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arXiv:1912.10851 · Physics Open

Solving a nonlinear analytical model for bosonic equilibration

N. Rasch, Georg Wolschin

Published Dec 18, 2019• 10 citations
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Abstract

An integrable nonlinear model for the time-dependent equilibration of a bosonic system that has been devised earlier is solved exactly with boundary conditions that are appropriate for a truncated Bose-Einstein distribution, and include the singularity at ε=μ. The buildup of a thermal tail during evaporative cooling, as well as the transition to the condensed state are accounted for. To enforce particle-number conservation during the cooling process with an energy-dependent density of states for a three-dimensional thermal cloud, a time-dependent chemical potential is introduced.

DOI
10.1016/j.physo.2019.100013

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An integrable nonlinear model for the time-dependent equilibration of a bosonic system that has been devised earlier is solved exactly with boundary conditions that are appropriate for a truncated Bose-Einstein distribution, and include the singularity at ε=μ. The buildup of a thermal tail during evaporative cooling, as well as the transition to the condensed state are accounted for. To enforce particle-number conservation during the cooling process with an energy-dependent density of states for a three-dimensional thermal cloud, a time-dependent chemical potential is introduced.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.
Abstract We calculate the time-dependent formation of Bose–Einstein condensates (BECs) in potassium vapours based on a previously derived exactly solvable nonlinear boson diffusion equation (NBDE). Thermalization following a sudden energy quench from an initial temperature TiT_\mathrm {i}Ti​ T i to a final temperature TfT_\mathrm {f}Tf​ T f below the critical value and BEC formation are accounted for using closed-form analytical solutions of the NBDE. The time-dependent condensate fraction is compared with available 39^{39}39 39 K data for various scattering lengths. Graphic Abstract
Time-dependent Bose-Einstein condensate formation in ultracold atoms is investigated in a nonlinear diffusion model. For constant transport coefficients, the model has been solved analytically. Here, we extend it to include energy-dependent transport coefficients and numerically solve the nonlinear equation. Our results are compared with the earlier analytical model for constant transport coefficients, and with recent deep-quench data for 39K^{39}\mathrm{K}39K at various scattering lengths. Some nonphysical predictions from the constant-coefficient model are resolved using energy-dependent drift and diffusion.
Bose–Einstein condensation is the anomalous accumulation of particles in the ground state of a system of bosons, when compared with the population in a system of particles obeying classical statistics. I use undergraduate-level statistical mechanics and a symbolic algebra computer program to study the occupation numbers of the energy levels of a finite number of noninteracting particles confined in a three-dimensional harmonic oscillator potential. I also calculate the heat capacity of the gas. The harmonic oscillator potential simplifies the calculations and approximates the conditions of the recent experiments achieving Bose–Einstein condensation in laser-cooled alkali vapors.