Analysis of deterministic swapping of photonic and atomic states through single-photon Raman interaction

The long-standing goal of deterministic quantum interactions between single photons and single atoms was recently realized in various experiments. Among these, an appealing demonstration relied on single-photon Raman interaction (SPRINT) in a three-level atom coupled to a single-mode waveguide. In essence, the interference-based process of SPRINT deterministically swaps the qubits encoded in a single photon and a single atom, without the need for additional control pulses. It can also be harnessed to construct passive entangling quantum gates, and can therefore form the basis for scalable quantum networks in which communication between the nodes is carried out only by single-photon pulses. Here we present an analytical and numerical study of SPRINT, characterizing its limitations and defining parameters for its optimal operation. Specifically, we study the effect of losses, imperfect polarization, and the presence of multiple excited states. In all cases we discuss strategies for restoring the operation of SPRINT.

Figure 1

(a) A two-level system coupled to a single-mode waveguide resulting in complete reflection of the probe. (b) A three-level $\mathrm{\Lambda }$ system in which reflection of a single photon enforces a Raman transfer from one ground state to the other.

Figure 2

Schematic of an atomic three-level $\mathrm{\Lambda }$ system inside a single-sided cavity.

Figure 3

Schematic of the theoretical model; in inset, the atomic $\mathrm{\Lambda }$ system.

Figure 4

Fidelity as a function of $r_{\sigma }$ or $r_{\pi }$ for optimized (solid line) and nonoptimized ${\kappa }_{\mathrm{ex}}$ (dashed line).

Figure 5

Illustration of the 12 contributions to the reflection coefficient $R$, associated with no atomic toggle (a), and an atomic toggle to the state $G_2$ (b) or to the state $G_0$ (c). The SPRINT term is shown in a box.

Figure 6

(a) Two excited states instead of one provide two pathways for SPRINT. Depending on the signs of the coupling strengths, the symmetry required for SPRINT can be either maintained or removed. (b) Actual level scheme of the ${}^{87}\mathrm{Rb}{D}_2$ line. Transitions that are weak due to polarization mismatch are shown in gray. Note that the transition $G_0\to e^{\prime }$ is forbidden, and that the effect of the excited state $F^{\prime }=2$ is negligible, due to its large detuning and low coupling strength to $F=1$.

Figure 7

Fidelity as a function of $h$ (a), $r_{\sigma }$ (b), and ${\kappa }_i$ (c) for different values of ${\kappa }_{\mathrm{ex}}$ written in the figure in units of $2\pi$ MHz. For (a) and (c), the insets give the value of ${\kappa }_{\mathrm{ex}}$ that maximizes the fidelity; the corresponding maximal fidelity is shown as a dashed line in the main frames. In (b), $\mathcal{F}$ is insensitive to the choice of ${\kappa }_{\mathrm{ex}}$.

Figure 8

Contour plots of (a) ${\kappa }_{\mathrm{ex}}^{\mathrm{opt}}$ that maximizes the fidelity and (b) the corresponding maximal fidelity $\mathcal{F}\left({\kappa }_{\mathrm{ex}}^{\mathrm{opt}}\right)$ as a function of both $h$ and ${\kappa }_i$.

Figure 9

Simulations of the fidelity maximized by tuning ${\kappa }_{\mathrm{ex}}$ and ${\delta }_C$, as a function of ${\kappa }_i$. In solid lines, simulations with a realistic Gaussian input pulse (left inset) taking into account the transient dynamics. In dashed lines, calculations only involving the steady state with a long input pulse (here exponentially decaying, right inset); simulations with a wide Gaussian pulse coincide. Labels 1–5 refer to the following:123, 3’45$\mathrm{\Lambda }$-system $F\to F^{\prime }$1 → 11 → 1 and 1 → 0(3) 1 → 1 (3’) 1 → 01 → 1actual level scheme$g$$\overline{g}$$\overline{g}$gaussian spread$\overline{g}$gaussian spread$h,r_{\sigma },r_{\pi }$000≠ 0≠ 0model from Sec.3a3e3d4aThe dashed line 1 becomes the dotted line 1 when the transient dynamics is considered, with ${\kappa }_s=0.1\times 2\pi$ MHz. In all cases, $\gamma =3\times 2\pi$ MHz.