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Journal of Physics B Atomic Molecular and Optical Physics

Cusp catastrophe of symmetry breaking in two-component Bose–Einstein condensates

Jae Gil Kim, Jae Weon Lee, Dong-Chul Ihm, Hye Jin Lee, Eok Kyun Lee

Published May 16, 2003• 2 citations
Publisher

Abstract

Bifurcation analysis is applied to the spontaneous spatial symmetry breaking occurring in the ground state of two-component Bose–Einstein condensates. The cusp catastrophe describing the supercritical pitchfork bifurcation associated with the symmetry breaking is derived via the identification of the local curvature of the Gross–Pitaevskii energy functional. The bifurcation diagram and universal scaling laws for the eigenvalue and energy are obtained from the catastrophe function. 1.

DOI
10.1088/0953-4075/36/11/311

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•Jan 1, 2017•1 citation
Bifurcation analysis is applied to the spontaneous spatial symmetry breaking occurring in the ground state of two-component Bose–Einstein condensates. The cusp catastrophe describing the supercritical pitchfork bifurcation associated with the symmetry breaking is derived via the identification of the local curvature of the Gross–Pitaevskii energy functional. The bifurcation diagram and universal scaling laws for the eigenvalue and energy are obtained from the catastrophe function. 1.
The characteristic features of the collapse of the ground state in trapped one-component attractive Bose-Einstein condensates are studied by applying the catastrophe theory. From numerically obtained stable and unstable solutions of the Gross-Pitaevskii equation, we derive the catastrophe function defining the stability of the stationary points on the Gross-Pitaevskii energy functional. The bifurcation diagram and the universal scaling laws stemming from the catastrophe function show quantitative agreement with the numerical results.
We examine the stability properties of the ground state of two-component Bose-Einstein condensates as a function of the interspecies interactions. A stability criterion is identified from the curvature matrix of the Gross-Pitaevskii energy functional subject to the normalization conditions. By analyzing the stability signature, the characteristic features of the spontaneous spatial symmetry breaking are verified in various types of traps. The details of the differences between the continuous symmetry breaking and discrete symmetry breaking are discussed.
Bose–Einstein condensates are described in a mean-field approach by the nonlinear Gross–Pitaevskii equation and exhibit phenomena of nonlinear dynamics. The stationary states can undergo bifurcations in such a way that two or more eigenvalues and the corresponding wavefunctions coalesce at critical values of external parameters. For example, in condensates without long-range interactions a stable and an unstable state are created in a tangent bifurcation when the scattering length of the contact interaction is varied. At the critical point, the coalescing states show the properties of an exceptional point. In dipolar condensates fingerprints of a pitchfork bifurcation have been discovered by Rau et al (2010 Phys. Rev. A 81 031605). We present a method to uncover all states participating in a pitchfork bifurcation, and investigate in detail the signatures of exceptional points related to bifurcations in dipolar condensates. For the perturbation by two parameters, namely the scattering length and a parameter breaking the cylindrical symmetry of the harmonic trap, two cases leading to different characteristic eigenvalue and eigenvector patterns under cyclic variations of the parameters need to be distinguished. The observed structures resemble those of three coalescing eigenfunctions obtained by Demange and Graefe (2012 J. Phys. A: Math. Theor. 45 025303) using perturbation theory for non-Hermitian operators in a linear model. Furthermore, the splitting of the exceptional point under symmetry breaking in either two or three branching singularities is examined. Characteristic features are observed when one, two or three exceptional points are encircled simultaneously.
An analytical approximation method to clarify the symmetry-breaking stationary solutions to the Gross–Pitaevskii equation with symmetric double-well external potential is presented. By this method, we can understand that each symmetry-breaking solution bifurcates from a symmetry-preserving solution as one varies the coupling constant of the nonlinear interaction, and predict the bifurcation point. Also, the method gives a good approximation to the energy eigenvalues of the lower states, symmetry-breaking as well as symmetry-preserving, as long as the nonlinear interaction is not too strong.
The consequences of spatial symmetry breaking in two-component Bose-Einstein condensates are explored for the 85Rb+87Rb{}^{85}{\mathrm{R}\mathrm{b}+}^{87}\mathrm{Rb}85Rb+87Rb system. It is shown that the mean-field critical number of 85Rb{}^{85}\mathrm{Rb}85Rb atoms depends strongly on whether the single-particle wave function is symmetry preserving or symmetry breaking. This system thus provides a uniquely straightforward experimental means of observing spatial symmetry breaking.
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.