**The term Bose-Einstein condensate describes a state of matter in which atoms or elementary particles combine into a single quantum mechanical object during extreme cooling. Science does not yet fully understand exactly how these macroscopic states - beyond the confines of traditional physics - develop from a thermal atomic cloud in just a few milliseconds and when statistical equilibrium is reached, according to Georg Wolschin. Using a new theoretical model he developed, the Heidelberg University physicist has now succeeded in calculating the time-dependent formation of a Bose-Einstein condensate in sodium atoms. **

When a Bose-Einstein condensate develops, bosonic atoms - with an integer spin - form a gas that condenses at extremely cold temperatures approaching -273.5 degrees Celsius, or absolute zero. This phenomenon in theoretical physics was postulated in 1924 by Albert Einstein based on a theory by Indian physicist Satyendra Nath Bose and proven experimentally in 1995. "To understand the processes involved in the formation of this condensate, the percentage of condensate atoms must be calculated as a function of time, which requires substantial mathematical effort," explains Prof. Wolschin.

The researcher developed a model that can be applied to sodium-23 and other bosonic atoms. It is based on the calculation of exact solutions of a non-linear differential equation. Solving these types of equations with the necessary physical boundary conditions analytically is possible only in rare cases. The solutions can be used to model how the phase transition to the condensate is achieved as a function of time through evaporative cooling, which removes high-velocity atoms. This cooling process is essential for the formation of Bose-Einstein condensates.

According to the calculations by Prof. Wolschin and Master’s student Alessandro Simon - now a doctoral candidate at the University of Tübingen - Bose-Einstein condensates in sodium atoms form within approximately 50 milliseconds once the critical temperature has been attained through cooling; statistical equilibrium in sodium atoms is reached after approximately 300 milliseconds. "Our result is largely in line with the experimental data from the Massachusetts Institute of Technology from 1998. However, it does suggest that in other atoms such as lithium or rubidium, which can have very different time scales, new and more precise measurements are needed to better understand time-dependent condensate formation and statistical equilibration," adds the physicist. Prof. Wolschin stresses that a comparison of theory and experimental data is critical to verify, refine, and further develop the theory.

The research results were published in the journal "Physica A: Statistical Mechanics and its Applications".

A. Simon, G. Wolschin: Time-dependent condensate fraction in an analytical model. Physica A: Statistical Mechanics and its Applications (17 March 2021)