OWU NSF REU/RET Program 2008: Project Descriptions

Project Descriptions

Coupled Nonlinear Systems: the Rich Physics of Josephson Junction Arrays and Micro- and Nanomechanical Oscillators: Dr. Brad R. Trees

This project studies the effect of coupling one or more current-biased Josephson junctions (JJs), which are basically superconducting tunnel junctions, to a nanomechanical oscillator. The focus is on two particular aspects of this coupling: the effect on synchronization of the voltage across two or more JJs, and the effect on tunneling and decoherence rates of a single JJ when biased so as to operate as a so-called phase qubit.

Systems of limit-cycle oscillators show a wealth of interesting behaviors. For example, synchronization due to coupling between all or a subset of the oscillators in a set has been experimentally observed in many systems in many scientific disciplines, e.g. physics, chemistry, and biology. On theoretical ground, the Kuramoto model (Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984)), describing the dynamics of N globally-coupled oscillators of phase φ_i , angular velocity ω_i, and coupling strength K:

has provided important insight into the nature of a synchronization transition itself. Thus, for many reasons, the study of synchronization has been of growing interest for at least a decade (A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, UK, 2001)). JJ arrays, which have been theoretically linked to the Kuramoto model in certain geometries (K. Wiesenfeld, P. Colet, and S. H. Strogatz, Phys. Rev. Lett. 76, 404 (1996)), are perhaps the quintessential nonlinear system for studying the causes and stability of synchronous behavior. Well-controlled, modern fabrication techniques allow the design of JJ arrays with precise geometries and junction parameters. The result is an exquisitely controlled “test bed” for the study of complex dynamical systems. This computational project involves the simulation of JJ arrays coupled both to each other as well as to external loads that could result in synchronous time-dependent behavior of the junctions. Specifically, the goal is to determine if coupling a nanomechanical oscillator to a JJ array can result in stable synchronization of the junctions in the array (see B. R. Trees, S. Natu, and D. Stroud, Phys. Rev. B 72, 214524 (2005)). Such coupled nanomechanical oscillator and JJ systems are currently of interest because of their potential use as quantum bits (see second aspect of project described below) and because of the expanding expertise in growing/fabricating high-Q mechanical oscillators on the microscopic scale (N. Cleland and M. R. Geller, Phys. Rev. Lett. 93, 070501 (2004)). This project is highly suitable to undergraduates because the model for a classical, current-biased JJ is equivalent to the equation of motion for a damped, driven, nonlinear pendulum. Such an analogy with classical mechanics means that students can quickly build up physical intuition about the behavior of single junctions. They can then focus on learning new physics, which is based on how multiple junctions behave when coupled.

When current biased appropriately, a single JJ behaves as a metastable quantum system called a phase qubit. Such phase qubits are under intense study as possible ingredients in quantum computers. The effects of a nanomechanical oscillator on the macroscopic quantum tunneling rate, the thermal hopping rate, and the decoherence rate in such a metastable system have not been fully explored but are important if the full potential of phase qubits is to be understood. This calculation would include the effect of damping on the JJ via the technique of Caldeira and Leggett (A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983)). This particular project would be best suited for a student with a strong math and physics background and would make heavy use of Mathematica^®.
Stellar Surface Imaging via Light-curve Inversion: Dr. Robert O. Harmon

Light-curve Inversion (LI) is a technique for mapping the surfaces of late-type stars from their rotational light curves. Even through the Hubble Space Telescope, stars appear to be featureless pinpoints. As a result, it is necessary to use indirect techniques in order to obtain information about their surface features. Of particular interest are “starspots,” which are analogous to sunspots on the Sun and are known to be present on certain classes of stars. Like sunspots, starspots are believed to be manifestations of stellar magnetic fields. The study of starspots can thus provide valuable insights into the physics of the magnetic dynamos operating in the Sun and other stars.

If there is a dark spot on the surface, then every time the star’s rotation carries the spot into view from Earth, there will be a dip in the star’s brightness. If we knew in detail the appearance of the star’s surface, a relatively straightforward calculation would allow us to predict the star’s brightness as a function of time, i.e., its light curve. With LI we attempt to go in the other direction: knowing the light curve, determine the appearance of the star’s surface. This is not a simple matter, because the problem is ill-posed, in that very different surfaces can give rise to nearly identical light curves. This arises because the effects of a large number of small bright and dark patches on the surface would nearly but not completely cancel, such that their presence would impart a low-amplitude, high-frequency “ripple” on the light curve as the star rotates. This ripple would look very similar to random noise, with the result that a straightforward attempt to find the surface which best replicates the observed light curve will produce a surface peppered with spurious bright and dark spots which are merely noise artifacts. LI circumvents this problem by constraining the solution so as to favor surfaces which are “smooth” and thus free of noise artifacts in a well-defined sense.

The student who works on this project will first search the literature to select a suitable spotted star for investigation. Following this, images of a star field surrounding the target star will be obtained using an SBIG ST-8E CCD camera and B, V, R and I photometric filters at OWU’s Perkins Observatory a few miles south of campus. Standard reductions (dark subtraction and flat fielding) will be performed on the images in order to reduce random noise and systematic errors. Then differential aperture photometry will be used to obtain the light curve (plot of intensity vs. time) of the target star as seen through each filter. Using multiple filters significantly improves the latitude resolution of the technique by taking advantage of the wavelength dependence of the limb darkening (center-to-edge dimming) of the stellar surface. The light curve data will then be analyzed via LI to produce maps of the stellar surface.

The algorithm used to perform LI is compute-intensive and thus benefits greatly from being run on a cluster. A mathematically inclined student with some background in programming could get involved in explicitly optimizing the program to take advantage of parallelization.
Shared Shorted Paths in Graphs: Dr. Sean McCulloch

A graph is a mathematical structure comprising of vertices and edges that connect the vertices together. Many graphs also define a cost for each edge, denoting the expense incurred by traversing the edge. Graphs can be used to represent many real-world structures, including computer networks, travel times and distances, connections of wires on a computer chip, and the search space for an artificially intelligent agent.

A common problem in the study of graphs is to find the path of least cost between any two vertices. This problem has a well-known solution. Our investigation expands upon this problem to consider the situation where we are given many “journeys” to connect on a single graph, and journeys that share an edge will split the cost of the edge between them. This situation is analogous to planning the location of wires in a network, and deciding how to split the cost of the network among the many connections that are made within the network structure.

Previous work in this area has developed an optimal solution for the case where there are only two journeys in the graph, as well as a set of heuristics to use to approximate the multiple-journey case. The proposed project will extend on this work in both the theoretical and applied areas.

In the theoretical areas, we will attempt to classify the difficulty of the multiple-journey problem using techniques from Game Theory, which is a field used to solve problems with multiple competing agents, each with their own strategies and goals. We will try to find algorithms to solve the general problem, as well as attempt to characterize types of graphs that are likely to have easily-found solutions or where heuristics could be helpful.

More practically, we will be developing and testing many sample graphs and heuristics to approach this problem. This will involve a large amount of computing time and resources, and will use our parallel cluster.
Nuclear Structure of Exotic Proton-Rich Nuclei: Dr. Robert Kaye

One method to explore nuclear structure involves stressing the nucleus near the limits of binding. Such systems offer unique opportunities to study the fundamental nucleon interaction because the interaction is known to be sensitive to the total number of nucleons. As part of the testing of nuclear models across the table of isotopes, the experimental study of heavy proton-rich systems is needed. Of particular interest are nuclei with masses between 60 and 100 nucleons since these are the heaviest for which protons and neutrons occupy the same energy levels. REU participants will have the opportunity to collaborate on experiments that will synthesize and study exotic (short-lived) proton-rich nuclei at the particle accelerator facilities located at Argonne National Laboratory, Florida State University, and/or Michigan State University. The resulting data will then be brought back to OWU for analysis using the methods of -ray spectroscopy. Particular emphasis will be placed on the determination of the evolution of shape and deformation with increasing angular momentum through mean lifetime measurements of the excited states in the nuclei of interest. Such studies will help to illuminate the interplay between collective and single-particle degrees of freedom as the nuclear system approaches the limit of binding in very proton-rich cases. The heavy proton-rich nuclei, especially the ones which have the same number of protons and neutrons, also provide an excellent laboratory for studying the possible effects of proton-neutron spin coupling, a phenomenon predicted by theory but which so far has proven elusive to determine experimentally.

Similar collaborative projects involving Dr. Kaye and his undergraduate research students at Purdue University Calumet have produced five publications in refereed journals and six presentations at conferences within the past five years (see Kaye biographical information for more detail). Some possible computation projects involved with the anticipated analyses include Monte Carlo simulations of nuclei slowing down and decaying in a target (for the determination of excited-state lifetimes), large-scale shell model calculations to compare theoretical excited-state energies with the measured ones, and total Routhian surface calculations used to predict nuclear shapes and deformations. Of particular general interest to all the REU participants will be the numerical integration techniques used in the Monte Carlo simulation.
Sampling Distribution of Regression Statistics with Data Subjected to Type II Censoring: Dr. Scott Linder

The sampling (probability) distributions for regression-associated statistics are well-known for simple cases – in particular, when observations are multivariate normal in distribution and when no data are censored. A sample of bivariate normal data has been subjected to Type II censoring when we may observe only pairs corresponding to predetermined order statistics associated with one of the variates. For example, 1000 students apply for admission to a college based on the scores to a standardized entrance exam. After one year of study, grade point averages and entrance exam scores are known only for the 200 students admitted. Censoring mechanisms impose probabilistic dependence on the observations, which often renders the sampling distributions of statistics mathematically intractable. Understanding these distributions is essential for basic inference. This project involves an investigation of these sampling distributions. Using intuition, simulation and analysis, we suggest approximations to these distributions. Using simulation, we investigate their goodness of fit. Based on adequate approximation, one can propose simple inferential methods, and simulation may again be used to investigate their statistical properties.
Comparison of Conditional and Unconditional Inferential Methods: Dr. Scott Linder

Commonly used inferential methods associated with simple linear regression are mathematically derived under the assumption of a fixed predictor variable and random response. Of course, in many applications both variates are random. Inference based on standard methods is, then, conditional – the inference is valid, given the observed predictor variates. Conditional inferential methods are often used simply because they are easier to apply, and because unconditional inferential methods are difficult to derive mathematically. In the case of simple linear regression, conditional and unconditional inferential methods generally vary little in outcome. In other cases, unconditional inference may yield a fundamentally different decision from conditional inference. Here, we consider the impact on decisions (power of a hypothesis test, width of a confidence interval, etc.) of using conditional inferential methods when suitable unconditional inferential methods are also available.
Coupled Mechanical Oscillators, Experiment and Computer Simulation: Dr. Barbara S. Andereck

The classical analog of coupled Josephson junctions is the coupling of macroscopic oscillators. Such coupling is described by the Kuramoto model, where synchronization of oscillators arises due to a mechanical coupling. (See Kuramoto equation in Brad Trees’ project description.) Coupling of two or more upright pendula (metronomes) was studied analytically and experimentally by James Pantaleone (J. Pantaleone, Am. J. Phys. 70 (2002) 992-1000). Our goal is ultimately to study the coupling of multiple (10-20) oscillators both experimentally and numerically in order to make connection between the physical parameters such as coupling and frequency and their mathematical counterparts in the Kuromoto model. The experimental results will provide a measure of the validity of the computer simulations which will enable more trustworthy extension of the simulations beyond the experimentally feasible realm.

The first step in understanding the coupling involves studying two coupled metronomes. The metronomes are place on a board that is suspended from the ceiling by strings. Metronomes of slightly different frequencies are started out of phase and as a result of the motion they produce in the board, settle into in-phase oscillations. The evolution of this phase shift can be predicted theoretically and is crucially dependent on the non-linear nature of the metronomes. By including vertical motion as well as the more obvious horizontal motion, we are able to very accurately match the observed relaxation to synchronization using the theoretical model.

The next step in this project, for summer 2008, will be increasing the number of metronomes in the system, both experimentally and theoretically. Much of the summer will probably be spent investigating the behavior of a three-metronome system. If the challenges of starting, monitoring, and modeling three metronomes can be addressed, then we can begin to increase the number of metronomes, to better simulate the Kuromoto model. Systems will be studied experimentally and theoretically to determine how starting angles, frequencies differences, and coupling strengths affect the time to synchronization.

Later studies may include the effect of noise or randomness (in position or natural frequency) in the path to synchronization, especially when systems of a dozen or more metronomes are feasible. Periodic boundary conditions might be possible if the metronomes are placed on a suspended circular platform.

Theoretical calculations will be done using Mathematica, initially, and extending to the computing cluster as needed. Numerical results for the same parameter ranges will be compared to the experimental behavior.

This project is particularly suited to participation by high school teachers as the concept of harmonic motion is central to physics and readily conceptualized by high school students. The equipment needed to demonstrate the coupling of the oscillators is minimal, so teachers could share their research visually with their classes. Mini-versions/extensions of the project could be done by students of the high school teacher in an honors physics course.