Memory imperfections in atomic-ensemble-based quantum repeaters

Quantum repeaters promise to deliver long-distance entanglement overcoming loss in realistic quantum channels. A promising class of repeaters, based on atomic ensemble quantum memories and linear optics, follows the proposal by L.-M. Duan et al., Nature (London) 414, 413 (2001). Here we analyze this protocol in terms of a very general model for the quantum memories employed. We derive analytical expressions for scaling of entanglement with memory imperfections, dark counts, loss, and distance, and we apply our results to two specific quantum memory protocols. Our methods apply to any quantum memory with an interaction Hamiltonian at most quadratic in the mode operators and are in principle extendible to more recent modifications of the original proposal of Duan, Lukin, Cirac, and Zoller.

Figure 1

(a) Entanglement generation. Two PDC’s (large boxes) emit two-mode squeezed states. One mode from each pair is stored in an atomic memory (circles), while the remaining modes are mixed on a 50/50 beam splitter and measured. Observation of a single click leads to entanglement between the atomic modes. The dashed elements are virtual beam splitters and PDC’s modeling losses and detector dark counts. (b) Entanglement connection is achieved by reading out the connecting ends of two entangled pairs and mixing on a 50/50 beam splitter. A single click heralds successful connection. For perfect memories, this setup is equivalent to the DLCZ setup [2].

Figure 2

Given two copies of a single-rail entangled state $\rho$, dual-rail entanglement is obtained by mixing and conditioning on a single click at each end. The half-filled circles denote excitations shared between two modes.

Figure 3

Entanglement connection with no transmission losses. Two pairs of entangled light modes, each in state ${\rho }_n$ extending over a distance $2^n{L}_0$, are connected. Detection of a single click heralds successful connection and leads to entanglement between the faraway ends of the pairs, leaving them in the state ${\rho }_{n+1}$ extending a distance $2^{n+1}{L}_0$. A full state transfer through the memories is included in the connection step.

Figure 4

Creation and propagation of errors. Filled circles denote excitations, empty circles denote vacuum. (a) During connection of two segments in $|{\Psi }^{+}⟩$, vacuum is read out but the connection is accepted due to a detector dark count leaving the remaining modes in the separable state $|11⟩$. (b) Once an error has occurred, it may propagate. Connecting $|11⟩$ with an ideal entangled state and requiring a single click leads to $|11⟩$ in the output (if no loss occurs). A vacuum state of the form $|00⟩$ propagates in the same fashion.

Figure 5

(Color online) The effect of finite initial squeezing. $S\left(L\right)$ is plotted for $r={10}^{-2}$ and transmission losses of $p_{\text{gen}}=0.9$, $p_{\text{con}}=0.1$ in the case of counting (dots) and noncounting (circles) detectors. Dashed lines show the analytical approximations.

Figure 6

(Color online) Effect of detector dark counts in entanglement connection. $S\left(L\right)$ is plotted for ${\overline{n}}_{\text{dc}}={10}^{-4}$ and transmission losses of $p_{\text{gen}}=0.9$, $p_{\text{con}}=0.1$ for counting (dots) and noncounting (circles) detectors. Dashed lines are analytical results. The inset shows the effect of dark counts in entanglement generation for the same parameter values.

Figure 7

Significance of perturbation cross terms. $S\left(L\right)$ is plotted with dark counts in all detectors for ${\overline{n}}_{\text{dc}}={10}^{-5}$ in the two cases $r^2={10}^{-4}$ (left) and $r^2={10}^{-6}$ (right). When ${\overline{n}}_{\text{dc}}>r^2$, a large discrepancy is observed between the numerical result and the analytical approximation omitting perturbation cross terms (dashed line).

Figure 8

(Color online) Setup for the two-pass memory. An atomic ensemble is placed in a magnetic field, with the spin of the ensemble strongly polarized along the field. The atoms are traversed twice in orthogonal directions in the plane perpendicular to the field by a light beam. As a result, the light polarization state can be stored in or retrieved from the atomic spin.

Figure 9

(Color online) Effect of two-pass memory imperfections. $S\left(L\right)$ is plotted for $\xi ={10}^{-4},{10}^{-3}$ and transmission losses of $p_{\text{gen}}=0.9$, $p_{\text{con}}=0.1$ for counting (dots) and noncounting (circles) detectors. Dashed lines are analytical results.

Figure 10

(Color online) Effect of one-pass memory imperfections. $S\left(L\right)$ is plotted for squeezing parameters (from the top down) $s={10}^5,{10}^3,10$ and transmission losses of $p_{\text{gen}}=0.9$, $p_{\text{con}}=0.1$ for counting (dots) and noncounting (circles) detectors. Dashed lines are analytical results.

Figure 11

(Color online) The rate for a fixed imperfection in $\mathit{S}$ of 5%. We plot the two cases of counting (circles) and noncounting (dots) detectors for $p_{\text{gen}}=0.9$ and $p_{\text{con}}=0.1$ (upper curves), $p_{\text{con}}=0.9$ (lower curves). The attenuation length is $L_{\text{att}}=20\text{km}$.

Figure 12

(Color online) Plot of the relative error (LHS-RHS)/LHS, where LHS and RHS refer to Eq. (E11).