Publications

2001
Tannenbaum, E. & Heller, E.J. Semiclassical Quantization Using Invariant Tori: A Gradient-Descent Approach. The Journal of Physical Chemistry A 105, 2803–2813 (2001). Publisher's VersionAbstract

This paper presents a PDE-based, gradient-descent approach (GDA) to the EBK quantization of nearly separable Hamiltonians in the quasi-integrable regime. The method does this by finding an optimal semiclassical basis of invariant tori which minimizes the angular dependence of the Hamiltonian. This representation of the Hamiltonian is termed an intrinsic resonance representation (IRR), and it gives the smallest possible basis obtainable from classical mechanics. Because our method is PDE-based, we believe it to be significantly faster than previous IRR algorithms, making it possible to EBK quantize systems of higher degrees of freedom than previously studied. In this paper we demonstrate our method by reproducing results from a two-degree-of-freedom system used to demonstrate the previous Carioli, Heller, and Moller (CHM) implementation of the IRR approach. We then go on to show that our method can be applied to higher dimensional Hamiltonians than previously studied by using it to EBK quantize a four- and a six-degree-of-freedom system.

Topinka, M.A., et al. Coherent branched flow in a two-dimensional electron gas. Nature 410, 183 (2001). Publisher's VersionAbstract
Presents a study to observe electron flow through a narrow constriction in a semiconductor nanostructure. Methods; Results; Conclusion that a branching of current flux is due to focusing of the electron paths by ripples in the background potential.
2000
M. A. Topinka, R. M. Westervelt,, et al. Imaging Coherent Electron Flow from a Quantum Point Contact. Science 289, 5488, 2323-2326 (2000). Publisher's VersionAbstract

Scanning a charged tip above the two-dimensional electron gas inside a gallium arsenide/aluminum gallium arsenide nanostructure allows the coherent electron flow from the lowest quantized modes of a quantum point contact at liquid helium temperatures to be imaged. As the width of the quantum point contact is increased, its electrical conductance increases in quantized steps of 2e 2/h, where e is the electron charge and h is Planck's constant. The angular dependence of the electron flow on each step agrees with theory, and fringes separated by half the electron wavelength are observed. Placing the tip so that it interrupts the flow from particular modes of the quantum point contact causes a reduction in the conductance of those particular conduction channels below 2e 2/h without affecting other channels.

Barnett, A., Cohen, D. & Heller, E.J. Deformations and dilations of chaotic billiards: Dissipation rate, and quasiorthogonality of the boundary wave functions. Physical review letters 85, 1412 (2000). Publisher's VersionAbstract

We consider chaotic billiards in d dimensions, and study the matrix elements Mnm corresponding to general deformations of the boundary. We analyze the dependence of |Mnm|2 on ω=(EnEm)/ħ using semiclassical considerations. This relates to an estimate of the energy dissipation rate when the deformation is periodic at frequency ω. We show that, for dilations and translations of the boundary, |Mnm|2 vanishes like ω4 as ω0, for rotations such as ω2, whereas for generic deformations it goes to a constant. Such special cases lead to quasiorthogonality of the eigenstates on the boundary.

Cohen, D. & Heller, E.J. Unification of perturbation theory, random matrix theory, and semiclassical considerations in the study of parametrically dependent eigenstates. Physical review letters 84, 2841 (2000). Publisher's VersionAbstract

We consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where x is a constant parameter. Specifically, we discuss a gas particle inside a cavity, where x controls a deformation of the boundary or the position of a "piston." The quantum eigenstates of the system are |n(x)>. We describe how the parametric kernel P(nmid R:m) = |<n(x)mid R:m(x(0))>|(2) evolves as a function of deltax = x-x(0). We explore both the perturbative and the nonperturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as random waves and random-matrix-theory considerations.

1999
Heller, E.J. Semiclassical wave packets. The Physics and Chemistry of Wave Packets; Yeazel, J.; Uzer, T. eds, Wiley: New York (1999).
Forrey, R.C., Balakrishnan, N., Dalgarno, A., Haggerty, M.R. & Heller, E.J. Quasiresonant energy transfer in ultracold atom-diatom collisions. Physical review letters 82, 2657 (1999). Publisher's VersionAbstract

Quantum and classical quasiresonant vibration-rotation energy transfer is investigated for ultracold HeH2 collisions. Classical trajectory computations show that extremely strong correlations between Δj and Δv persist at low energies, though the changes themselves are less than one quantum. Quantum computations show that quasiresonant transitions occur in the limit of zero collision energy but that threshold effects become important and that some quasiresonant channels close. The qualitative similarity between classical and quantum results suggests that they share a common mechanism.

1995
Grossmann, F. & Heller, E.J. A semiclassical correlation function approach to barrier tunneling. Chemical physics letters 241, 45–50 (1995). Publisher's VersionAbstract

A time domain approach employing the semiclassical approximation to the quantum mechanical propagator, as applied to Gaussian wavepackets, is used to study the barrier penetration problem. We have observed that qualitative agreement with the exact quantum calculations for the correlation function and the transmission probability can be achieved by considering only classically allowed trajectories. The results slowly tend to the classical step function at the barrier top as a function of the wavepacket center parameter, however. We suggest that a full semiclassical calculation, including nonclassical trajectories with complex energies, would improve the results.

1994
Rost, J.M. & Heller, E.J. Ionization of hydrogen by positron impact near the fragmentation threshold. Physical Review A 49, R4289 (1994). Publisher's VersionAbstract

The semiclassical approximation of Feynman’s path integral is used to calculate the S matrix for the positron-impact ionization of hydrogen. The formulation provides a full scattering amplitude, and more importantly does not require knowledge of the asymptotic three-body Coulomb state in the continuum. In the limit of vanishing excess energy, the results confirm Wannier’s classical model for fragmentation [Phys. Rev. 90, 817 (1953)]. The experimentally observable ratio of fragmentation versus total ionization (including positronium formation) is predicted.

Sepúlveda, M.A. & Heller, E.J. Semiclassical analysis of hierarchical spectra. The Journal of chemical physics 101, 8016–8027 (1994). Publisher's VersionAbstract

The semiclassical techniques developed in the previous paper are applied to the understanding of the hierarchical structure underlying the spectra. This organization, as analyzed by Davis with statistical models, is revealed by continuously changing the energy resolution of the spectra and noting the branching pattern of the peaks. We argue that the greater part of this hierarchical organization can be understood with classical events in the time domain.

Sepúlveda, M.A. & Heller, E.J. Semiclassical calculation and analysis of dynamical systems with mixed phase space. The Journal of chemical physics 101, 8004–8015 (1994). Publisher's VersionAbstract

A semiclassical method for the propagation of arbitrary wave packets in a multidimensional Hamiltonian is presented. The method is shown to be valid for treating Hamiltonian systems whose classical phase space is a combination of chaotic and quasiperiodic motion (mixed dynamics). The propagation can be carried out long enough for the nonlinearities of the system to be important. The nonlinear dynamics is reflected in spectra and correlation functions. We suggest this new semiclassical method can be a tool for analyzing the nonlinear aspects of the vibrational spectra.

1993
Tomsovic, S. & Heller, E.J. Long-time semiclassical dynamics of chaos: The stadium billiard. Physical Review E 47, 282 (1993). Publisher's VersionAbstract

In a recent Letter [Phys. Rev. Lett. 67, 664 (1991)] we found semiclassical propagation to be remarkably accurate in the chaotic stadium billiard long after classical fine structure had developed on a scale much smaller than ħ. We give a complete account of that work and derive an approximate time scale for the validity of the semiclassical approximation as a function of ħ.

Tomsovic, S. & Heller, E.J. Semiclassical construction of chaotic eigenstates. Physical review letters 70, 1405 (1993). Publisher's VersionAbstract

It has been a long-standing problem to understand the eigenfunctions of a system whose classical analog is strongly chaotic. We show that in some cases the eigenfunctions can be constructed by purely semiclassical calculations.

1992
O'Connor, P.W., Tomsovic, S. & Heller, E.J. Accuracy of semiclassical dynamics in the presence of chaos. Journal of statistical physics 68, 131–152 (1992). Publisher's VersionAbstract

We review some of the issues facing semiclassical methods in classically chaotic systems, then demonstrate the long-time accuracy of semiclassical propagation of a nonstationary wave packet using the quantum baker's map of Balazs and Voros. We show why some of the standard arguments against the efficacy of semiclassical dynamics for long-time chaotic motion are incorrect.

O'Connor, P.W., Tomsovic, S. & Heller, E.J. Semiclassical dynamics in the strongly chaotic regime: breaking the log time barrier. Physica D: Nonlinear Phenomena 55, 340–357 (1992). Publisher's VersionAbstract

We investigate the behavior of the quantum baker's transformation, a system whose classical analogue is completely chaotic, for time scales where the classical mechanics generates phase space structures on a scale smaller than Planck's constant (i.e., past the log time t ≈ ln ħ-1). Surprisingly, we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime.

Heller, E.J. & Tomsovic, S. Time domain approach to semiclassical dynamics: Breaking the log time barrier. Chaos 2, 105 (1992). Publisher's VersionAbstract
Outlines two methods that have been developed for performing the evaluation and demonstrate the quantitative agreement between the quantum and semiclassical dynamics for systems with nonintegrable underlying dynamics. Method for calculating the semiclassical and semiclassical propagation of initial states and correlation functions for nonlinear and chaotic dynamics.
1991
Heller, E.J. Wavepacket dynamics and quantum chaology. Session LII-Chaos and quantum physics (Les Houches, France, 1989) (1991).
Heller, E.J. Cellular dynamics: A new semiclassical approach to time-dependent quantum mechanics. The Journal of chemical physics 94, 2723–2729 (1991). Publisher's VersionAbstract

A new semiclassical approach that constructs the full semiclassical Green’s function propagation of any initial wave function directly from an ensemble of real trajectories, without root searching, is presented. Each trajectory controls a cell of initial conditions in phase space, but the cell area is not constrained by Planck’s constant. The method is shown to be accurate for rather long times in anharmonic oscillators, indicating the semiclassical time‐dependent Green’s function is clearly worthy of more study. The evolution of wave functions in anharmonic potentials is examined and a spectrum from the semiclassical correlation function is calculated, comparing with exact fast Fourier transform results.

Tomsovic, S. & Heller, E.J. Semiclassical dynamics of chaotic motion: unexpected long-time accuracy. Physical review letters 67, 664 (1991). Publisher's VersionAbstract

Chaos introduces essential complications into semiclassical mechanics and the conventional wisdom maintains that the semiclassical time-dependent Green’s function fails to describe the quantum dynamics once the underlying chaos has had time to develop much finer structure than a quantum cell (h). We develop a method to evaluate the semiclassical approximation and test it for the first time under these circumstances. The comparison of the quantum and semiclassical dynamics of the stadium billiard shows remarkable agreement despite the very intricate underlying classical dynamics.

Gauss, J. & Heller, E.J. A time-dependent semiclassical wavepacket method using a fast Fourier transform (FFT) algorithm. Computer Physics Communications 63, 375–388 (1991). Publisher's VersionAbstract

A new semiclassical propagator based on a local expansion of the potential up the wavepacket is proposed. Formulas for the propagator are derived and the implementation using grid and fast Fourier transform (FFT) methods is discussed. The semiclassical propagator can be improved up to the exact quantum mechanical limit by including anharmonic corrections using a split operator approach. Preliminary applications to the CH3I photodissociation problem show the applicability and accuracy of the proposed method.

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