  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 $^{40}$Ca$^+$ ion trapped in a harmonic potential, we
set a stringent limit of $5.4\times 10^{-12}$ on the magnitude of the unitless
scaling factor $\tilde{\epsilon}_{\gamma}$ for the predicted causal, non-linear
perturbation.
