Thermal versus quantum fluctuations of optical-lattice fermions

We show that, for fermionic atoms in a one-dimensional optical lattice, the fraction of atoms in doubly occupied sites is a highly nonmonotonic function of temperature. We demonstrate that this property persists even in the presence of realistic harmonic confinement, and that it leads to a suppression of entropy at intermediate temperatures that offers a route to adiabatic cooling. Our interpretation of the suppression is that such intermediate temperatures are simultaneously too high for quantum coherence and too low for significant thermal excitation of double occupancy thus offering a clear indicator of the onset of quantum fluctuations.

Figure 1

Double-occupancy fraction versus temperature for the uniform 1D Hubbard model at half filling for several $U/t$. The dashed line indicates the low-$T$ expansion (5); the solid line is the fourth-order high-$T$ expansion; the dot-dashed line is the exact result for a two-site Hubbard model in the grand canonical ensemble; and the dotted line is the same but with the number of particles set to exactly two. At low $T$, nonzero $d$ arises from coherent quantum fluctuations, but at intermediate temperatures this coherence is destroyed, leading to a suppression of $d\left(T\right)$ shown by the double-headed arrow.

Figure 2

Schematic of trap filling in three different regimes with decreasing $U$ from left to right. Left: Mott insulator with single occupancy, Center: Doubly occupied sites at the trap center flanked by edges with low filling. Right: A band insulator dominated by doubly occupied sites. These sketches ignore both quantum and thermal fluctuations.

Figure 3

The circles indicate double-occupancy fraction versus particle number in a trapped system ($L=50$) at $T=0$ for half filling at the trap center and $U/t=8$ computed using BA-LDA. The solid line is the bulk value in the Mott insulating ground state [Eq. (4)]. The insets, from left to right, show the corresponding density as a function of position in the trap for 100, 216, and 300 particles, respectively.

Figure 4

The circles and solid line show the same as Fig. 3 but for three values of $U$. The dashed (dotted) lines show results computed from an exact second-order high-temperature series expansion at $T=2t$ ($T=4t$). The plateau at finite $d$ indicates strong quantum fluctuations. The double arrow in the bottom panel indicates the observable effects of weak thermal fluctuations in a trap and corresponds to the double arrow in Fig. 1.