Bogoliubov theory of disordered Bose-Einstein condensates

Transcription

1 Bogoliubov theory of disordered Bose-Einstein condensates Christopher Gaul Universidad Complutense de Madrid BENASQUE 2012 DISORDER

2 Bogoliubov theory of disordered Bose-Einstein condensates Abstract The interplay of interaction, disorder, and Bose statistics is a long standing problem of condensed matter physics, known as the dirty boson problem. Here, we present a Bogoliubov theory for disordered Bose-Einstein condensates, i.e., the bosonic field operator is split into the (mean field) condensate and (quantum) fluctuations. The mean-field part consists in solving the Gross-Pitaevskii equation describing the deformed condensate wave function. The condensate, in turn, determines the Hamiltonian for the quantum fluctuations. Diagonalizing this Bogoliubov Hamiltonian is a difficult task. As it is not desirable anyway to solve the problem for a particular realization of disorder, we resort to disorder perturbation theory in terms of Green functions and compute quantities like the disorder-averaged sound velocity or the mean free path of Bogoliubov excitations. Beyond that, the Bogoliubov theory is used to count the number of particles that are excited out of the condensate, even at zero temperature. This depletion of the condensate is shown to remain small in presence of disorder, which validates a posteriori the Bogoliubov ansatz. References: C. Gaul & C.A. Müller, Phys. Rev. A, 83, (2011) C.A. Müller & C. Gaul, New J. Phys (2012)

3 Bogoliubov theory of disordered Bose-Einstein condensates Outline Bose statistics + Interaction + Disorder Dirty Boson Problem Experiments with ultracold quantum gases How is Bose-Einstein condensation affected by disorder? How to define the condensate in presence of inhomogeneity? Fraction of non-condensed particles How are the elementary excitations affected by disorder?

4 Classical: Bose statistics a b a b b a a b Bosons: indistinguishable, symmetric wf.: â 1â 2 0 = â 1â 2 0 Indistinguishable bosons tend to cluster n B (ε) = 1 exp( ε µb k B T ) 1 n µ B 1 2 µ F ε k BT Fermi: n F (ε) = 1 exp( ε µf k B T Boltzmann: n(ε) e ε k B T ) +1 n B diverges for ε µ B Bose-Einstein condensation (BEC) If: thermal de-broglie wavelength average particle distance

5 BEC in experiments How to reach critical phase space density Magneto-optical trapping Doppler cooling Evaporative cooling [wikipedia.org] How to see it Time-of-flight imaging momentum-space density: macroscopic occupation of single-particle orbital Φ(r) [

6 Optical potentials Optical lattices: [Greiner et al., nature (2008)] Speckle disorder Well-known statistics: V k V k = δ kk R(k) [Clément et al., New J. Phys., 8, 165 (2006)]

7 Penrose-Onsager criterion Starting point: bosonic many-body Hamiltonian E[ ˆΨ, ˆΨ ] = [ d d r ˆΨ 2 (r) 2m 2 + V(r) + g ] 2 ˆΨ (r) ˆΨ(r) µ ˆΨ(r) One-body denity matrix (OBDM): ˆΨ (r) ˆΨ(r ) BEC: many particles occupy condensate orbital Penrose & Onsager (1956): d d r ˆΨ (r) ˆΨ(r ) Φ(r ) = N }{{} c Φ(r) OBDM Condensate Φ(r) Number of condensed particles N c 1

8 Mean field and Bogoliubov theory Condensate and quantum fluctuations ˆΨ(r) = Φ(r) + δ ˆψ(r, t) Φ(r) 2 V(r) δˆn(r) Meanfield: Minimize E[Φ] Gross-Pitaevskii equation [ ] 2 2m 2 + g Φ(r) 2 + V(r) µ Φ(r) = 0 } interaction: g Φ(r) 2 = gn c kinetic energy: 2 k 2 /2m = 2 /2mξ 2 ξ 2 = 2 /(2mgn c ) ξ ξ σ σ

9 Effective Hamiltonian for quantum fluctuations E[Φ + δ ˆψ] E[Φ] + 1 d d r d d r ( δ 2 ˆψ (r ), δ ˆψ(r ) ) ( ) H(r δ ˆψ(r), r) δ ˆψ (r) }{{} Ĥ {[ ]( ) ( )} H = δ(r r Φ(r) 2 1 ) 2m 2 + V(r) µ + g 2 Φ(r) Φ (r) 2 Φ(r) 2 In terms of density and phase: Φ(r) + ˆψ(r) = e iδ ˆϕ(r) n c + δˆn(r) Fourier- & Bogoliubov trafo: bogolons ˆγ k = δˆn k /(2a k nc ) + ia k nc δ ˆϕ k a k = ε 0 k /ε k Ĥ = k ε kˆγ kˆγ k + ( ) ˆΓ k V ˆγk kk ˆΓk, ˆΓk = ˆγ k,k k

10 Homogeneous Bogoliubov problem 15 Ĥ (0) = ε kˆγ kˆγ k k Bogoliubov dispersion relation ε k = ε 0 k (2gn c + ε 0 k ), ε0 k = 2 k 2 2m ε 10 k µ [ξ 2 = 2 2mgnc ] Condensate depletion δn (0) = 1 L d k δ ˆψ k δ ˆψ k = 1 L k d v2 (3D) 1 k = 6 2π 2 ξ 3 ξ d kξ δ ˆψ k = u kˆγ k + v kˆγ k Relative depletion δn(0) n c (3D) = 8 3 π na 3 s dilute-gas parameter [Lee, Huang & Yang (1957)]

11 Bogolons in a disordered medium Hamiltonian Bogoliubov-Nambu spinor Ĥ = ε kˆγ kˆγ k + ( ) ˆΓ k V ˆγk kk ˆΓk, ˆΓk = ˆγ k k,k k ( ) W Y Vertex V = Y W W kpˆγ kˆγ p W (1) kp Anomalous scattering Y k, k ˆγ k ˆγ k + Y k,k ˆγ k ˆγ k + = w(1) kp V k p, W (2) kp =... Bogoliubov-Nambu vertex V, V =

12 Disorder-averaged effective medium How do Booliubov quasi-particles travel on average through the disordered medium? Matrix-valued (retarded) Green function G kk (t) = Θ(t) i [ˆΓk (t), ˆΓ k (0) ], Contains dispersion relation: [G 0 (k, ω)] 11 = [ ω ε k + i0 + ] 1 Expansion in terms scattering vertex V = V and G 0 = : G = G 0 + G 0 VG = + V + V V +...

13 Computing the disorder-averaged Green function = + V + V V +... = Disorder average: = 0, q = R(q) = (reducible) Dyson equation: self-energy Σ = + Σ Σ = (irreducible) Renormalized dispersion relation ω = ε k + Σ (2) 11 (k, ω) ImΣ finite mean free path

14 Mean free path E.g. Gaussian disorder V q V q = L d δ qq ( 2πσ) d (vgn c ) 2 e q2 σ 2 2 }{{} R(q) Finite mean free path (kl s ) 1 ImΣ 1 kl s v kσ = kξ d = 1 d = 2 d = 3 Related to localization of Bogoliubov quasiparticles [Lugan et al. PRA (2011)]

15 Disorder-renormalized speed of sound ReΣ renormalizes sound velocity c cv d = 3 d = d = σ/ξ ζ = σ/ξ v 2 δ = R(0)/(gn cξ d ) c/c σ ξ σ ξ d = 1 v 2 / v2 δ d = 2 v 2 /4 0 d = 3 v 2 / π v2 δ * * [Giorgini et al., PRB 1994]

16 Momentum distribution of fluctuations To compute: δn k = δ ˆψ k δ ˆψ k. We have: Hamiltonian for ˆγ k = δˆn k /(2a k nc ) + ia k nc δ ˆϕ k δ ˆψ(r) = δˆn(r)/[2φ(r)] + iφ(r)δ ˆϕ(r) Transformation δ ˆψ k = ( ) p u kpˆγ p v kpˆγ p, with u kp = 1 [ ] 2 N a 1 c p Φ k p + a p ˇΦk p, ˇΦk = [n c /Φ(r)] k v kp = 1 [ ] 2 N a 1 c p Φ k p a p ˇΦk p δn k = { δ pp v kp 2 + ( u kp u kp + v kp v ) } kp ˆγ pˆγ p (u kp v kp ˆγ pˆγ p +c.c.) p,p T = 0: only due to inhomogeneity V(r) homogeneous quantum depletion

17 Momentum distribution of fluctuations Pick second-order terms of δn k = {δ pp v kp 2 + ( u kp u kp + v kp v ) } kp ˆγ pˆγ p (u kp v kp ˆγ pˆγ p +c.c.) p,p ˆγ pˆγ p = ˆγ pˆγ p (0) + ˆγ pˆγ p (1) + ˆγ pˆγ p (2) +... u kp = u (0) kp + u(1) kp + u(2) kp +... δn (2) k = p M (2) kp V k p 2 a monstrous envelope function

18 Momentum distribution in a 2D lattice V(r) = j Vj cos(kj r) Condensate deformation Φ k 2 2 V (r) 1 0 gnc 1 2 yky π 0 0 π 3π 2π xkx nc(r) nc π Quantum fluctuations δn k Quantum depletion δn (0) Potential depletion δn (2) δn (0) k δn (2) k δn k 1 n ck N c 1 1 δn k k y K y k y K y k x /K x k x /K x

19 Condensate depletion due to Gaussian disorder δn (2) = L d k p M (2) kp V k p 2 δn (2) δn (0) σ ξ σ ξ d = v v 2 δ d = v 2 δ d = v v 2 δ 0.3 δn (2) δn (0) v V q 2 = L d ( 2πσ) d (vgn c ) 2 e q 2 σ 2 2 }{{} R(q) v 2 δ = R(0)/(gn cξ d ) σ/ξ σ ξ: depletion correction scales with v 2 δ R(0) ζ σ ξ: depletion coincides with local density approximation δn (2) TF δn d(d 2)v2 = (0) 8 d = 3 d = 1 d = 2

20 Take-home messages Hamiltonian for quantum excitations on top of deformed condensate Diagrammatic disorder perturbation theory Mean free path Renormalized speed of sound Calculation of the potential-induced condensate depletion Depletion remains small validates Bogoliubov ansatz References C. Gaul and C. A. Müller, Phys. Rev. A, 83, (2011) C. A. Müller and C. Gaul, New J. Phys (2012) Thanks! Cord A. Müller Funding: Moncloa Campus of International Excellence (UCM-UPM)