Superfluidity in bosonic systems

Transcription

1 Superfluidity in bosonic systems Rico Pires PI Uni Heidelberg

2 Outline Strongly coupled quantum fluids 2.1 Dilute Bose gases 2.2 Liquid Helium Wieman/Cornell A. Leitner, from wikimedia

3 When are quantum effects important?? λ DB = 2 2 π π mk T B λ DB n 1/3 nλ 3 DB 1 Pfau webpage

4 Critical Temperature f ( ε ) = 1 ( )/( kt ) e ε μ N = 1 N = N + h f( ε ( r, p)) d rd p= N + f( ε) ρ( ε) dε ε f( ε ) In a 3D box N N T = TC 0 1 3/2 In a harmonic potential T C N N T = TC /3 n 3 2π = mkb ξ (3 / 2) 3 nλ DB 2.61

5 Onset of condensation nr ( ) = h 3 f ( ε ) d 3 p Sudden jump of n(0) for T=T C Wieman/Cornell

6 87 Rb BEC ground state fraction Ensher et al., PRL 77 (1996) N N T = TC 0 1 3

7 Gross-Pitaevski Equation 2 2m 2 Δ+ Utrap () r + gψ0(,) r t ψ0(,) r t = i ψ0(,) r t t g = 4π 2 a Non-linear Schrödinger equation for the order parameter in zero temperature limit. The solution is the chemical potential!

8 The case of finite interaction C 1/3 δ T = c an T [1] diagrammatic perturbation theory, Phys. Rev. Lett. 83, c 1703 (1999) [2] Perturbation in canonical ensemble, J. Phys. B 33, L779 (2000) [3] Path integral Monte Carlo, Phys. Rev. Lett. 79, 3549 (1997) [4] Ursell operators, Eur. Phys. J. B 10, 739 (1999) [5] O(N) scalar field theory (NLO), Phys. Rev. A 62, (2000) [6] Path integral Monte Carlo simulation, Phys. Rev. Lett (1999) [7] O(N) scalar field theory (LO), Europhys. Lett. 49, 150 (2000) [8] O(N) scalar field theory linear δ expansion, condmat/ (unpublished). [9] Renormalization group, Phys. Rev. A 45, 8398 (1992) Arnold and Moore, PRL 87 (2001)

9 Lower Critical Temperature (MF approach) Critical temperature decreased in 87 Rb: Gerbier et al., PRL 92 (2004)

10 Higher critical temperature (beyond MF,non-perturbative) Only approximation: High modes of Bose quantum field can be approximated by classical field Davis, Blakie PRL 96 (2006)

11 Higher critical temperature (beyond MF,non-perturbative) Only approximation: High modes of Bose quantum field can be approximated by classical field Davis, Blakie PRL 96 (2006)

12 Gross-Pitaevski Equation 2 2m g 2 Δ+ Utrap () r + gψ0(,) r t ψ0(,) r t = i ψ0(,) r t t = 4π 2 a Linearize (Bogoliubov-DeGennes eqs.) = p ( p + 8 π an) 8π an 2m p c p 2 2 p p p + μ = + 4π 2m 2m 2 an 0

13 Sound wave propagation Blue detuned laser BEC Andrews et al.,prl 79 (1997) cr () = nr ()4π m 2 a

14 Collective excitations Perturbing the trapping potential of a 87 Rb BEC with a sine wave Ketterle Jin et al., PRL 77 (1996)

15 Superfluid behaviour Heating of the sample due to vortex formation only above v c Raman et al., PRL 83 (1999) Na BEC Blue detuned laser

16 Vortices From GPE we get: n + ( nv) = 0 t with: v ψ ψ ψ ψ = 2 2mi ψ use ψ ( r) = ψ ( r, z) e iϕ : h vϕ = l 2π mr Γ= v dr = = l n v 2 π 2 ln 2π l m b ξ v = m ϕ v = 0

17 Vortex lattices Ketterle et al., Science 292 (2001)

18 BEC-BCS crossover Ketterle Nature (2005)

19 Eulers equation for a perfect fluid Use trial function in GPE: p n V t mn 2 m 2m n m 2 2 v 1 v = + Quantum pressure Cf. Eulers eq. For a perfect fluid v v p V t mn 2 m 2 v 1 v 1 ( ) =

20 Superfluidity in Helium Yarmchuk et al., PRL 43 (1979) Bewley et al., University of Maryland, from aps.org (2006)

21 Superfluidity in Helium Yarmchuk et al., PRL 43 (1979) Bewley et al., University of Maryland, from aps.org

22 Why is Helium a quantum fluid? dhe ( ) = 0.265nm d( Ne) = 0.296nm Lennard-Jones type potential (approx.!) d d V() r = ε 2 r r ε ( He) = 1.03meV 0 ε ( Ne) = 3.94meV 0 λ λ DB DB ( He) 0.4nm > d ( Ne) 0.07nm < d

23 Helium phase diagram LTT Helsinki 3 E0 = ω0 7meV 70K 2

24 Specific heat capacity of 4 He Pulsed heat method compared to PIMC simulation C v α CT ( ) + A+ T TC for ( T> TC) = α CT ( ) + A T TC for ( T< TC) α Nissen/Israelsson Heat pulse method in space shuttle Ceperley, Rev. Mod. Phys. 67 (1995)

25 Discovery of superfluidity in Helium Allen, Misener, Kapitza (NP 1978) Experiments on flow through capillary ΔP v η L R 2

26 Theoretical understanding of superfluids London, Tisza and Landau j = j + j s n n S ν BT ( C T) T< T 0 T > T C C

27 Landaus dispersion curve Neutron scattering experiments ε = ε v( p p ) f i i f ε ( p) = cp 2 p 2m p p small large Donnelly et al., J. Low Temp. Phys. 44 (1981)

28 Grand canonical ensemble nn, ( N ) n Z = e β μ ( E N ) N = k T B ln Z μ ln Z U = H = μ N β Exact solutions for no interactions Solve with perturbation theory for weak interactions QMC simulations for strong interactions n 0.1n 0 Ceperley and Pollock, PRL 56 (1986) Path integral Neutron Scattering R. A. Cowley and A. D. B. Woods, Can. J. Phys. 49, 177 (1971) H. R. Glyde, J. Low Temp. Phys. 59, 561 (1985) R. N. Silver, Phys. Rev. B 37, 3794 (1988);ibid. 38, 2283 (1988) A. S. Rinat, Phys. Rev. B 42, 9944 (1990)

29 Correlation function and structure factor ρ ( r r ) = N ψ ( r, r,..., r ) ψ ( r, r,..., r ) d r... d r N N 2 N n lim ( r r ) = 0 ρ r r 1 1 ( ) 3 P p d p 3 Vd p nˆ 3 K = (2 π ) Ψ Ψ Ψ n Ψ = r e d r S k ikr 3 ˆ K ρ1( ) ( )

30 Static structure factor Woods et al.,rep. Prog. Phys. 36, 1135 (1973)

31 The End Thank you for your attention!