This thesis explores the effects singularities have on stationary and dynamical properties of many-body quantum systems. In papers I and II we find that the ground
state suffers a Z2 symmetry breaking phase transition (PT) when a single impurity
is added to a Bose-Einstein condensate (BEC) in a double well (bosonic Josephson
junction). The PT occurs for a certain value of the BEC-impurity interaction energy,
Λc . A result of the PT is the mean-field dynamics undergo chaotic motion in phase
space once the symmetry is broken. We determine the critical scaling exponents that
characterize the divergence of the correlation length and fidelity susceptibility at the
PT, finding that the BEC-impurity system belongs to the same universality class as
the Dicke and Lipkin-Meshkov-Glick models (which also describe symmetry breaking
PTs in systems of bosons).
In paper III we study the dynamics of a generic two-mode quantum field following a
quench where one of the terms in the Hamiltonian is flashed on and off. This model is
relevant to BECs in double wells as well as other simple many-particle systems found
in quantum optics and optomechanics. We find that when plotted in Fock-space plus
time, the semiclassical wave function develops prominent cusp-shaped structures after
the quench. These structures are singular in the classical limit and we identify them
as catastrophes (as described by the Thom-Arnold catastrophe theory) and show that
they arise from the coalescence of classical (mean-field) trajectories in a path integral
description. Furthermore, close to the cusp the wave function obeys a remarkable set
of scaling relations signifying these structures as examples of universality in quantum
dynamics. Within the cusp we find a network of vortex-antivortex pairs which are
phase singularities caused by interference. When the mean-field Hamiltonian displays
a Z2 symmetry breaking PT modelled by the Landau theory of PTs we calculate
scaling exponents describing how the separation distance between the members of
each pair diverges as the PT is approached. We also find that the cusp becomes
infinitely stretched out at the PT due to critical slowing down.
In paper IV we investigate in greater detail the morphology of the vortex network
found within cusp catastrophes in many-body wave functions following a quench. In
contrast to the cusp catastrophes studied so far in the literature, these structures live
in Fock space which is fundamentally granular. As such, these cusps represent a new
iii
type of catastrophe, which we term a ‘quantum catastrophe’. The granularity of Fock
space introduces a new length scale, the quantum length lq = N −1 which effectively
removes the vortex cores. Nevertheless, a subset of the vortices persist as phase
singularities as can be shown by integrating the phase of the wave function around
circuits in Fock-space plus time. Whether or not the vortices survive in a quantum
catastrophe is governed by the separation of the vortex-antivortex pairs lv ∝ N −3/4
in comparison to lq , i.e. they survive if lv
lq . When particle numbers are reached
such that lq ≈ lv the vortices annihilate in pairs. / Thesis / Doctor of Philosophy (PhD)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/21978 |
| Date | January 2017 |
| Creators | Mumford, Jesse |
| Contributors | O'Dell, Duncan, Physics and Astronomy |
| Source Sets | McMaster University |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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