How Wigner Functions Transform Under Symplectic Maps

It is shown that, while Wigner and Liouville functions transform in an identical way under linear symplectic maps, in general they do not transform identically for nonlinear symplectic maps. Instead there are ``quantum corrections'' whose hbar tending to zero limit may be very complicated. Examples of the behavior of Wigner functions in this limit are given in order to examine to what extent the corresponding Liouville densities are recovered.Comment: 8 pages, 6 figures [RevTeX/epsfig, macro included]. To appear in Proceedings of the Advanced Beam Dynamics Workshop on Quantum Aspects of Beam Physics (Monterey, CA 1998

Accurate Transfer Maps for Realistic Beamline Elements: Part I, Straight Elements

The behavior of orbits in charged-particle beam transport systems, including both linear and circular accelerators as well as final focus sections and spectrometers, can depend sensitively on nonlinear fringe-field and high-order-multipole effects in the various beam-line elements. The inclusion of these effects requires a detailed and realistic model of the interior and fringe fields, including their high spatial derivatives. A collection of surface fitting methods has been developed for extracting this information accurately from 3-dimensional field data on a grid, as provided by various 3-dimensional finite-element field codes. Based on these realistic field models, Lie or other methods may be used to compute accurate design orbits and accurate transfer maps about these orbits. Part I of this work presents a treatment of straight-axis magnetic elements, while Part II will treat bending dipoles with large sagitta. An exactly-soluble but numerically challenging model field is used to provide a rigorous collection of performance benchmarks.Comment: Accepted to PRST-AB. Changes: minor figure modifications, reference added, typos corrected

A Lie connection between Hamiltonian and Lagrangian optics

It is shown that there is a non-Hamiltonian vector field that provides a Lie algebraic connection between Hamiltonian and Lagrangian optics. With the aid of this connection, geometrical optics can be formulated in such a way that all aberrations are attributed to ray transformations occurring only at lens surfaces. That is, in this formulation there are no aberrations arising from simple transit in a uniform medium. The price to be paid for this formulation is that the Lie algebra of Hamiltonian vector fields must be enlarged to include certain non-Hamiltonian vector fields. It is shown that three such vector fields are required at the level of third-order aberrations, and sufficient machinery is developed to generalize these results to higher order

ADVANCED METHODS FOR THE COMPUTATION OF PARTICLE BEAM TRANSPORT AND THE COMPUTATION OF ELECTROMAGNETIC FIELDS AND MULTIPARTICLE PHENOMENA

Since 1980, under the grant DEFG02-96ER40949, the Department of Energy has supported the educational and research work of the University of Maryland Dynamical Systems and Accelerator Theory (DSAT) Group. The primary focus of this educational/research group has been on the computation and analysis of charged-particle beam transport using Lie algebraic methods, and on advanced methods for the computation of electromagnetic fields and multiparticle phenomena. This Final Report summarizes the accomplishments of the DSAT Group from its inception in 1980 through its end in 2011

ADVANCED METHODS FOR THE COMPUTATION OF PARTICLE BEAM TRANSPORT AND THE COMPUTATION OF ELECTROMAGNETIC FIELDS AND MULTIPARTICLE PHENOMENA

Since 1980, under the grant DEFG02-96ER40949, the Department of Energy has supported the educational and research work of the University of Maryland Dynamical Systems and Accelerator Theory (DSAT) Group. The primary focus of this educational/research group has been on the computation and analysis of charged-particle beam transport using Lie algebraic methods, and on advanced methods for the computation of electromagnetic fields and multiparticle phenomena. This Final Report summarizes the accomplishments of the DSAT Group from its inception in 1980 through its end in 2011

Advanced methods for the computation of particle beam transport and the computation of electromagnetic fields and beam-cavity interactions

The University of Maryland Dynamical Systems and Accelerator Theory Group carries out research in two broad areas: the computation of charged particle beam transport using Lie algebraic methods and advanced methods for the computation of electromagnetic fields and beam-cavity interactions. Important improvements in the state of the art are believed to be possible in both of these areas. In addition, applications of these methods are made to problems of current interest in accelerator physics including the theoretical performance of present and proposed high energy machines. The Lie algebraic method of computing and analyzing beam transport handles both linear and nonlinear beam elements. Tests show this method to be superior to the earlier matrix or numerical integration methods. It has wide application to many areas including accelerator physics, intense particle beams, ion microprobes, high resolution electron microscopy, and light optics. With regard to the area of electromagnetic fields and beam cavity interactions, work is carried out on the theory of beam breakup in single pulses. Work is also done on the analysis of the high behavior of longitudinal and transverse coupling impendances, including the examination of methods which may be used to measure these impedances. Finally, work is performed on the electromagnetic analysis of coupled cavities and on the coupling of cavities to waveguides

The Moyal-Lie Theory of Phase Space Quantum Mechanics

A Lie algebraic approach to the unitary transformations in Weyl quantization is discussed. This approach, being formally equivalent to the $\star$-quantization, is an extension of the classical Poisson-Lie formalism which can be used as an efficient tool in the quantum phase space transformation theory.Comment: 15 pages, no figures, to appear in J. Phys. A (2001

Quantum logic gates for coupled superconducting phase qubits

Based on a quantum analysis of two capacitively coupled current-biased Josephson junctions, we propose two fundamental two-qubit quantum logic gates. Each of these gates, when supplemented by single-qubit operations, is sufficient for universal quantum computation. Numerical solutions of the time-dependent Schroedinger equation demonstrate that these operations can be performed with good fidelity.Comment: 4 pages, 5 figures, revised for publicatio

Multilevel effects in the Rabi oscillations of a Josephson phase qubit

We present Rabi oscillation measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase qubit with a 100 um^2 area junction acquired over a range of microwave drive power and frequency detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the qubit dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi oscillations were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed oscillation frequencies with a simplified model constructed from the full phase qubit Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi frequency, and leakage to the higher excited states.Comment: 14 pages, 9 figures; minor corrections, updated reference

Advanced Computing for 21st Century Accelerator Science and Technology

Dr. Dragt of the University of Maryland is one of the Institutional Principal Investigators for the SciDAC Accelerator Modeling Project Advanced Computing for 21st Century Accelerator Science and Technology whose principal investigators are Dr. Kwok Ko (Stanford Linear Accelerator Center) and Dr. Robert Ryne (Lawrence Berkeley National Laboratory). This report covers the activities of Dr. Dragt while at Berkeley during spring 2002 and at Maryland during fall 2003