VORTEX MATTER IN ATOMIC BOSE-EINSTEIN CONDENSATES W. V. Pogosov, Institut des NanoSciences de Paris, Universite Paris VI – Pierre et Marie Curie, Paris; - презентация

1 VORTEX MATTER IN ATOMIC BOSE-EINSTEIN CONDENSATES W. V. Pogosov, Institut des NanoSciences de Paris, Universite Paris VI – Pierre et Marie Curie, Paris; Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow In collaboration with Prof. K. Machida, Department of Physics, Okayama University, Okayama, Japan

2 1. Introduction 2. Vortex formation: role of antivortices 3. Thermal melting of vortex clusters 4. Thermal fluctuations in spin-1 condensates 5. Thermal fluctuations in spin-2 condensates: kinks and roughening transition 6. Summary Outline

3 Introduction Vortices in Bose-condensates (W. Ketterles group, MIT).

4 Harmonic trapping potential Energy functional

5 Dimensionless form: Gas parameter: Variational approach:

6 How vortices nucleate, if the trap is rotated? - there is no well-defined boundary! What is the critical frequency for the vortex formation? Vortex nucleation: role of antivortices

7 Order parameter: Instability:

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11 3D case - Prediction: vortex loops nucleate instead of vortex-antivortex pairs - Bending of vortices in 3D case was observed in numerical simulations: A. Aftalion and I. Danaila, Phys. Rev. A68, (2003).

12 Quasi 2D condensates can be created experimentally Thermal effects are more important in low dimensional systems Aims: What are typical melting temperatures for vortex clusters in trapped atomic condensates? What is the effect of symmetry and vortex number quantization on melting temperatures? Thermal melting of vortex clusters

13 Structure of the vortex molecule: ground state Vortices are repealed by each other and by cloud boundaries, therefore, they form well-defined patterns At small number of vortices (low rotation speeds), one-shell vortex molecule. At large number of vortices (high rotation speeds), more and more shells. We concentrate on two-shell vortex clusters

14 p = 5

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16 Ground state configurations at coupling constant g = 5 (two-shells clusters) Total number of vortices In the inner shell 2334 In the outer shell 8899 Commensurability +-+- Rotation speed /

17 Number of vortices d, degrees At low temperatures, intershell melting is happening! Melting temperature (Lindemann criterion): n 2 is the number of vortices in the outer shell. The difference in melting temperatures is several orders of magnitude!

18 Quadrupolar potential: It fixes orientation of vortex molecule in the space What is the depinning temperature? Vortices in a trap with quadrupole deformation

19 2-vortex molecule 3-vortex molecule 4-vortex molecule Number of vortices Rotation speed

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21 Thermal fluctuations in spin-1 condensate

22 Nonsingular vortex Spin texture

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24 Fluctuations of transverse magnetization

25 Thermal fluctuations in spin-2 condensates: kinks and roughening transition Cyclic phase: Spin-mixing term (sine-Gordon):

26 Structure of a single kink in a homogeneous system

27 Roughening transition

28 Kinks destroy the LRO 2D: k T = energy of the critical nucleus 1D: no LRO in infinite systems! entropy BKT temperature

29 -Topological defects nucleate via vortex-antivortex pairs (2D) or vortex loops (3D). -Melting temperatures depend on vortex molecule symmetry (orders of magnitude) and can be very low. -In the trap with an quadrupole deformation, depinning temperature is extremely sensitive to the symmetry of the vortex molecule. -In spin-1 condensates with vortices thermal decoherence among order parameter components occurs at very low temperatures leading to fluctuations of spin textures -In spin-2 condensates we found a thermal roughening transition destroying coherence among order parameter components via proliferation of kink-antikink pairs Summary