Popular-science description

We toss two coins. What is the probability of getting one head and one tail? Anyone can perform this experiment independently – just toss a pair of coins many times, divide the number of outcomes with one head and one tail by the total number of tosses, and thus estimate this probability - which will be around 50%. This results from the fact that tossing two coins yields four possible outcomes: heads-heads, heads-tails, tails-heads, tails-tails. Indeed, half of them are “our” desired outcome. However, if we considered the two coins indistinguishable, then the outcomes heads-tails and tails-heads would be treated as one possibility. In that case, we would conclude that there are only 3 possible outcomes, which would change the answer to the originally posed question - to 33%.. In the 19th century, Leibniz stated that indistinguishable objects cannot exist. In our coin example – one can always assume that one coin is on the left, the other on the right, and thus distinguish them. Location is an additional property that makes objects distinguishable. Following this line of thinking, even identical objects are distinguishable. Reality turned out to be more surprising than “philosophers could dream.” Indian physicist S. Bose noticed that by assuming the indistinguishability of light particles, i.e., photons, one can explain Planck’s law. A. Einstein, inspired by Bose’s idea, developed a theory of gases, assuming that atoms are also indistinguishable. His method of calculating probabilities caused shock and criticism in the scientific community, but soon it turned out that the idea of indistinguishable atoms was correct. A solid mathematical formulation of quantum mechanics allowed a break from thinking in terms of particle positions – thereby altering the very foundations on which Leibniz's reasoning was based. One of the consequences of the indistinguishability of particles and the new statistics is the phenomenon of Bose-Einstein condensation. This phenomenon involves a significant portion of atoms suddenly transitioning into the same state after cooling the gas below a certain temperature, forming nearly motionless atoms that create a matter wave. In the case of an ideal gas approaching absolute zero temperature, on average all particles form a condensate. The keyword is on average – Einstein’s formalism would lead us to conclude that even at zero temperature, the condensate either does not appear at all or, quite the opposite, appears with a number of particles greater than the average number in the system. These huge fluctuations, called the fluctuation catastrophe, were pointed out by E. Schrödinger in criticism of Einstein’s formalism. In fact, a better understanding of statistical mechanics shows that condensate fluctuations strongly depend on the chosen model. Ultimately, experiment has the final word – in 2019, it was for the first time possible to increase measurement precision enough to detect fluctuations of a Bose-Einstein condensate [1]. A subsequent, more accurate measurement conducted on a larger number of cases showed that the fluctuations were smaller than predictions based on the canonical ensemble [2]. Both measurements were carried out by Jan Arlt’s group at Aarhus University and interpreted in collaboration with teams from Warsaw. In this project, I aim to study the statistical properties of an ultracold gas by analyzing the cooling process itself. I expect that depending on the experimental procedure, it is possible to generate gas with different statistical properties. I will compare the statistics resulting from the cooling method with models typical of quantum statistical mechanics. I intend to investigate how statistical details affect quantum correlations between atoms, particularly density fluctuations and quantum coherence in the system. Particle position correlations, in turn, influence particle loss rates and the accuracy of estimating gas parameters, such as its temperature. In the final stage of the project, I plan to examine how reducing condensate fluctuations translates into its practical applications in metrology.

Figure 1: The recent observation of BEC fluctuations shows the mismatch between measurements (blue points) and the predictions of the canonical ensemble (solid black line). We will improve the statistical description of ultracold gas and use it to refine predictions for correlations between atoms and practical consequences for gas parameters’ estimation and consequences for metrology and its finite-temperature limitations. Inset adapted from [2]. .