Quantum Processing Photonic States in Optical Lattices

The mapping of photonic states to collective excitations of atomic ensembles is a powerful tool which finds a useful application in the realization of quantum memories and quantum repeaters. In this work we show that cold atoms in optical lattices can be used to perform an entangling unitary operation on the transferred atomic excitations. After the release of the quantum atomic state, our protocol results in a deterministic two qubit gate for photons. The proposed scheme is feasible with current experimental techniques and robust against the dominant sources of noise.

Figure 1

Creation of a line of atoms in $|b⟩$. (a) A $\pi /2$ pulse is applied to the target atoms. (b) $|a⟩$ and $|b⟩$ components of the target qubits are separated spatially by a $|b⟩$ lattice shift along $-\widehat{z}$. (c) The $|b⟩$ lattice is further displaced along $-\widehat{z}$, such that the control atom in $|b⟩$ interacts successively with the $|a⟩$ part of target atoms along its path, each time leading to a collisional phase $\pi /2$. Both lattice shifts are reversed leaving all atoms in their original positions. (d) A $-\pi /2$ pulse is applied to the target qubits. Atoms which have interacted with the control atom are transferred to $|b⟩$.

Figure 2

Mapping of excitations in the bulk to the plane. (a) A $\pi /2$ pulse is applied to the plane. (b) The $|b⟩$ lattice is shifted along $\widehat{x}$ such that atoms in $|b⟩$ in the bulk interact with the $|a⟩$ part of the plane. The time spent after each single-site displacement is chosen such that a phase $\pi /2$ is accumulated if a collision occurs. Then this lattice movement is reversed. Thus the initial positions of the atoms are restored and each target atom which is located on the path of a control atom in $|b⟩$ experienced two collisions and picked up a total phase of $\pi$. (c) Finally the plane is subjected to a $-\pi /2$ pulse, which transfers most of the atoms back to $|a⟩$. Only atoms, which suffered a collision, are transferred to $|b⟩$.

Figure 3

Initialization of the lattice. (a) A control atom in $|b⟩$ is placed outside the ensemble. (b) The control qubit interacts successively with a row of target atoms in the ensemble, thus transferring them to $|b⟩$, as explained in Fig. 1. (c) The resulting line of atoms in $|b⟩$ is separated from the ensemble along $-\widehat{y}$. (d) The line is now used to create a plane of atoms in $|b⟩$. For this purpose a $\pi /2$ pulse is applied to the ensemble, collisions are induced by a $|b⟩$ lattice shift along $\widehat{y}$, and a $-\pi /2$ pulse is applied to the bulk. Since each control atom in the line leads to a line of atoms in $|b⟩$, which is aligned along $\widehat{y}$ we obtain a plane in the $\widehat{x}\widehat{y}$ plane. (e) The plane is separated from the ensemble by a $|b⟩$ lattice shift along $\widehat{x}$.

Figure 4

Quantum gate protocol transforming the input state $|{\Psi }^{\mathrm{in}}{⟩}_A=\alpha |0{⟩}_A+\beta |1{⟩}_A+\gamma |2{⟩}_A$ into $|{\Psi }^{\mathrm{out}}{⟩}_A=\alpha |0{⟩}_A+i\beta |1{⟩}_A+\gamma |2{⟩}_A$. (a)–(c) Excitations in the Mott insulator are successively mapped to structures of lower dimensionality resulting in a single atom being in state $|a⟩/|b⟩$ in case of an even/odd number of excitations in the Mott insulator. (d) A state-dependent phase is applied to the isolated particle such that $|1{⟩}_A\mapsto i|1{⟩}_A$. Subsequently steps (a)–(c) have to be reversed.