Quantum Spontaneous Stochasticity

Abstract

The quantum wave-function of a massive particle with small initial uncertainties (consistent with the uncertainty relation) is believed to spread very slowly, so that the dynamics is deterministic. This assumes that the classical motions for given initial data are unique. In fluid turbulence non-uniqueness due to "roughness" of the advecting velocity field is known to lead to stochastic motion of classical particles. Vanishingly small random perturbations are magnified by Richardson diffusion in a "nearly rough" velocity field so that motion remains stochastic as the noise disappears, or classical spontaneous stochasticity, . Analogies between stochastic particle motion in turbulence and quantum evolution suggest that there should be quantum spontaneous stochasticity (QSS). We show this for 1D models of a particle in a repulsive potential that is "nearly rough" with $V(x) \sim C|x|^{1+\alpha}$ at distances $|x|\gg \ell$ , for some UV cut-off $\ell$, and for initial Gaussian wave-packet centered at 0. We consider the WKB limit with $\hbar/m \to 0$, then position-spread $\sigma\to 0$. The limit of the position density is non-deterministic, with equal probabilities of the particle following either of two, non-unique classical solutions and with "Richardson-like" spreading of the wave-packet. The splitting of the wave-packet occurs in a very short time $t\sim(\ell^{1+\alpha}/C)^{1/2}$ for small $\ell$. QSS also occurs in the scattering of a wave-packet off the rough potential, for careful fine-tuning. We also consider other semi-classical limits by a numerical solution of the Schr\"odinger equation, observing QSS both in position- and momentum-space. Although the wave-function remains split into two widely separated branches in the classical limit, rapid phase oscillations within each branch prevent any coherent superposition. These effects should be observable in laboratory experiments.