Impurity in a quantum gas: exact diagonalization meets Bethe Ansatz

We examine stationary state properties of an impurity particle injected into a one-dimensional quantum gas. For equal masses of the impurity and host particles the problem reduces to a variant of the Gaudin-Young model, which is integrable and admits a formal solution based on the Bethe Ansatz. Obtaining physical observables from the formal solution is still non-trivial: the form-factor summation can be done via a stochastic enumeration based on the Metropolis algorithm.

For unequal masses, the problem is no longer integrable, and Bethe Ansatz approach breaks down. To account for integrability-breaking perturbations, we develop an exact diagonalization procedure in the (truncated) basis of the Bethe Ansatz states for an integrable model. We study steady-state properties for both cases, of a heavy and a light impurity.