Bose-Einstein Condensates with Strong Disorder: Replica Method

Transcription

1 Bose-Einstein Condensates with Strong Disorder: Replica Method January 6, 2014 New Year Seminar

2 Outline Introduction 1 Introduction 2 Model Replica Trick 3 Self-Consistency equations Cardan Method 4 Model Method 5

3 Introduction Superfluid Helium in Porous Media: Reppy et al., PRL 51, 666 (1983) Laser Speckles: Inguscio et al., PRL 95, (2005) Aspect et al., PRL 95, (2005) Wire Traps: Schmiedmayer et al., Phys. Rev. A 76, (2007) Localized Atomic Species: Gavish and Castin, PRL 95, (2005) Schneble et al., PRL 107, (2011) Incommensurate Lattices: Lewenstein et al., PRL 91, (2003) Ertmer et al., PRL 95, (2005)

4 Model System Introduction Model Replica Trick Action of a Bose gas β A = 0 { [ dτ dr ψ (r, τ) τ 2 2M + V (r) + U(r) µ ] ψ (r, τ) + g2 ψ (r, τ) 2 ψ (r, τ) 2 } Properties trap potential V (r) disorder potential U(r) chemical potential µ repulsive interaction g = 4π 2 a M periodic Bose fields ψ (r, τ + β) = ψ (r, τ)

5 Random Potential Introduction Model Replica Trick Disorder Ensemble Average = DU P [U], DUP [U] = 1 Assumption U(r) = 0 U(r 1 )U(r 2 ) = R (2) (r 1 r 2 ) Characteristic Functional { exp i } { i n drj(r)u (r) = exp n! n=2 dr 1 dr nr (n) (r 1 r n)j(r 1 ) J(r n) }

6 Grand-Canonical Potential Model Replica Trick Aim Z : partition function Problem Z = F = 1 β ln Z Dψ Dψe A[ψ,ψ]/ ln Z = ln Z Solution: Replica Trick F = 1 β G. Parisi, J. Phys. France 51, 1595 (1990) M. Mezard and G. Parisi, J. Phys. I France 1, 809 (1991) lim Z N 1 N 0 N

7 Replica Trick Introduction Model Replica Trick Disorder Averaged Partition Function { N } Z N = Dψα Dψ α e { N } N α=1 A[ψα,ψα]/ = Dψα Dψα e A(N ) / α =1 Replicated Action β N [ A (N ) = dτ dr {ψ α (r, τ) ] 0 τ 2 2M + V (r) µ ψ α (r, τ) α=1 + g ( ) 2 ψα (r, τ) 4} 1 1 n 1 β β + dτ 1 dτ n dr 1 dr n n! n=2 0 0 N N R (n) (r 1 r n) ψ α1 (r 1, τ 1 ) 2 ψ αn (r n, τ n) 2 α 1 =1 α n=1 Remarks: - In the replica limit N 0 Higher-order disorder cumulants are negligible: only R (2) (r) contributes - Disorder amounts to attractive interaction for n = 2 α=1

8 Model Replica Trick Assumptions Bogoliubov background method ψ α (r, τ) = Ψ α (r, τ) + δψ α (r, τ) Hartree-Fock theory Semiclassical approximations due to V (r) Replica symmetry Ψ α(r, τ) = n 0 (r) ( δψα (r, τ) δψ α r, τ ) = Q (r r, r+r 2, τ τ ) ( δ αα + q r+r 2, τ τ ) n (r) = Ψ α(r, τ)ψ α (r, τ) + δψα (r, τ) δψα (r, τ) Homogeneous case worked out in Ref. R. Graham and A. Pelster, Int. J. Bif. Chaos 19, 2745 (2009)

9 Self-Consistency equations Cardan Method Self-Consistency equations for finite temperature T ( ) M 3/2 n(r) = n 0 (r) + q(r) + 2πβħ 2 ς 3/2 (e ) β [µ d2 2gn(r) V (r)] q(r) = [ ( ) 3/2 ( ) ] d n(r) M 2πβħ ς 2 3/2 e β [µ d2 2gn(r) V (r)] d + µ + d 2 + 2gn(r) + V (r) { gn 0 (r) + [ ] } 2 µ + d 2 + 2gn(r) + V (r) + d ħ2 n0 2M (r) = 0 N = n(r)dr where R (2) (r 1 r 2 ) = Dδ(r 1 r 2 ), d = ( ) 3/2 πd M 2πħ 2

10 Self-Consistency equations Cardan Method Assumptions T = 0 V (r) = 1 2 MΩ2 r 2 Thomas-Fermi approximation length scale l = MΩ Energy scale µ 0 = 152/5 2 ( an l ) 2/5ħΩ Dimensionless quantities MΩ r = 2µ 0 r, ñ( r) = gn(r) µ 0, µ = µ d2 µ 0, d = d µ0

11 Self-Consistency equations Cardan Method ñ( r) = ñ 0 ( r) + q ( r) q ( r) = dñ( r) µ + 2ñ( r) + r 2 + d { ñ 0 ( r) + [ µ + 2ñ( r) + r 2 + d] 2 } ñ0 ( r) = 0

12 Self-Consistency equations Cardan Method Bose-glass Phase ñ 0 ( r) = 0 and q ( r) = ñ( r) 0 µ + 2ñ( r) + r 2 = 0 Superfluid Phase ñ 0 ( r) 0 and q ( r) 0 [ 2 ñ 0 ( r) = µ + 2ñ( r) + r 2 + d] q ( r) = d [ 2 µ + 2ñ( r) + r 2 + d] µ + 2ñ( r) + r 2 ñ ( r) = 1 [ 3 µ + 2ñ( r) + r 2 + d] µ + 2ñ( r) + r 2

13 Self-Consistency equations Cardan Method Solving a cubic equation Variable transformation Az 3 + Bz 2 + Cz + D = 0 gives normal form y = z B 3A y 3 + Py + Q = 0 where P = B2 + C 3A 2 A Q = B 27A ( 2B2 9C A 2 A ) + D A

14 Self-Consistency equations Cardan Method Solution ansatz y = u + v { u 3 + v 3 = Q u 3 v 3 = P3 27 u 3 and v 3 solve quadratic equation X 2 + QX P3 27 = 0 Discriminant δ = Q P3

15 Self-Consistency equations Cardan Method δ > 0 1 real solution + 2 complex solutions 3 Q+ δ + 3 Q δ B 2 2 3A e 2iπ 3 3 Q+ δ + e 2iπ 3 3 Q δ B 2 2 3A e 2iπ 3 3 Q+ δ + e 2iπ 3 3 Q δ B 2 2 3A δ < 0 3 real solutions 3 Q+i δ + 3 Q i δ 2 e 2iπ 3 Q+i δ 3 + e 2iπ e 2iπ 3 Q+i δ 3 + e 2iπ δ = 0 2 real solutions 3Q P B 3A 3Q 2P B 3A 2 B 3A (doubly degenerate) Q i δ 2 B 3A Q i δ 2 B 3A

16 Self-Consistency equations Cardan Method Application of Cardan method in superfluid phase for ñ( r) ( ) ( ) δ = 4 µ µ2 d µ d d 6 4 µ µ d 2 8 d 4 r µ d 2 r r 6 n r Realsolutions: 1Realsolution 3Realsolutions: 1 Real negative 1 n 0 r n r n r jumpsupwards 1 n r jumpsupwards solution andn r 1 2 n r toosmall 2 n r increases 3 n r negative 3 accepted r 1 r 2 r 3 r 0 0 0

17 Self-Consistency equations Cardan Method Application of Cardan method in superfluid phase for ñ( r) ( ) ( ) δ = 4 µ µ2 d µ d d 6 4 µ µ d 2 8 d 4 r µ d 2 r r 6 n r SF Bose glass r 1 r 2 r 3 r 0 0 0

18 Densities Introduction Self-Consistency equations Cardan Method 87 Rb, N = 10 6, d = 0.117, µ = 1.177, Ω = 200πHz and a = 5.29nm Cleancase:d 0 n r n 0 r q r r

19 Thomas-Fermi Radii Introduction Self-Consistency equations Cardan Method 87 Rb, N = 10 6, Ω = 200πHz and a = 5.29nm 1.2 TF Radii Condensate radius Cloudradius d

20 Thomas-Fermi Radii Introduction Self-Consistency equations Cardan Method 87 Rb, N = 10 6, Ω = 200πHz and a = 5.29nm 1.2 TF Radii SF Condensate radius Cloud radius Bose glass 0.2 Quantum phase trasition d

21 Model Introduction Model Method Gross-Pitaevskii equation for the ground state [ 2 2m µ + U(x) + V (x) + g ] 2 ψ (x)ψ(x) ψ(x) = 0 Assumptions One dimension { } Gaussian correlation function R(x) = ε2 2πλ exp x 2 2λ 2 Condensate depletion Particle density n(x) = ψ(x) 2 Condensate density n 0(x) = ψ(x) 2 Depletion q(x) = n(x) n 0(x)

22 Method Introduction Model Method Generating random potential U(x) = 1 N 1 [A n cos(k n x) + B n sin(k n x)] N n=0 A nb n = 0, A na m = B nb m = R(0)δnm and p(k n) = λ { exp λ2 k 2 } n 2π 2 J. Majda and P. Kramer, Phys. Rep. 314, 237 (1999) Sample potential: N = 100, ε = 1 and λ = U(x) x

23 Model Method Correlation U(x)U(0) and R(x): ε = 1 and λ = 1 N = 1000 N = (a) (b) x x C program for solving time-(in)dependent Gross-Pitaevskii equation in one space dimension A. Balaž et al., Comput. Phys. Commun. 183, 2021 (2012)

24 Introduction Model Method Particle density: N = and λ = ε=0 ε=0.01 ε=0.1 ε=0.5 ε=1 ε=5 ε= n(x) x

25 Introduction Model Method Condensate density: N = and λ = 1 Increasing disorder strength the global condensate density decreases ε=0 ε=0.01 ε=0.1 ε=0.5 ε=1 ε=5 ε=10 (a) (b) N0 /N n0 (x) x ε2 ( R N0 = n0 (x) dx R where N = n (x) dx

26 Introduction Model Method Condensate depletion: N = and λ = 1 Increasing disorder strength depletion increases ε=0.01 ε=0.1 ε=0.5 ε=1 ε=5 ε=10 (a) (b) q(x) Q/N e e-06 1e x where Q = R ε2 q (x) dx

27 Model Method Bose-glass phase: N = 10000, λ = 1 and ε = 10 Existence of a Bose-glass phase in the intermediate region

28 Comparison with Huang-Meng theory Perturbation method Beyond Thomas-Fermi approximation Anisotropic trap potential Extend the numerical study to 3 dimensions General interaction potential Finite temperature Time dependence of densities and Thomas-Fermi radii Replica Symmetry Breaking?

29 Thank You For Your Attention