Title: Experiments with linear, nonlinear, and topological excitations in a superfluid gas
1Experiments with linear, nonlinear, and
topological excitations in a superfluid gas
Eric Cornell, Peter Engels, Volker Schweikhard,
Shihkuang Tung, and illustrious forebears (JILA,
NIST/CU, Boulder) thank NSF, NIST
Thanks also to Mark Ablowitz, Mark Hoefer, Keith
Julien HAPPY 40TH BIRTHDAY
MARK!
2Experiments with linear, nonlinear, and
topological excitations in a superfluid gas
Eric Cornell, Peter Engels, Volker Schweikhard,
Shihkuang Tung, and illustrious forebears (JILA,
NIST/CU, Boulder) thank NSF, NIST
Thanks also to Mark Ablowitz, Mark Hoefer, Keith
Julien HAPPY 40TH BIRTHDAY
MARK!
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15Why are BECs so interesting?
QM Particle described by Schrödinger equation
BEC many weakly interacting particles ?
Gross-Pitaevskii equation
? Nonlinear atom optics! Solitons, 4 wave
mixing,
16Why are BECs so interesting?
QM Particle described by Schrödinger equation
BEC many weakly interacting particles ?
Gross-Pitaevskii equation
? Nonlinear atom optics! Solitons, 4 wave
mixing,
17Near-zero T pictures from here on in, and well
lose the goofy 3-d look
18Why are BECs so interesting?
QM Particle described by Schrödinger equation
BEC many weakly interacting particles ?
Gross-Pitaevskii equation
? Nonlinear atom optics! Solitons, 4 wave
mixing,
19NLSE?
Tell me something new.
20NLSE?
Tell me something new.
How about, very little damping?
210.Ground state big, fat, static cloud. The
Thomas-Fermi Approximation
22The Thomas-Fermi Approximation
BEC many weakly interacting particles ?
Gross-Pitaevskii equation
Ignore this term
23Ignore this term
0.Ground state big, fat, static cloud.
241.Linear excitations on a big, fat, static
cloud. Standing waves of soundor Coherent
states of Bogoliubov excitations
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262.Topological excitations on a big, fat, static
(actually, rotating) cloud. Vortex, and vortex
arrays
27NLSE?
Tell me something new.
How about, very little damping?
28Very little damping
29Connections to other fields
Type-II superconductors
Superfluid 4He, rotating bucket
Dilute gas BEC
Bell Labs
Quantum Hall Systems
JILA MIT ENS Oxford
E. J. Yarmchuk, M. J. V. Gordon, R. E.
PackardPhys. Rev. Lett. 43, 214 (1979)
30Floating blob of gas, rotating slowly.
Paul Haljan, Ian Coddington, Peter Engels and
E.A. Cornell cond-mat/0106362 Driving
Bose-Einstein condensate vorticity with a
rotating normal cloud
W
Sucking through straw inserted at north pole
decreases N, decreases E (faster) keeps L
fixed increases W, decreases T!
Condensates form, vortices nucleate, with Vext
not rotating only thermal cloud is rotating.
W
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39Vortex detection an expansion series
In trap
0. 3 ms
21. 3 ms
41. 3 ms
61. 3 ms
81. 3 ms
final diameter gt 1.5 mm
nearly two-dimensional expansion due to
centrifugal energy - Side view
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422.5 Topological excitations on a big, fat, static
cloud with SU(2) order parameter. Spin meets
rotation.
43Square vortex lattice in a two-component (spinor)
BEC
V042835 Wait 7000 ms RF 2.55MHz
Zoom-in local FFT
443.Linear excitations of an array of topological
excitations. Tkachenko waves or, who says
superfluids dont support shear waves?.
45Useful diagnostic for lattice properties
Tkachenko modes. Shear modes in the solid
lattice. Theory Anglin, Baym, Bigelow.
500 ms
1000 ms
1300 ms
2100 ms
1700 ms
46Tkachenko modes arise from resistance of lattice
to shear forces. Frequency of modes goes
as (shear modulus)1/2
Shear modulus in turn is useful probe of
microscopic physics
476
4
(1,0) mode for rotation ?0.95 ??
2
amplitude a.u.
0
-2
time s
0
1
2
3
4
5
- Very low frequency modes! e.g. (1,0) 0.6 Hz _at_
? 0.95
indicates very weak shear modulus!
- Compare to radial breathing mode 16.6 Hz
484.Nonlinear excitations on a big, fat, static
cloud. Superfluid blast, superfluid shock
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50Density peak, single particle
Single particle Schrödinger equation
x
0
...
51Density peak on top of a background, single
particle
Single particle Schrödinger equation
x
0
...
52Density peak, with interactions, no background
53Density peak on top of a background with
interactions
Many interacting particles? GP equation
x
0
g gt 0
nonlinearity
dispersion
Steep front like in classical shock, but why all
these ripples?? later (quantum pressure)
...
54GP, density dip
x
g gt 0
0
nonlinearity
dispersion
Balance of nonlinearity and dispersion
...
?Solitons!
55From GP to quantum hydrodynamics
How similar is quantum shock to classical
shock?? need a hydrodynamic description of the
BEC.
BEC Gross-Pitaevskii equation
Hydrodynamic equation for a BEC
continuity equation
56Classical vs. quantum hydrodynamics
Classical Navier Stokes
Viscosity
Quantum mechanics is irrotational fluid dynamics!
Quantum pressure
Quantum
Almost identical, except for viscosity vs.
quantum pressure!!!
57Quantum Reynolds Number
Is the quantum pressure important at all?
Classical case Viscoscity important when low
Reynolds number Re.
Quantum case Compare mean field to quantum
pressure.
L characteristic length, e.g. width of shock
front
? QP important when important lengthscale beomces
on order of healing length! (e.g. after
sufficient self-steepening).
58Momentarily pierce cloud with beam of laser
light, a repulsive potential.
59c) 0.415 mW
a) 0.221 mW
b) 0.304 mW
f) pulse during expansion
d) 0.460 mW
e) 0.515 mW
Fig. 1 Blast waves in static BECs.
60Shockwaves
How do we create supersonic flow?
Laser
BEC in magnetic trap
Let BEC expand into wall a repulsive laser
beam
Invert the trapping potential ? antitrapped
61Our expansion the details
Our workhorse Antitrapped 2D expansion
typical parameters for a nonrotating cloudRx,y
32 ?mRz 50 ?mNatoms 3.4 millioncsound in
center 1.2 mm/s
vedge(120 ms) 9.1 cm/s
Machnumber (Rtf/2, 100 ms) 3478
some subtle issues due to exponential expansion
Impact velocity
speed of sound
62Hunting for self-steepening and dispersion
63Watching a shock coming into existence
60 ms
V2003102770, 60 ms expansion
64Watching a shock coming into existence
70 ms
V2003102771,70 ms expansion
65Watching a shock coming into existence
80 ms
V2003102761, 80 ms expansion
66Watching a shock coming into existence
100 ms
V2003102763, 100 ms expansion
67Watching a shock coming into existence
110 ms
V2003102764, 110 ms expansion
68Steepening of the wavefront
70 ms
80 ms
110 ms
20 ?m
37.5?m
?heal 28.2 ?m
?heal 6.2 ?m
69Bow shock in a vortex lattice
70From normal to oblique shock
Place wall such that BEC is not hitting it
orthogonally any more.
200310289110 ms expansion.
2003102818 110 ms expansion
2003102825 110 ms expansion.
2003102838 110 ms expansion.
71Oblique shock
shock
All edges about 12-15 microns wide.Calculated
healing length approx. 17 microns.
Clearly self-steepening to a healing-length
feature!!!!!!
V2003102914 went 20 up,100 ms expansion.
72Oblique shock compression corner
73More complicated shock patterns
V2003102925 100 ms expandsion
74Bow shock crossing oblique shock
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77Cylinders, curved flow, curved shockfronts and
vorticity
78Vortices behind a cylinder
Guadalupe Island
Von- Karman street
79Vortices behind a cylinder
Why does an obstacle produce vortices???
Intuitive picture
fast fluid
wake, slow
phase slip
300 ms
0 ms
50 ms
100 ms
For a better simulation, would need
injecting/absorbing boundaries...
80Vortices behind a small cylinder
laser beam
vortices
V102275 poky sits 586/556
81Bigger beam now 12.4 pixel, 7.2 mW
using a bigger beam
82Now pokywaist 23 pixel, 2.6 mW
... and an even bigger beam
83and a really big beam.
V1024116 poky 589/546
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85Transsonic vs. supersonic
transonic
supersonic
Nobody ever heard the bullet that killed him
Theodore von Karman
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