John Delos' Research Program

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John Delos' Research Program

Our research involves nonlinear dynamics applied to atomic and molecular physics, and to a problem in medical physics.

Atomic and Molecular Systems

We study the structure and dynamics of high-energy states of atoms and molecules, especially to predict and interpret the results of measurements. We use modern developments in classical and semiclassical mechanics to construct quantum wave functions and other properties of microscopic systems, and we use quantum interference phenomena to uncover newly discovered aspects of classical mechanics. The construction of quantum wave functions from classical trajectories provides an intuitive picture and a depth of insight that cannot be obtained in other ways.

Fractals in Chaotic Escape of Electrons from Atoms

Chaotic transport, and the escape of trajectories from defined regions of phase space, has been an important topic in dynamics for many years, because it describes phenomena that occur in many branches of physics. For example, some meteorites that fell on Antarctica are believed to have come from Mars; how they escaped from Mars’ gravitational field is a problem in the theory of chaotic transport. The mixing of fluids is another such problem. At a smaller scale, one of the important topics in nanophysics is ballistic transport of electrons through a small junction: electrons enter a junction from one lead, bounce around within the junction following either regular or chaotic paths, and eventually find their way to an exit lead. A closely related problem is chaotic propagation of light rays in a distorted cylindrical glass bead. At the molecular level, we may think about the breakup of a temporarily bound complex, such as a He atom weakly bound to an I2 molecule.

At the atomic level, the ionization of an excited hydrogen atom in applied electric and magnetic fields is an ideal candidate for the laboratory study of chaotic transport. We predict that a hydrogen atom in parallel electric and magnetic fields, excited by a short laser pulse to an energy above a saddle in the potential energy, ionizes via a train of electron pulses. These pulses are a consequence of classical chaos induced by the magnetic field. We have connected the structure of this pulse train (e.g., pulse size and spacing) to fractal structure in the classical dynamics. This fractal displays a weak self-similarity, which we call ‘‘epistrophic self-similarity.’’ We have shown how this self-similarity is reflected in the pulse train.

Our group is continuing such calculations, seeking systems that can display the fractal structure.

FIGURE: PREDICTED PULSE TRAIN OF ELECTRONS ESCAPING FROM HYDROGEN

(a) For hydrogen in parallel fields, with B = 0.49 T, F = 19 V/cm and effective quantum number N = 80, we show the flux of electrons striking a detector as a function of time T after the excitation of the atom. The thin dark line models the outgoing electron wave packet by an ensemble of trajectories with precise energy and start time. The thick shaded line uses an ensemble given by a minimum uncertainty Gaussian wave packet with central energy corresponding to. One scaled unit of time T is equal to 52 ps. We used a wave packet of width T = 0.1 scaled units (5.2 ps).

(b) The time it takes a trajectory to strike the detector is plotted as a function of the initial angle θ at which the trajectory is launched. Rotating the picture on its side, we see a collection of “icicles”. These icicles have structure within structure – converging on the edge of every icicle is an infinite sequence of other icicles. We call each such sequence an “epistrophe”. For each icicle in (b) there corresponds a pulse in (a). The dashed lines connect some of the icicles to their corresponding pulses.

(c) The icicle graph is “rectified” by using a surface of section. Here the number of iterates required to escape is plotted as a function of θ. Each escape segment in (c) corresponds to an icicle in (b).

(d) The epistrophic structure of the first several escape segments is shown. The dashed lines connect segments within one epistrophe. The solid arrows show the creation of new epistrophes according to an epistrophe start rule. The asterisk denotes a segment which does not fit into the pattern of epistrophes.

References:

Chaos 13, 880, 892 (2003); Phys. Rev. Lett. 92, 073001 (2004); Phys. Rev. A 70, 043407 (2004); Few-body systems 38, 181 (2006); Physica D 170 (2006); Phys. Rev. E 73, 66226 (2006).

Interference in Escaping Orbits

Our group pioneered the Closed Orbit Theory of atomic spectra. (This is a variant of periodic orbit theory of the density of states.) When the spectrum of highly excited states of an atom is studied at low resolution, one finds that every closed orbit produces an interference structure that gives oscillations in the absorption rate as a function of energy.

Do comparable interferences occur in escaping orbits? Yes: if an electron can follow two or more paths from an atom to a detector, then waves travelling along these paths will interfere, and a pattern of rings will appear on a detector. This phenomenon was first predicted many years ago by Fabrikant and Ostrovsky and Kondratovich; it is now being observed by groups led by Blondel and by Bordas.

If photodetachment of an electron from a negative ion is observed in an electric field, then a pretty pattern is found in the classical trajectories, in the wave functions, and in the current observed on a detector. What happens for ionization in parallel fields? We already know that the family of trajectories can show a fractal pattern. What will happen to the interference structure?

FIGURE: TRAJECTORIES AND WAVE FUNCTIONS FOR AN ELECTRON EMITTED FROM A POINT SOURCE IN PARALLEL ELECTRIC AND MAGNETIC FIELDS

Motion of electrons emitted from a fixed energy, isotropic point source at the origin into parallel fields. (Parameters used: E = 10-4eV, F = 15V/m, B = 0.02T. All dimensions are in microns.) The classical trajectory field (top left panel, blue curves) traces out an intersecting set of caustic surfaces (red curves) that separate regions with different numbers of classical paths connecting the source to a given destination point. The maximum number (here, eight) occurs in the diamond-shaped entities along the symmetry axis. The trajectory field contains two closed orbits that return to the source: a “snake” orbit (green), and a “balloon” orbit (purple). --- The center panel depicts the density profile of the electrons in primitive semiclassical approximation (blue plot), superimposed onto the caustic surfaces (red) and the closed orbits. The caustics clearly delineate the features of the density distribution, and the interference pattern shows approximate symmetry with respect to the closed “snake” orbit. --- The quantum density profile is displayed in the top right panel. Its agreement with the semiclassical result (center panel) is remarkable. --- Bottom row: Profiles of the radially integrated electronic density distribution. The green curve in the left panel denotes the purely classical cross section. It is strongly modified by semiclassical interference (center panel). Both classical and semiclassical density profiles diverge at the positions of the caustics (marked by red arrows). Otherwise, the semiclassical density profile quantitatively matches the quantum result (blue curve, right panel).

References: Physics letters. A. 347, 62 (2005); Phys. Rev. Letters. 96,100404 (2006); Phys. Rev. A 73, 62114R (2006). [Also Phys. Rev. A 47, 3020 3036(1993); Phys. Rev. A 38, 5609 (1988).]

Hamiltonian Monodromy and its Consequences

Here is another recently-discovered phenomenon in classical mechanics that produces interesting consequences in quantum systems. Below is a set of links to a recent paper and associated movies.

Paper: Static and dynamical manifestions of Hamiltonian monodromy

(The movies work on my computers if I download and save them first, then play them, for example with Windows Media Player. My computers will not open them directly.)

Additional References: Phys. Rev. A 76, 013404 (2007); 77, 043422 2008

The Pleasure and Utility of Semiclassical Methods

Quantum mechanics is accepted as the correct description of nature at the atomic level, but classical and semiclassical methods not only give great insight, they also sometimes provide the easiest method for calculation of the behavior of atomic and molecular systems. The revival of semiclassical methods in recent years has even attracted the attention of historians of science. Wise and Brock connect recent research in semiclassical theory with the early debates between “intuitionists” and “mathematicalidealists.” “[Bohr’s] insistence on the priority of traditional mechanical categories . . . meant that classical forms of perception supplied an indispensable guide to knowledge of quantum states . . . .” In contrast, “Pauli, Jordan and even Heisenberg . . .wanted to jettison classical analogies as a crippling nostalgia for a lost material-mechanical reality. . . . [They] sought the essence of physical reality in the bare mathematical forms” of quantum theory. Recent developments in semiclassical theory do not challenge the standard formulation and interpretation of quantum theory, but they allow us to “get our physical intuition back.” The construction of quantum wave-functions from classical trajectories provides an intuitive picture and a depth of insight that cannot be obtained in other ways.

Furthermore, the ideas, the methods, and the results of these theories are directly connected to technological innovation. For more than a decade, a number of groups have seen applications in technology, and have filed patents and sought industrial partners. The group led by Noordam at FOM realized that they could detect the time-spectrum of electrons ionized from an atom in an electric field by using a streak camera. Then they realized that they could invent a better streak camera by using these ionized electrons to measure short pulses of infrared light.

Microjunctions with sizes of a few hundred nanometers can be fabricated today. When such microjunctions are cooled to millikelvin temperatures, quantum interference effects become important. The theory of these phenomena is closely related to Closed-Orbit Theory: interferences occur between waves traveling on various paths from entrance to exit of the junction. Cold microjunctions with such quantum effects might be useful as elements in a quantum computer.

One can use similar ideas to describe light bouncing around inside a small bead. Stone and his collaborators at Yale filed a patent for the use of such beads as optical resonators, which could be useful in the communications industry. Ideas associated with “quantum chaos” were an integral part of the research leading to this filing.

References: G.M.Lankhuijzen and L.D.Noordam, Phys.Rev.Lett. 76, 1784 (1996); E.J. Heller, Phys. Rev. Lett. 53, 1515 (1984); C.D. Schwieters and J.B.Delos, Phys. Rev. A 51, 1023, 1030 (1995); J.U. Nockel and A.D. Stone, Nature 385, 45 (1997).

The Dynamics of the Pacing System of the Heart

The pacing system of the heart is complex; a healthy heart is constantly integrating and responding to signals from the autonomic nervous and endocrine systems. This complexity is manifest in the heart rate (HR) in the form of high variability and perhaps even multifractality. It is impossible at present to account for all of the factors and feedback loops affecting the heart rate under normal conditions. However, in controlled laboratory situations, and in some pathological or age-related states, one finds that the dynamics can show reduced complexity, yielding behavior more readily described and modeled. Such events of reduced dynamical complexity may provide an early, noninvasive warning of illness, and may also provide an opportunity to understand elements of the pacing system of the human heart.

This project is being carried out together with Randall Moorman, MD and Abigail Flower, PhD student, biophysics, University of Virginia.

Background

For many years physicians have studied electrocardiograms, and used them to evaluate health or illness of the heart. Can these signals give a noninvasive monitor of other aspects of health or illness? Several years ago, Dr. Moorman and his colleagues began collecting data on heart rates of infants in the neonatal intensive care unit at University of Virginia. In general the infants have healthy hearts, but they are vulnerable to a variety of bacterial, viral or fungal infections (collectively called sepsis). The diagnosis of neonatal sepsis is difficult and time-consuming, so it would be good to have a continuous noninvasive monitor.

The graph below shows time between beats vs. beat number for two babies: one has a very high variability in its heart rate – the graph looks like that of the stock market – while the other has a heart that is ticking like a clock, with scarcely any variation in its rate. Which is the healthy baby?

In health the heart rate (or the time between beats) has high variability; Moorman and colleagues have shown the reduced variability is one warning sign of possible sepsis. They also had indirect evidence that “decelerations” provided a second warning sign. A deceleration is a transient slowing of the heart followed by a return to normal rhythm.

FIGURE: HEART RATES OF TWO INFANTS

FIGURE: FOUR SAMPLES OF HEART RATE RECORDS OF INFANTS

The figure above shows examples of approximately four minutes of continuous inter-beat intervals from four different NICU patients.

(a) Inter-beat interval series for a healthy NICU patient,

(b) Inter-beat interval series for a NICU patient showing reduced heart rate variability prior to diagnosis of sepsis,

(d) Inter-beat interval series showing decelerations near sepsis. Each peaked structure in (c) and (d) is termed a “deceleration.” (d) shows striking periodicity (period ~40 beats), which lasted for almost two days.

Questions

Can one develop an algorithm for detection of decelerations?

Are decelerations truly correlated with sepsis?

What other information could be used as a measure of complexity of the signal?

Can other measures provide a noninvasive monitor of health of NICU infants?

Is there a model that could account for periodic decelerations?

Future Work

The heart rate is controlled locally by a collection of cells called the sino-atrial node (SAN), which constitute the natural pacemaker of the heart. The SAN is governed internally by two coupled clocks, a transmembrane oscillator, and an internal calcium oscillator. A dynamical analysis of this pair of clocks is needed.

The SAN is also governed by a complex set of feedback loops, including, among other things, blood-pressure, O2 and CO2 detectors. Certain models of these feedback loops suggest the presence of bifurcations. Could such bifurcations change the behavior of the heart from steady pulsing to periodic oscillations?