quantum gases in disorder
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Quantum gases in disorder Gora Shlyapnikov LPTMS, Orsay, France - PowerPoint PPT Presentation

Quantum gases in disorder Gora Shlyapnikov LPTMS, Orsay, France University of Amsterdam, The Netherlands Russian Quantum Center, Moscow, Russia Introduction. Many-body localization-delocalization transition MBLDT for 1D disordered bosons


  1. Quantum gases in disorder Gora Shlyapnikov LPTMS, Orsay, France University of Amsterdam, The Netherlands Russian Quantum Center, Moscow, Russia Introduction. Many-body localization-delocalization transition MBLDT for 1D disordered bosons MBLDT in the AAH model Phase diagram Conclusions Collaborations B.L. Altshuler/I.L. Aleiner (Columbia Univ.), V. Michal (LPTMS, Orsay) ICTP , Italy, August 25, 2015 . – p.1/20

  2. 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 Palaiseau, LENS, Rice, Urbana experiments. More underway . – p.2/20

  3. 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.3/20

  4. Experiments with cold atoms BEC V z BEC in a harmonic + weak random potential | V ( z ) | ≪ ng ⇒ small density modulations of the static BEC. Switch off the harmonic trap, but keep the disorder ⇒ What happens? (Orsay, LENS, Rice) Orsay experiment L (mm) 100 0.8 8 Atom density (at/µm) a) b) 0.8 s 6 loc 1.0 s 4 0.6 2.0 s Localization length 2 0.4 10 8 6 0.2 4 2 0 -0.5 0 0.5 0 1 2 z (mm) t (s) . – p.4/20

  5. Quantum gases in disorder. What was not expected? One-dimensional disordered bosons at finite temperature DOGMA → No finite temperature phase transitions in 1D as 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.5/20

  6. 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 ( ε α ) that for a given state | α � there exist | β � , | α ′ � , | β ′ � such that | α, β � and | α ′ , β ′ � are in resonance: � α, β | H int | α ′ , β ′ � exceeds ∆ α ′ β ′ ≡ | ε α + ε β − ε α ′ − ε β ′ | αβ MBLDT criterion P ( ε α ) ∼ 1 α α | β β | a ε α ≈ ε α ′ ; ε β ≈ ε β ′ ⇒ Matrix element � α, β | H int | α ′ , β ′ � = UN β ζ max � � 1 1 1 Mismatch ∆ α ′ β ′ � � αβ = | ε α + ε β − ε α ′ − ε β ′ | ≈ ζ α ρ ( ε α ) + � ≈ � � ζ β ρ ( ε β ) ( ζρ ) min � . – p.6/20

  7. MBLDT criterion The probability that � α, β | H int | α ′ , β ′ � exceeds ∆ α ′ β ′ αβ a ( ζρ ) min P α ′ β ′ ≈ UN β αβ ζ max a ( ζρ ) min � P α ′ β ′ � P ( ε α ) = = U dε β ρ ( ε β ) ζ β N β αβ ζ max β,α ′ ,β ′ � − 1 �� a ( ζρ ) min Critical coupling strength U c ≈ dε β ρ ( ε β ) ζ β N β ζ max . – p.7/20

  8. 1D bosons Interacting 1D Bose gas. No disorder ⇒ Fluid phase Degenerate QuasiBEC Classical gas thermal gas T 0 γ 2 2 T T =h n /m d d γ = mg � 2 n = ng ≪ 1 → weakly interacting regime T d Disordered non-interacting 1D bosons All single-particle states are localized at any energy → Anderson insulator . – p.8/20

  9. 1D Bose gas in disorder I.L. Aleiner, B.L. Altshuler, G.S., 2010 � 1 / 3 � � U 0 σ 2 m m � ε ρ ( ε ) ≃ 2 π � 2 ε ; ζ ( ε ) ≃ ε > ε ∗ = U 0 m 1 / 2 ε 3 / 2 � 2 ∗ ( ε ) ρ ( ε ) ζ ε * ζ * ε ε Classical gas ⇒ T > T d ∼ � 2 n 2 /m ; µ = T ln n Λ T � ε ∗ � 1 / 2 ng c ∼ ε ∗ ≪ ε ∗ T Quantum decoherent gas ⇒ T d √ γ < T < T d ; µ ∼ T 2 /T d � 1 / 2 � ε ∗ T d ∼ 1 ng c ∼ ε ∗ T ≪ ε ∗ T 2 QuasiBEC ⇒ T < T d √ γ ; ng c ∼ ε ∗ . – p.9/20

  10. 1D Bose gas in disorder disorder �������������������������������������������������� �������������������������������������������������� Insulator �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� �������������������������������������������������� Fluid temperature ng c ε * Fluid Insulator T T T γ d . – p.10/20 d

  11. LENS experiment. What is expected? 1D quasiperiodic potential Single-particle state J ( ψ n +1 + ψ n − 1 ) + V cos(2 πκn ) ψ n = εψ n V > 2 J → all single-particle states are localized Aubry/Andre (1980) . – p.11/20

  12. LENS experiment Feshbach modification of the interaction for 39 K Observation of the fluid-insulator transition 10 insulator 8 6 /J 4 fluid 2 0 0 2 4 6 8 nU/J . – p.12/20

  13. AAH model Localization length for all eigenstates is ζ = a ln − 1 [ V/ 2 J ] (Aubry/Andre, 1980); ζ ≃ V a/ ( V − 2 J ) ≫ a for V close to 2 J √ Single-particle energy states for κ ≪ 1 ( κ = 2 / 20 and V = 2 . 05 J ) � Interacting bosons H int = U n j ( n j − 1) / 2 . – p.13/20 j

  14. MBLDT in the AAH model The number of clusters N 1 ≃ 1 /κ for κ ≪ 1 The width of a cluster Γ grows exp[onentially with energy For N 1 < ζ ζ/N 1 ⇒ number of states of a given cluster participating in MBLDT T ≪ 8 J → lowest energy cluster � Γ 0 dε ρ 2 ( ε ) ζn ε U c = 1 MBLDT criterion 0 Occupation number of particle states n ε = [exp( ε + Un ε /ζ − µ ) /T − 1] − 1 � Chemical potential → ρ ( ε ) n ε dε = ν . – p.14/20

  15. Critical coupling at T = 0 T = 0 ⇒ ε + Un ε /ζ ( ε ) = µ n ε = ζ ( µ 0 − ε ) /U ; ε < µ 0 n ε = 0; ε > µ 0 U c ν ≃ 2Γ 0 κζ Valid also at T ≪ ω . – p.15/20

  16. Critical coupling at finite temperatures n ε = ζ � � ( µ − ε ) 2 + 4 TU/ζ � ( µ − ε ) + if n ε ≫ 1 2 U n ε = exp − ( ε − µ ) /T if n ε � 1 ( ε > µ ) U c ( T ) � T � T �� U c (0) ≃ 1 + 8 νJ ln ; ω ≪ T ≪ 8 J ω Ab initio not expected. Anomalous temperature dependence! T → ∞ ⇒ n ε ≃ ν ; µ ≃ − T/ν U c ν ≃ Γ 0 U c ( ∞ ) U c (0) = 1 κ 2 ζ ; κ . – p.16/20

  17. Critical coupling κ close to 1/8 and V = 2 . 05 J Increase in temperature favors the insulator state. ”Freezing with heating” . – p.17/20

  18. Critical coupling √ Golden ratio κ = ( 5 − 1) / 2 and V = 2 . 1 J . – p.18/20

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