14
1 Eniko Madarassy Reconnections and Turbulence in atomic BEC with C. F. Barenghi Durham University, 2006

Eniko Madarassy Reconnections and Turbulence in atomic BEC with C. F. Barenghi

  • Upload
    trapper

  • View
    42

  • Download
    0

Embed Size (px)

DESCRIPTION

Eniko Madarassy Reconnections and Turbulence in atomic BEC with C. F. Barenghi. Durham University, 2006. Outline. Gross - Pitaevskii / Nonlinear Schrödinger Equation Vortices (phase, density, quantized circulation) Phase imprinting produces a soliton-like disturbance which - PowerPoint PPT Presentation

Citation preview

Page 1: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

1

Eniko Madarassy

Reconnections and Turbulence in atomic BECwith C. F. Barenghi

Durham University,

2006

Page 2: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

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

Page 3: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

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)

ztr LgVmt

i 222

2)(

ZL

D

NdV2

222 112

1)(, yxmrVpotentialtrappingV Yxtrtr

operatormomentumangularLZ

Page 4: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

4

Vortices

ldvS

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:

ie

Page 5: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

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)

Page 6: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

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 y > 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)

ie

Page 7: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

7

Case I. Snapshots of the density profileThe perturbation was created from the phase change

The original sound wave

The perturbation bends and decays into the vortex pairSound waves due to the decay of the perturbation

Page 8: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

8

Case I. (continued) The perturbation starts to move and bends because of the difference in the density

Higher velocitySound waves due to the vortex pair production

Five pairs of vortices

Three pairs go into boundary.Two pairs survive .

Page 9: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

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.

Page 10: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

10

Re

Imtan 1Phase

(Case I continued)

Phase:

Random phase region: 0 Large fluctuation of the phase: Im 0 Re 0

0,

0,´

´

ye

yi

imprinting

Page 11: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

11

Transfer of the energy from the vortices to the sound field

Divide the total energy into a component due to the sound field Es and a component due to the vortices Ev [8]

Procedure to find Ev 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 Ev.

Finally, the the sound energy is: Es = E – Ev

dxdytg

ttVtm

ET

422

2

)(2

)()()(2

[8] N.G. Parker and C.S. Adams, Phys. Rev. Lett. 95, 145301 (2005)

Page 12: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

12

Case II,

Phase imprinting applied to vortex lattice in rotating frame 00 and

Snapshots of the density and phase profileat the times:

KE VE

SE

200 208

200t

4.200t

95.201t

VSK EEE

dx

dxEK

2

Page 13: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

13

The sound energy in connection with the total energyDue to the new level of energy by the discontinuity, the total energy changes.

redES

Hzx 2192,1

18greenET

Dimensionless unit: (The time units is less than 1ms)

Time:

200 208

Page 14: Eniko  Madarassy   Reconnections and Turbulence in atomic BEC with C. F. Barenghi

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.

VST EEE