Gaussian Wave Packet Research Articles

Interference with Non-Interacting Free Particles and a Special Type of Detector

This paper demonstrates how a classical detector that collects non-interacting individual classical massive free particles can generate a quantum interference pattern. The proposed classical picture requires that particles carry the information of a phase equal to an action integral along their trajectory. At the point of their detection, a special type of detector collects the phases from all individual particles reaching it, adds them up over time as complex numbers, and divides them by the square root of their number. The detector announces a number of detections equal to the square of the amplitude of the resulting complex number. An interference pattern is gradually built from the collection of particle phases in the detection bins of the detector after several repetitions of the experiment. We obtain perfect agreement with three solutions of the Schrödinger equation for free particles: a Gaussian wavepacket, two Gaussian wavepackets approaching each other, and a Gaussian wavepacket reflecting off a wall. The main conclusion of the present work is that the interference of quantum mechanics is basically due to the detectors that collect the particles when there are macroscopic detectors operating as proposed. Finally, a simple physical experiment with a single-photon detector is proposed that will be able to test our theory.

Wheeler-DeWitt states of the AdS-Schwarzschild interior

We solve the Wheeler-DeWitt equation for the planar AdS-Schwarzschild interior in a minisuperspace approximation involving the volume and spatial anisotropy of the interior. A Gaussian wavepacket is constructed that is peaked on the classical interior solution. Simple observables are computed using this wavepacket, demonstrating the freedom to a choose a relational notion of ‘clock’ in the interior and characterizing the approach to the spacelike singularity. The Wheeler-DeWitt equation may be extended out through the horizon, where it describes the holographic renormalization group flow of the black hole exterior. This amounts to the Hamilton-Jacobi evolution of the metric component gtt from positive interior values to negative exterior values. The interior Gaussian wavepacket is shown to evolve into the Lorentizan partition function of the boundary conformal field theory over a microcanonical energy window.

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.

Correction to: A novel spectral method for the semiclassical Schrödinger equation based on the Gaussian wave-packet transform

Correction to: A novel spectral method for the semiclassical Schrödinger equation based on the Gaussian wave-packet transform

Dynamic skin effects in non-Hermitian systems

We study the time evolution processes of non-Hermitian systems under the open boundary condition and confirm that the dynamical skin effect exists in non-Hermitian systems analytically, and unveil the mechanism of its formation, which is caused by both the non-Hermitian skin effect and the Hermitian wave packet spreading. Furthermore, we find that in contrast to the uniform speed motion in Hermitian situations, the Gaussian wave packet can be accelerated and amplified during its time evolution in non-Hermitian systems. This additional motion is found to be responsible for the dynamic skin effect.

Relativistic effects on the Schrödinger-Newton equation

The Schr\"odinger-Newton model describes self-gravitating quantum particles, and it is often cited to explain the gravitational collapse of the wave function and the localization of macroscopic objects. However, this model is completely nonrelativistic. Thus, in order to study whether the relativistic effects may spoil the properties of this system, we derive a modification of the Schr\"odinger-Newton equation by considering certain relativistic corrections up to the first post-Newtonian order. The construction of the model begins by considering the Hamiltonian of a relativistic particle propagating on a curved background. For simplicity, the background metric is assumed to be spherically symmetric and it is then expanded up to the first post-Newtonian order. After performing the canonical quantization of the system, and following the usual interpretation, the square of the module of the wave function defines a mass distribution, which in turn is the source of the Poisson equation for the gravitational potential. As in the nonrelativistic case, this construction couples the Poisson and the Schr\"odinger equations and leads to a complicated nonlinear system. Hence, the dynamics of an initial Gaussian wave packet is then numerically analyzed. We observe that the natural dispersion of the wave function is slower than in the nonrelativistic case. Furthermore, for those cases that reach a final localized stationary state, the peak of the wave function happens to be located at a smaller radius. Therefore, the relativistic corrections effectively contribute to increase the self-gravitation of the particle and strengthen the validity of this model as an explanation for the gravitational localization of the wave function.

Quantum cosmology of the flat universe via closed real-time path integral

Quantum cosmology is crucial to understand the evolution of the early universe. Despite significant progress, challenges still remain. For example, the role of time in quantum cosmology is unclear. Furthermore, the influence of the environment on the evolution of the quantum universe is challenging. In this work, we studied the evolution of the quantum universe non-perturbatively using the closed real-time path integral. The environments coupled to the quantum universe being considered are the radiation, the non-relativistic matter, or the dark matter. We evaluated the influence functional of the massless scalar field coupled with the flat FRW universe. We studied the evolution of the quantum universe by setting the initial state of spacetime as a Gaussian wave packet. In different scenarios, we show that the classical trajectory of the universe is consistent with the quantum evolution of the wave packet. The coherence, the absolute quantum fluctuation and the Gibbs entropy all monotonically increase with time, yet the relative quantum fluctuation decreases with time. We show that for a given size of the radiation dominated universe, the lower temperature corresponds to a more quantum universe. We find that the minimal coupling of the free massless scalar field with the flat FRW spacetime generally gives rise to the memory characterized via non-Markovian correlations. Finally, we show that under higher radiation temperatures, a small universe has a higher chance of a transition to a bigger universe.

Scaling dynamics of the ultracold Bose gas

The large-scale expansion dynamics of quantum gases is a central tool for ultracold gas experiments and poses a significant challenge for theory. In this work we provide an exact reformulation of the Gross-Pitaevskii equation for the ultracold Bose gas in a coordinate frame that adaptively scales with the system size during evolution, enabling simulations of long evolution times during expansion or similar large-scale manipulation. Our approach makes no hydrodynamic approximations, is not restricted to a scaling ansatz, harmonic potentials, or energy eigenstates, and can be generalized readily to non-contact interactions via the appropriate stress tensor of the quantum fluid. As applications, we simulate the expansion of the ideal gas, a cigar-shaped condensate in the Thomas-Fermi regime, and a linear superposition of counter propagating Gaussian wavepackets. We recover known scaling for the ideal gas and Thomas-Fermi regimes, and identify a linear regime of aspect-ratio preserving free expansion; analysis of the scaling dynamics equations shows that an exact, aspect-ratio invariant, free expansion does not exist for nonlinear evolution. Our treatment enables exploration of nonlinear effects in matter-wave dynamics over large scale-changing evolution.

Role of spatially compact nucleon wave packets in an ab initio description of H3 within high-momentum antisymmetrized molecular dynamics

We found the important role of the spatially compact nucleon wave packets to fully describe the nucleon correlations within the framework of the high-momentum antisymmetrized molecular dynamics (HM-AMD), which is a variational method based on the antisymmetrized molecular dynamics. In HM-AMD, short-range and tensor correlations between nucleons are described by nucleon pairs with high momentum (high-momentum pairs), which are given by putting large centroids of the Gaussian wave packets in opposite signs. In this paper, we further improve momentum distribution of the high-momentum pairs by varying the width parameter of the Gaussian wave packets of nucleons. We show the reliability of this new scheme by applying it to the $^{3}\mathrm{H}$ nucleus with the bare Argonne v8' potential. It is found that the spatially compact nucleon wave packets give the important effect to lower the total energy, which brings sufficient high-momentum components in the nucleus. We also include the spin-parallel configuration of the three nucleons induced by the tensor interaction, which is necessary to converge the HM-AMD results. Finally, comparable results to the other theoretical calculations are obtained for the total energy and Hamiltonian components.

Quasiresonant diffusion of wave packets in one-dimensional disordered mosaic lattices

We investigate numerically the time evolution of wave packets incident on one-dimensional semi-infinite lattices with mosaic modulated random on-site potentials, which are characterized by the integer-valued modulation period $\ensuremath{\kappa}$ and the disorder strength $W$. For Gaussian wave packets with the central energy ${E}_{0}$ and a small spectral width, we perform extensive numerical calculations of the disorder-averaged time-dependent reflectance, $\ensuremath{\langle}R(t)\ensuremath{\rangle}$, for various values of ${E}_{0}, \ensuremath{\kappa}$, and $W$. We find that the long-time behavior of $\ensuremath{\langle}R(t)\ensuremath{\rangle}$ obeys a power law of the form ${t}^{\ensuremath{-}\ensuremath{\gamma}}$ in all cases. In the presence of the mosaic modulation, $\ensuremath{\gamma}$ is equal to 2 for almost all values of ${E}_{0}$, implying the onset of the Anderson localization, while at a finite number of discrete values of ${E}_{0}$ dependent on $\ensuremath{\kappa}, \ensuremath{\gamma}$ approaches 3/2, implying the onset of the classical diffusion. This phenomenon is independent of the disorder strength and arises in a quasiresonant manner such that $\ensuremath{\gamma}$ varies rapidly from 3/2 to 2 in a narrow energy range as ${E}_{0}$ varies away from the quasiresonance values. We deduce a simple analytical formula for the quasiresonance energies and provide an explanation of the delocalization phenomenon based on the interplay between randomness and band structure and the node structure of the wave functions. We explore the nature of the states at the quasiresonance energies using a finite-size scaling analysis of the average participation ratio and find that the states are neither extended nor exponentially localized, but critical states.

The Kostin Equation, the Deceleration of a Quantum Particle and Coherent Control

Fifty years ago Kostin (J Chem Phys 57(9):3589–3591, 1972. https://doi.org/10.1063/1.1678812) proposed a description of damping in quantum mechanics based on a nonlinear Schrödinger equation with the potential being governed by the phase of the wave function. We show for the example of a moving Gaussian wave packet, that the deceleration predicted by this equation is the result of the same non-dissipative, homogeneous but time-dependent force, that also stops a classical particle. Moreover, we demonstrate that the Kostin equation is a special case of the linear Schrödinger equation with three potentials: (i) a linear potential corresponding to this stopping force, (ii) an appropriately time-dependent parabolic potential governed by a specific time dependence of the width of the Gaussian wave packet and (iii) a specific time-dependent off-set. The freedom of the width opens up the possibility of engineering the final state by the time dependence of the quadratic potential. In this way the Kostin equation is a precursor of the modern field of coherent control. Motivated by these insights, we analyze in position and in phase space the deceleration of a Gaussian wave packet due to potentials in the linear Schrödinger equation similar to those in the Kostin equation.

Excitation of surface plasmon polaritons by diffraction-free and vector beams.

Surface plasmon polaritons (SPPs) are traditionally excited by plane waves within the Rayleigh range of a focused transverse-magnetic (TM) Gaussian beam. Here we investigate and confirm the coupling between SPPs and two-dimensional Gaussian and Bessel-Gauss wave packets, as well as one-dimensional light sheets and space-time wave packets. We encode the incoming wavefronts with spatially varying states of polarization; then we couple the respective TM components of radial and azimuthal vector beam profiles to confirm polarization-correlation and spatial-mode selectivity. Our results do not require material optimization or multi-dimensional confinement via periodically corrugated metal surfaces to achieve coupling at a greater extent, hereby outlining a pivotal, yet commonly overlooked, path towards the development of long-range biosensors and all-optical integrated plasmonic circuits.

Asymptotic approximations for the phase space Schrödinger equation

We consider semi-classical time evolution for the phase space Schrödinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the anisotropic Gaussian approximation, related to the nearby orbit approximation, which is identified with the phase space representation of the Littlejohn semi-classical propagator. By means of the semi-classical propagator, we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the complex WKB theory developed by Maslov, on the basis of a canonical system in double phase space, analogous to the Berezin–Shubin–Marinov Hamilton–Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in .

Gaussian-wave-packet model for single-photon generation based on cavity quantum electrodynamics under adiabatic and nonadiabatic conditions

For single-photon generation based on cavity quantum electrodynamics, we investigate a practical model assuming a Gaussian wavepacket. This model makes it possible to comprehensively analyze the temporal dynamics of an atom-cavity system with both adiabatic and nonadiabatic conditions using analytical expressions. These results enable us to clarify the relationship between pulse width and maximum success probability, over the full range of coupling regimes, for arbitrary detunings. We demonstrate how to achieve a high success probability while keeping a short pulse width by optimizing the cavity transmittance parameter. Our formulations provide a practical tool for efficient single-photon generation in a wide variety of experimental platforms.

Demonstration of speckle resistance using space–time light sheets

The capacity of self-healing fields to reconstruct after passing through scattering media may prove useful in reducing speckle formation. Here, we study the speckle response of the space–time (ST) light sheet compared to a Gaussian wave packet, Airy beam, and Bessel Gauss beam. We find that the Pearson’s correlation coefficient for the ST light sheet is 50%, 48% and 40% larger than that of the Gaussian, Airy beam and Bessel Gauss beams, respectively, demonstrating a strong correlation to an input beam that has not been speckled. These results suggest that the ST light sheet exhibits considerable resistance to speckle generation. We also investigate the speckle response of the ST light sheet at its second-harmonic frequency and observe a mean Pearson’s correlation coefficient close to 0.6, comparable to the second-harmonic Bessel Gauss beam, and 2.8 × the value obtained for the second-harmonic Gaussian beam. Our results lend themselves to a variety of applications including bioimaging, communications, and optical tweezers.

Propagation of controllable spatiotemporal Airy–Ince–Gaussian beams propagating in strongly nonlocal nonlinear media

The controllable self-accelerating and self-decelerating Airy–Ince–Gaussian (CAiIG) wave packets in strongly nonlocal nonlinear media are investigated theoretically and numerically by solving the (3 + 1)D Schrödinger equation in cylindric coordinates. Typical examples of the obtained solutions are based on the ratio of the input power and the critical power as well as the initial velocity and the ellipticity. The CAiIG wave packets are constructed by the Airy pulses with initial velocity in temporal domain and the controllable Ince–Gaussian (IG) beams in spatial domain. Self-decelerating and self-accelerating CAiIG wave packets are obtained by selecting different initial velocities. The CAiIG wave packets keep approximately nondispersion properties in temporal dimension. However, the width of packets periodically oscillate or keep steady in spatial dimension according to the ratio of the input power and the critical power. The transverse field patterns of the CAiIG wave packets are manipulated by the IG wave packets, meanwhile the Airy pulses affect their transverse amplitudes. For the self-accelerating CAiIG wave packets, the transverse intensity decays during the propagation. However, to the self-decelerating CAiIG wave packet, the transverse intensity initially vibrates then decays. The direction of the energy flow of CAiIG wave packets in spatial domain periodically changes from the propagating center if the ratio of the input power and the critical power is not equal to 1, which causes the CAiIG beams periodically oscillating during propagation. Their Poynting vector snapshots at different propagating distances are shown.

Black hole perturbations in higher-order scalar–tensor theories: initial value problem and dynamical stability

We propose a physically sensible formulation of initial value problem for black hole perturbations in higher-order scalar–tensor theories. As a first application, we study monopole perturbations around stealth Schwarzschild solutions in a shift- and reflection-symmetric subclass of degenerate higher-order scalar–tensor (DHOST) theories. In particular, we investigate the time evolution of the monopole perturbations by solving a two-dimensional wave equation and analyze the Vishveshwara’s classical scattering experiment, i.e., the time evolution of a Gaussian wave packet. As a result, we confirm that stealth Schwarzschild solutions in the DHOST theory are dynamically stable against the monopole perturbations with the wavelength comparable or shorter than the size of the black hole horizon. We also find that the damped oscillations at the late time do not show up unlike the ringdown phase in the standard case of general relativity. Moreover, we investigate the characteristic curves of the monopole perturbations as well as a static spherically symmetric solution with monopole hair.

Electron wave spin in a quantum well

The particle-wave duality of the electron poses a principle question of whether the spin is a property of the particle or the wave. In this paper, the wave nature of the spin is studied for an electron inside a two-dimensional quantum well. By solving the exact 4 − spinor eigen solution to the Dirac equation, we show that a stable circulating total current density exists inside the well with a donut shaped topography. A spin value is modified by the confining geometry of the well. Our analysis also shows that a free electron Gaussian wavepacket is unstable and experiences quick decoherence.

Time-domain simulation of charged particle diffraction by an electrostatically biased grating: Transmission tunability and shaping of the quantum point contact for protons

A numerical simulation of a two-dimensional Gaussian wave packet of charged particles has been performed to investigate the diffraction phenomena from a single-, double-, and multi-slit grating biased with an electrostatic potential (Ve0). The wave packet dynamics are obtained by solving the time-dependent Schrödinger’s equation using the generalized finite difference time domain (GFDTD-Q) method for quantum systems. The effect of Ve0 on transmission properties, fringe pattern, motion of the peaks, and wave number distribution in the diffracted wave has been studied. It is found that Ve0 changes the shape of the quantum point contact of diffracting constriction, which controls the allowed quantum states in the diffracted wave and the transmission coefficient Tc can be tuned by Ve0. It is observed that the number of peaks, their relative intensity, and quantization of lateral wavenumber depend upon Ve0. This study will be helpful in optimizing the parameters for material grating-based matter–wave interferometers employing charged particle such as proton beams.

Bio-inspired computing with magnetic skyrmions using deep learning

Spintronic systems demonstrate significant value due to variety of features that can be used in bio-inspired computing applications. Skyrmion is a magnetic nanostructure which can be used as an information carrier. In this paper, we simulate vortex-free Gaussian wave packet by incorporating spin–orbit coupling (SoC) into three-component Bose–Einstein condensates (BECs). We next observe that skyrmions can be naturally generated from vortex-free Gaussian wave packet, further these structurally asymmetric devices are constructed and initial device classifications are carried out. A spin device consisting of vertically-stacked multiple skyrmions is proposed termed as skyrmion spintronic synapse (SkSS) and skyrmion spintronic Neuron (SkSN). This paper depicts training the neural network with surrogate gradients with deep learning strategies using topologically nontrivial spin textures. The current popular Graphics Processing Unit (GPU)-based neural network computations are compared with the proposed method. This research paves a way for a neuromorphic computing systems on exotic topological solitons.