Breakdown of Tan's Relation in Lossy One-Dimensional Bose Gases

In quantum gases with contact repulsion, the distribution of momenta of the atoms typically decays as similar to 1/vertical bar p vertical bar(4) at large momentum p. Tan's relation connects the amplitude of that 1/vertical bar p vertical bar(4) tail to the adiabatic derivative of the energy with respect to the coupling constant or scattering length of the gas. Here it is shown that the relation breaks down in the one-dimensional Bose gas with contact repulsion, for a peculiar class of stationary states. These states exist thanks to the infinite number of conserved quantities in the system, and they are characterized by a rapidity distribution that itself decreases as 1/vertical bar p vertical bar(4). In the momentum distribution, that rapidity tail adds to the usual Tan contact term. Remarkably, atom losses, which are ubiquitous in experiments, do produce such peculiar states. The development of the tail of the rapidity distribution originates from the ghost singularity of the wave function immediately after each loss event. This phenomenon is discussed for arbitrary interaction strengths, and it is supported by exact calculations in the two asymptotic regimes of infinite and weak repulsion.