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Many-body physics with ultracold particles in disorder
Gora ShlyapnikovLPTMS, Orsay, France
University of Amsterdam, The Netherlands
Introduction.
Experiments with cold atoms in disorder
Many-body localization-delocalization transition
MBLDT for 1D disordered bosons
MBLDT in the AAH model
Conclusions
Collaborations B.L. Altshuler/I.L. Aleiner (Columbia Univ.), V. Michal(LPTMS, Orsay)Santa Barbara, USA, November 19, 2015
. – p.1/35
Many-body system in disorder
Many-particle system in disorder ⇒ Transport and localization properties
Anderson localization (P.W. Anderson, 1958)
Destructive interference in the scattering of a particle from random defects
Old question. How does the interparticle interaction influence localization?
Long standing problem. Crucial for charge transport in electronic systems
Appears in a new light for disordered ultracold bosons
. – p.2/35
What was known and expected?
What was done?
Anderson localization of
Light
Microwaves
Sound waves
Electrons in solids
What is expected?
Anderson localization of neutral atoms
. – p.3/35
Experiments with cold atoms
V
z
BEC
BEC in a harmonic + weak random potential |V (z)| ≪ ng ⇒ smalldensity modulations of the static BEC. Switch off the harmonic trap,but keep the disorder ⇒ What happens? (Orsay, LENS, Rice)
Orsay experiment
2
4
6
810
2
4
6
8100
-0.5 0 0.5
0.8 s 1.0 s 2.0 s
z (mm)
Ato
m d
en
sit
y (
at/
µm
)
t (s)
L (
mm
)lo
c
a) b)
Lo
ca
liza
tio
n le
ng
th
0.8
0.6
0.4
0.2
0210
. – p.4/35
Examples from other experiments
Aspect group (Palaiseau, France)Inguscio/Modugno group Florence, Italy)
De MArco group (Illinois, USA)Bloch group (Munich, Germany)
Esslinger group (Zurich, Switzerland)Hulet group (Rice, USA)
Labeyrie group (Nice, France)Rolston group (Maryland, USA)
Schnable group (Stony Brook, USA)
. – p.5/35
Coherent backscattering. Weak localization
Palaiseau (Aspect group), Nice (Labeyrie group)Palaiseau experiment. Quasi2D geometry
. – p.6/35
Coherent backscattering
. – p.7/35
2D disordered film
Zurich, Esslinger group
2D disordered film of Li2 Feshbach molecules (large interaction).Observation of the resistence of the film versus disorder strength
. – p.8/35
2D disordered film
. – p.9/35
3D Anderson localization of cold atoms
Urbana (De Marco group)Palaiseau (Aspect group)Florence (Inguscio/Modugno group)
Palaiseau experiment
. – p.10/35
3D Anderson localization of cold atoms
0
5
10
15
20
25
30
35
40
0 200 400 600 800 1000
3m<
D>
/− h
VR/h (Hz)
exp
th
exp
th
6789
101112
13x103
0 0.5 1 1.5 2
∆y2, ∆z
2 (
µm2)
t−ti (s)
z
y
. – p.11/35
3D Anderson localization of cold atoms
0
5
10
15
20
25
30
35
40
0 200 400 600 800 1000
f loc
(%)
VR/h (Hz)
exp
th 0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
∼ n(0,0
,t) / ∼ n
i(0,0
)
1/(t−ti) (s−1
)
expfit
. – p.12/35
LENS experiment. What is expected?
1D quasiperiodic potential
Single-particle state
J(ψn+1 + ψn−1) + V cos(2πκn)ψn = εψn
V > 2J → all single-particle states are localizedAubry/Andre (1980)
. – p.13/35
LENS experiment
Feshbach modification of the interaction for 39K
Observation of the fluid-insulator transition
0 2 4 6 80
2
4
6
8
10
/J
nU/J
insulator
fluid
. – p.14/35
Quantum gases in disorder. What was not expected?
One-dimensional disordered bosons at finite temperature
DOGMA → No finite temperature phase transitions in 1Das all spatial correlations decay exponentially
There is a non-conventional phase transition between two distinct states
Fluid and Insulator
Interaction-induced transition
I.L. Aleiner, B.L. Altshuler, GS, (2010). – p.15/35
Many-body localization-delocalization transition
(Aleiner, Altshuler, Basko 2006-2007)
How different states of two particles |α, β〉 hybridize due to the interaction?The probability P (εα) that for a given state |α〉 there exist |β〉, |α′〉, |β′〉
such that |α, β〉 and |α′, β′〉 are in resonance:〈α, β|Hint|α′, β′〉 exceeds ∆α
′β′
αβ ≡ |εα + εβ − εα′ − εβ′ |
MBLDT criterion P (εα) ∼ 1α
βα
β||
εα ≈ εα′ ; εβ ≈ εβ′ ⇒ Matrix element 〈α, β|Hint|α′, β′〉 = UNβa
ζmax
Mismatch ∆α′β′
αβ = |εα+ εβ −εα′ − εβ′ | ≈∣
∣
∣
∣
1
ζαρ(εα)+
1
ζβρ(εβ)
∣
∣
∣
∣
≈ 1(ζρ)min
. – p.16/35
MBLDT criterion
The probability that 〈α, β|Hint|α′, β′〉 exceeds ∆α′β′
αβ
Pα′β′
αβ ≈UNβa(ζρ)minζmax
P (εα) =∑
β,α′,β′
Pα′β′
αβ =U
∫
dεβρ(εβ)ζβNβa(ζρ)minζmax
Critical coupling strength Uc ≈[∫
dεβρ(εβ)ζβNβa(ζρ)minζmax
]−1
. – p.17/35
1D bosons
Interacting 1D Bose gas. No disorder ⇒ Fluid phase
T
dT =h n /m
22T γ
d
0
QuasiBECthermal gas
DegenerateClassical gas
γ =mg
~2n=ng
Td≪ 1 → weakly interacting regime
Disordered non-interacting 1D bosons
All single-particle states are localized at any energy → Anderson insulator. – p.18/35
1D Bose gas in disorder
I.L. Aleiner, B.L. Altshuler, G.S., 2010
ρ(ε) ≃√
m
2π~2ε; ζ(ε) ≃ ~ε
m1/2ε3/2∗
ε > ε∗ = U0
(
U0σ2m
~2
)1/3
ρ
ε
(ε)
ε*
ε
ζ
ζ
(ε)
*
Classical gas ⇒ T > Td ∼ ~2n2/m; µ = T lnnΛT
ngc ∼ ε∗(ε∗T
)1/2
≪ ε∗
Quantum decoherent gas ⇒ Td√γ < T < Td; µ ∼ T 2/Td
ngc ∼ ε∗(
ε∗TdT 2
)1/2
∼ 1T
≪ ε∗
QuasiBEC ⇒ T < Td√γ; ngc ∼ ε∗ . – p.19/35
1D Bose gas in disorder
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����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������
Fluid
Insulator
disorder
temperature
Insulator
T
ng c
ε*
T Td dγ
Fluid
. – p.20/35
Strongly interacting bosons
mg
~2n≫ 1
Map onto weakly (attractively) interacting fermions
HB = −~2
2m
∑Nj=1 ∂
2xj
+ g∑
1≤j
Strongly interacting bosons
Interaction between 1D fermions renomalizes the disorder
Degenerate fermions Tc ≈ Td(
Dγ
ln γ
)γ/(γ−8)
;γ
ln γ≪ 1
D
Tc vanishes at γ = γ0 ≈ 8consistent with Giamarchi/Schultz (K = 3/2, γ0 ≈ 7.9)
T ≫ Td ⇒ Single-particle picture Tc ∼ Td(Dγ)2/3
. – p.22/35
1D bosons. Phase diagram
. – p.23/35
1D bosons. Phase diagram
. – p.24/35
Atoms in quasiperiodic potential. AAH model
Localization length for all eigenstates is ζ = a ln−1[V/2J ]
(Aubry/Andre, 1980); ζ ≃ V a/(V − 2J) ≫ a for V close to 2JSingle-particle energy states for κ≪ 1 (κ =
√2/20 and V = 2.05J)
Interacting bosons Hint = U∑
j
nj(nj − 1)/2. – p.25/35
MBLDT in the AAH model
The number of clusters N1 ≃ 1/κ for κ≪ 1The width of a cluster Γ grows exp[onentially with energy
For N1 < ζ
ζ/N1 ⇒ number of states of a given cluster participating in MBLDT
T ≪ 8J → lowest energy cluster
MBLDT criterion∫ Γ0
0
dε ρ2(ε)ζnεUc = 1
Occupation number of particle statesnε = [exp(ε+ Unε/ζ − µ)/T − 1]−1
Chemical potential →∫
ρ(ε)nεdε = ν
. – p.26/35
Critical coupling at T = 0
T = 0 ⇒ ε+ Unε/ζ(ε) = µ
nε = ζ(µ0 − ε)/U ; ε < µ0nε = 0; ε > µ0
Ucν ≃2Γ0κζ
Valid also at T ≪ ω
. – p.27/35
Critical coupling at finite temperatures
nε =ζ
2U
{
(µ− ε) +√
(µ− ε)2 + 4TU/ζ}
if nε ≫ 1
nε = exp−(ε− µ)/T if nε . 1 (ε > µ)
Uc(T )
Uc(0)≃
[
1 +T
8νJln
(
T
ω
)]
; ω ≪ T ≪ 8J
Ab initio not expected. Anomalous temperature dependence!
T → ∞ ⇒ nε ≃ ν; µ ≃ −T/ν
Ucν ≃Γ0κ2ζ
;Uc(∞)Uc(0)
=1
κ
. – p.28/35
Critical coupling
κ close to 1/8 and V = 2.05J
Increase in temperature favors the insulator state. ”Freezing with heating”
. – p.29/35
Critical coupling
Golden ratio κ = (√5− 1)/2 and V = 2.1J
. – p.30/35
Conclusions
1D bosons is a promising system to study the many-bodylocalization-delocalization transition
Atoms in quasiperiodic potentials ⇒ Increasing temperaturemay favor localization
Thank you for attention!
. – p.31/35
New system
Random system of polar molecules in a lattice
. – p.32/35
Levy flights and eigenstates of dipolar excitations
H = −∑
i,j
1
|ri − rj |3(a†iaj + h.c.)
Spectrum of fractal dimensions f(α)
Iq =∑
i
|ψ(i)|2q ∝ N−τ(q)
τ(q) = qα− f(α); f ′(α) = q
. – p.33/35
1D and 2D
In 1D and 2D all eigenstates are (algebraically) localizedIn 2D the loc. length likely diverges at E → 0
0 1 2 3 40
0.2
0.4
0.6
0.8
1
α
f(α)
f(α,N)
infinite limitk ~ 0.35
. – p.34/35
3D
In 3D all states are extended
Transition from non-ergodic to ergodic states?
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
α
f(α)
−0.5 0 0.50.4
0.6
0.8
1
α
f(α,N)
infinite limit f(1+α) = f(1−α)+α
. – p.35/35
�lack �f small Many-body physics with ultracold particles in disordersmall Many-body system in disordersmall What was known and expected?small Experiments with cold atomssmall Examples from other experimentssmall Coherent backscattering. Weak localizationsmall Coherent backscatteringsmall 2D disordered film2D disordered filmsmall 3D Anderson localization of cold atomssmall 3D Anderson localization of cold atomssmall 3D Anderson localization of cold atomssmall LENS experiment. What is expected?small LENS experimentsmall Quantum gases in disorder. What was not expected?small Many-body localization-delocalization transitionsmall MBLDT criterionsmall 1D bosonssmall 1D Bose gas in disordersmall 1D Bose gas in disordersmall Strongly interacting bosonssmall Strongly interacting bosonssmall 1D bosons. Phase diagramsmall 1D bosons. Phase diagramsmall Atoms in quasiperiodic potential. AAH modelsmall MBLDT in the AAH modelsmall Critical coupling at $T=0$small Critical coupling at finite temperaturessmall Critical couplingsmall Critical couplingsmall Conclusionssmall New systemsmall Levy flights and eigenstates of dipolar excitationssmall 1D and 2Dsmall 3D