Superfluidity of Disordered Bose Systems: Numerical Analysis of the - PowerPoint PPT Presentation

Superfluidity of Disordered Bose Systems: Numerical Analysis of the Gross-Pitaevskii Equation with a Random Potential Michikazu Kobayashi and Makoto Tsubota Faculty of Science, Osaka City University, Japan

Motivation • How does disorder affect Bose-Einstein condensation and its Superfluidity? • What is the relation between BEC and superfluidity? • By adjusting with disorder, we may divide BEC from superfluidity and understand this relation! • We investigate this problem by the Gross- Pitaevskii(GP) equation with a random potential

Dynamics of two dimensional Bose system Bose field operator →macroscopic wave function    of BEC and its fluctuation ˆ ˆ Ψ = Φ + ϕ ( x ) ( x ) ( x ) neglecting the fluctuation ⇒ equation of the macroscopic wave function(GP equation)  2  ∂      2 2  Φ = − ∇ − µ + + Φ Φ i ( x , t )  U ( x ) g ( x , t )  ( x , t ) ∂ t 2 m    

U(x) : random potential Fourier transformation of U(x) One example of U(x) (we take 100 ensemble average) R 0 |U(k)| 2 /V 0 1 2 3 4 5 6 7 8 The average number of Wave number L/ λ wave is regulated to be Of course, U(k) decays above about 4 in a direction. the wave number 4

Ground state of GP equation Parameter: strength of the random potential R . 0 R / μ= 20 R / μ= 70 0 0 R / μ= 50 0 Wave function localizes as R increases 0

Parameter: coherence length ξ of the wave function. ξ =  µ µ / 2 m ( ~ gn ) λ: Characteristic width of the random potential ξ / λ= 3 ξ / λ= 1 ξ / λ= 2 Wave function localizes as ξ decreases

Ground state depends on the shape of the random potential even at same parameters R / μ= 50 0 ξ / λ= 2

The superfluid density of ground states 100 ensemble average at same R 0 The dependence on R The dependence on ξ 0 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 ρ / ρ ρ / ρ 0 0 ρ / ρ ρ / ρ s s 0 0 40 80 120 160 200 1 1.5 2 2.5 3 ξ / λ R / µ 0 Superfluidity is depressed as the ground states localizes

Dynamics of vortex pairs in applied velocity field µ  µ  t / t / v = = 0 . 16 0 amplitude 2 m = v 1 . 5 µ -π π -π π phase R / μ= 50 0 ξ / λ= 2 Phase becomes to be complicated

µ  µ  t / t / = 0 . 23 = 0 . 40 amplitude -π π -π π phase Vortex pair nucleates! (Branch cut of the phase)

Dynamics of vortex pairs The dependence on The dependence of the critical velocity the velocity field The average number of vortex pairs on the coherence length 1.4 2.5 1.4 ρ / ρ 0 1.2 1.2 χ / ρ 2 T n /n or χ (trans)/ ρ 1 1 c (2m/ µ ) 1/2 v 1.5 0.8 0.8 0.6 0.6 1 Vortex pairs 0.4 0.4 0.5 0 0.2 0.2 0 0 0 1 1.5 2 2.5 3 0.5 1 1.5 2 ξ/λ 2m/ µ ) 1/2 v (2m The critical velocity Nucleation of vortex pairs is small as the wave destroys superfluidity function localizes

Conclusion • By using GP equation with a random potential, the superfluid density in disordered system can be calculated. • The dynamics of vortex pairs in applied field can be calculated • We will expand this calculation to 3-dimension.