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