Abstract

Realizing the topological bands of helical states poses a challenge in studying ultracold atomic gases. Motivated by the recent experimental success in realizing chiral optical ladders, here we present a scheme for synthesizing topological quantum matter, especially the quantum spin Hall phase, in the chiral optical ladders. More precisely, we first establish the synthetic pseudo-spin-orbit coupling and Zeeman splitting in the chiral ladders. After analyzing the band structure of the ladders exposed to the bichromatic optical potentials, we report the existence of quantum spin Hall phase. We further identify a rich phase diagram of the bichromatic chiral ladders, illustrating that our proposal features a large space of system parameters exhibiting quantum phase transitions. Our scheme is within reach of the existing ladder optical lattices and hence provides a new method to engineer the elaborate topological bands for cold atomic gases.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  28. Y. E. Kraus and O. Zilberberg, “Quasiperiodicity and topology transcend dimensions,” Nat. Phys. 12(7), 624–626 (2016).
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  29. P. G. Harper, “Single Band Motion of Conduction Electrons in a Uniform Magnetic Field,” Proc. Phys. Soc., London, Sect. A 68(10), 874–878 (1955).
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    [Crossref]
  31. Y. Yang, Z. Xu, L. Sheng, B. Wang, D. Y. Xing, and D. N. Sheng, “Time-reversal-symmetry-broken quantum spin Hall effect,” Phys. Rev. Lett. 107(6), 066602 (2011).
    [Crossref]
  32. N. Goldman, W. Beugeling, and C. Smith, “Topological phase transitions between chiral and helical spin with spin-orbit coupling and a magnetic field-tight binding,” Europhys. Lett. 97(2), 23003 (2012).
    [Crossref]
  33. W. Beugeling, N. Goldman, and C. M. Smith, “Topological phases in a two-dimensional lattice: Magnetic field versus spin-orbit coupling,” Phys. Rev. B 86(7), 075118 (2012).
    [Crossref]
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    [Crossref]
  35. M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice,” Nat. Phys. 12(4), 350–354 (2016).
    [Crossref]
  36. S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
    [Crossref]
  37. M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553(7686), 55–58 (2018).
    [Crossref]
  38. O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553(7686), 59–62 (2018).
    [Crossref]
  39. L. Fu and C. Kane, “Time reversal polarization and a Z2 adiabatic spin pump,” Phys. Rev. B 74(19), 195312 (2006).
    [Crossref]
  40. S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck, and T. Vekua, “Spontaneous Increase of Magnetic Flux and Chiral-Current Reversal in Bosonic Ladders: Swimming against the Tide,” Phys. Rev. Lett. 115(19), 190402 (2015).
    [Crossref]
  41. M. C. Strinati, E. Cornfeld, D. Rossini, S. Barbarino, M. Dalmonte, R. Fazio, E. Sela, and L. Mazza, “Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders,” Phys. Rev. X 7(2), 021033 (2017).
    [Crossref]
  42. J. Jünemann, A. Piga, S. J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez, “Exploring Interacting Topological Insulators with Ultracold Atoms: The Synthetic Creutz-Hubbard Model,” Phys. Rev. X 7(3), 031057 (2017).
    [Crossref]

2019 (3)

N. R. Cooper, J. Dalibard, and I. B. Spielman, “Topological bands for ultracold atoms,” Rev. Mod. Phys. 91(1), 015005 (2019).
[Crossref]

H. Cai, J. Liu, J. Wu, Y. He, S. Y. Zhu, J. X. Zhang, and D. W. Wang, “Experimental Observation of Momentum-Space Chiral Edge Currents in Room-Temperature Atoms,” Phys. Rev. Lett. 122(2), 023601 (2019).
[Crossref]

J. H. Han, J. H. Kang, and Y. Shin, “Band Gap Closing in a Synthetic Hall Tube of Neutral Fermions,” Phys. Rev. Lett. 122(6), 065303 (2019).
[Crossref]

2018 (4)

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553(7686), 55–58 (2018).
[Crossref]

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553(7686), 59–62 (2018).
[Crossref]

J. H. Kang, J. H. Han, and Y. Shin, “Realization of a cross-linked chiral ladder with neutral fermions in an optical lattice by orbital-momentum coupling,” Phys. Rev. Lett. 121(15), 150403 (2018).
[Crossref]

D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, “Topological quantum matter with cold atoms,” Adv. Phys. 67(4), 253–402 (2018).
[Crossref]

2017 (4)

S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, “Spin-orbit-coupled fermions in an optical lattice clock,” Nature (London) 542(7639), 66–70 (2017).
[Crossref]

F. A. An, E. J. Meier, and B. Gadway, “Direct observation of chiral currents and magnetic reflection in atomic flux lattices,” Sci. Adv. 3(4), e1602685 (2017).
[Crossref]

M. C. Strinati, E. Cornfeld, D. Rossini, S. Barbarino, M. Dalmonte, R. Fazio, E. Sela, and L. Mazza, “Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders,” Phys. Rev. X 7(2), 021033 (2017).
[Crossref]

J. Jünemann, A. Piga, S. J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez, “Exploring Interacting Topological Insulators with Ultracold Atoms: The Synthetic Creutz-Hubbard Model,” Phys. Rev. X 7(3), 031057 (2017).
[Crossref]

2016 (5)

M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper, J. Ye, and A. M. Rey, “Synthetic Spin Orbit Coupling in an Optical Lattice Clock,” Phys. Rev. Lett. 116(3), 035301 (2016).
[Crossref]

L. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, and L. Fallani, “Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition,” Phys. Rev. Lett. 117(22), 220401 (2016).
[Crossref]

M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice,” Nat. Phys. 12(4), 350–354 (2016).
[Crossref]

S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
[Crossref]

Y. E. Kraus and O. Zilberberg, “Quasiperiodicity and topology transcend dimensions,” Nat. Phys. 12(7), 624–626 (2016).
[Crossref]

2015 (4)

M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, “Observation of chiral edge states with neutral fermions in synthetic Hall ribbons,” Science 349(6255), 1510–1513 (2015).
[Crossref]

B. Stuhl, H. I. Lu, L. Aycock, D. Genkina, and I. Spielman, “Visualizing edge states with an atomic Bose gas in the quantum Hall regime,” Science 349(6255), 1514–1518 (2015).
[Crossref]

M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbéne, N. R. Cooper, I. Bloch, and N. Goldman, “Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms,” Nat. Phys. 11(2), 162–166 (2015).
[Crossref]

S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck, and T. Vekua, “Spontaneous Increase of Magnetic Flux and Chiral-Current Reversal in Bosonic Ladders: Swimming against the Tide,” Phys. Rev. Lett. 115(19), 190402 (2015).
[Crossref]

2014 (3)

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515(7526), 237–240 (2014).
[Crossref]

M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Observation of chiral currents with ultracold atoms in bosonic ladders,” Nat. Phys. 10(8), 588–593 (2014).
[Crossref]

D. Hügel and B. Paredes, “Chiral ladders and the edges of quantum Hall insulators,” Phys. Rev. A 89(2), 023619 (2014).
[Crossref]

2013 (5)

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of Topological Phase Transitions in Photonic Quasicrystals,” Phys. Rev. Lett. 110(7), 076403 (2013).
[Crossref]

Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal,” Phys. Rev. Lett. 111(22), 226401 (2013).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 111(18), 185301 (2013).
[Crossref]

H. Miyake, G. Siviloglou, C. Kennedy, W. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices,” Phys. Rev. Lett. 111(18), 185302 (2013).
[Crossref]

M. Atala, M. Aidelsburger, J. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

2012 (4)

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological States and Adiabatic Pumping in Quasicrystals,” Phys. Rev. Lett. 109(10), 106402 (2012).
[Crossref]

L. J. Lang, X. Cai, and S. Chen, “Edge States and Topological Phases in One-Dimensional Optical Superlattices,” Phys. Rev. Lett. 108(22), 220401 (2012).
[Crossref]

N. Goldman, W. Beugeling, and C. Smith, “Topological phase transitions between chiral and helical spin with spin-orbit coupling and a magnetic field-tight binding,” Europhys. Lett. 97(2), 23003 (2012).
[Crossref]

W. Beugeling, N. Goldman, and C. M. Smith, “Topological phases in a two-dimensional lattice: Magnetic field versus spin-orbit coupling,” Phys. Rev. B 86(7), 075118 (2012).
[Crossref]

2011 (1)

Y. Yang, Z. Xu, L. Sheng, B. Wang, D. Y. Xing, and D. N. Sheng, “Time-reversal-symmetry-broken quantum spin Hall effect,” Phys. Rev. Lett. 107(6), 066602 (2011).
[Crossref]

2010 (1)

N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, “Realistic Time-Reversal Invariant Topological Insulators with Neutral Atoms,” Phys. Rev. Lett. 105(25), 255302 (2010).
[Crossref]

2009 (1)

M. Modugo, “Exponential localization in one-dimensional quasi-periodic optical lattices,” New J. Phys. 11(3), 033023 (2009).
[Crossref]

2008 (1)

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose Einstein condensate,” Nature 453(7197), 895–898 (2008).
[Crossref]

2006 (2)

L. Fu and C. Kane, “Time reversal polarization and a Z2 adiabatic spin pump,” Phys. Rev. B 74(19), 195312 (2006).
[Crossref]

S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M. Duan, “Spin Hall Effects for Cold Atoms in a Light-Induced Gauge Potential,” Phys. Rev. Lett. 97(24), 240401 (2006).
[Crossref]

1983 (1)

D. J. Thouless, “Quantization of particle transport,” Phys. Rev. B 27(10), 6083–6087 (1983).
[Crossref]

1976 (1)

D. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B 14(6), 2239–2249 (1976).
[Crossref]

1955 (1)

P. G. Harper, “Single Band Motion of Conduction Electrons in a Uniform Magnetic Field,” Proc. Phys. Soc., London, Sect. A 68(10), 874–878 (1955).
[Crossref]

Abanin, D.

M. Atala, M. Aidelsburger, J. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Aidelsburger, M.

M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice,” Nat. Phys. 12(4), 350–354 (2016).
[Crossref]

M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbéne, N. R. Cooper, I. Bloch, and N. Goldman, “Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms,” Nat. Phys. 11(2), 162–166 (2015).
[Crossref]

M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Observation of chiral currents with ultracold atoms in bosonic ladders,” Nat. Phys. 10(8), 588–593 (2014).
[Crossref]

M. Atala, M. Aidelsburger, J. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 111(18), 185301 (2013).
[Crossref]

An, F. A.

F. A. An, E. J. Meier, and B. Gadway, “Direct observation of chiral currents and magnetic reflection in atomic flux lattices,” Sci. Adv. 3(4), e1602685 (2017).
[Crossref]

Atala, M.

M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbéne, N. R. Cooper, I. Bloch, and N. Goldman, “Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms,” Nat. Phys. 11(2), 162–166 (2015).
[Crossref]

M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Observation of chiral currents with ultracold atoms in bosonic ladders,” Nat. Phys. 10(8), 588–593 (2014).
[Crossref]

M. Atala, M. Aidelsburger, J. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 111(18), 185301 (2013).
[Crossref]

Aycock, L.

B. Stuhl, H. I. Lu, L. Aycock, D. Genkina, and I. Spielman, “Visualizing edge states with an atomic Bose gas in the quantum Hall regime,” Science 349(6255), 1514–1518 (2015).
[Crossref]

Barbarino, S.

M. C. Strinati, E. Cornfeld, D. Rossini, S. Barbarino, M. Dalmonte, R. Fazio, E. Sela, and L. Mazza, “Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders,” Phys. Rev. X 7(2), 021033 (2017).
[Crossref]

Barreiro, J.

M. Atala, M. Aidelsburger, J. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

Barreiro, J. T.

M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbéne, N. R. Cooper, I. Bloch, and N. Goldman, “Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms,” Nat. Phys. 11(2), 162–166 (2015).
[Crossref]

M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Observation of chiral currents with ultracold atoms in bosonic ladders,” Nat. Phys. 10(8), 588–593 (2014).
[Crossref]

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, “Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 111(18), 185301 (2013).
[Crossref]

Bermudez, A.

J. Jünemann, A. Piga, S. J. Ran, M. Lewenstein, M. Rizzi, and A. Bermudez, “Exploring Interacting Topological Insulators with Ultracold Atoms: The Synthetic Creutz-Hubbard Model,” Phys. Rev. X 7(3), 031057 (2017).
[Crossref]

N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, “Realistic Time-Reversal Invariant Topological Insulators with Neutral Atoms,” Phys. Rev. Lett. 105(25), 255302 (2010).
[Crossref]

Beugeling, W.

W. Beugeling, N. Goldman, and C. M. Smith, “Topological phases in a two-dimensional lattice: Magnetic field versus spin-orbit coupling,” Phys. Rev. B 86(7), 075118 (2012).
[Crossref]

N. Goldman, W. Beugeling, and C. Smith, “Topological phase transitions between chiral and helical spin with spin-orbit coupling and a magnetic field-tight binding,” Europhys. Lett. 97(2), 23003 (2012).
[Crossref]

Bloch, I.

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553(7686), 55–58 (2018).
[Crossref]

M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice,” Nat. Phys. 12(4), 350–354 (2016).
[Crossref]

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N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, “Realistic Time-Reversal Invariant Topological Insulators with Neutral Atoms,” Phys. Rev. Lett. 105(25), 255302 (2010).
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S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
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M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553(7686), 55–58 (2018).
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M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper, J. Ye, and A. M. Rey, “Synthetic Spin Orbit Coupling in an Optical Lattice Clock,” Phys. Rev. Lett. 116(3), 035301 (2016).
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M. C. Strinati, E. Cornfeld, D. Rossini, S. Barbarino, M. Dalmonte, R. Fazio, E. Sela, and L. Mazza, “Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders,” Phys. Rev. X 7(2), 021033 (2017).
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N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, “Realistic Time-Reversal Invariant Topological Insulators with Neutral Atoms,” Phys. Rev. Lett. 105(25), 255302 (2010).
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S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck, and T. Vekua, “Spontaneous Increase of Magnetic Flux and Chiral-Current Reversal in Bosonic Ladders: Swimming against the Tide,” Phys. Rev. Lett. 115(19), 190402 (2015).
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M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553(7686), 55–58 (2018).
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M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice,” Nat. Phys. 12(4), 350–354 (2016).
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M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbéne, N. R. Cooper, I. Bloch, and N. Goldman, “Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms,” Nat. Phys. 11(2), 162–166 (2015).
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M. C. Strinati, E. Cornfeld, D. Rossini, S. Barbarino, M. Dalmonte, R. Fazio, E. Sela, and L. Mazza, “Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders,” Phys. Rev. X 7(2), 021033 (2017).
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M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, “Observation of chiral edge states with neutral fermions in synthetic Hall ribbons,” Science 349(6255), 1510–1513 (2015).
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M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of Topological Phase Transitions in Photonic Quasicrystals,” Phys. Rev. Lett. 110(7), 076403 (2013).
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H. Miyake, G. Siviloglou, C. Kennedy, W. Burton, and W. Ketterle, “Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices,” Phys. Rev. Lett. 111(18), 185302 (2013).
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N. Goldman, W. Beugeling, and C. Smith, “Topological phase transitions between chiral and helical spin with spin-orbit coupling and a magnetic field-tight binding,” Europhys. Lett. 97(2), 23003 (2012).
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N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, “Realistic Time-Reversal Invariant Topological Insulators with Neutral Atoms,” Phys. Rev. Lett. 105(25), 255302 (2010).
[Crossref]

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M. C. Strinati, E. Cornfeld, D. Rossini, S. Barbarino, M. Dalmonte, R. Fazio, E. Sela, and L. Mazza, “Laughlin-like States in Bosonic and Fermionic Atomic Synthetic Ladders,” Phys. Rev. X 7(2), 021033 (2017).
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B. Stuhl, H. I. Lu, L. Aycock, D. Genkina, and I. Spielman, “Visualizing edge states with an atomic Bose gas in the quantum Hall regime,” Science 349(6255), 1514–1518 (2015).
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S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
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S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
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S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
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Troyer, M.

S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
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G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515(7526), 237–240 (2014).
[Crossref]

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S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, U. Schollwöck, and T. Vekua, “Spontaneous Increase of Magnetic Flux and Chiral-Current Reversal in Bosonic Ladders: Swimming against the Tide,” Phys. Rev. Lett. 115(19), 190402 (2015).
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M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of Topological Phase Transitions in Photonic Quasicrystals,” Phys. Rev. Lett. 110(7), 076403 (2013).
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Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological States and Adiabatic Pumping in Quasicrystals,” Phys. Rev. Lett. 109(10), 106402 (2012).
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S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, “Spin-orbit-coupled fermions in an optical lattice clock,” Nature (London) 542(7639), 66–70 (2017).
[Crossref]

M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper, J. Ye, and A. M. Rey, “Synthetic Spin Orbit Coupling in an Optical Lattice Clock,” Phys. Rev. Lett. 116(3), 035301 (2016).
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Y. Yang, Z. Xu, L. Sheng, B. Wang, D. Y. Xing, and D. N. Sheng, “Time-reversal-symmetry-broken quantum spin Hall effect,” Phys. Rev. Lett. 107(6), 066602 (2011).
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H. Cai, J. Liu, J. Wu, Y. He, S. Y. Zhu, J. X. Zhang, and D. W. Wang, “Experimental Observation of Momentum-Space Chiral Edge Currents in Room-Temperature Atoms,” Phys. Rev. Lett. 122(2), 023601 (2019).
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S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, “Topological Thouless pumping of ultracold fermions,” Nat. Phys. 12(4), 296–300 (2016).
[Crossref]

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O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553(7686), 59–62 (2018).
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S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M. Duan, “Spin Hall Effects for Cold Atoms in a Light-Induced Gauge Potential,” Phys. Rev. Lett. 97(24), 240401 (2006).
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H. Cai, J. Liu, J. Wu, Y. He, S. Y. Zhu, J. X. Zhang, and D. W. Wang, “Experimental Observation of Momentum-Space Chiral Edge Currents in Room-Temperature Atoms,” Phys. Rev. Lett. 122(2), 023601 (2019).
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Y. Yang, Z. Xu, L. Sheng, B. Wang, D. Y. Xing, and D. N. Sheng, “Time-reversal-symmetry-broken quantum spin Hall effect,” Phys. Rev. Lett. 107(6), 066602 (2011).
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Y. Yang, Z. Xu, L. Sheng, B. Wang, D. Y. Xing, and D. N. Sheng, “Time-reversal-symmetry-broken quantum spin Hall effect,” Phys. Rev. Lett. 107(6), 066602 (2011).
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D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, “Topological quantum matter with cold atoms,” Adv. Phys. 67(4), 253–402 (2018).
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Y. Yang, Z. Xu, L. Sheng, B. Wang, D. Y. Xing, and D. N. Sheng, “Time-reversal-symmetry-broken quantum spin Hall effect,” Phys. Rev. Lett. 107(6), 066602 (2011).
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S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, “Spin-orbit-coupled fermions in an optical lattice clock,” Nature (London) 542(7639), 66–70 (2017).
[Crossref]

M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper, J. Ye, and A. M. Rey, “Synthetic Spin Orbit Coupling in an Optical Lattice Clock,” Phys. Rev. Lett. 116(3), 035301 (2016).
[Crossref]

Zaccanti, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, “Anderson localization of a non-interacting Bose Einstein condensate,” Nature 453(7197), 895–898 (2008).
[Crossref]

Zhang, D. W.

D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, “Topological quantum matter with cold atoms,” Adv. Phys. 67(4), 253–402 (2018).
[Crossref]

Zhang, J. X.

H. Cai, J. Liu, J. Wu, Y. He, S. Y. Zhu, J. X. Zhang, and D. W. Wang, “Experimental Observation of Momentum-Space Chiral Edge Currents in Room-Temperature Atoms,” Phys. Rev. Lett. 122(2), 023601 (2019).
[Crossref]

Zhang, S.-C.

S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M. Duan, “Spin Hall Effects for Cold Atoms in a Light-Induced Gauge Potential,” Phys. Rev. Lett. 97(24), 240401 (2006).
[Crossref]

Zhang, X.

S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, “Spin-orbit-coupled fermions in an optical lattice clock,” Nature (London) 542(7639), 66–70 (2017).
[Crossref]

M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper, J. Ye, and A. M. Rey, “Synthetic Spin Orbit Coupling in an Optical Lattice Clock,” Phys. Rev. Lett. 116(3), 035301 (2016).
[Crossref]

Zhao, Y. X.

D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, “Topological quantum matter with cold atoms,” Adv. Phys. 67(4), 253–402 (2018).
[Crossref]

Zhu, S. L.

D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, “Topological quantum matter with cold atoms,” Adv. Phys. 67(4), 253–402 (2018).
[Crossref]

Zhu, S. Y.

H. Cai, J. Liu, J. Wu, Y. He, S. Y. Zhu, J. X. Zhang, and D. W. Wang, “Experimental Observation of Momentum-Space Chiral Edge Currents in Room-Temperature Atoms,” Phys. Rev. Lett. 122(2), 023601 (2019).
[Crossref]

Zhu, S.-L.

S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M. Duan, “Spin Hall Effects for Cold Atoms in a Light-Induced Gauge Potential,” Phys. Rev. Lett. 97(24), 240401 (2006).
[Crossref]

Zhu, Y. Q.

D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, “Topological quantum matter with cold atoms,” Adv. Phys. 67(4), 253–402 (2018).
[Crossref]

Zilberberg, O.

O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4D quantum Hall physics,” Nature 553(7686), 59–62 (2018).
[Crossref]

M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4D quantum Hall physics with a 2D topological charge pump,” Nature 553(7686), 55–58 (2018).
[Crossref]

Y. E. Kraus and O. Zilberberg, “Quasiperiodicity and topology transcend dimensions,” Nat. Phys. 12(7), 624–626 (2016).
[Crossref]

M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A Thouless Quantum Pump with Ultracold Bosonic Atoms in an Optical Superlattice,” Nat. Phys. 12(4), 350–354 (2016).
[Crossref]

Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal,” Phys. Rev. Lett. 111(22), 226401 (2013).
[Crossref]

M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of Topological Phase Transitions in Photonic Quasicrystals,” Phys. Rev. Lett. 110(7), 076403 (2013).
[Crossref]

Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological States and Adiabatic Pumping in Quasicrystals,” Phys. Rev. Lett. 109(10), 106402 (2012).
[Crossref]

Zoller, P.

M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, “Observation of chiral edge states with neutral fermions in synthetic Hall ribbons,” Science 349(6255), 1510–1513 (2015).
[Crossref]

Adv. Phys. (1)

D. W. Zhang, Y. Q. Zhu, Y. X. Zhao, H. Yan, and S. L. Zhu, “Topological quantum matter with cold atoms,” Adv. Phys. 67(4), 253–402 (2018).
[Crossref]

Europhys. Lett. (1)

N. Goldman, W. Beugeling, and C. Smith, “Topological phase transitions between chiral and helical spin with spin-orbit coupling and a magnetic field-tight binding,” Europhys. Lett. 97(2), 23003 (2012).
[Crossref]

Nat. Phys. (6)

M. Atala, M. Aidelsburger, J. Barreiro, D. Abanin, T. Kitagawa, E. Demler, and I. Bloch, “Direct measurement of the Zak phase in topological Bloch bands,” Nat. Phys. 9(12), 795–800 (2013).
[Crossref]

M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbéne, N. R. Cooper, I. Bloch, and N. Goldman, “Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms,” Nat. Phys. 11(2), 162–166 (2015).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic representation of the two-leg ladder. This optical lattice has a double-well structure in the $x$ direction, while unlimitedly extends in the $y$ direction. $A$ and $B$ (the red and blue colors) label the two different degrees of freedom of legs, denoting the two different pseudo-spin states. $J$ and $K$ are the hopping amplitudes along the legs and rungs, respectively. An artificial magnetic flux $2\phi$ penetrates the plaquettes. An additional potential difference $2\Delta$ is imposed between the legs. (b)-(d) Band structures of the chiral ladders for various $K$’s and $\Delta$’s. Energy is set in units of $2J$. The color of the lines specifies the spin magnetization of the Bloch state. The magnetic flux is set as $2\phi =0.6\pi$. Other parameters: (b)$K=0,\; \Delta =0$ (c) $K=1,\; \Delta =0$, (d) $K=1,\; \Delta =1$.
Fig. 2.
Fig. 2. Energy spectrum for $2\phi =0.6\pi$, $\Lambda =1.5$, $\beta =1/3$, $\Delta =1$, and $K=0.1$. The energy is set in units of $J$. (a) Energy spectrum $E=E (k_y, \theta )$ for the infinite BCL. The integers near the graph label the spin Chern numbers of different bandgaps. (b) Energy spectrum obtained from a finite ladder. As a function of $\theta$, the spectrum is composed of the bulk bands (solid lines) and dispersion curves that traverse the gaps (dashed lines). $\blacksquare$ ($\bullet$) is the markers for the states at the same Fermi energy.
Fig. 3.
Fig. 3. Mode amplitudes of the gapless states in Fig. 2(b). (a-d) correspond to the states marked by $\blacksquare$, while (e,f) to the states $\bullet$. The spin component $\Psi _{n\uparrow }$ (respectively, $\Psi _{n\downarrow }$) is represented in red (respectively, blue).
Fig. 4.
Fig. 4. $E_{\textrm {Fermi}}$-$\Delta$ phase diagram. Here, the bandgaps (bands) are designated by the white (shaded) regions, respectively. The pairs of integers indicate the Chern numbers of the bandgaps for spin up and down, distinguishing the topological regimes of the model.

Equations (6)

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H = J n ( e i ϕ c n + 1 , A c n , A + e i ϕ c n + 1 , B c n , B + h . c . ) + Δ n ( c n , A c n , A c n , B c n , B ) K n ( c n , A c n , B + c n , B c n , A ) .
H = n Ψ n [ Δ σ ^ z K σ ^ x ] Ψ n J n Ψ n + 1 e ı ϕ σ ^ z Ψ n + h . c . ,
H = k y Ψ k y M ( k y ) Ψ k y
H 1 D ( n , θ ) = n Ψ n [ Λ cos ( 2 π β n + θ ) ] Ψ n + n Ψ n [ Δ σ ^ z K σ ^ x ] Ψ n n J Ψ n + 1 e ı ϕ σ ^ z Ψ n + h . c . .
H 2 D ( n , k z ) = n , k z Ψ n , k z [ 2 t z cos ( 2 π β n + k z ) ] Ψ n , k z + n , k z Ψ n , k z [ Δ σ ^ z K σ ^ x ] Ψ n , k z n , k z t y Ψ n + 1 , k z e ı ϕ σ ^ z Ψ n , k z + h . c . .
H 2 D ( n , m ) = t z n , m e ı 2 π β n Ψ n , m + 1 Ψ n , m + h . c . + n , m Ψ n , m [ Δ σ ^ z K σ ^ x ] Ψ n , m n , m t y Ψ n + 1 , m e ı ϕ σ ^ z Ψ n , m + h . c . .

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