1. 1 Enikö Madarassy Vortex motion in trapped Bose-Einstein condensate Durham University, March, 2007

2. 2 Outline Gross - Pitaevskii / Nonlinear Schrödinger Equation Vortex - Antivortex Pair (Without Dissipation and with Dissipation) - Sound Energy, Vortex Energy - Trajectory - Translation Speed One vortex (Without Dissipation and With Dissipation) - Trajectory - Frequency of the motion - Connection between dissipation and friction constants in vortex dynamics Conclusions

3. 3 This work is part of my PhD project with Prof. Carlo F. Barenghi We are grateful to Brian Jackson and Andrew Snodin for useful discussions. Notations: : initial position of the vortex from the centre of the condensate ( = 0.0 ) : initial separation distance between the vortex-antivortex pair ( = 0.0 ) : friction constants : model of dissipation in atomic BEC :period of the vortex motion; :frequency of the vortex motion

4. 4 The Gross-Pitaevskii equation also called Nonlinear Schrödinger Equation The GPE governs the time evolution of the (macroscopic) complex wave function :Ψ(r,t) Boundary condition at infinity: Ψ(x,y) = 0 The wave function is normalized: = wave function = reduced Planck constant = dissipation [1] = chemical potential m = mass of an atom g = coupling constant [1] Tsubota et al, Phys.Rev. A65 023603-1 (2002)

5. 5 Vortex-antivortex pair (Without dissipation) Fig. 1 Fig.1, t = 87.2 Fig.2, t = 93.0 Fig.3, t = 98.8 Fig.4, t = 104.4 Fig.5, t = 110.2 Fig.6, t = 116.0 Period = 28.8 = 0.8 The first vortex has sign +1 and the second sign -1 Levels: 0.012…….0.002

6. 6 Transfer of the energy from the vortices to the sound field Divide the kinetic energy (E) into a component due to the sound field E s and a component due to the vortices E v [2] Procedure to find E v at a particular time: 1. Compute the kinetic energy. 2. Take the real-time vortex distribution and impose this on a separate state with the same a) potential and b) number of particles 3. By propagating the GPE in imaginary time, the lowest energy state is obtained with this vortex distribution but without sound. 4. The energy of this state is E v. Finally, the the sound energy is: E s = E – E v [2]M Kobayashi and M. Tsubota, Phys. Rev. Lett. 94, 065302 (2005)

7. 7 The sound energy and the vortex energy The sound is reabsorbed Sound energy Vortex energy Correlation between vortex energy and sound energy The corelation coefficient: -0.844 which means anticorrelation Sound Energy Vortex Energy

8. 8 The period and frequency of motion for vortex – antivortex pair The period of motion The frequency of motion Triangle with Circle with

9. 9 The translation speed for different separation distance The translation speed for vortex-antivortex pair: In our case: and The trajectory for one of the vortices in the pair In a homogeneous superfluid Circle: with the formula, Triangle: with numerical calculation

10. 10 The trajectory for the one of the vortices in the pair and for one vortex The trajectory for one of the vortices in the pair (xy) x - coordinate vs time y - coordinate vs time The trajectory for one vortex (xy) x - coordinate vs time y - coordinate vs time are: 0.00 (purple); 0.01 (red); 0.07 (green); 0.10 (blue) (xy) are: 0.00 (purple); 0.01 (red); 0.07 (green); 0.10 (blue) (xt and yt) are: 0.00 (purple); 0.01 (red); 0.07 (green); 0.10 (blue) (xy) are: 0.01 (purple); 0.07 (green); 0.10 (blue) (xt and yt)

11. 11 Two vortices without dissipation and with dissipation =0.01 Density of the condensate with two vortices The initial separation distance d = 1.00

12. 12 The trajectory for one vortex set off-centre Varying initial position and dissipation The trajectory for one vortex (xy) x - coordinate vs time y - coordinate vs time are: 0.90 (y = 0.0) and 1.30 (y = 0.0) are: 0.00 (purple) ; 0.01 (red) ; 0.07 (green) ; 0.10 (blue) (xy) are: 0.01 (red) ; 0.07 (green) ; 0.10 (blue) (xt and yt) =1.30 =0.90

13. 13 The x- and y-component of the trajectory for one vortex (same initial position) = - 2.00 = 0.030 (purple) ; 0.010 (blue) ;0.003 (aquamarine) ; and 0.000 (red)

14. 14 The x- and y-component of the trajectory for one vortex (same dissipation) = -0.90 (green) and - 2.00 (red) = 0.001

15. 15 The trajectory for one vortex (same initial position) = - 2.00 = 0.000 (red) and 0.003 (green) = - 2.00 = 0.030 (red) and 0.010 (green)

16. 16 The radius of the trajectory for one vortex (same initial position) = - 0.90 = 0.030 (red) ; 0.010 (purple) ; 0.003 (blue) and 0.001 (green) = - 2.0 = 0.030 (green) ; 0.010 (purple), ; 0.003 (blue), 0.001 (aquamarine) and 0.000 (red)

17. 17 The frequency of the motion for one vortex as a function of the initial position [3] B.Jackson, J. F. McCann, and C. S. Adams, Phys.Rev. A 61 013604 (1999)

18. 18 The friction constants for one vortex as a function of the dissipation and initial position The friction constant for :0.90 (blu) and 2.00 (red), : 0.001; 0.003; 0.010 and 0.030 The friction constant for 0.90 (blu) and 2.00 (red), : 0.001; 0.003; 0.010 and 0.030

19. 19 Conclusions: Inhomogeneity of the condensate induces vortex cyclical motion. With dissipation the vortex spirals out to the edge of the condensate. The cyclical motion of the vortex produces acoustic emissions. The sound is reabsorbed. Relation between (in GP equation) and (in vortex dynamics).