  The order parameter of a quantum-coherent many-body system can include a
phase degree of freedom, which, in the presence of an electromagnetic field,
depends on the choice of gauge. Because of the relationship between the phase
gradient and the velocity, time-of-flight measurements reveal this gradient.
Here, we make such measurements using initially trapped Bose-Einstein
condensates (BECs) subject to an artificial magnetic field. Vortices are
nucleated in the BEC for artificial field strengths above a critical value,
which represents a structural phase transition. By comparing to
superfluid-hydrodynamic and Gross-Pitaevskii calculations, we confirmed that
the transition from the vortex-free state gives rise to a shear in the released
BEC's spatial distribution, representing a macroscopic method to measure this
transition, distinct from direct measurements of vortex entry. Shear is also
affected by an artificial electric field accompanying the artificial magnetic
field turn-off, which depends on the details of the physical mechanism creating
the artificial fields, and implies a natural choice of gauge. Measurements of
this kind offer opportunities for studying phase in less-well-understood
quantum gas systems.
