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1 Eniko Madarassy Reconnections and Turbulence in atomic BEC with C. F. Barenghi Durham University, 2006

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2 Outline Gross - Pitaevskii / Nonlinear Schrödinger Equation Vortices (phase, density, quantized circulation) Phase imprinting produces a soliton-like disturbance which decays into vortices Sound energy and Kinetic energy Conclusions

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3 The Gross-Pitaevskii equation in a rotating system 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 = rotation frequency of the trap = centrifugal term g = coupling constant [1] Tsubota et al, Phys.Rev. A65 023603-1 (2002)

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4 Vortices Vortex : a flow involving rotation about an axis = Madelung transformation = Density = 0, on the axis = Phase: changes from 0 to 2π going around the axis Quantized circulation:

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5 Aim / motivations Creation of mini-turbulent vortex system Large scale turbulence of quantized vortices is studied in superfluid 3He-B and 4He. Disadvantage of turbulence in BEC: small system and few vortices Advantage: relatively good visualization of individual vortices, more detail Particularly: can study detail of transformation of kinetic energy into acustic energy [2], (which occurs in liquid helium too). Because of: 1) vortex reconnection [3] 2) vortex acceleration [4] [2] C. Nore, M. Abid, and M.E. Brachet., Phys. Rev. Lett. 78, 3896 (1997 ) [3] M.Leadbeater, T. Winiecki, D.S. Samuels, C.F. Barenghi, C.S. Adam, Phys. Rev. Lett. 86, 1410 (2001) [4] N.G. Parker, N.P. Proukakis, C.F. Barenghi and C.S. Adams, Phys. Rev. Lett. 92, 160403-1 (2004)

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6 Decay of soliton-like perturbation into vortices Dark solitons are observed in BECs [5],[6], they are produced with the ” Phase Imprinting ” method [7]. For example: We imprint the phase in two ways: Case I: Case II: in upper two quadrants in upper left quadrant (x 0) and bottom right quadrant (x > 0 and y < 0) In both cases soliton-like perturbations are produced. Solitary waves in matter waves are characterized by a particular local density minimum and a sharp phase gradient of the wave function at the position of the minimum. [5] S. Burger et al., Phys.Rev. Lett. 83,5198 (1999); J. Denschlag et al., Science 287, 97, (2000) [6] N.P. Proukakis, N.G. Parker, C.F. Barenghi, C.S. Adams, Phys. Rev. Lett. 93, 130408-1, (2004) [7] L. Dobrek et al., Phys. Rev. A 60, R3381 (1999)

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7 Case I. Snapshots of the density profile The perturbation was created from the phase change The original sound wave The perturbation bends and decays into the vortex pair Sound waves due to the decay of the perturbation

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8 Case I. (continued) The perturbation starts to move and bends because of the difference in the density Higher velocity Sound waves due to the vortex pair production Five pairs of vortices Three pairs go into boundary. Two pairs survive.

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9 (Case I. Continued) Another view Sound waves due to the decay of the perturbation. The perturbation bends and starts to move. The perturbation decays into the vortex pair. The soliton like perturbation.

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10 ( Case I continued) Phase: Random phase region: 0 Large fluctuation of the phase: Im 0 Re 0 imprinting

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11 Transfer of the energy from the vortices to the sound field Divide the total energy into a component due to the sound field E s and a component due to the vortices E v [8] Procedure to find E v at a particular time: 1. Compute the total 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 [8] N.G. Parker and C.S. Adams, Phys. Rev. Lett. 95, 145301 (2005)

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12 Case II, Phase imprinting applied to vortex lattice in rotating frame Snapshots of the density and phase profile at the times:

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13 The sound energy in connection with the total energy Due to the new level of energy by the discontinuity, the total energy changes. Dimensionless unit: ( The time units is less than 1ms) Time:

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14 Conclusions: By generating a discontinuity in the phase, the system tries to smooth out this change and generate a soliton-like perturbation, which decays into vortices. We observe transformation of kinetic energy into sound energy. The sound energy is the biggest contribution to the change of the total energy. Two contributions to the sound energy. First, from the phase change and second from the interaction between vortex-antivortex.

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System and definitions In harmonic trap (ideal): er.

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