Defect-Suppressed Atomic Crystals in an Optical Lattice

We present a coherent filtering scheme which dramatically reduces the site occupation number defects for atoms in an optical lattice by transferring a chosen number of atoms to a different internal state via adiabatic passage. With the addition of superlattices it is possible to engineer states with a specific number of atoms per site (atomic crystals), which are required for quantum computation and the realization of models from condensed matter physics, including doping and spatial patterns. The same techniques can be used to measure two-body spatial correlation functions.

Figure 1

Plots showing avoided crossings in the energy eigenvalues for $N=1,2,3$ (note the vertical scale variation). We must choose ${\delta }_i$ and ${\delta }_f$ (dashed vertical lines) so that we cross only from $|N,0⟩\to |N-1,1⟩$, as illustrated in the inset.

Figure 2

(a) The parameter range for which appropriate values of ${\delta }_i$ and ${\delta }_f$ can be found when $N_{\mathrm{m}\mathrm{a}\mathrm{x}}=4$, in the adiabatic limit (light shading) and from numerical simulations with a smoothed rectangular pulse which rises and falls with a ${sin}^2(t/\tau )$ shape, $\tau =100/U_b$ and max $\Omega =0.3U_b$, giving a transfer error $ϵ<1\%$ (dark shading). (b) Transfer errors ${ϵ}_N$ as a function of $\Omega$ for $U_a=U_{ab}=0.2U_b$ and $\tau =200/U_b$ (the dashed lines show the corresponding results from the Landau-Zener formula). (c) Maximum transfer errors from $N=1,2,3$ as a function of $\tau$, for example, values of $U_a$ and $U_{ab}$ (parameters are shown in units of $U_b$). (d) Initial and final state fidelities, ${\mathcal{F}}_a$ and ${\mathcal{F}}_b$, as a function of temperature, $T$, for an initial MI state described at each site using the GCE. The inset shows values of $1-{\mathcal{F}}_a$ (solid line) and $1-{\mathcal{F}}_b$ for ${\overline{N}}_a=2$ (dashed line) and ${\overline{N}}_a=3$ (dotted line) on a logarithmic scale.

Figure 3

The production of (c) a BCS state at exactly half-filling in an optical lattice from (a) a degenerate Fermi gas using pattern loading techniques described in the text ($a\to b$). The 3D plots show the pair correlation function ${\rho }_{i,j,0}$, illustrating the crossover from (d) localized pairs [$V=10J$, $U=-10J$, MI regime (b)] to (e) delocalized pairs [$V=0$, $U=-0.1J$, BCS regime (c)]. Information about these correlations can be obtained from ${\gamma }_l=\sum _i{\rho }_{i-l,i+l}$. These results are from numerical diagonalization of (2) for eight sites and half-filling with periodic boundary conditions.

Figure 4

Numerical values of ${\kappa }_l$, which describes the pair correlation length. These results are examples from numerical diagonalization of (2) for 40 sites and 1 particle of each spin type and 16 sites and 2 particles of each spin type (inset) with (a) $V=0$, $U=-10J$, and (b) $V=0$, $U=-0.1J$.