Engineering and Probing Topological Bloch Bands in Optical Lattices

Transcription

1 Engineering and Probing Topological Bloch Bands in Optical Lattices Monika Aidelsburger, Marcos Atala, Michael Lohse, Christian Schweizer, Julio Barreiro Christian Gross, Stefan Kuhr, Manuel Endres, Marc Cheneau, Takeshi Fukuhara, Peter Schauss, Sebastian Hild, Johannes Zeiher Ulrich Schneider, Simon Braun, Philipp Ronzheimer, Michael Schreiber, Tim Rom, Sean Hodgman Ulrich Schneider, Monika Schleier-Smith, Lucia Duca, Tracy Li, Martin Reitter, Josselin Bernadoff, Henrik Lüschen Ahmed Omran, Martin Boll, Timon Hilker, Michael Lohse, Thomas Reimann, Alexander Keesling, Christian Gross Simon Fölling, Francesco Scazza, Christian Hofrichter, Pieter de Groot, Moritz Höfer Christoph Gohle, Tobias Schneider, Nikolaus Buchheim, Zhenkai Lu Humboldt Research Awardees: N. Cooper, C. Salomon, W. Ketterle Max-Planck-Institut für Quantenoptik Ludwig-Maximilians Universität funding by MPG, European Union, DFG $ DARPA (OLE)

2 Outline Realizing Artificial Gauge Fields Realizing the Hoftstadter & Quantum Spin Hall Hamiltonian Meissner -currents in bosonic flux ladders Outline Realizing Artificial Gauge Fields Realizing the Hoftstadter & Quantum Spin Hall Hamiltonian Meissner -currents in bosonic flux ladders Probing Topological Features of Bloch Bands 3 Probing Zak Phases in Topological Bands 4 An Aharonov Bohm Interferometer for measuring Berry curvature

3 Realizing Artificial Gauge Fields in Optical Lattices Gauge Fields Quantum Hall Effect in D Electron Gases Integer Quantum Hall Effect E s xy = ne /h n Integer LLL c Chern Insulators (w/o magnetic field see D. Haldane 988) Fractional Quantum Hall Effect Laughlin state at = /3 flux quantum =h/ec electron

4 Gauge Fields State of the Art ) Rotation In rapidly rotating gases, Coriolis force is equivalent to Lorentz force. ) Raman Induced Gauge Fields F L = qv B F C = mv W rot K. Madison et al., PRL () J.R. Abo-Shaeer et al. Science () Spatially dependent optical couplings lead to a Berry phase analoguous to the Aharonov-Bohm phase Y. Lin et al., Nature (9) Gauge Fields State of the Art ) Rotation ) Raman Induced Gauge Fields In rapidly rotating gases, Coriolis force is equivalent to Lorentz force. F L = qv B Problem in both cases: small B-fields (large ν> for now), heating... F C = mv W rot K. Madison et al., PRL () J.R. Abo-Shaeer et al. Science () Spatially dependent optical couplings lead to a Berry phase analoguous to the Aharonov-Bohm phase Y. Lin et al., Nature (9)

5 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 RâR+dx + J â RâR+dy + h.c. K e i d x d y Magnetic flux through a plaquette: Z f = BdS = j j y x J K e i J D. Jaksch & P. Zoller, New J. Phys. (3) F. Gerbier & J. Dalibard, New J. Phys. () N. Cooper, PRL () E. Mueller, Phys. Rev. A (4) L.-K. Lim et al. Phys. Rev. A () A. Kolovsky, Europhys. Lett. () see also: lattice shaking E. Arimondo, PRL(7), K. Sengstock, Science (), M. Rechtsman & M. Segev, Nature (3) 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 RâR+dx + J â RâR+dy + h.c. K e i d x d y Magnetic flux through a plaquette: Z f = BdS = j j y x J K e i J D. Jaksch & P. Zoller, New J. Phys. (3) F. Gerbier & J. Dalibard, New J. Phys. () N. Cooper, PRL () E. Mueller, Phys. Rev. A (4) L.-K. Lim et al. Phys. Rev. A () A. Kolovsky, Europhys. Lett. () see also: lattice shaking E. Arimondo, PRL(7), K. Sengstock, Science (), M. Rechtsman & M. Segev, Nature (3)

6 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 RâR+dx + J â RâR+dy + h.c. K e i d x d y Magnetic flux through a plaquette: Z f = BdS = j j y x J K e i J D. Jaksch & P. Zoller, New J. Phys. (3) F. Gerbier & J. Dalibard, New J. Phys. () N. Cooper, PRL () E. Mueller, Phys. Rev. A (4) L.-K. Lim et al. Phys. Rev. A () A. Kolovsky, Europhys. Lett. () see also: lattice shaking E. Arimondo, PRL(7), K. Sengstock, Science (), M. Rechtsman & M. Segev, Nature (3) 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 RâR+dx + J â RâR+dy + h.c. y x J d x Fluxes corresponding to severals thousand Tesla Z d y magnetic field strength f = are BdS = j j possible! K e i K e i J Magnetic flux through a plaquette: D. Jaksch & P. Zoller, New J. Phys. (3) F. Gerbier & J. Dalibard, New J. Phys. () N. Cooper, PRL () E. Mueller, Phys. Rev. A (4) L.-K. Lim et al. Phys. Rev. A () A. Kolovsky, Europhys. Lett. () see also: lattice shaking E. Arimondo, PRL(7), K. Sengstock, Science (), M. Rechtsman & M. Segev, Nature (3)

7 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. B4, 39 (976) see alo Y. Avron, D. Osadchy, R. Seiler, Physics Today 38, 3 Artificial magnetic fields Experimental method Atoms in a D lattice Tunneling inhibited along one direction using energy offsets e.g. by using a linear potential y J x J x x M. Aidelsburger et al., PRL (); M. Aidelsburger et al., Appl. Phys. B (3)

8 Artificial magnetic fields Experimental method Atoms in a D lattice Tunneling inhibited along one direction using energy offsets!! = /~ 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 M. Aidelsburger et al., PRL (); M. Aidelsburger et al., Appl. Phys. B (3) 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, k, y Lattice modulation: VKcos(!t + (r)) with spatial-dependent phase (r) = k r x k = k k! =!! Realization of time-dependent Hamiltonian, where tunneling is restored Discretization of the phase due to underlying lattice m,n M. Aidelsburger et al., PRL (); M. Aidelsburger et al., Appl. Phys. B (3)

9 Artificial magnetic fields Time-dependent Hamiltonian: Ĥ(t) = X J x â m+,nâm,n m,n Experimental method J y â m,n+âm,n +h.c. + X m,n m + V K cos(!t + m,n ) ˆn m,n Artificial magnetic fields Time-dependent Hamiltonian: Ĥ(t) = X J x â m+,nâm,n m,n Experimental method J y â m,n+âm,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+,nâm,n J â m,n+âm,n +h.c. To avoid excitations to higher bands has to be smaller than the band gap ~! F. Grossmann and P. Hänggi, EPL (99) M. Holthaus, PRL (99) A. Kolovsky, EPL (); A. Eckardt, PRL (5) A. Eckhardt, EPL (7); P. Hauke, PRL () A. Bermudez, PRL (); A. Bermudez, NJP ()

10 Artificial magnetic fields Time-dependent Hamiltonian: Ĥ(t) = X J x â m+,nâm,n m,n Experimental method J y â m,n+âm,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 see e.g. time-averaged N. Goldman & time-independent J. Dalibard arxiv: Hamiltonian for ~! J x,j y,u & related work A. Polkovnikov Ĥ eff = X m,n Ke i m,n â m+,nâm,n J â m,n+âm,n +h.c. To avoid excitations to higher bands has to be smaller than the band gap ~! F. Grossmann and P. Hänggi, EPL (99) M. Holthaus, PRL (99) A. Kolovsky, EPL (); A. Eckardt, PRL (5) A. Eckhardt, EPL (7); P. Hauke, PRL () A. Bermudez, PRL (); A. Bermudez, NJP () Artificial magnetic fields Effective coupling strength Effective coupling strength: J (x) : Bessel-functions of the first kind and K = J x J (x) J = J y J (x) x = f( )V K K/Jx J/Jy, x : Phase difference of the modulation between neighboring bonds.5 f () see also: H. Lignier et al. PRL (7)

11 Artificial magnetic fields Experimental setup Experimental parameters: d d k, y / k ' k = d m,n = (m + n) k, V K V K / x Artificial magnetic fields Experimental setup Experimental parameters: d d k, y / k ' k = d m,n = (m + n) k, Flux through one unit cell: V K = m,n+ m,n = V K / x depends only on phase difference along y!

12 Artificial magnetic fields Experimental setup Experimental parameters: d d k, y / k ' k = d m,n = (m + n) k, Flux through one unit cell: V K = m,n+ m,n = V K / x depends only on phase difference along y! The value of the flux is fully tunable by changing the geometry of the driving-beams! Uniform flux Laser-assisted tunneling Study laser-assisted tunneling in the presence of a magnet field gradient 87 Initial state: atoms ( Rb) in 3D lattice only populate even sites even odd B x M. Aidelsburger et al., PRL, 853 (3) similar work: H. Miyake et al., PRL, 853 (3)

13 Uniform flux Laser-assisted tunneling Study laser-assisted tunneling in the presence of a magnet field gradient 87 Initial state: atoms ( Rb) in 3D lattice only populate even sites even odd B x M. Aidelsburger et al., PRL, 853 (3) similar work: H. Miyake et al., PRL, 853 (3) Uniform flux Laser-assisted tunneling Study laser-assisted tunneling in the presence of a magnet field gradient 87 Initial state: atoms ( Rb) in 3D lattice only populate even sites even odd B x Atom population in odd sites vs. modulation frequency.5 n odd () (khz) M. Aidelsburger et al., PRL, 853 (3) similar work: H. Miyake et al., PRL, 853 (3)

14 Uniform flux Laser-assisted tunneling Study laser-assisted tunneling in the presence of a magnet field gradient 87 Initial state: atoms ( Rb) in 3D lattice only populate even sites even odd res/() (khz) 8 4 B x 3 4 B (mg/m) Atom population in odd sites vs. modulation frequency.5 n odd () (khz) Large range of values accessible! M. Aidelsburger et al., PRL, 853 (3) similar work: H. Miyake et al., PRL, 853 (3) Uniform flux Hamiltonian Realization of the Hofstadter-Harper Hamiltonian Ĥ = Ke i m,nâ m+,nâm,n + Jâ m,n+âm,n +h.c. X m,n k, k, J + _ Scheme allows for the realization of an effective uniform flux of = / K B x M. Aidelsburger et al., PRL, 853 (3) similar work: H. Miyake et al., PRL, 853 (3)

15 Uniform flux Local observable Classical: Charged particle in magnetic field e - B 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 of a tilted plaquette. A B

16 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 of a tilted plaquette. A B Switch on running-wave to induce tunneling J D A K e i K C J B Uniform flux A Lattice of Plaquettes Using two superlattices, we realize a lattice whose elementary cell is a 4-site plaquette.

17 Uniform flux Mean atom position Site resolved detection along one direction S. Fölling et al., Nature (7); J. Sebby-Strabley et al., PRL (7) Uniform flux Mean atom position Site resolved detection along one direction bandmapping bandmapping S. Fölling et al., Nature (7); J. Sebby-Strabley et al., PRL (7) Site resolved detection in plaquettes D C A B

18 Uniform flux Mean atom position Site resolved detection along one direction bandmapping S. Fölling et al., Nature (7); J. Sebby-Strabley et al., PRL (7) Site resolved detection in plaquettes D C A B Mean atom position along x and y hxi = N A + N B + N C N D d x N and hy i = N A N B + N C + N D d y N Uniform flux Cyclotron orbit Quantum analogue of cyclotron orbit. D A v C B Ydy -. Time (ms) Xd x Parameters: J/h =.5kHz K/h =.3kHz D/h = 4.5kHz

19 Uniform flux Uniformity of the flux Observation of the uniformity of the effective flux: Superlattice potential shifted by one lattice constant Uniform flux Uniformity of the flux Observation of the uniformity of the effective flux: Superlattice potential shifted by one lattice constant

20 Uniform flux Uniformity of the flux Observation of the uniformity of the effective flux: Superlattice potential shifted by one lattice constant. Ydy -. Time (ms) Xd x Uniform flux Uniformity of the flux Observation of the uniformity of the effective flux: Superlattice potential shifted by one lattice constant Ydy. -. Time (ms) Xd x Ydy. -. Time (ms) Xd x

21 Uniform flux Uniformity of the flux Observation of the uniformity of the effective flux: Superlattice potential shifted by one lattice constant Ydy. -. Time (ms) Xd x Same chirality!! Ydy. -. Time (ms) Xd x Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =,m F = i B-field gradient tilted lattice potential

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

23 Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =,m F = i #i = F =,m F = i 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 Uniform flux Quantum Spin Hall Hamiltonian Value of the flux depends on the internal state of the atom "i = F =,m F = i #i = F =,m F = i 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: = / () = /

24 Uniform flux Quantum Spin Hall Hamiltonian Time-reversal-symmetric quantum spin Hall Hamiltonian: Ĥ ",# = Ke ±i m,nâ m+,nâm,n + Jâ m,n+âm,n X m,n +h.c. Bernevig and Zhang, PRL 96, 68 (6); N. Goldman et al., PRL () Uniform flux Quantum Spin Hall Hamiltonian Time-reversal-symmetric quantum spin Hall Hamiltonian: Ĥ ",# = Ke ±i m,nâ m+,nâm,n + Jâ m,n+âm,n X m,n +h.c. k, k, J + _ K B x " = / Bernevig and Zhang, PRL 96, 68 (6); N. Goldman et al., PRL ()

25 Uniform flux Quantum Spin Hall Hamiltonian Time-reversal-symmetric quantum spin Hall Hamiltonian: Ĥ ",# = Ke ±i m,nâ m+,nâm,n + Jâ m,n+âm,n X m,n +h.c. k, k, J + _ - _ K B x B x " = / # = / Bernevig and Zhang, PRL 96, 68 (6); N. Goldman et al., PRL () Uniform flux Spin-dependent cyclotron orbit Spin up: D C A B v "i =( Ai + Di)/ p Ydy. -. Time (ms) Xd x

26 Uniform flux Spin-dependent cyclotron orbit Spin up: D C A B v "i =( Ai + Di)/ p Ydy. -. Time (ms) Xd x Spin down: D C A v B #i =( Bi + Ci)/ p Uniform flux Spin-dependent cyclotron orbit Spin up: D C A B v "i =( Ai + Di)/ p Ydy. -. Time (ms) Xd x Spin down:. D C A B v #i =( Bi + Ci)/ p Ydy Time (ms) Xd x

27 Uniform flux Spin-dependent cyclotron orbit Spin up:. D C A B v "i =( Ai + Di)/ p Ydy -. Time (ms) Xd x Opposite chirality! Spin down:. D C A B v #i =( Bi + Ci)/ p Ydy Time (ms) Xd x Observation of chiral currents in bosonic flux ladders M. Atala et al., arxiv:4.89

28 Flux Ladder Flux ladder: experimental realization i i 3 i i k, i i resonant laser-assisted tunneling:!! = /~ Spatial dependent phase factors k, y J 3 i i i i K i i 3 i i 3 i i i i Uniform flux n = n / = / x Experiment: M. Atala et al., arxiv:4.89 (4) Theory: D. Hügel, B. Paredes, PRA 89, 369 (4) E. Orignac & T. Giamarchi PRB 64, 4455 () A. Tokuno & A. Georges arxiv:43.43 R. Wei & E. Mueller arxiv:45.3 Flux Ladder Flux ladder: experimental realization k, y J i i 3 i i i i K 3 i i i i 3 i i k, i i i Minimal 3 System for orbital magnetism i i i resonant laser-assisted tunneling:!! = /~ Spatial dependent phase factors Uniform flux n = n / = / x Experiment: M. Atala et al., arxiv:4.89 (4) Theory: D. Hügel, B. Paredes, PRA 89, 369 (4) E. Orignac & T. Giamarchi PRB 64, 4455 () A. Tokuno & A. Georges arxiv:43.43 R. Wei & E. Mueller arxiv:45.3

29 Flux Ladder Flux Ladder Hamiltonian Hamiltonian of the system written in a simpler theory gauge d x L R d y J e -i J e -i J e -i J e i J e i J e i H = J X` K X` e i`' â `+;Lâ`;L + e i`' â `+;Râ`;R â `;Lâ`;R +h.c. J e -i J e i J e -i J e i Flux: =' K Flux Ladder Flux Ladder Hamiltonian Hamiltonian of the system written in a simpler theory gauge d x L R d y J e -i J e -i J e -i J e i J e i J e i H = J X` K X` e i`' â `+;Lâ`;L + e i`' â `+;Râ`;R â `;Lâ`;R +h.c. J e -i J e i J e -i J e i Flux: =' Define: K â q;µ = X` e iq`â`;µ, and solve for the ansatz q i =( q â q;l + qâ q;r ) i

30 Flux Ladder Ladder Band Structure q =Jcos(q)cos(') ± q K 4J sin (')sin (q) d x L R d y J e -i J e i K Flux Ladder Ladder Band Structure q =Jcos(q)cos(') ± Two energy bands q K 4J sin (')sin (q) d x L R d y J e -i J e i K

31 Flux Ladder Ladder Band Structure q =Jcos(q)cos(') ± Two energy bands q K 4J sin (')sin (q) =' = / K/J= K/J=. d y L d x R Energy (J) q (/d y ) Energy (J) q (/d y ) K/J=.8 K/J= J e -i K J e i Energy (J) - - Energy (J) q (/d y ) q (/d y ) Flux Ladder Ladder Band Structure q =Jcos(q)cos(') ± Two energy bands q K 4J sin (')sin (q) =' = / K/J= K/J=. d y L d x R Energy (J) Energy (J) Critical point at K/J = p q (/d y ) q (/d y ) - K/J=.8 K/J= J e -i K J e i Energy (J) - - Energy (J) q (/d y ) q (/d y )

32 Flux Ladder Probability Currents in Ladder d x L R d y Current along the legs: ĵ ỳ ;µ = i â `+;µâ`;µh`!`+;µ ~ h.c µ = (L=left,R=right) J e -i K J e i In the experiment total current is measured j L = N leg Âj y l;l l Chiral current: j C = j L j R Flux Ladder Phase Diagram Chiral current j C (a.u.) Meissner phase K/J.5.5 Vortex phase Flux see E. Orignac & T. Giamarchi PRB 64, 4455 ()

33 Flux Ladder Phase Diagram Chiral current j C (a.u.) Meissner phase K/J.5.5 Vortex phase j C (a.u.) K/J= Flux C Flux see E. Orignac & T. Giamarchi PRB 64, 4455 () Flux Ladder Phase Diagram Chiral current j C (a.u.) Meissner phase Experiment K/J.5 (K/J)C.5 Vortex phase j C (a.u.) K/J= Flux j C (a.u.) C Flux see E. Orignac & T. Giamarchi PRB 64, 4455 ()

34 Flux Ladder Phase Diagram Chiral Chiral current current j C (a.u.) j C (a.u.) Experiment.5 Meissner Meissner phase phase Experiment 4 Meissner phase L R K/J K/J Vortex Vortex phase phase (K/J)C (K/J)C K/J 3 Vortex phase j C (a.u.) j C (a.u.) K/J= Flux K/J=..4 C C Flux.6.8 Flux j C (a.u.) j C (a.u.) Atom density (a.u.) see E. Orignac & T. Giamarchi PRB 64, 4455 () Flux ladder Spin-orbit coupling - short digression The flux ladder Hamiltonian can be mapped into a spin-orbit coupled system Left right legs are mapped into pseudo-spins: â`;r! â`;# â`;l! â`;" Spin-Momentum locking: D. Hügel, B. Paredes, PRA 89, 369 (4) Continuum: I. B. Spielman Nature 47, 83 ()

35 Flux ladder Spin-orbit coupling - short digression The flux ladder Hamiltonian can be mapped into a spin-orbit coupled system Left right legs are mapped into pseudo-spins: â`;r! â`;# â`;l! â`;" k, L = "i R = #i k, Spin-Momentum locking: D. Hügel, B. Paredes, PRA 89, 369 (4) Continuum: I. B. Spielman Nature 47, 83 () Flux ladder Spin-orbit coupling - short digression The flux ladder Hamiltonian can be mapped into a spin-orbit coupled system Left right legs are mapped into pseudo-spins: â`;r! â`;# â`;l! â`;" k, k, L = "i R = #i L = "i R = #i k, k, Spin-Momentum locking: D. Hügel, B. Paredes, PRA 89, 369 (4) Continuum: I. B. Spielman Nature 47, 83 ()

36 Spin-orbit coupling - short digression Flux ladder The flux ladder Hamiltonian can be mapped into a spin-orbit coupled system Left right legs are mapped into pseudo-spins: a `;R! a `;# a `;L! a `;" q K/J -qk/j Spin-Momentum locking: D. Hügel, B. Paredes, PRA 89, 369 (4) Continuum: I. B. Spielman Nature 47, 83 () Current Measurements: Sequence Flux Ladder How to measure currents in our setup? project the state into isolated double wells S. Trotzky et al. Nature Physics 8, 35 () S. Kessler & F. Marquardt, arxiv: () J K Groundstate Josephson oscillations in double wells

37 Flux Ladder Current Measurements: Sequence How to measure currents in our setup? project the state into isolated double wells S. Trotzky et al. Nature Physics 8, 35 () S. Kessler & F. Marquardt, arxiv: () J Project Isolated doublewells K Groundstate Josephson oscillations in double wells Flux Ladder Current Measurements: Sequence How to measure currents in our setup? project the state into isolated double wells S. Trotzky et al. Nature Physics 8, 35 () S. Kessler & F. Marquardt, arxiv: () J Project Isolated doublewells Hold in double wells for a time t K Groundstate Josephson oscillations in double wells

38 Flux Ladder Current Measurements: Sequence How to measure currents in our setup? project the state into isolated double wells S. Trotzky et al. Nature Physics 8, 35 () S. Kessler & F. Marquardt, arxiv: () Project Hold in double wells for a time t J Isolated doublewells Measure even-odd population in double well K Groundstate Josephson oscillations in double wells Flux Ladder Double well oscillations - currents In the experiment we measure the average of all the oscillations on either side of the ladder: n even;µ (t) = [ + (n even;µ() n odd;µ ())cos(!t) j µ J/~ sin(!t)] L R Even site Odd site Even site Odd site

39 Flux Ladder Oscillations in double wells Prepare ground state of the flux ladder with K/J= and project into isolated double wells.6 n odd;l n odd;r Time (ms) Flux Ladder Oscillations in double wells Prepare ground state of the flux ladder with K/J= and project into isolated double wells n odd;l n odd;r Time (ms) n odd;l n odd;r Reversed flux Time (ms) When inverting the flux the current gets reversed

40 Zero flux ladders Prepare ground state of the ladder with zero flux project into isolated double wells.6 n odd;l n odd;r Time (ms) Flux Ladder Extracting the Chiral current The chiral current can be reliably calculated by n even;µ (t) = [ + (n even;µ() n odd;µ ())cos(!t) j µ J/~ sin(!t)] n even;l (t) n even;r (t) = j C J/~ sin(!t)

41 Flux Ladder Extracting the Chiral current The chiral current can be reliably calculated by n even;l (t) n even;r (t) = j C J/~ sin(!t) Chiral current j C (a.u.)..8.4 Vortex phase Meissner phase 3 K/J Flux Ladder Experimental Results - Momentum Distribution.5 Vortex phase Meissner phase Peak separation (/d y ) Momentum k y (/d y ) K/J 3 3 K/J

42 Summary and Outlook New detection method for probability currents Measurement of Chiral Edge States in Ladders Identification of Meissner-like effect in bosonic ladder Outlook: Entering the strongly correlated regime Chiral Mott Insulators Spin Meissner effect Connection of chiral ladder states to Hoftstadter model edge states Spin-Orbit Coupling in D E. Orignac & T. Giamarchi PRB 64, 4455 () Dhar, A et al., PRA 85, 46 () Petrescu, A. & Le Hur, K. PRL, 56 (3) A Tokuno & A Georges, arxiv:43.43 Probing Band Topology

43 Measuring the Zak-Berry s Phase of Topological Bands M. Atala et al., Nature Physics (3) Berry Phase Berry Phase in Quantum Mechanics (R)! e i(' Berry+' dyn ) (R) Adiabatic evolution through closed loop I ' Berry = C I ' Berry = I A n (R)dR = i hn(r) r R n(r)i dr C A n (R) da Berry Phase M.V. Berry, Proc. R. Soc. A (984) Berry connection A n (R) =ihn(r) r R n(r)i Berry curvature n,µ µ A n,µ Example: Spin-/ particle in magnetic field

44 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 kx Mention Problem with going on a line is generally NOT A LOOP IN PARAMETER SPACE! 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 kx kx Mention Problem with going on a line is generally NOT A LOOP IN PARAMETER SPACE!

45 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 kx kx kx Mention Problem with going on a line is generally NOT A LOOP IN PARAMETER SPACE! 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 n Chern = I BZ kx A k dk = Z BZ kx k d k xy = n Chern e /h Chern Number (Topological Invariant) Quantized Hall Conductance Thouless, Kohmoto, den Nijs, and Nightingale (TKNN), PRL 98 Kohmoto Ann. of Phys. 985 Mention Problem with going on a line is generally NOT A LOOP IN PARAMETER SPACE!

46 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 n Chern = I BZ kx A k dk = Z BZ kx k d k xy = n Chern e /h Chern Number (Topological Invariant) Quantized Hall Conductance Thouless, Kohmoto, den Nijs, and Nightingale (TKNN), PRL 98 Kohmoto Ann. of Phys. 985 What is the extension to D? Mention Problem with going on a line is generally NOT A LOOP IN PARAMETER SPACE! Berry Phase Zak Phase D Brillouin Zone ky kx going straight means going around! Band structure has torus topology!

47 Berry Phase Zak Phase D Brillouin Zone ky kx going straight means going around! Band structure has torus topology! ' Zak = i Z k +G k hu k u k i dk Zak Phase - the D Berry Phase J. Zak, Phys. Rev. Lett. 6, 747 (989) Berry Phase Zak Phase D Brillouin Zone ky kx going straight means going around! Band structure has torus topology! ' Zak = i Z k +G k hu k u k i dk Zak Phase - the D Berry Phase J. Zak, Phys. Rev. Lett. 6, 747 (989) Non-trivial Zak phase: Topological Band Edge States (for finite system) Domain walls with fractional quantum numbers R. Jackiw and C. Rebbi, Phys. Rev. D 3, 3398 (976) J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (98)

48 Berry Phase Su-Shrieffer-Heeger Model (SSH) Polyacetylene W. P. Su, J. R. Schrieffer & A. J. Heeger Phys. Rev. Lett. 4, 698 (979). H SSH = Â n Jâ n ˆb n + J â n ˆb n + h.c. Berry Phase Su-Shrieffer-Heeger Model (SSH) Polyacetylene W. P. Su, J. R. Schrieffer & A. J. Heeger Phys. Rev. Lett. 4, 698 (979). H SSH = Â n Jâ n ˆb n + J â n ˆb n + h.c. Two topologically distinct phases: D: J>J D: J >J J J J J J J J J J J

49 Berry Phase Su-Shrieffer-Heeger Model (SSH) Polyacetylene W. P. Su, J. R. Schrieffer & A. J. Heeger Phys. Rev. Lett. 4, 698 (979). H SSH = Â n Jâ n ˆb n + J â n ˆb n + h.c. Two topologically distinct phases: D: J>J D: J >J J J J J J J J J J J ' Zak = ' D Zak ' D Zak = Topological properties: domain wall features fractionalized excitations Zak phase difference ' Zak is gauge-invariant Berry Phase SSH Energy Bands - Eigenstates V(x) x d a n- b n- a n b n a n+ b n+ J J J J J J J

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

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

52 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/ Eigenstates k, = p ± e i k k, /4 Energy k /4 /d /d /d /d Adiabatic evolution in momentum space Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/ Eigenstates k, = p ± e i k k, /4 Energy k /4 /d /d /d /d ' Zak = i Z k +G k ( k@ k k + k@ k k ) dk Zak Phase SSH Model

53 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/ Eigenstates k, = p ± e i k k, /4 Energy k /4 /d /d /d /d ' Zak = Z G+k k k dk Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/ Eigenstates k, = p ± e i k k, /4 D: Energy k /4 /d /d /d /d D: J>J ' D Zak =

54 Berry Phase SSH Energy Bands - Eigenstates...ABABA... Lattice Structure... X x = X ( e ikx k x x k e ikd/ Eigenstates k, = p ± e i k k, /4 D: Energy k /4 D: /d /d /d /d D: J>J ' D Zak = D: J >J ' D Zak = Berry Phase Realization with ultracold atoms H SSH = Â n Ja nb n + J a nb n + h.c. 767 nm 534 nm D: J>J D: J >J Phase shift J J J J J J J J J J ' Zak = ' D Zak ' D Zak =

55 Berry Phase D: J>J Measuring the Berry-Zak Phase (SSH Model) Spin-dependent Bloch oscillations + Ramsey interferometry /4 Energy k /4 /d /d /d /d Prepare BEC in state,ki = #, i, with =", # Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J /4 Energy k /4 /d /d /d /d MW /-pulse Create coherent superposition p ( ", i + #, i)

56 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J /4 Energy k /4 /d /d /d /d MW /-pulse Create coherent superposition p ( ", i + #, i) Spin components with opposite magnetic moments! Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J /4 Energy k /4 /d /d /d /d Closed Loop in k-space Apply magnetic field gradient! adiabatic evolution in momentum space

57 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J /4 Energy k /4 /d /d /d /d Closed Loop in k-space Apply magnetic field gradient! adiabatic evolution in momentum space Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J /4 Energy k /4 /d /d /d /d ' Zak = ' D Zak + ' dyn + ' Zeeman

58 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J /4 Energy k /4 /d /d ' Zak = ' D Zak + ' X dyn + ' Zeeman /d /d Z ' dyn = E(t)/~ dt E(k) =E( k) Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J D: J >J /4 Energy k /4 /d /d /d /d MW -pulse Apply Spin-Echo pulse + dimerization change

59 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J>J D: J >J /4 Energy k /4 /d /d /d /d MW -pulse Apply Spin-Echo pulse + dimerization change Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J >J /4 Energy k /4 /d /d /d /d Apply magnetic field gradient! adiabatic evolution in momentum space

60 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J >J /4 Energy k /4 /d /d /d /d Apply magnetic field gradient! adiabatic evolution in momentum space Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J >J /4 Energy k /4 /d /d /d /d ' Zak = ' D Zak ' D Zak + ' Zeeman

61 Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J >J /4 Energy k /4 /d /d /d /d ' Zak = ' D Zak Spin-Echo pulse ' D Zak + ' Zeeman X Berry Phase Measuring the Berry-Zak Phase (SSH Model) D: J >J /4 Energy k /4 /d /d /d /d MW /-pulse, with phase MW Detect phase difference with Ramsey interferometry ' Zak = ' D Zak ' D Zak

62 Berry Phase Reference measurement Phase of reference fringe: Fraction in / 3/ Microwave pulse MW () ' 6= Average of five individual measurements Exp. imperfections: - Small detuning of the MW-pulse - Magnetic field drifts Berry Phase Measuring the Zak Phase (SSH Model) Measured Topological invariant: Zak phase difference ' D Zak ' D Zak = Fraction in / 3/ Microwave pulse MW () ' Zak =.97() obtained from 4 independent measurements

63 Berry Phase Measuring the Zak Phase (SSH Model) Measured Topological invariant: Zak phase difference ' D Zak ' D Zak = Fraction in Zak / 3/ Microwave pulse MW () 5 5 Data set 4 Occurences ' Zak =.97() obtained from 4 independent measurements Geometric Phase Fractional Zak Phase Zak Phase becomes fractional for heteropolar dimerization!

64 SSH Measuring Fractional Charge Probability Density of Eigenstates Lattice Topology Mode function Topologically Trivial Lattice site Mode function Lattice site Topologically Non-Trivial SSH Fractional Charge /4 Energy k /4 /d /d /d /d R. Rajaraman & J. Bell, Phys. Lett B 98, Nucl. Phys. B 983

65 SSH Fractional Charge /4 Energy k /4 /d /d /d /d R. Rajaraman & J. Bell, Phys. Lett B 98, Nucl. Phys. B 983 SSH Fractional Charge /4. Energy k Density.5 /d /d / /d Lattice Site /d R. Rajaraman & J. Bell, Phys. Lett B 98, Nucl. Phys. B 983

66 SSH Fractional Charge /4. Energy k Density.5 /d Always one atom /4 on the Lattice Site /dedge! /d /d R. Rajaraman & J. Bell, Phys. Lett B 98, Nucl. Phys. B 983 Aharonov-Bohm Interferometer for Measuring Berry Curvature

67 Band Topology Lattice: A and B degenerate sublattices H = H JÂ R 3 Â i= â R ˆb R+d i + h.c. Hexagonal Lattices 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. 8, 9 (9) cold atoms: hexagonal - K. Sengtsock (Hamburg), brick wall - T. Esslinger (Zürich) k x Band Topology Scalar & Geometric Features Band structure characterized by scalar & geometric features! Eigenstates: Bloch waves y q,n (r)=e iqr u q,n (r) Scalar Features Dispersion relation E q,n Geometric Features Berry connection A n (q)=ihu q,n q u q,n i Berry curvature W n (q)= q A n (q) e z k y ' k x

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

69 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 D f (q) H(q)= f (q) D Expanding momenta close to K Dirac point q H( q)= x + i q y q x i q y ' Eigenstates u ± K, q = e iq(q)/ ± e iq(q)/ 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

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

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

72 Band Topology The Interferometer k y k x /,, /, MW t Forces applied by lattice acceleration and magnetic gradients! Band Topology Interferometry Results 3 k y k x

73 Band Topology Interferometry Results 3 k y k x Band Topology Interferometry Results 3 k y k x

74 Band Topology Stückelberg Interferometry Lattice acceleration allows for arbitrary path choice Has allowed us to detect off-diagonal Berry connection through Wilson loops! 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?

75 Gauge Field Team From left to right: Christian Schweizer Monika Aidelsburger I.B. Michael Lohse Marcos Atala Julio Barreiro D Berry Curvature Interferometer Team Lucia Duca Tracy Li Martin Reitter Monika Schleier-Smith IB Ulrich Schneider

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