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



Van{\'ıček, J.{\'ı & Heller, E.J. Uniform semiclassical wave function for coherent two-dimensional electron flow. Physical Review E 67, 016211 (2003). Publisher's VersionAbstract

We find a uniform semiclassical (SC) wave function describing coherent branched flow through a two-dimensional electron gas (2DEG), a phenomenon recently discovered by direct imaging of the current using scanned probed microscopy [M.A. Topinka, B.J. LeRoy, S.E.J. Shaw, E.J. Heller, R.M. Westervelt, K.D. Maranowski, and A.C. Gossard, Science 289, 2323 (2000)]. The formation of branches has been explained by classical arguments [M.A. Topinka, B.J. LeRoy, R.M. Westervelt, S.E.J. Shaw, R. Fleischmann, E.J. Heller, K.D. Maranowski, and A.C. Gossard, Nature (London) 410, 183 (2001)], but the SC simulations necessary to account for the coherence are made difficult by the proliferation of catastrophes in the phase space. In this paper, expansion in terms of “replacement manifolds” is used to find a uniform SC wave function for a cusp singularity. The method is then generalized and applied to calculate uniform wave functions for a quantum-map model of coherent flow through a 2DEG. Finally, the quantum-map approximation is dropped and the method is shown to work for a continuous-time model as well.

Shaw, S. & Heller, E.J. Branching and fringing in microstructure electron flow. International Journal of Modern Physics B 17, 3977-3987 (2003). Publisher's VersionAbstract


Recent experimental work in the Westervelt laboratory at Harvard has succeeded in directly imaging electron flow in two degree of freedom electron gasses formed in semiconductor microstructures. Here, we give a brief account of the unexpected high resolution of the resulting images, the surprising branching of the flow which was observed, and the survival of quantum fringing beyond where it was thought to have been obliterated by thermal effects.


Van Voorhis, T. & Heller, E.J. Nearly real trajectories in complex semiclassical dynamics. Physical Review A 66, 050501 (2002). Publisher's VersionAbstract

We introduce a very general approximation to the quantum propagator that is based on the assumption that the most important contributions to the complex semiclassical propagator evolve from real classical trajectories that almost satisfy the desired boundary conditions. Our results for two systems — the autocorrelation function for the quartic anharmonic oscillator and the photodissociation spectrum of CO2 — show that these nearly real contributions yield an excellent approximation to the quantum propagator for quite long times. The approach taken here is applicable to problems with many (e.g., several hundred) degrees of freedom, and hence promises to provide an accurate and useful representation of the quantum dynamics for a wide variety of physically interesting systems.

Tannenbaum, E., Higgins, K.J., Klemperer, W., Segev, B. & Heller, E.J. A perturbative approach to vibrational predissociation rates: Application to ArHF. The Journal of Physical Chemistry B 106, 8100–8107 (2002). Publisher's VersionAbstract

This paper presents a perturbative model for the vibrational predissociation dynamics of inert gas hydrogen halide (RgHX) complexes. The predissociation is modeled as a Fermi Golden Rule (FGR) process from a bound state residing on a two-dimensional potential energy surface (PES) obtained by averaging over the HX vibrational state of interest to a series of one-dimensional exit channels obtained by averaging over the HX rovibrational state of interest. This model is applied to ArHF, for which a high-quality ab initio interaction potential is available. In particular, we focus on the vv − 1 transition for the bound states (1000), (2000), (2110), (3000), and (3110). We confirm the experimental observation that the product HF tends to come off at the highest accessible j state, which is j = 13 for this system. This results from a strong angular anisotropy in the ArHF interaction potential that couples low-j and high-j HF states. The basic mechanism for this high-j preference is determined to be the suppression of the low-j exit channels arising from highly oscillatory low-j outgoing wave functions. We also observed that the tails of the bound-state wave functions, in the inner wall region of the interaction potential, gave the main contribution to the predissociation rate, indicating that the vibrational predissociation process is due to tunneling and is therefore a purely quantum effect. The calculations also confirm the strong v-dependence of the predissociation rates, as well as the stabilization of the complex that occurs when energy is placed into the HF bending mode. For the (1000) and (2110) states, we obtain rates well below 1600 s-1, which is consistent with the observation by Miller that the vibrational predissociation rates are too slow to be measured. The (2000) state does give a measurable rate, with a computed decay into the j = 13 exit channel of 14 000 s-1. The (3000) state gives a corresponding rate of 200 000 s-1, in good agreement with the overall dissociation rate of 250 000 s-1. The (3110) rate is slower, with a value of 12 000 s-1. Though this rate seems somewhat small, given that the lifetime of the (3110) state was measured to only be twice as long as that of the (3000) state, this rate and all our rates are within an order of magnitude of the measured rates.

Mody, A., Haggerty, M., Doyle, J.M. & Heller, E.J. No-sticking effect and quantum reflection in ultracold collisions. Physical Review B 64, 085418 (2001). Publisher's VersionAbstract

We provide a general and nonperturbative theoretical basis for quantum reflection of an ultracold atom incident on a cold or warm surface. Sticking is identified with the formation of a long-lived resonance, from which it emerges that the physical reason for not sticking is that the many internal degrees of freedom of the target serve to decohere the incident one body wave function, thereby upsetting the delicate interference process necessary to form a resonance. We then explore the transition to the post-threshold behavior, when sticking prevails at higher incident energies. Studying the WKB wave functions of the atom provides a quick understanding of our results even in the ultracold regime where WKB is not applicable. Explicit examples are examined in detail and we predict the temperatures required to reach the various regimes.

Cohen, D., Barnett, A. & Heller, E.J. Parametric evolution for a deformed cavity. Physical Review E 63, 046207 (2001). Publisher's VersionAbstract

We consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where (Q,P) describes a particle moving inside a cavity, and x controls a deformation of the boundary. The quantum eigenstates of the system are \|n(x)>. We describe how the parametric kernel P(n\|m)=\|<n(x)\|m(x0)>\|2, also known as the local density of states, evolves as a function of δx=x-x0. We illuminate the nonunitary nature of this parametric evolution, the emergence of nonperturbative features, the final nonuniversal saturation, and the limitations of random-wave considerations. The parametric evolution is demonstrated numerically for two distinct representative deformation processes.

Stöckmann, H.-J. & Heller, E.J. Quantum chaos: an introduction. Physics Today 54, 49 (2001). Publisher's Version
Heller, E.J. Quantum physics: Air juggling and other tricks. Nature 412, 33–34 (2001). Publisher's VersionAbstract

Quantum tunnelling breaks the rules of classical physics — and leads to ghost-like transfer of matter through barriers. Demonstrations of a new type of quantum tunnelling have the ghosts taking new liberties.

Barnett, A., Cohen, D. & Heller, E.J. Rate of energy absorption for a driven chaotic cavity. Journal of Physics A: Mathematical and General 34, 413 (2001). Publisher's VersionAbstract

We consider the response of a chaotic cavity in d dimensions to periodic driving. We are motivated by older studies of one-body dissipation in nuclei, and also by anticipated mesoscopic applications. For calculating the rate of energy absorption due to time-dependent deformation of the confining potential, we introduce an improved version of the wall formula. Our formulation takes into account that a special class of deformations causes no heating in the zero-frequency limit. We also derive a mesoscopic version of the Drude formula, and explain that it can be regarded as a special example of our calculations. Specifically we consider a quantum dot driven by an electro-motive force which is induced by a time-dependent homogeneous magnetic field.

Van{\'ıček, J.{\'ı & Heller, E.J. Replacement manifolds: A method to uniformize semiclassical wave functions. Physical Review E 64, 026215 (2001). Publisher's VersionAbstract

We present a semiclassical technique that relies on replacing complicated classical manifold structure with simpler manifolds, which are then evaluated by the usual semiclassical rules. Under circumstances where the original manifold structure gives poor or useless results semiclassically the replacement manifolds can yield remarkable accuracy. We give several working examples to illustrate the theory presented here.

Fiete, G.A., et al. Scattering theory of Kondo mirages and observation of single Kondo atom phase shift. Physical review letters 86, 2392 (2001).Abstract

We explain the origin of the Kondo mirage seen in recent quantum corral Scanning Tunneling Microscope (STM) experiments with a scattering theory of electrons on the surfaces of metals. Our theory combined with experimental data provides the first direct observation of a single Kondo atom phase shift. The Kondo mirage observed at the empty focus of an elliptical quantum corral is shown to arise from multiple electron bounces off the corral wall adatoms in a manner analagous to the formation of a real image in optics. We demonstrate our theory with direct quantitive comparision to experimental data. *This research was supported by the National Science Foundation under Grant No. CHE9610501 and by ITAMP.