Mesoscopic transport of fermions through an engineered optical lattice connecting two reservoirs

We study transport of fermions in a system composed of a short optical lattice connecting two finite atomic reservoirs at different filling levels. The average equilibration current through the optical lattice, for strong lattice-reservoir coupling and finite temperatures, is calculated within the Landauer formalism using a nonequilibrium Green's functions approach. We moreover determine quantum and thermal fluctuations in the transport and find significant shot-to-shot deviations from the average equilibration current. We show how to control the atomic current by engineering specific optical lattice potentials without requiring site-by-site manipulations and suggest the realization of a single level model. Based on this model we discuss the blocking effect on the atomic current resulting from weak interactions between the fermions.

Figure 1

Fermions confined to an optical lattice hop from the left reservoir $L$ through a short coherent part $C$ (sites 1 to $m$) into the right reservoir $R$. The hopping parameters $J_{i,j}$, the on-site energies ${\varepsilon }_j$, and the couplings $J_L$ and $J_R$ may take arbitrarily engineered values, whereas the hopping parameter in the reservoirs $J_0$ is held constant.

Figure 2

Filling level and fluctuations in the right reservoir (initially empty) for a constant transmission ${\mathcal{T}}_0$. The average filling level $\langle N_R\left(t\right)\rangle /M$ (solid line) increases with time $t/t_{eq}$ and saturates at the equilibrium value $1/2$. The standard deviation $\sqrt{\langle \delta N_R^2\left(t\right)\rangle }/M$ from the average, due to thermal fluctuations, is indicated by the orange (gray) band. The Fano factor $F=\langle \delta N_R^2\left(t\right)\rangle /\langle N_R\left(t\right)\rangle$ (dashed line) decreases with time and approaches a constant value in the regime $t/t_{eq}\gg 1$. The parameters are $k_{B}T/J_{0}M=1/10$ for the filling level and $k_{B}T/J_0=1$ for the Fano factor.

Figure 3

The engineered transmission $\mathcal{T}\left(\varepsilon \right)$ as a function of the energy $\varepsilon /J_0$ for different modulations of an optical lattice with length $m=10$. (a) A single centered beam with waist $\sigma =2$ and increasing depths $V/J_0=1,2,4$ (dotted, dashed, and full line) shifts the energies out of the reservoir band and reduces the transmission to a peak at the upper band limit. (b) Two beams isolate a few central lattice sites and create a single resonant level at $\varepsilon /J_0\approx 1$ for sufficiently strong intensities. The beams are positioned at ${\nu }_1=3$, ${\nu }_2=8$ with waist $\sigma =1$, depths $V_2=1,3,5$ (dotted, dashed, and full line) and $V_1/V_2=1/2$.

Figure 4

Filling level and fluctuations in the right reservoir (initially empty) for a lattice modulated by a single beam. (a) The average filling level $\langle N_R\left(t\right)\rangle /M$ (solid line) as a function time $t/t_{eq}$ (with $t_{eq}=\hslash M/2J_0$) for the modulation strength $V/J_0=2$. Different bands indicate the standard deviation $\sqrt{\langle \delta N_R^2\left(t\right)\rangle }/M$ due to quantum fluctuations (blue or light gray) and total fluctuations, i.e., quantum plus thermal (orange or dark gray). (b) The same quantities for a stronger modulation strength $V/J_0=4$. The lower transmission results in a smaller current between the reservoirs and enhanced quantum fluctuations in comparison to (a). In both plots damping has not been taken into account and the parameters are $k_{B}T/J_0=1$, $m=10$, and $M=50$.

Figure 5

The effect of interactions on a single level with original ($U=0$) position ${\varepsilon }_0=0$ and the relative chemical potentials ${\mu }_L/J_0=1$ and ${\mu }_R/J_0=-1$. The occupation $\langle n\rangle$ (blue or dark gray) and the energy shift $U\langle n\rangle /J_0$ (orange or light gray) as a function of the interaction $U/J_0$ are plotted for the couplings $\Gamma /J_0=0.1$ (solid) and $\Gamma /J_0=1$ (dashed). For $U>0$ the level is depleted and shifted toward ${\mu }_L$. For $U<0$ the level is almost completely occupied and shifted below ${\mu }_R$ for sufficiently strong interactions. The current is blocked if the single level at ${\varepsilon }_0+U\langle n\rangle$ leaves the energy window $[{\mu }_L,{\mu }_R]$.