Evolution Of Gaussian Wave Packets Research Articles

Non-perturbative dynamics of flat-band systems with correlated disorder

Abstract We develop a numerical method for the time evolution of Gaussian wave packets on flat-band lattices in the presence of correlated disorder. To achieve this, we introduce a method to generate random on-site energies with prescribed correlations. We verify this method with a one-dimensional (1D) cross-stitch model, and find good agreement with analytical results obtained from the disorderdressed evolution equations. This allows us to reproduce previous findings, that disorder can mobilize 1D flat-band states which would otherwise remain localized. As explained by the corresponding disorder-dressed evolution equations, such mobilization requires an asymmetric disorder-induced coupling to dispersive bands, a condition that is generically not fulfilled when the flat-band is resonant with the dispersive bands at a Dirac point-like crossing. We exemplify this with the 1D Lieb lattice. While analytical expressions are not available for the two-dimensional (2D) system due to its complexity, we extend the numerical method to the 2D α-T 3 model, and find that the initial flat-band wave packet preserves its localization when α = 0, regardless of disorder and intersections. However, when α ≠ 0, the wave packet shifts in real space. We interpret this as a Berry phase controlled, disorder-induced wave-packet mobilization. In addition, we present density functional theory calculations of candidate materials, specifically Hg1-xCdxTe. The flat-band emerges near the Γ point (k =0) in the Brillouin zone.

Spin-polarized transport properties in diluted-magnetic-semiconductor/semiconductor superlattices under light-field assisted

Based on the single electron effective mass approximation theory and the transfer-matrix method, the spin polarized transport properties of electrons in a diluted-magnetic-semiconductor/semiconductor superlattice are studied. The influence of a light-field and a magnetic-field on spin polarized transport and the tunneling time in the superlattice structure are discussed in more detail. The results show that, due to the sp-d electron interaction between conduction band electrons and doped Mn ions, giant Zeeman splitting occurs. It is shown that a significant spin-dependent transmission and the position and width of the resonant-transmission-band of spin-dependent electron can be manipulated by adjusting the magnetic- and light-field. Considering the light field irradiation, the resonance band of electron is deformed and broadened with the increase of the light field intensity. For the case of a strong magnetic field, the transmission coefficient (TC) in the low-energy region is almost zero when the light field is not added, but with the increase of light intensity, the TC increased significantly in the zone increases significantly, that is, a quasi-bound band appears. These features are due to the energy exchange between electrons and the light field when electrons tunnel through the superlattice structure under light irradiation. In addition, light and magnetic fields can significantly change the spin polarization of electrons. Under a certain magnetic field intensity (<i>B</i> = 2 T), the light field significantly changes the spin polarization of electrons, the main effect is that the width of the spin polarization platform narrows and oscillatory peaks are accompanied on both sides of the platform. This effect is strengthened with the increase of the light field intensity. However, when the magnetic field is stronger (<i>B</i> = 5 T), the opposite is true. These show that the spin polarization can be modulated by the light field. Finally, the tunneling time of spin-up and spin-down electrons is studied by the evolution of Gaussian wave packets in the structure. The results show that the tunneling time depends on a spin of electrons, and it can be seen that the tunneling time of the spin-down electron is shorter than that of the spin-up electron in the superlattice structure. These remarkable properties of spin polarized transport may be beneficial for the devising tunable spin filtering devices based on diluted magnetic semiconductor/semiconductor superlattice structure.

Evolution of Gaussian wave packets in capillary jets.

A temporal analysis of the evolution of Gaussian wave packets in cylindrical capillary jets is presented through both a linear two-mode formulation and a one-dimensional nonlinear numerical scheme. These analyses are normally applicable to arbitrary initial conditions but our study focuses on pure-impulsive ones. Linear and nonlinear findings give consistent results in the stages for which the linear theory is valid. The inverse Fourier transforms representing the formal linear solution for the jet shape is both numerically evaluated and approximated by closed formulas. After a transient, these formulas predict an almost Gaussian-shape deformation with (i) a progressive drift of the carrier wave number to that given by the maximum of the Rayleigh dispersion relation, (ii) a progressive increase of its bell width, and (iii) a quasiexponential growth of its amplitude. These parameters agree with those extracted from the fittings of Gaussian wave packets to the numerical simulations. Experimental results are also reported on near-Gaussian pulses perturbing the exit velocity of a 2-mm diameter water jet. The possibility of controlling the breakup location along the jet and other features, such as pinch-off simultaneity, are demonstrated.

Visualizing topological orders through Zitterbewegung in ultracold atoms

Topological Haldane model recently has been experimentally realized with ultracold atoms. The low energy Hamiltonian in the model can be formally described by the Dirac equation. In this paper, we explore the Zitterbewegung effect of the Dirac quasiparticles in the vicinity of the two nonequivalent Dirac points in the Haldane model. Through analytical and numerical calculation of the evolution of Gaussian wave packets, we find that the Zitterbewegung patterns could be used to characterize the topological phases of the Haldane model. Since the patterns can be detected through standard time-of-flight measurements, our proposal provides a promising scheme to directly visualize the topological phases in ultracold atoms.

Time-evolution of quantum systems via a complex nonlinear Riccati equation. II. Dissipative systems

In our former contribution (Cruz et al., 2015), we have shown the sensitivity to the choice of initial conditions in the evolution of Gaussian wave packets via the nonlinear Riccati equation. The formalism developed in the previous work is extended to effective approaches for the description of dissipative quantum systems. By means of simple examples we show the effects of the environment on the quantum uncertainties, correlation function, quantum energy contribution and tunnelling currents. We prove that the environmental parameter γ is strongly related with the sensitivity to the choice of initial conditions.

A split-step numerical method for the time-dependent Dirac equation in 3-D axisymmetric geometry

A numerical method is developed to solve the time-dependent Dirac equation in cylindrical coordinates for 3-D axisymmetric systems. The time evolution is treated by a splitting scheme in coordinate space using alternate direction iteration, while the wave function is discretized spatially on a uniform grid. The longitudinal coordinate evolution is performed exactly by the method of characteristics while the radial coordinates evolution uses Poisson's integral solution, which allows to implement the radial symmetry of the wave function. The latter is evaluated on a time staggered mesh by using Hermite polynomial interpolation and by performing the integration analytically. The cylindrical coordinate singularity problem at r=0 is circumvented by this method as the integral is well-defined at the origin. The resulting scheme is reminiscent of non-standard finite differences. In the last step of the splitting, the remaining equation has a solution in terms of a time-ordered exponential, which is approximated to a higher order than the time evolution scheme. We study the time evolution of Gaussian wave packets, and we evaluate the eigenstates of hydrogen-like systems by using a spectral method. We compare the numerical results to analytical solutions to validate the method. In addition, we present three-dimensional simulations of relativistic laser–matter interactions, using the Dirac equation.

Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling

The validation and parallel implementation of a numerical method for the solution of the time-dependent Dirac equation is presented. This numerical method is based on a split operator scheme where the space–time dependence is computed in coordinate space using the method of characteristics. Thus, most of the steps in the splitting are calculated exactly, making for a very efficient and unconditionally stable method. We show that it is free from spurious solutions related to the fermion-doubling problem and that it can be parallelized very efficiently. We consider a few simple physical systems such as the time evolution of Gaussian wave packets and the Klein paradox. The numerical results obtained are compared to analytical formulas for the validation of the method.

Markovian evolution of Gaussian states in the semiclassical limit

We derive an approximate Gaussian solution of the Lindblad equation in the semiclassical limit, given a general Hamiltonian and linear coupling with the environment. The theory is carried out in the chord representation and describes the evolved quantum characteristic function, which gives direct access to the Wigner function and the position representation of the density operator by Fourier transforms. The propagation is based on a system of non-linear equations taking place in a double phase space, which coincides with Heller's theory of unitary evolution of Gaussian wave packets when the Lindbladian part is zero. The example of a double well is worked out.

COMPARATIVE STUDY OF SEMICLASSICAL APPROACHES TO QUANTUM DYNAMICS

Quantum states can be described equivalently by density matrices, Wigner functions, or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance, and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.

Space-time evolution of Gaussian wave packets through superlattices containing left-handed layers

We study the space-time evolution of Gaussian electromagnetic wave packets moving through (L/R)n superlattices, containing alternating layers of left and right-handed materials. We show that the time spent by the wave packet moving through arbitrary (L/R)n superlattices are well described by the phase time. We show that in the particular case where the thicknesses dL,R and indices nl,r of the layers satisfy the condition dL|nL| = dRnR, the usual band structure becomes a sequence of isolated and equidistant peaks with negative phase times.

Semiclassical Propagation of Gaussian Wave Packets

We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.

Path integrals and wavepacket evolution for damped mechanical systems

Damped mechanical systems with various forms of damping are quantized using the path integral formalism. In particular, we obtain the path integral kernel for the linearly damped harmonic oscillator and a particle in a uniform gravitational field with linearly or quadratically damped motion. In each case, we study the evolution of Gaussian wave packets and discuss the characteristic features that help us distinguish between different types of damping. For quadratic damping we show the connection of the action and equation of motion to a zero-dimensional version of a scalar field theory. We also demonstrate that the equation of motion for quadratic damping can be identified as a geodesic equation in a fictitious two-dimensional space.

Velocity in Lorentz-violating fermion theories

We consider the role of the velocity in Lorentz-violating fermionic quantum theory, especially emphasizing the nonrelativistic regime. Information about the velocity will be important for the kinematical analysis of scattering and other problems. Working within the minimal standard model extension, we derive new expressions for the velocity. We find that generic momentum and spin eigenstates may not have well-defined velocities. We also demonstrate how several different techniques may be used to shed light on different aspects of the problem. A relativistic operator analysis allows us to study the behavior of the Lorentz-violating Zitterbewegung. Alternatively, by studying the time evolution of Gaussian wave packets, we find that there are Lorentz-violating modifications to the wave packet spreading and the spin structure of the wave function.

`Nonclassical' states in quantum optics: a `squeezed' review of the first 75 years

Seventy five years ago, three remarkable papers by Schrödinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called `nonclassical states' (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics.

The hamilton formalism of gaussian wave-packet dynamics

Gaussian wave packets have been used extensively to describe short-time localised processes in small molecular systems. We present a formalism to compute approximately the quantal time evolution of Gaussian wave packets for N-body systems. The wave-packet dynamics are obtained by invoking the time-dependent variational principle in Lagrangian form. It is then shown that the parametrisation of a Gaussian wave packet can be chosen so as to yield a Hamilton-like set of evolution equations for the wave-packet parameters. The resulting formalism proves to be energy conserving. For those parts of the system for which a classical description is appropriate, a localisation limit recovers the classical Hamilton equations. Also a mixed classical-quantal description of an N-body system is feasible.

Spreading and recurrence in anharmonic quantum-mechanical systems

The dynamical evolution of gaussian wave packets initiating on the soft wall and on the hard wall of an anharmonic potential of one degree of freedom is studied. In particular, it is found that the coherent-state representation (CSR) is a unique representation for monitoring the spreading, the breaking-up, and the regrouping (recurrence) of these wave packets.