Wave packet broadening in usual quantum mechanics is a consequence of dispersion behavior of the medium which the wave propagates in it. In this paper, we consider the problem of wave packet broadening in the framework of Generalized Uncertainty Principle(GUP) of quantum gravity. New dispersion relations are derived in the context of GUP and it has been shown that there exists a gravitational induced dispersion which leads to more broadening of the wave packets. As a result of these dispersion relations, a generalized Klein-Gordon equation is obtained and its interpretation is given.
We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classically and quantum mechanically. Thus, analogous to Ehrenfest's theorem, nonspreading wave packets offer another view point directly connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such …
The dynamics of quantum systems can be approximated by the time propagation of Gaussian wave packets. Applying a time dependent variational principle, the time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations. Unfortunately, the set of equations is ill-behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. We present a general method for regularization based on applying adequate nonholonomic inequality constraints to the evolut…
Unbound wave packets propagating to macroscopic space and time coordinates become proportional to their (Fourier transform) momentum distribution at earlier times whereby the asymptotic coordinates and the initial momenta are connected everywhere by appropriate classical trajectories. This asymptotic imaging theorem is relevant to every quantum reaction involving macroscopic extraction and detection of fragments emerging from a microscopic volume. It justifies the usual assumption of classical particle motion used in the design of particle detectors. We illustrate this quantum to semiclassical…
We find a set of new exact solutions of a quantum harmonic oscillator, which describes some wave-packet trains with average energy being proportional to both the quantum level and classical energy of the oscillator. Center of the wave-packet trains may oscillate like a classical harmonic oscillator of frequency $ω$. Width and highness of the trains may change simultaneously with frequency $2 ω$ as an array of breathers. Under some perturbations the wave-packet trains could transit between the states of different quantum numbers. We demonstrate analytically and numerically that the wave-packet …
The conditions for optimal reflection-free complex-absorbing potentials (CAPs) are discussed. It is shown that the CAPs as derived from the smooth-exterior-scaling transformation of the Hamiltonian,[J. Phys. B. 31, 1431 (1998)], serve as optimal reflection-free CAPs (RF-CAPs) in wave-packet propagation calculations of open systems. The initial wave packet, can be located in the interaction region (as in half collision experiments) where the CAPs have vanished or in the asymptote where the absorbing potential non zero. As we show the optimal CAPs can be introduced also in the region where the p…
A general solution to the "shutter" problem is presented. The propagation of an arbitrary initially bounded wavefunction is investigated, and the general solution for any such function is formulated. It is shown that the exact solution can be written as an expression that depends only on the values of the function (and its derivatives) at the boundaries. In particular, it is shown that at short times ($t\ll2mx^2/\hbar$, where $x$ is the distance to the boundaries) the wavefunction propagation depends only on the wavefunction's values (or its derivatives) at the boundaries of the region. Finall…
The concept of Lorenz invariant mass and mean propagation velocity have been investigated in detail for various multiphoton wave-packet states of light. Based on photodetection theory and straightforward kinematics, we presented a physically reasonable and at the same time rigorous proof that mean propagation velocity is consistent with the Lorentz-invariant mass concept for an arbitrary multiphoton wave-packet state. We argued that mean propagation velocity is less than the speed of light constant in vacuum and is governed by geometric properties of state's amplitude in wave-vector space for …
Space-time wave packets are propagation-invariant pulsed beams that travel in free space without diffraction or dispersion by virtue of tight correlations introduced into their spatio-temporal spectrum. Such correlations constitute an embodiment of classical entanglement between continuous degrees of freedom. Using a measure of classical entanglement based on the Schmidt number of the field, we demonstrate theoretically and experimentally that the degree of classical entanglement determines the diffraction-free propagation distance of ST wave packets. Reduction in the degree of classical entan…
In this paper we will study for the first time how the wave-packet of a self-gravitating meso-scopic system spreads in theories beyond Einstein's general relativity. In particular, we will consider a ghost-free infinite derivative gravity, which resolves the $1/r$ singularity in the potential - such that the gradient of the potential vanishes within the scale of non-locality. We will show that a quantum wave-packet spreads faster for a ghost-free and singularity-free gravity as compared to the Newtonian case, therefore providing us a unique scenario for testing classical and quantum properties…
We study phase contributions of wave functions that occur in the evolution of Gaussian surface gravity water wave packets with nonzero initial momenta propagating in the presence and absence of an effective external linear potential. Our approach takes advantage of the fact that in contrast to matter waves, water waves allow us to measure both their amplitudes and phases.
The tunneling of Gaussian wave packets has been investigated by numerically solving the one-dimensional Schrödinger equation. The shape of wave packets interacting with a square barrier has been monitored for various values of the barrier width, height and initial width of the wave packet. Compared to the case of free propagation, the maximum of a tunneled wave packet exhibits a shift, which can be interpreted as an enhanced velocity during tunneling.
In this paper we will discuss how to localise a quantum wave-packet due to self-gravitating meso-scopic object by taking into account gravitational self-interaction in the Schrödinger equation beyond General Relativity. In particular, we will study soliton-like solutions in infinite derivative ghost free theories of gravity, which resolves the gravitational $1/r$ singularity in the potential. We will show a unique feature that the quantum spread of such a gravitational system is larger than that of the Newtonian gravity, therefore enabling us a window of opportunity to test classical and quant…
This article investigates some solutions of the time-dependent free Dirac equation. Visualizations of these solutions immediately reveal strange phenomena that are caused by the interference of positive- and negative-energy waves. The effects discussed here include the Zitterbewegung, the opposite direction of momentum and velocity in negative-energy wave packets, and the superluminal propagation of the wave packet's local maxima.
A practical search technique for finding the complex saddle points used in wave packet or coherent state propagation is developed which works for a large class of Hamiltonian dynamical systems with many degrees of freedom. The method can be applied to problems in atomic, molecular, and optical physics, and other domains. A Bose-Hubbard model is used to illustrate the application to a many-body system where discrete symmetries play an important and fascinating role. For multidimensional wave packet propagation, locating the necessary saddles involves the seemingly insurmountable difficulty of s…
The electron transfer between an $F^{-}$ ion and $Cu(111)$ and $Ag(111)$ surfaces is studied by the wave packet propagation method in order to determine specifics of the charge transfer interaction between the negative ion and the metal surface due to the projected band gap. A new modeling of the $F^{-}$ ion is developed that allows one to take into account the six quasi-equivalent electrons of $F^{-}$ which are {\it a priori} active in the charge transfer process. The new model invokes methods of constrained quantum dynamics. The six-electron problem is transformed to two one-electron problem…
An analytical treatment of a propagating wave packet incident on a transient barrier reveals a counterintuitive quantum mechanical effect in which, for a particular time interval, the time-varying transmission probability {\it exceeds} (`superarrival') that for the free propagation of the wave packet. It is found that the speed with which the information about the barrier perturbation propagates across the wave packet can exceed the group velocity of the wave packet. An interesting implication of this effect regarding information transfer is analyzed by showing one-to-one correspondence betwee…
Quantum mechanical real-time tunneling through general scattering potentials is studied in the semiclassical limit. It is shown that the exact path integral of the real-time propagator is dominated in the long time sector by quasi-stationary fluctuations associated with caustics. This leads to an extended semiclassical propagation scheme for wave packet dynamics which accurately describes deep tunneling through static and, for the first time, driven barrier potentials.
Space-Time Electron Wave Packets with Tunable Skyrmion Textures
Nanyang Technological University（南洋理工大学）；School of Electrical and Electronics Engineering, Nanyang Technological University（南洋理工大学电气与电子工程学院）；Centre of Disruptive Photonics Technologies, School of Physical and Mathematical Sciences（物理与数学学院破坏性光子技术研究中心）
We investigate the transient phenomenon or property of the propagation of an optical probe field in a medium consisting of many $Λ$-type three-level atoms coupled to this probe field and an classical driven field. We observe a hidden symmetry and obtain an exact solution for this light propagation problem by means of the spectral generating method. This solution enlightens us to propose a practical protocol implementing the quantum memory robust for quantum decoherence in a crystal. As an transient dynamic process this solution also manifests an exotic result that a wave-packet of light will s…
The wave packet approach to neutrino oscillations provides an enlightening description of quantum decoherence induced, during propagation, by localization effects. Within this approach, we show that a deeper insight into the dynamical aspects of particle mixing can be obtained if one investigates the behavior of quantum correlations associated to flavor oscillations. By identifying the neutrino three-flavor modes with (suitably defined) three-qubit modes, the exploitation of tools of quantum information theory for mixed states allows a detailed analysis of the dynamical behavior of flavor enta…
The most obvious obstacle behind a direct test of Quantum Gravity (QG) is its energy scale ($10^{19}$ GeV), which remains well outside of any human made machine. The next best possible approach is to provide indirect tests on effective theories of QG which can be performed in a lower energy scale. This paper is aimed in this direction, and shows a promising path to test the existence of the fundamental minimal length scale of Nature by measuring the dispersion of free, large molecular wave-packets. The existence of the minimal length is believed to be the reason for a modified commutation rela…
The propagation of an initially Gaussian wave packet of width $Δ_0$ in a cubic non-linear Schrodinger equation with a negative coupling constant for the nonlinear term is considered . It is predicted analytically and verified numerically that for a free particle if $Δ_0$ is less than a critical value $Δ_c$, then the packet will propagate in time with linearly growing width but for $Δ>Δ_c$, the packet will start becoming narrow and cease to be a Gaussian . For a simple harmonic oscillator, we find that for $Δ_0$ smaller than a critical value, there always exist a coupling strength for which the…
Wave packet revivals and fractional revivals are striking quantum interference phenomena that can occur under suitable conditions in a system with a nonlinear spectrum. In the framework of a specific model (the propagation of an initially coherent wave packet in a Kerr-like medium), it is shown that distinctive signatures of these revivals and fractional revivals are displayed by the time evolution of the expectation values of physical observables and their powers, i.e., by experimentally measurable quantities. Moreover, different fractional revivals can be selectively identified by examining …
Semiclassical methods are extremely important in the subjects of wave packet and coherent state dynamics. Unfortunately, these essentially saddle point approximations are considered nearly impossible to carry out in detail for systems with multiple degrees of freedom due to the difficulties of solving the resulting two-point boundary value problems. However, recent developments have extended the applicability to a broader range of systems and circumstances. The most important advances are first to generate a set of real reference trajectories using appropriately reduced dimensional spaces of i…
We study the dynamics of superposed wave packets in a specific nonlinear Hamiltonian which models the wave packet propagation in Kerr-like media and the dynamics of Bose-Einstein condensates. We show the dependence of initial wave packet superposition on fractional revival times using analysis based on the expectation values, Rényi entropy and Wigner function. We also show how the selective identification of fractional revivals using moments of appropriate observables depends on the number of wave packets present in the initial state.
The weak equivalence principle of gravity is examined at the quantum level in two ways. First, the position detection probabilities of particles described by a non-Gaussian wave-packet projected upwards against gravity around the classical turning point and also around the point of initial projection are calculated. These probabilities exhibit mass-dependence at both these points, thereby reflecting the quantum violation of the weak equivalence principle. Secondly, the mean arrival time of freely falling particles is calculated using the quantum probability current, which also turns out to be …
We study the output coupling of atoms from a magnetic trap into a linear potential slope of gravity using a weak radio-frequency field. We present a one-dimensional wave-packet model based on a continuous loading of a continuous spectrum of generalised eigenstates to describe the scenario. Analyzing the model, we show how the interference of the classical coupling fields maps to the interference of the resulting atomic streams.
We present a method to calculate exact dynamics of a wave-packet in a quantum two-state problem with Dirac delta coupling. The advantage of our method is that the calculations are done in the time domain. Hence inverting the solutions from other domains (Laplace or Fourier domain) do not intervene us from presenting the solution in the time domain. The initial wave packet is considered to be a Gaussian function. The wave propagation in the quantum state is governed by time-dependent Schrödinger equation and the coupling is given by non-diagonal element in the matrix representation. We present …
The numerical analysis of the diffraction features rendered by transmission electron microscopy (TEM) typically relies either on classical approximations (Monte Carlo simulations) or quantum paraxial tomography (the multislice method and any of its variants). Although numerically advan- tageous (relatively simple implementations and low computational costs), they involve important approximations and thus their range of applicability is limited. To overcome such limitations, an alternative, more general approach is proposed, based on an optimal combination of wave-packet propagation with the on…
The Jaynes-Cummings model, with and without the rotating wave approximation, is expressed in the conjugate variable representation and solved numerically by wave packet propagation. Both cases are then cast into systems of two coupled harmonic oscillators, reminiscent of coupled bound electronic potential curves of diatomic molecules. Using the knowledge of such models, this approach of the problem gives new insight of the dynamics. The effect of the rotating wave approximation is discussed. The collapse-revival phenomenon is especially analyzed in a non-standard manner. Extensions of the meth…
We study the effect of gravity on neutrino flavor oscillations when each mass eigenstate is described by a wave packet instead of a plane wave. Two different approaches for implementing the wave packet formalism in the study of neutrino flavor oscillations in curved spacetime are examined. We work with a general static and spherically symmetric spacetime before applying our results to a few spacetime metrics of interest. We first focus on general relativity by examining the effect of the exterior and interior Schwarzschild solutions, as well as the de Sitter-Schwarzschild metric, and then we e…
We report a new set of Laguerre-Gaussian wave-packets that propagate with periodical self-focusing and finite beam width in weakly guiding inhomogeneous media. These wave-packets are solutions to the paraxial form of the wave equation for a medium with parabolic refractive index. The beam width is defined as a solution of the Ermakov equation associated to the harmonic oscillator, so its amplitude is modulated by the strength of the medium inhomogeneity. The conventional Laguerre-Gaussian modes, available for homogenous media, are recovered as a particular case.
We consider the gravitational effect of quantum wave packets when quantum mechanics, gravity, and thermodynamics are simultaneously considered. Under the assumption of a thermodynamic origin of gravity, we propose a general equation to describe the gravitational effect of quantum wave packets. In the classical limit, this equation agrees with Newton's law of gravitation. For quantum wave packets, however, it predicts a repulsive gravitational effect. We propose an experimental scheme using superfluid helium to test this repulsive gravitational effect. Our studies show that, with present techno…
We consider systems described by the two-dimensional Dirac equation where the Fermi velocity is inhomogeneous as a consequence of mechanical deformations. We show that the mechanical deformations can lead to deflection and focusing of the wave packets. The analogy with known reflectionless quantum systems is pointed out. Furthermore, with the use of the qualitative spectral analysis, we discuss how inhomogeneous strains can be used to create waveguides for valley polarized transport of partially dispersionless wave packets.
Operational Limits of Gravity-Mediated Entanglement
University of Chinese Academy of Sciences（中国科学院大学）；Ningbo University（宁波大学）；Beihua University（北华大学）；Nantong University（南通大学）；Institute for Research in Fundamental Sciences (IPM)（伊朗基础科学研究所）
We consider a discrete-time dynamical process on graphs, firstly introduced in connection with a protocol for controlling large networks of spin 1/2 quantum mechanical particles [Phys. Rev. Lett. 99, 100501 (2007)]. A description is as follows: each vertex of an initially selected set has a packet of information (the same for every element of the set), which will be distributed among vertices of the graph; a vertex v can pass its packet to an adjacent vertex w only if w is its only neighbour without the information. By mean of examples, we describe some general properties, mainly concerning ho…
Dark matter or violations of the Einstein equivalence principle influence the motion of atoms, their internal states as well as electromagnetic fields, thus causing a signature in the signal of atomic detectors. To model such new physics, we introduce dilaton fields and study the modified propagation of light used to manipulate atoms in light-pulse atom interferometers. Their interference signal is dominated by the matter's coupling to gravity and the dilaton. Even though the electromagnetic field contributes to the phase, no additional dilaton-dependent effect can be observed. However, the li…
We present an ab initio approach to solve the time-dependent Schrödinger equation to treat electron and photon impact multiple ionization of atoms or molecules. It combines the already known time scaled coordinate method with a new high order time propagator based on a predictor-corrector scheme. In order to exploit in an optimal way the main advantage of the time scaled coordinate method namely that the scaled wave packet stays confined and evolves smoothly towards a stationary state the modulus square of which being directly proportional to the electron energy spectra in each ionization chan…
We consider wave packets of free particles with a general energy-momentum dispersion relation $E(p)$. The spreading of the wave packet is determined by the velocity $v = \p_p E$. The position-velocity uncertainty relation $Δx Δv \geq {1/2} |< \p_p^2 E >|$ is saturated by minimal uncertainty wave packets $Φ(p) = A \exp(- αE(p) + βp)$. In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation $E(p) = p^2/2m$, analytic calculations are presented for the spreading of wave packets with minimal position-velocity uncertainty product for the la…
We consider semiclassical higher-order wave packet solutions of the Schrodinger equation with phase vortices. The vortex line is aligned with the propagation direction, and the wave packet carries a well-defined orbital angular momentum (OAM) $\hbar l$ ($l$ is the vortex strength) along its main linear momentum. The probability current coils around momentum in such OAM states of electrons. In an electric field, these states evolve like massless particles with spin $l$. The magnetic-monopole Berry curvature appears in momentum space, which results in a spin-orbit-type interaction and a Berry/Ma…
