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Physical Review A

Theory of a collective atomic recoil laser

Hong Yuan Ling, Han Pu, Leslie O. Baksmaty, N. P. Bigelow

Published Apr 17, 2001• 13 citations
Publisher

Abstract

We perform a study of a collective atomic recoil laser (CARL) that goes beyond the initial growth period. The study is based on a theory that treats both internal and external degrees of atomic freedom quantum mechanically but regards the laser light as a classical field obeying Maxwell's equations. We introduce the concepts of momentum families and diffraction groups and organize the matter wave equations in terms of diffraction groups. The steady-state lasing conditions are discussed in connection with the probe gain in the recoil-induced resonances. The nontrivial steady states and the linear stability analysis of the steady states are both carried out by the method of two-dimensional continued fractions. Both stable and unstable nontrivial steady states are calculated and discussed in the context of regarding the CARL as multiwave mixing involving many modes of matter waves and two optical fields.

DOI
10.1103/physreva.63.053810

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Steady states and instabilities of collective atomic recoil laser

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Summary form only given. The collective atomic recoil laser (CARL) was proposed as an atomic analog of the free electron laser. Recently, there has been an increased interest in the CARL primarily because it can be regarded as a parametric amplifier for the momentum side modes. Our CARL model consists of a cavity CARL signal of wave number k/sub 1/, and an opposing non-depletion pump field of k/sub 2/, and a steady flow of an ultra-cold two-level atomic beam along a direction normal to the propagation of the fields. Both fields are assumed to be far-off resonance. A momentum change of /spl planck/K, where K = k/sub 1/ + k/sub 2/, takes place in each virtual two-photon process. This allows us to divide the total momentum space into different momentum families, and each momentum family into different diffraction groups.

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A cloud of ultra-cold atoms is loaded into the attractive potential of a light wave that is generated by two counter-propagating modes of a high-finesse ring resonator. The two modes are coupled by the atoms due to coherent Rayleigh scattering and generate a potential which acts back on the motion of the atoms. This feedback leads to a new frequency component and can be described in terms of the long time proposed collective atomic recoil laser (CARL). This model is investigated experimentally and extended by introducing an optical friction force acting on the atoms. This allows for steady state operation of the CARL. Furthermore, it leads to a threshold behaviour of the CARL that translates into a novel type of phase transition: while passing the threshold the initially homogeneous atomic distribution is bunched in space and velocity. With this behaviour the system turns out to acquire some of the main features of the so-called Kuramoto model which provides a very general description of a network of limit cycle oscillators.

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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 ${E}_{2}(t)$ and the gain depends linearly on the atomic density, while the CARL limit is one in which ${\mathrm{dE}}_{2}/dt={\ensuremath{\int}}_{0}^{t}{f(t,t}^{\ensuremath{'}}{)E}_{2}{(t}^{\ensuremath{'}}{)dt}^{\ensuremath{'}},$ 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.

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N. Piovella, Mary M. Cola, R. Bonifacio

A collective atomic recoil laser (CARL) realized with a Bose-Einstein condensate offers the possibility to investigate new effects in the coherent interaction between optical and matter waves. This paper discusses some aspects of the nonlinear evolution of scattered radiation and the matter-wave field in the high-Q cavity and superradiant CARL regimes.

Journal of Modern Optics•Apr 1, 2004•1 citation
We perform a study of a collective atomic recoil laser (CARL) that goes beyond the initial growth period. The study is based on a theory that treats both internal and external degrees of atomic freedom quantum mechanically but regards the laser light as a classical field obeying Maxwell's equations. We introduce the concepts of momentum families and diffraction groups and organize the matter wave equations in terms of diffraction groups. The steady-state lasing conditions are discussed in connection with the probe gain in the recoil-induced resonances. The nontrivial steady states and the linear stability analysis of the steady states are both carried out by the method of two-dimensional continued fractions. Both stable and unstable nontrivial steady states are calculated and discussed in the context of regarding the CARL as multiwave mixing involving many modes of matter waves and two optical fields.
Summary form only given. The collective atomic recoil laser (CARL) was proposed as an atomic analog of the free electron laser. Recently, there has been an increased interest in the CARL primarily because it can be regarded as a parametric amplifier for the momentum side modes. Our CARL model consists of a cavity CARL signal of wave number k/sub 1/, and an opposing non-depletion pump field of k/sub 2/, and a steady flow of an ultra-cold two-level atomic beam along a direction normal to the propagation of the fields. Both fields are assumed to be far-off resonance. A momentum change of /spl planck/K, where K = k/sub 1/ + k/sub 2/, takes place in each virtual two-photon process. This allows us to divide the total momentum space into different momentum families, and each momentum family into different diffraction groups.
We present a wave atom optics theory of the Collective Atomic-Recoil Laser, where the atomic center-of-mass motion is treated quantum mechanically. It extends the previous ray atom optics theory, which treated the center-of-mass atomic motion classically, to the realm of ultracold atoms. For the case of a far off resonant pump laser we derive an analytical solution which gives the linear response of the CARL system for both the quantum and classical regimes. A linear stability analysis reveals significant qualitative differences between these two regimes, which arise from the effects of diffraction on the atomic center-of-mass motion.
We formulate a wave-atom-optics theory of the collective atomic recoil laser (CARL) where the atomic center-of-mass motion is treated quantum mechanically. By comparing the predictions of this theory with those of the ray-atom-optics theory, which treats the center-of-mass atomic motion classically, we show that for the case of a far off-resonant pump laser the ray-optics model fails to predict the linear response of the CARL when the temperature is of the order of the recoil temperature or less. This is due to the fact that in this temperature regime one can no longer ignore the effects of matter-wave diffraction on the atomic center-of-mass motion.
A cloud of ultra-cold atoms is loaded into the attractive potential of a light wave that is generated by two counter-propagating modes of a high-finesse ring resonator. The two modes are coupled by the atoms due to coherent Rayleigh scattering and generate a potential which acts back on the motion of the atoms. This feedback leads to a new frequency component and can be described in terms of the long time proposed collective atomic recoil laser (CARL). This model is investigated experimentally and extended by introducing an optical friction force acting on the atoms. This allows for steady state operation of the CARL. Furthermore, it leads to a threshold behaviour of the CARL that translates into a novel type of phase transition: while passing the threshold the initially homogeneous atomic distribution is bunched in space and velocity. With this behaviour the system turns out to acquire some of the main features of the so-called Kuramoto model which provides a very general description of a network of limit cycle oscillators.
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, dE2/dt,{\mathrm{dE}}_{2}/dt,dE2​/dt, depends locally on E2(t){E}_{2}(t)E2​(t) and the gain depends linearly on the atomic density, while the CARL limit is one in which ${\mathrm{dE}}_{2}/dt={\ensuremath{\int}}_{0}^{t}{f(t,t}^{\ensuremath{'}}{)E}_{2}{(t}^{\ensuremath{'}}{)dt}^{\ensuremath{'}},$ 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.