  Dissipative systems can often exhibit wavelength-dependent loss rates. One
prominent example is Rydberg polaritons formed by electromagnetically-induced
transparency, which have long been a leading candidate for studying the physics
of interacting photons and also hold promise as a platform for quantum
information. In this system, dissipation is in the form of quantum diffusion,
i.e., proportional to $k^2$ ($k$ being the wavevector) and vanishing at long
wavelengths as $k\to 0$. Here, we show that one-dimensional condensates subject
to this type of loss are unstable to long-wavelength density fluctuations in an
unusual manner: after a prolonged period in which the condensate appears to
relax to a uniform state, local depleted regions quickly form and spread
ballistically throughout the system. We connect this behavior to the
leading-order equation for the nearly-uniform condensate -- a dispersive
analogue to the Kardar-Parisi-Zhang (KPZ) equation -- which develops
singularities in finite time. Furthermore, we show that the wavefronts of the
depleted regions are described by purely dissipative solitons within a pair of
hydrodynamic equations, with no counterpart in lossless condensates. We close
by discussing conditions under which such singularities and the resulting
solitons can be physically realized.
