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|Singularities in a BEC in a double well potential
|Physics and Astronomy
|quantum;phase transition;catastrophe theory;many-body systems
|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.
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