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 (1D) Bose gases and two-dimensional (2D) Bose gases below the Berezinskii-Kosterlitz-Thouless (BKT) 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.
Deep Dive into Density ripples in expanding low-dimensional gases as a probe of correlations.
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 (1D) Bose gases and two-dimensional (2D) Bose gases below the Berezinskii-Kosterlitz-Thouless (BKT) 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
Quantum correlations can be used to identify and study interesting quantum phases and regimes in ultracold atomic systems. Recent experimental advances include detection of the Mott insulator phase of bosonic [1] and fermionic [2] atoms in optical lattices, production of correlated atom pairs in spontaneous four-wave mixing of two colliding Bose-Einstein condensates [3], studies of dephasing [4] and interference distribution functions [5] in coherently split one-dimensional (1D) atomic quasicondensates (QC), observation of the Berezinskii-Kosterlitz-Thouless (BKT) transition [6,7] in twodimensional (2D) quasicondensates [8], and Hanbury-Brown-Twiss correlation measurements for nondegenerate (ND) metastable 4 He [9] and 3 He atoms [10], bosonic [11] and fermionic [12] atoms in optical lattices, and in atom lasers [13]. In one-dimensional atomic gases [14,15,16,17,18,19,20], in situ measurements of correlations have been attained by means of photoassociation spectroscopy [21] or by measuring the three-body inelastic decay [22], using the proportionality of the corresponding rates to the zero-distance two-particle and three-particle correlation functions, respectively [23].
Recently it was demonstrated that one can detect sin-gle neutral atoms in a tight trap or guide [24,25,26,27,28,29]. However, direct (not inferred from any kind of atomic loss rate [21,22]) observation of interatomic correlations at short distances in trapped ultracold atomic gases is hindered in many cases by either the finite spatial resolution of the optical detection technique or the very low detection efficiency of the scanning electron microscope [27]. Therefore one needs to release ultracold atoms from the trap, diluting the atomic cloud in the course of expansion.
In this paper we address the question of how the correlations in the low-dimensional system evolve during the time-of-flight expansion, and discuss how the density variations in the time-of-flight images relate to the properties of the original trapped quantum gas. These “density ripples” in the expanding gas reflect the original thermal or quantum phase fluctuations existing in the cloud under confinement. Such phase fluctuations are already present in three-dimensional (3D) Bose-condensed clouds under an external confinement with large aspect ratio [30]. Their effect on density ripples of expanding clouds has been observed [31,32,33], but quantitative analysis of such experiments was complicated since one had to take into account interactions in the course of expansion. However, for sufficiently strong transverse confinement reached in current experiments with lowdimensional gases (chemical potential of the order of the transverse confinement frequency), the gas expands rapidly in the transverse direction so interactions during the expansion stage can be safely neglected. Then one can develop a simple analytical theory, which directly relates the spectrum of the density ripples after the expansion to the correlation functions of the original fluctuating condensates. Similar question has been considered for 3D clouds expanding in the gravitational field but only for noninteracting atoms [34]. We also note the density ripples we discuss are different from the density modulations which appear due to interactions during expansion and have been studied in Refs. [35] and [36].
We consider one-or two-dimensional atomic gases released from a tight trap formed by a scalar potential as realized on atom chips or in optical lattice experiments. We consider the situation when free expansion takes place in all three dimensions. This should be contrasted to the expansion of such a gas inside a waveguide [16,37,38,39,40,41,42,43,44,45], with the transverse confinement being permanently maintained. In the latter case, the nonlinear atomic coupling constant
where ω ⊥ is the transverse trapping frequency and a s is the atomic s-wave scattering length, remains the same. While a bosonic gas rarifies during such expansion, collisions remain important. For example, in the 1D case dynamics asymptotically reaches the limiting Tonks-Girardeau (TG) [46] regime of impenetrable bosons. In our case, if the fundamental frequency of the potential of the transverse confinement is much larger than the initial chemical potential of the atoms, the expansion in the transverse directions is determined mainly by the kinetic energy stored in the initial localized state of the transverse motion. Interatomic collisions play almost no role in the expansion. Moreover tight transverse confinement decouples the motion of trapped atoms in the longitudinal and transverse directions. Thus when analyzing density ripples we can reduce the problem to the same number of dimensions as the initial trap (see discussion below in Sec. II). For a 1D trap we consider a one-dimensional spectrum of density ripples, and for atoms which were originally confined in a pancake trap we analyze two-dimensional density ripples. Before
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