Abstract
We theoretically investigate the effect of atom losses in the one-dimensional (1D) Bose gas with repulsive contact interactions, a famous quantum integrable system also known as the Lieb-Liniger gas. The generic case of KK-body losses (K=1,2,3,\dotsK=1,2,3,…) is considered. We assume that the loss rate is much smaller than the rate of intrinsic relaxation of the system, so that at any time the state of the system is captured by its rapidity distribution (or, equivalently, by a Generalized Gibbs Ensemble). We give the equation governing the time evolution of the rapidity distribution and we propose a general numerical procedure to solve it. In the asymptotic regimes of vanishing repulsion – where the gas behaves like an ideal Bose gas – and hard-core repulsion – where the gas is mapped to a non-interacting Fermi gas –, we derive analytic formulas. In the latter case, our analytic result shows that losses affect the rapidity distribution in a non-trivial way, the time derivative of the rapidity distribution being both non-linear and non-local in rapidity space.
Highlights
Several early works on atom losses – predating the ones on the absence of thermalization – have focused on the calculation of g3 in the ground state of the Lieb-Liniger gas [18, 19] and g2 in thermal states [20,21,22,23]. These results were soon extended to excited states of the gas [24,25], culminating in general expressions for gK valid for arbitrary rapidity distributions [26, 27]
The latter is a local quantity since q is local. [ H is not a local operator, it is an integral of local operators so its commutator with the local operator q is local.] One can use the Generalized Gibbs Ensemble (GGE) density matrix ρGGE to represent its average, and we find that the contribution of this term vanishes since [H, ρGGE] = 0
We compute F numerically using Eq (7), and we find that the results are in good agreement with formula (10)
Summary
Nowadays, trapped ultracold atoms offer versatile platforms for the study of isolated quantum many-body dynamics. The time-evolution of the system can be computed [6, 7] Such an analysis is not valid for one-dimensional (1D) bosons with point-like repulsive interactions, known as the Lieb-Liniger gas [8]. Several early works on atom losses – predating the ones on the absence of thermalization – have focused on the calculation of g3 in the ground state of the Lieb-Liniger gas [18, 19] and g2 in thermal states [20,21,22,23] These results were soon extended to excited states of the gas [24,25], culminating in general expressions for gK valid for arbitrary rapidity distributions [26, 27].
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