Neutron optical tests of nonlinear wave mechanics

We analyze the free-space propagation of matter waves with a view to placing an upper limit on the strength of possible nonlinear terms in the Schrödinger equation. Such additional terms of the form $\psi F\left({\left|\psi \right|}^2\right)$ were introduced by Bialynicki-Birula and Mycielski in order to counteract the spreading of wave packets, thereby allowing solutions which behave macroscopically like classical particles. For the particularly interesting case of a logarithmic nonlinearity of the form $F=-\mathit{b}\ln {\left|\psi \right|}^2$, we find that the free-space propagation of slow neutrons places a very stringent upper limit on the magnitude of $\mathit{b}$. Precise measurements of Fresnel diffraction with slow neutrons do not give any evidence for nonlinear effects and allow us to deduce an upper limit for $\mathit{b}<3.3\mathrm{}\times {10}^{-15\mathrm{}}$ eV about 3 orders of magnitude smaller than the lower bound proposed by the above authors.