In the context of cavity QED, most simulations are performed in one of two gauges, either the Coulomb gauge or the dipole gauge, where the term "gauge" refers to the specific representation of the vector potential Â. Expressing Â = Â∥ + Â⊥ , with its longitudinal part Â∥ that is curl free ∇ × Â∥ = 0, and the transverse part Â⊥ that is divergence-free ∇ • A ⊥ = 0. In principle, one can do gauge transformations that change the longitudinal part Â∥ , because the physically observed quantities will not change, (e.g. the magnetic field, since B = ∇ × Â = ∇ × Â⊥ ). One often refers to fixing a gauge by choosing the value of ∇ × Â such that the gauge transformation is effectively adding an additional ∇χ component to Â∥ , which is purely longitudinal because when χ is a scalar function in space, ∇χ is curl-free (∇ × ∇χ = 0).