Anderson Localization - Theoretical description and ...albrecht/AL09ca...Anderson Localization Theoretical description and experimental observation in Bose–Einstein-condensates Conrad

Anderson LocalizationTheoretical description and experimental observation in

Bose–Einstein-condensates

Conrad Albrecht

ITP Heidelberg

22.07.2009

Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 1 / 28

Outline

1 Introduction and a first qualitative picture of the Anderson localization

2 quantitative criteria for Anderson localizationa) return probability and inverse participation numberb) decay of the wavefunction in the spatial limit to infinity

i) transfer matrix methodii) phase formalism and speckle potential

3 experimental observation and quantitative understanding of AL inBose–Einstein–condensates

4 theoretical background (additional, not included in the handout version)

a) second quantizationb) tight binding approximation and Anderson–model

5 conclusion and references


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 acertain physical principle (e.g. least action δS = δ

∫dtL = 0)

• Wave: object that assigns some values to the whole space. Theevolution in time follows the same physical principle like for particles.→Crucial feature: interference, i.e. superposition of solutions due tolinear equations1

QM states that the possible values of the dynamic variables (position, mo-mentum, energy, ...) of a particle follow a probability distribution |Ψ|2 thatevolves in time according to a (complex) wave equation.

1in cases of non-linearities (e.g. Gross–Pitaevskii equation) my followingargumentation is of course not valid! A more careful treatment is necessary.Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 3 / 28

Introduction

first qualitative statement

The Anderson localization phenomenon is an interference effect of thewavefunction Ψ that yields non-classical (unexpected) behaviour for an el-ementary particle moving through a disordered potential.

Anderson’s original definition [Anderson1958]

”Absence of Diffusion in Certain Random Lattices”


quantitative criteria for Anderson localizationa) 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) = limT→∞1T

∫ T0 pdt

↪→ p(x) =∣∣∣〈x | 1e iHt1 |x〉

∣∣∣2 =

∣∣∣∣∣∑n,m

e iEmtφ∗n(x)φm(x)δn,m

∣∣∣∣∣2

P(x) =∑m,n

|φm|2 |φn|2 limT→∞

1

T

∫ T

0dte i(Em−En)t

=∑n

|φn(x)|4

• inverse participation number : P−1n =

∑j |φn(xj)|4 , xj+1 = xj +4x

notation: 1 =∑

n |n〉〈n|, H |n〉 = En |n〉, φn(x) = 〈x |n〉Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 6 / 28

quantitative criteria for Anderson localizationa) return probability and inverse participation number

Interpretation of P−1

• φ(x) equal distributed/maximal extended:

P−1 = limN→∞

N∑i=1

1/N2 = limN→∞

1/N = 0

• φ(x) maximal localized: P−1 =∑∞

i=1 δ2k,i ⇒ 0 ≤ P−1 ≤ 1

• further illustration: |φ|44x � |φ|24x↪→ P−1 ’measures’ amount of space (in units of 4x where |φ|2 issignificantly larger than zero)


quantitative criteria for Anderson localizationb) decay of the wavefunction in the spatial limit to infinity [Sanchez-Palencia et al.2008]

• question2: |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 Schrodinger equation with random potential)

2the following refers to the simplest case of one-imensional, non-interacting andstationary single-particle physicsConrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 8 / 28

quantitative criteria for Anderson localizationb.1) transfer matrix method [van Tiggelen1998]

• second quantization Hamiltonian under consideration3:H =

∑m εma†mam + J

∑〈m,n〉 a

†man + h.c .

• time evolution of the system:i∂tam = [a,H] = ...︸︷︷︸

[a,a†]=1

= εnan + J(an+1 + an−1)

• stationary case: an(t) = e−iEt ⇒ Ean = ∂tan

↪→(

an+1

an

)=

(E−εn

J −11 0

)︸︷︷︸

tn

(an

an−1

)→

(an+1

an

)=

N∏i=1

ti︸︷︷︸TN

(a1

a0

)

3(discrete) tight binding approximation using Wannier–eigenfunctions: a†i creates a

particle at site i , convention: ~ = 1Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 9 / 28

quantitative criteria for Anderson localizationb.1) transfer matrix method

proven facts:

Furstenberg theorem

|y | = |TN(E )x | ≤ |TN(E )| |x | , det ti = 1⇒ for almost everya y : |TN | ∝ eN/λ(E), λ−1 > 0

arestricts the result to a number δn of realizations x with zero measure (δn/N = 0with respect to the total number N)

Ossedelec theorem

Furstenberg theorem ⇒(∃! x ⇒ |TN | ∝ e−N/λ

)


quantitative criteria for Anderson localization(my first simulation results for the Anderson model)

trapped situation:E3 ≈ 0.00

εn ∈ [0, 1] = ∆J = 10(κ = |∆| /J � 1)

note:H |n〉 = En |n〉 with n = 1 . . . 2000, i.e. 2000 latice sites correspond to thesystem’s length

κ is the only ’free’ parameter of the system (scaling invariance of H)



barely trapped:E189 ≈ 0.86

εn ∈ [0, 1] = ∆J = 10κ � 1



classically free:E275 ≈ 1.83

εn ∈ [0, 1] = ∆J = 10κ � 1



border effects:E450 ≈ 4.47

εn ∈ [0, 1] = ∆J = 10κ � 1


quantitative criteria for Anderson localizationb.2) phase formalism [Sanchez-Palencia et al.2008]

• starting point:[−~2

2m ∂2x + V (x)

]φ(x) = Eφ(x)

with a disordered potential V (x)

• Solution of second-order ODE via standard reduction to first-ordersystem of ODEs and transforming φ and φ′ to physical meaningfulquantities4 (E = ~2k2/2m):

φ(x) = r(x) sin θ(x), φ′(x) = kr(x) cos θ(x)

↪→ resulting differential equations:

θ′(x) = k − k

EV (x) sin2 θ(x) (1)

r ′

r(x) =

2k

EV (x) sin[2θ(x)] (2)

4envelope r and phase θ of the wavefunctionConrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 15 / 28

quantitative criteria for Anderson localizationb.2) phase formalism

• for a weak disordered potential V � 1 we can treat the phase θaccording to its differential equation (1) as slowly variing with respectto x , i.e. θ(x) = θ0 + kx + δθ(x) in the language of theBorn–approximation with δθ(x) = −

∫ x0 dz k

E V (z) sin2(θ0 + kz)

• the integration of (2) yields a factorization5 to

ln [r(x)/r(0)] = |x | /λ(k) with λ−1 =m

4~E

∫ ∞

−∞dxC (x) cos(2kx)

(3)in the limit |x | → ∞ that demonstrates the exponential decay of thewavefunction

5this result needs some steps of calculation, C(x) = 〈V (x − x ′)V (x)〉 − 〈V 〉2Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 16 / 28

quantitative criteria for Anderson localizationb.2) phase formalism

crucial point

from equation (3) it becomes clear that the correlation function of thedisordered potential V determines the localization lenght of thewavefunction, i.e. for weak disorder there is Anderson-localization

→ in the case of a speckle potential we arrive at:

λ ∝ k2max

V 20 σc(1− kmaxσc)

6

6the correlation length σc is defined via the decay of the correlation function C(x)Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 17 / 28

experimental observation of AL in BECs

speckle potential7 V:

experimentalist’s language:V > 0 ... blue detuned speckleV < 0 ... red detuned speckle

7image source [Lye et al.2005]Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 18 / 28

experimental observation of AL in BECs(speckle potential)

properties of speckle potentials8 V:

• P(ν) = e−ν−1, ν + 1 ≥ 0

• c(x) =(

sin(x)x

)2

experimental realization:

• V ... light intensity with sign from the polarizability α(ω)

Vem(x) = −α(ω)2

⟨E2(x, t)

⟩t, α(ω) ∝ (ωr − ω)−1

ωr is the BEC atoms’ resonancy frequency → red/blue tuning

• C ... determined by the transmission function of the diffuse plate

8V (x) = V0ν(x/σc), C(x) = V 20 c(x/σc), C ... correlation function


experimental observation of AL in BECs [Sanchez-Palenciaet al.2008], [Billy et al.2008]

experimental setup9:

9images from [Billy et al.2008], modifiedConrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 20 / 28


experimental conditions vs. theoretical model:

• kinetic energy of BEC atoms � V0 (disorder magnitude)↪→ no classical trapping

• low atom density of atoms in the condensate’s wings↪→ almost no mutual interaction

• cutoff in k− wavevectors at kmax

↪→ observing wavefunction decay transition at β = kmaxσc = 1(for β > 1 algebraic and for β < 1 exponential wings)


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 ∝ k2max

V 20 σc (1−kmaxσc )

⇒ transition at β = 1



qualitative understanding

Due to the low density of the condensate in the wings the atoms can betreated as almost independent. The wavefunction of each single atomlocalizes in the random potential V and the superposition of alllocalizations lenghts yields the resulting decay of the condensate’s wings.


Conclusion

1 The wavecharacter of quantum mechanics yields non-classicalbehaviour of matter, e.g. the phenomenon of Andersonlocalizations

2 The quantitative formulation of the AL apears along with severalmeasures, but the exponential-decaying-wavefunction-criterion is acommon and useful property which can be explored via differentanalytical techniques.

3 The experiments with BECs give a direct acess to observe |Ψ|2 whichwas impossible with solid probes before.

4 A careful treatment of the background theory is essential for theresult’s interpretation.


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.,Clement, D., Sanchez-Palencia, L., Bouyer, P., and Aspect, A. (2008).Direct observation of anderson localization of matter waves in acontrolled disorder.nature letters, 453:10.1038.

Lye, J. E., Fallani, L., Modugno, M., Wiersma, Fort, C., and Inguscio,M. (2005).Bose-einstein condensates in a random potential.Phys.Rev. Lett., 95:070401.


References

Mahan, G. D. (2000).Many–Particle Physics, chapter 1.2.Kluwer Academic/Plenum Publishers, 3. edition.

Sanchez-Palencia, L., Clement, D., Lugan, P., Bouyer, P., and Aspect,A. (2008).Disorder-induced trapping versus anderson localization inbose–einstein condensates expanding in disordered potentials.njp, 10:045019.

van Tiggelen, B. (1998).Localization of waves.online at www.andersonlocalization.com.lecture notes, chapter 2.1.


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Anderson Localization - Theoretical description and ...albrecht/AL09ca...Anderson Localization Theoretical description and experimental observation in Bose–Einstein-condensates Conrad