Anderson Localization Theoretical description and experimental observation in Bose Einstein-condensates

Transcription

1 Anderson Localization Theoretical description and experimental observation in Bose Einstein-condensates Conrad Albrecht ITP Heidelberg Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

2 Outline 1 Introduction and a first qualitative picture of the Anderson localization 2 quantitative criteria for Anderson localization a) return probability and inverse participation number b) decay of the wavefunction in the spatial limit to infinity i) transfer matrix method ii) phase formalism and speckle potential 3 experimental observation and quantitative understanding of AL in Bose Einstein condensates 4 theoretical background (additional, not included in the handout version) a) second quantization b) tight binding approximation and Anderson model 5 conclusion and references Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

3 Introduction Quantum mechanics (QM) merges two basic concepts of physics: particles and waves. Particle picture: assigns characteristic properties like mass, charge, flavour etc. to a space-time point and describe its dynamics with a certain physical principle (e.g. least action δs = δ dtl = 0) Wave: object that assigns some values to the whole space. The evolution in time follows the same physical principle like for particles. Crucial feature: interference, i.e. superposition of solutions due to linear equations 1 QM states that the possible values of the dynamic variables (position, momentum, energy,...) of a particle follow a probability distribution Ψ 2 that evolves in time according to a (complex) wave equation. 1 in cases of non-linearities (e.g. Gross Pitaevskii equation) my following argumentation is of course not valid! A more careful treatment is necessary. Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

4 Introduction first qualitative statement The Anderson localization phenomenon is an interference effect of the wavefunction Ψ that yields non-classical (unexpected) behaviour for an elementary particle moving through a disordered potential. Anderson s original definition [Anderson1958] Absence of Diffusion in Certain Random Lattices Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

5 quantitative criteria for Anderson localization a) return probability and inverse participation number [van Tiggelen1998] time evolution of spatial eigenvector x : x(t) = e iht x probability to measure eigenvalue x again: p[x(t)] = x x(t) 2 expectation value of p (time average): P(x) = lim T 1 T p(x) = P(x) = m,n = n x 1e iht 1 x 2 = φ m 2 φ n 2 lim T φ n (x) 4 T 0 pdt 2 e iemt φ n(x)φ m (x)δ n,m n,m 1 T dte i(em En)t T 0 inverse participation number: P 1 n notation: 1 = n n n, H n = E n n, φ n (x) = x n = j φ n(x j ) 4, x j+1 = x j + x Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

6 quantitative criteria for Anderson localization a) return probability and inverse participation number Interpretation of P 1 φ(x) equal distributed/maximal extended: P 1 = lim N N i=1 1/N 2 = lim N 1/N = 0 φ(x) maximal localized: P 1 = i=1 δ2 k,i 0 P 1 1 further illustration: φ 4 x φ 2 x P 1 measures amount of space (in units of x where φ 2 is significantly larger than zero) Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

7 quantitative criteria for Anderson localization b) decay of the wavefunction in the spatial limit to infinity [Sanchez-Palencia et al.2008] question 2 : φ(x) 2 e x /λ, λ... localization length 1/λ = lim x ln φ(x) 2 x we now consider two methods to investigate λ: transfer matrix method (tight binding approximation) phase formalism (single particle Schrödinger equation with random potential) 2 the following refers to the simplest case of one-imensional, non-interacting and stationary single-particle physics Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

8 quantitative criteria for Anderson localization b.1) transfer matrix method [van Tiggelen1998] second quantization Hamiltonian under consideration 3 : H = m ɛ ma ma m + J m,n a ma n + h.c. time evolution of the system: i t a m = [a, H] = }{{}... [a,a ]=1 = ɛ n a n + J(a n+1 + a n 1 ) stationary case: a n (t) = e iet Ea n = t a n ( an+1 a n ) = ( E ɛn J ) 1 } 1 {{ 0 } t n ( an a n 1 ) ( an+1 a n ) = N i=1 t i }{{} T N ( a1 a 0 ) 3 (discrete) tight binding approximation using Wannier eigenfunctions: a i creates a particle at site i, convention: = 1 Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

9 quantitative criteria for Anderson localization b.1) transfer matrix method proven facts: Fürstenberg theorem y = T N (E)x T N (E) x, det t i = 1 for almost every a y : T N e N/λ(E), λ 1 > 0 a restricts the result to a number δn of realizations x with zero measure (δn/n = 0 with respect to the total number N) Ossedelec theorem Fürstenberg theorem (! x T N e N/λ) Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

10 quantitative criteria for Anderson localization (my first simulation results for the Anderson model) trapped situation: E ɛ n [0, 1] = J = 10 (κ = /J 1) note: H n = E n n with n = , i.e latice sites correspond to the system s length κ is the only free parameter of the system (scaling invariance of H) Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

11 quantitative criteria for Anderson localization (my first simulation results for the Anderson model) barely trapped: E ɛ n [0, 1] = J = 10 κ 1 Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

12 quantitative criteria for Anderson localization (my first simulation results for the Anderson model) classically free: E ɛ n [0, 1] = J = 10 κ 1 Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

13 quantitative criteria for Anderson localization (my first simulation results for the Anderson model) border effects: E ɛ n [0, 1] = J = 10 κ 1 Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

14 quantitative criteria for Anderson localization b.2) phase formalism [Sanchez-Palencia et al.2008] [ ] starting point: 2 2m 2 x + V (x) φ(x) = Eφ(x) with a disordered potential V (x) Solution of second-order ODE via standard reduction to first-order system of ODEs and transforming φ and φ to physical meaningful quantities 4 (E = 2 k 2 /2m): φ(x) = r(x) sin θ(x), φ (x) = kr(x) cos θ(x) resulting differential equations: θ (x) = k k E V (x) sin2 θ(x) (1) r r 2k (x) = V (x) sin[2θ(x)] (2) E 4 envelope r and phase θ of the wavefunction Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

15 quantitative criteria for Anderson localization b.2) phase formalism for a weak disordered potential V 1 we can treat the phase θ according to its differential equation (1) as slowly variing with respect to x, i.e. θ(x) = θ 0 + kx + δθ(x) in the language of the Born approximation with δθ(x) = x 0 dz k E V (z) sin2 (θ 0 + kz) the integration of (2) yields a factorization 5 to ln [r(x)/r(0)] = x /λ(k) with λ 1 = m dxc(x) cos(2kx) 4 E (3) in the limit x that demonstrates the exponential decay of the wavefunction 5 this result needs some steps of calculation, C(x) = V (x x )V (x) V 2 Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

16 quantitative criteria for Anderson localization b.2) phase formalism crucial point from equation (3) it becomes clear that the correlation function of the disordered potential V determines the localization lenght of the wavefunction, i.e. for weak disorder there is Anderson-localization in the case of a speckle potential we arrive at: λ k 2 max V 2 0 σ c(1 k max σ c ) 6 6 the correlation length σ c is defined via the decay of the correlation function C(x) Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

17 experimental observation of AL in BECs speckle potential 7 V: experimentalist s language: V > 0... blue detuned speckle V < 0... red detuned speckle 7 image source [Lye et al.2005] Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

18 experimental observation of AL in BECs (speckle potential) properties of speckle potentials 8 V: P(ν) = e ν 1, ν c(x) = ( sin(x) x ) 2 experimental realization: V... light intensity with sign from the polarizability α(ω) V em (x) = α(ω) 2 E 2 (x, t) t, α(ω) (ω r ω) 1 ω r is the BEC atoms resonancy frequency red/blue tuning C... determined by the transmission function of the diffuse plate 8 V (x) = V 0ν(x/σ c), C(x) = V 2 0 c(x/σ c), C... correlation function Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

19 experimental observation of AL in BECs [Sanchez-Palencia et al.2008], [Billy et al.2008] experimental setup 9 : 9 images from [Billy et al.2008], modified Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

20 experimental observation of AL in BECs experimental conditions vs. theoretical model: kinetic energy of BEC atoms V 0 (disorder magnitude) no classical trapping low atom density of atoms in the condensate s wings almost no mutual interaction cutoff in k wavevectors at k max observing wavefunction decay transition at β = k max σ c = 1 (for β > 1 algebraic and for β < 1 exponential wings) Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

21 experimental observation of AL in BECs (measurement results [Billy et al.2008]) exponential wings algebraic ( z r, r 2) wings (β < 1) (β > 1) remember: localization length λ V 2 0 k 2 max σc (1 kmax σc ) transition at β = 1 Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

22 experimental observation of AL in BECs qualitative understanding Due to the low density of the condensate in the wings the atoms can be treated as almost independent. The wavefunction of each single atom localizes in the random potential V and the superposition of all localizations lenghts yields the resulting decay of the condensate s wings. Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

23 Conclusion 1 The wavecharacter of quantum mechanics yields non-classical behaviour of matter, e.g. the phenomenon of Anderson localizations 2 The quantitative formulation of the AL apears along with several measures, but the exponential-decaying-wavefunction-criterion is a common and useful property which can be explored via different analytical techniques. 3 The experiments with BECs give a direct acess to observe Ψ 2 which was impossible with solid probes before. 4 A careful treatment of the background theory is essential for the result s interpretation. Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

24 References Anderson, P. (1958). Absence of diffusion in certain random lattices. Phys.Rev., 109:5. Billy, J., Josse, V., Zuo, Z., Bernard, A., Hambrecht, B., Lugan, P., Clément, D., Sanchez-Palencia, L., Bouyer, P., and Aspect, A. (2008). Direct observation of anderson localization of matter waves in a controlled disorder. nature letters, 453: Lye, J. E., Fallani, L., Modugno, M., Wiersma, Fort, C., and Inguscio, M. (2005). Bose-einstein condensates in a random potential. Phys.Rev. Lett., 95: Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28

25 References Mahan, G. D. (2000). Many Particle Physics, chapter 1.2. Kluwer Academic/Plenum Publishers, 3. edition. Sanchez-Palencia, L., Clément, D., Lugan, P., Bouyer, P., and Aspect, A. (2008). Disorder-induced trapping versus anderson localization in bose einstein condensates expanding in disordered potentials. njp, 10: van Tiggelen, B. (1998). Localization of waves. online at lecture notes, chapter 2.1. Conrad Albrecht (ITP Heidelberg) Anderson Localization / 28