Prospects for storage and retrieval of a quantum-dot single photon in an ultracold ${}^{87}$Rb ensemble

Epitaxially grown quantum dots (QDs) are promising sources of nonclassical states of light such as single photons and entangled photons. However, in order for them to be used as a resource for long-distance quantum communication, distributed quantum computation, or linear optics quantum computing, these photons must be coupled efficiently to long-lived quantum memories as part of a quantum repeater network. Here, we theoretically examine the prospects for efficient storage and retrieval of a QD-generated single photon with a 1-ns lifetime in a multilevel atomic system. We calculate using an experimentally demonstrated optical depth of 150 that the storage (total) efficiency can exceed 46$\%$ (28$\%$) in a dense, ultracold ensemble of ${}^{87}$Rb atoms. Furthermore, we find that the optimal control pulse required for storage and retrieval can be obtained using a diode laser and an electro-optic modulator rather than a mode-locked, pulsed laser source. Increasing the optical depth, for example, by using Bose-condensed ensembles or an optical cavity, can increase the efficiencies to near unity. Aside from enabling a high-speed quantum network based on QDs, such an efficient optical interface between an atomic ensemble and a QD can also lead to entanglement between collective spin-wave excitations of atoms and the spin of an electron or hole confined in the QD.

Figure 1

Simple schematic of photon storage and retrieval in a $\Lambda$ system. (a) A quantum field $\mathcal{E}\left(\tau \right)$ and a classical control field $\Omega \left(\tau \right)$ with a common excited state detuning $\delta$ impinge on an ensemble of $\Lambda$-type atomic systems from the left. All atoms are initialized into the state $|g\rangle$. (b) The quantum field is transformed and stored into a ground-state coherence (spin wave) $S(\tilde{z},\tau )$. (c) The spin wave is converted back into a photon by exciting the system with another classical control field from the right. (d) The single photon exits the ensemble propagating to the left.

Figure 2

Diagram of the level structure considered for photon storage. The quantum field $\mathcal{E}\left(\tau \right)$ oscillates at frequency ${\nu }_{eg}={\omega }_{eg}+{\delta }_g$ while the classical control field $\Omega \left(\tau \right)$ oscillates at frequency ${\nu }_{es}={\omega }_{es}+{\delta }_s$.

Figure 3

Results of the numerical calculation using gradient ascent for storage and retrieval of a quantum-dot photon in an idealized Rb gas. The storage (total) efficiency is plotted as a function of the optical depth $d$ in blue (red). Solid curves are guides to the eye.

Figure 4

Optimal control pulses for quantum-dot photon storage with $d=\{10.0,26.4,78.0,230,678\}$ determined using gradient ascent. The Rabi frequencies are scaled by $\gamma =2\pi \times 3.035$ MHz.

Figure 5

Diagram of the level structure considered for photon storage in a four-level system. The structure is the same as Fig. 2 except an additional excited state $|2\rangle$ is included at an energy ${\Delta }_e$ above $|1\rangle$.

Figure 6

Optimal control pulses for quantum-dot photon storage in a three-level system (short-dotted green), the 4L$+$ four-level scenario (solid blue), and the 4L$-$ four-level scenario (long-dotted, red).

Figure 7

The storage efficiency after gradient ascent optimization as a function of detuning $\delta ={\delta }_g={\delta }_s$. The slight asymmetry about $\delta /{\Delta }_e=0.5$ is due to the finite accuracy of the numerics (see Appendix ).

Figure 8

Optical transitions of ${}^{87}$Rb that are eligible for implementation of a quantum memory.

Figure 9

Optimized control pulses for the configurations detailed in Table .

Figure 10

Optimized control pulses for the configurations detailed in Table .

Figure 11

Optimized control pulses for configurations 2 and 3 detailed in Table .

Figure 12

Storage and total efficiencies for configuration 4 of $D_2$ storage with ground states $|F=2,m_F=0\rangle$ and $|F=1,m_F=0\rangle$ as a function of ${\delta }_g$. All other parameters are kept fixed.

Figure 13

Storage and total efficiencies for configuration 4 of $D_2$ storage with ground states $|F=2,m_F=0\rangle$ and $|F=1,m_F=0\rangle$ as a function of $\Delta {\omega }_{\mathrm{add}}$ resulting from spectral wandering.

Figure 14

Average storage (a) and total (b) efficiencies for configuration 4 of $D_2$ storage with ground states $|F=2,m_F=0\rangle$ and $|F=1,m_F=0\rangle$ as a function of $D_{\varphi }=\Delta {\omega }_{\mathrm{add}}$ resulting from pure dephasing (red, open circles). Uncertainties in $\eta$ are given by the standard deviation of the ensemble. For comparison, results from spectral wandering (Fig. 13) are shown as blue lines.

Figure 15

Optimized control pulses and ${\mathcal{E}}_{\mathrm{in}}$ for the Heaviside model and for the three-level (3L) model ($T_L=10$ ps) of the QD photon.

Figure 16

Three-level system including cross-coupling for four-wave mixing (FWM).