VOLUME 86, NUMBER 20 PHYSICAL REVIEW LETTERS 14 MAY 2001 Phase Transition in a Radiation-Matter Interaction with Recoil and Collisions M. Perrin, 1 G. L. Lippi, 1 and A. Politi 1,2 1 Institut Non Linéaire de Nice, UMR 6618 CNRS, Université de Nice-Sophia Antipolis, 1361 Route des Lucioles, F-06560 Valbonne, France 2 Istituto Nazionale di Ottica Applicata, Largo E. Fermi 6, 50125 Firenze, Italy (Received 19 January 2001) The standard model introduced to describe the collective atomic recoil of an ensemble of atoms inter- acting with a strong electromagnetic field has been here extended by the inclusion of collisions with a buffer gas. As a result, we find that in the thermodynamic limit the coherent emission of radiation ex- hibits a continuous phase transition upon increasing the pump intensity. The output laser field is strictly larger than 0 only above a critical value. We find that the transition is not associated with the onset of spatial ordering but rather with the onset of a synchronization between the polarization phase and spatial position. A coherence parameter is introduced to characterize the phase transition. DOI: 10.1103/PhysRevLett.86.4520 PACS numbers: 42.50.Vk, 05.45.Xt, 05.65.+b, 42.65.Sf A few years ago, a theoretical model, describing the interaction between a strong electromagnetic field and quasiresonant two-level atoms, predicted the occurrence of an instability. Under appropriate conditions, the initially disordered atomic sample would organize itself along the direction of propagation of the (pump) field and form a periodic structure in space. As a result, part of the incident field would be reflected back, giving rise to what has been called a collective atomic recoil laser (CARL) [1]. The new ingredient contained in that model is the self-consistent treatment of the momentum transfer in each atom-field interaction to account for its effect on the system’s global behavior. Experimental indications suggesting the existence of such an ordering transition have been reported in Refs. [2,3]. However, the simplifications of the model are so strong that neither experiment [2,3] could be conducted under conditions that matched the theory [1]. An alternative interpretation of the experimental results was proposed, which did not make use of atomic recoil [4]. A simple polarization grating, as demonstrated experi- mentally in potassium vapor [5], could produce some of the features observed in the experiments [2,3]. However, applying those same ideas to one of the experimental sys- tems showed that removing recoil from the interaction [6] produced results that were no longer in agreement with the original observations [2]. Very recently, the collective interaction between matter and radiation has been experimentally demonstrated on a Bose-Einstein condensate [7]. This result renews the in- terest in collective atom-field problems, in spite of the fact that the physical picture is much simpler in this case. In- deed, since the matter is already in a collective state, the appearance of a spatial modulation at the optical wave- length is not too surprising. The aim of this Letter is to reexamine the problem for noncondensed atoms by removing one of the simplifica- tions of the model which is at the basis of a strong dis- agreement with the experimental conditions. In [1] the atoms were considered to move with (nearly) the same (vectorial) velocity and to be free of collisions. While such a representation holds well in an accelerated beam (e.g., for the free electron laser [8] from which the idea for [1] came), the atomic density values required for the instability to occur are so high that a supersonic beam [3] or —even better — a cell [2,3] is necessary. The cell, however, presents the two following features that strongly contrast with the model’s approximations: (1) the atomic motion is of thermal nature and does not occur with the same (vectorial) velocity for all atoms; (2) collisions are present in the vapor (and become im- portant when a buffer gas is used [2,3]). Both effects are expected to destroy any spatial ordering that may ap- pear in the system. Thermal motion will displace the atoms so that a “density grating” should be prevented from appearing (or would be washed out, if it existed some- how). On the other hand, even if a periodic structure were created, collisions would bump the atoms out of their privileged positions, homogenizing the atomic dis- tribution in space [9]. It is therefore legitimate to harbor doubts as to whether the CARL effect should still ex- ist in the presence of collisions and thermally distributed atoms. A positive answer was provided in [10], where both Doppler broadening and collisions have been taken into account. However, the further simplifications and as- sumptions introduced to derive analytic expressions do not yet allow clear-cut conclusions. A deeper understanding of CARL passes necessarily through the investigation of more realistic regimes. The model introduced in Ref. [1] involves four variables to describe each atom: the complex polarization S j , the population inversion D j , the position u j , and the momen- tum P j . Additionally, there is a single equation for the output field A 1 , while the dynamics of the input (pump) field A 2 is neglected, as the model equations are derived under the approximation of a small response. With refer- ence to the standard adimensional variables introduced in Ref. [1], the equations read as 4520 0031-9007 01  86(20)  4520(4)$15.00 © 2001 The American Physical Society