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arXiv:math-ph/0410037 · Journal of Physics A Mathematical and General

Models for equilibrium BEC superradiance

Joseph V Pulé, André F. Verbeure, Valentin Anatol'evich Zagrebnov

Published Jul 1, 2004• 6 citations
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Abstract

We study the effect of electromagnetic radiation on the condensate of a Bose gas. In an earlier paper we considered the problem for two simple models showing the cooperative effect between Bose-Einstein condensation and superradiance. In this paper we formalise the model suggested by Ketterle et al in which the Bose condensate particles have a two level structure. We present a soluble microscopic Dicke type model describing a thermodynamically stable system. We find the equilibrium states of the system and compute the thermodynamic functions giving explicit formulae expressing the cooperative effect between Bose-Einstein condensation and superradiance.

DOI
10.1088/0305-4470/37/28/l03

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We study the effect of electromagnetic radiation on the condensate of a Bose gas. In an earlier paper we considered the problem for two simple models showing the cooperative effect between Bose-Einstein condensation and superradiance. In this paper we formalise the model suggested by Ketterle et al in which the Bose condensate particles have a two level structure. We present a soluble microscopic Dicke type model describing a thermodynamically stable system. We find the equilibrium states of the system and compute the thermodynamic functions giving explicit formulae expressing the cooperative effect between Bose-Einstein condensation and superradiance.
Motivated by recent experiments with superradiant Bose-Einstein Condensate (BEC) we consider simple microscopic models describing rigorously the interference of the two cooperative phenomena, BEC and radiation, in thermodynamic equilibrium. Our resuts in equilibrium confirm the presence of the observed superradiant light scattering from BEC: (a) the equilibrium superradiance exists only below a certain transition temperature; (b) there is superradiance and matter-wave (BEC) enhancement due to the coherent coupling between light and matter.
In this paper, we consider two models which exhibit equilibrium BEC superradiance. They are related to two different types of superradiant scattering observed in recent experiments. The first one corresponds to the amplification of matter waves due to Raman superradiant scattering from a BE condensate, when the recoiled and the condensed atoms are in different internal states. The main mechanism is stimulated Raman scattering in two-level atoms, which occurs in a superradiant way. Our second model is related to the superradiant Rayleigh scattering from a BE condensate. This again leads to a matter-wave amplification but now with the recoiled atoms in the same state as the atoms in the condensate. Here the recoiling atoms are able to interfere with the condensate at rest to form a matter-wave grating (interference fringes ) which is observed experimentally.
In this paper we consider two models which exhibit equilibrium BEC superradiance. They are related to two different types of superradiant scattering observed in recent experiments. The first one corresponds to the amplification of matter-waves due to Raman superradiant scattering from a cigar-shaped BE condensate, when the recoiled and the condensed atoms are in different internal states. The main mechanism is stimulated Raman scattering in two-level atoms, which occurs in a superradiant way. Our second model is related to the superradiant Rayleigh scattering from a cigar-shaped BE condensate. This again leads to a matter-waves amplification but now with the recoiled atoms in the same state as the atoms in the condensate. Here the recoiling atoms are able to interfere with the condensate at rest to form a matter-wave grating (interference fringes) which is observed experimentally.
The features of Bose condensation in an equilibrium ideal gas consisting of two types of charged fermions and their bound states—hydrogenlike atoms—in the presence of equilibrium between the photons and matter are investigated. It is shown that under such conditions the main influence on the Bose–Einstein condensation comes from the existence of levels concerned with the hyperfine splitting of the ground state of the hydrogenlike atom. The critical temperature and condensate density as functions of magnetic field are determined by considering effects due to the additional splitting of the levels of the hyperfine structure of the ground state in an external uniform static magnetic field (the Zeeman and Paschen–Back effects). It is found that under conditions of total statistical equilibrium in the system, a condensate is formed only by atoms found in the lowest energy state. It is shown that in the absence of equilibrium between radiation and matter, in the region of ultralow temperatures and low densities, the system can be treated as a multicomponent ideal gas of hydrogenlike atoms. The existence of a hierarchy of individual transition temperatures of each of the samples to the state with Bose–Einstein condensation is established. Expressions are found for the critical temperatures and number densities of particles in the condensate for each of the system components.
The Dicke model and the superradiance of two-level systems in a radiation field have many applications. Recently, a Dicke quantum phase transition has been realized with a Bose-Einstein condensate in a cavity. We numerically solve the many-body Schrödinger equation and study correlations in the ground state of interacting bosons in a cavity as a function of the strength of a driving laser. Beyond a critical strength, the bosons occupy multiple modes macroscopically while remaining superradiant. This fragmented superradiance can be detected by analyzing the variance of single-shot measurements.
The atom - electromagnetic field interaction is studied in the Dicke model, wherein a single field mode is interacting with a collection of two level atoms at thermal equilibrium. It is found that in the superradiant phase of the system, wherein the Bose-Einstein condensation of photons takes place, the notion of photon as an elementary electromagnetic excitation ceases to exist. The phase and intensity excitations of the condensate are found to be the true excitations of electromagnetic field. It is found that in this phase, the atom interacts with these excitations in a distinct coherent transition process, apart from the known stimulated emission/absorption and spontaneous emission processes. In the coherent transition it is found that while the atomic state changes in course of the transition process, the state of electromagnetic field remains unaffected. It is found that the transition probability of such coherent transition process is macroscopically large compared to other stimulated emission/absorption and spontaneous emission processes.