Generation of entangled matter qubits in two opposing parabolic mirrors

We propose a scheme for the remote preparation of entangled matter qubits in free space. For this purpose, a setup of two opposing parabolic mirrors is considered, each one with a single ion trapped at its focus. To get the required entanglement in this extreme multimode scenario, we take advantage of the spontaneous decay, which is usually considered as an apparent nuisance. Using semiclassical methods, we derive an efficient photon-path representation to deal with this problem. We also present a thorough examination of the experimental feasibility of the scheme. The vulnerabilities arising in realistic implementations reduce the success probability, but leave the fidelity of the generated state unaltered. Our proposal thus allows for the generation of high-fidelity entangled matter qubits with high rate.

Figure 1

Scheme of the setup, including the postselection procedure: Two ${}^{171}\mathrm{Yb}{}^{+}$ ions are trapped at the foci of two parabolic mirrors. Entanglement between the two ions is mediated by a circularly polarized photon ($\sigma$) emitted by ion 1 and absorbed by ion 2. Successful entanglement is probed by the dispersive interaction of weak linearly polarized coherent states ($\pi$) with the ions. Only if an ion resides in one of the desired entangled states, a phase shift is imprinted onto the coherent state. Probe pulses are coupled into the parabolic mirrors by means of beam splitters. For simplicity, the coherent pulses used for dispersive state detection are indicated for only one of the two ions.

Figure 2

Hyperfine level scheme of a ${}^{171}{\mathrm{Yb}}^{+}$ ion: The states of the logical qubit, depicted with lighter colors, are defined by the electronic levels $|6^2{S}_{1/2},F=1,m=-1\rangle$ and $|6^2{S}_{1/2},F=1,m=1\rangle$.

Figure 3

Sequence of the processes which are the building blocks for the remote entanglement preparation: The states of the logical qubit, depicted with lighter colors, are defined by the electronic levels $|6^2{S}_{1/2},F=1,m=-1\rangle$ and $|6^2{S}_{1/2},F=1,m=1\rangle$. Optical transitions which are necessary for the entanglement generation are indicated by solid arrows, whereas the undesired transitions are indicated by dashed arrows. The three columns correspond to three phases. The first column shows the possible decay channels of the first ion's initially prepared state $|6^2{P}_{1/2},F=1,m=0\rangle$; the second column shows the possible excitation procedures of the second ion which was initially prepared in the state $|6^2{S}_{1/2},F=1,m=0\rangle$ followed by the spontaneous decay processes used to accomplish the state transfer from the $6^2{P}_{1/2}$ manifold to the radiatively stable $6^2{S}_{1/2}$ manifold; the last column shows the optical transitions used to perform the postselection procedure based on hyperfine splitting and off-resonant matter-field interactions.

Figure 4

Time evolution of the excitation probability $P$ in the case of ion 1 (dashed line) and of ion 2 (solid line): The interaction time $t$ is plotted in units of the time $\tau =(4f+d)/c$ which a photon needs to travel from the first ion to the second ion. $f$ is the focal length of the parabolas and $d$ is the distance between the foci. We set the spontaneous decay $\Gamma \tau =3$ and the Zeeman splitting was neglected.

Figure 5

Error probability $P$ in determining the phase of a coherent pulse probing the $\pi$ transitions from $S_{1/2}$ to $P_{1/2}$ plotted over the beam-splitter reflectivity $R$. Solid line: ion 1, dashed line: ion 2. In both cases, the relative detuning to resonance is two linewidths, corresponding to a phase shift of $0.14\pi$. The length of the pulse is ${10}^4$ upper-state lifetimes or 81 $\mu \mathrm{s}$, respectively. The amplitude of the coherent state incident onto the ion is chosen such that the probability to excite the respective upper state is $5\times {10}^{-4}$, as marked by the thin dotted line. The calculation for ion 2 accounts for the threshold reflectivity found for ion 1, which is marked by the crossing of the solid and the dotted line.