Dynamical oscillator-cavity model for quantum memories

We propose a dynamical approach to quantum memories using an oscillator-cavity model. This overcomes the known difficulties of achieving high quantum input-output fidelity with long storage times compared to the input signal duration. We use a generic model of the memory response, which is applicable to any linear storage medium ranging from a superconducting device to an atomic medium. The temporal switching or gating of the device may either be through a control field changing the coupling, or through a variable detuning approach, as in more recent quantum memory experiments. An exact calculation of the temporal memory response to an external input is carried out. This shows that there is a mode-matching criterion which determines the optimum input and output pulse time evolution. This optimum pulse shape can be modified by changing the gate characteristics. In addition, there is a critical coupling between the atoms and the cavity that allows high fidelity in the presence of long storage times. The quantum fidelity is calculated both for the coherent state protocol, and for a completely arbitrary input state with a bounded total photon number. We show how a dynamical quantum memory can surpass the relevant classical memory bound, while retaining long storage times.

Figure 1

(Color online) Proposed dynamical atom-cavity memory scheme. The cavity couples effectively to only one external incoming and outgoing mode, labeled here as $u_0^{\mathit{in}}$ and $u_0^{\mathit{out}}$, respectively. This implies an optimal pulse shape necessary for efficient imprinting and retrieval of the quantum information, as represented by the mode $a_0$, onto and from the atomic medium internal to the cavity. Storage is achieved through modulation of the atom-cavity coupling $g$ or detuning $\Delta$.

Figure 2

(Color online) A quantum memory involves three stages: writing, reading, and storing. The interaction is turned on, then off, then on, in a controllable way.

Figure 3

(Color online) Fidelity ${\overline{F}}_{\overline{n}}^c$ (dashed) and corresponding beam-splitter efficiency $\sqrt{{\eta }_M}$ (solid) required for a quantum memory as a function of $\overline{n}$ for coherent input states. High mean photon numbers $\overline{n}$ will require a high efficiency ${\eta }_M$ to claim of a quantum memory, whereas for low $\overline{n}$, a quantum memory is achievable for lower efficiencies.

Figure 4

(Color online) Average fidelity vs beam-splitter efficiency of a quantum memory for arbitrary input states with up to $n=1$ (solid line) and $n=2$ (dashed line) photons. The horizontal black lines indicate the fidelity required to surpass the classical limit in each case, while the vertical black lines give the corresponding efficiency threshold.

Figure 5

(Color online) $Q$-switched cavity input (dashed blue line) and output (solid red line) amplitudes with $\kappa =1$, ${\kappa }_S=0.1$, $T=2.0$. The input mode shape is mode matched to the time-reversed cavity decay, and vice versa.

Figure 6

(Color online) Cavity input (dashed blue line) and output amplitudes with $\gamma ∕\kappa =0.01∕4$ (solid black line), $\gamma ∕\kappa =0.05∕4$ (dash-dotted green line) for the zero detuning strategy $\Delta =0$. Corresponding dashed thin lines represent the information $b\left(t\right)$ stored in the atom. Here $C\simeq 100$ and $T_s=16∕\kappa$, and the critically damped case applies.

Figure 7

(Color online) Cavity input (dashed blue line) and output (solid red line) amplitudes in the zero detuning $\Delta =0$ case with $\kappa =4$, $g=2$, $\gamma =0.01$, $T_S=4.0$, using direct numerical integration. Input mode shape is mode matched to the critical cavity decay.

Figure 8

(Color online) Cavity input (dashed blue line) and output (solid black line) amplitudes with $\gamma ∕\kappa =0.01∕4$ and different storage time $T_S=4,8,15$. Input mode shape is mode matched to the critical cavity decay. Here $\kappa =4$, $g=2$.

Figure 9

(Color online) Cavity input (dashed blue line) and output amplitudes with $\gamma ∕\kappa =0.01∕4$ (solid red line). Input mode shape is mode matched to the critical cavity decay. $\kappa =4$, $g=(\kappa -\gamma )∕2$, $T_S=4.0$, ${\Delta }_L=27\pi$.