Theory of phase space transport and symbolic dynamics: This line of research is largely motivated by the question: How does one classify the tremendous complexity of behaviors seen in chaotic systems? For the simplest systems (e.g. mappings of the unit interval to itself) this question has been resolved by Milnor and Thurston. However, for most systems this remains an open question, and there exists no general framework (in the sense of symbolic dynamics) capable of classifying the diversity of chaotic behavior exhibited by physical systems. Our central contribution to this field is the technique of “homotopic lobe dynamics” to describe phase space transport using the topology of stable and unstable manifolds. We have developed this into a nonlinear dynamics toolbox that can extract an accurate (symbolic dynamics) model for the structure of chaotic transport in Hamiltonian systems with two degrees of freedom. This model can be made arbitrarily precise, even for systems exhibiting a mixture of chaos and regularity. Though originally motivated by problems in atomic physics, our techniques can be applicable to a wide variety of applications, including microfluidic mixing, celestial dynamics, chemical reaction rates, and transport through microjunctions.
Symbolic dynamics for systems in higher dimensions. We have succeeded in extending our theory of homotopic lobe dynamics to maps in three dimensions, an important frontier in nonlinear dynamics. Three-dimensional maps have direct applications to problems in fluid mixing and plasma physics, and they serve as an important stepping stone to higher-dimensional Hamiltonian systems.
Applications of nonlinear dynamics to AMO physics: Nonlinear dynamics has historically played a fundamental role in explaining diverse and complex atomic processes, a role which in turn has stimulated numerous theoretical advances in classical and quantum chaos. This trend continues as advances in atomic and optical techniques provide an unprecedented level of control and precision for experimental studies of chaos in atomic systems. For example, the time evolution of highly localized initial states (e.g. ultrashort optical pulses, Rydberg wavepackets, and localized ensembles of ultracold atoms), can be measured as they evolve and disperse within a chaotic potential, thereby probing the detailed fractal structure in the chaotic phase space. We have exploited new advances in chaotic dynamics to motivate, guide, and interpret experiments capable of probing chaotic phase space with unprecedented resolution. These include the chaotic transport, ionization, and control of Rydberg wavepackets as well as the elucidation of novel chaotic pathways for the mixing and loss of ultracold atoms in optical traps.
Chaos-induced pulse trains in the ionization of hydrogen. These papers predict that the pulsed ionization of hydrogen in applied electric and magnetic fields would result in a series of electron pulses that would bear the fingerprint of the underlying fractal structure of the dynamics. This paper stimulated a full quantum analysis by Francis Robicheaux’s group at Auburn.
Fractal structures in the escape dynamics of trapped ultracold atoms. These papers propose that the loss of ultracold or Bose-condensed atomic ensembles from suitably tailored chaotic traps could serve as high precision laboratory models for the observation of fractal dynamical structures. The first paper grew out of a collaboration with Dan Steck, a cold-atom experimentalist at the University of Oregon. The second paper is coauthored with an applied math colleague, Boaz Ilan, at UC Merced. This paper studies the full quantum theory of Bose condensate evolution, using both the Schrödinger and Gross-Pitaevskii models. The most important observation of this paper is that attractive atom-atom interactions (a focusing nonlinearity) can significantly enhance the observed fractal resolution. In the masters work of a former student, we also explored chaotic soliton motion that can arise with long-range dipolar interactions.
The role of phase space turnstiles in the ionization of kicked Rydberg atoms. Refs. [5, 11, 13]: These papers formed the core of my student Korana Burke’s doctoral dissertation. We studied the ionization of a highly excited Rydberg atom subjected to a periodic sequence of external electric field pulses. Such systems are important laboratory and theoretical models of classical and quantum chaos. Furthermore, a good theoretical understanding of this chaos is important for controlling electronic states in atoms, as has been demonstrated in a series of remarkable experiments on the creation and control of electronic wavepackets in the group of Barry Dunning at Rice. In the first paper, we developed a theoretical analysis of the chaotic ionization of a hydrogen atom based on the observation that ionization is due to a turnstile in phase space. The geometry of the lobes associated with the turnstile provides a clear geometric picture of how the ionization process proceeds. For the second two papers, we collaborated with Barry Dunning. Dunning’s lab produced remarkably clear experimental confirmations of our predictions, resulting in a publication in Phys. Rev. Lett. and a follow up publication in Phys. Rev. A.
Chaotic escape from a vase-shaped cavity. Ref. [18]: This work proposed that the chaos-induced pulse trains could also be experimentally visible in the ray dynamics of cavities, e.g. in microwave or optical cavities. This work was further developed in the doctoral thesis of Jaison Novick, a student at William and Mary, and has led to a series of successful acoustics experiments by Len Keeler at the University of Minnesota, Morris.
Quantum pumping. Quantum pumping has been predicted to occur in solid-state devices for a long time, but has been difficult to achieve in practice. These articles resulted from a collaboration with researchers at the College of William and Mary and at Kutztown University of Pennsylvania on the dynamics of quantum pumping protocols in BECs, as an alternative to solid-state devices.
The geometry of front propagation in fluid flows. Refs. [1, 2, 4, 7, 10]: Many physical phenomena evolve via fronts, separating distinct phases that propagate through a flowing fluid medium. Examples of both fundamental and applied interest include chemical reactions in microfluidic devices, plankton blooms in ocean currents, waves within plasmas, epidemics in migrating populations, flame fronts in combustion, growth of the atmospheric ozone hole, and phase transitions in liquid crystals. Our group, in collaboration with the experimental lab of Tom Solomon, has made important advances in understanding such front propagation by establishing that invariant manifold theory, familiar in the analysis of passive advective mixing, is applicable to the analysis of front propagation in fluids. We have coined the term burning invariant manifolds, or BIMs, for these structures. The most important property of BIMs is that they form time-invariant, or time-periodic, one-way barriers to front propagation in fluid flows that are themselves either time-invariant or time-periodic. The unstable BIMs are attracting structures, in that an initial reaction stimulated near such a BIM ultimately creates a reaction front in the fluid that converges upon the BIM. In the most striking cases, this front persists for arbitrarily long times, forming a frozen or pinned front, whose profile follows the BIM.
A few specific highlights of our work on BIMs include the following. We have shown that BIMs lead to a theory of “turnstiles” for front propagation in time-periodic fluid flows in analogy with the well-known turnstiles of passive advection. We have also shown that unstable BIMs are the fundamental geometric objects defining frozen, or pinned, reaction fronts, in which no reacted fluid makes it past the BIM. Similarly, we have demonstrated that stable BIMs form basin boundaries, dividing fluid regions within which a given initial autocatalytic stimulation will give rise to a given frozen front. We have developed algorithms to determine all potential frozen fronts and their basins within a given disordered flow. In a manner similar to pinning, we have shown how BIMs give rise to mode-locking of reaction fronts to an external drive. Finally, perhaps the most significant result for applications to naturally occurring biological, geophysical, and even turbulent flows is the extension of BIM ideas to flows defined over a finite-time window, with no assumption of temporal independence or periodicity. Importantly, this work extends the concepts of Lagrangian Coherent Structures (LCSs), a prominent tool in the analysis of passive advection, to the front propagation context.
Doctoral research. My doctoral thesis concerned the geometric (or Berry) phase that results from coupling the rotational degrees of freedom of a freely rotating body to its internal, or shape, degrees of freedom. An important application of this theory is to molecular dynamics and spectroscopy, where strong coupling between the molecular vibrational and rotational degrees of freedom can have a strong impact on molecular spectra and the time evolution of molecular states. From the viewpoint of mathematical physics, this work involved extensive use of gauge theory (the theory of connections on fiber bundles), differential and Riemannian geometry, and group theory. In another project from my thesis, I derived the Hamiltonian obtained when restricting a quantum system to a lower-dimensional constraint manifold.