vortices in bose-einstein condensates
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VORTICES IN BOSE-EINSTEIN CONDENSATES. TUTORIAL. IVW 10, TIFR, MUMBAI. 8 January 2005. R. Srinivasan. Raman Research Institute, Bangalore. - PowerPoint PPT PresentationTRANSCRIPT
VORTICES IN BOSE-EINSTEIN
CONDENSATES TUTORIAL
R. Srinivasan
IVW 10, TIFR, MUMBAI
8 January 2005
Raman Research Institute, Bangalore
ORDER PARAMETER (r,t) OF THE CONDENSATE IS A COMPLEX QUANTITY GIVEN BY
(r,t) = ( n(r,t))½ exp (S(r,t))
IT SATISFIES THE GROSS-PITAEVSKI EQUATION IN THE MEAN FIELD APPROXIMATION:
Dalfovo et al. Rev. Mod. Phys. (1999),71,463
h (r,t)/t =
h 2/2m) 2 + Vext + g r,t)|2} (r,t)
Vext (r) = ½ m [2x x2 + 2
y y2 + 2z z2 ]
g = 4h2 a / m IS THE INTERACTION TERM
a IS THE s WAVE SCATTERING LENGTH
WHICH IS A FEW NANO-METRES
FOR STEADY STATE
(r,t) = (r) exp ( / h )t)
h 2/2m) 2 + Vext + g r)|2} (r) = (r)
WEAK INTERACTION: n a3 << 1
WHEN gn (r) >> h 2/2m) 2 (r)}, WE
HAVE THE THOMAS-FERMI APPROXIMATION
IN THIS APPROXIMATION
n(r) = [Vext (r)]/ g
SUBSTITUTING FOR IN TERMS OF n AND S
n/t + [n(( h/m) grad S)] = 0
hS/t + (1/2m) ( h grad S)2+ Vext + g n
( h2/2m)(1/n)2(n) = 0
CURRENT DENSITY
j = h /2m) [ * * ]= n(h /m) S
SO v = ( h/ m) S; Curl v = 0
THE CONDENSATE IS A SUPERFLUID
COLLECTIVE EXCITATIONS OF THE CONDENSATE
(r,t) = expt/ h )(r) +u(r)exp(t)
+ v*(r) exp(t)]
SUBSTITUTE IN GP EQUATION AND KEEP TERMS
LINEAR IN u AND v
h u = [ H0 g|2] u + g | 2 v
h v = [ H0 g|2] v + g | 2 u
H0 = (h2 / 2m) 2 + Vext
FOR A SPHERICAL TRAP
n(r) = P l(2nr)
(r/R) rl Ylm(,)
nr, l) = 2nr2+ 2nrl+3nr+l]
Stringari S., PRL, (1996), 77, 2360
SURFACE MODES HAVE NO RADIAL NODES
nr = 0
IN THE HYDRODYNAMIC APPROXIMATION
FOR AXIALLY SYMMETRIC TRAPS
2l = 2
l
SURFACE MODES ARE IMPORTANT FOR
VORTEX NUCLEATION.
DALFOVO et al. PHYS.REV.A(2000),63, 11601
GROSS-PITAEVSKI EQUATION IN A ROTATING
FRAME: HR = HL
IS THE ANGULAR VELOCITY OF ROTATION
AND L IS THE ANGULAR MOMENTUMTHE LOWEST EIGENSTATE OF HR IS THE VORTEX FREE STATE WITH L = 0 TILL REACHES A CRITICAL VELOCITY C. THEN A STATE WITH .L = h HAS THE LOWEST ENERGY. THIS IS A VORTEX STATE.
C vdr = ( h /m) C grad S.dr = (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
v = (h/m )
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 8na)½. FOR THE CONDENSATES
THIS AMOUNTS TO A FRACTION OF A m.
CRITICAL VELOCITY FOR PRODUCING A
VORTEX WITH CIRCULATION (h/m) is
DEFINED AS
c = ( h) 1[
IS THE ENERGY OF THE SYSTEM IN THE
LAB FRAME WHEN EACH PARTICLE HAS AN
ANGULAR MOMENTUM h
FOR AN AXIALLY SYMMETRIC TRAP LUNDH etal
DERIVED THE FOLLOWING EXPRESSION FOR
THE CRITICAL ANGULAR VELOCITY c FOR
c = {5h /2mR2} ln{0.671 R
Lundh et al. Phys. Rev.(1997) A 55,2126
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.
SINCE THE CORE RADIUS IS A FRACTION OF A m, IT WILL BE DIFFICULT TO RESOLVE IT BY IN SITU OPTICAL IMAGING.
K.W.Madison et al. PRL(2000),84,806.
VORTICES CAN BE CREATED BY
¶ PHASE IMPRINTING ON THE CONDEN-
SATE.
¶ BY ROTATING THE TRAP ABOVE TC
SIMULTANEOUSLY COOLING THE
CLOUD BELOW TC.
¶ BY STIRRING THE CONDENSATE WITH
AN OPTICAL SPOON.
VORTICES DETECTED BY
¶ RESONANT OPTICAL IMAGING AFTER
BALLISTIC EXPANSION
¶ 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 2AROUND A VORTEX
Haljan et al. P.R.L. (2001),86,2922
Around a vortex there is a phase winding of 2If
a moving condensate interferes with a condensate
with a vortex the interference pattern is distorted
Fork like dislocations are seen when a vortex is present
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.
Anderson et al. P.R.L., (2000), 85, 2857