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arXiv:1305.1210 · Physical Review A

Bose-Einstein condensation of photons in an ideal atomic gas

Alexander Kruchkov, Yu. V. Slyusarenko

Published Jul 9, 2013• 23 citations
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Abstract

We study peculiarities of Bose-Einstein condensation of photons that are in thermodynamic equilibrium with atoms of noninteracting gases. General equations of the thermodynamic equilibrium of the system under study are obtained. We examine solutions of these equations in the case of high temperatures, when the atomic components of the system can be considered as nondegenerated ideal gases of atoms, and the photonic component can form a state with the Bose condensate. Transcendental equation for transition temperature and expression for the density of condensed photons in the considered system are derived. We also obtain analytical solutions of the equation for the critical temperature in a number of particular cases. The existence of two regimes of Bose condensation of photons, which differ significantly in nature of transition temperature dependence on the total density of photons pumped into the system, is revealed. In one case, this dependence is a traditional fractional-power law, and in another one it is the logarithmic law. Applying numerical methods, we determine boundaries of existence and implementation conditions for different regimes of condensation depending on the physical parameters of the system under study. We also show that for a large range of physical systems that are in equilibrium with photons (from ultracold gases of alkali metals to certain types of ideal plasma), the condensation of photons should occur according to the logarithmic regime.

DOI
10.1103/physreva.88.013615

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We study peculiarities of Bose-Einstein condensation of photons that are in thermodynamic equilibrium with atoms of noninteracting gases. General equations of the thermodynamic equilibrium of the system under study are obtained. We examine solutions of these equations in the case of high temperatures, when the atomic components of the system can be considered as nondegenerated ideal gases of atoms, and the photonic component can form a state with the Bose condensate. Transcendental equation for transition temperature and expression for the density of condensed photons in the considered system are derived. We also obtain analytical solutions of the equation for the critical temperature in a number of particular cases. The existence of two regimes of Bose condensation of photons, which differ significantly in nature of transition temperature dependence on the total density of photons pumped into the system, is revealed. In one case, this dependence is a traditional fractional-power law, and in another one it is the logarithmic law. Applying numerical methods, we determine boundaries of existence and implementation conditions for different regimes of condensation depending on the physical parameters of the system under study. We also show that for a large range of physical systems that are in equilibrium with photons (from ultracold gases of alkali metals to certain types of ideal plasma), the condensation of photons should occur according to the logarithmic regime.
We have studied conditions of photon Bose-Einstein condensate formation that is in thermodynamic equilibrium with ideal gas of two-level Bose atoms below the degeneracy temperature.Equations describing thermodynamic equilibrium in the system were formulated; critical temperatures and densities of photonic and atomic gas subsystems were obtained analytically.Coexistence conditions of these photonic and atomic Bose-Einstein condensates were found.There was predicted the possibility of an abrupt type of photon condensation in the presence of Bose condensate of ground-state atoms: it was shown that the slightest decrease of the temperature could cause a significant gathering of photons in the condensate.This case could be treated as a simple model of the situation known as "stopped light" in cold atomic gas.We also showed how population inversion of atomic levels can be created by lowering the temperature.The latter situation looks promising for light accumulation in atomic vapor at very low temperatures.
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.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2L^2L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01
Abstract The original Bose–Einstein condensation (BEC) was theoretically established in the 1920s, for the case of an ideal gas of boson particles (a consequence of the symmetry of the wave function). In the 1990s, it was realized in systems of atomic alkali gases. At the beginning of the 21st century, investigators considered its emergence in solid‐state matter, and a BEC‐like phenomenon was observed in the case of bosonic quasiparticles (elementary excitations in solids). Both types of BEC can occur in a condensation in conditions of thermodynamic equilibrium. A third type of what has been dubbed a BEC in non‐equilibrium conditions can occur in systems of bosonic quasiparticles (phonons, magnons, excitons); this phenomenon (which can be considered as cases of systems with complex behavior) follows after a certain threshold of intensity of an external pumping source is achieved. In this review, after brief comments on the two types of BEC in equilibrium, a description is presented on how a BEC‐like phenomenon arises in systems of quasiparticles that are driven away from thermodynamic equilibrium, and several examples are described. This description is given within a unique theoretical framework, and a comparison with experimental results is performed; in addition, the possible technological applications are outlined.
Bose–Einstein condensation of an ideal relativistic Bose gas trapped in a generic power-law potential is investigated. The analytical expressions for some important parameters such as the critical temperature, ground-state fraction and heat capacity are derived. The general criteria on the occurrence of Bose–Einstein condensation and the discontinuity of heat capacity at the critical temperature are obtained. The results obtained here present a unified description for the Bose–Einstein condensation of a class of ideal Bose systems so that many important conclusions in the literature are included in this paper.