When a Bose-Einstein condensate (BEC) is rotated sufficiently fast, it nucleates vortices. The system is only stable if the rotational velocity is lower than a critical value . Experiments show that as approaches , the condensate nucleates more and more vortices, which become periodically arranged. We present here a mathematical study of this limit. Using Bargmann transform and an analogy with semi-classical analysis in second quantization, we prove that the system necessarily has an infinite number of vortices and provide an ansatz for the solution. This summarizes two joint works, with A. Aftalion (LJLL, Univ. Paris 6) and J. Dalibard (LKB, Ecole Normale Supérieure), on the one hand, and with A. Aftalion and F. Nier (IRMAR, Univ. Rennes I) on the other hand.
@article{SEDP_2005-2006____A5_0, author = {Blanc, Xavier}, title = {Fast rotating {Bose-Einstein} condensates and {Bargmann} transform}, journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"}, note = {talk:5}, pages = {1--18}, publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2005-2006}, mrnumber = {2276071}, language = {en}, url = {http://www.numdam.org/item/SEDP_2005-2006____A5_0/} }
TY - JOUR AU - Blanc, Xavier TI - Fast rotating Bose-Einstein condensates and Bargmann transform JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:5 PY - 2005-2006 SP - 1 EP - 18 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2005-2006____A5_0/ LA - en ID - SEDP_2005-2006____A5_0 ER -
%0 Journal Article %A Blanc, Xavier %T Fast rotating Bose-Einstein condensates and Bargmann transform %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:5 %D 2005-2006 %P 1-18 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2005-2006____A5_0/ %G en %F SEDP_2005-2006____A5_0
Blanc, Xavier. Fast rotating Bose-Einstein condensates and Bargmann transform. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 5, 18 p. http://www.numdam.org/item/SEDP_2005-2006____A5_0/
[1] J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science 292, pp 476–479, 2001.
[2] A.A. Abrikosov, Magnetic properties of group II Superconductors, J. Exptl. Theoret. Phys. (USSR) 32(5), pp 1147–1182, 1957.
[3] A. Aftalion, Vortices in Bose Einstein condensates, Progress in nonlinear differential equations, Vol 67, Birkhauser, 2006. | MR | Zbl
[4] A. Aftalion, X. Blanc Vortex lattices in rotating Bose Einstein condensates, to appear in SIAM J. Math. Anal. | MR | Zbl
[5] A. Aftalion, X. Blanc, F. Nier, Vortex distribution in the lowest Landau level, Phys. Rev. A 73, p 011601(R), 2006.
[6] A. Aftalion, X. Blanc, F. Nier, Lowest Landau level and Bargmann transform in Bose-Einstein condensates, in preparation.
[7] A. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemann, E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269, p 198, 1995.
[8] A. Aftalion, X. Blanc and J. Dalibard, Vortex patterns in fast rotating Bose Einstein condensates, Phys. Rev. A 71, p 023611, 2005.
[9] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, pp 187–214, 1961. | MR | Zbl
[10] G. Baym, C.J. Pethick, Vortex core structure and global properties of rapidly rotating Bose-Einstein condensates, Phys. Rev. A 69, 2004.
[11] V. Bretin, S. Stock, Y. Seurin, J. Dalibard, Fast Rotation of a Bose-Einstein Condensate, Phys. Rev. Lett. 92, 050403, 2004.
[12] E. Carlen, Some Integral Identities and Inequalities for Entire Functions and Their Application to the Coherent State Transform, J. Funct. Analysis 97, 1991. | MR | Zbl
[13] K. Chandrasekharan Elliptic Functions Grundlehren der mathematischen Wissenschaften 281, Springer, 1985. | MR | Zbl
[14] N. R. Cooper, S. Komineas, N. Read, Vortex lattices in the lowest Landau level for confined Bose-Einstein condensates, Phys. Rev. A 70, p 033604, 2004.
[15] J.M. Delort FBI transformation. Second microlocalization and semilinear caustics. Lect. Notes in Math. 1522, Springer-Verlag, 1992. | MR | Zbl
[16] P. Engels et al, Observation of Long-lived Vortex Aggregates in Rapidly Rotating Bose-Einstein Condensates, Phys. Rev. Lett. 90, p 170405, 2003.
[17] E. A. Cornell, W. Ketterle, C. E. Wiemann, Les Prix Nobel. The Nobel Prizes 2001, Editor Tore Frängsmyr, [Nobel Foundation], Stockholm, 2002.
[18] U.R. Fischer, G. Baym, Vortex states of rapidly rotating dilute Bose-Einstein condensates, Phys. Rev. Lett. 90, p 140402, 2003.
[19] G.B Folland Harmonic Analysis in Phase Space, Princeton University Press, 1989. | MR | Zbl
[20] L. Gross Hypercontractivity on complex manifolds, Acta Math. 182, 1999. | MR | Zbl
[21] T. L. Ho, Bose-Einstein Condensates with Large Number of Vortices, Phys. Rev. Lett. 87, p 060403, 2001.
[22] L. Hörmander. Lectures on Nonlinear Hyperbolic Differential Equations. Mathématiques et Applications 26, Springer, 1997. | MR | Zbl
[23] W.H. Kleiner, L.M. Roth, S.H Autler, Bulk Solutions of Ginzburg-Landau Equations for type II Superconductors: Upper Critical Fiel Region, Phys. Rev. 133(5A), pp 1226–1227, 1964. | Zbl
[24] L. D. Landau,E. M. Lifschitz, Quantum Mechanics (Oxford: Pergamon), 1965.
[25] N. Lerner The Wick calculus of pseudo-differential operators and energy estimates, New trends in microlocal analysis (Tokyo 1995), Springer, Tokyo, 1997. | MR | Zbl
[26] E. H. Lieb, R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Arxiv:math-ph/0504042, 2005. | MR
[27] E. H. Lieb, R. Seiringer, J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A 61, p 0436021, 2000.
[28] E. H. Lieb, R. Seiringer, J. Yngvason, A rigourous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys. 224, pp 17–31, 2001. | MR | Zbl
[29] K. Lu, X. B. Pan, Gauge invariant eigenvalue problems in and in . Trans. Amer. Math. Soc. 352, no. 3, pp 1247–1276, 2000. | MR | Zbl
[30] K. Madison K, Chevy F, Bretin V, Dalibard J, Vortex Formation in a Stirred Bose-Einstein Condensate, Phys. Rev. Lett. 84, p 806, 2000
[31] A. Martinez An Introduction to Semiclassical Analysis and Microlocal Analysis, Universitext, Springer-Verlag, 2002. | MR | Zbl
[32] M.R. Matthews et al., Vortices in a Bose-Einstein Condensate, Phys. Rev. Lett. 83, p 2498, 1999.
[33] S. Nonnenmacher, A. Voros, Chaotic Eigenfunctions in Phase Space, J. Stat. Phys. 92, pp 431–518, 1998. | MR | Zbl
[34] C. Pethick, H. Smith, Bose-Einstein condensation in dilute gases, Cambridge University Press, 2001.
[35] L. Pitaevskii, S. Stringari, Bose-Einstein condensation, Oxford University Press, 2003. | MR | Zbl
[36] V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, E. A. Cornell, Rapidly Rotating Bose-Einstein Condensates in and near the Lowest Landau Level, Phys. Rev. Lett. 92, p 040404 2004.
[37] K. Schnee, J. Yngvason, Bosons in Disc-Shaped Traps: From 3D to 2D, Erwin Schrödinger Institute of Vienna, preprint n 1716, 2005. | MR | Zbl
[38] D. E. Sheehy, L. Radzihovsky, Vortices in Spatially Inhomogeneous Superfluids, Phys. Rev. A 70, p 063620 2004.
[39] J. Sjöstrand Singularités analytiques microlocales. Soc. Math. France, Astérisque 95, pp 1–166, 1982. | Numdam | MR | Zbl
[40] S. Stock, V. Bretin, F. Chevy, J. Dalibard, Shape oscillation of a rotating Bose-Einstein condensate, Europhys. Lett. 65, p 594, 2004.
[41] O. Törnkvist, Ginzburg-Landau theory of the electroweak phase transition and analytical results, arXiv:hep-ph/9204235, 1992.
[42] G. Watanabe, G. Baym and C. J. Pethick, Landau levels and the Thomas-Fermi structure of rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett. 93, p 190401 2004.