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Many-body physics with ultracold particles in disorder Gora Shlyapnikov LPTMS, 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

Gora Shlyapnikov LPTMS, Orsay, France Many-body localization-delocalization …online.kitp.ucsb.edu/online/mbl-c15/shlyapnikov/pdf/... · 2015. 11. 25. · 1D quasiperiodic potential

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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(

    ln γ

    )γ/(γ−8)

    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