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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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VORTICES IN BOSE-EINSTEIN CONDENSATES TUTORIAL IVW 10, TIFR, MUMBAI 8 January 2005 R. Srinivasan Raman Research Institute, Bangalore
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
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
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 << 1 WHEN gn F(r) >> {-( h 2/2m) 2 F(r)}, WE HAVE THE THOMAS-FERMI APPROXIMATION
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 hS/t + (1/2m) ( h grad S)2+ Vext + g n - ( h2/2m)(1/n)2(n) = 0
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)]
SUBSTITUTE IN GP EQUATION AND KEEP TERMS LINEAR IN u AND v h w u = [ H0- m + 2g|F|2] u + g | F|2 v - h w v = [ H0- m + 2g|F|2] v + g | F|2 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 [ 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 w2l = w2 l SURFACE MODES ARE IMPORTANT FOR VORTEX NUCLEATION.
GROSS-PITAEVSKI EQUATION IN A ROTATING FRAME: HR = H- WL 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.
C vdr = ( 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)
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.
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
FOR AN AXIALLY SYMMETRIC TRAP LUNDH etal DERIVED THE FOLLOWING EXPRESSION FOR THE CRITICAL ANGULAR VELOCITY Wc FOR k = 1 Wc = {5h /2mR2} ln{0.671 R/x } Lundh et al. Phys. Rev.(1997) A 55,2126
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.
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 2p AROUND A VORTEX
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
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.
