Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 1/20

Lecture:Dynamics of a quantum gas

5.1 Collective modes of a trapped quantum gas

5.2 Sound5.3 Measuring Bogoliubov excitations5.4 Solitons5.5 Quantized Vortices

- creating and observing vortices- vortex lattice- Critical Rotation

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 2/20

Shaking the BEC

10 msec. per frame

Sloshing motion

5 milliseconds per frame

“Non-destructive” observation of a time-dependent wave function

Quadrupole oscillations

a very sensitive measurement tool: any change in the potential will change the oscillation frequency

application: Atom-Surface interaction, Van deer Waals and Casimir Polder interaction

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 3/20

Shak

ing

the

BEC

tem

pera

ture

and

den

sity

depe

nden

ce

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 4/20

Scissors ModeO.M. Marago et al. Phys. Rev. Lett. 84, 2056 - 2059 (2000)

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 5/20

Sound = propagating density perturbationsSound propagationM.R. Andrews PRL79, 553 (1997)

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 6/20

high energies:

particle like excitations coeff of Bogoliubov transf:

cross over

low energies:

excitations: sound waves sppeed of sound: c

coeff of Bogoliubov transf:

mpng

mpp

22

22

1 pp uv

cppmngp

pmcvu pp 2

Bogoliubov Excitation Spectrum

222

2

mpp

mngp

22

2mc

mp

2

24man

mgnc

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 7/20

Bragg Spectroscopy

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 8/20

Bragg Spectroscopy

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 9/20

Excitation Spectrum of a Bose-Einstein Condensate

Dynamic structure factor S(k,w): response to excitation

Static structure factor S(k): Fourier transform of the density correlation function

PRL 88, 120407 (2002)

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 10/20

SolitonsSoliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed GP equation: Soliton solutions for repulsive interaction: dark solitonstationary solution

moving solution

velocity u is related to the density ratio nmin/n0

change of phase though the soliton

soliton solutions for attractive interaction: bright soliton

self bound states localized in space

02

with2

tanhmgn

xx

0

minarccos2)()(nnxx

mgns

mgnu

nn

su 02min2

0

min2

2

or or

2

2min0min

)/(1 with

2tanh)(,

su

utxnnntxn uu

221

2/ )0( with

/||2cosh

1)0(.

gxm

etx ti

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 11/20

Solitons

solitons as solutions of nonlinear differential equations which • represent waves of permanent form; • are localised, so that they decay or approach a constant at infinity; • can interact strongly with other solitons, but they emerge from the collision unchanged

apart from a phase shift. Many exactly solvable models have soliton solutions, including the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and the sine-Gordon equation. The mathematical theory of these equations is a broad and very active field of mathematical research.

John Scott Russell (Scottish engineer 1808-1882) September 1844: ``I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation''.

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 12/20

Dark Solitons in a BEC

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 13/20

Excitation Spectrum

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 14/20

Structure of a Vortex

axial symmetry: cylindrical coordinates

from he velocity field we get an additional term in the energycylindrical GP equation for f

scaled variablesuniformmedium

approximate solution

numerical solutionenergy per vortex

b

mfv 464.1ln

22

0

iezf ),(r

2

222

2 f

m

fgffzVfmdz

fdddf

dd

m

3

2

22

2

22

),(2

12

fFx / and /

01 32

xdxdx

dxd

x

22 x

x

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 15/20

Making Vortices

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 16/20

Vortex Formation

vortices are created by surface instabilities:

Landau criterion: above a critical velocity, the flow at the surface becomes turbulent

and breaks apart into vortices

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 17/20

3-dim Structure

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 18/20

Large number of Vortices

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 19/20

Crystallization of the Vortex lattice

Degenerate Quantum Gases SS 2011 I. Mazets, J. Schmiedmayer Lecture: Dynamics of Quantum Gas: experimental demonstrationsFolie: 20/20

Rotating harmonic trap: transform into the co rotating frame

formally equivalent to a particle with charge q* moving in an effective magnetic field B*

the Hamiltonian is there for reminiscent to the Quantum Hall effect: filling factor n

Fast RotationQuantum Hall states in rotating BEC

220

22220

2

220

2

))((21

2ˆ

ˆ21

2

zyxmm

m

mm

H

wwww

www

rzp

przrp

zrzB ˆ)2()/ˆ( *** qmqm ww

vNN

mh

AN

Bqh

AN

w

n2**

To achieve these states the rotation frequency ahs to be ~ trap frequencyThen the centrifugal potential exactly compensates the trapping potential -> looks like a free 2d system with a coupling to vector potential like in electro magnetism