Entanglement fidelity limits of photonically-networked atomic qubits from recoil and timing

Abstract

The remote entanglement of two atomic quantum memories through photonic interactions is accompanied by atomic momentum recoil. When the interactions occur at different times, such as from the random emission over the lifetime of the atomic excited state, the difference in recoil timing can expose ``which-path'' information and ultimately lead to decoherence. Time-bin encoded photonic qubits can be particularly sensitive to asynchronous recoil timing. In this paper we study the limits of entanglement fidelity in atomic systems due to recoil and other timing imbalances and show how these effects can be suppressed or even eliminated through proper experimental design.

Figures (3)

• Figure 1: Schematics of photon generation, interference and detection. (a) Atom-like emitter $q$ is bound in 3D with harmonic frequencies $\omega_{qi}$. (Extension to normal modes of a chain of atoms is straighforward and discussed in the text.) Atom is excited with probability $P_q$ by a laser pulse having wavevector $\mathbf{k}'_q$ and emits a single photon with wavevector $\mathbf{k}_q$ with probability $p_q$. (b) A laser pulse (blue) excites atom $q$ from state $\vert{g}\rangle_q$ to $\vert{e}\rangle_q$ (radiative lifetime $\tau$), followed by emission (red) to an equal superposition of atomic qubit states $\mid\downarrow_q\rangle$ and $\mid\uparrow_q\rangle$, correlated with photonic qubit state $\mu$ and $\nu$, respectively. (c) Photons from atomic qubits $q=A$ and $q=B$ are mode-matched onto the input ports of a beamsplitter (BS) that has field transmission and reflection coefficients $\mathfrak{t}$ and $\mathfrak{r}$ with $\mathfrak{t}^2+\mathfrak{r}^2=1$. After interference, the BS output ports are relabeled by $q' \in A',B'$ as shown. Finally, the photon qubit states are separated and detected. The figure shows an example for polarization qubits, where additional polarizing beamsplitters (PBS) direct each qubit state to independent detectors labeled by the photonic qubit $\mu$ or $\nu$ and output spatial mode $A'$ or $B'$.
• Figure 2: Phase space recoil displacements of a single harmonic motional mode $i$ of atom $q$ in a frame rotating at frequency $\omega_{qi}$, with initial coherent state amplitude $\alpha_{qi}$ (lowest black dots in each plot). (a) excitation and recoil with scaled momentum kick of magnitude $\eta'_{qi}$ (blue line), then after time $t_\mu$ and $t_\nu$ emission and recoil (red lines) to coherent states $\beta_{qi}(t_\mu)$ and $\beta_{qi}(t_\nu)$ (red dots) given by Eq. \ref{['eqn:alpha1']}. (b) For time-bin encoding with preset delay $T$ between excitation pulses, there is additional evolution. In this case, the excitation recoils have different phases (blue lines), corresponding to final coherent states (red dots) $\beta_{qi}(t_\mu)$ and $\beta^T_{qi}(t_\nu)$ from Eqs. \ref{['eqn:alpha1']}-\ref{['eqn:alpha2']}.
• Figure 3: Integration time window (dark shaded region) of two-photon events appearing on the $\mu$ and $\nu$ detectors at times $t_\mu$ and $t_\nu$, respectively. Each detector has an independent time window of $(0,T_D)$, and also the time difference between detections is restricted to be $|t_\nu - t_\mu| < T_{{\Delta}}$.