Publications

2009
Ruiz, A., Palao, J.P. & Heller, E.J. Classical and quantum analysis of quasiresonance in grazing atom-surface collisions. Physical Review A 79, 052901 (2009). Publisher's VersionAbstract

Quasiresonance is a general effect that may arise from the coupling between approximately resonant degrees of freedom in a system perturbed by some transient interaction. In a process induced by a slowly switching on and off of the coupling interaction, quasiresonance is characterized by the existence of significant ranges of initial states in the perturbed system over which some very specific and efficient transfer of energy between the approximately resonant degrees of freedom occurs. This work presents a classical and quantum analysis of quasiresonant processes in grazing incident angle atom-surface collisions. The momentum transfer between the normal components to an index direction is investigated. For fast atoms with grazing angle of incidence there is an interval of azimuthal angles around the index directions, the quasiresonance region, in which the energy transfer can be very efficient. This effect is reflected in quantum diffraction patterns with large nonspecular peaks, associated with the parallel to the surface and normal to the index direction momentum component. We demonstrate the essentially classical underlying mechanism for the persistence of a pattern of diffraction peak intensities for incidence close to an index direction. The analysis also shows that the size of the quasiresonance region is approximately equal to the spectral width of the diffraction pattern.

Sanders, S.N., Mintert, F. & Heller, E.J. Coherent scattering from a free gas. Physical Review A 79, 023610 (2009). Publisher's VersionAbstract

We investigate decoherence in atom interferometry due to scattering from a background gas and show that the supposition that residual coherence is due to near-forward scattering is incorrect. In fact, the coherent part is completely unscattered, although it is phase shifted. This recoil-free process leaves both the atom and the gas in an unchanged state, but allows for the acquisition of a phase shift. This is essential to understanding decoherence in a separated-arm atom interferometer, where a gas of atoms forms a refractive medium for a matter wave. Our work elucidates the actual microscopic, many-body, quantum-mechanical scattering mechanism that gives rise to prior phenomenological results for the phase shift and decoherence.

Ruiz, A., Palao, J.P. & Heller, E.J. Nearly resonant multidimensional systems under a transient perturbative interaction. Physical Review E 80, 066606 (2009). Publisher's VersionAbstract

We analyze the response of a classical system with N2 internal degrees of freedom satisfying R(N1) approximated resonance conditions to an external perturbative transient interaction. Under certain assumptions on the system internal frequencies and on the coupling interaction, we show the precise NR adiabatic invariants and obtain an estimate of the span of the domain defined by the intersecting resonances. The results are illustrated considering a system of three anharmonic oscillators transiently coupled by an explicitly time-dependent interaction, and applied to the low energy vibro-rotationally inelastic collisions between two diatomic molecules.

Landry, B.R., Wasserman, A. & Heller, E.J. Semiclassical Ground-State Energies of Many-Electron Systems. Physical review letters 103, 066401 (2009). Publisher's VersionAbstract

A new semiclassical method is proposed to obtain accurate ground-state energies for many-electron systems. The method borrows its semiclassical character from Thomas-Fermi (TF) theory, but improves upon it by including exchange-correlation effects, at least approximately. We illustrate our method (correlated TF method) on simple models of 1D-interacting electrons, showing that it yields dramatic improvements over TF theory, particularly in the strongly correlated regime.

Mintert, F. & Heller, E.J. Simulation of open quantum systems. EPL (Europhysics Letters) 86, 50006 (2009). Publisher's VersionAbstract

We present an approach for the semiclassical treatment of open quantum systems. An expansion into localized states allows restriction of a simulation to a fraction of the environment that is located within a predefined vicinity of the system. Adding and dropping environmental particles during the simulation yields an effective reduction of the size of the system that is being treated.

2008
Kramer, T., Heller, E.J. & Parrott, R.E. An efficient and accurate method to obtain the energy-dependent Green function for general potentials. Journal of Physics: Conference Series 99, 012010 (2008). Publisher's VersionAbstract

Time-dependent quantum mechanics provides an intuitive picture of particle propagation in external fields. Semiclassical methods link the classical trajectories of particles with their quantum mechanical propagation. Many analytical results and a variety of numerical methods have been developed to solve the time-dependent Schrödinger equation. The time-dependent methods work for nearly arbitrarily shaped potentials, including sources and sinks via complex-valued potentials. Many quantities are measured at fixed energy, which is seemingly not well suited for a time-dependent formulation. Very few methods exist to obtain the energy-dependent Green function for complicated potentials without resorting to ensemble averages or using certain lead-in arrangements. Here, we demonstrate in detail a time-dependent approach, which can accurately and effectively construct the energy-dependent Green function for very general potentials. The applications of the method are numerous, including chemical, mesoscopic, and atomic physics.

Wasserman, A., Maitra, N.T. & Heller, E.J. Investigating interaction-induced chaos using time-dependent density-functional theory. Physical Review A 77, 042503 (2008). Publisher's VersionAbstract

Systems whose underlying classical dynamics are chaotic exhibit signatures of the chaos in their quantum mechanics. We investigate the possibility of using the linear response formalism of time-dependent density functional theory (TDDFT) to study the case when chaos is induced by electron-interaction alone. Nearest-neighbor level-spacing statistics are in principle exactly and directly accessible from TDDFT. We discuss how the TDDFT linear response procedure can reveal information about the mechanism of chaos induced by electron-interaction alone. A simple model of a two-electron quantum dot highlights the necessity to go beyond the adiabatic approximation in TDDFT.

Heller, E.J. & Tomsovic, S. Postmodern quantum mechanics. Physics Today 46, 38–46 (2008). Publisher's VersionAbstract

 

Postmodern movements are well known in the arts. After a major artistic revolution, and after the “modern” innovations have been assimilated, the threads of premodern thought are always reconsidered. Much of value may be rediscovered and put to new use. The modern context casts new light on premodern thought, which in turn shades perspectives on modernism.

Recent progress in semiclassical theory has overcome barriers posed by classical chaos and cast light on the correspondence principle. Semiclassical ideas have also become central to new experiments in atomic, molecular, microwave and mesoscopic physics.

 

Heller, E.J. Surface physics: Electron wrangling in quantum corrals. Nature Physics 4, 443 - 444 (2008). Publisher's VersionAbstract
The article reports on the discovery of image electron waves' ability to propagate along a metal surface by Don Eigler, Christopher Lutz, and Michael Crommie at IBM Almaden in 1990. Eigler, Lutz, and Crommie noticed unexpectedly long and periodic undulations in the scanning tunnelling microscope (STM) conductance signal as a function of tip position. The undulations were recognized as the signature of surface confined electron waves whose de Broglie wavelength is bigger than the lattice.
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]
2007
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]

2006
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.

2005
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.

2004
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.

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