Photon echoes from an atomic frequency comb affected by population relaxation

The atomic frequency comb (AFC) is a promising approach for quantum memory, yet creating an ideal AFC only in the ground-state population remains a challenge. We studied an imperfect AFC initially created in the population inversion between the excited and ground states and redistributed due to population relaxation in a three-level system where a long-lived metastable state plays a crucial role. A theoretical model has been developed to treat the storage of a weak input pulse and retrieval in the form of a photon echo. By comparing theory with experimental results we have studied the echo efficiency and overall attenuation of an input pulse in the AFC affected by both its initial spectral structure and the population relaxation.

Figure 1

(a). Squared function as an example of a single tooth $\xi \left(\mathrm{\Delta }\right)$, Dirac delta function $d\left(\mathrm{\Delta }\right)$, and convolution $d\left(\mathrm{\Delta }\right)\otimes \xi \left(\mathrm{\Delta }\right)$. (b). Population inversion $w\left(\mathrm{\Delta }\right)$ with a peak value of −1 and background $b=0$. (c) AFC with a peak value of −1 and background at maximum ($b=1$) and minimum ($b=-1$).

Figure 2

(a). Three-level system. $T_e$, excited-state lifetime; $T_m$, metastable state lifetime; $\lambda$, $|e\rangle \to |m\rangle$ relaxation branching fraction. (b) Sketch of experiment setup. AOM, acousto-optical modulator; WM, wavelength meter; F-P, Fabry-Perot spectrum analyzer. Two crystals are cooled to $\sim 4\mathrm{K}$. Two optical isolators, various beam splitters, and mirrors make an experiment path ending at a photodetector, an optical feedback loop for frequency stabilization, and a monitoring port leading to the WM and F-P.

Figure 3

(a). The amplitude and frequency of eight WURST pulses (20 μs each). (b) Transmission of an 800-μs-long pulse from a linear frequency chirp of 40 MHz centered on the AFC. (c) Transmitted input pulse and echo of 5.2- and 9.6-MHz bandwidth.

Figure 4

Measured ratio ($R$) of the echo power to the transmitted input as a function of delay time ($T_d$) for 5.2- (diamonds) and 9.6-MHz (triangles) bandwidth, and fitting to theory (solid curve).

Figure 5

Transmittance for 5.2- (diamonds) and 9.6-MHz (triangles) input pulses and their fitting to linear functions.

Figure 6

Comparison of measured values with theory. Circles, measured transmittance of input pulse; squares, measured efficiency of forward propagating echo; solid line, corresponding theoretical curves; dashed line, theoretical efficiency of backward propagating echo.