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**Vortices in Bose-Einstein condensates.**
*(English)*
Zbl 1129.82004

Progress in Nonlinear Differential Equations and their Applications 67. Boston, MA: Birkhäuser (ISBN 0-8176-4392-3/hbk). xii, 203 p. (2006).

A Bose-Einstein condensate (BEC) is a quantum state of a gas of non-interacting particles, in which a macroscopic fraction of the atoms occupy the state of lowest energy. Predicted by Einstein in 1925 on the basis of a paper by Bose, BECs have been discovered experimentally in 1995 in alkali gases. This monograph by Amandine Aftalion presents rigorous mathematical models of some experiments on BECs which display vortices.

Employing energy estimates, Gamma convergence, homogenization techniques and Bargmann transforms, the analysis yields information on vortex shape, number and location.

The book sets out with two introductory chapters; one of them is dedicated to the experimental and theoretical physics of BECs, the other is an overview of the main theorems used in the mathematical modeling. The formalism is based on the mean field description by the Gross-Pitaevskii energy, whose minimizers are analyzed.

The next five chapters dive into specific problems. Each of these includes an introductory paragraph, which is followed by well structured sections with detailed derivations of the main results. The objective of each section is concisely stated right at the beginning, resulting in a remarkable readability of the book in spite of its rather technical style.

Chapter 3 deals with a two-dimensional (2D) model of a rotating condensate in the Thomas-Fermi regime, when the kinetic energy is small in comparison to the trapping and interaction energies, allowing for a small parameter expansion. The energy formulation used here relies on a 2D reduction and a bounded-domain reduction.

In order to make a closer contact with recent experiments, Chapter 4 extends the analysis of the previous one to the context of more realistic trapping potentials. Two of these are treated in detail: a non-radial harmonic potential and a quartic potential.

Chapter 5 turns to the case of high rotational velocity, which gets close to the trapping frequency. The small parameter of this theory is the deviation of their ratio from 1. It is conjectured and argued that the Gross-Pitaevskii energy reduces to contributions of states belonging to the lowest Landau level, and the vortex lattice configuration is found which minimizes this contribution. To this end, homogenization techniques and double-scale convergence is used. A dense triangular Abrikosov lattice is found with a characteristic size of vortices of the same order of magnitude as their interdistance.

The 3D Thomas-Fermi regime is tackled down in Chapter 6 with the goal of explaining experimental observations of global vortex shapes. Planar U vortices are found as global minimizers of the energy. Critical points of the energy are identified, which correspond to planar and non-planar S-shaped vortices. The U vortex has a roughly straight central part and the extremities point normally to the condensate boundary. The analytical study of 3D vortex shapes is augmented with numerical simulations.

Chapter 7 is dedicated to dissipationless flow around an obstacle: a stirring laser beam that penetrates the condensate. The problem is studied in 2D, and in the hydrodynamic formulation, a critical velocity of the stirrer is identified above which the flow becomes dissipative. Finally, the full 3D analysis is performed numerically using a factorization anzatz and rigorous analytical results are derived in the low velocity regime.

Chapters 3 to 7 end with sections dedicated to open questions related to their specific topic, whereas Chapter 8 supplements the rich list of open questions with some general ones. Precisely identified open problems turn the book into a valuable tool in the hands of PhD students or researchers willing to enter this fascinating endeavor of understanding quantum vortices.

Employing energy estimates, Gamma convergence, homogenization techniques and Bargmann transforms, the analysis yields information on vortex shape, number and location.

The book sets out with two introductory chapters; one of them is dedicated to the experimental and theoretical physics of BECs, the other is an overview of the main theorems used in the mathematical modeling. The formalism is based on the mean field description by the Gross-Pitaevskii energy, whose minimizers are analyzed.

The next five chapters dive into specific problems. Each of these includes an introductory paragraph, which is followed by well structured sections with detailed derivations of the main results. The objective of each section is concisely stated right at the beginning, resulting in a remarkable readability of the book in spite of its rather technical style.

Chapter 3 deals with a two-dimensional (2D) model of a rotating condensate in the Thomas-Fermi regime, when the kinetic energy is small in comparison to the trapping and interaction energies, allowing for a small parameter expansion. The energy formulation used here relies on a 2D reduction and a bounded-domain reduction.

In order to make a closer contact with recent experiments, Chapter 4 extends the analysis of the previous one to the context of more realistic trapping potentials. Two of these are treated in detail: a non-radial harmonic potential and a quartic potential.

Chapter 5 turns to the case of high rotational velocity, which gets close to the trapping frequency. The small parameter of this theory is the deviation of their ratio from 1. It is conjectured and argued that the Gross-Pitaevskii energy reduces to contributions of states belonging to the lowest Landau level, and the vortex lattice configuration is found which minimizes this contribution. To this end, homogenization techniques and double-scale convergence is used. A dense triangular Abrikosov lattice is found with a characteristic size of vortices of the same order of magnitude as their interdistance.

The 3D Thomas-Fermi regime is tackled down in Chapter 6 with the goal of explaining experimental observations of global vortex shapes. Planar U vortices are found as global minimizers of the energy. Critical points of the energy are identified, which correspond to planar and non-planar S-shaped vortices. The U vortex has a roughly straight central part and the extremities point normally to the condensate boundary. The analytical study of 3D vortex shapes is augmented with numerical simulations.

Chapter 7 is dedicated to dissipationless flow around an obstacle: a stirring laser beam that penetrates the condensate. The problem is studied in 2D, and in the hydrodynamic formulation, a critical velocity of the stirrer is identified above which the flow becomes dissipative. Finally, the full 3D analysis is performed numerically using a factorization anzatz and rigorous analytical results are derived in the low velocity regime.

Chapters 3 to 7 end with sections dedicated to open questions related to their specific topic, whereas Chapter 8 supplements the rich list of open questions with some general ones. Precisely identified open problems turn the book into a valuable tool in the hands of PhD students or researchers willing to enter this fascinating endeavor of understanding quantum vortices.

Reviewer: Adrian Neagu (Timisoara)