Quantum ratchets for quantum communication with optical superlattices

We propose to use a quantum ratchet to transport quantum information in a chain of atoms trapped in an optical superlattice. The quantum ratchet is created by a continuous modulation of the optical superlattice which is periodic in time and in space. Though there is zero average force acting on the atoms, we show that indeed the ratchet effect permits atoms on even and odd sites to move along opposite directions. By loading the optical lattice with two-level bosonic atoms, this scheme permits us to perfectly transport a qubit or entangled state imprinted in one or more atoms to any desired position in the lattice. From the quantum computation point of view, the transport is achieved by a smooth concatenation of perfect swap gates. We analyze setups with noninteracting and interacting particles and in the latter case we use the tools of optimal control to design optimal modulations. We also discuss the feasibility of this method in current experiments.

Figure 1

(Color online) An optical superlattice arises from a combination of two potentials with different periods. By modulating the depths and displacements of these potentials we can raise and lower the tunneling rates between odd and even pairs of sites (red arrows). This way, by means of perfect swaps, the state of a particle can be transported along the lattice.

Figure 2

(Color online) (a) Model case of two lattice sites disconnected from the rest because $J_2=0$. By fixing the hopping between wells to a precise value $J$ for a time $T$, it is possible to swap the atoms. Note that the doubly occupied sites have more energy due to the interaction $U$ and a possible inhomogeneity of the lattice potential $\Delta$. (b) Combining the solution for a pair of sites we obtain one possible modulation of the hoppings $J_1$ (solid) and $J_2$ (dashed), namely, $J_1$ and $J_2$ are square waves in antiphase. This produces the quantum transport. (c) Values of $J$ and $T$ for which perfect transport is achieved. Each circle is a solution; the solid line joins the solutions with smallest hopping and the dashed line is the solution for noninteracting particles.

Figure 3

(Color online) (a) Hopping and (b) fidelity of the gate $\left(F={\left|⟨01|U\left(t\right)|10⟩\right|}^2\right)$ with the perfect transport for half a period $\left[0,T\right)$ during which $J_1\left(t\right)=J\left(t\right)$ and $J_2=0$. We find two solutions, one for $T=2\pi ∕U$ (solid) and another one for $T=4\pi ∕U$ (dashed), with three and two modes, respectively. In (c) we plot the corresponding modulations of the superlattice.

Figure 4

(Color online) On the upper figure we plot a solution of $J_1\left(t\right)$ (solid) and $J_2\left(t\right)$ (dashed) for a perfect transport of the qubit. In the lower figure we plot the average position of the qubit on the lattice as transported by these modulations. These plots have been computed using $T=4\pi ∕U$.

Figure 5

(Color online) Parameters of the double well potential as a function of the superlattice modulation $\Delta V=\left(V_x+V_2\right)∕2$. We plot the effective hopping between wells $J$ (dash-dot, right axis), the energy gap to higher bands $\Delta E$ (dashed), and the on-site interaction $u$ (solid). Everything is expressed in units of the recoil energy $E_r$.