Quantum Monte Carlo study of the visibility of one-dimensional Bose-Fermi mixtures

The study of ultracold optically trapped atoms has opened new vistas in the physics of correlated quantum systems. Much attention has now turned to mixtures of bosonic and fermionic atoms. A central puzzle is the disagreement between the experimental observation of a reduced bosonic visibility ${\mathcal{V}}_b$, and quantum Monte Carlo (QMC) calculations which show ${\mathcal{V}}_b$ increasing. In this paper, we present QMC simulations which evaluate the density profiles and ${\mathcal{V}}_b$ of mixtures of bosons and fermions in one-dimensional optical lattices. We resolve the discrepancy between theory and experiment by identifying parameter regimes where ${\mathcal{V}}_b$ is reduced, and where it is increased. We present a simple qualitative picture of the different response to the fermion admixture in terms of the superfluid and Mott-insulating domains before and after the fermions are included. Finally, we show that ${\mathcal{V}}_b$ exhibits kinks which are tied to the domain evolution present in the pure case, and also additional structure arising from the formation of boson-fermion molecules, a prediction for future experiments.

Figure 1

(Color online) Comparison of the density profiles at $U_{bb}=8.3t$ and $U_{bf}=-5.0t$ for $N_f=0,3,5$ fermions on an 80-site chain with 40 bosons. The Mott insulator at the trap center for the pure bosonic case is destroyed by the addition of fermions. This drives the increase in the visibility.

Figure 2

(Color online) (a) Bosonic visibility ${\mathcal{V}}_b$ as a function of the number of bosons for $N_f=0$ and 9 fermions with fixed $U_{bb}=12.0t$ and $U_{bf}=-5.0t$. (b) Bosonic density profiles for $N_b=26$ bosons and $N_f=0,9$ fermions. The addition of the fermions induces Mott-insulating behavior in the bosons. The key consequence is a decrease in ${\mathcal{V}}_b$ for $N_f=9$ relative to $N_f=0$, similar to that seen in the experiments.

Figure 3

(Color online) (a) Bosonic and fermionic visibilities and bosonic $S_{\max }$ as functions of $U_{bb}∕t$ for a system with 40 bosons, 3 fermions, and $U_{bf}=-5.0t$. The “plateau” regions where the rate of reduction of $\mathcal{V}$ is reduced are due to freezing of the density profiles (see text). The fast decrease after $U_{bb}∕t\approx 9.3$ is due to the formation of a Mott-insulating region in the central core, which is fully formed and indicated by the arrow at $U_{bb}∕t=9.6$. (b) Boson density profiles at five different values of $U_{bb}∕t$.

Figure 4

(Color online) (a) Trapping $\left(E_{\mathrm{trap}}\right)$ and interaction $\left(E_{\mathrm{int}}\right)$ energies as functions of $U_{bb}∕t$ for the system of Fig. 3 (b) Bosonic $\left(E_{\mathrm{kin}}^b\right)$ and fermionic $\left(E_{\mathrm{kin}}^f\right)$ kinetic energies.

Figure 5

(Color online) Comparison of (a) bosonic visibility ${\mathcal{V}}_b$, (b) the trapping energy, and (c) the interaction energy for $N_f=0$, 3, and 5 fermions on an 80-site chain with 40 bosons and fixed $U_{bf}=-5.0t$. The arrows in panels (a), (c), and (b), respectively, denote the locations of the kinks (the onset of rapid change in energy and visibility) for $N_f=0$, 3, and 5 fermions.

Figure 6

(Color online) Bosonic and fermionic density profiles for (a) $U_{bb}=6.3t$, (b) $U_{bb}=7.2t$, (c) $U_{bb}=8.6t$, and (d) $U_{bb}=9.5t$ with 5 fermions, 40 bosons, and fixed $U_{bf}=-5.0t$. The fermionic density is offset and the dashed gray line indicates zero density. The densities match in the center of the trap, with the region of coincidence decreasing as $U_{bb}$ increases.