Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Real space mean-field theory of a spin-1 Bose gas in synthetic dimensions



Ian B. Spielman, Justin H. Wilson, Hilary Hurst, Jed Pixley, Stefan Natu


The internal degrees of freedom provided by ultracold atoms give a route for realizing higher dimensional physics in systems with limited spatial dimensions. Non-spatial degrees of freedom in these systems are dubbed “synthetic dimensions”. This connection is useful from an experimental standpoint but complicated by the fact that interactions alter the condensate ground state. Here we use the Gross-Pitaevskii equation to study ground state properties of a spin-1 Bose gas under the combined influence of an optical lattice, spin-orbit coupling, and interactions at the mean field level. The associated phases depend on the sign of the spin-dependent interaction parameter and the optical lattice potential. We find “charge” and spin density wave phases which are directly related to helical spin order in real space and a↵ect the behavior of edge currents in the synthetic dimension. We determine the resulting phase diagram as a function of the spin-orbit coupling and spin dependent interaction strength, considering both attractive (ferromagnetic) and repulsive (polar) spin dependent interactions. Our results are applicable to current and future experiments, specifically with 87Rb, 7Li, 41K, and 23Na.
Physical Review A


Spin orbit coupling Bose Einstein condensate


Spielman, I. , Wilson, J. , Hurst, H. , Pixley, J. and Natu, S. (2016), Real space mean-field theory of a spin-1 Bose gas in synthetic dimensions, Physical Review A, [online], (Accessed April 17, 2024)
Created December 15, 2016, Updated June 2, 2021