Ground-state cooling of atoms in optical lattices

We propose two schemes for cooling bosonic and fermionic atoms that are trapped in a deep optical lattice. The first scheme is a quantum algorithm based on particle number filtering and state dependent lattice shifts. The second protocol alternates filtering with a redistribution of particles by means of quantum tunnelling. We provide a complete theoretical analysis of both schemes and characterize the cooling efficiency in terms of the entropy. Our schemes do not require addressing of single lattice sites and use a novel method, which is based on coherent laser control, to perform very fast filtering.

Figure 1

Spatial dependence of the average particle number $⟨n⟩$ and entropy $S$ before (left) and after (right) the application of the filter operation $F_1$. The final particle distribution can be well described by Eq. (4) [dashed line in (b)]. Numerical parameters for initial state, $N_i=65$, $s_i=1$, and $U∕b=700$ ($\beta U=4.5$, $\mu ∕U=1$). Figures of merit, $s_f∕s_i=0.56$ and $N_f∕N_i=0.80$.

Figure 2

Effective description of thermal states in the no-tunnelling limit in terms of noninteracting fermions occupying two energy bands. The dispersion relations are ${\epsilon }_{\mathrm{I}}=b{k}^2$ and ${\epsilon }_{\mathrm{II}}=b{k}^2+U$, where $k$ denotes the lattice site and $U$ is the interaction energy. Increasing the harmonic trap strength from $b$ to $b^{\prime }$ increases the chemical potential to ${\mu }^{\prime }$ so that the population of the upper band becomes energetically favorable. In the bosonic picture this process corresponds to the formation of doubly occupied sites.

Figure 3

(Color online) Entropy per particle $S∕N$ (left) and normalized number of particles $N∕N_0$ (right) as a function of the number of filtering iterations for initially $N_0=100$ particles. As discussed in the text, we distinguish between the scenarios (i) (black line), (iia) (red line) and (iib) (blue and cyan).

Figure 4

(Color online) Illustration of the protocol. The state is initialized with the filter operation $F_2$. (a) Particles from doubly occupied sites are transferred to state $\mid b⟩$ (red) using operation $U_{2,0}^{1,1}$. (b) The lattice $\mid b⟩$ has been shifted $2k_{\epsilon }$ sites to the right, so that the two distributions barely overlap. (c) Density distribution after $k_s$ lattice shifts. After each shift doubly occupied sites have been emptied. Afterwards lattice $\mid b⟩$ is shifted $4k_{\epsilon }-k_s$ sites to the left and an analogous filter sequence is applied. (d) The final distribution of atoms in state $\mid a⟩$ is sharper compared to the initial distribution (dotted). Numerical parameters, $N_i=65$, $s_i=1$, $U∕b=300$, $k_{\epsilon }=21$, $k_s=20$, $N_f=30.2$, $s_f=0.31$ (after equilibration).

Figure 5

(Color online) Schematic description of the initial state for lattice sites $k\ge 0$: Two identical density distributions for hard-core bosons, belonging to different species $A$ (black) and $B$ (blue), are shifted $2k_{\epsilon }$ lattice sites apart. The region of noninteger filling has width ${\delta }_{\epsilon }$. In the course of the protocol the lattice of species $B$ is shifted $k_s=3{\delta }_{\epsilon }$ sites to the left.

Figure 6

(Color online) Analytical (solid) and numerical (dots) results for the entropy per particle $s=S∕N$ versus the inverse temperature $\beta U$ for fixed trap strength $b∕U=1∕2500$ and fixed $\mu =U$. We plot the initial value (black line) and the values after filtering with $F_1$ [Eq. (17), brown line]. This is compared to more elaborate cooling protocols: entropy after two iterations of sequential filtering including equilibration [Eq. (22), red line], minimal entropy after sequential filtering without equilibration (cyan), and entropy after algorithmic cooling [Eq. (32), blue line]. For the algorithmic protocol we have chosen $\epsilon =0.03$ and $k_s=2{\delta }_{\epsilon }$. Note that this protocol creates classical correlations between lattice sites. The numerics are therefore based on a representation of classical density matrices in terms of matrix-product states [35].

Figure 7

Operation error $ϵ$ vs time $t{U}_a$ for different interaction energies, $U_b=0.2U_a$, $U_{ab}=0.2U_a$, $\delta =0.8$ (solid line); $U_b=U_a$, $U_{ab}=0.6U_a$, $\delta =0.8$ (dashed line); $U_b=U_a$, $U_{ab}=0.2U_a$, $\delta =1.6$ (dotted line); $U_b=U_a=U_{ab}$, $\delta =0$ (dashed-dotted line). We optimize a sequence of $M=10$ pulses (34).