$d$-Wave Resonating Valence Bond States of Fermionic Atoms in Optical Lattices

We study controlled generation and measurement of superfluid $d$-wave resonating valence bond (RVB) states of fermionic atoms in 2D optical lattices. Starting from loading spatial and spin patterns of atoms in optical superlattices as pure quantum states from a Fermi gas, we adiabatically transform this state to an RVB state by a change of the lattice parameters. Results of exact time-dependent numerical studies for ladders systems are presented, suggesting generation of RVB states on a time scale smaller than typical experimental decoherence times.

Figure 1

Optical potentials generated by two counterpropagating laser beams with wave vector $k=2\pi /\lambda$. (a) $V=V_0\sin (kx{)}^2$; (b) $V=V_0\sin (kx{)}^2+V_1\sin (kx/2{)}^2$; (c) $V=V_0\sin (kx{)}^2+V_1\sin (kx/2{)}^2+V_{2}x$; (d) $V=V_0\sin (kx{)}^2+V_{2}x$.

Figure 2

Schematic illustration of the adiabatic protocol that generates RVB states on two decoupled plaquettes with four atoms (left) and two atoms (right). The protocol sequence is (1) $t\to 1$; (2) $\mu \to 0$; (3) ${\mu }_{\perp }\to 0$. Optical potentials are sketched for the $x$ and $y$ directions, and the large (intermediate, small) circles indicate two (one, zero) atoms on a site.

Figure 3

Signature for $d$-wave RVB pairing when decoupling two plaquettes with six atoms for varying on-site repulsion $U/t$. (b) For $U/t\lesssim 4.5$, pairs are formed with a small binding energy shown in the inset. In the final state after sufficiently slow decoupling, the hole pair is located on one plaquette, while for large repulsion $U/t\gtrsim 4.5$ the unpaired holes separate into three atoms on each plaquette (right panel). (a) Projection of the final state onto the subspace with three atoms on each plaquette.

Figure 4

Singlet gaps for the path that couples and decouples two plaquettes illustrated by the optical potentials on top. Two plaquettes with four and two atoms are coupled for $U/t=2.5$ with a chemical potential gradient $\mu$ being applied along the $x$ direction (first three panels). The decoupling of plaquettes is shown for bound (fourth panel) and unbound pairs (right panels). Dashed (dotted) lines indicate states to which a transition from the ground state is forbidden due to symmetry constraints (particle conservation on the individual plaquettes). The $x$ and $y$ parities are given by “$+$” (even) and “$-$” (odd).

Figure 5

Ramping times to achieve a fidelity larger than 0.999 for an adiabatically evolved wave function as the chemical potential shift $\mu$ at predefined lattice sites is reduced. The adiabatic protocol is performed in small sequences $\mu \to \mu -\delta \mu$, with $\delta \mu =0.1$. Doping $\delta =1/8$. On the right: (a) decoupled plaquettes; (b) comblike structure; (c) ladder.