Shaking the BEC
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Transcript of Shaking the BEC
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