Uniform ring condensate ($v=n\hslash /(mR)$)

A one-dimensional condensate on a ring of circumference $2\pi R$ has a uniform winding state $\psi =\sqrt{N/(2\pi R)}{e}^{in\theta }$ when the order parameter is normalized to atom number. Its angular momentum per atom is $n\hslash$, while the flow contribution to its total chemical potential is ${\hslash }^2{n}^2/(2m{R}^2)$.

For equal ring halves with potentials $V_0$ and zero, the zero-current Thomas–Fermi bulk solution has $\mu ={\mu }_0+V_0/2$ and populations $N_{L,R}=N(1\mp V_0/2{\mu }_0)/2$ while both halves are occupied. For $V_0>2{\mu }_0$ the raised half is depleted and $\mu =2{\mu }_0$ in this approximation.