Hubbard Model for Atomic Impurities Bound by the Vortex Lattice of a Rotating Bose-Einstein Condensate

We investigate cold bosonic impurity atoms trapped in a vortex lattice formed by condensed bosons of another species. We describe the dynamics of the impurities by a bosonic Hubbard model containing occupation-dependent parameters to capture the effects of strong impurity-impurity interactions. These include both a repulsive direct interaction and an attractive effective interaction mediated by the Bose-Einstein condensate. The occupation dependence of these two competing interactions drastically affects the Hubbard model phase diagram, including causing the disappearance of some Mott lobes.

Figure 1

Hubbard description of atoms trapped in a vortex lattice. (a) The density $N_b|{\psi }^{\{0\}}(\mathbf{r}){|}^2$ of a harmonically trapped and rotating BEC. Inset: the phase structure of the central vortex. (b) The BEC $N_b|{\psi }^{\{1\}}(\mathbf{r}){|}^2$ (top left) and impurity $|w^{\{1\}}(\mathbf{r}){|}^2$ (top right) densities for a single impurity localized in the central vortex. Bottom: the $x$-axis cross section of these densities, together with the unperturbed BEC density $N_b|{\psi }^{\{0\}}(\mathbf{r}){|}^2$ (dashed line). The parameters for (a) and (b) are $N_b=500$, $\mathrm{\Omega }/{\mathrm{\Omega }}_b=0.98$, $m_b{g}_b/{\hslash }^2=1$, $m_b/m_a=1$, $g_a/g_b=1.1$, and $g_{ab}/g_b=6$. Quantities are expressed in terms of the characteristic length $a_0=\sqrt{\hslash /m_b{\mathrm{\Omega }}_b}$ of the trap. (c) An illustration of the Hubbard physics. Impurities at the vortex site $\mathbf{R}$ are described by the wave function $w_{\mathbf{R}}^{\{n_{\mathbf{R}}\}}(\mathbf{r})$ that, along with the deformation of the BEC wave function $|\psi (\mathbf{r})|$ in the vicinity of $\mathbf{R}$, is dependent on the occupation $n_{\mathbf{R}}$. This leads to an occupation-dependent energy ${\epsilon }^{\{n_{\mathbf{R}}\}}$ per impurity at the site and an occupation-dependent hopping energy $J_{{\mathbf{R}}^{\prime }\mathbf{R}}^{\{n_{{\mathbf{R}}^{\prime }}{n}_{\mathbf{R}}\}}$ between sites. (d) The Mott lobes (shaded regions) of the Hubbard phase diagram over different chemical potentials ${\mu }_a$ and hopping energies $|J|$ of the impurities, found within the Gutzwiller ansatz [62]. If ${\epsilon }^{\{n\}}$ increases monotonically then all Mott lobes exist; otherwise, some are missing. Here, we show both cases, corresponding to $g_a/g_b=1.5$ and $g_{ab}/g_b=3$ (left), and $g_a/g_b=1.1$ and $g_{ab}/g_b=6$ (right). The other parameters are $N_b=10$, $m_b{g}_b/{\hslash }^2=1$, and $m_b/m_a=1$.

Figure 2

Strong coupling. (a) The BEC wave function $|{\psi }^{\{n\}}(r)|$, normalized over the unit cell $0\le r\le \ell$, and the impurity wave function $w^{\{n\}}(r)$, normalized over all space, for different numbers of impurities $n$. The energy per impurity ${\epsilon }^{\{n\}}$, for different $n$, as (b) ${\eta }_a=g_a/g_b$ and (c) ${\eta }_{ab}=g_{ab}/g_b$ is varied. Crosses mark data points and lines guide the eye. Unless stated otherwise, $N_b=10$ is the number of bosons per unit cell, $m_b{g}_b/{\hslash }^2=1$, $m_b/m_a=1$, $g_a/g_b=1.1$, and $g_{ab}/g_b=6$.