Spontaneous symmetry breaking in spinor Bose-Einstein condensates

Abstract

We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state m F = 0 (|0 ) can be understood by approximating the effective trapping potential for the state |±1 with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given.

Spin dynamics in a two-dimensional quantum gas

“…Typically, ring-shaped density distributions in such timeof-flight images indicate the presence of orbital angular momentum [22], and the appearance of density modulations around the circumference could be interpreted as the matterwave interference pattern of two or more angular momentum eigenstates [11,23]. For large |q|, one may obtain a good understanding of the ring formed in a simplified, "free-space" picture.…”

“…Accordingly, atoms in |±1 experience a flat potential bottom plus a small parabolic repulsion U 1 n 0 [26]. Outside the |0 BEC, the potential rises steeply and the effective potential may be approximated by a cylindrical box [11]. Figure 3(a) shows the effective potential experienced by |±1 and the cylindrical box approximation.…”

“…In particular, highly asymmetric optical traps provide a way to reduce the dimensionality of a trapped quantum gas, both with respect to the motional and spin degrees of freedom [10]. Thus, tailored confining potentials offer new avenues for exploiting spin dynamics, e.g., the generation of correlated pairs of atoms in well-defined motional states [11], similar to work on four-wave mixing of ultracold atoms in an optical lattice [12][13][14].In this Rapid Communication, we investigate spin dynamics in a quantum gas confined to two dimensions (2D) by an optical lattice. We show how the spin excitation modes in the 2D potential lead to ring-shaped density distributions with a superimposed angular density modulation in time-of-flight images.…”

“…The state is given by ψ = ν nl c nl ϕ nl + c nl ϕ n−l , where the complex amplitude is c nl = √ P nl (t)e iθ , and c nl and c nl are sampled separately. The first factor, √ P nl (t), is the square root of the probability to obtain N atoms in mode (n,l) in |±1 ; i.e., a two-mode Fock state [11]. The second factor, e iθ , establishes a random phase between l and −l modes, and between different lattice sites.…”

“…In light of earlier work [11,26], we form a general superposition state for one spin state (e.g., |+1 ) in each lattice site, consisting of positive and negative angular momentum components. The expansion coefficients are sampled from appropriate probability distributions for each realization of the simulation [11]. The state is given by ψ = ν nl c nl ϕ nl + c nl ϕ n−l , where the complex amplitude is c nl = √ P nl (t)e iθ , and c nl and c nl are sampled separately.…”