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Physical Review A

Reduced model for the description of radiation-matter interaction including atomic recoil

J. Javaloyes, Gian Luca Lippi, Antonio Politi

Published Sep 12, 2003• 5 citations
Publisher

Abstract

We show that a model for the collective atomic recoil laser, previously introduced to include collisions with an external buffer gas, can be reduced to a single dynamical equation for the probe amplitude. This is the result of a clever adiabatic elimination of the atomic variables and of the assumption of a negligible effect of the probe field onto the atomic motion. This reduced model provides a fairly accurate description of the phase diagram of the original set of equations and allows for the investigation of more realistic regimes, where the direct simulation of the full model would be otherwise unfeasible. As a result, we find that the onset of a coherent field can be either described by a second- or first-order transition, the former scenario being observable only below a given temperature. Moreover, the first-order transition is accompanied by an intrinsic optical bistability region.

DOI
10.1103/physreva.68.033405

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We show that a model for the collective atomic recoil laser, previously introduced to include collisions with an external buffer gas, can be reduced to a single dynamical equation for the probe amplitude. This is the result of a clever adiabatic elimination of the atomic variables and of the assumption of a negligible effect of the probe field onto the atomic motion. This reduced model provides a fairly accurate description of the phase diagram of the original set of equations and allows for the investigation of more realistic regimes, where the direct simulation of the full model would be otherwise unfeasible. As a result, we find that the onset of a coherent field can be either described by a second- or first-order transition, the former scenario being observable only below a given temperature. Moreover, the first-order transition is accompanied by an intrinsic optical bistability region.
We consider here a model previously introduced to describe the collective behavior of an ensemble of cold atoms interacting with a coherent electromagnetic field. The atomic motion along the self-generated spatially periodic force field can be interpreted as the rotation of a phase oscillator. This suggests a relationship with synchronization transitions occurring in globally coupled rotators. In fact, we show that whenever the field dynamics can be adiabatically eliminated, the model reduces to a self-consistent equation for the probability distribution of the atomic "phases." In this limit, there exists a formal equivalence with the Kuramoto model, though with important differences in the self-consistency conditions. Depending on the field-cavity detuning, we show that the onset of synchronized behavior may occur through either a first- or second-order phase transition. Furthermore, we find a secondary threshold, above which a periodic self-pulsing regime sets in, that is immediately followed by the unlocking of the forward-field frequency. At yet higher, but still experimentally meaningful, input intensities, irregular, chaotic oscillations may eventually appear. Finally, we derive a simpler model, involving only five scalar variables, which is able to reproduce the entire phenomenology exhibited by the original model.
The standard model introduced to describe the collective atomic recoil of an ensemble of atoms interacting 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 exhibits 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.
With the help of a microscopic model we investigate the effects of collisions and atomic recoil on the behavior of absorbing atoms placed inside a bidirectional resonator and driven by an external injected field. This model complements and generalizes earlier studies of the collective atomic recoil laser and of optical bistability. According to our model, even in the presence of collisions, the resonator can support bidirectional propagation. In particular, for appropriate selection of the parameters, we predict the existence of stationary solutions such that the cavity field that copropagates with the injected signal is locked in frequency with the external source, while the counterpropagating field is frequency shifted from both, even in steady state. The early stage of growth of the counterpropagating field is accompanied by a spatial modulation (grating structure) in the density of the medium, but this modulation decays away in a time roughly of the order of the average interval between collisions.
Our model, presented earlier, which predicts the circumstances under which intrinsic mirrorless optical bistability (OB) can occur due to atomic pair correlation in a small volume, is outlined and the results presented. These results, which Predict a first-order phase transition in steady state for an externally-driven collection of a large number of atoms far removed from thermodynamic equilibriun, forms the motivation for a detailed microscopic examination of the dynamical behavior of atonic pair correlation in the presence of externally-applied coherent radiation. 4 model is presented and results are discussed for the transient dynamic evolution of two, two-level atoms separated from each other by a distance r in the presence of an externally-applied coherent radiation field. The results predict collective radiation reaction, frequency shifts, relaxation in terms of the atomic seoaration r (assumed much larger than single atom dimensions), the externally-applied field intensity and spacial uniformity of the field with respect to the inter-atomic volume.
We have studied collective recoil lasing by a cold atomic gas, scattering photons from an incident laser into many radiation modes in free space. The model consists of a system of classical equations for the atomic motion of NNN atoms where the radiation field has been adiabatically eliminated. We performed numerical simulations using a molecular dynamics code pepc (Pretty Efficient Parallel Coulomb Solver) to track the trajectories of the atoms. These simulations show the formation of an atomic density grating and collective enhancement of scattered light, both of which are sensitive to the shape and orientation of the atomic cloud. In the case of an initially circular cloud, the dynamical evolution of the cloud shape plays an important role in the development of the density grating and collective scattering. The ability to use efficient molecular dynamics codes will be a useful tool for the study of the multimode interaction between light and cold gases.
We carry out a systematic adiabatic elimination of the atomic degrees of freedom from the quantum-mechanical master equation for the single-mode laser. We represent the reduced density operator of the field mode by various quasiprobabilities and construct the respective equations of motion. The generators of infinitesimal time translations are obtained as series in powers of two parameters, the smallness of which defines the adiabatic limit. Our "adiabatic" expansion treats that part of the atom-field interaction as a zeroth-order effect which describes the action of the field on the atoms, while the reaction of the atoms is treated perturbatively. The adiabatic equilibrium thus assigned to the atomic variables at all times is a conditional one, contingent on the current state of the field mode. As a result, saturation effects in the atoms are fully accounted for in low orders of our expansion. In second order, especially, we obtain a Fokker-Planck equation for the Wigner function of the field mode which is valid for arbitrary pump strengths below, near, and above threshold. We compare our results with those of previous theories.