Three-Dimensional Quasi-Tonks Gas in a Harmonic Trap

We analyze the macroscopic dynamics of a Bose gas in a harmonic trap with a superimposed two-dimensional optical lattice, assuming a weak coupling between different lattice sites. We consider the situation in which the local chemical potential at each lattice site can be considered as that provided by the Lieb-Liniger solution. Because of the weak coupling between sites and the form of the chemical potential, the three-dimensional ground-state density profile and the excitation spectrum acquire remarkable properties different from both 1D and 3D gases. We call this system a quasi-Tonks gas. We discuss the range of applicability of this regime, as well as realistic experimental situations where it can be observed.

Figure 1

Density profile as a function of the dimensionless radius ${\tilde{r}}^2=\sum _i(r_i/R_i{)}^2$ (see text). The solid, dotted, and dashed lines correspond to $\log A=-3$, $0$, $3$, respectively. The resulting fit with a power law dependence $(1-{\tilde{r}}^2{)}^s$ provides $s=0.54$, $0.72$, $0.91$, respectively, being an indicator of the transition from the MF regime to the strongly interacting one.

Figure 2

Parameter $f$ as a function of $\log A$ (see text).

Figure 3

Frequencies of the breathing mode (solid line), quadrupole $1$ (dashed line), and quadrupole $2$ (dash-dotted line) as a function of $V_0/E_R$, for the particular case of ${}^{87}\mathrm{R}\mathrm{b}$ atoms, $N=2\times {10}^5$, ${\omega }_z=2\pi \times 4\mathrm{H}\mathrm{z}$, ${\omega }_{\perp }=2\pi \times 40\mathrm{H}\mathrm{z}$, and $d=0.5\mu \mathrm{m}$. Note that $V_0/E_R\simeq 25$, $w_z={\tilde{\omega }}_{\perp }$, and hence ${\omega }_{Q1}={\omega }_{Q2}$.