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 + V ext + g  r,t)| 2 }  (r,t) V ext (r) = ½ m [  2 x x 2 +  2 y y 2 +  2 z z 2 ] g = 4  h 2 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 + V ext + g  r)| 2 }  (r) =  (r) WEAK INTERACTION: n a 3 << 1 WHEN gn  (r) >>  h 2 /2m)  2  (r)}, WE HAVE THE THOMAS-FERMI APPROXIMATION

IN THIS APPROXIMATION n  (r) = [  V ext (r)]/ g SUBSTITUTING FOR  IN TERMS OF n AND S  n/  t +  [n(( h/m) grad S)] = 0 h  S/  t + (1/2m) ( h grad S) 2 + V ext + g n  ( h 2 /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) = exp  t/ h )  (r) +u(r)exp(  t) + v*(r) exp(  t)]

SUBSTITUTE IN GP EQUATION AND KEEP TERMS LINEAR IN u AND v h  u = [ H 0  g|  2 ] u + g |  2 v  h  v = [ H 0  g|  2 ] v + g |  2 u H 0 = (  h 2 / 2m)  2 + V ext FOR A SPHERICAL TRAP  n(r) = P l (2 n r ) (r/R) r l Y l m ( ,  )  n r, l) =  2  n r n r l+3 n r +l] Stringari S., PRL, (1996), 77, 2360

SURFACE MODES HAVE NO RADIAL NODES n r = 0 IN THE HYDRODYNAMIC APPROXIMATION FOR AXIALLY SYMMETRIC TRAPS  2 l =  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: H R = H  L  IS THE ANGULAR VELOCITY OF ROTATION AND L IS THE ANGULAR MOMENTUM THE LOWEST EIGENSTATE OF H R 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 v  dr = ( 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  8  na) 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 /2mR 2  } 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 T C SIMULTANEOUSLY COOLING THE CLOUD BELOW T C.

¶ 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 2  AROUND A VORTEX

Haljan et al. P.R.L. (2001),86,2922

Around a vortex there is a phase winding of 2  If 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