Spinor Bose-Einstein condensates

Abstract

Bose-Einstein (BE) condensation is a phenomenon in which a macroscopic number of particles occupy a single-particle state. Once a system undergoes BE condensation, a significant fraction of particles behave in lockstep, leading to a number of spectacular macroscopic quantum phenomena. This article delineates fundamental aspects of BE condensates with an emphasis on spin degrees of freedom. Here the gauge and spin degrees of freedom are coupled to produce a rich variety of spinor superfluidity and topological excitations.

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No potential conflict of interest was reported by the author.

Notes

1 This relation can be derived as follows. The completeness relation of the spin-1 states can be expressed in terms of magnetic sublevels $\{|m〉\}$ as $\begin{array}{r}\sum _{m=-1}^1|m〉〈m|=I,\end{array}$ where I is the identity operator. For two particles, the completeness relation reads $\begin{array}{rl}{I}_1\otimes I_2& =\sum _{m_1=-1}^1|m_1〉〈m_1|\otimes \sum _{m_2=-1}^1|m_2〉〈m_2|\\ & =:\sum _{m_1,m_2=-1}^1|m_1,m_2〉〈m_1,m_2|.\end{array}$ By operating this completeness relation from the right side of $P_0+P_2$, we have $\begin{array}{r}{P}_0+P_2=\sum _{m_1,m_2}^1(P_0+P_2)|m_1,m_2〉〈m_1,m_2|.\end{array}$ The total spin of two identical bosons must be either 0 or 2, so that $P_0+P_2$ should act as the identity operator for any spin state of two bosons: $(P_0+P_2)|m_1,m_2〉=|m_1,m_2〉$. We thus obtain $\begin{array}{r}{P}_0+P_2=\sum _{m_1,m_2=-1}^1|m_1,m_2〉〈m_1,m_2|=I_1\otimes I_2.\end{array}$

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Masahito Ueda

Masahito Ueda is Professor in the Department of Physics at the University of Tokyo. He obtained his PhD from the University of Tokyo in 1991. After that he worked for six years at NTT Basic Research Laboratories, for six years at Hiroshima University, and then for eight years at Tokyo Institute of Technology, until he joined the University of Tokyo in 2008. He is interested in atomic, molecular and optical physics, and the foundational problems in quantum mechanics, thermodynamics and machine learning. In particular, he has developed ways to study information thermodynamics and non-Hermitian physics.