  We show that the spreading of the center-of-mass density of ultracold
attractively interacting bosons can become superballistic in the presence of
decoherence, via single-, two- and/or three-body losses. In the limit of weak
decoherence, we analytically solve the numerical model introduced in [Phys.
Rev. A 91, 063616 (2015)]. The analytical predictions allow us to identify
experimentally accessible parameter regimes for which we predict superballistic
spreading of the center-of-mass density. Ultracold attractive Bose gases form
weakly bound molecules; quantum matter-wave bright solitons. Our
computer-simulations combine ideas from classical field methods ("truncated
Wigner") and piecewise deterministic stochastic processes. While the truncated
Wigner approach to use an average over classical paths as a substitute for a
quantum superposition is often an uncontrolled approximation, here it predicts
the exact root-mean-square width when modeling an expanding Gaussian wave
packet. In the superballistic regime, the leading-order of the spreading of the
center-of-mass density can thus be modeled as a quantum superposition of
classical Gaussian random walks in velocity space.
