Local density approximation for a perturbative equation of state

Knowledge of a series expansion of the equation of state provides a deep insight into the physical nature of a quantum system. Starting from a generic “perturbative” equation of state of a homogeneous ultracold gas we make predictions for the properties of the gas in the presence of harmonic confinement. The local density approximation is used to obtain the chemical potential, total and release energies, Thomas-Fermi size, and density profile of a trapped system in three-, two-, and one-dimensional geometries. The frequencies of the lowest breathing modes are calculated using scaling and sum-rule approaches and could be used in an experiment as a high-precision tool for obtaining the expansion terms of the equation of state. The derived formalism is applied to dilute Bose and Fermi gases in different dimensions and to integrable one-dimensional models. The physical meaning of the expansion terms in a number of systems is discussed.

Figure 1

Square of the axial mode frequency in cigar-shaped trap for the Monte Carlo (thick solid line) [29] and the mean-field BCS (thin line) [22, 23, 24] equation of states as a function of ${\Lambda }_{3D}$. Experimental results (symbols) are taken from [13]. The dashed lines show the expansions at unitarity (short dashed line) and in the BEC regime (long-dashed line) taken from Table .

Figure 2

Frequency of the breathing mode in a number of one-dimensional bosonic systems. The frequencies for the Lieb-Liniger model were obtained numerically in [9]. The collective excitations of a gas of hard rods were discussed in [36] in their relation to the excitations of a gas in the super-Tonks regime. The dashed lines correspond to the expressions given in Table .

Figure 3

Frequencies of the breathing mode of a two-component Fermi gas in a quasi-one-dimensional geometry. The solid line shows results obtained by solving numerically the LDA with an exact equation of state [40]. The dashed lines show the small- and large-density expansions from Table . The upper curve corresponds to attraction between atoms of opposite spin, while the lower curve corresponds to repulsion.