Next: Frontmatter

Dissipation in Deforming Chaotic Billiards

Alexander Harvey Barnett
October 2000

A thesis presented to the Department of Physics
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Physics

Harvard University
Advisor: Eric J. Heller

Frontmatter
- Abstract
- Acknowledgments
- Citations
- Dedication
Chapter 1: Introduction and summary
- Structure of this thesis
- Chapters 2,3 and 4: Dissipation rate and deformations of chaotic billiards
- Chapters 5 and 6: Improved billiard quantization methods
- Chapter 7: Quantum point contact conductance and scattering in the half-plane
- Chapter 8: Waveguides for neutral atoms using evanescent light fields
Chapter 2: Quantum and classical theories of ergodic dissipation
- Review of classical dissipation in a general system
- Review of the linear response theory of quantum dissipation
- Quantum-classical correspondence
Chapter 3: Dissipation rate in chaotic billiards, and `special' deformations
- The cavity system
  - Form of correlation spectrum and timescales
  - Conversion of time averages to averages over collision parameters
- The white noise approximation (WNA)
  - Relation to random wave approximation
- `Special' deformations
  - Band profile power laws
  - The quantum band profile case
- The WNA revisited--cavity shape effects
Chapter 4: Improving upon the white noise approximation: a new `wall formula'
- Decomposition of general deformations
- Addition of deformations: numerical tests
  - Symmetry effects
- Beyond the WNA
  - Numercial tests of the improved formula
  - Historical remarks and conclusion
Chapter 5: Improved sweep methods for billiard quantization
- Introduction and history of quantum eigenproblems
- Definition of the billiard problem
- Representation by a Helmholtz basis
- The choice of norm matrix
  - Estimation by interior points--relation to Heller's PWDM
  - Exact form on the boundary and Dirichlet approximation
- The hunt for local tension minima as a function of
- Conclusion and discussion
Chapter 6: The scaling method of Vergini and Saraceno
- The basic scaling method
- Higher-order correction and normalization
  - Correction of the eigenwavenumbers
  - `Automatic' normalisation of states
- Sources of error in the method
- Application: local density of states at finite deformations
  - Deformations and eigenstate computation
  - Computation of overlap matrix $P(n\vert m)$ and its profile
- Discussion
Chapter 7: Conductance of quantum point contacts and half-plane scattering
- Conductance in terms of cross section
- Partial-wave channel modes for a 2-terminal system
- Idealized `slit' aperture point contact
- Point contact coupled to a resonator
- What is the maximum conductance of a single quantum channel?
- Reciprocity and `conductance' of atom waves
- Conclusions
Chapter 8: Substrate-based atom waveguide using two-color evanescent light fields
- Introduction
- Trap Concept
- Trap properties
- Numerical solution of the light fields
- Further decoherence and loss mechanisms
  - Effects of surface interactions
  - Bending the waveguide
- Conclusion
Appendix A: General transformation of the 1D Fokker-Planck equation
Appendix B: Numerical evaluation of the classical band profile in billiards
- Trajectories--the issue of exponential divergence
- General system--windowed estimation of correlation spectrum
- Considerations in a billiard system
Appendix C: Numerical evaluation of quantum band profile in billiards
- Deformation matrix elements
- Smoothing
Appendix D: How Many Special Deformations Are There?
- Requirement on time-integral of ${\mathcal{F}}(t)$
- General potential case
- Scaling potentials and the billiard case
Appendix E: Cross correlations I: general-special
Appendix F: Cross correlations II: normal-general
Appendix G: Numerical evaluation of wavefunction boundary integrals
- Convergence with number of sample points--theory
- Convergence in practice
Appendix H: Boundary evaluation of matrix elements of Helmholtz solutions
- Overlaps: $\hat{O} = 1$
  - Differing wavenumbers $k_a \neq k_b$
  - Equal wavenumber
- Matrix trick for pushing integrals onto the boundary
  - Volume integrals of scalar quantities
  - Volume integrals of vector quantities
- Other matrix elements
Appendix I: Scaling expansion of eigenfunctions and tension matrix
- Expansion of the dilation operator
- Curvilinear boundary coordinates
- Applying boundary conditions and simplifying
- Tension matrix expansion
- Useful geometric boundary integrals
Appendix J: Helmholtz basis functions for two-dimensional billiards
- Basis `badness' $\gamma$
- Real plane waves (RPWs)
- Adding evanescent plane waves (EPWs)
- Symmetrization and reduction of effort
Appendix K: Transmission cross section in the narrow slit limit
Appendix L: Explicit relation of cross section to Landauer formula
Bibliography
About this document ...

Alex Barnett 2001-10-03