Band Gap Closing in a Synthetic Hall Tube of Neutral Fermions
BBand Gap Closing in a Synthetic Hall Tube of Neutral Fermions
Jeong Ho Han, Jin Hyoun Kang, and Y. Shin ∗ Department of Physics and Astronomy and Institute of Applied Physics, Seoul National University, Seoul 08826, KoreaCenter for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Korea
We report the experimental realization of a synthetic three-leg Hall tube with ultracold fermionicatoms in a one-dimensional optical lattice. The legs of the synthetic tube are composed of threehyperfine spin states of the atoms, and the cyclic inter-leg links are generated by two-photon Ra-man transitions between the spin states, resulting in a uniform gauge flux φ penetrating each sideplaquette of the tube. Using quench dynamics, we investigate the band structure of the Hall tubesystem for a commensurate flux φ = 2 π/
3. Momentum-resolved analysis of the quench dynamicsreveals that a critical point of band gap closing as one of the inter-leg coupling strengths is varied,which is consistent with a topological phase transition predicted for the Hall tube system.
Ultracold atoms in optical lattices have become aunique platform for studying condensed matter physicsin a clean and controllable environment [1, 2]. Over thepast decade, many experimental techniques have beendemonstrated to generate artificial gauge potentials forneutral atoms, providing an interesting opportunity forexploring topologically nontrivial states of matter [3].The Hofstadter-Harper (HH) Hamiltonian, which is theessential model for quantum Hall physics, was realized intwo-dimensional (2D) optical lattice systems using laser-assisted tunneling [4–7]. Recently, ladder systems withthe HH Hamiltonian, dubbed Hall ribbons, were demon-strated in the synthetic dimension framework [8, 9]; inthis framework, the internal degrees of freedom of atomssuch as hyperfine spins [10, 11] and clock states [12] areexploited as a virtual lattice dimension and the hoppingalong the dimension is provided by laser-induced cou-plings between the internal states. The framework wasfurther extended with the external degrees of freedomof atoms such as momentum states [13–15] and latticeorbitals [16].The key advantage of using synthetic lattice dimen-sions is versatile boundary manipulation. Sharp edgescan be defined and individually detected with state-sensitive imaging, thus allowing for experimental inves-tigation of various phenomena such as chiral edge cur-rents [10, 11], topological solitons at interfaces [13], andmagnetic reflection [14, 15]. Furthermore, nontrivial lat-tice geometries can be created in synthetic dimensions,which are hardly achievable with conventional optical lat-tices but may give rise to novel topological states [17, 18].A remarkable example is a ladder geometry with a pe-riodic boundary condition (PBC), which can be realizedby cyclically connecting the synthetic lattice sites. It isunder a PBC that a Hall lattice system exhibits a truefractal structure of the single-particle energy spectrum,called Hofstadter’s butterfly [19]. Additionally, Laugh-lin’s pump, which is an ideal manifestation of quantizedHall conductivity and corresponding Chern number, hasbeen proposed for a torus geometry [20–22].In this paper, we report the experimental realization ofa synthetic Hall lattice system of a tube geometry with ultracold fermionic atoms. In our scheme, the neutralfermions are confined in a one-dimensional (1D) opticallattice and three hyperfine spin states are employed as asynthetic dimension to form a three-leg tube structure.The cyclic links between the legs are created by spin-momentum couplings via two-photon Raman transitionsbetween the spin states, and a uniform gauge flux φ =2 π/ Yb atoms in the | F = 5 / , m F = − / (cid:105) hyperfine spin state of the S ground energy level [23].The typical atom number is N ≈ . × and the tem-perature is T /T F ≈ .
3, where T F is the Fermi temper-ature of the trapped sample. The atoms are adiabati-cally loaded in a three-dimensional optical lattice poten-tial generated by superposing three orthogonal standingwaves with periodicity d x,z = λ L / d y = λ L / √ λ L = 532 nm is the laser wavelength. The finallattice depths are ( V x , V y , V z ) = (5 , , E L ,α , where E L ,α = h / (8 md α ) for α ∈ { x, y, z } , h is the Planck con-stant, and m is the atomic mass. Because tunneling alongthe y and z directions is highly suppressed by large latticedepths, our lattice system is effectively 1D. The tunnel-ing amplitude is t x = 2 π ×
264 Hz, and the characteristicfilling factor is estimated to be ≈ .
75 with trapping fre-quencies of ( ω x , ω y , ω z ) = 2 π × (58 , , z tolift the spin degeneracy of the S energy level.The three lowest spin states, which we denote | (cid:105) ≡ | m F = − / (cid:105) , | (cid:105) ≡ | m F = − / (cid:105) , and | (cid:105) ≡| m F = − / (cid:105) , are employed for the three legs of the syn-thetic tube system. To generate inter-leg couplings, threelinearly polarized Raman laser beams R , , are irradi- a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p (a) (b)(c) S ( F =5/2) R1R R R B l L LB ( s- ) Yb xyzR ( s+ ) OL ( p ) δω R ( p ) +R ( s+ ) ω = ωω = ω+δωω = ω-2δωm F = -5/2 -3/2 -1/2R R Φ R e a l d i m e n s i o n ( x ) Ω e iΦj j+1j-1 jj-2 j+2 Φ t x Φ OL ( p ) FIG. 1. Realization of a synthetic Hall tube with neutralatoms. (a) Schematic of the experimental setup. Fermionic
Yb atoms are confined in an optical lattice and illuminatedby three Raman laser beams R , , . A magnetic field B andan additional laser light (LB) are applied along ˆ z to controlthe energy levels of the spin states. (b) The three lowest spinstates of Yb are coupled to each other via two-photon Ra-man transitions by R , , . (c) Synthetic three-leg Hall tubewith a uniform gauge flux φ on each side plaquette. Thethree legs are formed by the three spin states of the atoms inthe 1D optical lattice (black lines) and the inter-leg tunnelingwith complex amplitude (gray lines) is provided by the cyclicRaman couplings between the spin states. The flux φ is con-trolled by the Raman beam angle θ in (a) (see the text fordetails). ated on the sample [Fig. 1(a)], where the wave vectors ofthe laser beams are given by (cid:126)k r = k R (cos θ ˆ x +sin θ ˆ y ) and (cid:126)k r = (cid:126)k r = k R ˆ x , respectively, and the polarization direc-tions are horizontal for R , and vertical for R to the xy plane. The laser frequencies of R , , are set to ω = ω , ω = ω + δω , and ω = ω − δω , respectively, where ω isthe laser frequency blue-detuned by 1 .
97 GHz from the (cid:12)(cid:12) S , F = 5 / (cid:11) → (cid:12)(cid:12) P , F (cid:48) = 7 / (cid:11) transition line. When δω is tuned to half of the energy difference between | (cid:105) and | (cid:105) , the three spin states {| (cid:105) , | (cid:105) , | (cid:105)} can be res-onantly coupled to each other in a cyclic manner bytwo-photon Raman transitions, as described in Fig. 1(b).Thus, a three-leg synthetic tube is constructed with thefermions in the 1D optical lattice [Fig. 1(c)].In the synthetic tube system, the Raman coupling be-tween the spin states | s (cid:105) and | s (cid:48) (cid:105) is described by inter-legtunneling with complex amplitude Ω ss (cid:48) e iφj , where Ω ss (cid:48) isthe Rabi frequency of the corresponding two-photon Ra-man transition and j is the site index for the real lattice.The spatial phase modulations of the tunneling ampli-tude originate from the momentum transfer (cid:126) ∆ (cid:126)k of thetwo-photon transition, yielding φ = (∆ (cid:126)k · ˆ x ) d x [7]. In our experimental setup, ∆ (cid:126)k = (cid:126)k r ,r − (cid:126)k r = k R [(1 − cos θ )ˆ x − sin θ ˆ y ] for all the cyclic inter-leg couplings and φ = 2 πk R d x (1 − cos θ ) regardless of spin state. Whena fermion hops around any side plaquette of the tube,it acquires a uniform net phase of φ , thus realizing theHH Hamiltonian in the tube geometry. In this work, weset the Raman beam angle θ ≈ . ◦ to have φ = 2 π/ σ - σ transition (∆ m F =2) for the | (cid:105) - | (cid:105) coupling isrelatively weak, the intensity ratio of R , , is adjustedto create a symmetric coupling structure. We measureΩ = Ω ≈ . t x . Here, Ω / Ω is fixed because the π - σ (∆ m F =1) transitions for the | (cid:105) – | (cid:105) and | (cid:105) – | (cid:105) cou-plings are created by the same pair of Raman beams, andthe ratio is nearly unity within 1.2%.In realizing the three-leg Hall tube, careful control ofthe energy levels of the spin states is necessary to sup-press the optical transitions to the other spin states, | (cid:105) ≡ | m F = 1 / (cid:105) and | (cid:105) ≡ | m F = 3 / (cid:105) . The energylevel ν s of spin state | s (cid:105) is determined by the sum ofthe magnetic Zeeman shift and the total AC Stark shiftdue to laser radiation. To generate sufficiently large dif-ferential AC Stark shifts, we apply to the sample anadditional laser radiation along ˆ z [25], which is σ − –polarized and detuned by −
70 MHz with respect to the (cid:12)(cid:12) S , F = 5 / (cid:11) → (cid:12)(cid:12) P , F (cid:48) = 7 / (cid:11) transition line. Underthe final experimental condition, the energy level differ-ences between the spin states are spectroscopically mea-sured [26] and ( ξ , ξ , ξ , ξ , ξ ) ≈ (0 , − . , , − , .
7) Ω ,where ξ s = ( ν s − ν ) − ( s − δω and δω = 2 π × . ξ s is the detuning of | s (cid:105) from the energy staircase formedby two-photon Raman processes with a step unit of δω .The atom loss rate into | (cid:105) and | (cid:105) is measured to be ≈ . .The Bloch Hamiltonian of the three-leg Hall tube sys-tem is given byˆ H q / (cid:126) = − t x cos ( q − φ ) Ω / / / ξ − t x cos ( q ) Ω / / / − t x cos ( q + φ ) , (1)where q is the quasimomentum of the lattice tube sys-tem normalized by d − x [26]. For a symmetric case withΩ = Ω = Ω and ξ = 0, it is well known that theHamiltonian ˆ H q for φ = 2 π/ ss (cid:48) and thetopological state survives ξ (cid:54) = 0, featuring a nonzero Zakphase Z = 1 of its lowest band [28]. In ˆ H q , time-reversalsymmetry, particle-hole symmetry and chiral symmetryare broken, which corresponds to the symmetry class A(unitary) of the Altland-Zirnbauer classification [29, 30].When the lowest band is completely filled, the systemrepresents a topologically insulating state analogous to (a) S p i n c o m po s i t i on (c) (d) (b) k C ( t ) , ( k L ) j j+1j-1j-2 j+2 s x Φ t i m e ( m s ) k ( k L ) n ( k , t ) n ( k , t ) n ( k , t )2-2 0 2-2 0 2-2 0 OD0 max
FIG. 2. Quench dynamics of the three-leg Hall tube for φ = 2 π/
3. (a) Illustration of the atomic motion in the Halltube. Atoms are initially prepared in the spin- | (cid:105) leg and theinter-leg couplings are suddenly activated. (b) Time evolu-tion of the lattice momentum distribution n ( k, t ) of the sam-ple, n ( k, t ) of the atoms in | (cid:105) , and n ( k, t ) of the atoms in | (cid:105) . Time evolution of (c) the fractional spin populations, (d)the average lattice momentum (cid:104) k (cid:105) of the sample, and the dif-ference C ( t ) = (cid:104) k (cid:105) − (cid:104) k (cid:105) between the momenta of the twolegs | (cid:105) and | (cid:105) . Each data point comprises five measurementsof the same experiment, and the error bar is their standarddeviation. The solid and dashed lines in (d) show the numer-ical simulation results for C and (cid:104) k (cid:105) , respectively, includingphenomenological damping [26]. the integer quantum Hall state [31].To demonstrate the presence of a gauge flux on pla-quettes, we investigate the quench dynamics of the syn-thetic Hall system. Atoms are initially prepared in theleg | (cid:105) , and then the inter-leg couplings are suddenly ac-tivated by turning on the Raman laser beams. After avariable hold time, the spin composition of the sampleis measured by imaging with optical Stern-Gerlach spinseparation [32], and separately, the lattice momentumdistribution n ( k ) of the sample is measured using an adi-abatic band-mapping technique [24, 26]. Note that in theband-mapping process, the quasimomentum state with q is transformed into a superposition of free-space momen-tum states of the three spin states in the first Brillouinzone (BZ), where the momentum k s of spin state | s (cid:105) isrelated to q as k s d x = [ q + ( s − φ ] modulo 2 π and − k L < k s ≤ k L with k L = π/d x . The momentum dis-tribution n s ( k ) of the atoms in | s (cid:105) is also measured byspin-selective imaging [Fig. 2(b)] [23].The measurement results of the time evolution of thequenched synthetic Hall tube system are displayed inFigs. 2(c) and 2(d). At the early time t < µ s,when the atoms start transferring to the legs | (cid:105) and | (cid:105) , the average lattice momentum of the sample, (cid:104) k (cid:105) = (a) (b)(c)(e) (d)(f)(h) Φ open 2-leg s x Φ open 3-leg s x S p i n c o m po s i t i on (g) -110 1-10time (ms)(sites) x (sites) x sk ( k L ) FIG. 3. Quench dynamics of (a) two-leg and (b) three-legladders with open boundaries for φ = 2 π/
3. Time evolutionof (c,d) the fraction spin populations and (e,f) the averagelattice momentum (cid:104) k (cid:105) . The solid lines display the numericalsimulation results for (cid:104) k (cid:105) [26]. Each data point comprisesfive measurements of the same experiment. (g,h) Trajectoriesof the ladder systems in the plane of the spin and real latticepositions (cid:104) s (cid:105) and (cid:104) x (cid:105) . (cid:104) s (cid:105) = ˜ n − ˜ n , where ˜ n s is the fractionalpopulation of spin component | s (cid:105) and (cid:104) x (cid:105) is calculated from (cid:104) k (cid:105) using the knowledge of band dispersion [10]. (cid:82) k L − k L kn ( k ) dk/ (cid:82) k L − k L n ( k ) dk , shows no significant varia-tions; however, the difference between the momenta ofthe atoms transferred into | (cid:105) and | (cid:105) , C ( t ) = (cid:104) k (cid:105) − (cid:104) k (cid:105) ,where (cid:104) k s (cid:105) = (cid:82) k L − k L kn s ( k ) dk/ (cid:82) k L − k L n s ( k ) dk , increases no-ticeably. This means that the atoms in the legs | (cid:105) and | (cid:105) move in positive and negative directions of the real lat-tice, respectively, which is understandable based on theclassical motion of a charged particle moving in the tubein the presence of a magnetic field [Fig. 2(a)]. At latertimes, the spin composition and C ( t ) show damped oscil-lations, which are reasonably accounted for by a numeri-cal simulation for ˆ H q including phenomenological damp-ing [26]. The asymmetry between | (cid:105) and | (cid:105) and thesmall oscillations of (cid:104) k (cid:105) result from nonzero ξ .The quench evolution of the Hall system is further ex-amined for open ladder geometries [Figs. 3(a) and 3(b)].The structure modification is achieved by deactivatingtwo or one of the inter-leg links; by shifting ω ( ω ) by2 π ×
400 ( − (cid:104) k (cid:105) are displayed in Figs. 3(c)–3(f). In contrastto the Hall tube case, (cid:104) k (cid:105) shows relatively large oscilla-tions because the atoms are initially prepared at an edgeof the ladder. Interestingly, (cid:104) k (cid:105) changes its sign duringthe oscillations, and the behavior is well captured by thenumerical simulations [Fig. 3(e) and 3(f)]. In the openthree-leg ladder case, we attribute the behavior mainlyto the large gauge flux φ > π/ (cid:104) k (cid:105) was not observed in a previous experiment for a smallergauge flux [10]. In Figs. 3(g) and 3(h), the semiclassi-cal trajectories of the ladder systems are displayed in theplane of spin and real lattice positions. The open two-legladder case shows damped cyclotron motion truncated bythe ladder edge, and the three-leg case exhibits bounc-ing motions due to the Bloch oscillations in the courseof cyclotron motion. These observations corroborate thepresence of a gauge flux on the side plaquettes of thesynthetic tube.In Fig. 4(a), we present the phase diagram of the Halltube system for φ = 2 π/ and Ω .The topological phase with Z = 1 exists in a region ofΩ − < Ω < Ω + , where the boundaries are given byΩ ± = ± t x − ξ + (cid:112) (3 t x ∓ ξ ) + Ω . Our current sys-tem with Ω = Ω ≈ . t x is located in the topologicalregime and its transition to a topologically trivial phasewith Z = 0 can be driven by, for example, decreasingΩ below the critical value of Ω − = 11 . t x [18]. InFig. 4(b), the band dispersions of the Hall tube systemare displayed for various Ω , showing that the topolog-ical phase transition at Ω = Ω − occurs with closingthe energy gap between the first and second bands atquasimomentum q c = ± π [35]. According to the bulk-edge correspondence, band gap closing is a generic andnecessary feature of the topological phase transition of asymmetry-preserving system [36].The critical point of band gap closing is probed viamomentum-resolved analysis of the quench dynamics. Asthe band gap closes, the dynamic evolution for q = q c isgoverned by a single energy scale that is determined bythe energy difference between the third band and thetwo touching lowest bands. Therefore, the gap closingwould be characteristically reflected in the quench evo-lution of the spin composition at q c . The momenta ofthe spin states | (cid:105) and | (cid:105) corresponding to q c = ± π are k c = − k L and k c = − k L /
3, respectively, and wemeasure the quench evolution of n ( k c ) and n ( k c ) forvarious Ω ≤ Ω [Fig. 4(c)]. When Ω is decreased bydecreasing the intensity of R , the resulting reduction ofthe AC Stark shift is compensated for by applying an- (d) t , ( m s ) Ω /t x O D ( a r b . un i t ) (c)(a) (b) E ( q ) /t x -100 q ( p )1-1 010 T opo l og i c a l ( Z = ) T r i v i a l ( Z = ) Z=1Z=01 23s /t x T o p o l o g i c a l ( Z = ) Ω /t x t Ω /t x =12.3 t Ω /t x =5.9 t t FIG. 4. Observation of band gap closing in the three-legHall tube. (a) Phase diagram of the system. The shadedarea indicates a topologically nontrivial phase with a nonzeroZak phase Z = 1 of the lowest band. The solid dots indi-cate the parameter positions explored in the experiment. (b)Band structures calculated for various Ω with Ω = Ω ≈ . t x . A topological phase transition occurs together withband gap closing at q c = ± π . (c) Quench evolution of n ( k c )and n ( k c ), where k c and k c are the lattice momenta of | (cid:105) and | (cid:105) , respectively, corresponding to q c . Each data pointis obtained by averaging five measurements of the same ex-periment and the error bar is their standard deviation. Thetime τ ( τ ) for the first maximum of n ( k c ) ( n ( k c )) is de-termined by fitting the experimental data to an asymmetricparabola function (solid line). (d) τ and τ as functions ofΩ . The red and blue dashed lines show the numerical re-sults for τ and τ , respectively. The crossing of τ and τ reveals the critical point of band gap closing. other off-resonant laser light with the same polarizationas R . To obtain the characteristic time scales of the spincomposition oscillations, we determine the times τ and τ at which n ( k c ) and n ( k c ) reach their first max-ima, respectively, by fitting the experimental data to anasymmetric parabolic function [37].Figure 4(d) shows the measurement results of the timescales as functions of Ω . At Ω = Ω , τ is smallerthan τ and increases faster than τ as Ω decreases.The crossing of τ and τ occurs at Ω ≈ . t x in thevicinity of the expected critical point Ω − . The numericalsimulation reproduces the observed crossing behavior ofthe two time scales and yields τ = τ at Ω = Ω − , whichvalidates our experimental approach using the time scalesof quench dynamics to probe band gap closing. The de-viation of the measured critical value from the predictedΩ − is not clearly understood. This might be due to im-perfection in spin-selective imaging, the damping, or theinteraction effects in the quench dynamics, which are ne-glected in our numerical simulations. We note that forour experimental parameters, the on-site interaction en-ergy is estimated to be U/ (cid:126) ≈ . t x .In conclusion, we realize a synthetic three-leg Hall tubewith φ = 2 π/ φ can be controlled by θ , and we expect an immediateexpansion of this work to study fractal band structureswith varying magnetic fluxes from commensurate to in-commensurate values. Further studies may include in-teratomic interactions [38], which are expected to showfractional charge behavior [21], using the recently imple-mented orbital Feschbach resonance [39, 40].We thank Moosong Lee for early contributions to thiswork and Seji Kang for experimental assistance. Thiswork is supported by the Institute for Basic Science(IBS-R009-D1) and the National Research Foundationof Korea (Grant Nos. NRF-2018R1A2B3003373, 2014-H1A8A1021987). ∗ [email protected][1] D. Jaksch and P. Zoller, Ann. Phys. (Amsterdam) ,52 (2005).[2] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[3] J. Dalibard, F. Gerbier, G. Juzeli¯unas, and P. ¨Ohberg,Rev. Mod. Phys. , 1523 (2011).[4] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro,B. Paredes, and I. Bloch, Phys. Rev. Lett. , 185301(2013).[5] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur-ton, and W. Ketterle, Phys. Rev. Lett. , 185302(2013).[6] Y.-J. Lin, K. Jim´enez-Garc´ıa, and I. B. Spielman, Nature , 83 (2011).[7] N. Goldman, G. Juzeli¯unas, P. ¨Ohberg, and I. B. Spiel-man, Rep. Prog. in Phys. , 126401 (2014).[8] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein,Phys. Rev. Lett. , 133001 (2012).[9] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B.Spielman, G. Juzeli¯unas, and M. Lewenstein, Phys. Rev.Lett. , 043001 (2014).[10] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider,J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte,and L. Fallani, Science , 1510 (2015).[11] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I.B. Spielman, Science , 1514 (2015).[12] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati,M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio,and L. Fallani, Phys. Rev. Lett. , 220401 (2016).[13] E. J. Meier, F. A. An, and B. Gadway, Nat. Commun. ,13986 (2016).[14] F. A. An, E. J. Meier, and B. Gadway, Sci. Adv. ,e1602685 (2017).[15] F. A. An, E. J. Meier, and B. Gadway, Nat. Commun. ,325 (2017).[16] J. H. Kang, J. H. Han, and Y. Shin, arXiv:1807.01444 [cond-mat.quant-gas] (2018).[17] O. Boada, A. Celi, J. Rodr´ıguez-Laguna, J. I. Latorre,and M. Lewenstein, New J. Phys. , 045007 (2015).[18] S. Barbarino, M. Dalmonte, R. Fazio, and G. E. Santoro,Phys. Rev. A , 013634 (2018).[19] D. R. Hofstadter, Phys. Rev. B , 2239 (1976).[20] R. B. Laughlin, Phys. Rev. B , 5632 (1981).[21] T.-S. Zeng, C. Wang, and H. Zhai, Phys. Rev. Lett ,095302 (2015).[22] L. Taddia, E. Cornfeld, D. Rossini, L. Mazza, E. Sela,and R. Fazio, Phys. Rev. Lett , 230402 (2017).[23] M. Lee, J. H. Han, J. H. Kang, M.-S. Kim, and Y. Shin,Phys. Rev. A , 043627 (2017).[24] M. K¨ohl, H. Moritz, T. St¨oferle, K. G¨unter, and T.Esslinger, Phys. Rev. Lett. , 080403 (2005).[25] B. Song, L. Zhang, C. He, T. F. J. Poon, E. Hajiyev, S.Zhang, X.-J. Liu, and G.-B. Jo, Sci. Adv. , eaao4748(2018).[26] See Supplemental Material for the details of experimen-tal sequence, measurement of energy level difference, thetight-binding model description, and numerical simula-tions.[27] H. L. Nourse, I. P. McCulloch, C. Janani, and B. J. Pow-ell, Phys. Rev. B , 214418 (2016).[28] J. Zak, Phys. Rev. Lett , 2747 (1989).[29] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142(1997).[30] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Rev. Mod. Phys. , 035005 (2016).[31] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-wig, Phys. Rev. B , 195125 (2008).[32] S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki, T. Tsuji-moto, R. Murakami, and Y. Takahashi, Phys. Rev. Lett. , 190401 (2010).[33] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H.Zhai, and J. Zhang, Phys. Rev. Lett. , 095301 (2012).[34] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah,W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. ,095302 (2012).[35] For Ω = Ω , the ˆ H q is mirror symmetric along spindimension with respect to | (cid:105) , and therefore the quasimo-mentum for gap closing should be q = 0 or ± π . Here, thetopological transition at Ω = Ω + shows the gap closingat q = 0.[36] M. Ezawa, Y. Tanaka, and N. Nagaosa, Sci. Rep. , 2790(2013).[37] The fitting function is given by f ( t ) = α ( t − τ s ) + β for t ≤ t s and α ( t − τ s ) + β for t > t s with four fittingparameters, α , , β , and τ s .[38] S. Barbarino, L. Taddia, D. Rossini, L. Mazza, and R.Fazio, Nat. Commun. , 8134 (2015).[39] G. Pagano, M. Mancini, G. Cappellini, L. Livi, C. Sias,J. Catani, M. Inguscio, and L. Fallani, Phys. Rev. Lett. , 265301 (2015).[40] M. H¨ofer, L. Riegger, F. Scazza, C. Hofrichter, D. R.Fernandes, M. M. Parish, J. Levinsen, I. Bloch, and S.F¨olling, Phys. Rev. Lett. , 265302 (2015).[41] J. Heinze, S. G¨otze, J. S. Krauser, B. Hundt, N.Fl¨aschner, D.-S. L¨uhmann, C. Becker, and K. Sengstock,Phys. Rev. Lett , 135303 (2011). Supplemental Material
Experimental sequence
A schematic of the experimental sequence is presentedin Fig. S1. First, we prepare a degenerate Fermi gas of
Yb in the | (cid:105) ≡ | F = 5 / , m F = − / (cid:105) hyperfineground state in a crossed optical dipole trap (ODT) us-ing forced evaporative cooling and optical pumping tech-niques [23]. The total atom number is N ≈ . × ,and the sample temperature is T /T F ≈ .
3, where T F isthe Fermi temperature of the trapped sample. The frac-tional population of the atoms in the other spin stateswith m F (cid:54) = − / d x = d z = λ L / d y = λ L / √
3, where λ L = 532 nm is the laser wave-length. The lattice potential is exponentially ramped upin 70 ms to the target depth ( V x , V y , V z )=(5 , , E L ,α ,where E L ,α = h / md α is the lattice recoil energy for the α ∈ { x, y, z } direction. The adiabaticity of the latticeloading is confirmed by the fact that the sample temper-ature is not significantly altered even after reversing theloading sequence. V α is calibrated by a modulation spec-troscopy method [41]. During the lattice ramp-up, wereduce the ODT depth to counteract the increase in theoverall trapping potential due to the OL and apply anexternal magnetic field of 153 G along ˆ z to lift the spindegeneracy of the S ground level, resulting in a Zeemanenergy splitting of h × . N (cid:81) α d α ζ α ≈ .
75, where ζ α = (cid:112) (cid:126) t α / ( mω α ), t α is the tunneling amplitude, and ω α is the trappingfrequency of the harmonic trapping potential.Next, we turn on the σ − –polarized laser beam, whichis referred to as a Lift beam (LB), and after 200 µ s, weswitch on the Raman laser beams. The role of the Liftbeam is to generate differential AC stark shifts of the spinstates, which is necessary for suppressing unwanted Ra-man transitions, in particular, to the | (cid:105) ≡ | m F = 1 / (cid:105) spin state. The Lift laser beam is detuned by −
70 MHzfrom the (cid:12)(cid:12) S , F = 5 / (cid:11) → (cid:12)(cid:12) P , F (cid:48) = 7 / (cid:11) transitionline and its intensity is 8.5 mW/cm . In the quenchexperiment of the three-leg Hall tube, the fractional spinpopulations of | (cid:105) and | (cid:105) ≡ | m F = 3 / (cid:105) are measuredto be less than 13% and 7%, respectively, after 1 ms evo-lution. The beam waists of the Lift and Raman beamsare 150 µ m, much larger than the in situ sample radiusof 15 µ m; thus, the confining effect due to the inhomoge-neous intensity distributions of the laser beams is negli-gible. Under the Lift beam, the lifetime of the atoms inthe optical lattice is measured to be ≈
250 ms, which isapproximately four times shorter than that of the atoms
BlastLift, RamanMagnetic field ImagingOLODT
70 0.2 1.5Variablehold time 15
20 (in ms) time
FIG. S1. Schematic of the experimental sequence. The figureis not to scale. without the Lift beam. As the atoms are illuminated bythe Raman laser beams, their lifetime is further reducedto ≈
20 ms in the open three-leg ladder case and evendown to ≈ S → P transition. For spin-selective imaging, the atoms notin the target spin state are removed by applying shortpulses of laser light resonant with the (cid:12)(cid:12) S , F = 5 / (cid:11) → (cid:12)(cid:12) P , F (cid:48) = 7 / (cid:11) transition within the initial 5 ms of thefree expansion. The removal process causes inter-spincollisions, which results in atom position blurring in theabsorption image [Fig. 2(a)]. Measurement of energy level differences
We measure the energy level differences between thespin states by two-photon Raman spectroscopy. For aspin-polarized atomic cloud prepared in the presence ofthe Lift beam but without the optical lattice, we ap-ply a short pulse of the Raman laser beams with apulse duration of t = 50 µ s and we measure the frac-tional population of the atoms transferred to the targetspin state using an optical Stern-Gerlach spin separationmethod as a function of the frequency difference δω r ofthe two Raman laser beams associated with the tran-sition. Here, the frequency of the other Raman laserbeam which is not involved in the target transition isset to be far detuned to prevent Raman transitions toother spin states, while its AC Stark shift effect is main-tained. Figure S2 shows a typical Raman spectrum forthe | (cid:105) → | (cid:105) transition, where | (cid:105) ≡ | m F = − / (cid:105) . Thecenter frequency δω r,c is determined by fitting a Gaus- A t o m nu m be r f r a c t i on p x kHz)-60 -50-70 -40-80 FIG. S2. Raman spectrum of a spin-polarized sample in | (cid:105) ≡ | m F = − / (cid:105) . The fractional population of the atomsin | (cid:105) ≡ | m F = − / (cid:105) is plotted as a function of the frequencydifference of the Raman laser beams. The Raman beam pulseduration is t = 50 µ s. The solid line indicates a Gaussiancurve fit to the data. sian function to the spectrum, and taking into accountthe kinetic energy contribution, the energy level differ-ence ν − ν between the two spin states is obtained as ν − ν = δω r,c − (cid:126) m [2 k R sin( θ/ , where k R is the wavenumber of the Raman beams and θ is the angle betweenthe two Raman beams. In determining the energy level ν ( ν ) of the spin state | (cid:105) ( | (cid:105) ), we use a spin-polarizedatomic sample in | (cid:105) . For our experimental condition, wemeasure ( ξ , ξ , ξ , ξ , ξ ) ≈ (0 , − . , , − , .
7) Ω , where ξ s = ( ν s − ν ) − ( s − δω , δω ≡ ν / π × . = 2 π × . is the Rabi frequency ofthe | (cid:105) – | (cid:105) Raman coupling ( | (cid:105) ≡ | m F = − / (cid:105) ). Tight-binding model
In a rotating wave approximation, the tight-bindingmodel Hamiltonian for our synthetic three-leg Hall tubesystem is given byˆ H/ (cid:126) = (cid:88) j (cid:88) s =1 (cid:16) − t x ˆ c † j +1 ,s ˆ c j,s + h . c . (cid:17) + (cid:88) j (cid:88) s =1 (cid:18) Ω s, ( s +1) e iφj ˆ c † j,s +1 ˆ c j,s + h . c . (cid:19) + (cid:88) j (cid:18) Ω e iφj ˆ c † j, ˆ c j, + h . c . (cid:19) + (cid:88) j (cid:88) s =1 ( ξ s + (cid:15) j / (cid:126) ) ˆ c † j,s ˆ c j,s + U (cid:126) (cid:88) j (cid:88) s (cid:54) = s (cid:48) ˆ n j,s ˆ n j,s (cid:48) , (S1)where ˆ c j,s (ˆ c † j,s ) is the annihilation (creation) operator fora fermion in the Wannier state | j, s (cid:105) localized at the reallattice site j = 1 , ..., L x with spin s = 1 , ,
3. The first term represents tunneling in the real lattice; the secondand third terms describe the inter-leg couplings gener-ated by the Raman laser beams, where Ω ss (cid:48) is the Rabifrequency of the two-photon Raman transition betweenthe spin states | s (cid:105) and | s (cid:48) (cid:105) and the position-dependentcomplex phase factor e iφj results from the momentumimparted by the Raman transition; the fourth term isthe on-site energy in the rotating frame, including theexternal trapping potential contribution, (cid:15) j ; and the lastterm is the on-site interaction energy with number oper-ator ˆ n j,s ≡ ˆ c † j,s ˆ c j,s .Under a unitary transformation ˆ U ˆ c j,s ˆ U † = e iφ ( s − j ˆ c (cid:48) j,s ,the Hamiltonian is re-expressed asˆ H (cid:48) / (cid:126) = (cid:88) j (cid:88) s =1 (cid:16) − t x e − iφ ( s − ˆ c (cid:48)† j +1 ,s ˆ c (cid:48) j,s + h . c . (cid:17) + (cid:88) j (cid:18) Ω c (cid:48)† j, ˆ c (cid:48) j, + Ω c (cid:48)† j, ˆ c (cid:48) j, + h . c . (cid:19) + (cid:88) j (cid:18) Ω e i φj ˆ c (cid:48)† j, ˆ c (cid:48) j, + h . c . (cid:19) + (cid:88) j (cid:88) s =1 ξ s ˆ c (cid:48)† j,s ˆ c (cid:48) j,s , (S2)where the external potential and interaction terms areneglected. When φ = 2 π/
3, the complex phase factor e i φj in the third term becomes unity and j -independent,and via a transformation ˆ c (cid:48) q,s = √ L x (cid:80) j e iqj ˆ c (cid:48) j,s , ˆ H (cid:48) canbe represented in momentum space by the 3-by-3 BlochHamiltonianˆ H q / (cid:126) = ξ − t x cos ( q − φ ) Ω / / / ξ − t x cos ( q ) Ω / / / ξ − t x cos ( q + φ ) . (S3) Numerical simulation
We perform a numerical simulation of the quench dy-namics by solving the Bloch equation, i (cid:126) ∂∂t c ( q, t ) c ( q, t ) c ( q, t ) = ˆ H q c ( q, t ) c ( q, t ) c ( q, t ) . (S4)The atomic density n s ( k s , t ) for spin s and momentum k s is calculated as n s ( k s ) = | c s ( q ) | , where k s d x =[ q + ( s − φ ] modulo 2 π and − k L < k s ≤ k L with k L = π/d x . The initial conditions for c s at t = 0 areset as c ( q,
0) = (cid:112) n ( k ,
0) and c ( q,
0) = c ( q,
0) = 0,where n ( k ,
0) is obtained by averaging the experimen-tally measured lattice momentum distributions of the ini-tial spin-polarized samples. (a) (b) time (ms) time (ms) (c) (d)(e) (f) F r a c t i ona l s p i n c o m po s i t i on FIG. S3. Calculated quench evolution of the fractional spincomposition for various boundary conditions: (a) three-legHall tube, (c) open two-leg ladder, and (e) open three-legladder. (b, d, f) Corresponding experimental data shown inFig. 2(c), Fig. 3(c) and 3(d), respectively. (a) Ω /t x =0 time (ms) 0 0.5time (ms)0.01.00.01.0 0 0.25 0.250.5 0.01.00.01.0 F r a c t i ona l s p i n c o m po s i t i on (b) Ω /t x =5(c) Ω /t x =10 (d) Ω /t x =15 t t t t t t t t FIG. S4. Calculated quench evolution of the fractional spincomposition at q c = ± π for various values of Ω /t x in thesynthetic three-leg Hall tube system. t x = 2 π ×
264 Hz andΩ = 12 . t x as in the experiment present in Fig. 4. τ s isthe time when the fractional population in the spin s statereaches its first maximum. Figure S3 displays the numerical results of the quenchdynamics for the various boundary conditions of the ex-periment. We observe that spin oscillations show damp-ing in the three-leg Hall tube and three-leg open laddercases [Figs. S3(a) and S3(e)], whereas those in the two-leg open ladder case are not damped [Fig. S3(c)]. Wefind that the effective damping originates from ξ beingnonzero in the numerical simulations. In the experiment,we also observe that damping is enhanced in the syn-thetic Hall tube and open-three leg ladder cases. In thecalculations of (cid:104) k (cid:105) ( t ) and C ( t ) for the three-leg cases inFigs. 2(d) and Fig. 3(f), we include the damping effectphenomenologically as g e ( t ) = (cid:0) g ( t ) − ¯ g (cid:1) e − t/τ d + ¯ g, (S5)where g is (cid:104) k (cid:105) or C directly obtained from the numericalsimulation and ¯ g is the mean value determined from theexperiment. We find that τ d = 0 .
15 ms for the syntheticthree-leg Hall tube and τ d = 0 . q c = ± π , i.e., { k c , k c , k c } = { / , ± , − / } k L for various values of Ω . The timescales of spin oscillations are characterized with τ s atwhich spin population in | s (cid:105) reaches its first maximum.At the critical point Ω = Ω − of the topological phasetransition, τ = τ3