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arXiv:0706.2505 · Physical Review Letters

Inert States of Spin- S Systems

H. Mäkelä, Kalle‐Antti Suominen

Published Nov 9, 2007• 45 citations
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Abstract

We present a simple but efficient geometrical method for determining the inert states of spin-S systems. It can be used if the system is described by a spin vector of a spin-S particle and its energy is invariant in spin rotations and phase changes. Our method is applicable to an arbitrary S and it is based on the representation of a pure spin state of a spin-S particle in terms of 2S points on the surface of a sphere. We use this method to find candidates for some of the ground states of spinor Bose-Einstein condensates.

DOI
10.1103/physrevlett.99.190408

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We present a simple but efficient geometrical method for determining the inert states of spin-S systems. It can be used if the system is described by a spin vector of a spin-S particle and its energy is invariant in spin rotations and phase changes. Our method is applicable to an arbitrary S and it is based on the representation of a pure spin state of a spin-S particle in terms of 2S points on the surface of a sphere. We use this method to find candidates for some of the ground states of spinor Bose-Einstein condensates.
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.
This thesis concentrates on the topological defects of spin-1 and spin-2 Bose-Einstein condensates, the ground states of spin-3 condensates, and the inert states of spinor condensates with arbitrary spin. Our work is based on the description of a spinor condensate of spin-S atoms in terms of a state vector of a spin-S particle. The results of the homotopy theory are used to study the existence and structure of the topological defects in spinor condensates. We construct examples of defects, study their energetics, and examine how their stability is affected by the presence of an external magnetic field. The ground states of spin-3 condensates are calculated using analytical and numerical means. Special emphasis is put on the ground states of a chromium condensate, whose dependence on the magnetic dipole-dipole interaction is studied. A simple geometrical method for the calculation of inert states of spinor condensates is presented. This method is used to find candidates for the ground states of spin-S condensates.
In this paper, we consider spinor Bose–Einstein condensates with spin f = 5 and f = 6 in the presence and absence of an external magnetic field at the mean field level. We calculate all of the so-called inert states of these systems. Inert states are a very unique class of stationary states because they remain stationary while Hamiltonian parameters change. Their existence comes from Michel’s theorem. For illustration of symmetry properties of the inert states we use a method that allows for the classification of the systems as a polyhedron with 2 f vertices proposed by Barnett et al (2006 Phys. Rev. Lett. 97 180412).
The objective of this project is to use the Majorana description of a spin-1 system to examine many of its properties. In the Majorana approach, an arbitrary pure state of a spin-1 system is represented by a pair of points on the unit sphere, or a pair of unit vectors. An expression is derived for the squared overlap of two states in terms of their Majorana (or M-) vectors and used to work out all the orthogonalities and bases that are possible for the three geometrically distinct types of spin-1 states. Finally, a geometrical construction is given of a set of mutually unbiased bases for a spin-1 system. The applications of this result to problems in quantum information are mentioned.
In "Part I: Vector Analysis of Spinors", the author studied the geometry of two component spinors as points on the Riemann sphere in the geometric algebra of three dimensional Euclidean space. Here, these ideas are generalized to apply to four component Dirac spinors on the complex Riemann sphere in the complexified geometric algebra of spacetime, which includes Lorentz transformations. The development of generalized Pauli matrices eliminate the need for the traditional Dirac gamma matrices. We give the discrete probability distribution of measuring a spin 1/2 particle in an arbitrary spin state, assuming that it was prepared in a given state immediately prior to the measurement, independent of the inertial system in which measurements are made. The Fierz identities between the physical observables of a Dirac spinor are discussed.