Gaussian Wave Packet Research Articles

Mimicking an expanding universe by optical interference in a helicoid waveguide.

According to modern cosmology, expansion of the universe is due to the metric changing of spacetime itself. Here, we propose to mimic an expanding universe by utilizing optical interference and helicoid waveguides. The evolution of interference pattern in the helicoid waveguide is investigated theoretically and experimentally. For precise measurements, we design an air helicoid waveguide which allows us to investigate the wave front of laser beams from the waveguide. Redshift of a Gaussian wave packet in the expanding universe is demonstrated with high precision, showing that the helicoid waveguide acts as a parabolic gradient index lens exactly. The proposed waveguide structure can be used as an efficient waveguide adapter.

Tunneling, reflection and gravitational weak equivalence principle in the continuous transition from quantum to classical mechanics

In an effort to describe quantum and classical mechanics with the same language, a wave equation for a continuous transition from quantum to classical mechanics has been proposed. Furthermore, the equivalence of this nonlinear equation with a linear one, known as the scaled equation, which is just the Schrodinger equation with the scaled Planck constant instead of the usual one, has been proved. Using this equation, we'll study three interesting phenomena; these include tunneling through a rectangular barrier, total reflection from a hard wall, and the gravitational weak equivalence principle in quantum, transition and classical regions. Time-independent scaled equation for the stationary states is derived and solved for a flux of particles incident on the barrier. The relations show that tunneling probability is exactly zero in the classical regime. For the other problems, we use a Gaussian wavepacket to calculate the expectation value of the position operator in reflection from the hard wall and to estimate the detection probability and arrival time in the problem of the gravitational weak equivalence.

Dark energy effects in the Schrödinger-Newton approach

The Schr\"odinger-Newton equation is a proposed model to explain the localization of macroscopic particles by suppressing quantum dispersion with the particle's own gravitational attraction. On cosmic scales, however, dark energy also acts repulsively, as witnessed by the accelerating rate of universal expansion. Here, we introduce the effects of dark energy in the form of a cosmological constant $\Lambda$, that drives the late-time acceleration of the Universe, into the Schr\"odinger-Newton approach. We then ask in which regime dark energy dominates both canonical quantum diffusion and gravitational self-attraction. It turns out that this happens for sufficiently delocalized objects with an arbitrary mass and that there exists a minimal delocalization width of about $67$ m. While extremely macroscopic from a quantum perspective, the value is in principle accessible to laboratories on Earth. Hence, we analyze, numerically, how the dynamics of an initially spherical Gaussian wave packet is modified in the presence of $\Lambda > 0$. A notable feature is the gravitational collapse of part of the wave packet, in the core region close to the center of mass, accompanied by the accelerated expansion of the more distant shell surrounding it. The order of magnitude of the distance separating collapse from expansion matches analytical estimates of the classical turnaround radius for a spherically symmetric body in the presence of dark energy. However, the time required to observe these modifications is astronomical. They can potentially be measured only in physical systems simulating a high effective cosmological constant, or, possibly, via their effects on the inflationary universe.

Generalized coherent-squeezed-state expansion for the super-radiant phase transition

We develop a systematic variational coherent-squeezed-state expansion for the ground state of the quantum Rabi model, which includes an additional squeezing effect with comparisons to previous coherent-state approach. For finite large ratio between the atomic and field frequency, the essential feature of the ground-state wave function in the super-radiant phase appears, which has a structure of two delocalized wake packets. The single-peaked wave function with one coherent-squeezed state works well even around the critical regime, exhibiting the advantage over the coherent-state method. As the coupling increases to form strong correlations physics in the vicinity of phase transition, we develop an improved wave function with a structure of two Gaussian wave packets, which is a linear superposition of two coherent-squeezed state. The ground-state energy and the average photon number agree well with numerical ones even in the strong-correlated regimes, exhibiting a substantial improvement over the coherent-state expansion. The advantage of the coherent-squeezed-state expansion lies in the inclusion of the second coherent-squeezed state and the additional squeezed deformation of the wave function, providing a useful tool for multi-modes spin-boson coupling systems of greater complexity.

Pseudo-Invariant Approach for a Particle in a Complex Time-Dependent Linear Potential

The Lewis and Riesenfeld method has been investigated, by Ramos et al in Ref.[1], for quantum systems governed by time-dependent PT symmetric Hamiltonians and particularly where the quantum system is a particle submitted to action of a complex time-dependent linear potential. We discuss the method they used and propose an alternative one which leads to physically acceptable uncertainty product and to complex x and p expectation values but describe the classical motion. We used, for this situation, a linear pseudo hermitian invariant operator which allow us to solve analytically the time-dependent Schr\"odinger equation for this problem and to construct a Gaussian wave packet solution. The normalization condition for the invariant eigenfunctions with the Dirac delta function is correctly obtained, contrary to what is stated in Ref.[1].

Dissipative quantum backflow

Dissipative backflow is studied in the context of open quantum systems. This theoretical analysis is carried out within two frameworks, the effective time-dependent Hamiltonian due to Caldirola-Kanai (CK) and the Caldeira-Leggett (CL) one where a master equation is used to describe the reduced density matrix in presence of dissipation and temperature of the environment. Two examples are considered, the free evolution of one and two Gaussian wave packets as well as the time evolution under a constant field. Backflow is shown to be reduced with dissipation and temperature but never suppressed. Interestingly enough, quantum backflow is observed when considering both one and two Gaussian wave packets within the CL context. Surprisingly, in both cases, the backflow effect seems to be persistent at long times. Furthermore, the constant force $ m g\geq 0 $ behaves against backflow. However, the classical limit of this quantum effect within the context of the classical Schr\"odinger equation is shown to be present. Backflow is also analyzed as an eigenvalue problem in the Caldirola-Kanai framework. In the free propagation case, eigenvalues are independent on mass, Planck constant, friction and its duration but, in the constant force case, eigenvalues depend on a factor which itself is a combination of all of them as well as the force constant.

Single-electron quantum dynamics in high-harmonic generation spectrum from LiH molecule: Analysis of potential energy surfaces for electrons constructed from a model of localized Gaussian wave packets with valence-bond spin-coupling.

A high-harmonic generation (HHG) spectrum from a LiH molecule induced by an intense laser pulse is computed and analyzed with potential energy surfaces for electron motion (ePES) constructed from a model of localized electron wave packets with valence-bond spin-coupling. The molecule has two valence ePES with binding energies of 0.39 hartree and 1.1 hartree. The HHG spectrum from an electron dynamics on the weaker bound valence ePES, virtually assigned to Li 2s, exhibits a dominant peak at the first harmonic without plateau and cutoff. This compares with the free electron spectrum under an oscillating laser field and is comprehensive with the shape and depth of the ePES. The other valence ePES, assigned to H 1s, is deeper bound such that the overall profile of the wave function is well approximated by a Gaussian of the width comparable to the Li-H bond length. However, a small fraction, less than 10-3, of the probability density amplitude tunnels out from the bound potential with high wave number and spreads over tens of nm with parts recombining to the molecule due to the laser field oscillation. This minor portion of the electronic wave function is the major origin of the HHG extending up to 50 harmonic orders. Nonlinear dynamics within the potential well induced by the laser field oscillation also contributes to the HHG up to 30 harmonic orders.

Hamiltonian dynamics of semiclassical Gaussian wave packets in electromagnetic potentials

We extend our previous work on symplectic semiclassical Gaussian wave packet dynamics to incorporate electromagnetic interactions by including a vector potential. The main advantage of our formulation is that the equations of motion derived are naturally Hamiltonian. We obtain an asymptotic expansion of our equations in terms of and show that our equations have corrections to those presented by Zhou, whereas ours also recover the equations of Zhou in the case of a linear vector potential and quadratic scalar potential. One and two dimensional examples of a particle in a magnetic field are given and numerical solutions are presented and compared with the classical solutions and the expectation values of the corresponding observables as calculated by the Egorov or initial value representation (IVR) method. We numerically demonstrate that the correction terms improve the accuracy of the classical or Zhou’s equations for short times in the sense that our solutions converge to the expectation values calculated using the Egorov/IVR method faster than the classical solutions or those of Zhou as .

Quantum traversal time across a potential well

We consider the quantum traversal time of an incident wave packet across a potential well using the theory of quantum time of arrival (TOA)-operators. This is done by constructing the corresponding TOA-operator across a potential well via quantization. The expectation value of the potential well TOA-operator is compared to the free particle case for the same incident wave packet. The comparison yields a closed-form expression of the quantum well traversal time which explicitly shows the classical contributions of the positive and negative momentum components of the incident wave packet and a purely quantum mechanical contribution significantly dependent on the well depth. An incident Gaussian wave packet is then used as an example. It is shown that for shallow potential wells, the quantum well traversal time approaches the classical traversal time across the well region when the incident wave packet is spatially broad and approaches the expected quantum free particle traversal time when the wave packet is localized. For deep potential wells, the quantum traversal time oscillates from positive to negative implying that the wave packet can be advanced or delayed.

Nondispersive analytical solutions to the Dirac equation

This paper presents new analytic solutions to the Dirac equation employing a recently introduced method that is based on the formulation of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. A first family of solutions describe the shape-preserving translation of a wavepacket along any desired trajectory in the x-y plane. In particular, we show that the dispersionless motion of a Gaussian wavepacket along both elliptical and circular paths can be achieved with rather simple electromagnetic field configurations. A second family of solutions involves a plane electromagnetic wave and a combination of generally inhomogeneous electric and magnetic fields. The novel analytical solutions of the Dirac equation given here provide important insights into the connection between the quantum relativistic dynamics of electrons and the underlying geometry of the Lorentz group.

Three-dimensional chirped Airy Complex-variable-function Gaussian vortex wave packets in a strongly nonlocal nonlinear medium.

Three-dimensional chirped Airy Complex-variable-function Gaussian vortex (CACGV) wave packets in a strongly nonlocal nonlinear medium (SNNM) are studied. By varying the distribution parameter, CACGV wave packets can rotate stably in a SNNM in different forms, including dipoles, elliptic vortices, and doughnuts. Numerical simulation results for the CACGV wave packets agree well with theoretical analysis results under zero perturbation. The Poynting vector related to the physics of the rotation phenomenon and the angular momentum as a torque corresponding to the force are also presented. Finally, the radiation forces of CACGV wave packets acting on a nanoparticle in a SNNM are discussed.

Controllable quantum interface between itinerant optical photon and microwave photonic memory

We propose a scheme for controlling the quantum interface between the travelling optical wave packet and microwave photonic memory. The effective coupling between the optical and microwave modes is mediated by the rare earth doped crystal as first proposed by Williamson et al. [L.A. Williamson, Y.H. Chen, J.J. Longdell, Phys. Rev.Lett. 113, 203601 (2014)], and can be modulated by a time-dependent coherent driving field. With the effective model, we obtain the fidelity of the write process for arbitrary input wave packet as well as driving field, and investigate its optimization in two cases. In the first case of constant coupling, the optimal input wave packet is analytically derived to maximize the fidelity, while in the latter case of tunable coupling, the sequence of coherent driving field is numerically optimized for arbitrary given input wave packet. The write process for the Gaussian input wave packet can be explicitly characterized in the tunable parameter regime of real experiments, revealing that the fidelity can reach 0.974η (η being the coupling efficiency of the optical cavity). Based on these advantages, the proposal will have potential applications for the investigation of quantum networks.

Automated Analysis of DP-systems Using Timed-Arc Petri Nets via TAPAAL Tool

Timed-Arcs Petri nets (TaPN-nets) are a time extension of Petri nets that allows assigning clocks to tokens. System of dynamic points on a metric graph (DP-systems) is another dynamical model that is studied in discrete geometry dynamics and motivated by study of localized Gaussian wave packets scattering on thin structures; as well, DP-systems could be utilized to overapproximate the dynamics of messages scattering in distributed systems. In the latter case, time-temporal properties of DP-systems become a matter of interest. However, there are no tools that enable us to analyse them. In this work, we provide a new approach to automated analysis of DP-systems using the translation of a DP-system into a TaPN-net which is implemented as a TAPAAL plugin. The translation let us use the comprehensive tool support for TaPN-nets (TAPAAL/UPPAAL) to analyze DP-systems dynamical characteristics expressed in TCTL language. We demonstrated how to express some of them and verify time-temporal properties of a DP-system using the suggested approach, and performed experiments to obtain empirical estimates of the tool performance.

Neutrino decoherence in presence of strong gravitational fields

We explore the impact of strong gravitational fields on neutrino decoherence. To this aim, we employ the density matrix formalism to describe the propagation of neutrino wave packets in curved spacetime. By considering Gaussian wave packets, we determine the coherence proper time, neglecting the effect of matter outside the compact object. We show that strong gravitational fields nearby compact objects significantly influence neutrino coherence.

Origin of continuous curves and dotted spots in laser transverse modes with geometric structures.

The transverse structures of geometric modes in degenerate laser resonators with large astigmatism are systematically investigated by starting from the Gaussian wave packet formulation with ray-wave connections. The overlapping degree D between Gaussian wave packets is derived as an analytic expression that manifests the critical overlapping with D=1. The transverse patterns are confirmed to display a structure of continuous curves or dotted spots, depending on D>1 or D<1, respectively. A thorough comparison between experimental results and numerical calculations is performed to validate the theoretical analysis.

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.

Quantics: A general purpose package for Quantum molecular dynamics simulations

Quantics is a general purpose program package to simulate the time-evolution of a molecular system by solving the time-dependent Schrödinger equation. The main code is based on the multi-configurational time-dependent Hartree (MCTDH) algorithm in various variants, including the powerful multilayer-MCTDH algorithm that has been used to propagate a wavefunction for up to 1000 degrees of freedom. MCTDH uses a contraction of traditional discrete basis set representations of the Hamiltonian and wavefunction, and Quantics includes a range of variable representation (DVR) grid basis sets and collocation methods. Input is via ascii text files and for molecules with analytical potential functions no programming is required. A library of potential functions is included to treat more complicated cases, and more functions can be added as required by the user. The code also includes the variational multi-configurational Gaussian (vMCG) method that is based on a Gaussian wavepacket expansion of the wavefunction. vMCG can be run in a “direct” manner (DD-vMCG), calculating the potential energy surfaces on-the-fly using a number of quantum chemistry programs. In addition to wavepacket propagation, Quantics can solve the time-independent Schrödinger equation for small systems and can solve the Liouville–von-Neumann equation to propagate density matrices. The Package includes auxiliary programs to help set up calculations and analyse the output. Quantics is a community code of the UK Collaborative Computational Project for Quantum Dynamics (CCPQ) and the European E-CAM project, an e-infrastructure for software development run by the Centre Européen de Calcul Atomique et Moléculaire (CECAM). Through this it has become a framework for general dynamics codes, for example enabling an external surface hopping code to use the Quantics input and operator interfaces. Program summaryProgram Title: QuanticsProgram Files doi:http://dx.doi.org/10.17632/x9dcpc2r5c.1Licensing provisions: LGPLv3Programming language: Fortran90. Some Fortran77, Fortran2003, C and python.Nature of problem: Solving the time-dependent Schrödinger equation for a set of nuclei allows a range of physical processes to be studied including all quantum effects. This allows an experimental signal to be given a molecular interpretation. Typical applications are scattering cross-sections or time-resolved spectra, but also rate constants and other transport properties are possible. The exact problem to be solved is defined by the Hamiltonian, which must be provided by the user, and the initial wavepacket, again defined by the user. The final analysis of the evolving wavepacket then provides the experimental signal or molecular property.Solution method: A range of methods are possible for solving the time-evolution of a wavepacket (see main text). These can be broadly described as basis-set methods, in which the wavepacket and Hamiltonian are expanded in a set of functions. Various functions are possible, including grid-based sets (DVRs and collocation), and Gaussian wavepackets. The wavepacket can then be propagated using a variety of algorithms depending on the representation chosen. These include the full numerically-exact solution, various versions of the multi-configurational time-dependent Hartree method and approximate methods such as trajectory surface hopping. Full details are given in the documentation provided with the package and in a book and a number of review articles [1,2,3].Additional comments including restrictions and unusual features: The code has been tested on a number of linux distributions and compilers. It works best with a bash environment and a gnu gcc / gfortran compiler greater than version 4.8. The code is parallelised in parts using either OpenMP or MPI. There is a suite of test calculations to test an installation.

Basis function method for numerical loop quantum cosmology: the Schwarzschild black hole interior

Loop quantum cosmology is a symmetry-reduced application of loop quantum gravity that has led to the resolution of classical singularities such as the big bang, and those at the center of black holes. This can be seen through numerical simulations involving the quantum Hamiltonian constraint that is a partial difference equation. The equation allows one to study the evolution of sharply-peaked Gaussian wave packets that generically exhibit a quantum ‘bounce’ or a non-singular passage through the classical singularity, thus offering complete singularity resolution. In addition, von-Neumann stability analysis of the difference equation—treated as a stencil for a numerical solution that steps through the triad variables—yields useful constraints on the model and the allowed space of states. In this paper, we develop a new method for the numerical solution of loop quantum cosmology models using a set of basis functions that offer a number of advantages over computing a solution by stepping through the triad variables. We use the Corichi and Singh model for the Schwarzschild interior as the main case study in this effort. The main advantage of this new method is computational efficiency and the ease of parallelization. In addition, we also discuss how the stability analysis appears in the context of this new approach.

Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having 𝒫𝒯 symmetry

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.

Gaussian wave packet transform based numerical scheme for the semi-classical Schrödinger equation with random inputs

In this work, we study the semi-classical limit of the Schrödinger equation with random inputs, and show that the semi-classical Schrödinger equation produces O(ε) oscillations in the random variable space. With the Gaussian wave packet transform, the original Schrödinger equation is mapped to an ordinary differential equation (ODE) system for the wave packet parameters coupled with a partial differential equation (PDE) for the quantity w in rescaled variables. Further, we show that the w equation does not produce ε dependent oscillations, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, i.e. simulating the w equation, it is sufficient to use ε independent samples. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm, and hopefully shed light on possible future directions.