We show that beating effect also manifests itself in the free Klein-Gordon and Dirac…
We investigate Klein tunneling through finite potential barriers with space-time resolved solutions to relativistic quantum field equations. We find that no particle actually tunnels through a finite supercritical barrier, even in the case of resonant tunneling. The transmission is instead mediated by modulations in pair production rates, at each edge of the barrier, caused by the incoming electron. We further examine the effect of the barrier's width on the numbers of produced pairs in the fermionic case (characterized by saturation) and in the bosonic case (characterized by exponential super…
The absence of backscattering in metallic nanotubes as well as perfect Klein tunneling in potential barriers in graphene are the prominent electronic characteristics of carbon nanostructures. We show that the phenomena can be explained by a peculiar supersymmetry generated by a first order Hamiltonian and zero order supercharge operators. Like the supersymmetry associated with second order reflectionless finite-gap systems, it relates here the low-energy behavior of the charge carriers with the free particle dynamics.
The problem of the Klein tunneling across a potential barrier in bi-layer graphene is addressed. The electron wave functions are treated as massive chiral particles. This treatment allows us to compute the statistical complexity and Fisher-Shannon information for each angle of incidence. The comparison of these magnitudes with the transmission coefficient through the barrier is performed. The role played by the evanescent waves on these magnitudes is disclosed. Due to the influence of these waves, it is found that the statistical measures take their minimum values not only in the situations of…
The effect of a parallel magnetic field $B$ and a potential substrate on the transmission, conductivity, and Fano factor of a biased ABC-stacked $pn$ and $pnp$ trilayer graphene junction (ABC-TLG) is investigated theoretically at low energy. We discovered that, in the presence of a high magnetic field $B=1400$T, the ABC-TLG can exhibit new Klein tunneling at two new incidence $k_y$ values, in addition to the usual value of $k_y=0$ and whatever the energy value is. Indeed, the transmission of ABC-TLG via $ pn $ and $ pnp $ junctions is transformed into a three-separate transmission of single-li…
When a single-layer graphene sheet is contacted with metallic electrodes, tunnel barriers are formed as a result of the doping of graphene by the metal in the contact region. If the Fermi energy level is modulated by a gate voltage, the phenomenon of Klein tunneling results in specific features in the conductance and noise. Here we obtain analytically exact solutions for the transmission and reflection probability amplitudes using a trapezoidal potential barrier, allowing us to calculate the differential conductance and the Fano factor for a graphene sheet in the ballistic regime. We put in ev…
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.
The "particle in a box" problem is investigated for a relativistic particle obeying the Klein-Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for "shallow" wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wavefunction outside the well vanish…
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. …
A few properties of the nonminimal vector interactions in the Duffin-Kemmer-Petiau theory are revised. In particular, it is shown that the space component of the nonminimal vector interaction plays a peremptory role for confining bosons whereas its time component contributes to the leakage. Scattering in a square step potential with proper boundary conditions is used to show that Klein's paradox does not manifest in the case of a nonminimal vector coupling.
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…
We propose a new two--qubit phase gate for ultra--cold atoms confined in an experimentally realized tilted double--well optical lattice [Sebby--Strabley et al., Phys. Rev. A {\bf 73} 033605 (2006)]. Such a lattice is capable of confining pairs of atoms in a two--dimensional array of double--well potentials where control can be exercised over the barrier height and the energy difference of the minima of the two wells (known as the ``tilt''). The four lowest single--particle motional states consist of two pairs of motional states in which each pair is localized on one side of the central barrier…
We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external pot…
The electron nondipole dynamics in tunneling ionization in an elliptically polarized laser field is investigated theoretically using a relativistic Coulomb-corrected strong-field approximation (SFA) based on the eikonal approximation of the Klein-Gordon equation. We calculate attoclock angle-resolved light-front momentum distributions at different ellipticities of the laser field in quasistatic and nonadiabatic regimes and analyze them with an improved Simpleman model. The nondipole correlations between longitudinal and transverse momentum components are examined. Deviations of the photoelectr…
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.
