# Vortices in Bose Einstein condensates

*Symposium organized by Amandine Aftalion, Yvan Castin and Yves Pomeau from 3 to 8 June, 2002.*

**Participants **

Charles S. Adams (University of Durham, Royaume-Uni), Amandine Aftalion (Université Pierre et Marie Curie, Paris), Yvan Castin (École Normale Supérieure, Paris), Myriam Comte (Université Pierre et Marie Curie, Paris), Franco Dalfovo (Catholic University, Brescia, Italie), Ionut Danaila (Université Pierre et Marie Curie, Paris), Qiang Du (The Pennsylvania State University, USA), Alexandre Fetter (Stanford University, California, USA), Frédéric Recht (Université Pierre et Marie Curie, Paris), Christophe Josserand (Université Paris VI, Paris), Yves Pomeau (CNRS, École Normale Supérieure, Paris), Ludovic Pricoupenko (Université Pierre et Marie Curie, Paris), Sergio Rica (CNRS, Santiago, Chili), Etienne Sandiеr (Université Paris XII, Créteil), Robert Seiringer (Princeton University, New Jersey, USA), Gora Shliapnikov (École Normale Supérieure, Paris), Subhasis Sinha (Université Paris Sud, Orsay), Didier Smets (Université Pierre et Marie Curie, Paris).

**Report**

Bose Einstein condensates (BEC) owe their name to the prediction of Bose and Einstein in 1925 that for a gas of non interacting particles at vert’ 1ow temperature, a macroscopic fraction of the gas is in the state of lower energy, that is condensed. At that time, this idea was only theoretical. The first experimental realization of atomic BEC was obtained in 1995 by American teams and was awarded the Nobel Prize in 2001. Since then, a lot of properties of these systems have been studied both experimentally and theoretically, in particular at the Ecole Normale Supérieure (ENS) in Paris. The problem is made non trivial by the existence of interactions between the particles, which were neglected by Einstein but which play a crucial role in the experiments.

The aim of this meeting was highly interdisciplinary : bring together applied mathematicians and physicists. We wanted to understand the problems at the core of international research in physics, describe results that have been obtained and derive new mathematical open problems.

We have addressed four different types of questions : in relation with the experiments at the Ecole Normale Supérieure where a BEC is rotated in a harmonic trap, we have studied the vortex nucleation and various vortex configurations. One of the observations is that the vortices are not straight along the axis of rotation but bending. The second topic deals with vortex arrays and giant vortices when the trap is rotated faster or is more confining than the harmonic trap. The third topic was to address the effects of finite temperature and try to find good models for it. Finally, we studied the motion of an object through a BEC, accompanied by vortex shedding and radiation of sound.

The focus in all the talks was both to understand the physical phenomena, find theoretical models and relate to numerical and mathematical questions. The workshop ended with a session on open problems. New collaborations and exciting developments are expected. We hope to create an interdisciplinary group on these topics and meet again in about two years.

**Vortices****in****rotating****harmonic****traps***Stability and dynamics of vortices in a trapped BEC*

Alexander Fetter

The basic physics of dilute trapped atomic gases reflects both the interparticle interactions aid the quantum dеgeneraсy (which occurs when the thermal De Broglie wavelength becomes comparable with the interparticle spacing). In the simplest case that most of the particles are in the condensate, the time-dependent Gross-Pitaevskii equation describes well the dynamical evolution (this is formally equivalent to a nonlinear Schrodinger equation). The stability of a vortex in a rotating condensate can be studied in at least three distinct wаys :

— the first examines how the energy changes as the vortex is displaced from the central position and predicts the onset of metastability at a critical angular velocity (below this value, the central position is a local maximum of the energy, whereas above this value, the central position becomes a local minimum).

— A more direct dynamiсal approach considers the small-amplitude perturbations with the Bogoliubov equations and finds a negative frequency if the angular velocity is smaller than that for onset of metastability (this behavior indicates an instability of the Landau type).

— The mort physical approach derives the local velocity of each

element of the vortex, confirming the previous analyses. In addition, this latter method predicts large-amplitude periodic tipping orbits in a nearly spherical condensate. Experiments confirm these latter motions and also the predictions about the precession of a nearly straight vortex in an аxisymmetric condensate.

*Vortex nucleation in rotating BEC*

*Franco Dalfovo*

This talk is a brief summary of the work recently done in Trento on rotating Bose-Einstein condensates. The talk starts showing how one can transfer angular momentum to a condensate by distorting its shape with an external rotating field, provided the rotational frequency is larger than a critical frequency fixed by the energy and angular momentum of the excited states of the system. By using the Gross-Pitaevskii equation and sur rules, one can explore the dependence of such a critical frequency on the multipolarity of the excitations and the asymmetry of the confining potential. With a similar mechanism one can also generate deformed configurations which are stable in the rotating frame. These configurations have been studied by solving the hydrodynamiс equations of superfluids, with the irrotationality constraint for the velocity field. One finds also an overcritical branch where the system rotates with angular velocity larger thon the oscillator frequencies (overcritical rotation). One can show that in the case of isotropic trapping the system exhibits a bifurcation from an axisymmetric to a triaxial configuration, as a consequence of the interatomic forces. The dynamical stability of the rotational motion with respect to the dipole and quadrupole oscillations is explicitly discussed. Finally, one use the critical frequency for shape deformations and the existence of deformed configurations in the rotating frame to provide a picture for vortex nucleation. For sufficiently high angular velocities of the trap, the tendency of the system to exhibit spontaneous deformation is shown to lower the barrier which inhibits the nucleation of vortices. The experimental data seem to confirm this picture.

*Dynamic instability and nucleation of vortices in a rotating Bose-Einstein condensate*

*Subhasis Sinha*

We consider a Bose-Einstein condensate subject to a rotating harmonic potential, in connection with recent experiments leading to the formation of vortices. Here we consider the dynamics of rotating condensate. We use the classical hydrodynamic approximation to the non-linear Schrodinger equation to determine almost analytically the evolution of the condensate. Next we consider the linear stability analysis of such vortex free dynamical state of the rotating condensate. Wc predict that this evolution can exhibit dynamical instabilities, for the stirring procedure prеviously demonstrated at ENS and for a new stirring procedure that we put forward. These instabilities take place within the range of stirring frеquency and amplitude for which vortices are produced experimentally. Within this range of rotation frequencies where the sуstem is dynamically unstable, any small fluctuation can grow exponentially fast and this vortex free state breaks down and vortices enter in the condensate.

They provide therefore *an *initiating mechanism for vortex nucleation. We also checked ; by numerical integration of the full non-linear Schrodinger equation, that the vortices appear in the range of rotation frequency where the hydrodynamic equation shows dynamical instabilitу.

*Mathematical mode) for rotating BEC, vortex energy, vortex bending Amandine Aftalion*

In *an *experiment achieved at ENS, a laser beam is imposed on the magnetic trap holding the atoms to create a harmonic anisotropie rotating potential. For sufficiently large angular velocities, vortices are detected in the system. Experimentally, it has been observed that when the vortex is nucleated, the contrast is not 100 % which means that the vortex line is not straight but bending. Numerical computations solving the Gross-Pitaevskii equation have shown that there is a range of velocities for which the vortex lire is indeed bending. The aim of this talk is to justify these observations thеoretically in the Thomas Fermi regime.

We define an asymptotic parameter which is small in the Thomas Fermi regime and approximate the Gross-Pitaevskii energy to obtain a simpler form of the energy which only depends on the number and shape of the vortex lies. This simpler form of the energy has a term which depends on the vortex contribution and one due to rotation. Then we check numerically and analytically that our characterization leads to solutions with a bent vortex for a range of values of the rotational velocity which are consistent with the experiments and the oies obtained numerically by Castin, Modugno, Pricoupenko. We prove that when the trap has a cigar shape, as in the ENS experiment, the straight vortex is unstable at 1ow rotational velocities and stable at larger velocities. On the other hand, for pancake shape condensates, the minimizer of the energy is the straight vortex and there is no bending.

*Bose-Einstein condensates with a bent vortex in a rotating trap*

*Yvan Castin*

A Bose-Einstein condensate of spinless bosons is characterized by a wavefunction that is by a complex function of the three spatial coordinates. This wavefunction can support one or several vortex lies, that is nodal lies with a non trivial winding number of the phase around each fine.

In convection with recent experiments at ENS, Paris, we have studied the case of a steady state condensate in a rotating harmonic trap, assuming that the condensate wavefunction is a local minimum of the GrossPitaevskii energy functional taking into account the effect of particle interactions at the mean field level. When the trap is cigar shaped, that is when the oscillation frequency of the particles along the rotating axis is much smaller than the oscillation frequencies in the transverse plane, one fends numerically that the vortex lie can be bent in equilibrium configuration, even when a single vortex is present inside the typical radius of the condensate. This is a beautiful illustration of spontaneous symmetry breaking in a nonlinear system.

We have developed a simple analytical approach to understand why the vortex fine is bent : one is able to calculate approximately the enеrgy of the single vortex configuration, one finds that it is the sur of the energies of a continuous collection of 2D condensates, each one corresponding to a slice of the 3D condensate in a plane orthogonal to the rotation axis. The 2D case was well studied, and is characterised in particular by a critical rotatio frequency aboye which the single vortex configuration has a lower energy than the vortex free configuration.

In the limit of an ultra-cigar shaped condensate one then expects that the vortex line minimizes its full 3D energy by minimizing its 2D enеrgy inside each slice, that is by staying on the rotation axis where the trap rotation frequency is larger than the local 2D critical rotation frequency, and by bending and moving out of the condensate in the plane where the trap rotation frequency is equal to the local 2D critical velocity. This explains quantitatively the numerical results in a physically simple way.

*Numerical computation of vortices in Bose-Einstein condensate*

*Qiang Du*

In this talk, we will present some asymptotic energy expansion of the two dimensional Gross-Pitaevskii free еnergy for a Bose-Einstein condensate in a rotating trap. This expansion can be rigorously justified through mathematical analysis, and it allows us to provide accurate estimates of the critical angular vеloсity for the nuleation of vortices as well as the structure of the vortex lattices near the critical velocity. Our results covers the case of symmetric and asymmetric traps and generalizes earlier results of Castin-Dum and Fetter. Some strikingly similarities based on the phenomenological Ginzburg-Landau models for the vortex state in type-II superconductors will be illustrated. We also present some effective numerical algorithms for the computation of vortices based on the popular finite element methods. These methods have been widely used in various scientific and engineering applications, thеу have also been extensively used in the numerical simulation of the vortex dynamics in super- conductors. We will also give numerical examples that closely resembles the experimental pictures.

*Symmetry breaking in the Gross-Pitaevskii mode) of a rotating Bose**gas*

*Robert Seiringe**r*

In the first part of the talk, we study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state ; more precisely, for any value of the angular velocity and for large enough values of the interaction strength, the ground state of the functional is not an eigenfunction of the angular momentum. This has interesting consequences on the Bose gas with spin; in particular, the ground state energy depends non-trivially on the number of spin components, and the different components do not have the same wave function.

The second part of the talk is devoted to the N-body problem associated with the Bose gas. Recently a proof of Bose-Einstein condensation was established rigorously for the first time (in joint work with E.H. Lieb). We give an account of this result that applies to the ground state of a dilute Bose gas in the limit where the Gross-Pitaevskii formula is applicable.

**Vortex****arrays****and****giant vortices***Dense vortex**arrays**in**иotating dilute**Bose**condensates*

Alexande*r**Fetter*

In the limit of a large condensate, the vortex core size is small соmpared to the harmonic oscillator length aid to the dimensions of the condensate. When the condensate rotates rapidly, the repulsive centrifugal energy tends to cancel the attractive oscillator trap potential, leading to a reduced effective confining frequency that vanishes as the rotation frequency approaches the trap frequency. For moderate rotation rates, the radial kinetic energy is negligible and the condensate can be described with a straightforward generalization of the theory for a nonrotating condensate. In this case, the centrifugal forces expand the radial dimension and shrink the axial dimension, with a corresponding reduction of the chemical potential (this latter effect ultimately expands the vortex-core radius, but such behavior occurs only very near the loss of confinement). For rapid rotation speeds, in contrast, it is essential to include the radial kinetic energy, and Ho has introduced a mode! based on the analogy with the behavior of a charged particle in a uniform magnetic field. In the limit that only the lowest Landau level is occupied, the thеоry predicts a radial expansion and axial shrinkage that are similar to those found in the previous analуsis. Consequently, a direct experimental distinction between the two different descriptions may well be difficult.

*Giant vortices in BEC*

*Christophe Josse**r**and*

The recent achievement of Bose-Einstein condensates (BEC) have open large areas of research on quantum liquids. One of the striking properties of these quantum gases is the quantization of the vorticity. Such vortices have been observed in many different experiments these last yеars. The standard theory of infinite and homogenous condensate predicts only single charged vortices. Thus the question I would like to address here is whether in the confined BEC multiple vortex can exist (that we call giant vortex later on). For BEC in harmonic trap, the usual experimental situation, it can be shown that only single vortices are present. That appears eventually to be a peculiar property of harmonic confinement and it suggests that nothing would prevent the stability of giant vortices for stronger trap potentials.

It would be somehow a great challenge to achieve such vortices experimentally, thanks for example to Laguerre-Gaussian laser beam which allows strong trap potentials. The goal of mу presentation was to show how giant vortices become stable when stronger than harmonic potentials are taken in a rotating system. I used numerical simulations of the Gross-Pitaevskii (G-P) equation that models the dynamics. Within this context, we were able to follow a solution without vortex to one with vortices as the angular velocity, increases. Then we observed that for a large enough frequency, the giant vortex configuration appears versus single vortices. Some analysis were also performed to deduce the different regime where these giant vortices are stable.

*Limiting vorticity for Ginzburg-L**а**ndau**е**qu**а**tion*

*Etienne S**а**ndier*

Our purpose here is to describe minimizers and critical points of the Ginzburg-Landau functional of superconductivity in terms of vortices. A vortex often designates a radially symmetric solution to the variational equations of some functional carrying a topological charge, in our case the Ginzburg-Landau functional of superconductivity.

When the Ginzburg-Landau parameter is large, these vortex solutions are localized, then a solution of the equation is said to have a vortex at a point if it behaves near this point like a vortex solution.

The study of vortex solutions in this generalized sense goes back at least to the original work of Abrikosov. A first approach is to assume a vortex solution exists and assumes a certain form to derive a posteriori information about the location and number of vortices. Techniques employеd here include energy expansions and matched asymptotics. Numerical analysis is another option.

A new approach was initiated by Е Bethuel, H. Brezis and F. Hélein, which allows to aсtually prove — taking the Ginzburg-Landau functional as a starting point — that actual solutions of the equation indeed exhibit vortices and have a certain behaviour near them, justifying a posteriori the ansatz that serves as a basis to the formal approaches. Recently, the author and S. Serfaty have developped tools that allow to deal with a number of vortices possibly diverging as the Ginzburg-Landau parameter goes to zero, which is relevant to many physical situations. It is also a case where the ansatz of well-separated vortices which is the basis of formal calculations needs more justification. This allows to асtually compute limiting vorticity measures which describe the average density of vortices in the limit of a large Ginzburg-Landau parameter, as well as other results.

**III. ****Finite ****temperature effects**

*Thermodynamics and normal gas/superfluid vortex inte**r**action*

Yves Pomeau

Although the concept of quantized vortex is well understood at zero temperature, as a property of Gross-Pitaevskii dynamics, it is far less obvious when normal gas is present at finite temperature. In particular, there is no generally accepted extension of the familiar Kelvin equations taking into account the forces (drag and Magnus) arising from the scattering of thermal phonons. Ву analysing the vortex motion in the wave field of a single phonon beyond the linear approximation, one shows that already at the quadratic order a damping appears. Interactions in the normal gas yield at the end a consistent set of equations interpolating between the Kelvin dynamics for vortices in a perfect (super) fluid and the normal fluid Navier-Stokes like dynaenics. This gives a fully consistent relaxation dynamics without relying on a formal, and unproved, Aristotelean principle of minimization.

*Finite**temperature effects for a single vortex in a rotating BEC*

*Ludovic Pricoupenko, in collaboration wth Yvan Castin and Michele*

*Modugno*

In this talk we explain how to describe temperature effects in Bose Einstein Condensates and apply the formalism to the case of a single vortex configuration. First, we recall the context of this study : the experiments at ENS. Unfortunately, in this experimental configuration as explained in the talk of Yvan Castin, the vortex bends. This behaviour complicates a lot the numerical approach as one need to diagonalise very big matrices. Our strategy has been then to reduce the cоmplеxity of the problem by considering an axi-symmetriс trap such that the vortex is straight at zero temperature. In this situation, we can use the rotational symmetry with respect to the azimutal angle, so that one can handle the numerical approach on standard computers.

The first part of the talk is an introduction to the basic tools of the physicist for studying quantum Bose gases. We introduce the powerful notion of Bose atomic field for describing the system at a quantum level. We point out the problem of divergencies arising because of the modelisation of the interaction between two atoms by a simple Dirac distribution and explain the need of a regularisation bу using, for example, the pseudo-potential approach. Then, we briefly describe the Bogoliubov theory in the canonical set (in this formulation due to Castin and Dur, the number of atoms is kept constant). Finally, we present the temperature dependence for the mean values of observables such as the « out of condensate density » and the contrast (ratio between the density on the vortex line and the maximum density). These results permit to understand the pion contrast observed in ENS experiments.

In the second part of the talk, we insist on the fact that the previous approach do not lead to a complete description of the experimental results as we have averaged over the different realisation of the many-body density operator. As an example, it is nit bard to imagine that the density profile in a single measurement breaks the azimutal symmetry of the problem, whereas if one averages over all possible realisation, the symmetry is restored. Hence, to have a clear link with experiments, we have decided to mimic a single realisation of the many-body density operator. For this purpose, we use the so-called Glauber-P representation. This technique borrowed from quantum optics is especially well suited for describing thermal fluctuations in the low density regime as it is the case here. As a conclusion, we present the results for typical realisation, showing large fluctuations in the phase (the thermal occupation of surface modes leads to the apparition of satellite vortices at the border of the condensate) and also in the number of atoms in the core of the line (the line fluctuates around the rotation axis because of the thermal occupation of the kelvon modes described also in the talk of Gora Shlyapnikov).

*Finite temperature effects in vortex condensates*

*Gora. V. Sh**lу**a**р**nik**о**v*

I discuss finite-temperature effects in trapped Bose-Einstein condensates containing a vortex. First, I develop a non-linear approach for finding the contrast of a straight vortex in an idealized (infinitely long) cylindrical trap. The contrast deviates from 100 % because of oscillations of the vortex line (kelvins). The oscillation amplitude increases with temperature and I show how this provides a non-linear temperature dependence of the contrast.

I then turf to the dissipative dynamics of a straight vortex in an idealized non-rotating cylindrical trap. Assuming a thermal cloud to be at rest, there is a friction force acting on the vortex line due to the scattering of thermal excitations from the vortex core. The radial component of the friction force induces the motion of the vortex line towards the border of the condensate. I describe this process resulting in the vortex dесay at the border via the emission of a bunch of excitations.

In the last part of my presentation I address the dissipative dynamics of a curved vortex in the presence of axial confinement. This problem is direсtly related to the recent experiment at ENS. I discuss how the vortex line rives to the border and becomes more and more bended due to the interaction with a thermal cloud.

IV Motion of an object through a BEC

*Limit speed and vo**r**tex nucleation**in the**nonlinear Schrodinger equation*

*Sergio Rica*

The discovery of Bose-Einstein condensation in atomic vapors opens the way to test in a rather detailed fashion some predictions of the equi- librium and nonequilibrium quantum statistical mechanics. As those vapours are weakly interacting systems there is hope to compare the predictions of the Gross-Pitaevskii thеоry with experimental results, in particular the sound propagation was tested successfully. More recently it was observed by Ketterle & Co. some evidence of the existence of a superflow around an obstacle with a well defined critical speed (around one third the speed of sound).

Some years ago in collaboration with T. Frisch and Y. Pomeau we have shown that the criterion for vortex nucleation in the Gross-Pitaevskü equation (the nonlinear Schrodinger equation, NLS later) is closely related to the so-called transonic transition.Indeed NLS could be trans- formed into a set of « hydrodynamical » equations for compressible fluid with a ballistic state equation for the pressure in terms of the density in the long wave limit. It appears that the vortices are nucleated when the flowbecomes locally (at the edge of the disk) supersonic. Later, in collaboration with C. Josserand, we have studied in detail the transonic transition at threshold with the help of the Euler — Tricomi equation. The analysis show that thе vortex are nucleated via a saddle node bifurcation.

*Vortex-sound interaction*

*Charles Adams*

Fluids are remarkably complex and our models of their behaviour are only approximation, Partly for this reason, there has been considerable excitation in the scientific community about the realisation of a new kind of fluid known as a dilute Bose-Einstein condensate (BEC), because from a theoretical viewpoint one can argue that this is the simplest fluid with physical relevance. The physiсs of dilute BECS becomes even simpler in the limit of 1ow temperature where most of the thermal degrees of free-dom can be frozen out. Howevеr, even in this limit the fluid still displays a fascinating range of interesting phenomena.

In Durham we have been studying the motion of an object through the fluid and observed how the object can produce vortex rings and become trapped inside the core of the ring. We have also been studying how the motion of vortices and collisions between vortices produce sound radiation. This conversion of vortex energy into sound energy is the main dissipation mechanism in the limit of 1ow temperature and consеquently is important in developing the understanding of turbulence in 1ow temperature fluids.

*Dissipative flow in the Painleve boundary layer of a Bose Einstein**condensate*

*Qiand Du*

Raman et al. have found experimental evidence for a critical velocity under which there is no dissipation when Inc rives a detuned laser beam in a Bose-Einstein condensate. In continuation of the work of Frisch, Pomeau, Rica, we analyze the origin of this critical veloсity in the 1ow density region close to the boundary layer of the cloud, which *can *be described by a Painleve equation. This is a joint work with A. Aftalion and Y. Pomeau.

*Quantized vortices in Helium nanodroplets*

*Franco Dalfovo*

Helium nanodroplets and trapped Bose-Einstein condensates in dilute atomic gases offer complementary views of fundamental aspects of quantum many-body systеms. Using a density functional method, we investigate the properties of liquid Helium droplets doped with atoms (Ne and Xe) and molecules (SF_6 and HCN). We consider the case of droplets having a quantized vortex pinned to the dopant. A liquid drop formula is proposed that accurately describes the total energy of the complex and allows one to extrapolate the density functional results to large droplet numbers N. For a given impurity, we find that the formation of a dopant + vortex + 4He_N complex is energetically favored below a critical size N_cr. Our result support the possibility to observe quantized vortices in helium droplets by means of spectroscopic techniques. This work has been done in collaboration with M. Barranco, M. Pi and R. Мaуol, Universitat de Barcelon, Spain.

OpenEdition vous propose de citer ce billet de la manière suivante :

ldiebold (3 août 2002). Vortices in Bose Einstein condensates. *Les carnets de la Fondation des Treilles*. Consulté le 22 juillet 2024 à l’adresse https://doi.org/10.58079/quvo