Mixture of bosonic and spin-polarized fermionic atoms in an optical lattice

We investigate the properties of trapped Bose-Fermi mixtures for experimentally relevant parameters in one dimension. The effect of the attractive Bose-Fermi interaction onto the bosons is to deepen the parabolic trapping potential, and to reduce the bosonic repulsion in higher order. This leads for many situations to an increase in bosonic coherence. The opposite effect was observed in ${}^{87}\text{R}\text{b-}{}^{40}\text{K}$ experiments, most likely due to a sharp rise in temperature. We also discuss low-temperature features, such as a bosonic Mott insulator transition driven by the fermion concentration, and the formation of composite particles such as polarons and molecules.

Figure 1

(Color online) (a) Shift in critical $U_{\text{BB}}^c$ of the Mott transition for attractive $U_{\text{BF}}$. The system consists of 13 fermions and 64 bosons on a homogeneous lattice of 64 sites. Both species have unit hopping and the inverse temperature is $\beta =64/J_{\text{B}}$. The critical value $U_{\text{BB}}^c/J_{\text{B}}=3.28\pm 0.04$ at $U_{\text{BF}}/J_{\text{B}}=0$ is taken from Ref. [25]. At finite $U_{\text{BF}}$ the transition is located where the Green function has the same algebraic decay as in the purely bosonic case. The solid curve is a parabolic fit. (b) Comparison between the calculation in the presence of fermions and their approximation by a site-dependent potential for a trapped system of 60 sites, 40 bosons, 8 fermions, $\beta =1/J_{\text{B}}$, and optical potentials $V_0=6E_R$ $\left(U_{\text{BB}}/J_{\text{B}}=11.89\right)$. Here, $E_R={\hslash }^2{k}^2/2m_{\text{Rb}}$ is the bosonic recoil energy. $n_{\text{B}\left(\text{F}\right)}$ denotes the bosonic (fermionic) density for the mixture, $n_0$ is the density obtained in the approximation. Analogous for the density fluctuations ${\kappa }_{\text{B}}$ and ${\kappa }_0$.

Figure 2

(Color online) Dependence of bosonic (red, upper curve) and fermionic density (blue, lower curve) profiles on the interspecies interaction strength. In the system there are 60 sites, 50 bosons, 20 fermions, optical potential is $V_0=3E_R$ $\left(U_{\text{BB}}/J_{\text{B}}=4.26\right)$, and the inverse temperature $\beta =4.26/J_{\text{B}}$.

Figure 3

(Color online) The bosonic visibility $\nu$ as a function of fermionic number for a system of $L=60$ sites, $N_{\text{B}}=40$ bosons and inverse temperature $\beta =1/J_{\text{B}}$ for optical potentials $V_0=3,6,8E_R$ (or $U_{\text{BB}}/J_{\text{B}}=4.26,11.89,21.58$, respectively). The inset shows the bosonic density profiles at $V_0=6E_R$ for $N_f=0,40,4,20,16$ bottom to top in the trap center.

Figure 4

(Color online) Change in bosonic visibility when inverse temperature $\beta$ is increased for a system of 100 sites, 50 bosons, and 30 fermions.

Figure 5

(Color online) (a) Fermionic $⟨{\rho }_{\text{F}}\left(k\right)⟩$ and molecular Bose-Fermi $⟨{\rho }_P\left(k\right)⟩$ momentum profiles for the same system. Error bars are shown, but are smaller than the point size. (b) Bosonic and fermionic ground-state density profiles for a system with parameters $L=60,N_{\text{B}}=N_{\text{F}}=8,V_0=4E_R\left(U_{\text{BB}}/J_{\text{B}}=6.13\right)$ determined using DMRG (QMC results agree within error bars). The bosonic and fermionic density profiles almost coincide.