Thermalisation and temporal relaxation in closed quantum systems
Author(s)
Genway, Sam
Type
Thesis
Abstract
This thesis approaches questions concerning the thermalisation of subsystems of
closed quantum systems, prepared in pure states of definite energy but far from
equilibrium, under exact unitary evolution. Taking motivation from experiments in
the field of ultracold atoms, an extensive study of relaxation to a thermal state in
the Hubbard model is presented. The study of small local subsystems in Hubbard-model
lattice clusters has led to some interesting findings. Explored are the effects
of interactions between fermions, the initial-state energy and the energy uncertainty
in the initial state and their effects on relaxation dynamics and thermalisation.
The most significant finding is that while subsystem thermalisation is seen for a
large range of subsystem-bath coupling strengths, the temporal form of the relaxation
varies markedly from exponential decay for weak couplings with a crossover
to Gaussian behaviour with increased coupling strength. This is found to hold more
generally for random couplings between the subsystem and bath and for bosons as
well as fermions, thus demonstrating generality. As well as being demonstrated numerically,
this behaviour is derived for a generic class of bi-partite quantum systems
which may be described with the use of random matrices. A Brownian motion model
is employed to show the exponential to Gaussian crossover when the subsystem-bath
coupling matrix takes a banded form. This result agrees well with numerical
Hubbard-model results, and yields identical results at short times to those from
straight-forward perturbative methods. It is demonstrated that the non-Markovian
Gaussian behaviour should also be observable in the limit of macroscopic baths.
closed quantum systems, prepared in pure states of definite energy but far from
equilibrium, under exact unitary evolution. Taking motivation from experiments in
the field of ultracold atoms, an extensive study of relaxation to a thermal state in
the Hubbard model is presented. The study of small local subsystems in Hubbard-model
lattice clusters has led to some interesting findings. Explored are the effects
of interactions between fermions, the initial-state energy and the energy uncertainty
in the initial state and their effects on relaxation dynamics and thermalisation.
The most significant finding is that while subsystem thermalisation is seen for a
large range of subsystem-bath coupling strengths, the temporal form of the relaxation
varies markedly from exponential decay for weak couplings with a crossover
to Gaussian behaviour with increased coupling strength. This is found to hold more
generally for random couplings between the subsystem and bath and for bosons as
well as fermions, thus demonstrating generality. As well as being demonstrated numerically,
this behaviour is derived for a generic class of bi-partite quantum systems
which may be described with the use of random matrices. A Brownian motion model
is employed to show the exponential to Gaussian crossover when the subsystem-bath
coupling matrix takes a banded form. This result agrees well with numerical
Hubbard-model results, and yields identical results at short times to those from
straight-forward perturbative methods. It is demonstrated that the non-Markovian
Gaussian behaviour should also be observable in the limit of macroscopic baths.
Date Issued
2011-06
Date Awarded
2011-12
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Ho, Andrew
Lee, Derek
Creator
Genway, Sam
Publisher Department
Physics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)