Interacting atoms and molecules in triangular lattices with varying geometry. Dan Stamper-Kurn, UC Berkeley

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1 Interacting atoms and molecules in triangular lattices with varying geometry Dan Stamper-Kurn, U erkeley

2 perspective on atoms in optical lattices Gedanken laboratory for study of single-, few- and many-body quantum physics Simplicity Precision Newly deep view wide range of new materials to study, probably with some very interesting properties

3 Lattice with triangular motif: simplicity ee 0 ee 4ππππ/3? ee 2ππππ/3 Minimal (simple) building block for Two dimensions most interesting dimension (critical for short-range interactions) Examples: Mermin-Wagner thm., KT transition, quantum Hall Rotation (and flux) Geometric frustration route to interesting phases

4 lattice characterization via matter-wave crystallography Signatures of spatial inversion asymmetry of an optical lattice observed in matter-wave diffraction, PR 93, (2016)

5 Inversion asymmetric hexagonal lattice EE EE σσ σσ + σσ + σσ σσ + σσ EE VV llllllllllllll = VV ss + VV aa VV ss = scalar ac Stark shift = inversion symmetric VV aa = vector ac Stark shift = inversion anti-symmetric

6 Inversion asymmetric hexagonal lattice P. Soltan-Panahi, et al., Nature Physics 7, 434 (2011) VV llllllllllllll = VV ss + VV aa VV ss = scalar ac Stark shift = inversion symmetric VV aa = vector ac Stark shift = inversion anti-symmetric

7 Nature Physics 8, 71 (2012)

8 Twisted superfluid? not expected in mean-field theory (Feynman) ħ2 2mm VV ±1 + gg ±1,1 ψψ +1 + gg ±1, 1 ψψ 1 μμ ±1 ψψ ±1 = 0 if ψψ +1 ψψ 1 is solution, so is ψψ +1 ; therefore, ψψ +1 ψψ 1 ψψ 1 is real

$Kaptiza-Dirac diffraction from asymmetric honeycomb lattice similar$

9 Kaptiza-Dirac diffraction from asymmetric honeycomb lattice similar inversion asymmetry observed symmetry breaking term VV ss VV aa < 2% +

10 symmetric diffraction in Raman-Nath regime Scattering matrix elements of potentials (Fourier transforms) VV ss GG = VV ss GG = ββ SS VV aa GG = VV aa GG = ii ββ SS

11 Origin of asymmetric time-of-flight of superfluid Equilibrium in lattice: constant chemical potential Lattice off: leaves asymmetric honeycomb lattice of interaction energy (coherent matter-wave mixing)

12 kagome lattice for ultracold atoms Ultracold atoms in a tunable optical kagome lattice, PRL 108, (2012)

13 Geometric frustration of the kagome lattice Triangular? F Ising: frustrated F Heisenberg: 3 sub-lattice Neel state

14 Geometric frustration of the kagome lattice Kagome recipe: eliminate every 4 th site from short-length lattice eliminated sites form long-length triangular lattice

15 Quantum magnetism in the kagome lattice Very frustrated quantum magnetism Quantum spin (s=1/2) antiferromagnet What is the ground state? Long-standing question Good candidate for a quantum spin liquid but which type? Z2? hiral? Dirac? e.g. Yan, Huse, White, Science 332, 1173 (2011); Iqbal, Poilblanc, ecca, PR 89, (R) (2014); H. J. Liao PRL 118, (2017) Solid state realizations are complicated: Herbertsmithite (Znu 3 (OH) 6 l 2 ): substitutional disorder, etc. Volborthite (u 3 V 2 O 7 (OH) 2 2H2O): Dzyaloshinsky-Moriya anisotropy

16 Orbital frustration extended wavefunction, tight binding: what is the highest energy state? HH kkkkkkkkkkkkkk = JJ cos(φφ ii φφ jj ) Flat and φφ φφ φφ

17 onnection to quantum Hall physics

18 Scheme: two-color optical superlattice Short wavelength lattice, with light at λλ SSSS = 532 nm lue detuned: atoms attracted to locations of lowest intensity Yields triangular lattice geometry 709 nm D Triangular: VV,,,DD = 0 high intensity = high potential

19 Scheme: two-color optical superlattice dding long-wavelength lattice with light at λλ LLWW = 1064 nm Red detuned: atoms attracted to locations of highest intensity 709 nm position (xx, yy) of LW potential maximum high intensity = low potential

20 Scheme: two-color optical superlattice Overlaying both at kagome alignment Tuning knobs: long-wavelength lattice depth VV LLLL and position (xx, yy) 709 nm D kagome: VV DD VV,, = VV > 0

21 phase transition driven by change in lattice geometry & quantitative test of ose-hubbard model Mean-field scaling of the superfluid to Mott insulator transition in a two-dimensional optical superlattice, PRL 119, (2017)

Structural and electronic (transport) transitions Solid

$diffraction study of metallic VO2, McWhan et al.$

22 Structural and electronic (transport) transitions Solid state: Simultaneous structural and metal -insulator transitions in various materials e.g. vanadium dioxide Metal-insulator transition in vanadium dioxide, Zylbersztejn and Mott, PR 11, 4383 (1975) X-ray diffraction study of metallic VO2, McWhan et al., PR 10, 490 (1974) which causes which?

23 Superfluid to Mott insulator transition ose-hubbard model: spinless bosons, in lattice, on-site interactions HH = JJ aa nn ii (nn ii 1) ii aa jj + UU 2 <ii,jj> ii superfluid long-range coherence Mott insulator only short-range coherence UU/zzzz UU/zzzz cc zz = coordination number number of nearest neighbors even next-neighbor coherence suppressed as UU zzzz

24 Structural transition of the optical lattice Vary relative intensities: turn up long-wavelength lattice; kagome triangular lattice zz = 6 nearest neighbors kagome lattice zz = 4 nearest neighbors Hubbard model parameters UU and JJ nearly constant

25 Superfluid Mott insulator transition driven by lattice geometry 87 Rb atoms in many 2D layers, central filling 1/site, momentum-space focusing triangular lattice, VV DD = 0 VV SSSS /h = 54 khz 66 khz 74 khz 82 khz kagome lattice, VV DD = h 15 khz Deeper short-wavelength lattice

Precision experimental test of ose-hubbard model theory superfluid

$UU/zzzz cc condensate fraction interference peak visibility/width closing of$ PRL (2010) Trotzky et al. Nature (2010) Endres et al.

26 Precision experimental test of ose-hubbard model theory superfluid long-range coherence Mott insulator only short-range coherence UU/zzzz UU/zzzz cc condensate fraction interference peak visibility/width closing of the Higgs gap Spielman et al. PRL (2008) Jiménez-García et al. PRL (2010) Trotzky et al. Nature (2010) Endres et al. Nature (2012) Tests achieve moderate precision (10% at best) Interpretation is complicated by finite temperature & inhomogeneity

Scaling hypothesis in the -H equation of state Originates in mean-field (local) theory ρρ(tt) ρρ ii (TT) ii equation of state: nn = particle # nn ss = condensate # ss = entropy per site μμ = ff αα

27 Scaling hypothesis in the -H equation of state Originates in mean-field (local) theory ρρ(tt) ρρ ii (TT) ii equation of state: nn = particle # nn ss = condensate # ss = entropy per site μμ = ff αα zzzz, UU zzzz, kk TT zzzz scaling hypothesis valid throughout phase diagram valid for all lattice geometries pplying to trapped gas under (local) density approximation NN ss = condensate # NN = particle # SS = entropy total μμ 0 = KK /zzzzdd μμ nn αα μμ, UU zzzz, kk TT zzzz μμ 0 zzzz μμ KK = # occupied sites

28 Precise comparison of two lattice geometries atom number NN entropy SS triangular zz = 6 Scaling Hypothesis: at same UU/zzzz (and KK) all measured properties should be the same kagome zz = 4 NN ss = condensate # NN = particle # SS = entropy total μμ 0 = KK /zzzzdd μμ nn αα μμ, UU zzzz, kk TT zzzz μμ 0 zzzz μμ

29 Superfluid Mott insulator transition driven by lattice geometry Extract coherent fraction from the peaks in momentum-space images UU/JJ (Lattice depth)

$oherent fraction: Test of scaling hypothesis coherent fraction triangular kagome H.E.$ ) such a genius. What else did you expect? FKE SIENE! or Physics Expert @MRad - Feb 23 UU zzzz 5.

) such a genius. What else did you expect? FKE SIENE! or Physics Expert @MRad - Feb 23 UU zzzz 5.

30 oherent fraction: Test of scaling hypothesis coherent fraction triangular kagome H.E. - Feb 23 Oh yes, of course, that mean-field model is so great. Fisher is (are?) such a genius. What else did you expect? FKE SIENE! or Physics - Feb 23 UU zzzz 5.8 Oh yes, of course, everybody knows mean-field is horrible in 2D. Kagome geometry is tricky b/c of flat band. What else did you expect? Sad!

31 oherent fraction: Test of scaling hypothesis ζζ ssssssss = zz tttttt zz kkkkkk = 1.5 ζζ eeeeee = 1.6(1) scale U/J axis for kagome data by ζζ to fit best to triangular data Why does mean-field-derived scaling prediction apply so well??

32 trimerized kagome lattice arter, Leung and Okano, in progress

33 eyond the kagome lattice short wavelength λλ = 532 nm long wavelength λλ = 1064 nm optical potential high + = low polarization in plane polarization in plane

34 eyond the kagome lattice short wavelength λλ = 532 nm long wavelength λλ = 1064 nm optical potential high + = low polarization in plane polarization out of plane HH = JJ aa ii aa jj JJ aa ii aa jj + UU 2 nn ii nn ii 1 uuuu dddddddd

35 Trimerized kagome lattice inversion-asymmetric ose-hubbard system optical potential high HH = JJ aa ii aa jj JJ aa ii aa jj + UU 2 nn ii nn ii 1 uuuu dddddddd low

Detecting bond order/nearest-neighbor correlations Momentum space image = momentum structure factor = direct image of spatial correlations These modulations reveal phase

36 Detecting bond order/nearest-neighbor correlations Momentum space image = momentum structure factor = direct image of spatial correlations These modulations reveal phase correlations between neareset neighbors SS kk = ww kk 2 ii,jj ee ii kk rr ii rr jj bb jj bb ii ww kk αα cos(kk rr nnnn ) + αα sin(kk rr nnnn ) + ββ + γγ + nnnn nnnn

37 Nearest-neighbor correlations: non-trimerized vs. trimerized JJ JJ JJ JJ ee ii kk rr bb bb + bb bb ee ii kk rr bb bb + bb bb as weak bond JJ 0: bb jj bb ii nnnn JJ UU 0

38 Nearest-neighbor correlations: non-trimerized vs. trimerized trimerized kagome lattice with three constant settings of JJ/UU JJ /UU primitive triangular lattice used as one with no dimerization

39 Nearest-neighbor correlations: detecting inversion asymmetry JJ JJ add phase offset φφ JJ JJ ee ii kk rr ee iiii bb bb + ee iiii bb bb ee ii kk rr ee iiii bb bb + ee iiii bb bb amplitude of modulation changes phase of modulation changes

40 Nearest-neighbor correlations: detecting inversion asymmetry ττ = 0 φφ = VV ττ/ħ JJ JJ ττ = 25 μμs consistent with 100% asymmetry in MI regime ττ = 50 μμs

41 Ideas for future work Few-body quantum states in triangular plaquettes E.g. Two hard-core bosons, excited to rotating state Site basis: Ψ pp=2 1,1,0 + ee 2ππππ 3 0,1,1 + ee 4ππππ 3 1,0,1 ngular momentum basis: Ψ pp=2 = 1 2 ee4ππππ 3 1,0,1 pp 0,2,0 pp Laughlin νν = 1/2 state: Ψ LL zz 1 zz 2 2 = zz zz zz 1 zz 2 Orbital quantum magnetism

42 Things to ask me about a. magnetic excitations in spinor gases, coherent magnon optics and interferometry, atoms in. space b. collective atomic motion and spin dynamics in driven optical cavity c. Li + Rb quantum gas mixtures and molecules Looking for excellent postdocs and students Masayuki Okano laire Thomas Tom arter Zephy Leung $$ Thanks! NSF, RO MURI

Ultracold toms in Space old tom Laboratory (L) US NS/JPL ose-einstein condensation aboard the International Space Station (ISS) Launch: mid 2018? Operation: 2018 2022?

43 Ultracold toms in Space old tom Laboratory (L) US NS/JPL ose-einstein condensation aboard the International Space Station (ISS) Launch: mid 2018? Operation: ? ose-einstein ondensation old tom Lab (EL) US NS/Germany DLR US Science Definition Team: Lundblad, Müller, Stamper-Kurn (hair), Stuhl German SDT: ecker, Krutzik, Rasel, Schleich (hair) genda: 1) tom interferometry, 2) tom optics, 3) Scalar Es, 4) Spinor Es and mixtures, 5) Strongly interacting atoms + molecules, 6) Quantum optics Launch 2022? Operation ?

Cold atoms in optical lattices

Cold atoms in optical lattices www.lens.unifi.it Tarruel, Nature Esslinger group Optical lattices the big picture We have a textbook model, which is basically exact, describing how a large collection of