Heller, E.J., Kaplan, L. & Pollmann, F. Inflationary dynamics for matrix eigenvalue problems. Proceedings of the National Academy of Sciences of the United States of America 105, 7631 - 7635 (2008). Publisher's VersionAbstract
Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the acoustic modes of a concert hall, or hundreds of other physical quantities. Often only the few eigenpairs with the lowest or highest frequency (extremal solutions) are needed. Methods that have been developed over the past 60 years to solve such problems include the Lanczos algorithm. Jacobi-Davidson techniques, and the conjugate gradient method. Here, we present a way to solve the extremal eigenvalue/eigenvector problem, turning it into a nonlinear classical mechanical system with a modified Lagrangian constraint. The constraint induces exponential inflationary growth of the desired extremal solutions. [ABSTRACT FROM AUTHOR]
Petitjean, C., Bevilaqua, D.V., Heller, E.J. & Jacquod, P. Displacement echoes: classical decay and quantum freeze. Physical review letters 98, 164101 (2007). Publisher's VersionAbstract

Motivated by neutron scattering experiments, we investigate the decay of the fidelity with which a wave packet is reconstructed by a perfect time-reversal operation performed after a phase-space displacement. In the semiclassical limit, we show that the decay rate is generically given by the Lyapunov exponent of the classical dynamics. For small displacements, we additionally show that, following a short-time Lyapunov decay, the decay freezes well above the ergodic value because of quantum effects. Our analytical results are corroborated by numerical simulations.

Walls, J.D. & Heller, E.J. Spin-orbit coupling induced interference in quantum corrals. Nano letters 7, 3377–3382 (2007). Publisher's VersionAbstract

Lack of inversion symmetry at a metallic surface can lead to an observable spin−orbit interaction. For certain metal surfaces, such as the Au(111) surface, the experimentally observed spin−orbit coupling results in spin rotation lengths on the order of tens of nanometers, which is the typical length scale associated with quantum corral structures formed on metal surfaces. In this work, multiple scattering theory is used to calculate the local density of states (LDOS) of quantum corral structures composed of nonmagnetic adatoms in the presence of spin−orbit coupling. Contrary to previous theoretical predictions, spin−orbit coupling induced modulations are observed in the theoretical LDOS, which should be observable using scanning tunneling microscopy.

Heller, E.J. & Landry, B.R. Statistical properties of many particle eigenfunctions. Journal of Physics A: Mathematical & Theoretical 40, 9259 - 9274 (2007). Publisher's VersionAbstract

Wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N? ?. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries and collisions between particles can be included. [ABSTRACT FROM AUTHOR]

Heller, E.J. Freak ocean waves and refraction of Gaussian seas. Extreme Events in Nature and Society 189–210 (2006). Publisher's VersionAbstract

Rogue or freak waves sink ships at an alarming rate — estimated at one large ship every few weeks worldwide. It is thought that vulnerable ships (light cargo ships) simply break in two when they plough into a 60 foot wave preceded by a 40 foot hole in the sea, as some sailors that have survived such experiences have called it. Wave refraction due to current eddies (which are ubiquitous in the oceans) has long been suspected to play a role in concentrating wave energy into rogue waves. Existing theories have been based on refraction of plane waves, not the stochastic Gaussian seas one finds in practice. Gaussian seas ruin the dramatic focal caustic concentration of energy, and this fact has discouraged further investigations. Although it was thought that chaos, or the extreme sensitivity to initial conditions displayed by individual ray trajectories would quickly wipe out all significant fluctuations, we show that this is incorrect, and the fluctuations are “structually stable” entities. Significant “lumps” of energy survive the averaging over wave directions and wavelengths. We furthermore demonstrate that the probability of freak waves increases dramatically in the presence of these lumps, even though most parameters, such as the significant wave height, are unchanged. We show here that a single dimensionless parameter determines the potential for freak waves; this is the “freak index” of the current eddies — a typical angular deflection in one focal distance, divided by the initial angular uncertainty of the incoming waveset. If the freak index is greater than 2 or so, truly spectacular enhancements of freak index waves can result, even though the caustics are washed out by the Gaussian nature of the impinging sea.

Walls, J.D., Huang, J., Westervelt, R.M. & Heller, E.J. Multiple-scattering theory for two-dimensional electron gases in the presence of spin-orbit coupling. Physical Review B 73, 035325 (2006). Publisher's VersionAbstract

In order to model the phase-coherent scattering of electrons in two-dimensional electron gases in the presence of Rashba spin-orbit coupling, a general partial-wave expansion is developed for scattering from a cylindrically symmetric potential. The theory is applied to possible electron flow imaging experiments using a moveable scanning probe microscope tip. In such experiments, it is demonstrated theoretically that the Rashba spin-orbit coupling can give rise to spin interference effects, even for unpolarized electrons at nonzero temperature and no magnetic field.

Ruiz, A. & Heller, E.J. Quasiresonance. Molecular Physics 104, 127–145 (2006). Publisher's VersionAbstract

The concept of quasiresonance was introduced in connection with inelastic collisions between one atom and a vibro-rotationally excited diatomic molecule. In its original form, the collisions induce quasiresonant transfer of energy between the internal degrees of freedom of the diatom: there is a surprisingly accurate low order rational value for the ratio of the changes in the vibrational and rotational classical actions, provided the vibrational and rotational frequencies of the diatom are approximately related by low order rational values, and the collision was longer than the rotational period of the molecule. In this paper, we show that quasiresonance can be extended to many other processes and systems, and that it may be understood in terms of the adiabatic invariance theory and the method of averaging.

Heller, E. Eric Heller. ACM SIGGRAPH 2005 Electronic Art and Animation Catalog 78–79 (2005).
Heller, E. Freak waves: just bad luck, or avoidable?. Europhysics News 36, 159–162 (2005). Publisher's Version
Fallahi, P., et al. Imaging a single-electron quantum dot. Nano letters 5, 223–226 (2005). Publisher's VersionAbstract

Images of a single-electron quantum dot were obtained in the Coulomb blockade regime at liquid He temperatures using a cooled scanning probe microscope (SPM). The charged SPM tip shifts the lowest energy level in the dot and creates a ring in the image corresponding to a peak in the Coulomb-blockade conductance. Fits to the line shape of the ring determine the tip-induced shift of the energy of the electron state in the dot. SPM manipulation of electrons in quantum dots promises to be useful in understanding, building, and manipulating circuits for quantum information processing.

Cohen, D., Lepore, N. & Heller, E.J. Consolidating boundary methods for finding the eigenstates of billiards. Journal of Physics A: Mathematical and General 37, 2139 (2004). Publisher's VersionAbstract

The plane-wave decomposition method, a widely used means of numerically finding eigenstates of the Helmholtz equation in billiard systems is described as a variant of the mathematically well-established boundary integral method (BIM). A new unified framework encompassing the two methods is discussed. Furthermore, a third numerical method, which we call the gauge freedom method is derived from the BIM equations. This opens the way to further improvements in eigenstate search techniques.

M. A. Topinka, R. M. Westervelt, & Heller, E.J. Imaging Electron Flow. Physics Today 47 (2004).
Westervelt, R.M., et al. Imaging electron waves. Physica E: Low-dimensional Systems and Nanostructures 24, 63 - 69 (2004). Publisher's VersionAbstract

One can image the coherent flow of electron waves through a quantum point contact (QPC) into a two-dimensional electron gas by using scanning probe microscopy. A negatively charged tip depletes the electron gas below, backscatters electron waves, and reduces the QPC conductance. By raster scanning the tip over the sample, an image of electron flow is obtained. Images at liquid He temperatures show the individual quantum modes of the QPC. At greater distances, the electron flow forms narrow branches caused by small-angle scattering. Interference fringes in the images demonstrate the coherence of electron flow. An electron interferometer that acts as a quantum phase shifter was constructed by adding a gate to reflect electron waves back to the QPC, producing a V-shaped path for interfering electron waves with the apex at the QPC. When the length of one leg of the V is altered by changing the reflector gate voltage, the fringes at the other end of the V, under the tip, shift by the same distance. The interferometer is sensitive to transit time differences as small as ∼0.1 ps between the two electron paths. These observations are in good agreement with theoretical simulations of electron flow.

Fiete, G.A. & Heller, E.J. Colloquium: Theory of quantum corrals and quantum mirages. Reviews of Modern Physics 75, 933 (2003). Publisher's VersionAbstract

Quantum corrals are two-dimensional structures built atom by atom on an atomically clean metallic surface using a scanning tunneling microscope (STM). These two-dimensional structures “corral” electrons in the surface states of noble metals, leading to standing-wave patterns in the electron density inside the quantum corral. The authors review the physics of quantum corrals and relate the signal of the STM to the scattering properties of substrate electrons from atomic impurities supported on the surface. The theory includes the effects of incoherent surface-state electron scattering at the impurities and quantitively describes nearly all of the current STM data on quantum corrals, including the recent quantum mirage experiments with Kondo effect. The physics underlying the recent mirage experiments is discussed, as are some of the outstanding questions regarding the Kondo effect from impurities in nanoscale structures on metallic surfaces. The authors also summarize recent work on variations of “quantum” corrals: Optical corrals and acoustical corrals.

Tomsovic, S. & Heller, E.J. Comment on “Ehrenfest times for classically chaotic systems”. Physical Review E 68, 038201 (2003). Publisher's VersionAbstract

In a recent Rapid Communication [P. G. Silvestrov and C. W. J. Beenakker, Phys. Rev. E 65, 035208(R) (2002)], the authors, Silvestrov and Beenakker, introduce a way to lengthen the Ehrenfest time τ for fully chaotic systems. We disagree with several statements made in their paper, and address the following points essential to their conclusions: (1) it is not true that all semiclassical approximations for chaotic systems fail at a so-called “log time” τln(ħ), differing only by a numerical coefficient; and (2) the limitation of the semiclassical approximation as expressed in the authors’ Eq. (8) is not limited by their argument leading to Eq. (12).

Topinka, M.A., Westervelt, R.M. & Heller, E.J. Imaging electron flow. Physics Today 56, 47–52 (2003).
Li, S. & Heller, E.J. Quantum multiple scattering: Eigenmode expansion and its applications to proximity resonance. Physical Review A 67, 032712 (2003).Abstract

We show that for a general system of N s-wave point scatterers, there are always N eigenmodes. These eigenmodes or eigenchannels play the same role as spherical harmonics for a spherically symmetric target—they give a phase shift only. In other words, the T matrix of the system is of rank N, and the eigenmodes are eigenvectors corresponding to nonzero eigenvalues of the T matrix. The eigenmode expansion approach can give insight to the total scattering cross section; the position, width, and superradiant or subradiant nature of resonance peaks; the unsymmetric Fano line shape of sharp proximity resonance peaks based on the high-energy tail of a broadband; and other properties. Off-resonant eigenmodes for identical proximate scatterers are approximately angular-momentum eigenstates.

Van{\'ıček, J.{\'ı & Heller, E.J. Semiclassical evaluation of quantum fidelity. Physical Review E 68, 056208 (2003). Publisher's VersionAbstract

We present a numerically feasible semiclassical (SC) method to evaluate quantum fidelity decay (Loschmidt echo) in a classically chaotic system. It was thought that such evaluation would be intractable, but instead we show that a uniform SC expression not only is tractable but it also gives remarkably accurate numerical results for the standard map in both the Fermi-golden-rule and Lyapunov regimes. Because it allows Monte Carlo evaluation, the uniform expression is accurate at times when there are 1070 semiclassical contributions. Remarkably, it also explicitly contains the “building blocks” of analytical theories of recent literature, and thus permits a direct test of the approximations made by other authors in these regimes, rather than an a posteriori comparison with numerical results. We explain in more detail the extended validity of the classical perturbation approximation and show that within this approximation, the so-called “diagonal approximation” is automatic and does not require ensemble averaging.

Fiete, G.A. & Heller, E.J. Semiclassical theory of coherence and decoherence. Physical Review A 68, 022112 (2003). Publisher's VersionAbstract

A general semiclassical approach to quantum systems with system-bath interactions is developed. We study system decoherence in detail using a coherent-state semiclassical wave-packet method which avoids singularity issues arising in the usual Green’s function approach. We discuss the general conditions under which it is approximately correct to discuss quantum decoherence in terms of a “dephasing” picture and we derive semiclassical expressions for the phase and phase distribution. Remarkably, an effective system wavefunction emerges whose norm measures the decoherence and is equivalent to a density-matrix formulation.

Van Voorhis, T. & Heller, E.J. Similarity transformed semiclassical dynamics. The Journal of chemical physics 119, 12153–12162 (2003). Publisher's VersionAbstract