A numerical immersion in a quantum cloud

How can vortices be produced in a cloud of ultra-cold atoms? This question is of particular interest to physicists specialising in Bose-Einstein condensates (BECs). This state of matter, predicted in 1925 by Satyendra Nath Bose and Albert Einstein, was created in the laboratory for the first time in 1995. What distinguishes it is that the atoms adopt a collective behaviour, which can be described from the point of view of quantum physics as that of a single entity.

In quantum physics, particles have no clearly defined position or energy level. These properties are described by a “wave function”, which indicates the set of states that are possible simultaneously for an object. The different properties of a particle are not independent of each other: its location depends on its energy, and vice versa. This phenomenon can lead to the formation of Bose-Einstein condensates.

Recipe for a condensate

In practice, as an atom cools, its energy level decreases until it reaches a clearly defined state: its “fundamental” level, its lowest possible energy. At the same time, this increases the undetermined nature of its location: the atom’s wave function becomes increasingly extended in space. When this principle is applied to a cloud of atoms, their individual locations are extended to the point where they merge, so that they can be described collectively by the same wave function.

Over the past thirty years, experiments on Bose-Einstein condensates have provided a better understanding of matter. At the University of Lille, research on this topic is being carried out at PhLAM (Laboratory for Laser, Atom and Molecule Physics). The laboratory has a system capable of producing these condensates: potassium atoms are cooled to a few millikelvins using lasers, restricting their movement. Among the experiments being carried out, the scientists want to study turbulence in this environment.

When a liquid is agitated, vortices can be observed: the same applies to Bose-Einstein condensates when the gas is set in motion by a magnetic field. The experiment aims to analyse the formation and evolution of these vortices. Radu Chicireanu, a physicist at PhLAM, wants to carry out the experiment with his team on a condensate trapped in the form of a ring. “This is a relatively recent project, which still requires further technical and experimental development”, he explains.

Approximations for an efficient model

This is where numerical simulation comes in. While waiting on the practical experiment, the phenomenon can be modelled. “The aim is to know what to expect”, says Quentin Chauleur, a researcher on the Paradyse project team at the Inria centre at the University of Lille. “The experiment involves a huge number of physical parameters: the number of atoms, the intensity of the lasers, the intensity of the magnetic field, and many others,” he explains. How do these different parameters influence the formation of vortices in the condensate? This question was the subject of his postdoctoral research between 2022 and 2024.

The evolution of the cloud of atoms can be described by a formula well known to physicists: the Gross-Pitaevskii equation - also known as the “nonlinear Schrödinger equation” to mathematicians. “This equation provides an approximation of the actual movement, which is probably much more complex”, says Quentin Chauleur. However, to be workable in simulation, the equation must be simplified. “That’s where we started”, he recalls.

One of these simplifications concerns the dimensions of the condensate. In reality, the gas ring is three-dimensional. To simplify the calculations, the researchers reduced it to two dimensions – an approximation that has proven effective for similar problems. “This saves a huge amount of computation time”, emphasises Quentin Chauleur. “It also simplifies the codes that need to be written, and the result is easier to visualise”. In three dimensions, the vortices appear as lines. In two dimensions, they become points.

A realistic numerical model

“At this stage, we are faced with a problem of numerical analysis, which is the core of the Paradyse project team’s work”, continues the researcher. The aim is to create an effective numerical model. To do this, the spatial area of the model has to be discretised - i.e. divided into small areas (triangles), to which the equation is applied. The challenge is to strike the right balance between computation time and accuracy. The smaller the triangles, the closer the result will be to reality - at the cost of longer calculations. By contrast, dividing the model into triangles that are too large, which saves on calculations, may not be realistic enough. 

The same logic is applied to the simulation process: the experiment is divided into successive stages, or “steps”, which are more or less closely spaced. This is because the computer cannot calculate the evolution of a CBE continuously over time. It must therefore perform calculations one after the other, as if sampling the experiment. It is therefore necessary to find the optimal time interval between “steps”, again in order to preserve both accuracy and calculation time.

But how can we know whether these simplifications ensure that the behaviour of the simulation is consistent? “To do this, we monitor changes in certain physical quantities”, explains the mathematician. “We know, for example, that the total mass of our gas must remain constant. If the result of our calculations does not respect this parameter, this means that we need to fine-tune our division of space or reduce the duration of our time steps.”

In search of vortices

Once the simulation is working correctly, the next step is to detect the formation of vortices. In a vortex, matter is pushed outwards. There is a void at the centre, which means that the mass is zero. “Physicists have a method for calculating “vorticity””, says Quentin Chauleur. “This is a quantity that is zero in the presence of matter, but becomes very large when a vortex is encountered”. The computer then calculates the vorticity for each triangle at each time step. When this value moves away from zero, it means that a vortex has formed.

“The key challenge in this work was to write an efficient code so that the calculations could be performed quickly enough” emphasises the researcher. There are well-known methods for doing this: by identifying which mathematical operations are computationally onerous, it is possible to make approximations by replacing them with others that can be calculated more quickly. Supported by tests, the aim here is once again to find the best balance between accuracy and computation time.

This means that “by using fairly rough time steps and triangles, I can obtain results with my code in a few minutes on my desktop computer”, says Quentin Chauleur. This programme makes it possible to explore different scenarios, varying the parameters to identify the most interesting cases. These can then be modelled in greater detail and calculated on a dedicated, more powerful machine. This represents the equivalent of several years of work if the same thing had to be done through experiments!

New questions opened

“A number of interesting questions were raised following analysis of the numerical results”, says Radu Chicireanu. “For example, the variation in the number of vortices as a function of rotation speed seems to be a less trivial problem than we initially assumed. Another question concerns the mechanism of vortex generation when the condensate is set in rotation. Can it be demonstrated in a simulation? Would the result be relevant in the experimental context?” The Paradyse project team and PhLAM are currently working on this issue with researchers from San Diego State University and the University of Massachusetts in Armhest, in the United States.

The results open up many possibilities. A PhD thesis will begin at PhLAM in September 2025 to work on implementing the experiment. In terms of numerical analysis, there is still room for improvement in the current simulation: “The ideal would be to have an adaptive network of triangles: one that is rougher in areas where nothing is happening, and finer around the vortices”, explains Quentin Chauleur. “We are working on this with Guillaume Ferrière, another researcher in the Paradyse project team, and Julien Moatti, associate professor at Bordeaux INP.” Ultimately, these optimisation techniques could lead to a three-dimensional simulation of the phenomenon. Modelling reality is a lifelong endeavour!