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Time-dependent light-matter interactions are a widespread means by which todescribe controllable experimental operations. They can be viewed as anapproximation in which a third system - the control system - is treated asexternal within the Hamiltonian. We demonstrate that this results innon-equivalence between gauges. We provide a physical example in which eachdifferent non-equivalent model coincides with a gauge-invariant descriptionapplied in a different experimental situation. The qualitative final-timepredictions obtained from these models, including entanglement and photonnumber, depend on the gauge within which the time-dependent coupling assumptionis made. This occurs whenever the interaction switching is sufficiently strongand non-adiabatic even if the coupling vanishes at the preparation andmeasurement stages of the protocol, at which times the subsystems are uniqueand experimentally addressable.

Ultrastrong time-dependent light-matter interactions are gauge-relative