Entangling Strings of Neutral Atoms in 1D Atomic Pipeline Structures

We study a string of neutral atoms with nearest neighbor interaction in a 1D beam splitter configuration, where the longitudinal motion is controlled by a moving optical lattice potential. The dynamics of the atoms crossing the beam splitter maps to a 1D spin model with controllable time dependent parameters, which allows the creation of maximally entangled states of atoms by crossing a quantum phase transition. Furthermore, we show that this system realizes protected quantum memory, and we discuss the implementation of one- and two-qubit gates in this setup.

Figure 1

External beam splitter: (a) Atoms before (I) and after (II) the separation. The nearest neighbor interaction is denoted by $W$ and $J^x$ is the hopping matrix element between the two states $|a⟩$ and $|b⟩$ of the transverse trapping potential. Internal beam splitter: (b) Atoms in two different internal states $|a⟩$ and $|b⟩$ enter the beam splitter. These internal states are coupled by a Raman transition [see (c)] with a Rabi frequency $J^x=\Omega$. A laser excited Rydberg state $|r⟩$ realizes the off site interaction $W$ with $w$ the width of the interaction zone.

Figure 2

Homogeneous setup $J_l^x\equiv J^x$, $W_l\equiv W$: (a) Elementary excitations ${\epsilon }_{\nu }$ for $N=25$ against $J^x/W$. The dashed horizontal line indicates the ground state energy. (b) Upper bound and lower bound for the time $T$ yielding a fidelity of $F=95\%$ as a function of $N$. The dashed line illustrates the $N^2$ scaling predicted analytically. Inhomogeneous setup $J_l^x$, $W_l$: (c) Instantaneous eigenenergies $E_n$ for $N=6$. (d) Upper and lower bounds for the infidelity $1-F$ against $N$ for constant sweeping speed $v=0.01\lambda W$ and different interaction zone widths $w=0.1\lambda$ (diamonds), $w=0.2\lambda$ (solid circles), and $w=0.4\lambda$ (open circles) and similar for $J^x$.

Figure 3

Collectively enhanced interactions between two strings of atoms 1 and 2. (a) Antiferromagnetic setup: $N/2$ particles of each chain interact with strength $W^{\prime }$ only if they are in states $|0{⟩}_1|1{⟩}_2$ or $|1{⟩}_1|0{⟩}_2$ yielding a phase gate between the two qubits implemented by those chains. (b) Ferromagnetic setup: Entanglement creation between two chains of atoms via interactions $W^{\prime }$ in the state $|1{⟩}_1|0{⟩}_2$. Implementations with optical lattices or atom chips, for instance, offer the scalability of the scheme.

Figure 4

Illustration of the Hadamard gate for $N=8$ by adiabatically changing $J^z$ and $J^x$ (unprotecting the quantum memory). We follow the lowest two eigenstates (with energies given by the solid curves) transforming as $|0⟩+|1⟩\to |0⟩$ and $|0⟩-|1⟩\to |1⟩$ (up to a dynamical phase) when changing $J^x$, $J^z$ in three steps (1), (2), (3) [followed by turning off $J^z$ in step (4)] as described in the text. Note that if the condition $J^z<W/(N-1)$ is not fulfilled we get unwanted crossings and the first excited state after step (3) will not be of the form $|1⟩$. The dashed curve shows the third eigenenergy of the system and the inset the path in the $J^x-J^z$ plane.