Quantum wave packet transforms with compact frequency support: Implementations for wavelets and Gabor atoms

We argue that two specific wave packet families---curvelets and wave atoms---provide powerful tools for representing linear systems of hyperbolic differential equations with smooth, time-independent coefficients. In both cases, we prove that the matrix representation of the Green's function is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e., faster than any negative polynomial), and well organized in the sense that the very few nonnegligible entries occur near a few shifted diagonals, whose location is predicted by geometrical optics. This result holds only when the basis elements obey a precise parabolic balance between oscillations and support size, shared by curvelets and wave atoms but not wavelets, Gabor atoms, or any other such transform. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles. We also provide fast digital implementations of tight frames of curvelets and wave atoms in two dimensions. In both cases the complexity is O(N² log N) flops for N-by-N Cartesian arrays, for forward as well as inverse transforms. Finally, we present a geometric strategy based on wave atoms for the numerical solution of wave equations in smoothly varying, 2D time-independent periodic media. Our algorithm is based on sparsity of the matrix representation of Green's function, as above, and also exploits its low-rank block structure after separation of the spatial indices. As a result, it becomes realistic to accurately build the full matrix exponential using repeated squaring, up to some time which is much larger than the CFL timestep. Once available, the wave atom representation of the Green's function can be used to perform 'upscaled' timestepping. We show numerical examples and prove complexity results based on a priori estimates of sparsity and separation ranks. They beat the O(N^3) bottleneck on an N-by-N grid, for a wide range of physically relevant situations. In practice, the current wave atom solver can become competitive over a pseudospectral method in the regime when the wave equation should be solved several times with different initial conditions, as in reflection seismology.

The generalized Hartmann effect (GHE) predicts a strict inequality between the traversal times across a contiguous and a separated double-barrier system. This is compared to the implications of the time-of-arrival (TOA) operator approach to barrier traversal time [E. A. Galapon, Phys. Rev. Lett. 108, 170402 (2012)]. It is shown that, for initial wave packets with compact supports in the far incident side of the barrier system, the expectation value of the traversal time is independent of the separation between the barriers. On the other hand, for wave packets with supports extending inside the first barrier, the contribution of the barrier separation to the traversal time exponentially increases with the barrier height. Our result shows that if the support of the incident wave packet is far from the barrier region, the GHE inequality is violated. However, if the support of the wave packet extends inside the barrier region, the GHE inequality is consistent with the TOA operator approach, but only when the particle's incident energy is very small.

The Kuramoto model of a network of coupled phase oscillators exhibits a first-order phase transition when the distribution of natural frequencies has a finite flat region at its maximum. First-order phase transitions including hysteresis and bistability are also present if the frequency distribution of a single network is bimodal. In this study we are interested in the interplay of these two configurations and analyze the Kuramoto model with compact bimodal frequency distributions in the continuum limit. As of yet, a rigorous analytic treatment has been elusive. By combining Kuramoto's self-consistency approach, Crawford's symmetry considerations, and exploiting the Ott-Antonsen ansatz applied to a family of rational distribution functions that converge towards the compact distribution, we derive a full bifurcation diagram for the system's order parameter dynamics. We show that the route to synchronization always passes through a standing wave regime when the bimodal distribution is compounded by two unimodal distributions with compact support. This is in contrast to a possible transition across a region of bistability when the two compounding unimodal distributions have infinite support.

The success of wavelet transforms for seismic data compression leads naturally to the idea of “compressing” numerical wavefield propagators. In spite of numerous efforts (i.e. Beylkin, 1992, Dessing and Wapenaar, 1995, Mosher, Foster, and Wu, 1996) application of orthogonal wavelet bases to wavefield propagation has had limited success. In this work, we examine implementations of phase shift migration using orthogonal wavelet transforms as an illustration of the limitations that orthogonality places on wavelet domain propagators. Since wave propagation has a simple representation in the frequency domain, frequency domain wavelet transforms provide a useful framework for studying wave propagation. In particular, we describe phase shift extrapolators for 2-dimensional wavefields that have been Fourier transformed over time and wavelet transformed over space. The wavelet transform over the space axis is implemented in the wavenumber-frequency domain by complex multiplication of low and high pass wavenumber filter functions to form wave packet trees. Spacewavenumber-frequency transforms are usually referred to as ‘beamlet transforms’ (Wu and Chen 2001), and are closely related to Gaussian beams (Hill, 2001, Albertin et al, 2001). The interaction of beamlet transform filter banks and phase shift wavefield extrapolators are simple complex multiplications. Wavefield propagation in the beamlet domain is complicated, however, by the digital implementation of decimation and upsampling operators used in orthogonal wavelet transforms. Unlike the filter functions, which can be viewed as diagonal matrix operators, the decimation and upsampling operators have significant off-diagonal terms. Since these operators do not commute with the filter and phase shift operators, the effects of the off-diagonal terms must be accounted for in the application of wave propagation operators. Use of filters designed for simple shape and compact support in the wavenumber domain reduces the domain of the interactions, resulting in implementations of phase shift extrapolators that have computational complexity comparable to traditional Fourier approaches. Compact support in the wavenumber domain, however, corresponds to poor localization in the space domain. Use of orthogonal wavelet bases for beam-based wavefield propagators results in a trade-off between computational complexity when the wavelet transform filters overlap, and poor localization in space when the overlap is limited. These results suggest that non-orthogonal transforms may provide a better domain for wave propagation. Introduction

Wavelet transforms have a simple representation in the frequency domain (Daubchies, 1992; Veterlli and Herley, 1992; Mosher and Foster, 1995). Since wave propagation also has a simple representation in the frequency domain, frequency domain wavelet transforms provide a useful framework for studying the nature of wave propagation in the wavelet domain. In this paper, we study phase shift extrapolators for 2-dimensional wavefields that have been Fourier transformed over time and wavelet transformed over space. The wavelet transform over the space axis is implemented in the wavenumber-frequency domain by complex multiplication of low and high pass wavenumber filter functions to form wave packet trees. To differentiate this operation from time-frequency wavelet transforms, we refer to the space-wavenumber-frequency transform as the 'beamlet transform.' The interaction of beamlet transform filter banks and phase shift wavefield extrapolators are simple complex multiplications. Wavefield propagation in the beamlet domain is complicated, however, by the digital implementation of decimation and upsampling operators used in orthogonal wavelet transforms. Unlike the filter functions, which can be viewed as diagonal matrix operators, the decimation and upsampling operators have significant off-diagonal terms. Since these operators do not commute with the filter and phase shift operators, the effects of the non-diagonal operators must be accounted for in the application of wave propagation operators. A simple (but unsatisfying) solution would be to apply forward-inverse transforms at each extrapolation step. Beamlet transforms with compact support in the wavenumber domain (Mosher and Foster, 1995) provide an alternate solution. Analysis of phase shift migration in the beamlet domain yields a simple matrix representation defining the interaction of filters, phase operators, and decimation/upsampling. The effects of decimation/upsampling are represented by simple folding operations. Use of filters designed for simple shape and compact support in the wavenumber domain reduces the domain of the interactions, resulting in efficient implementations of phase shift extrapolators that compare favorably with traditional Fourier approaches. Coupled with data compression, implementations of phase shift migration with multi-dimensional wavelet/beamlet transforms that exceed traditional implementations in computational efficiency may be possible.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Much of the wavelet literature is focussed on wavelets with compact support in the time domain. For many geophysical applications, compact support in the frequency domain is desirable. For these applications, simple window functions can be used to construct appropriate filter banks in the frequency domain. Convolution with filter coefficients in the time domain is replaced with a Fourier transform and multiplication by window functions in the frequency domain. Given the dual nature of the Fourier transform, the time and frequency variables can be exchanged to produce a time windowing algorithm for computing wave packet transforms. Taken together, frequency-windowed and time-windowed wave packet transforms provide a comprehensive tool set for constructing new geophysical applications that take advantage of simultaneous access to time and frequency. Depending on the application, frequency windowing or time windowing may be more desirable.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Wavelet decomposition is pivotal for underwater image processing, known for its ability to analyse multi-scale image features in the frequency and spatial domains. In this paper, we propose a new biorthogonal cubic special spline wavelet (BCS-SW), based on the Cohen–Daubechies–Feauveau (CDF) wavelet construction method and the cubic special spline algorithm. BCS-SW has better properties in compact support, symmetry, and frequency domain characteristics. In addition, we propose a K-layer network (KLN) based on the BCS-SW for underwater image enhancement. The KLN performs a K-layer wavelet decomposition on underwater images to extract various frequency domain features at multiple frequencies, and each decomposition layer has a convolution layer corresponding to its spatial size. This design ensures that the KLN can understand the spatial and frequency domain features of the image at the same time, providing richer features for reconstructing the enhanced image. The experimental results show that the proposed BCS-SW and KLN algorithm has better image enhancement effect than some existing algorithms.

Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems. VI. Asymptotics in the two-cluster region

We study the phase dynamics of a chain of autonomous oscillators with a dispersive coupling. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons: solitary waves with a compact support and kovatons which are compact formations of glued together kink-antikink pairs that may assume an arbitrary width. These robust objects seem to collide elastically and, together with wave trains, are the building blocks of the dynamics for typical initial conditions. Numerical studies of the complex Ginzburg-Landau and Van der Pol lattices show that the presence of a nondispersive coupling does not affect kovatons, but causes a damping and deceleration or growth and acceleration of compactons.

Quantum Wave Packet Transforms with Compact Frequency Support

We present analytical expressions for the eigenstates and eigenvalues of electrons confined in a graphene monolayer in the presence of a disclination. The calculations are performed in the continuum limit approximation in the vicinity of the Dirac points, solving Dirac equation by freezing out the carrier radial motion. We include the effect of an external magnetic field and show the appearence of Aharonov-Bohm oscillation and find out the conditions of gapped and gapless states in the spectrum. We show that the gauge field due to a disclination lifts the orbital degeneracy originating from the existence of two valleys. The broken valley degeneracy has a clear signature on quantum oscillations and wave packet dynamics.

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.

Non-dispersive wave packets in periodically driven quantum systems

Singularities, which are commonplace in general relativity, are indicated by causal geodesic incompleteness in otherwise maximal spacetimes. Can such singularities be healed (or “resolved”) quantum mechanically? Geodesics are in effect the paths of classical particles, which led Horowitz and Marolf to propose that they be replaced by quantum wave packets. Then a singularity is healed if the quantum wave operator can be shown to be essentially self-adjoint. We explore here a generalized Horowitz–Marolf approach for conformally static spacetimes in the context of the Klein–Gordon operator in the vicinity of singularities in the self-similar spacetimes introduced by Brady. We show that it fails the self-adjointness test for an entire generic class of spacetimes that have asymptotically power-law metric coefficients near the classically-singular origin, so the singularities in these spacetimes cannot be healed using quantum wave packets.

The application of femtosecond pump-probe photoelectron spectroscopy to directly observe vibrational wave packets passing through an avoided crossing is investigated using quantum wave packet dynamics calculations. Transfer of the vibrational wave packet between diabatic electronic surfaces, bifurcation of the wave packet, and wave packet construction via nonadiabatic mixing are shown to be observable as time-dependent splittings of peaks in the photoelectron spectra.