Disordered Ultracold Gases

Transcription

1 Disordered Ultracold Gases 1. Ultracold Gases: basic physics 2. Methods: disorder 3. Localization and Related Measurements Brian DeMarco, University of Illinois

2 Localization & Related Measurements Diffusion Coherence Excitation spectrum Transport / Relaxation

3 Localization & Related Measurements Anderson Localization (1D, 3D) Disordered Bose Hubbard / Dirty boson Disordered Fermi-Hubbard Many-body localization (1D, 2D, 3D) This talk is not inclusive! Many other results, theory.

4 Diffusion Anderson Localization

5 Diffusion 1D: first experiments with disordered ultracold atoms Note: earliest work by Ertmer, Inguscio, Aspect; some confusion about AL ( ) Bouyer / Aspect, Nature 453, 891 (2008) 87 Rb BEC Initial expansion driven by interaction energy Speckle Inguscio, Nature 453, 985 (2008) 39 K BEC a=0 via Feshbach Resonance Incommensurate lattices

6 Weak localization: Theory of non-interacting waves MFP wavelength D D δd

7 Weak localization: Theory of non-interacting waves Bergmann and Langer & Neal Both paths are phase coherent: Interference! W coh A + A 2 = 4 A 2 W incoh A 2 + A 2 = 2 A 2 D D δd

8 Weak localization: Theory of non-interacting waves MFP wavelength ξ: localization length ψ e x/ξ D = 0

9 Scaling theory 1D, 2D: all states localized for any Δ 3D: only states with energy less than Caveats! mobility edge are localized For infinite system, at infinite time, assumptions about disorder

10 Observation of 1D AL 0.75 s Bouyer / Aspect, Nature 453, 891 (2008) Inguscio, Nature 453, 985 (2008)

11 Observation of 1D AL 0.75 s Bouyer / Aspect, Nature 453, 891 (2008) Inguscio, Nature 453, 985 (2008)

12 Aspect: effective 1D speckle mobility edge k max σ R < 1 k max σ R > 1 ħ 4 4 k max Born approximation, PRL 98, (2007) L loc = 2 πm 2 V 2 R σ 4 R 1 k max σ R k max σ R < 1 V R Δ (Sanchez-Palencia) σ R : cutoff in Fourier spectrum of speckle for uniformly illuminated lens

13 3D Anderson Localization DeMarco, Science 334, 66 (2011) Bouyer / Aspect, Nat. Phys 8, 398 (2012) Spin-polarized 40 K gas 87 Rb BEC Non-interacting T/T F 1, broad energy distribution 50 ms pre-expansion, disorder suddenly turned on

14 3D Anderson Localization Later observed out to 1s (with gravity compensated) DeMarco, Science 334, 66 (2011) Bouyer / Aspect, Nat. Phys 8, 398 (2012)

15 3D Anderson Localization Mobility edge DeMarco, Science 334, 66 (2011) Bouyer / Aspect, Nat. Phys 8, 398 (2012)

16 Anderson Localization Many more measurements & a lot of theory in 1D More measurements & more theory in 3D speckle Challenges: 2D Theory in 3D Dynamics

17 Localization with Interactions

18 Disordered Bose-Hubbard / Dirty Bosons Key model for understanding how interactions affect AL; dirty superfluids & superconductors Disordered Bose- Hubbard ground-state phase diagram now known in 1D, 2D, 3D Some things also known about T 0 DeMarco, Nat. Phys. 12, 646 (2016)

19 Disordered Bose-Hubbard / Dirty Bosons 1D: one example from Inguscio / Modugno 39 K atoms Incommensurate lattices Interacting Aubrey- Andre model H = t U i 2 i b i b i+1 + hc + Δ cos 2 πβi n i + n i n i 1 + α 2 i i 2 0 n i i i trap Inguscio / Modugno, PRL 113, (2014)

20 Coherence TOF image Peaks generated by interference of atoms from different sites Longer coherence length Γ k Shorter coherence length U/J Inguscio / Modugno, PRL 113, (2014)

21 Transport δp Apply impulse, measure resulting momentum Analog of conductivity Inguscio / Modugno, PRL 113, (2014)

22 Excitation Spectrum Modulate primary lattice amplitude U High U, Mott gap Strong Δ, BG Inguscio / Modugno, PRL 113, (2014)

23 1D Disordered Bosons Compare: impurity disorder & incommensurate lattice Impurity disorder: H = t σ i b i b i+1 + hc + U 2 σ i n i n i 1 + σ i n i δε i +trap Schneble, PRL 107, (2011)

24 1D Disordered Bosons Excitation Spectrum Clean Incommensurate lattice Impurities Schneble, PRL 107, (2011)

25 Impulse response Coherence impurities Incommensurate lattice Schneble, PRL 107, (2011)

26 2D Dirty Bosons 87 Rb atoms, interactions are present Continuum model + speckle potential Atoms strongly trapped in speckle Rolston, NJP 14, (2012)

27 2D Dirty Bosons TOF coherence measurement: Atoms from the two layers interfere Rolston, NJP 14, (2012)

28 2D Dirty Bosons Rolston, NJP 14, (2012) temperature See also Esslinger (diffusion, classical trapping)

29 2D Disordered Bose Hubbard 87 Rb atoms Projected disorder U H = t ij b i b i σ i n i n i 1 + σ i δε i n i +trap Gross / Bloch, Science 352, 1548 (2016)

30 2D Disordered Bose Hubbard Mott insulator, heated through lattice quench Claim: MBL Gross / Bloch, Science 352, 1548 (2016)

31 3D Disordered Bose Hubbard D s 1 H n t b b b b U n n 1 i i ij i j j i i i i i ij 2 i DeMarco, 87 Rb atoms, speckle disorder

32 3D Disordered Bose Hubbard Transport measurement: velocity following impulse Insulator (BG) SF MI m* D=3 E R D=0.75 E R D=0 E R DeMarco, Nat. Phys. 6, 677 (2010) Coherence also vanishes at transition See also disorder quench: Nat. Phys. 12, 646 (2016)

33 1D Disordered Fermions Two hyperfine states of 40 K atoms Incommensurate lattice H = t i,σ c i,σ c i+1,σ + hc + Δ σ i,σ cos 2 πβi n i,σ + U σ i n i, n i, +trap Schneider / Bloch, Science 349, 842 (2015)

34 1D Disordered Fermions Prepare CDW, Measure relaxation Claim: MBL Schneider / Bloch, Science 349, 842 (2015)

35 3D Disordered Fermi-Hubbard D s DeMarco, two hyperfine states of 40 K atoms H = i,σ n i,σ ε i ij,σ t ij c i,σ c j,σ + i U i n i, n i,

36 3D Disordered Fermi-Hubbard Transport Measurement Δ = 0 s U/t Δ c Δ large DeMarco, Phys. Rev. Lett. 114, (2015)

37 disorder 3D Disordered Fermi-Hubbard lattice insulator metal interactions

38 3D Disordered Fermi-Hubbard Measure temperature dependence Consistent with BAA predictions for MBL DeMarco, Phys. Rev. Lett. 114, (2015)

39 Conclusions Anderson Localization (1D, 3D) Disordered Bose Hubbard / Dirty boson Disordered Fermi-Hubbard Many-body localization (1D, 2D, 3D)

40 Outstanding Questions Distinguish localization from sub-diffusion? Distinguish MBL from non-exponential relaxation / glassiness? Relaxation dynamics in strongly correlated clean systems? Unified low T, high T picture?