Analytical framework for non-equilibrium phase transition to Bose–Einstein condensate

V. Yu. Shishkov1,2,3,4, E. S. Andrianov1,2,3,4, and Yu. E. Lozovik1,5,6

1Dukhov Research Institute of Automatics (VNIIA), 22 Sushchevskaya, Moscow 127055, Russia;
2Moscow Institute of Physics and Technology, 9 Institutskiy pereulok, Dolgoprudny 141700, Moscow region, Russia;
3Center for Photonics and Quantum Materials, Skolkovo Institute of Science and Technology, Moscow, Russian Federation
4Laboratories for Hybrid Photonics, Skolkovo Institute of Science and Technology, Moscow, Russian Federation
5Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia;
6Institute for Spectroscopy RAS, 5 Fizicheskaya, Troitsk 142190, Russia;

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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 develop a framework for the analytical description of a non-equilibrium phase transition to Bose–Einstein condensate (BEC) that, in contrast to previously developed approaches, takes into account the infinite number of continuously distributed states.
We obtain an analytical expression for the full density matrix of a non-equilibrium ideal BEC in the fast thermalization limit.
We find the temperature dependence of the ground state occupation and second-order coherence function.
We show that the buildup of coherence of the ground state strongly depends on the pumping scheme.
We also show that the Schawlow–Townes law holds for BEC in 3D and does not hold for BEC in 2D.

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