Quantum state transfer in the presence of nonhomogeneous external potentials

Heisenberg-type spin models in the limit of a low number of excitations are useful tools to study basic mechanisms in strongly correlated and magnetic systems. Many of these mechanisms can be experimentally tested using ultracold atoms. Here, we discuss the implementation of a quantum state transfer protocol in a tight-binding chain in the presence of an inhomogeneous external potential. We show that it can be used to extend the parameter range in which high-fidelity state transfer can be achieved beyond the well established weak-coupling regime. Among the class of mirror-reflecting potentials that allow for high fidelity quantum state transfer, the harmonic case is especially relevant because it allows us to formulate a proposal for the experimental implementation of the protocol in the context of optical lattices.

Figure 1

Schematic diagram of the physical implementation of the proposed QST protocol. The spins are arranged in a nonhomogeneous potential obtained by applying a site-dependent external field. The goal of the protocol is to transfer the “up” spin (blue) from the first site to the last one.

Figure 2

$\mathcal{F}$ as a function of $p$ and the ratio $J_1/J$. The potential depth is $a=1/2$. Away from the weak-coupling regime, the presence of the inhomogeneous potential greatly enhances the QST quality.

Figure 3

Dynamics of ${\left|\langle 1|1\left(t\right)\rangle \right|}^2$ [red (gray)] and ${\left|\langle N|1\left(t\right)\rangle \right|}^2$ (black) for a chain of eight sites. The Hamiltonian parameters are $p=2$, $a=0.5$, and $J_1/J=1$. The fast oscillations represent the deviation of the real dynamics from the two-level approximation.

Figure 4

Upper panel: QST time $t^{*}$ [red (gray) line] as a function of $p$ in the weak-coupling limit ($J_1/J={10}^{-2}$) with $a=0.5$ for 12 spins. Here, $t^{*}$ is defined as the first time in which the receiver's site population probability has a relevant maximum (fast oscillations are washed out by averaging the signal over a few periods). The excitation probability at the receiver's site [$P_N\left(t^{*}\right)$] is plotted in black. As explained in the main text, the dips around $p\left(t^{*}\right)$ are caused by the appearance of two oscillation frequencies with unbalanced weights. Lower panel: spectrum of $H$. The two almost degenerate eigenvalues responsible for high-fidelity QST are the continuous black line, while all the other eigenvalues are in drawn in red (gray).

Figure 5

$\mathcal{F}$ as a function of $p$ and $a$ in the constant coupling regime ($J_1=J$) for $N=8$.

Figure 6

QST time $t^{*}$ for $J_1=J$, $p=2$, and for eight sites. The red (gray) line represents the first time ${\left|\langle N|1\left(t\right)\rangle \right|}^2$ reaches the threshold value of $0.95$, while the black line is the theoretical estimation $\pi |E_{+}-E_{-}{|}^{-1}$ of a Rabi oscillation period.