Momentum relaxation of a mobile impurity in a one-dimensional quantum gas

Abstract

We investigate the time evolution of the momentum of an impurity atom injected into a degenerate TonksGirardeau gas. We establish that given an initial momentum p 0 the impurity relaxes to a steady state with a nonvanishing momentum p ∞ . The nature of the steady state is found to depend drastically on whether the masses of the impurity and the host are equal. This is due to multiple coherent scattering processes leading to a resonant interaction between the impurity and the host in the case of equal masses. The dependence of p ∞ on p 0 remains nontrivial even in the limit of vanishing interaction between the impurity and host particles. In this limit p ∞ (p 0 ) is found explicitly.

Perpetual motion of a mobile impurity in a one-dimensional quantum gas

“…The analytical derivation does not exploit any particular form of interaction (and thus is equally valid both arXiv:1308.6260v3 [cond-mat.quant-gas] 4 Mar 2014 for nonintegrable and integrable systems) and does not resort to any approximations. Thus, our result complements previous works, in which a pointlike interaction was studied and which relied on numerical methods [5,6], on perturbation theory [7][8][9] or/and on the Bethe anzats solution [5,7], the latter being applicable in the integrable case only.…”

“…These observations were further supported in [6] by an analytical argument based on the Hellmann-Feynman theorem: it was shown that the average momentum of the impurity in the ground state of an impurity-host system with a fixed total momentum is nonzero. Recently the infinitetime momentum of the impurity for the family of models studied in [5] was calculated analytically in the week coupling limit [7][8][9]. The results of the latter calculations confirm the absence of complete momentum relaxation in both integrable and non-integrable cases.…”

“…The price is twofold. First, the bound (9) is valid for a narrower class of potentials (which includes, however, an important pointlike repulsive potential studied in [5][6][7], see below). Second, it is not as tight as the bound (8), since the difference p 0 − p ∞ is bounded by a term linear in the coupling strength.…”

Impurity states in the one-dimensional Bose gas

“…Furthermore, by substituting these eight terms in Eq. (2.10) one concludes that only five contribute to the self-energy with the result: 24) where the symmetric functions D ± (k, s) read…”

“…A very interesting platform for the theoretical research in the Bose polaron problem is the case of onedimensional environments [19][20][21][22][23][24][25][26]. It is well-known that due to highly non-trivial physics in the one spatial dimension [27,28] these systems possess unexpected behavior which very often obstructs their analysis.…”