Efficiency and fidelity of photon-echo quantum memory in an atomic system with longitudinal inhomogeneous broadening

We have performed a theoretical analysis of the photon-echo-based quantum memory realized recently in A. L. Alexander et al., Phys. Rev. Lett. 96, 043602 (2006) in the medium with linear correlation between atomic spatial coordinates and frequency detunings of the inhomogeneously broadened transition. A more general vision on the physical picture of temporal and spatial light-atoms dynamics has been obtained in the analytical solutions, taking into account the atomic phase relaxation, inhomogeneous broadening, and arbitrary parameters of the probe light pulses. We have evaluated preferable experimental conditions for the storage of incoming data light field modes where quantum memory efficiency and fidelity have been found. Physics of high efficiency in the forward geometry of the quantum memory and advantages for the realization of more universal long-lived storage on this basis are clarified.

Figure 1

(Color online) The temporal scheme of the photon-echo-based QM. The first left blue arrow corresponds to the data light pulse; the two red arrows are the first and second laser control pulses; and the second blue arrow is an echo pulse.

Figure 2

(Color online) Behavior of the data light field in the medium with the gradient of the resonant atomic frequencies $(\pi \beta ∕\chi =\pi )$. The frequency gradient $\Delta (\tau <{\tau }_1)=\chi z$ is parallel to the $z$ axis along the propagation of the data light field. The gradient is switched off at $t=100$, which leads to a weak long-lived stationary light field near $z=0$.

Figure 3

(Color online) Real part of the echo pulse in Eq. (21) as a function of the interaction time $t_1$ [data pulse envelope is ${\cosh }^{-1}(t∕T_1)$]. The echo pulse demonstrates an additional phase modulation for short interaction time $t_1$.

Figure 4

(Color online) Storage and reconstruction of the data light pulse with the pulse area of 18° $\Delta (\tau <{\tau }_1)=\chi z$, $\Delta (\tau >{\tau }_1)=\chi z$ $(1<\pi \beta ∕\chi <\pi )$. The sign of the gradient was changed at $\tau =100$.

Figure 5

(Color online) Efficiency $E_{\mathrm{Ch}}^{\mathrm{int}}∕f({\gamma }_{31}{t}_{21},\pi \beta ∕\chi )$ as a function of two variables $t_1∕T_1$: 0–0.1 and$t_1∕T_1$: 0–7.

Figure 6

(Color online) Difference of efficiencies between $E_{\exp }^{\mathrm{int}}∕f(t_{21},\pi \beta ∕\chi )$ and $E_{\mathrm{Ch}}^{\mathrm{int}}∕f(t_{21},\pi \beta ∕\chi )$ for the two data pulse envelopes $\{t_1:\mathrm{1,250}\}$, $\{\delta t:\mathrm{1,200}\}$, ${\gamma }_{21}=0.0005$, $\beta =1$, $\chi =1$. ($E_{\exp }^{\mathrm{int}}>E_{\mathrm{Ch}}^{\mathrm{int}}$ for a small time delay $t_1⩽T_1$).

Figure 7

Map of the efficiencies: pictures (a) and (b) correspond to $E_{\mathrm{Ch}}^{\mathrm{int}}$ and similar pictures (c) and (d) correspond to $E_{\exp }^{\mathrm{int}}$. Down white areas in (a) and (c) correspond to the efficiency: $1>E_{\mathrm{Ch}}^{\mathrm{int}}$, $E_{\exp }^{\mathrm{int}}>0.9$; the efficiency is smaller on the 0.1 step for each next area. Down white areas in (b) and (d) correspond to $1>E_{\mathrm{Ch}}^{\mathrm{int}}$, $E_{\exp }^{\mathrm{int}}>0.99$; each next step is smaller on 0.01. Efficiency $E_{\exp }^{\mathrm{int}}$ in (c) and (d) demonstrate a sharper behavior and higher efficiency in comparison with $E_{\mathrm{Ch}}$ for short time delays $t_1∕T_1<3$.

Figure 8

(Color online) Fidelity as a function of optical density $\pi \beta ∕\chi$ for different values of inhomogeneous broadening width in units of data light spectrum width (small interaction time: $t_1∕T_1=2$): red (solid) line—$T_1\chi L=8$; orange (small dashed) line—$T_1\chi L=10$; green (dotted-dashed) line—$T_1\chi L=20$, blue (dashed) line—$T_1\chi L=50$. It is seen that increasing inhomogeneous broadening decreases the fidelity due to the enhancement of the nonlinear phase modulation.

Figure 9

(Color online) It is seen that increasing the interaction time $t_1∕T_1$ from 2–15 increases the fidelity for small inhomogeneous broadening (a) $T_1\chi L=10$ and (b) $T_1\chi L=20$.

Figure 10

(Color online) Fidelity as a function of optical density $\pi \beta ∕\chi$ and inhomogeneous broadening $T_1\chi L$ for interaction time $t_1∕T_1=5$. Increasing $T_1\chi L$ above decreases the fidelity The decrease is enhanced with an increase of optical density $\beta ∕\chi$.

Figure 11

(Color online) Normalized spatial excitation of atoms ${\rho }_{22}(z^{\prime };{\omega }^{\prime })={\mid r_{12}(z^{\prime };{\omega }^{\prime })\mid }^2$ arising due to the interaction with light field mode ${\tilde{A}}_1({\omega }^{\prime },z_0)$.