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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
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 2 / 28
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”
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 4 / 28
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)
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 7 / 28
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
)
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 10 / 28
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)
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 11 / 28
quantitative criteria for Anderson localization(my first simulation results for the Anderson model)
barely trapped:E189 ≈ 0.86
εn ∈ [0, 1] = ∆J = 10κ � 1
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 12 / 28
quantitative criteria for Anderson localization(my first simulation results for the Anderson model)
classically free:E275 ≈ 1.83
εn ∈ [0, 1] = ∆J = 10κ � 1
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 13 / 28
quantitative criteria for Anderson localization(my first simulation results for the Anderson model)
border effects:E450 ≈ 4.47
εn ∈ [0, 1] = ∆J = 10κ � 1
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 14 / 28
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
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 19 / 28
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 observation of AL in BECs
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)
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 21 / 28
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
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 22 / 28
experimental observation of AL in BECs
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.
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 23 / 28
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.
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 26 / 28
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.
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 27 / 28
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.
Conrad Albrecht (ITP Heidelberg) Anderson Localization 22.07.2009 28 / 28