Theoretical foundations for understanding two-dimensional quantum gases are detailed in a series of three lectures delivered at the São Paulo ICTP-SAIFR IVth School on Light and Cold Atoms. This course enables thorough description of phenomena including Bose-Einstein condensation, superfluidity, scaling symmetry, and vortex lattice behaviour in rotating systems. A detailed educational resource on two-dimensional quantum gases has been developed, explaining complex behaviours like superfluidity through mechanisms involving phase fluctuations and scaling symmetry.

Specifically, the course details how superfluids can emerge via the Berezinskii, Kosterlitz, Thouless mechanism, a key process for understanding materials exhibiting low viscosity flow and ultracold atomic systems. These materials cover fundamental phenomena such as Bose, Einstein condensation, where many particles simultaneously occupy the lowest energy state, akin to a crowded concert moving in unison, superfluidity, and scaling symmetry within these systems.

The course explains how superfluids can emerge through the Berezinskii-Kosterlitz-Thouless mechanism, which allows for order even with imperfections present at small scales, much like maintaining overall structure when crumpling a flat sheet. The team also examines rotating gases exhibiting organised whirlpool patterns known as vortex lattices, resembling swirling cream in coffee; however, understanding precisely how thermal affects disrupt this ordered arrangement remains an open question explored further in their detailed notes.

Variational techniques optimise atomic stability through iterative wavefunction refinement

The team employed variational methods, a technique where approximate solutions are refined iteratively to describe many-body quantum systems. This approach sidestepped directly solving complex equations governing numerous interacting particles which is often computationally impossible. Initially, they sought single particle wave functions, termed |φ⟩, that would minimise overall energy under specific constraints relating to atom number and finding the most stable configuration for individual atoms within the collective system.

The resulting minimisation process led to a modified equation known as the Gross, Pitaevskii equation, describing how these atomic waves behave and interact with each other via contact interactions; forces act only when atoms touch. This work expands upon initial findings by detailing interaction between atomic waves through contact occurring solely on physical contact, allowing modelling of more intricate systems. Further research will focus on applying it to investigate exotic states of matter like superfluids and Bose-Einstein condensates.

Scaling Symmetry and Quasi Long-Range Order in Two-Dimensional Bose Gases

The researchers de physique des lasers CNRS and Université Sorbonne Paris Nord have detailed a course exhibiting major expansion upon existing knowledge. Specifically, two-dimensional quantum Bose gases are now thoroughly addressed with scaling symmetry explained in detail, previously absent from comparable educational resources. This advancement allows for deeper understanding of superfluidity via the Berezinskii, Kosterlitz, Thouless mechanism; this is vital to describing phase transitions within these systems but was inaccessible without such focused instruction.

The three-lecture series provides an accessible resource covering topics ranging from ideal gas behaviour through interactions and rotating system dynamics. Quasi long-range order emerges within two-dimensional gases due to enhanced phase fluctuations, enabling analysis beyond traditional perturbative methods. Analysis of vortex lattices demonstrates thermal melting occurs at specific rotational speeds and temperatures while detailed explanation of how the Berezinskii, Kosterlitz, Thouless mechanism governs transitions via unbinding of vortex pairs offers a complete overview for students alongside examinations of ideal gas behaviour and rotating system dynamics.

Understanding Quasi Long-Range Order in Two Dimensional Quantum Gases

Laboratoire de physique des lasers CNRS and Université Sorbonne Paris Nord recently detailed a course clarifying complex behaviours exhibited by two-dimensional quantum gases; this builds upon decades of work exploring Bose, Einstein condensation and superfluidity as fundamental states of matter with potential technological applications. Fully capturing these phenomena requires navigating subtle theoretical challenges surrounding quasi long-range order, a state exhibiting correlation without strict periodicity, and how it emerges from enhanced phase fluctuations within such systems. The instructional series clarifies concepts like Bose, Einstein condensation where atoms behave collectively, alongside the phenomenon of superfluidity. In particular, building on existing knowledge, the course clarifies complex behaviours in rotating Bose gases including thermal melting observed in vortex lattices; organised whirlpool patterns form when such gases rotate providing students with an advanced perspective beyond conventional approaches.

The lectures detailed behaviour exhibited by two-dimensional quantum gases, covering ideal gas properties and superfluidity. Understanding how interactions enhance phase fluctuations is important because this leads to a state called quasi long-range order, differing from traditional periodic arrangements. The series also explained the Berezinskii, Kosterlitz, Thouless mechanism governing transitions via unbinding of vortex pairs and described thermal melting within rapidly rotating systems. These materials aim to provide accessible instruction on these complex topics for students studying light and cold atoms.

👉 More information
🗞 Course on two-dimensional quantum gases
✍️ Hélène Perrin
🧠 ArXiv: https://arxiv.org/abs/2609.17097

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