Tunable Flux through a Synthetic Hall Tube of Neutral Fermions
TTunable Flux through a Synthetic Hall Tube of Neutral Fermions
Xi-Wang Luo, Jing Zhang,
2, 3, ∗ and Chuanwei Zhang † Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R.China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, P. R. China
Hall tube with a tunable flux is an important geometry for studying quantum Hall physics, butits experimental realization in real space is still challenging. Here, we propose to realize a syntheticHall tube with tunable flux in a one-dimensional optical lattice with the synthetic ring dimensiondefined by atomic hyperfine states. We investigate the effects of the flux on the system topology andstudy its quench dynamics. Utilizing the tunable flux, we show how to realize topological chargepumping. Finally, we show that the recently observed quench dynamics in a synthetic Hall tubecan be explained by the random flux existing in the experiment.
Introduction.—
Ultracold atoms are emerging as apromising platform for the study of condensed matterphysics in a clean and controllable environment [1, 2].The capability of generating artificial gauge fields andspin-orbit coupling using light-matter interaction [3–18]offers new opportunity for exploring topologically non-trivial states of matter [19–25]. One recent notableachievement was the realization of Harper-HofstadterHamiltonian, an essential model for quantum Hallphysics, using laser-assisted tunneling for generating ar-tificial magnetic fields in two-dimensional (2D) opticallattices [26–28]. Moreover, synthetic lattice dimensiondefined by atomic internal states [29–37] provides a newpowerful tool for engineering new high-dimensional quan-tum states of matter with versatile boundary manipula-tion [32, 33].Nontrivial lattice geometries with periodic boundaries(such as a torus or tube) allow the study of many in-teresting physics such as the Hofstadter’s butterfly [38]and Thouless pump [39–44], where the flux through thetorus or tube is crucially important. In a recent exper-iment [37], a synthetic Hall tube has been realized in a1D optical lattice and interesting quench dynamics havebeen observed, where the flux effect was not considered.More importantly, the flux through the tube, determinedby the relative phase between Raman lasers, is spatiallynon-uniform and random for different iterations of the ex-periment, yielding major deviation from the theoreticalprediction. The physical significance and experimentalprogress raise two natural questions: can the flux in thesynthetic Hall tube be controlled and tuned? If so, cansuch controllability lead to the observation of nontrivialtopological phases and dynamics?In this Letter, we address these important questions byproposing a simple scheme to realize a controllable fluxΦ through a three-leg synthetic Hall tube and studyingits quench dynamics and topological pumping. Our mainresults are:i) We use three hyperfine ground spin states, each ofwhich is dressed by one far-detuned Raman laser, to re- 𝜃 𝜃 ℰ⃗ ℰ⃗ ℰ⃗ 𝐵 𝑦 𝑥 𝑧 ℰ ℰ ℰ ℰ ℰ 𝑆 𝑃 𝐹 = 72𝐹 = 92 ℰ ℰ ℰ ℰ ℰ (a) (b) (c) |1⟩ |3⟩ |2⟩ |3⟩ |2⟩ |1⟩ |1⟩ |2⟩ |3⟩ 𝑗 𝑗 + 1 𝑗 − 1 𝛷 𝐽 𝑥 𝜙 (d) FIG. 1: (a) Schematic of the experimental setup for tunableflux through the synthetic Hall tube. Three Raman lasers E , E and E generate the couplings along the synthetic dimen-sion spanned by atomic hyperfine states. (b,c) Three hyper-fine ground states and the corresponding two-photon Ramantransitions for alkaline-earth(-like) atoms (b) and for alkaliatoms (c). (d) Synthetic Hall tube with a uniform flux φ oneach side plaquette and Φ through the tube. alize the synthetic ring dimension of the tube. The fluxΦ can be controlled simply by varying the polarizationsof the Raman lasers [11]. The scheme can be applied toboth Alkali (e.g., potassium) [11–15] and Alkaline-earth(-like) atoms (e.g., strontium, ytterbium) [45].ii) The three-leg Hall tube is characterized as a 2Dtopological insulator with Φ playing the role of the mo-mentum along the synthetic dimension. The system re-duces to a 1D topological insulator at Φ = 0 and π ,where the winding number is quantized and protected bya generalized inversion symmetry.iii) The tunable Φ allows the experimental observationof topological charge pumping in the tube geometry.iv) We study the quench dynamics with a tunable fluxand show that the experimental observed quench dynam-ics in [37] can be better understood using a random flux a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b existing in the experiment. The model.—
We consider an experimental setup withcold atoms trapped in 1D optical lattices along the x -direction, where transverse dynamics are suppressed bydeep optical lattices, as shown in Fig. 1a. The bias mag-netic field is along the z direction to define the quantiza-tion axis. Three far-detuned Raman laser fields (cid:126) E s , prop-agating in the x - y plane, are used to couple three atomichyperfine ground spin states, with each state dressed byone Raman laser, as shown in Figs. 1b and 1c for alkaline-earth(-like) (e.g., strontium, ytterbium) and alkali (e.g.,potassium) atoms, respectively. The three spin statesform three legs of the synthetic tube system as shown inFig. 1d, and the tight-binding Hamiltonian is written as H = (cid:88) j ; s (cid:54) = s (cid:48) (cid:101) Ω ss (cid:48) ; j c † j,s c j,s (cid:48) − (cid:88) j ; s ( Jc † j,s c j +1 ,s + H.c. ) , (1)where (cid:101) Ω ss (cid:48) ; j = Ω ss (cid:48) e iφ j ; ss (cid:48) , c † j,s is the creation operatorwith j , s the site and spin index. J and Ω ss (cid:48) are the tun-neling rate and Raman coupling strength, respectively.For alkaline-earth(-like) atoms, we use three states in the S ground manifold to define the synthetic dimension.The long lifetime P or P levels are used as the inter-mediate states for the Raman process (see Fig. 1b) suchthat δm F = ± | F, m F (cid:105) , | F, m F − (cid:105) and | F − , m F (cid:105) from the ground-state manifold to avoid δm F = ± z -polarization (responsiblefor π transition) and in-plane-polarization (responsiblefor σ transition) components, which can be written as (cid:126) E s = ˆ e π E πs + ˆ e σ E σs . For alkaline-earth(-like) atoms, wechoose E π = 0 so that (cid:101) Ω j ∝ E σ E π ∗ , (cid:101) Ω j ∝ E π E σ ∗ and (cid:101) Ω j ∝ E σ E σ ∗ . The corresponding Raman couplingphases are φ j ;21 = jφ + ϕ π − ϕ σ , where ϕ π,σs are the( π, σ )-component phases of the s -th Raman laser at site j = 0, and φ = k R d x cos( θ ) gives rise to the magneticflux penetrating the side plaquette of the tube, with k R the recoil momentum of the Raman lasers and d x the lat-tice constant. Similarly, we have φ j ;32 = jφ + ϕ σ − ϕ π , φ j ;13 = − jφ + ϕ σ − ϕ σ . To obtain a uniform magneticflux for each side plaquette of the tube, we choose theincident angle θ such that φ = 2 π/
3. The phases ϕ σ,πs determine the flux through the tube Φ = ∆ ϕ − ∆ ϕ ,where ∆ ϕ s = ϕ πs − ϕ σs is the phase difference betweentwo polarization components of the s -th Raman laser.The phase differences ∆ ϕ s can be simply controlled andtuned using wave plates, and they do not depend on thetransverse positions y and z . Ω δ N T N (a) 𝐸 𝐸 𝐸 𝛷 𝛷 𝑘 (b) (d) (c) Edge states Edge states Closed at 𝑘 = 𝜋
Red:
𝛷 = 𝜋
Blue:
𝛷 = 0
Closed at 𝑘 = 0 𝐶 = 1 𝐶 = −2 𝐶 = 1 𝑊 𝜋 = 𝑊 , 𝜋 =± FIG. 2: (a) Band structures in the topological phase. (b)Phase diagram in the Ω - δ plane, with solid (dashed) linesthe boundary between topological (T) and normal (N) phasesfor the upper (lower) gap. (c) and (d) Band structures atthe phase boundary (blue lines) with δ = 0 and Ω = 1.Common parameters: Ω = 2 with energy unit J . For alkali atoms, we choose E σ = 0, yielding (cid:101) Ω j = α E π E σ ∗ , (cid:101) Ω j = α E π E π ∗ and (cid:101) Ω j = β E σ E π ∗ + α E π E σ ∗ , with α s,s (cid:48) , β s,s (cid:48) determined by the transi-tion dipole matrix. We further consider ( E π , E σ ) (cid:28) ( E σ , E π ) (cid:28) E π , thus (cid:101) Ω j (cid:39) α E π E σ ∗ with ampli-tudes Ω ∼ Ω ∼ Ω . Similar as the alkaline-earth(-like) atoms, we obtain uniform magnetic flux φ = 2 π/ k R d x cos( θ ) = 2 π/
3. Theflux through the tube becomes Φ = ∆ ϕ + ∆ ϕ , whichcan also be tuned at will through the polarization con-trol. With proper gauge choice, we can set the tunnelingphases as φ j ;21 = φ j ;32 = jφ and φ j ;13 = jφ + Φ, asshown in Fig. 1d. Phase diagram.—
The Bloch Hamiltonian in the basis[ c k, , c k, , c k, ] T reads H k = − J cos( k − φ ) Ω Ω e − i Φ Ω − J cos( k ) Ω Ω e i Φ Ω − J cos( k + φ ) , (2)with k the momentum along the real-space lattice. ForΩ = Ω = Ω , the above Hamiltonian is nothingbut the Harper-Hofstadter Hamiltonian with Φ the ef-fective momentum along the synthetic dimension, and φ = 2 π/ C n = i π (cid:90) dkd Φ (cid:104) ∂ Φ u n | ∂ k | u n (cid:105) − (cid:104) ∂ k u n | ∂ Φ | u n (cid:105) , (3)where | u n (cid:105) is the Bloch states of the n -th band, satis-fying H k (Φ) | u n (Φ , k ) (cid:105) = E n (Φ , k ) | u n (Φ , k ) (cid:105) . In Fig. 2a,we plot the band structures as a function of Φ with anopen boundary condition along the real-lattice direction.There are three bands, and two gapless edge states (one (a) (b) (c) 𝑗 ۃ𝑗ۄ 𝑗 𝑛̃ 𝑗 ,2 𝑛̃ 𝑗 ,1 𝑛̃ 𝑗 ,3 𝛷 𝛷 𝛷 𝛷 𝛷 𝜏 p /𝐽 −1 ۃ Δ 𝑗ۄ 𝑓 = 1 𝑓 = 2 𝑓 = 3 𝑛 𝑗 FIG. 3: (a) Total density distribution during one pump cycle.Inset shows the non-adiabatic effects on the center-of-massshift. (b) Center-of-mass (red line) and the rotated-spin con-tributions (blue lines) during one pump cycle. The blue linesare (cid:80) j j (cid:101) n j,f ( t ) /N with f = 1 , , at each ends) in each gap. The two edge states crossonly at Φ = 0 and Φ = π , where the tube belongs to a1D topological insulator with quantized winding number(Zak phase) [47] W ,πn = 1 π (cid:73) dk (cid:104) u n | ∂ k | u n (cid:105) (cid:12)(cid:12) Φ=0 ,π . (4)The winding number is protected by a generalized in-version symmetry I H k I − = H − k , where the inversionsymmetry I swaps spin states | (cid:105) and | (cid:105) [48].The Chern number and winding number are still welldefined even when the Raman couplings have detuningsand/or the coupling strength Ω ss (cid:48) are nonequal. Thechanges in these Raman coupling parameters drive thephase transition from topological to trivial insulators.The detuning can be introduced by including additionalterms (cid:80) j ; s δ s c † j,s c j,s in the Hamiltonian Eq. 1. Here-after we will fix Ω = Ω ≡ Ω and δ = δ = 0for simplicity. The phase diagram in the Ω - δ planeis shown in Fig. 2b. The solid (dashed) lines are thephase boundaries corresponding to the gap closing be-tween two lower (upper) bands, with topological phasesbetween two boundaries. At the phase boundaries, thecorresponding band gaps close at Φ = 0 (Φ = π ) for thetwo lower (upper) bands, as shown in Fig. 2c. In addi-tion, for the two lower (upper) bands, the gap closes at k = 0 and k = π ( k = π and k = 0) on the right andleft boundaries, respectively, as shown in Fig. 2d. Thegaps reopen in the trivial phase with the disappearanceof edge states. Topological pumping.—
The three-leg Hall tube is aminimal Laughlin’s cylinder. When the flux through thetube is adiabatically changed by 2 π , the shift of Wannier-function center is proportional to the Chern number of the corresponding band [39–42]. Therefore all particlesare pumped by C site (with C the total Chern number ofthe occupied bands) as Φ changes by 2 π , i.e., C particlesare pumped from one edge to another. Given the abilityof controlling the flux through the tube, we can mea-sure the topological Chern number based on topologicalpumping by tuning the flux Φ adiabatically (comparedto the band gaps).Here we consider the Fermi energy in the first gap withonly the lowest C = 1 band occupied, and study thezero temperature pumping process (the pumped parti-cle is still well quantized for low temperature compar-ing to the band gap) [49]. The topological pumpingeffect can be identified as the quantized center-of-massshift of the atom cloud [49–52] in a weak harmonic trap V trap = v T j . The harmonic trap strength v T = 0 . J and the atom number N = 36 are chosen such that theatom cloud has a large insulating region (correspondingto one atom per unit-cell) at the trap center. For sim-plicity, we set Ω ss (cid:48) = J and δ s = 0 for all s, s (cid:48) , choose thegauge as φ j ;21 = φ j ;32 = jφ , φ j ;13 = jφ − Φ, and changeΦ slowly (compared to the band gap) as Φ( t ) = πtτ p .In Figs. 3a and 3b, we plot the total density distribu-tion n j ( t ) and the center-of-mass (cid:104) j ( t ) (cid:105) ≡ (cid:80) j jn j ( t ) /N shift during one pumping circle with τ p = 40 J − , andwe clearly see the quantization of the pumped atom (cid:104) ∆ j (cid:105) ≡ (cid:104) j ( τ p ) (cid:105) − (cid:104) j (0) (cid:105) = 1. The atom cloud shifts asa whole with n j ( t ) = 1 near the trap center. The in-set in Fig. 3a shows non-adiabatic effect (finite pumpingduration τ p ) on the pumped atom.The atoms are equally distributed on the three spinstates [i.e., n j,s ( t ) ≡ (cid:104) c † j,s c j,s (cid:105) = n j ( t )]. To see thepumping process more clearly, we can examine the spindensities in the rotated basis (cid:101) n j,f ( t ) ≡ (cid:104) (cid:101) c † j,f (cid:101) c j,f (cid:105) , with (cid:101) c j,f = √ (cid:80) s c j,s e is fπ − Φ3 . The Hamiltonian in these ba-sis reads H = (cid:80) f =1 H f , where H f = 2 J cos( K j,f, Φ ) (cid:101) c † j,f (cid:101) c j,f + ( J (cid:101) c † j,f (cid:101) c j,f +1 + H.c. ) (5)is the typical Aubry-Andr´e-Harper (AAH) Hamilto-nian [53, 54] with K j,f, Φ = π ( f + j ) − Φ3 . The bulk-atomflow during the pumping can be clearly seen from thespin densities (cid:101) n j,f ( t ), as shown in Fig. 3c. Each spincomponent contributes exactly one third of the quantizedcenter-of-mass shift (see Fig. 3b). The quantized pump-ing can also be understood by noticing that the AAHHamiltonians H s are permutated as H → H → H → H after one pump circle. Each H f returns to itself afterthree pump circles with particles pumped by three sites(since the lattice period of H f is 3). Therefore, for thetotal Hamiltonian H , particles are pumped by one siteafter one pump circle. The physics for different values ofΩ ss (cid:48) and δ s are similar, except that the rotated basis (cid:101) c j,f may take different forms.In the presence of strong interaction, which is long- (c) (b) (d) (a) 𝜏 𝜏 𝜏 𝜏 𝑛 𝑠 𝑛 𝑠 𝜋 𝑛 𝑠 𝜋 𝑡/𝐽 −1 𝑡/𝐽 −1 𝑡/𝐽 −1 𝑡/𝐽 −1 ( 𝜋 ) FIG. 4: Quench dynamics for Φ = 0 (solid lines) and aver-aged over random Φ (dashed lines). Time evolution of spinpopulations (a) and averaged momenta (b) with Ω = Ω.Time evolution of spin populations at k = π with Ω = 5 in(c) and Ω = 7 . n πs = n s ( π ) (cid:80) s (cid:48) n s (cid:48) ( π ) , τ and τ cross atΩ c = 5 .
8. Both gaps are topological in (a) and (b), and onlythe upper (lower) gap is topological in (c) [(d)]. Commonparameters: Ω = 6 . δ = − .
2Ω with energy unit J . range in the synthetic dimension, our scheme offersan ideal platform for studying exotic fractional quan-tum Hall phases and topological fractional charge pump-ing [43]. Quench dynamics.—
Besides topological pumping, thequench dynamics of the system can also be used todemonstrate the presence of gauge field φ and detectthe phase transitions [37]. Here we study how Φ affectsthe quench dynamics by considering that all atoms areinitially prepared in state | (cid:105) , then the inter leg couplingsare suddenly activated by turning on the Raman laserbeams. In Figs. 4a and 4b, we show the time evolutionof the fractional spin populations n s = N (cid:82) dkn s ( k ), aswell as the momenta (cid:104) k (cid:105) = (cid:80) s (cid:104) k s (cid:105) and (cid:104) ∆ k (cid:105) = (cid:104) k (cid:105)−(cid:104) k (cid:105) (both can be measured by time-of-flight imaging) for Φ =0, where n s ( k ) = (cid:104) c s ( k ) † c s ( k ) (cid:105) and (cid:104) k s (cid:105) = N (cid:82) kn s ( k ) dk with N the total atom number. We find that the timeevolutions show similar oscillating behaviors for differentΦ, but with different frequencies and amplitudes. Thedifference between the momenta of atoms transferred tostate | (cid:105) and | (cid:105) increase noticeably at early time as a re-sult of the magnetic flux φ penetrating the surface of thetube [37], which does not depend on the flux Φ throughthe tube.The quench dynamics can also be used to measure thegap closing at phase boundaries. Similar as Ref. [37], weintroduce two times τ and τ , at which the spin- | (cid:105) andspin- | (cid:105) populations at k = 0 (or k = π ) reach their firstmaxima, to identify the phase boundary. As we change δ or Ω across the phase boundary (one gap closes and thedynamics is characterized by a single frequency), τ and τ cross each other. Notice that above discussions onlyapply to Φ = 0 , π where the gap closing occurs. We findthat τ and τ are Φ-dependent [55] and would cross eachother even when no gap closing occurs for other valuesof Φ, and the crossing point is generally away from thephase boundaries. Therefore, the measurement of gap (a) ℰ + ℰ ℰ Lattice Lattice 𝜃 𝐵 𝑥 𝑦 𝑧 ℰ ℰ ℰ ℰ ℰ δ 𝑆 (b) |1⟩ |2⟩ |3⟩ FIG. 5: (a) Schematic of the experimental setup in [37]. (b)The three hyperfine ground states and the corresponding two-photon Raman transitions. closing based on quench dynamics is possible only if Φcan be controlled. As an example, we consider Φ = 0 andplot the time evolution of the spin populations at k = π with Ω around the left phase boundary Ω c , as shownin Figs. 4c and 4d.The ability to lock and control the flux Φ is crucial forthe study of both topological properties and quench dy-namics. We notice that in the experiment in [37], Φ can-not be controlled and may vary from one experimental re-alization to another. It is also different for different tubeswithin one experimental realization. The experimentalsetup in [37] is shown in Fig. 5a, where three linearly po-larized Raman lasers are used to couple three hyperfinespin states of Yb atoms (Fig. 5b). It is straightfor-ward to show that the flux Φ = 3 ϕ σ − ϕ π − ϕ σ , whichcannot be controlled since the Raman lasers propagatealong different paths, not to mention that their wave-lengths are generally not commensurate with the lattice.Moreover, in realistic experiments, arrays of independentfermionic synthetic tubes are realized simultaneously dueto the transverse atomic distributions in the y , z direc-tions [36, 37], and the synthetic tubes at different y wouldhave different Φ due to the y -dependent ϕ σ . For the pa-rameters in Ref. [37], the difference of Φ between neigh-bor tubes in y direction is about 2 π × . . J and δ = − . T /T F = 0 .
3, with initial Fermi temperature T F given bythe difference between the Fermi energy E F and the ini-tial band minimum − J (i.e., T F = E F + 2 J ) [37]. Weuse T F = 2 J to get the similar initial filling and Fermidistribution as those in the experiment (the results areinsensitive to T F ). Moreover, τ and τ , which cross eachother at different values of Ω for different Φ with aver-aged value ¯Ω = Ω (cid:54) = Ω c , may not be suitable to iden-tify the phase boundaries (i.e., gap closings) for a randomflux Φ. Other experimental imperfections such as spin-selective imaging error may also affect the measured spindynamics, and the final observed phase boundary in [37]is smaller than Ω and Ω c . Conclusion.—
In summary, we propose a simplescheme to realize a controllable flux Φ through the syn-thetic Hall tube that can be tuned at will, and study theeffects of the flux Φ on the system topology and dynam-ics. The quench dynamics averaged over the random fluxmay better explain previous experimental results wherethe flux is not locked. Our results provide a new platformfor studying topological physics in a tube geometry withtunable flux and may be generalized with other syntheticdegrees of freedom, such as momentum states [56–58] andlattice orbitals [59, 60].
Acknowledgements : XWL and CZ are supported byAFOSR (FA9550-16-1-0387), NSF (PHY-1806227), andARO (W911NF-17-1-0128). JZ is supported by the Na-tional Key Research and Development Program of China(2016YFA0301602). ∗ Corresponding author.Email: [email protected] † Corresponding author.Email: [email protected][1] D. Jaksch and P. Zoller, The cold atom Hubbard toolbox,Ann. Phys. (Amsterdam) , 52 (2005).[2] I, Bloch, J. Dalibard, and W. Zwerger, Many-body physicswith ultracold gases, Rev. Mod. Phys. , 885, (2008).[3] Y.-J. Lin, K. Jim´enez-Garc´ıa, and I. B. Spielman, Spin-orbit-coupled Bose-Einstein condensates, Nature (Lon-don) , 83, (2011).[4] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du,B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen,and J.-W. Pan, Collective dipole oscillations of a spin-orbit coupled Bose-Einstein condensate, Phys. Rev. Lett. , 115301, (2012).[5] C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels, Ob-servation of zitterbewegung in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A , 021604, (2013).[6] S.-C. Ji, J.-Y. Zhang, L. Zhang, Z.-D. Du, W. Zheng, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Experimentaldetermination of the finite-temperature phase diagram ofa spinorbit coupled Bose gas, Nat. Phys. , 314 (2014).[7] A. Olson, S. Wang, R. Niffenegger, C. Li, C. Greene,and Y. Chen, Tunable landau-zener transitions in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A ,013616, (2014).[8] P. Wang, Z. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai,and J. Zhang, Spin-orbit coupled degenerate fermi gases,Phys. Rev. Lett. , 095301, (2012).[9] L. Cheuk, A. Sommer, Z. Hadzibabic, T. Yefsah, W. Bakr,and M. Zwierlein, Spin-injection spectroscopy of a spin-orbit coupled fermi gas, Phys. Rev. Lett. , 095302,(2012).[10] Z. Wu, L. Zhang, W. Sun, X. Xu, B. Wang, S. Ji, Y. Deng, S. Chen, X.-J. Liu, and J. Pan, Realizationof two-dimensional spin-orbit coupling for Bose-Einsteincondensates, Science , 83, (2016).[11] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu,C. Zhang, P. Wang, and J. Zhang, Experimental observa-tion of a topological band gap opening in ultracold Fermigases with two-dimensional spin-orbit coupling, Phys.Rev. Lett. , 235304, (2016).[12] L. Huang, Z. Meng, P. Wang, P. Peng, S. Zhang, L. Chen,D. Li, Q. Zhou, and J. Zhang, Experimental realization oftwo-dimensional synthetic spin-orbit coupling in ultracoldfermi gases, Nat. Phys. , 540, (2016).[13] L. Huang, P. Peng, D. Li, Z. Meng, L. Chen, C. Qu,P. Wang, C. Zhang and J. Zhang, Observation of Floquetbands in driven spin-orbit-coupled Fermi gases, Phys. Rev.A , 013615 (2018).[14] S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall,G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, andJ. Ye, Spin–orbit-coupled fermions in an optical latticeclock, Nature , 66 (2017).[15] S. L. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar,A. Safavi-Naini, M. L. Wall, C. Salomon, A. M. Rey, andJ. Ye, Dynamics of interacting fermions under spin–orbitcoupling in an optical lattice clock, Nat. Phys. , 399(2018).[16] D. Campbell, R. Price, A. Putra, A. Vald´es-Curiel,D. Trypogeorgos, and I. B. Spielman, Magnetic phasesof spin-1 spin-orbit-coupled Bose gases, Nat. Commun. ,10897 (2016).[17] X. Luo, L. Wu, J. Chen, Q. Guan, K. Gao, Z.-F. Xu,L. You and R. Wang, Tunable atomic spin-orbit couplingsynthesized with a modulating gradient magnetic field,Sci. Rep. , 18983 (2016).[18] A. Vald´es-Curiel, D. Trypogeorgos, Q.-Y. Liang,R. P. Anderson, and I. Spielman, Unconventional topol-ogy with a Rashba spin-orbit coupled quantum gas,arXiv:1907.08637.[19] J. Dalibard, F. Gerbier, G. Juzeli¯unas, and P. ¨Ohberg,Colloquium: Artificial gauge potentials for neutral atoms,Rev. Mod. Phys. , 1523, (2011).[20] N. Goldman, G. Juzeli¯unas, P. ¨Ohberg, and I. B. Spiel-man, Light-induced gauge fields for ultracold atoms, Rep.Prog. Phys. , 126401 (2014).[21] V. Galitski and I. B. Spielman, Spin-orbit coupling inquantum gases, Nature (London) , 49 (2013).[22] H. Zhai, Spin-orbit coupled quantum gases, Int. J. Mod.Phys. B , 1230001 (2012).[23] H. Zhai, Degenerate quantum gases with spin-orbit cou-pling: a review, Rep. Prog. Phys. , 026001 (2015).[24] J. Zhang, H. Hu, X.-J. Liu, and H. Pu, Fermi gases withsynthetic spin–orbit coupling, Annu. Rev. Cold At. Mol.
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Quench dynamics with different values of Φ . As wediscussed in the main text, even when no gap closing oc-curs for Φ away from 0 and π , τ and τ would cross eachother as we increase Ω . For different Φ, the crossingpoints are different and distributed around the averagedvalue Ω c = Ω. In Fig. S1, we plot the time evolution ofthe spin populations at k = π for different values of Φ. Quench dynamics with a harmonic trap.
In realistic ex-periments, a harmonic trap is usually applied to confinethe atoms. Here we consider its effects on the quenchdynamics studied in the main text. From the experi-mental parameters [1] with a harmonic trap frequency ω x = 2 π ×
57 Hz and J = 2 π ×
264 Hz, we obtain theharmonic trap as V trap = v T j with the trap strength v T (cid:39) . J . In Fig. S2, we plot the time evolutionof spin populations with all other parameters the sameas in Fig. 3a in the main text. We see that the quenchdynamics are hardly affected by the harmonic trap. 𝑛 𝑠 𝜋 𝑛 𝑠 𝜋 𝑛 𝑠 𝜋 𝑛 𝑠 𝜋 (a) (b) (c) (d) 𝑡/𝐽 −1 𝑡/𝐽 −1 𝑡/𝐽 −1 𝑡/𝐽 −1 𝛷 = 0
𝛷 = 0.1𝜋
𝛷 = −0.1𝜋
𝛷 = 𝜋
FIG. S1: (a)-(d) Time evolution of the spin populations at k = π for different values of Φ. All other parameters are thesame as Fig. 3c in the main text. 𝑛 𝑠 𝑡/𝐽 −1 FIG. S2: Quench dynamics in the presence of a harmonictrap V trap = v T j with trap strength v T (cid:39) . J . Allother parameters are the same as in Fig. 3a in the main text. ∗ Corresponding author.Email: [email protected] † Corresponding author.Email: [email protected][1] J. H. Han, J. H. Kang, and Y. Shin, Band Gap Closingin a Synthetic Hall Tube of Neutral Fermions, Phys. Rev.Lett.122