An ideal equilibrium Bose--Einstein condensate (BEC) is usually considered in the grand canonical ($μV T$) ensemble, which implies the presence of the chemical equilibrium with the environment. However, in most experimental scenarios, the total amount of particles in BEC is determined either by the initial conditions or by the balance between dissipation and pumping. As a result, BEC may possess the thermal equilibrium but almost never the chemical equilibrium. In addition, many experimentally achievable BECs are non-ideal due to interaction between particles. In the recent work [10.1103/PhysRevLett.128.065301], it has been shown that invariant subspaces in the system Hilbert space appear in non-equilibrium BEC in the fast thermalization limit. In each of these subspaces, Gibbs distribution is established with a certain number of particles that makes it possible to investigate properties of non-ideal non-equilibrium BEC independently in each invariant subspace. In this work, we analyze the BEC stability due to change in dispersion curve caused by non-ideal interactions in BEC. Generally, non-ideal interactions lead to the redshift or blueshift of the dispertion curve and to the change in the effective mass of the particles. We show that the redshift of the dispersion curve can lead to the negative compressibility of BEC, whereas the change in the effective mass always makes BEC more stable. We find the explicit condition for the particle density in BEC, at which the negative compressibility appears.
An ideal equilibrium Bose--Einstein condensate (BEC) is usually considered in the grand canonical (μVT) ensemble, which implies the presence of the chemical equilibrium with the environment. However, in most experimental scenarios, the total amount of particles in BEC is determined either by the initial conditions or by the balance between dissipation and pumping. As a result, BEC may possess the thermal equilibrium but almost never the chemical equilibrium. In addition, many experimentally achievable BECs are non-ideal due to interaction between particles. In the recent work [10.1103/PhysRevLett.128.065301], it has been shown that invariant subspaces in the system Hilbert space appear in non-equilibrium BEC in the fast thermalization limit. In each of these subspaces, Gibbs distribution is established with a certain number of particles that makes it possible to investigate properties of non-ideal non-equilibrium BEC independently in each invariant subspace. In this work, we analyze the BEC stability due to change in dispersion curve caused by non-ideal interactions in BEC. Generally, non-ideal interactions lead to the redshift or blueshift of the dispertion curve and to the change in the effective mass of the particles. We show that the redshift of the dispersion curve can lead to the negative compressibility of BEC, whereas the change in the effective mass always makes BEC more stable. We find the explicit condition for the particle density in BEC, at which the negative compressibility appears.
An ideal equilibrium Bose-Einstein condensate (BEC) is usually considered in the grand canonical (μVT) ensemble, which implies the presence of the chemical equilibrium with the environment. However, in most experimental scenarios, the total amount of particles in BEC is determined either by the initial conditions or by the balance between dissipation and pumping. As a result, BEC may possess the thermal equilibrium but almost never the chemical equilibrium. In addition, many experimentally achievable BECs are non-ideal due to interaction between particles. In the recent work [Shiskov et al., Phys. Rev. Lett. 128, 065301 (2022)0031-900710.1103/PhysRevLett.128.065301], it has been shown that invariant subspaces in the system Hilbert space appear in non-equilibrium BEC in the fast thermalization limit. In each of these subspaces, Gibbs distribution is established with a certain number of particles that makes it possible to investigate properties of non-ideal non-equilibrium BEC independently in each invariant subspace. In this work, we analyze the BEC stability due to change in dispersion curve caused by non-linearity in BEC. Generally, non-linearity leads to the redshift or blueshift of the dispersion curve and to the change in the effective mass of the particles. We show that the redshift of the dispersion curve can lead to the negative compressibility of BEC and onset of instability, whereas the change in the effective mass always makes BEC more stable. We find the explicit condition for the particle density in BEC, at which the negative compressibility appears. We show that the appearance of BEC instability is followed by the formation of stable and spatially inhomogeneous BEC.
Based on grand canonical assembles statistics,the Bose Einstein Condensation (BEC) in non interacting bound atomic system is studied.Some basic concepts of BEC in conventional infinite Bose system are extended.The condition of chemical potential μ ≤0 will not be guarantted for the present finite bound system,and the vapour with μ 0,the state without condensation at any eigenstate,might be possible in principle even below a critical temperature T C .Gibbs functions for both the gas and the BEC state at temperature TT C have been calculated.The condition for BEC to appear is presented.The relation of BEC temperature T C with the density of particles is estimated and the result is qualitatively in good agreement with the experimental results.
The theoretical description of non-equilibrium Bose–Einstein condensate (BEC) is one of the main challenges in modern statistical physics and kinetics. The non-equilibrium nature of BEC makes it impossible to employ the well-established formalism of statistical mechanics. We develop a framework for the analytical description of a non-equilibrium phase transition to BEC that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states. We consider the limit of fast thermalization and obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC which also covers the equilibrium case. For the particular cases of 2D and 3D, we investigate the non-equilibrium formation of BEC by finding the temperature dependence of the ground state occupation and second-order coherence function. We show that for a given pumping rate, the macroscopic occupation of the ground state and buildup of coherence may occur at different temperatures. Moreover, the buildup of coherence strongly depends on the pumping scheme. We also investigate the condensate linewidth and show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.
We report on the experimental observation of stimulated cooling in the non-equilibrium Bose-Einstein condensate (BEC) of weakly interacting exciton-polaritons from approximately room temperature down to 20K. By resolving the condensate in energy-momentum space and performing interferometric measurements, we distinguish the condensate from thermalized particles yet occupying excited states macroscopically. In contrast to the analytical quantum theories of non-equilibrium BEC [Shishkov et al., Phys. Rev. Lett. 128, 065301 (2022)], we observe segmentation of the particle density along the excited states into two fractions both following Bose-Einstein distribution, albeit with different effective temperatures and chemical potentials. Our results indicate that the temperature of the weakly interacting Bose gas is universally set by the density-dependent chemical potential, revealing a defining property of non-equilibrium BECs. Finally, we demonstrate that the stimulated nature of the cooling process directly governs the emergence of quantum coherence of the condensate and shapes the dissipative properties of the excited states.
In a recent paper [Int. J. Mod. Phys. B 14, 405 (2000)] we discussed the Bose-Einstein condensation (BEC) in the framework of Tsallis’s nonextensive statistical mechanics. In particular, we studied an ideal gas of bosons in a confining harmonic potential. In this memoir we generalize our previous analysis by investigating an ideal Bose gas in a generic power-law external potential. We derive analytical formulas for the energy of the system, the BEC transition temperature and the condensed fraction. PACS numbers: 05.30-d; 03.75.Fi 1 Recently we analyzed the consequences of weak nonextensivity on Bose-Einstein Condensation for an ideal Bose gas 1 by using Tsallis’s nonextensive statistical mechanics (NSM). 2 From the nonextensive Bose-Einstein distribution 3−5 we derived the BEC transition temperature, the condensed fraction and the energy per particle in three different cases: the homogeneous gas, the gas in a harmonic trap and the relativistic homogenous gas. In this short contribution we generalize our previous results by considering the ideal Bose gas in a generic power-law potential. All the calculations are performed by assuming a D-dimensional space. The NSM predicts that, for an ideal quantum gas of identical bosons in the grand canonical ensemble, the weak nonextensivity formula of the average number of particles with energy ǫ is given by 〈n(ǫ)〉q = 1
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.