Density ripples in expanding low-dimensional gases as a probe of correlations

Abstract

We investigate theoretically the evolution of the two-point density correlation function of a low-dimensional ultracold Bose gas after release from a tight transverse confinement. In the course of expansion thermal and quantum fluctuations present in the trapped systems transform into density fluctuations. For the case of free ballistic expansion relevant to current experiments, we present simple analytical relations between the spectrum of "density ripples" and the correlation functions of the original confined systems. We analyze several physical regimes, including weakly and strongly interacting one-dimensional ͑1D͒ Bose gases and twodimensional ͑2D͒ Bose gases below the Berezinskii-Kosterlitz-Thouless ͑BKT͒ transition. For weakly interacting 1D Bose gases, we obtain an explicit analytical expression for the spectrum of density ripples which can be used for thermometry. For 2D Bose gases below the BKT transition, we show that for sufficiently long expansion times the spectrum of the density ripples has a self-similar shape controlled only by the exponent of the first-order correlation function. This exponent can be extracted by analyzing the evolution of the spectrum of density ripples as a function of the expansion time.

Density fluctuations in a quasi-one-dimensional Bose gas as observed in free expansion

“…However, there is the second branch of excitations which are dark solitons [20] which is neglected in the considerations in Ref. [28]. Hence, the origin of the discrepancy between the two approaches is, in our opinion, related to the presence of the second type of excitations in one-dimensional systems, i.e.…”

“…[28]. This model allows to calculate the two-point density correlation function of very elongated ultracold Bose gas after its release from the trap.…”

“…One has to remember, however, that the validity of Ref. [28]'s model is restricted to such temperatures that k B T /µ K/π, where K = π n 1D /(mc) is the Luttinger liquid parameter with c being the speed of sound. In Fig.…”

“…This kind of excitations are included in the model of Ref. [28]. However, there is the second branch of excitations which are dark solitons [20] which is neglected in the considerations in Ref.…”

“…Earlier studies of the second order correlation function in expansion considered ballistic expansion of noninteracting atomic cloud [27] showing that the correlation function observed is related to the correlation in situ by some scaling only. In [28] the authors showed that density-density correlations of the interacting uniform gas manifest themselves in the expansion in a form of density ripples. The spectrum of the ripples is related to density-density correlations.…”