Reduced Order Models for Open Quantum Systems

• Main Author: Hopkins, Asa Sies
• Format: Others
• Published: 2009
• Online Access: https://thesis.library.caltech.edu/4599/1/00AsaHopkinsFullThesis.pdf; https://thesis.library.caltech.edu/4599/2/01IntroductionBackground.pdf; https://thesis.library.caltech.edu/4599/3/02ProjectionFilters.pdf; https://thesis.library.caltech.edu/4599/4/03ProperOrthogonalDecomposition.pdf; https://thesis.library.caltech.edu/4599/5/04LTSAManifolds.pdf; https://thesis.library.caltech.edu/4599/6/05ConclusionBibliography.pdf; Hopkins, Asa Sies (2009) Reduced Order Models for Open Quantum Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/KPS7-9C05. https://resolver.caltech.edu/CaltechETD:etd-11182008-113904 <https://resolver.caltech.edu/CaltechETD:etd-11182008-113904>

Many quantum mechanical systems require large (potentially infinite) numbers of variables to exactly describe their state. In this thesis, I examine two approaches to develop simple, approximate models for such systems, which capture their essential dynamics. I use two bistable regimes of the Jaynes-Cummings model of cavity quantum electrodynamics as example systems to evaluate the effectiveness of each approach. In the phase bistable regime (which occurs with large driving field, and which I study in an on-resonance "bad cavity" regime to make numerical simulations tractable), the cavity field switches between two states with identical amplitude but opposite phase. In the absorptive bistable regime (which I study with small driving field in an on-resonance "good cavity"'), two stable regions of state space differ in cavity field amplitude as well as their shape and qualitative behavior. After introducing these two regimes and their dynamics, I give a short introduction to projecting dynamical equations onto linear subspaces. Proper Orthogonal Decomposition (POD) allows the algorithmic construction of subspaces onto which the dynamics may be projected. I demonstrate that the application of POD to phase bistability results in effective approximate filters, while the asymmetry of the absorptive bistable case requires extensions to POD, developed in this thesis, to create a functional filter. Local Tangent Space Alignment is one of a class of unsupervised manifold learning algorithms which use the local geometry of high-dimensional data, such as quantum trajectories, to calculate the coordinates of that data on a low-dimensional manifold. I show how this algorithm functions, and characterize the manifolds that result from phase and absorptive bistability. I fit the 3-dimensional phase bistable manifold with a small set of system observables, and create a three-dimensional set of equations (similar to the semi-classical Maxwell-Bloch equations) which perform very well as a filter. Absorptive bistability again proves to be more complicated, but I am able to show that the underlying manifold is small, and make some progress on characterizing its relations with system observables.