Publications

2019
Keski-Rahkonen, J., Ruhanen, A., Heller, E.J. & Räsänen, E. Quantum Lissajous scars. Physical Review Letters (2019). Publisher's VersionAbstract
The enhancement of a quantum state in the vicinity of a classical periodic orbit, or a quantum scar, is a fundamental phenomenon connecting quantum and classical mechanics. Here we demonstrate that some of the eigenstates of the perturbed two-dimensional anisotropic (elliptic) harmonic oscillator are strongly scarred by the Lissajous orbits of the unperturbed classical counterpart. In particular, we show that the occurrence and geometry of these quantum Lissajous scars are connected to the anisotropy of the harmonic confinement, but unlike the classical Lissajous orbits the scars survive under a small perturbation of the potential. This Lissajous scarring is caused by the combined effect of the quantum (near) degeneracies in the unperturbed system and the localized character of the perturbation. Furthermore, we discuss experimental schemes to observe this perturbation-induced scarring.
Heller, E.J., Fleischmann, R. & Kramer, T. Branched Flow. arXiv 1910.07086 (2019). Publisher's VersionAbstract
In many physical situations involving diverse length scales, waves or rays representing them travel through media characterized by spatially smooth, random, modest refactive index variations. “Primary” diffraction (by individual sub-wavelength features) is absent. Eventually the weak refraction leads to imperfect focal “cusps”. Much later, a statistical regime characterized by momentum diffusion is manifested. An important intermediate regime is often overlooked, one that is diffusive only in an ensemble sense. Each realization of the ensemble possesses dramatic ray limit structure that guides the waves (in the same sense that ray optics is used to design lens systems). This structure is a universal phenomenon called branched flow. Many important phenomena develop in this intermediate regime. Here we give examples and some of the physics of this emerging field.
Schram, M.C. & Heller, E.J. Hitting a Ball on a Spring: A Simple Model for Understanding Decoherence with Wavefunctions. European Journal of Physics (2019). Publisher's VersionAbstract
The typical approach to studying decoherence begins by examining a combined system and interacting environment (termed bath), and then deriving a master equation to examine the behavior of the density matrix after tracing over the bath, typically by invoking the Markovian approximation. This approach is quite successful for a large variety of systems however is frequently a non-intuitive picture that can mask the fundamental behavior of the system. We examine here a simple model that captures the essence of decoherence and study it using the wavefunction picture, invoking the analytic concepts of the correspondence principle, the method of images, the phase jitter concept, and the scattering matrix formalism. We complement this with a numerical study using the split operator FFT method, the forced harmonic oscillator method, and the thawed Gaussian approximation.
Mohanty, V. & Heller, E.J. Lazy electrons in graphene. PNAS (2019). Publisher's VersionAbstract
Within a tight-binding approximation, we numerically determine the time evolution of graphene electronic states in the presence of classically vibrating nuclei. There is no reliance on the Born–Oppenheimer approximation within the p-orbital tight-binding basis, although our approximation is “atomically adiabatic”: the basis p-orbitals are taken to follow nuclear positions. Our calculations show that the strict adiabatic Born–Oppenheimer approximation fails badly. We find that a diabatic (lazy electrons responding weakly to nuclear distortions) Born–Oppenheimer model provides a much more accurate picture and suggests a generalized many-body Bloch orbital-nuclear basis set for describing electron–phonon interactions in graphene.
Tomiya, M., Sakamoto, S. & Heller, E.J. Periodic orbit scar in wavepacket propagation. International Journal of Modern Physics C 30, 4, 1950026 (2019). Publisher's VersionAbstract
This study analyzed the scar-like localization in the time-average of a time-evolving wavepacket on a desymmetrized stadium billiard. When a wavepacket is launched along the orbits, it emerges on classical unstable periodic orbits as a scar in stationary states. This localization along the periodic orbit is clarified through the semiclassical approximation. It essentially originates from the same mechanism of a scar in stationary states: piling up of the contribution from the classical actions of multiply repeated passes on a primitive periodic orbit. To achieve this, several states are required in the energy range determined by the initial wavepacket.
Heller, E.J. & Kim, D. Schrödinger Correspondence Applied to Crystals. J. Phys. Chem. A (2019). Publisher's VersionAbstract
In 1926, E. Schrödinger published a paper solving his new time dependent wave equation for a displaced ground state in a harmonic oscillator (now called a coherent state). He showed that the parameters describing the mean position and mean momentum of the wave packet obey the equations of motion of the classical oscillator while retaining its width. This was a qualitatively new kind of correspondence principle, differing from those leading up to quantum mechanics. Schrödinger surely knew that this correspondence would extend to an N-dimensional harmonic oscillator. This Schrödinger Correspondence Principle is an extremely intuitive and powerful way to approach many aspects of harmonic solids including anharmonic corrections.
2018
Soley, M.B. & Heller, E.J. Classical approach to collision complexes in ultracold chemical reactions. Phys. Rev. A 98, 052702 (2018). Publisher's VersionAbstract
Inspired by Wannier’s threshold law, we recognize that collision complex decay meets the requirements of quantum-classical correspondence in sufficiently exothermic ultracold reactions. We make use of this correspondence to elucidate the classical foundations of ultracold reactions and to help bring calculations currently beyond the capabilities of quantum mechanics within reach. A classical method with a simplified model of many-body interactions is provided for determination of the collision complex lifetime and demonstrated for a reduced-dimensional system, as preliminary to the calculation of collision complex lifetimes in the full-dimensional system.
Schram, M.C. & Heller, E.J. Approach to coherent interference fringes in helium-surface scattering. Phys. Rev. A 98, 022137 (2018). Publisher's VersionAbstract
The conventional notion of elastic, coherent atom-surface scattering originates from the scattering particles acting as a quantum mechanical matter wave, which coherently interfere to produce distinct Bragg peaks which persist at finite temperature. If we introduce inelastic scattering to this scenario, the result is that the surface particles become displaced by the scattering atoms, resulting in emission or absorption of phonons that shift the final energy and momentum of the scatterer. As the lowest lying phonons are gapless excitations, the ability to measure these phonons is very difficult and this difficulty is exacerbated by the roughly 1eV resolution found in high energy helium scattering experiments. Even though the surface has, in effect, measured the presence of the scatterer which decoheres the particle, we retain the diffraction spots which are referred to as coherent scattering. How do we reconcile these disparate viewpoints? We propose a new experiment to more precisely examine the question of coherence in atom-surface scattering. We begin with an initially coherent superposition of helium particles with equal probabilities of interacting with the surface or not interacting with the surface. The beams are directed so that after the scattering event, the atoms are recombined so that we can observe the resulting interference pattern. The degree to which phonons are excited in the lattice by the scattering process dictates the fringe contrast of the interference pattern of the resulting beams. We use semiclassical techniques to simulate and test the viability of this experiment, and show that for a wide range of conditions, despite the massive change in the momentum perpendicular to the surface, we can still expect to have coherent (in the superposition sense) scattering.
2017
Keski-Rahkonen, J., Luukko, P.J.J., Kaplan, L., Heller, E.J. & Räsänen, E. Controllable quantum scars in semiconductor quantum dots. Phys. Rev. B 96, 094204 (2017). Publisher's VersionAbstract
Quantum scars are enhancements of quantum probability density along classical periodic orbits. We study the recently discovered phenomenon of strong perturbation-induced quantum scarring in the two-dimensional harmonic oscillator exposed to a homogeneous magnetic field. We demonstrate that both the geometry and the orientation of the scars are fully controllable with a magnetic field and a focused perturbative potential, respectively. These properties may open a path into an experimental scheme to manipulate electric currents in nanostructures fabricated in a two-dimensional electron gas.
Yang, Y., Kolesov, G., Kocia, L. & Heller, E.J. Graphene Terahertz Absorption. Nano Letters 17, 10, 6077-6082 (2017). Publisher's VersionAbstract
The unique terahertz properties of graphene has been identified for novel optoelectronic applications. In a graphene sample with bias voltage added, there is an enhanced absorption in the far infrared region and a diminished absorption in the infrared region. The strength of enhancement(diminishment) increases with the gate voltage, and the enhancement compensates the diminishment. We find that it is the coherence length of electrons in graphene that allows pure electronic transitions between states differing by small momentums and makes intraband transition possible, is responsible for the far infrared enhancement. Phonon assisted processes are not necessary and would not in any case contribute to a sum rule. This naturally leads to results obeying the general sum-rule in optical absorptions. Our prediction of the strength of enhancement(diminishment) in terms of the bias agrees with experiments. This is the first direct calculation we are aware of, since the prior phonon assisted model for indirect transition should not obey a sum rule.
Yang, Y., Kolesov, G., Kocia, L. & Heller, E.J. Reassessing Graphene Absorption and Emission Spectroscopy. arxiv arXiv:1704.07500 (2017). Publisher's VersionAbstract

We present a new paradigm for understanding optical absorption and hot electron dynamics experiments in graphene. Our analysis pivots on assigning proper importance to phonon assisted indirect processes and bleaching of direct processes. We show indirect processes figure in the excess absorption in the UV region. Experiments which were thought to indicate ultrafast relaxation of electrons and holes, reaching a thermal distribution from an extremely non-thermal one in under 5-10 fs, instead are explained by the nascent electron and hole distributions produced by indirect transitions. These need no relaxation or ad-hoc energy removal to agree with the observed emission spectra and fast pulsed absorption spectra. The fast emission following pulsed absorption is dominated by phonon assisted processes, which vastly outnumber direct ones and are always available, connecting any electron with any hole any time. Calculations are given, including explicitly calculating the magnitude of indirect processes, supporting these views.

Vardhan, S., Tomasi, G.D., Heyl, M., Heller, E.J. & Pollmann, F. Characterizing time-irreversibility in disordered fermionic systems by the effect of local perturbations. Physical Review Letters 119, 1, 016802 (2017). Publisher's VersionAbstract

We study the effects of local perturbations on the dynamics of disordered fermionic systems in order to characterize time-irreversibility. We focus on three different systems, the non-interacting Anderson and Aubry-Andr\'e-Harper (AAH-) models, and the interacting spinless disordered t-V chain. First, we consider the effect on the full many-body wave-functions by measuring the Loschmidt echo (LE). We show that in the extended/ergodic phase the LE decays exponentially fast with time, while in the localized phase the decay is algebraic. We demonstrate that the exponent of the decay of the LE in the localized phase diverges proportionally to the single-particle localization length as we approach the metal-insulator transition in the AAH model. Second, we probe different phases of disordered systems by studying the time expectation value of local observables evolved with two Hamiltonians that differ by a spatially local perturbation. Remarkably, we find that many-body localized systems could lose memory of the initial state in the long-time limit, in contrast to the non-interacting localized phase where some memory is always preserved.

2016
Luukko, P.J.J., et al. Strong quantum scarring by local impurities. Sci. Rep. 6, 37656 (2016). Publisher's VersionAbstract

We discover and characterize strong quantum scars, or eigenstates resembling classical periodic orbits, in two-dimensional quantum wells perturbed by local impurities. These scars are not explained by ordinary scar theory, which would require the existence of short, moderately unstable periodic orbits in the perturbed system. Instead, they are supported by classical resonances in the unperturbed system and the resulting quantum near-degeneracy. Even in the case of a large number of randomly scattered impurities, the scars prefer distinct orientations that extremize the overlap with the impurities. We demonstrate that these preferred orientations can be used for highly efficient transport of quantum wave packets across the perturbed potential landscape. Assisted by the scars, wave-packet recurrences are significantly stronger than in the unperturbed system. Together with the controllability of the preferred orientations, this property may be very useful for quantum transport applications.

Pittman, S.M., Tannenbaum, E. & Heller, E.J. Dynamical tunneling versus fast diffusion for a non-convex Hamiltonian. J. Chem. Phys. 145, 054303 (2016). Publisher's VersionAbstract

This paper attempts to resolve the issue of the nature of the 0.01-0.1 cm−1 peak splittings observed in high-resolution IR spectra of polyatomic molecules. One hypothesis is that these splittings are caused by dynamical tunneling, a quantum-mechanical phenomenon whereby energy flows between two disconnected regions of phase-space across dynamical barriers. However, a competing classical mechanism for energy flow is Arnol’d diffusion, which connects different regions of phase-space by a resonance network known as the Arnol’d web. The speed of diffusion is bounded by the Nekhoroshev theorem, which guarantees stability on exponentially long time scales if the Hamiltonian is steep. Here we consider a non-convex Hamiltonian that contains the characteristics of a molecular Hamiltonian, but does not satisfy the Nekhoroshev theorem. The diffusion along the Arnol’d web is expected to be fast for a non-convex Hamiltonian. While fast diffusion is an unlikely competitor for longtime energy flow in molecules, we show how dynamical tunneling dominates compared to fast diffusion in the nearly integrable regime for a non-convex Hamiltonian, as well as present a new kind of dynamical tunneling.

Heller, E.J., et al. Theory of Graphene Raman Scattering. ACS Nano 10, 2, 2803–2818 (2016). Publisher's VersionAbstract

Raman spectroscopy plays a key role in studies of graphene and related carbon systems. Graphene is perhaps the most promising material of recent times for many novel applications, including electronics. In this paper, the traditional and well established Kramers-Heisenberg-Dirac (KHD) Raman scattering theory (1925-1927) is extended to crystalline graphene for the first time. It demands different phonon production mechanisms and phonon energies than does the popular "double resonance" Raman scattering model. The latter has never been compared to KHD. Within KHD, phonons are produced instantly along with electrons and holes, in what we term an electron-hole-phonon triplet, which does not suffer Pauli blocking. A new mechanism for double phonon production we name "transition sliding" explains the brightness of the 2D mode and other overtones, as a result of linear (Dirac cone) electron dispersion. Direct evidence for sliding resides in hole doping experiments performed in 2011 \cite{chenCrommie}. Whole ranges of electronic transitions are permitted and may even constructively interfere for the same laser energy and phonon q, explaining the dispersion, bandwidth, and strength of many two phonon Raman bands. Graphene's entire Raman spectrum, including dispersive and fixed bands, missing bands not forbidden by symmetries, weak bands, overtone bands, Stokes anti-Stokes anomalies, individual bandwidths, trends with doping, and D-2D band spacing anomalies emerge naturally and directly in KHD theory.

2015
Baker, S.M. & Heller, E.J. The Degree of Ergodicity of Ortho-and Para-aminobenzonitrile in an Electric Field. The Journal of Physical Chemistry A (2015). Publisher's VersionAbstract

We study the dynamics of the two molecules ortho-aminobenzonitrile (OABN) and para-aminobenzonitrile (PABN). They are structural isomers, with differing asymmetries and dipole moments. In this paper, we show that the dynamics of the system strongly depends on the region of phase space of the initial rotational state, the asymmetry of the molecule, and on the direction of the dipole. We also show that the ergodicity of the system varies gradually with energy, except where the rotational energy of the initial state is much less than the Stark interaction. In this regime, the projection of the dipole along the lab-frame z-axis varies linearly with increasing energy and follows the microcanonical ergodic estimate. Both molecules are far from full chaos for total angular momentum quanta J ∈ [0,45]. However, the initial rotational states in OABN access much more of the available phase space than in PABN. We show that this is a likely cause for the experimental discrepancies in molecular beam deflection experiments.

Kocia, L. & Heller, E.J. Directed HK propagator. The Journal of Chemical Physics 143, 124102 (2015). Publisher's Version PDF
Heller, E.J., Yang, Y. & Kocia, L. Raman Scattering in Carbon Nanosystems: Solving Polyacetylene. ACS Central Science (2015). Publisher's VersionAbstract

Polyacetylene has been a paradigm conjugated organic conductor since well before other conjugated carbon systems such as nanotubes and graphene became front and center. It is widely acknowledged that Raman spectroscopy of these systems is extremely important to characterize them and understand their internal quantum behavior. Here we show, for the first time, what information the Raman spectrum of polyacetylene contains, by solving the 35-year-old mystery of its spectrum. Our methods have immediate and clear implications for other conjugated carbon systems. By relaxing the nearly universal approximation of ignoring the nuclear coordinate dependence of the transition moment (Condon approximation), we find the reasons for its unusual spectroscopic features. When the Kramers–Heisenberg–Dirac Raman scattering theory is fully applied, incorporating this nuclear coordinate dependence, and also the energy and momentum dependence of the electronic and phonon band structure, then unusual line shapes, growth, and dispersion of the bands are explained and very well matched by theory.

2014
Li, Z., Krems, R.V. & Heller, E.J. Collision dynamics of polyatomic molecules containing carbon rings at low temperatures. The Journal of chemical physics 141, 104317 (2014). Publisher's VersionAbstract

We explore the collision dynamics of complex hydrocarbon molecules (benzene, coronene, adamantane, and anthracene) containing carbon rings in a cold buffer gas of 3He. For benzene, we present a comparative analysis of the fully classical and fully quantum calculations of elastic and inelastic scattering cross sections at collision energies between 1 and 10 cm−1. The quantum calculations are performed using the time-independent coupled channel approach and the coupled-states approximation. We show that the coupled-states approximation is accurate at collision energies between 1 and 20 cm−1. For the classical dynamics calculations, we develop an approach exploiting the rigidity of the carbon rings and including low-energy vibrational modes without holonomic constraints. Our results illustrate the effect of the molecular shape and the vibrational degrees of freedom on the formation of long-lived resonance states that lead to low-temperature clustering.

Kocia, L. & Heller, E.J. Communication: HK propagator uniformized along a one-dimensional manifold in weakly anharmonic systems. The Journal of chemical physics 141, 181102 (2014). Publisher's Version PDF

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