Quantized circulation ($\oint \mathbf{u}\cdot d\mathbf{l}=2\pi \hslash q/m$)

Single-valuedness of the condensate field gives total complex argument change $2\pi q$ on a closed loop avoiding zeros, where $q$ is an integer winding number. Integrating the superfluid velocity around that loop gives the displayed circulation. It is unchanged by smooth deformations of the loop that do not cross a quantum vortex.

If a condensate order parameter stays nonzero along a closed loop, its phase changes by an integer multiple of $2\pi$. Broken rotational symmetry does not destroy this single-valuedness constraint. In an inhomogeneous ring the winding integer need not label an angular-momentum eigenstate.

An event changing the phase winding of a condensate around a loop. A continuous nonzero order parameter cannot change its winding integer; a phase-slip path must pass through a locally vanishing amplitude or an equivalent vortex-crossing process. This use concerns a superfluid field, rather than only the relative phase of two classical oscillators.