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arXiv:1202.5775 · Physical Review A

Searching for non-Abelian phases in the Bose-Einstein condensate of dysprosium

Biao Lian, Tin-Lun Ho, Hui Zhai

Published May 24, 2012• 45 citations
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Abstract

The recently realized Bose-Einstein condensate of dysprosium will become a spin-8 spinor condensate at low magnetic fields. In such a high-spin condensate, many phases with different symmetries can exist. Among them the most interesting ones are those with non-Abelian point group symmetry. In this Rapid Communication we discuss the variety of symmetry phases in a spin-8 condensate resulting from numerical solutions of the Hamiltonian. We show that these symmetries can be determined uniquely from the measurements of density population on each spin component in an ultralow magnetic field, together with the measurements of the collective modes in the zero-field limit. This method can also be applied to Bose-Einstein condensates of other magnetic atoms, such as Cr and Er.

DOI
10.1103/physreva.85.051606

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The recently realized Bose-Einstein condensate of dysprosium will become a spin-8 spinor condensate at low magnetic fields. In such a high-spin condensate, many phases with different symmetries can exist. Among them the most interesting ones are those with non-Abelian point group symmetry. In this Rapid Communication we discuss the variety of symmetry phases in a spin-8 condensate resulting from numerical solutions of the Hamiltonian. We show that these symmetries can be determined uniquely from the measurements of density population on each spin component in an ultralow magnetic field, together with the measurements of the collective modes in the zero-field limit. This method can also be applied to Bose-Einstein condensates of other magnetic atoms, such as Cr and Er.
We propose a method for systematically finding ground states of spinor Bose-Einstein condensates by utilizing the symmetry properties of the system. By this method, we can find not only an inert state, whose symmetry is maximal in the manifold under consideration, but also a noninert state, which has lower symmetry and depends on the parameters in the Hamiltonian. We establish the symmetry-classification method for the spin-1, 2, and 3 cases at zero magnetic field, and find an additional phase in the last case. The properties of the vortices in the spin-3 system are also discussed.
We analyze the physics of spin-3 Bose-Einstein condensates, and, in particular, the new physics expected in ongoing experiments with condensates of chromium atoms. We first discuss the ground-state properties, which, depending on still unknown chromium parameters, and for low magnetic fields, can present various types of phases. We also discuss the spinor dynamics in chromium spinor condensates, which present significant qualitative differences when compared to other spinor condensates. In particular, dipole-induced spin relaxation may lead for low magnetic fields to transfer of spin into angular momentum similar to the well-known Einstein-de Haas effect. Additionally, a rapid large transference of population between distant magnetic states also becomes possible.
We study spin correlations in Bose-Einstein condensates of spin 1 bosons with scatterings dominated by a total spin equal 222 channel. We show that the low energy spin dynamics in the system can be mapped into an o(n)o(n)o(n) nonlinear sigma model. n=3n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3n=3 at the zero magnetic field limit and n=2n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2n=2 in the presence of weak magnetic fields. In an ordered phase, the ground state has a discrete Z2{Z}_{2}Z2​ symmetry and is degenerate under the group $[U(1)\ifmmode\times\else\texttimes\fi{}{S}^{n\ensuremath{-}1}]{/Z}_{2}$. We explore consequences of the discrete symmetry and propose some measurements to probe it.
In spinor Bose Einstein condensates (BEC) gases, a fraction of its thermally excited atoms can still interact with the condensate ground state, leading to spin-spin interactions that can modify the main features of its spin-phase diagrams. In this work we study the spin-phase diagram of a BEC of general spin-fff and fully characterize its noncondensate thermal fraction. The latter provided that the condensate ground state lies within a spin phase with rotational symmetry. The study is based in the Hartree-Fock approximation in conjunction with the Majorana stellar representation approach for pure and mixed quantum states and the use of point-group symmetry arguments. The method allows us to study the phase diagram of spinorial BECs with usual point-group symmetries, including those with some exotic phases associated to the platonic solids (f=2,3,4,f=2,3,4,f=2,3,4, and 6), which are known to lead to non-Abelian topological excitations. In addition, we explore the temperature effects on the admissible spin-phase domains for general spin values, as well as its physical implications on their multipolar magnetic moments.
We demonstrate that dynamical symmetry plays a crucial role in determining the structure of the eigenspectra of spinor Bose-Einstein condensates (BECs). In particular, the eigenspectra of spin-1 and spin-2 BECs in the single-mode approximation are shown to be completely determined by dynamical symmetries, where a spin-2 BEC corresponds to the U(5) limit of the interacting boson model in nuclear physics. The eigenspectrum of a spin-3 BEC is solved analytically for a specific class of coupling constants, while it is shown that dynamical symmetry alone is not sufficient to determine the spectrum for arbitrary coupling constants. We also study the low-lying eigenspectra of spin-1 and spin-2 BECs in the absence of external magnetic field, and find, in particular, that the quasidegenerate spectra emerge for antiferromagnetic and cyclic phases. This implies that these systems are highly susceptible to external perturbations and may undergo symmetry-breaking transitions to other states upon increasing the system's size.