Gaussian Wave Packet Research Articles

Super-ballistic diffusion induced by nonlinear interactions in a one-dimensional quasiperiodic lattice

We numerically investigate the spreading dynamics of wave packets in a one-dimensional lattice with quasiperiodic disorders and nonlinear interactions. Under moderate disorder strength, we reveal an intriguing super-ballistic diffusion driven by increasing the interaction strength from the ballistic diffusion in the non- and weakly interacting cases. We obtain a wide parameter region for such a faster-than-ballistic spreading of wave packets in the quasiperiodic lattice, although the nonlinear self-trapping is dominated under strong interactions. The super-ballistic diffusion is further analyzed based on the point source model. We also show the interaction-induced sub-diffusive spreading in the Anderson localized regime. The super-ballistic and sub-diffusive dynamics are exhibited in the spreading that starts from both the single-site and Gaussian wave packets. Our results showcase the interaction-enabled super-ballistic diffusion, which is scarce and experimentally realizable in artificial atomic or photonic lattices.

Relativistic single-electron wavepacket in quantum electromagnetic fields: quantum coherence, correlations, and the Unruh effect

Conventional formulation of QED since the 50s works very well for stationary states and for scattering problems, but with newly arisen challenges from the 80s on, where real time evolution of particles in a nonequilibrium setting are required, and quantum features such as coherence, dissipation, correlation and entanglement in a system interacting with its quantum field environment are sought after, new ways to formulate QED suitable for these purposes beckon. In this paper we present a linearized effective theory using a Gaussian wavepacket description of a charged relativistic particle coupled to quantum electromagnetic fields to study the interplay between single electrons and quantum fields in free space, at a scale well below the Schwinger limit. The proper values of the regulators in our effective theory are determined from the data of individual experiments, and will be time-dependent in the laboratory frame if the single electrons are accelerated. Using this new theoretical tool, we address the issues of decoherence of flying electrons in free space and the impact of Unruh effect on the electrons. Our result suggests that vacuum fluctuations may be a major source of blurring the interference pattern in electron microscopes. For a single electron accelerated in a uniform electric field, we identify the Unruh effect in the two-point correlators of the deviations from the electron’s classical trajectory. From our calculations we also bring out some subtleties, involving the bosonic versus fermionic spectral functions.

Propagation dynamics of self-accelerating second-order Hermite complex-variable-function Gaussian wave packets in a harmonic potential

This paper investigates the evolutionary dynamics of self-accelerating second-order Hermite complex-variable-function Gaussian (SSHCG) wave packets in a harmonic potential. The periodic variation of the wave packets is discussed via theoretical analysis and numerical simulation. The control variables method is applied to manipulate the distribution factor, cross-phase factor, potential depth, and chirp parameter, enabling the realization of unique propagation dynamics. In three-dimensional models, the SSHCG wave packets exhibit rotational states, featuring butterfly shape, three peaks shape, two polarity shape, elliptical shape, and ring-shaped double-vortex structures. Furthermore, the energy flow and the angular momentum of the wave packets are investigated. Additionally, the performance of the radiation force on a Rayleigh dielectric particle is studied. This investigation results in the emergence of distinct SSHCG wave packet propagation dynamics, and potential applications in optical communications and optical trapping are presented.

Quantum dynamics of excited state proton transfer in green fluorescent protein.

Photoexcitation of green fluorescent protein (GFP) triggers long-range proton transfer along a "wire" of neighboring protein residues, which, in turn, activates its characteristic green fluorescence. The GFP proton wire is one of the simplest, most well-characterized models of biological proton transfer but remains challenging to simulate due to the sensitivity of its energetics to the surrounding protein conformation and the possibility of non-classical behavior associated with the movement of lightweight protons. Using a direct dynamics variational multiconfigurational Gaussian wavepacket method to provide a fully quantum description of both electrons and nuclei, we explore the mechanism of excited state proton transfer in a high-dimensional model of the GFP chromophore cluster over the first two picoseconds following excitation. During our simulation, we observe the sequential starts of two of the three proton transfers along the wire, confirming the predictions of previous studies that the overall process starts from the end of the wire furthest from the fluorescent chromophore and proceeds in a concerted but asynchronous manner. Furthermore, by comparing the full quantum dynamics to a set of classical trajectories, we provide unambiguous evidence that tunneling plays a critical role in facilitating the leading proton transfer.

High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics.

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but they still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multidimensional, nonseparable coupled Morse potential.

Quantum analysis of a Mach–Zehnder interferometer with propagating a single photon Gaussian wave packet

In this paper, a Mach–Zehnder interferometer is analyzed with a single photon Gaussian wave packet at one of its input ports and vacuum state at the other input port. Various noises arising in the Mach–Zehnder interferometer such as photon flux noise and quadrature noise are mathematically analyzed. The effect of variation in device parameters on the power spectral density at the output ports of the Mach–Zehnder interferometer is studied. It is observed that the power spectral density of photon flux noise autocorrelation at the output ports of a Mach–Zehnder interferometer is a function of phase shift, whereas the power spectral density of quadrature noise autocorrelation at the output ports of a Mach–Zehnder interferometer is a function of frequency of a single photon wave packet and the phase shift. The analysis carried out in this paper will be helpful in understanding the quantum fluctuations in the design of quantum logic devices and circuits where the Mach–Zehnder interferometer is a core building block.

Energy densities in quantum mechanics

Quantum mechanics does not provide any ready recipe for defining energy density in space, since the energy and coordinate do not commute. To find a well-motivated energy density, we start from a possibly fundamental, relativistic description for a spin-12 particle: Dirac's equation. Employing its energy-momentum tensor and going to the non-relativistic limit we find a locally conserved non-relativistic energy density that is defined via the Terletsky-Margenau-Hill quasiprobability (which is hence selected among other options). It coincides with the weak value of energy, and also with the hydrodynamic energy in the Madelung representation of quantum dynamics, which includes the quantum potential. Moreover, we find a new form of spin-related energy that is finite in the non-relativistic limit, emerges from the rest energy, and is (separately) locally conserved, though it does not contribute to the global energy budget. This form of energy has a holographic character, i.e., its value for a given volume is expressed via the surface of this volume. Our results apply to situations where local energy representation is essential; e.g. we show that the energy transfer velocity for a large class of free wave-packets (including Gaussian and Airy wave-packets) is larger than its group (i.e. coordinate-transfer) velocity.

State dependence of tunneling processes and thermonuclear fusion

We discuss the sensitivity of tunneling processes to the initial preparation of the quantum state. We compare the case of Gaussian wave packets of different positional variances using a generalized Woods-Saxon potential for which analytical expressions of the tunneling coefficients are available. Using realistic parameters for barrier potentials we find that the usual plane wave approximation underestimates fusion reactivities by an order of magnitude in a range of temperatures of practical relevance for controlled energy production.

Superquantum effects on physical systems from a hydrodynamic perspective

Employing a nonlinear time-dependent Schrödinger equation that describes the transition between quantum and classical regimes, we explore superquantum effects on physical systems from a hydrodynamic point of view. The degree of quantumness, a pivotal parameter in this transition equation, gauges the impact of quantum effects arising from the quantum potential. As this parameter varies continuously between zero and one, it facilitates the gradual transformation of systems from purely classical to entirely quantum regimes. We extend our analysis by setting the degree of quantumness greater than one, thereby investigating superquantum phenomena. Notably, this transition equation can be reframed as a scaled time-dependent Schrödinger equation with a rescaled Planck constant. Through computational simulations derived from the scaled Schrödinger equation, we examine superquantum implications for two systems, including Gaussian wave packet scattering from an Eckart barrier and metastable state decay. This research suggests that the hydrodynamic formulation of the quantum–classical transition equation offers a profound understanding of superquantum influences on physical systems.

Bloch oscillations probed quantum phases in HgTe quantum wells

The semiconductor quantum well based on mercury telluride is characterized by two distinct phases: conventional insulating phase and topological insulating phase with helical edge states. The system undergoes a topological quantum phase transition from one phase to the other, tuned by the critical geometric parameters of the quantum well. It is shown that the quantum states in each phase exhibit distinct flavors of Bloch oscillations, depending strongly on the geometric parameters and crystal momentum of the system. In particular, the group and Berry velocities and the real-space trajectories exhibit pronounced Bloch oscillations. Interestingly, the x- and y-components of the group velocity are interchanged by interchanging their corresponding components of the crystal momentum. In addition, a Gaussian wave packet undergoes distinct time evolution in each quantum phase of the HgTe quantum well. Moreover, the effects of applied in-plane electric and transverse magnetic fields are determined within the framework of Newtonian mechanics, leading to the geometric visualization of such an oscillatory motion. We find that in the presence of both applied in-plane electric and transverse magnetic fields simultaneously, the system undergoes a dynamic phase transition between confined and de-confined states, tuned by the relative strength of the fields. It is argued that the distinct Bloch oscillations originate from the peculiar band structure of HgTe quantum wells in each quantum phase. Furthermore, we find that the direct-current drift velocity in each quantum phase exhibits negative differential conductivity, a hallmark of the Bloch oscillation regime.

Potential scatterings in the [formula omitted] space : (2) Rigorous scattering probability of wave packets

In this study, potential scatterings are formulated in experimental setups with Gaussian wave packets in accordance with a probability principle and associativity of products. A breaking of an associativity is observed in scalar products with stationary scattering states in a majority of short-range potentials. Due to the breaking, states of different energies are not orthogonal and their superposition is not suitable for representing a normalized isolate state. Free wave packets in perturbative expansions in coupling strengths keep the associativity, and give a rigorous amplitude that preserves manifest unitarity and other principles of the quantum mechanics. An absolute probability is finite and comprises cross sections and new terms of unique properties. The results also demonstrate an interference term displaying unique behavior at an extreme forward direction.

Wheeler DeWitt states of a charged AdS4 black hole

We solve the Wheeler DeWitt equation for the planar Reissner-Nordström-AdS black hole in a minisuperspace approximation. We construct semiclassical Wheeler DeWitt states from Gaussian wavepackets that are peaked on classical black hole interior solutions. By using the metric component gxx as a clock, these states are evolved through both the exterior and interior horizons. Close to the singularity, we show that quantum fluctuations in the wavepacket become important, and therefore the classicality of the minisuperspace approximation breaks down. Towards the AdS boundary, the Wheeler DeWitt states are used to recover the Lorentzian partition function of the dual theory living on this boundary. This partition function is specified by an energy and a charge. Finally, we show that the Wheeler DeWitt states know about the black hole thermodynamics, recovering the grand canonical thermodynamic potential after an appropriate averaging at the black hole horizon.

The exotic behavior of the wave evolution in Lévy crystals within a fractional medium

We investigate a traveling Gaussian wave packet transport through a rectangular quantum barrier of lévy crystals in fractional quantum mechanics formalism. We study both standard and fractional Schrödinger equations in linear and nonlinear regimes by using a split-step finite difference (SSFD) method. We evaluate the reflection, trapping, and transmission coefficients of the wave packet and the wave packet spreading by using time-dependent inverse participation ratio (IPR) and second moment. By simultaneously adjusting the fractional and nonlinear terms, we create sharp pulses, which is an essential issue in optoelectronic devices. We illustrate that the effects of barrier height and width on the transmission coefficient are strangely different for the standard and fractional Schrödinger equations. We observe fortunately soliton-like localized wave packets in the fractional regime. Thus, we can effectively control the behavior of the wave evolution by adjusting the available parameters, which can excite new ideas in optics.

Quantum mechanical Gaussian wavepackets of single relativistic particles

We study the evolutions of selected quasi-(1+1) dimensional wavepacket solutions to the Klein–Gordon equation for a relativistic charged particle in uniform motion or accelerated by a uniform electric field in Minkowski space. We explore how good the charge density of a Klein–Gordon wavepacket can be approximately described by a Gaussian state with the single-particle interpretation. We find that the minimal initial width of a wavepacket for a good Gaussian approximation in position space is about the Compton wavelength of the particle divided by its Lorentz factor at the initial moment. This can be considered as a manifestation of relativistic length contraction, which is also observed numerically in the spreading of the wavepacket’s charge density.

Quantum uncertainty as an intrinsic clock

In quantum mechanics, a classical particle is raised to a wave-function, thereby acquiring many more degrees of freedom. For instance, in the semi-classical regime, while the position and momentum expectation values follow the classical trajectory, the uncertainty of a wave-packet can evolve and beat independently. We use this insight to revisit the dynamics of a 1d particle in a time-dependent harmonic well. One can solve it by considering time reparameterizations and the Virasoro group action to map the system to the harmonic oscillator with constant frequency. We prove that identifying such a simplifying time variable is naturally solved by quantizing the system and looking at the evolution of the width of a Gaussian wave-packet. We further show that the Ermakov-Lewis invariant for the classical evolution in a time-dependent harmonic potential is actually the quantum uncertainty of a Gaussian wave-packet. This naturally extends the classical Ermakov-Lewis invariant to a constant of motion for quantum systems following Schrödinger equation. We conclude with a discussion of potential applications to quantum gravity and quantum cosmology.

Neutrino decoherence and violation of the strong equivalence principle

We analyze the dynamics of neutrino Gaussian wave-packets, the damping of flavor oscillations and decoherence effects within the framework of extended theories of gravity. We show that, when the underlying description of the gravitational interaction admits a violation of the strong equivalence principle, the parameter quantifying such a violation modulates the wave-packet spreading, giving rise to potentially measurable effects in future neutrino experiments.

Schrödinger dynamics in length-scale hierarchy: from spatial rescaling to Huygens-like proliferation of Gaussian wavepackets

Studying possible laws, rules, and mechanisms of time-evolution of quantum wavefunctions leads to deeper understanding about the essential nature of the Schrödinger dynamics and interpretation on what the quantum wavefunctions are. As such, we attempt to clarify the mechanical and geometrical processes of deformation and bifurcation of a Gaussian wavepacket of the Maslov type from the viewpoint of length-scale hierarchy in the wavepacket size relative to the range of relevant potential functions. Following the well-known semiclassical view that (1) Newtonian mechanics gives a phase space geometry, which is to be projected onto configuration space to determine the basic amplitude of a wavefunction (the primitive semiclassical mechanics), our study proceeds as follows. (2) The quantum diffusion arising from the quantum kinematics makes the Gaussian exponent complex-valued, which consequently broadens the Gaussian amplitude and brings about a specific quantum phase. (3) The wavepacket is naturally led to bifurcation (branching), when the packet size gets comparable with or larger than the potential range. (4) Coupling between the bifurcation and quantum diffusion induces the Huygens-principle like wave dynamics. (5) All these four processes are collectively put into a path integral form. We discuss some theoretical consequences from the above analyses, such as (i) a contrast between the δ-function-like divergence of a wavefunctions at focal points and the mesoscopic finite-speed shrink of a Gaussian packet without instantaneous collapse, (ii) the mechanism of release of the zero-point energy to external dynamics and that of tunneling, (iii) relation between the resultant stochastic quantum paths and wave dynamics, and so on.

Quantizing the quantum uncertainty

The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the sl2 Lie algebra formed by the operators xˆ2, pˆ2 and xp̂. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?

Decoherence in neutrino oscillation between 3D Gaussian wave packets

There is renewed attention to whether we can observe the decoherence effect in neutrino oscillation due to the separation of wave packets with different masses in near-future experiments. As a contribution to this endeavor, we extend the existing formulation based on a single 1D Gaussian wave function to an amplitude between two distinct 3D Gaussian wave packets, corresponding to the neutrinos being produced and detected, with different central momenta and spacetime positions and with different widths. We find that the spatial widths-squared for the production and detection appear additively in the (de)coherence length and in the localization factor for governing the propagation of the wave packet, whereas they appear as the reduced one (inverse of the sum of inverse) in the momentum conservation factor. The overall probability is governed by the ratio of the reduced to the sum.

Burr Puzzle-Inspired Seismic Metamaterials

Two novel seismic metamaterials models, trophy and apple core, inspired by an ancient Chinese puzzle are proposed in this study. Bandgaps below 15[Formula: see text]Hz are achieved by exclusively using concrete as the base material in the designs. Vibration modes are investigated to clarify the bandgap formation. Through comparison, we found that under the same volume, the apple core model can provide a lower frequency bandgap than the trophy model. Results from transient simulations and band structures are in good agreement, validating the accuracy of the simulation results. To broaden the bandwidth, the radii of the apple core models are varied and arranged in a gradient pattern. Transient analyses using a Gaussian wave packet and Taiwan’s Chi-Chi earthquake dataset are conducted to verify the effectiveness of the gradient apple core seismic metamaterial.