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

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