LECTURE 3 - Artificial Gauge Fields

Transcription

1 LECTURE 3 - Artificial Gauge Fields SSH model - the simplest Topological Insulator Probing the Zak Phase in the SSH model - Bulk-Edge correspondence in 1d - Aharonov Bohm Interferometry for Measuring Band Geometry - Berry connection/berry curvature - pi-flux Singularity in Graphene - Stückelberg Interferometry (non-abelian Berry connection, Wilson loops) Realizing Staggered Flux, Hofstadter & QSH Hamiltonian Hall Response and Chern Number in Hofstadter Bands

2 Measuring the Zak-Berry s Phase of Topological Bands M. Atala et al., Nature Physics (213)

3 Berry Phase Berry Phase in Quantum Mechanics (R)! e i(' Berry+' dyn ) (R) Adiabatic evolution through closed loop ' Berry = ' Berry = I C I A n (R)dR = i A n (R) da I C hn(r) r R n(r)i dr Berry Phase M.V. Berry, Proc. R. Soc. A (1984) Berry connection A n (R) =ihn(r) r R n(r)i Example: Spin-1/2 particle in magnetic field Berry curvature n,µ µ A n,µ

4 Berry Phase Berry Phase for Periodic Potentials k(r) =e ikr u k (r) Bloch wave in periodic potential Adiabatic motion in momentum space generates Berry phase! ky ky ky Berry phase is fundamental to characterize topology of energy bands kx kx kx n Chern = 1 2 I BZ A k dk = 1 2 Z BZ k d 2 k xy = n Chern e 2 /h Chern Number (Topological Invariant) Quantized Hall Conductance Thouless, Kohmoto, den Nijs, and Nightingale (TKNN), PRL 1982 Kohmoto Ann. of Phys Mention Problem with going on a line is generally NOT A LOOP IN PARAMETER SPACE!

5 Berry Phase Berry Phase for Periodic Potentials k(r) =e ikr u k (r) Bloch wave in periodic potential Adiabatic motion in momentum space generates Berry phase! ky ky ky Berry phase is fundamental to characterize topology of energy bands kx kx kx n Chern = 1 2 I BZ A k dk = 1 2 Z BZ k d 2 k xy = n Chern e 2 /h Chern Number (Topological Invariant) Quantized Hall Conductance Thouless, Kohmoto, den Nijs, and Nightingale (TKNN), PRL 1982 Kohmoto Ann. of Phys What is the extension to 1D? Mention Problem with going on a line is generally NOT A LOOP IN PARAMETER SPACE!

6 Berry Phase Zak Phase 2D Brillouin Zone ky kx going straight means going around! Band structure has torus topology!

7 Berry Phase Zak Phase 2D Brillouin Zone ky kx going straight means going around! Band structure has torus topology! ' Zak = i Z k +G hu k u k i dk Zak Phase - the 1D Berry Phase k J. Zak, Phys. Rev. Lett. 62, 2747 (1989)

8 Berry Phase Zak Phase 2D Brillouin Zone ky kx going straight means going around! Band structure has torus topology! ' Zak = i Z k +G hu k u k i dk Zak Phase - the 1D Berry Phase k J. Zak, Phys. Rev. Lett. 62, 2747 (1989) Non-trivial Zak phase: Topological Band Edge States (for finite system) Domain walls with fractional quantum numbers Polarization of solids R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976) J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981) R. King-Smith and D. Vanderbilt, Phys. Rev. B 46, 1651 (1993)

9 Berry Phase Su-Shrieffer-Heeger Model (SSH) Polyacetylene W. P. Su, J. R. Schrieffer & A. J. Heeger Phys. Rev. Lett. 42, 1698 (1979). H SSH = Â n Jâ n ˆb n + J â n ˆb n 1 + h.c.

10 Berry Phase Su-Shrieffer-Heeger Model (SSH) Polyacetylene W. P. Su, J. R. Schrieffer & A. J. Heeger Phys. Rev. Lett. 42, 1698 (1979). H SSH = Â n Jâ n ˆb n + J â n ˆb n 1 + h.c. Two topologically distinct phases: D1: J>J D2: J >J J J J J J J J J J J

11 Berry Phase Su-Shrieffer-Heeger Model (SSH) Polyacetylene W. P. Su, J. R. Schrieffer & A. J. Heeger Phys. Rev. Lett. 42, 1698 (1979). H SSH = Â n Jâ n ˆb n + J â n ˆb n 1 + h.c. Two topologically distinct phases: D1: J>J D2: J >J J J J J J J J J J J ' Zak = ' D1 Zak ' D2 Zak = Topological properties: domain wall features fractionalized excitations Zak phase difference ' Zak is gauge-invariant

12 Berry Phase SSH Energy Bands - Eigenstates V(x) x d a n-1 b n-1 a n b n a n+1 b n+1 J J J J J J J

13 Berry Phase SSH Energy Bands - Eigenstates V(x) x d a n-1 b n-1 a n b n a n+1 b n+1...ababa... Lattice Structure... J J J J J J J X x x = X x e ikx ( k k e ikd/2

14 Berry Phase SSH Energy Bands - Eigenstates V(x) x d a n-1 b n-1 a n b n a n+1 b n+1...ababa... Lattice Structure... J J J J J J J X x x = X x e ikx ( k k e ikd/2 2x2 Hamiltonian: apple k k k k = k k k with k = Je ikd/2 + J e ikd/2 = k e i k

15 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/2 Eigenstates k, k, = 1 p 2 ±1 e i k /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k

16 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/2 Eigenstates k, k, = 1 p 2 ±1 e i k /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k Adiabatic evolution in momentum space

17 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/2 Eigenstates k, k, = 1 p 2 ±1 e i k /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k Adiabatic evolution in momentum space

18 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/2 Eigenstates k, k, = 1 p 2 ±1 e i k /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k ' Zak = i Z k +G k ( k@ k k + k@ k k ) dk Zak Phase SSH Model

19 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/2 Eigenstates k, k, = 1 p 2 ±1 e i k /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k ' Zak = 1 2 Z G+k k k dk

20 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/2 Eigenstates k, k, = 1 p 2 ±1 e i k /4 D1: Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k D1: J>J ' D1 Zak = 2

21 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/2 Eigenstates k, k, = 1 p 2 ±1 e i k /4 D1: Energy k /4 D2: /d Quasimomentum k /d /d /d Quasimomentum k D1: J>J ' D1 Zak = 2 D2: J >J ' D2 Zak = 2

22 Berry Phase Realization with ultracold atoms H SSH = Â n Ja nb n + J a nb n 1 + h.c. 767 nm 1534 nm D1: J>J D2: J >J Phase shift J J J J J J J J J J ' Zak = ' D1 Zak ' D2 Zak =

23 Berry Phase D1: J>J Measuring the Berry-Zak Phase (SSH Model) Spin-dependent Bloch oscillations + Ramsey interferometry /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k Prepare BEC in state, ki = #, i, with =", #

24 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D1: J>J /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k MW /2-pulse Create coherent superposition 1 p ( ", i + #, i) 2

25 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D1: J>J /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k MW /2-pulse Create coherent superposition 1 p ( ", i + #, i) 2 Spin components with opposite magnetic moments!

26 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D1: J>J /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k Closed Loop in k-space Apply magnetic field gradient! adiabatic evolution in momentum space

27 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D1: J>J /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k Closed Loop in k-space Apply magnetic field gradient! adiabatic evolution in momentum space

28 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D1: J>J /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k ' Zak = ' D1 Zak + ' dyn + ' Zeeman

29 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D1: J>J /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k Z ' dyn = E(t)/~ dt ' Zak = ' D1 Zak + ' X dyn + ' Zeeman E(k) =E( k)

30 Berry Phase Reference measurement Phase of reference fringe: 1 Fraction in /2 3 /2 2 Microwave pulse MW ( ) ' 6= Average of five individual measurements Exp. imperfections: - Small detuning of the MW-pulse - Magnetic field drifts

31 Berry Phase Measuring the Zak Phase (SSH Model) Measured Topological invariant: Zak phase difference ' D1 Zak ' D2 Zak = 1 Fraction in /2 3 /2 2 Microwave pulse MW ( ) ' Zak =.97(2) obtained from 14 independent measurements

32 Berry Phase Measuring the Zak Phase (SSH Model) Measured Topological invariant: Zak phase difference ' D1 Zak ' D2 Zak = 1 2 Fraction in Zak /2 3 /2 2 Microwave pulse MW ( ) Data set 2 4 Occurences ' Zak =.97(2) obtained from 14 independent measurements

33 Geometric Phase Fractional Zak Phase Zak Phase becomes fractional for heteropolar dimerization!

34 SSH Measuring Fractional Charge Probability Density of Eigenstates Lattice Topology Mode function Topologically Trivial Lattice site Mode function Lattice site Topologically Non-Trivial

35 SSH Fractional Charge /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k R. Rajaraman & J. Bell, Phys. Lett B 1982, Nucl. Phys. B 1983

36 SSH Fractional Charge /4 Energy k /4 /d Quasimomentum k /d /d /d Quasimomentum k R. Rajaraman & J. Bell, Phys. Lett B 1982, Nucl. Phys. B 1983

37 SSH Fractional Charge /4 1. Energy k Density.5 /d Quasimomentum k /d / /d Lattice Site /d Quasimomentum k R. Rajaraman & J. Bell, Phys. Lett B 1982, Nucl. Phys. B 1983

38 SSH Fractional Charge /4 1. Energy k Density.5 /d Always one atom /4 on the Lattice Site /dedge! /d /d Quasimomentum k Quasimomentum k R. Rajaraman & J. Bell, Phys. Lett B 1982, Nucl. Phys. B 1983

39 An Aharonov Bohm Interferomer for Determining Bloch Band Topology -Magnetic Flux -Berry Flux Real space Crystal momentum space D. Abanin E. Demler D. Abanin et al. PRL 11, (213)

40 AB Aharonov-Bohm Effect Real Space j AB = q h y x B I Z A(r)dr = q h C Z j AB = q h BdS = 2p F/F S A(r) d 2 r Y. Aharonov D. Bohm..., contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. Y. Aharonov & D. Bohm Phys. Rev.(1959) W. Ehrenberg & R. Siday Proc. Phys. Soc B (1949) Exp: A. Tonomura, et al. Phys. Rev. Lett. (1986) Aharonov-Bohm Phase

41 Band Topology Hexagonal Lattices Lattice: A and B degenerate sublattices H = H JÂ R 3 Â â R ˆb R+d i + h.c. i=1 Real Space Lowest energy bands: Reciprocal Space Dirac points at the corners of the first BZ k y A. Castro Neto et al., Rev. Mod. Phys. 81, 19 (29) cold atoms: hexagonal - K. Sengtsock (Hamburg), brick wall - T. Esslinger (Zürich) k x

42 Scalar & Geometric Features Band Topology Band structure characterized by scalar & geometric features! Eigenstates: Bloch waves Scalar Features Dispersion relation Eq,n yq,n (r) = eiqr uq,n (r) Geometric Features Berry connection An (q) = ihuq,n q uq,n i Berry curvature Wn (q) = q An (q) ez ky ' kx

43 Mapping'the'dispersion'relaon' Preliminary'.8'E r' Mapped'path' Energy'difference'(kHz)'' 2.5'E r' Energy'difference'at'the'K'point'<1%'of'la?ce'bandwidth!'

44 Band Topolgy Aharonov Bohm Interferometer in Momentum Space Real Space y B j AB = q h x I Z A(r)dr = q h C Z j AB = q h BdS = 2p F/F Aharonov-Bohm Phase S A(r) d 2 r

45 Band Topolgy Aharonov Bohm Interferometer in Momentum Space Real Space Momentum Space B y q y j AB = q h x I Z A(r)dr = q h C Z j AB = q h BdS = 2p F/F Aharonov-Bohm Phase S A(r) d 2 r j Berry = I C q x A n (q)dq = Z Z S q A n (r) ds q j Berry = W n (q)ds q Berry Phase

46 Band Topology Berry Curvature in Hexagonal Lattices Berry curvature concentrated to Dirac cones, alternating in sign! Breaking time reversal or inversion symmetry gaps Dirac cones and spreads Berry curvature out Hexagonal Lattice Hamiltonian H(q)= D f (q) f (q) D Expanding momenta close to K Dirac point H( q)= q x + i q y q x i q y ' Eigenstates e iq(q)/2 ± e iq(q)/2 u ± K, q = 1 2

47 Band Topology Berry Phases in Graphene Berry Phase around K-Dirac cone I j Berry,K = C A(q)dq = p Berry Phase around K -Dirac cone j Berry,K = p k y K K ' k x

48 Stückelberg Accelerating the Lattice Arbitrary accelerations in any direction can be applied!

49 Band Topology The Interferometer k y k x /2,, /2, MW t Forces applied by lattice acceleration and magnetic gradients!

50 Band Topology The Interferometer k y k x /2,, /2, MW t Forces applied by lattice acceleration and magnetic gradients!

51 Band Topology The Interferometer k y k x /2,, /2, MW t Forces applied by lattice acceleration and magnetic gradients!

52 Band Topology The Interferometer k y k x /2,, /2, MW t Forces applied by lattice acceleration and magnetic gradients!

53 Band Topology The Interferometer k y k x /2,, /2, MW t Forces applied by lattice acceleration and magnetic gradients!

54 Stückelberg Interferometry

55 Stückelberg Accelerating the Lattice Arbitrary accelerations in any direction can be applied!

56 Bloch Oscillations Stückelberg Bloch oscillations induced by accelerating the lattice weaker force "us" 1"us" 2"us" 3"us" 4"us" 5"us" 6"us" 7"us" 8"us" 9"us" 1"us" 11"us" quasi-momentum distribution stronger force "us" 5"us" 1"us" 15"us" 2"us" 25"us" 3"us" 35"us" 4"us" 45"us" 5"us" 55"us" 6"us" 65"us" 7us" 75us"

57 Stückelberg Probing the Scalar Band Structure Force" "us" 5"us" 1"us" 15"us" 4"us" 2"us" 25"us" 3"us" 45"us" 5"us" 55"us" 6"us" 65"us" 7us" 35"us" 75us" Landau Zener acts as beam splitter between bands! Interband transitions at edge

58 Stückelberg Stückelberg oscillations: Double Landau Zener j 2 E# LZ# q# j 2 (t hold ) j 1 ##E### j 1 (t hold ) Final band populations encode j = j 1 j 2 q# ##E### LZ# ##q### Stückelberg, Helv. Phys. Acta 5, 369 (1932), Shevchenko et al., Phys. Rep. 492, 1 (21), Zanesini et al., PRA 82, 6561, (21), Weitz PRL 15, (21)

59 Stückelberg Stückelberg oscillations: Double Landau Zener j 2 E# LZ# q# j 2 (t hold ) j 1 ##E### j 1 (t hold ) Final band populations encode j = j 1 j 2 q# n =(E 2 E 1 )/h ##E### LZ# ##q### Stückelberg, Helv. Phys. Acta 5, 369 (1932), Shevchenko et al., Phys. Rep. 492, 1 (21), Zanesini et al., PRA 82, 6561, (21), Weitz PRL 15, (21)

60 Stückelberg Mapping the Dispersion Relation Frequency control on all three beams allows for arbitrary accelerations! Mapped path Energy$difference$(kHz)$$.8$E r$ 2.5$E r$

61 Interferometer & Loops j 1 j = j 1 j 2 j j 2 Interferometer Phase = Phase of closed loop

62 Stückelberg Geometric Phases Berry s Phase Cyclic adiabatic evolution in parameter space of H(R) F(T )i = e i(j dyn+j Berry ) F()i j Berry = i Z C hf R FidR Aharonov-Anandan Phase Assume evolution that is cyclic in state Y(T )i = e ij Y()i j = j 1 (T )+j AA j dyn = 1 h Z T E(t)dt : geometric phase, independent of evolution speed j AA : geometric phase, independent of evolution speed Y. Aharonov & J. Anandan, Phys. Rev. Lett. 58, 1593 (1987)

63 Stückelberg Closed Loops Y(q x,q y )i Closed loop within band

64 Stückelberg Closed Loops Y(q x,q y )i Closed loop within band Y + (q x,q y )i Closed loop between bands Y (q x,q y )i

65 Stückelberg Geometric Phases in Stückelberg Interferometry j 2 j = j i j 1 = j AA i j AA 1 purely geometrical! j 1 j i = j dyn i + j AA i j dyn i = const.

66 Stückelberg Geometric Phases in Stückelberg Interferometry j 2 j = j i j 1 = j AA i j AA 1 purely geometrical! j 1 j i = j dyn i + j AA i j dyn i = const.

67 Stückelberg Path without Dirac Cone

68 Stückelberg Path without Dirac Cone

69 Band Topology Outlook ToDo: Measure amplitude of off-diagonal Berry connection momentum resolved Lattice acceleration allows for arbitrary path choice Ramsey & Stückelberg allow for a full determination of band structure Dispersion relation + full geometric tensor

70 Band Topology Outlook ToDo: Measure amplitude of off-diagonal Berry connection momentum resolved Lattice acceleration allows for arbitrary path choice Ramsey & Stückelberg allow for a full determination of band structure Dispersion relation + full geometric tensor Next: Turn band topological! Turn on interactions! Haldane PRL (1988) Lindner et al. Nat. Phys. (211) Kitagawa et al. PRB (211) Rechtsman et al. Nature (213) Aidelsburger PRL (213)

71 Band Topology Outlook ToDo: Measure amplitude of off-diagonal Berry connection momentum resolved Lattice acceleration allows for arbitrary path choice Ramsey & Stückelberg allow for a full determination of band structure Dispersion relation + full geometric tensor Next: Turn band topological! Turn on interactions! Or: Turn off interactions! Haldane PRL (1988) Lindner et al. Nat. Phys. (211) Kitagawa et al. PRB (211) Rechtsman et al. Nature (213) Aidelsburger PRL (213) Veselago lens, Klein tunneling,... Cheianov, Fal ko, Altshuler, Science (27)

72 Hall Response & Chern Number Measurement in Hofstadter Bands M. Aidelsburger et al. (in preparation) N. Goldman N. Cooper

73 Gauge Fields Artificial B-Fields with Ultracold Atoms Controlling atom tunneling along x with Raman lasers leads to effective tunnel coupling with spatially-dependent Peierls phase j(r) Ĥ = Â R Ke ij(r) â + J â + h.c. RâR+dx RâR+dy K e i 2 d x d y Magnetic flux through a plaquette: Z f = BdS = j 1 j 2 y x J K e i 1 J D. Jaksch & P. Zoller, New J. Phys. (23) F. Gerbier & J. Dalibard, New J. Phys. (21) N. Cooper, PRL (211) E. Mueller, Phys. Rev. A (24) L.-K. Lim et al. Phys. Rev. A (21) A. Kolovsky, Europhys. Lett. (211) see also: lattice shaking E. Arimondo, PRL(27), K. Sengstock, Science (211), M. Rechtsman & M. Segev, Nature (213)

74 Gauge Fields Artificial B-Fields with Ultracold Atoms Controlling atom tunneling along x with Raman lasers leads to effective tunnel coupling with spatially-dependent Peierls phase j(r) Ĥ = Â R Ke ij(r) â + J â + h.c. RâR+dx RâR+dy K e i 2 d x d y Magnetic flux through a plaquette: Z f = BdS = j 1 j 2 y x J K e i 1 J D. Jaksch & P. Zoller, New J. Phys. (23) F. Gerbier & J. Dalibard, New J. Phys. (21) N. Cooper, PRL (211) E. Mueller, Phys. Rev. A (24) L.-K. Lim et al. Phys. Rev. A (21) A. Kolovsky, Europhys. Lett. (211) see also: lattice shaking E. Arimondo, PRL(27), K. Sengstock, Science (211), M. Rechtsman & M. Segev, Nature (213)

75 Gauge Fields Artificial B-Fields with Ultracold Atoms Controlling atom tunneling along x with Raman lasers leads to effective tunnel coupling with spatially-dependent Peierls phase j(r) Ĥ = Â R Ke ij(r) â + J â + h.c. RâR+dx RâR+dy K e i 2 d x d y Magnetic flux through a plaquette: Z f = BdS = j 1 j 2 y x J K e i 1 J D. Jaksch & P. Zoller, New J. Phys. (23) F. Gerbier & J. Dalibard, New J. Phys. (21) N. Cooper, PRL (211) E. Mueller, Phys. Rev. A (24) L.-K. Lim et al. Phys. Rev. A (21) A. Kolovsky, Europhys. Lett. (211) see also: lattice shaking E. Arimondo, PRL(27), K. Sengstock, Science (211), M. Rechtsman & M. Segev, Nature (213)

76 Gauge Fields Artificial B-Fields with Ultracold Atoms Controlling atom tunneling along x with Raman lasers leads to effective tunnel coupling with spatially-dependent Peierls phase j(r) Ĥ = Â R Ke ij(r) â + J â + h.c. RâR+dx RâR+dy Magnetic flux through a plaquette: magnetic field strength f = are BdS = j 1 j 2 K e i 2 d x Fluxes corresponding to severals thousand Tesla d y Z y x J K e i 1 J possible! D. Jaksch & P. Zoller, New J. Phys. (23) F. Gerbier & J. Dalibard, New J. Phys. (21) N. Cooper, PRL (211) E. Mueller, Phys. Rev. A (24) L.-K. Lim et al. Phys. Rev. A (21) A. Kolovsky, Europhys. Lett. (211) see also: lattice shaking E. Arimondo, PRL(27), K. Sengstock, Science (211), M. Rechtsman & M. Segev, Nature (213)

77 Gauge Fields Harper Hamiltonian and Hofstadter Butterfly Harper Hamiltonian: J=K and φ uniform E/J The lowest band is topologically equivalent to the lowest Landau level. D.R. Hofstadter, Phys. Rev. B14, 2239 (1976) see alo Y. Avron, D. Osadchy, R. Seiler, Physics Today 38, 23

78 Artificial magnetic fields Experimental method Atoms in a 2D lattice Tunneling inhibited along one direction using energy offsets e.g. by using a linear potential y J x J x x

79 Artificial magnetic fields Experimental method Atoms in a 2D lattice Tunneling inhibited along one direction using energy offsets 2! 2! 1 = /~ 1 e.g. by using a linear potential y J x J x x Induce resonant tunneling with a pair of far-detuned running-wave beams Reduced heating due to spontaneous emission compared to Raman-assisted tunneling! Independent of the internal structure of the atom

80 Artificial magnetic fields Experimental method Interference creates a running-wave that modulates the lattice The phase of the modulation depends on the position in the lattice k 1, 1 k 2, 2 y Lattice modulation: VKcos(!t + (r)) with spatial-dependent phase (r) = k r x k = k 2 k 1! =! 2! 1 Realization of time-dependent Hamiltonian, where tunneling is restored Discretization of the phase due to underlying lattice m,n M. Aidelsburger et al., PRL (211); M. Aidelsburger et al., Appl. Phys. B (213)

81 Artificial magnetic fields Experimental method Time-dependent Hamiltonian: Ĥ(t) = X J x â m+1,nâm,n m,n J y â m,n+1âm,n +h.c. + X m,n m +V K cos(!t + m,n ) ˆn m,n

82 Artificial magnetic fields Experimental method Time-dependent Hamiltonian: Ĥ(t) = X J x â m+1,nâm,n m,n J y â m,n+1âm,n +h.c. + X m,n m +V K cos(!t + m,n ) ˆn m,n Can be mapped on an effective time-averaged time-independent Hamiltonian for ~! J x,j y,u Ĥ eff = X m,n Ke i m,n â m+1,nâm,n J â m,n+1âm,n +h.c. To avoid excitations to higher bands has to be smaller than the band gap ~! F. Grossmann and P. Hänggi, EPL (1992) M. Holthaus, PRL (1992) A. Kolovsky, EPL (211); A. Eckardt, PRL (25) A. Eckhardt, EPL (27); P. Hauke, PRL (212) A. Bermudez, PRL (211); A. Bermudez, NJP (212)

83 Artificial magnetic fields Experimental method Time-dependent Hamiltonian: Ĥ(t) = X J x â m+1,nâm,n m,n J y â m,n+1âm,n +h.c. + X m,n m +V K cos(!t + m,n ) ˆn m,n Note: Corrections could be important! Can be mapped on an effective time-averaged time-independent Hamiltonian for ~! J x,j y,u see e.g. N. Goldman & J. Dalibard arxiv: & related work A. Polkovnikov Ĥ eff = X m,n Ke i m,n â m+1,nâm,n J â m,n+1âm,n +h.c. To avoid excitations to higher bands has to be smaller than the band gap ~! F. Grossmann and P. Hänggi, EPL (1992) M. Holthaus, PRL (1992) A. Kolovsky, EPL (211); A. Eckardt, PRL (25) A. Eckhardt, EPL (27); P. Hauke, PRL (212) A. Bermudez, PRL (211); A. Bermudez, NJP (212)

84 Uniform flux Hamiltonian Realization of the Hofstadter-Harper Hamiltonian Ĥ = Ke i m,nâ m+1,nâm,n + Jâ m,n+1âm,n X m,n +h.c. k 2, 2 k 1, 1 J + _ 2 Scheme allows for the realization of an effective uniform flux of = /2 K B x M. Aidelsburger et al., PRL 111, (213) H. Miyake et al., PRL 111, (213)

85 Uniform flux Local observable Classical: Charged particle in magnetic field e - B

86 Uniform flux Local observable Classical: Charged particle in magnetic field e - Quantum Analogue: B Initial State: Single Atom in the ground state J D C J y i = Ai + Di p 2 of a tilted plaquette. A B

87 Uniform flux Local observable Classical: Charged particle in magnetic field e - Quantum Analogue: B Initial State: Single Atom in the ground state J D C J y i = Ai + Di p 2 of a tilted plaquette. A B Switch on running-wave to induce tunneling J D A K e i K C J B

88 Uniform flux A Lattice of Plaquettes Using two superlattices, we realize a lattice whose elementary cell is a 4-site plaquette.

89 Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =1,m F = 1i B-field gradient tilted lattice potential

90 Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =1,m F = 1i #i = F =2,m F = 1i B-field gradient B-field gradient tilted lattice potential tilted lattice potential

91 Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =1,m F = 1i #i = F =2,m F = 1i B-field gradient B-field gradient tilted lattice potential tilted lattice potential Spin-dependent optical potential: ()

92 Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =1,m F = 1i #i = F =2,m F = 1i B-field gradient B-field gradient tilted lattice potential tilted lattice potential Spin-dependent optical potential: () Spin-dependent complex tunneling amplitudes: Ke i mn () Ke i mn

93 Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =1,m F = 1i #i = F =2,m F = 1i B-field gradient B-field gradient tilted lattice potential tilted lattice potential Spin-dependent optical potential: () Spin-dependent complex tunneling amplitudes: Ke i mn () Ke i mn Spin-dependent effective magnetic field: = /2 () = /2

94 Uniform flux Quantum Spin Hall Hamiltonian Time-reversal-symmetric quantum spin Hall Hamiltonian: Ĥ ",# = Ke ±i m,nâ m+1,nâm,n + Jâ m,n+1âm,n X m,n +h.c. Bernevig and Zhang, PRL 96, 1682 (26); N. Goldman et al., PRL (21)

95 Uniform flux Quantum Spin Hall Hamiltonian Time-reversal-symmetric quantum spin Hall Hamiltonian: Ĥ ",# = Ke ±i m,nâ m+1,nâm,n + Jâ m,n+1âm,n X m,n +h.c. k 2, 2 k 1, 1 + _ 2 J K B x " = /2 Bernevig and Zhang, PRL 96, 1682 (26); N. Goldman et al., PRL (21)

96 Uniform flux Quantum Spin Hall Hamiltonian Time-reversal-symmetric quantum spin Hall Hamiltonian: Ĥ ",# = Ke ±i m,nâ m+1,nâm,n + Jâ m,n+1âm,n X m,n +h.c. k 2, 2 k 1, 1 + _ 2 - _ 2 J K B x B x " = /2 # = /2 Bernevig and Zhang, PRL 96, 1682 (26); N. Goldman et al., PRL (21)

97 Uniform flux Spin-dependent cyclotron orbit Spin up: D C A B v "i =( Ai + Di)/ p 2 Y dy Time (ms) X d x

98 Uniform flux Spin-dependent cyclotron orbit Spin up: D C A B v "i =( Ai + Di)/ p 2 Y dy Time (ms) X d x Spin down: D C A B #i =( Bi + Ci)/ p 2 v

99 Uniform flux Spin-dependent cyclotron orbit Spin up: D C A B v "i =( Ai + Di)/ p 2 Y dy Time (ms) X d x Spin down:.1 D A C B Y dy -.1 #i =( Bi + Ci)/ p 2 v -.2 Time (ms) X d x

100 Uniform flux Spin-dependent cyclotron orbit Spin up:.1 D C A B v "i =( Ai + Di)/ p 2 Y dy -.1 Time (ms) X d x Opposite chirality! Spin down:.1 D A C B Y dy -.1 #i =( Bi + Ci)/ p 2 v -.2 Time (ms) X d x

101 Uniform Flux New Flux Rectification Scheme a r2 b2 b a r1 a 3 /2 3 /2 3 /2 y b1 x J y J x J /2 J /2 /2 red & blue bonds can be modulated independently! Flux rectification possible (related Gerbier & Dalibard NJP (21))

102 Uniform Flux Loading and Probing Hofstadter Bands a y x b 1 Energy (a.u.).5 k y ( /a) k x ( /a).5 k y ( /a) Topologically trivial.5 k x ( /a) k y ( /a) k y ( /a) Detuning (J) k x ( /a).5 Topologically non-trivial k x ( /a) 3 =1 2 =-2 1 =1.5 Hofstadter model for /2 Key insight for adiabatic loading/probing: keep Brillouin zone of topologically trivial & non-trivial phase matched! Flat bands realized! E gap /E bw ' 8

103 Uniform Flux Hall Response & Anomalous Velocity F y v x 26 h dk c dt = e f(r c )+ dr c dt B(r c ) dr c dt = 1 h k c e n (k c ) dk c dt W(k c )ẑ Karplus & Luttinger, Phys. Rev. (1954) Sundaram & Niu, Phys. Rev. B (1999) anomalous velocity

104 Uniform Flux Cloud Deflection & Chern Number F y v x v(k c )= dr c dt = 1 h k c e n (k c ) dk c dt W(k c )ẑ

105 Uniform Flux Cloud Deflection & Chern Number F y v x v(k c )= dr c dt = 1 h k c e n (k c ) dk c dt W(k c )ẑ Assume uniformly filled band and rational flux p/q: hv x i = 1 N at Z = 1 N at 2p = (2p) 2 = F y qa 2 h r(k)v x (k)d 2 k N at dk x dk y Fy h 1 2p Z 1 (2p/a)(2p/qa) Fy h n W(k)d 2 k n H. Price & N. Cooper PRA (212) A. Dauphin & N. Goldman PRL (213)

106 Uniform Flux Understanding the Displacement Only lowest band homogenously populated x(t)= 4a2 F h n 1t Higher bands also populated (short times) x(t)= 4a2 g F h n 1 t, with g = h 1 h 2 + h 3 Filling factor (make use of particle-hole symmetry in Hofstadter model & Â µ Higher bands also populated (full time evolution) Z t x(t)= n 1 4a 2 F h g(t )dt n µ = But how to measure band-population???

107 Topology Conclusions First realization of topological Bloch bands in 1d & 2d (SSH, Hoftstadter, Quantum Spin Hall) First direct measurements of topological invariants in 1d & 2d Aharonov-Bohm interferometric determination of band topology Stückelberg Interferometry for dispersion relation Extensions to non-abelian Berry connection (Wilson loops) Framework for full determination of band geometry and experimental implementation! Questions: interactions, heating, adiabatic loading,...

108 Outlook Rectified Flux, Hofstadter Butterfly Novel Correlated Phases in Strong Fields, Transport Measurements Adiabatic loading schemes Spectroscopy of Hoftstadter bands Novel Topological Insulators Image Edge States - directly/spectroscopically Measure spatially resolved full current distribution Non-equilibrium dynamics in gauge fields Thermalization?

109 Gauge Field Team From left to right: Christian Schweizer Monika Aidelsburger I.B. Michael Lohse Marcos Atala Julio Barreiro

110 2D Berry Curvature Interferometer Team Lucia Duca Tracy Li Martin Reitter Monika Schleier-Smith IB Ulrich Schneider

111 Single Atom Team

112 Venice VIU 214

113