Dynamics of a mobile impurity in a one-dimensional quantum fluid

Consider a mobile impurity particle injected in a one-dimensional quantum fluid with some initial velocity, $v_0$. What will be the relaxation dynamics of the impurity? Numerical and seminumerical studies of finite systems ($N<50$, where $N$ is the number of particles of the fluid) revealed a highly nontrivial dynamics: The impurity's velocity experienced oscillations superimposed on a slowdown; finally the velocity apparently saturated at some non-zero value, $v_f$ [1,2]. These studies, while producing much excitement, left unanswered basic questions on the nature of the effects discovered. It was even unclear whether the the incomplete relaxation was a finite-size effect or an effect present in the thermodynamic limit.
We present a detailed analytical study of the anomalous relaxation dynamics of an impurity particle injected in the one-dimensional quantum fluid [3-7]. In particular, we rigorously prove that the impurity particle of finite mass never stops completely, even in the thermodynamic limit [3,4]. This should be contrasted with the well-known absence of superfluidity in one dimension. These two facts can be reconciled since $v_f$ depends on the mass of the particle and vanishes for the infinite mass, which is equivalent to the absence of superfluid flow through a static constriction. We also find analytical dependence of the final velocity, $v_f$, on the initial velocity, $v_0$, for particular quantum fluids, the one-dimensional Fermi gas and the gas of impenetrable bosons [5-7].