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1 THERMAL MELTING OF VORTEX MOLECULES IN 2D BOSE-EINSTEIN CONDENSATES W. Pogosov and K. Machida Okayama University, Japan 1. Motivation 2. Model 3. Intershell melting of vortex clusters 4. Vortices in a trap with quadrupole deformation 5. Summary

2 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? Motivation

3
We consider low temperatures, T<

4 The order parameter Gaussian ansatz [D. Butts and D. S. Rokhsar, Nature 1999] Accurate at small coupling constant g; Variational parameters are degrees of freedom for the system

5 The energy as a function of variational parameters; Ground state values of variational parameters by numerical minimization of the energy; Equilibrium positions of vortices; Deviation of vortices from equilibrium positions as functions of deviations of variational parameters from the equilibrium; Deviation of energy from the ground state as a quadratic form of the deviation of variational parameters; Diagonalization of this quadratic form; Calculation of the statistical sum and average deviations of vortices due to thermal fluctuations

6 Structure of the vortex molecule: ground state Vortices are repealed by each other and by cloud boundaries, therefore, they form well-defined patterns At small number of vortices (low rotation speeds), one-shell vortex molecule.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 Intershell melting

7 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 /

8 At low temperatures, intershell melting is happening! Mutual thermal displacements of vortex shells

9 Number of vortices , degrees Melting temperature (Lindemann criterion): n 2 is the number of vortices in the outer shell.

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

12 2-vortex molecule 3-vortex molecule 4-vortex molecule Number of vortices Rotation speed

13 t = 0.1

14 Melting of two-shell vortex molecules in quasi 2D condensate can be observed at realistic parameters, low temperatures and large numbers of atoms. Melting temperatures depend strongly on vortex molecule symmetry (several orders of magnitude), i.e. commensurability between the numbers of vortices in shells. In the trap with an additional quadrupole deformation, depinning temperature of the molecule is very sensitive to the symmetry of the molecule. Summary

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