Abstract
Kaplan and Rajendran have recently demonstrated that non-linear and state-dependent terms can be consistently added to quantum field theory to yield causal non-linear time evolution in quantum mechanics. Causal non-linear theories have the unavoidable feature that their quantum effects are dramatically sensitive to the full physical spread of the quantum state of the system. As a result, such theories are not well tested by conventional atomic and nuclear spectroscopy. By using a well-controlled superposition of vibrational modes of a ⁴⁰Ca⁺ ion trapped in a harmonic potential, we set a stringent limit of 5.4× 10⁻¹² on the magnitude of the unitless scaling factor ε_γ for the predicted causal, non-linear perturbation.