VORTICES IN BOSE-EINSTEIN

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VORTICES IN BOSE-EINSTEIN
CONDENSATES
TUTORIAL
IVW 10, TIFR, MUMBAI
8 January 2005
R. Srinivasan
Raman Research Institute, Bangalore
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ORDER PARAMETER F(r,t) OF THE CONDENSATE IS A
COMPLEX QUANTITY GIVEN BY F
(r,t) ( n(r,t))½ exp (iS(r,t))
IT SATISFIES THE GROSS-PITAEVSKI EQUATION IN THE
MEAN FIELD APPROXIMATION
Dalfovo et al. Rev. Mod. Phys. (1999),71,463
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i h ?F(r,t)/?t
-( h 2/2m) ?2 Vext g F(r,t)2
F(r,t)
Vext (r) ½ m w2x x2 w2y y2 w2z z2
g 4 p h2 a / m IS THE INTERACTION TERM
a IS THE s WAVE SCATTERING LENGTH WHICH IS A FEW
NANO-METRES
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FOR STEADY STATE F(r,t) F (r) exp ( i
(m / h )t)
-( h 2/2m) ?2 Vext g F(r)2 F(r) m F(r)
WEAK INTERACTION n a3 ltlt 1
WHEN gn F(r) gtgt -( h 2/2m) ?2 F(r), WE HAVE
THE THOMAS-FERMI APPROXIMATION
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IN THIS APPROXIMATION
n (r) m - Vext (r)/ g
SUBSTITUTING FOR F IN TERMS OF n AND S
?n/?t ??n(( h/m) grad S) 0
h?S/?t (1/2m) ( h grad S)2 Vext g n
- ( h2/2m)(1/?n)?2(?n)
0
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CURRENT DENSITY j i (h /2m) F ?F
- F ?F n(h /m) ?S
SO v ( h/ m) ?S Curl v 0
THE CONDENSATE IS A SUPERFLUID
COLLECTIVE EXCITATIONS OF THE CONDENSATE
F (r,t) exp(-i m t/ h )F(r) u(r)exp(-iwt)
v(r)
exp(iwt)
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SUBSTITUTE IN GP EQUATION AND KEEP TERMS LINEAR
IN u AND v
h w u H0 - m 2gF2 u g F2 v -
h w v H0 - m 2gF2 v g F2 u H0
(- h2 / 2m) ?2 Vext

FOR A SPHERICAL TRAP dn(r)
P l(2nr) (r/R) rl Ylm(q,f) w(nr,
l) w 2 nr2 2nrl3nrl
Stringari S., PRL, (1996), 77, 2360
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SURFACE MODES HAVE NO RADIAL NODES
nr 0
IN THE HYDRODYNAMIC APPROXIMATION FOR AXIALLY
SYMMETRIC TRAPS
w2l w2? l
SURFACE MODES ARE IMPORTANT FOR
VORTEX NUCLEATION.
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DALFOVO et al. PHYS.REV.A(2000),63, 11601
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GROSS-PITAEVSKI EQUATION IN A ROTATING FRAME
HR H- W?L W IS THE ANGULAR
VELOCITY OF ROTATION AND L IS THE ANGULAR MOMENTUM
THE LOWEST EIGENSTATE OF HR IS THE VORTEX FREE
STATE WITH L 0 TILL W REACHES A CRITICAL
VELOCITY WC. THEN A STATE WITH .L h HAS THE
LOWEST ENERGY. THIS IS A VORTEX STATE.
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?C v?dr ( h /m) ?C grad S.dr k (h/m)
THE CIRCULATION AROUND A VORTEX IS QUANTISED
WITH THE QUANTUM OF VORTICITY h/m.
AROUND A VORTEX WITH AXIS ALONG Z, THE VELOCITY
FIELD IS GIVEN BY
vf (h/2pm r)
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THE DENSITY OF THE CONDENSATE AT THE CENTRE OF A
VORTEX IS ZERO. THE DEPLETED REGION IS CALLED
THE VORTEX CORE.
CORE RADIUS IS OF THE ORDER OF HEALING LENGTH x
(8pna)½. FOR THE CONDENSATES THIS AMOUNTS TO
A FRACTION OF A mm.
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CRITICAL VELOCITY FOR PRODUCING A VORTEX WITH
CIRCULATION k (h/m) is DEFINED AS
Wc ( k h) -1 e(k) - e(0) e(k) IS THE ENERGY
OF THE SYSTEM IN THE LAB FRAME WHEN EACH
PARTICLE HAS AN ANGULAR MOMENTUM k h
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FOR AN AXIALLY SYMMETRIC TRAP LUNDH etal DERIVED
THE FOLLOWING EXPRESSION FOR THE CRITICAL ANGULAR
VELOCITY Wc FOR k 1 Wc
5h /2mR2? ln0.671 R?/x
Lundh et al. Phys. Rev.(1997) A 55,2126
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SINCE THE CORE RADIUS IS A FRACTION OF A mm, IT
WILL BE DIFFICULT TO RESOLVE IT BY IN SITU
OPTICAL IMAGING.
SO THE TRAP IS SWITCHED OFF AND THE ATOMS ARE
ALLOWED TO MOVE BALLIS-TICALLY OUTWARDS FOR A FEW
MILLI-SECONDS. THE CORE DIAMETER INCREASES TEN
TO FORTY TIMES AND CAN BE SEEN BY ABSORPTION
IMAGING.
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K.W.Madison et al. PRL(2000),84,806.
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VORTICES CAN BE CREATED BY
PHASE IMPRINTING ON THE CONDEN- SATE.
BY ROTATING THE TRAP ABOVE TC
SIMULTANEOUSLY COOLING THE CLOUD BELOW TC.

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BY STIRRING THE CONDENSATE WITH AN
OPTICAL SPOON.
VORTICES DETECTED BY
RESONANT OPTICAL IMAGING AFTER BALLISTIC
EXPANSION
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BY DETECTING THE DIFFERENCE IN SURFACE
MODE FREQUENCIES FOR THE l 2, m 2 AND m
-2 MODES.
BY INTERFERENCE SHOWING A PHASE WINDING OF 2p
AROUND A VORTEX
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Haljan et al. P.R.L. (2001),86,2922
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Around a vortex there is a phase winding of 2p.
If a moving condensate interferes with a
condensate with a vortex the interference pattern
is distorted
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Fork like dislocations are seen when a vortex is
present
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A VORTEX MAY BE CREATED SLIGHTLY OFF AXIS. IN
SUCH A CASE DUE TO THE TRANSVERSE DENSITY
GRADIENT A FORCE ACTS ON THE VORTEX AND MAKES IT
PRECESS ABOUT THE AXIS. SUCH A PRECESSION HAS
BEEN DETECTED.
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Anderson et al. P.R.L., (2000), 85, 2857