Comparison of recoil-induced resonances and the collective atomic recoil laser

The theories of recoil-induced resonances (RIR) [J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, Phys. Rev. A 46, 1426 (1992)] and the collective atomic recoil laser (CARL) [R. Bonifacio and L. De Salvo, Nucl. Instrum. Methods Phys. Res. A 341, 360 (1994)] are compared. Both theories can be used to derive expressions for the gain experienced by a probe field interacting with an ensemble of two-level atoms that are simultaneously driven by a pump field. It is shown that the underlying formalisms of the RIR and CARL are equivalent. Differences between the RIR and CARL arise because the theories are typically applied for different ranges of the parameters appearing in the theory. The RIR limit is one in which the time derivative of the probe field amplitude, ${\mathrm{dE}}_2/dt,$ depends locally on $E_2\mathrm{}\left(t\right)\mathrm{}$ and the gain depends linearly on the atomic density, while the CARL limit is one in which ${\mathrm{dE}}_2/dt={\int }_0^t{f(t,t}^{\prime }{)E}_2{(t}^{\prime }{)dt}^{\prime },$ where f is a kernel, and the gain has a nonlinear dependence on the atomic density. Validity conditions for the RIR or CARL limits are established in terms of the various parameters characterizing the atom-field interaction. The probe gain for a probe-pump detuning equal to zero is analyzed in some detail, in order to understand how gain arises in a system which, at first glance, appears to have a symmetry that would preclude the possibility for gain. Moreover, it is shown that these calculations, carried out in perturbation theory, have a range of applicability beyond the recoil problem. Experimental possibilities for observing CARL are discussed.