Singularities in nearly-uniform 1D condensates due to $k^2$ loss

Certain dissipative systems have one-body loss proportional to $k^2$ ($k$ being momentum). 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 hold promise as quantum information platforms. Here we show that one-dimensional condensates having such $k^2$ 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, localized regions quickly dissipate and the depleted zones spread throughout the system. We connect this behavior to the leading-order equation for the nearly-uniform condensate, which develops singularities in finite time. Furthermore, the fronts which form can be described by novel dissipative solitons unrelated to the standard solitons in dissipation-free condensates. We close by discussing conditions under which such singularities and the resulting solitons can be observed experimentally.