Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms

Aidelsburger, Monika; Lohse, Michael; Schweizer, C.; Atala, Marcos; Barreiro, Julio T.; Nascimbène, Sylvain; Cooper, N. R.; Bloch, Immanuel; Goldman, Nathan

doi:10.1038/nphys3171

Cited by 999 publications

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References 37 publications

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“…Other values of ϕ are also realizable. In contrast to previous setups [4][5][6][7], here the artificial magnetic field is not a static field imprinted by external laser beams, but dynamically emerges due to off-resonant scattering of pump light into the cavity mode. The resulting feedback mechanism between the rung tunneling process and the cavity field population leads to a selforganization of the system into a state in which the atoms are subjected to a strong artificial magnetic field.…”

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confidence: 80%

“…From the x-direction two different contributions can play an important role depending on the exact geometry of the optical lattices [4][5][6][7]. The first one is given by the direct overlap integral between the Wannier functions of the two wells and the cavity mode function cos (k c x):…”

Section: Bogolioubov Transformationmentioning

confidence: 99%

“…The second contribution is given by the time-periodic modulation of the energy offset between the sites of the double well [7]. Its amplitude is related to the on-site overlap integrals φ m,m ⊥ = dx cos (k c (x + md ⊥ )) w m (x)w * m (x).…”

Section: Bogolioubov Transformationmentioning

confidence: 99%

“…Such artificial magnetic fields have been used in quantum gases confined to optical lattices to realize two showcase models of topologically insulating phases, the Hofstadter model in two-dimensions [4][5][6][7] or on a ladder geometry [8] and the Haldane model [9].…”

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confidence: 99%

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Ultracold Fermions in a Cavity-Induced Artificial Magnetic Field

et al. 2016

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We show how a fermionic quantum gas in an optical lattice and coupled to the field of an optical cavity can self-organize into a state in which the spontaneously emerging cavity field amplitude induces an artificial magnetic field. The fermions form either a chiral insulator or a chiral liquid carrying edge currents. The feedback mechanism via the cavity field enables robust and fast switching of the edge currents and the cavity output can be employed for non-destructive measurements of the atomic dynamics. The controlled generation of topologically non-trivial quantum phases is of greatest interest since they possess special properties such as extended edge states that can be well protected against destructive environmental effects[1]. Thus, materials with topological non-trivial properties have the prospect to be utilized for quantum devices and quantum computing. Well known are topological insulators with protected edge currents created in quantum Hall devices [2] by strong magnetic fields.For neutral atoms strong artificial magnetic fields can be created [3] which act similarly as magnetic fields for charged particles. Such artificial magnetic fields have been used in quantum gases confined to optical lattices to realize two showcase models of topologically insulating phases, the Hofstadter model in two-dimensions [4][5][6][7] or on a ladder geometry [8] and the Haldane model [9].Yet, the dynamic control and detection of topologically non-trivial quantum states remains a great challenge. In order to overcome this difficulty, in this work the dynamical feedback between atoms and an optical cavity is employed to reach a self-organization of topologically non-trivial phases. One fascinating example for a selforganization of a coupled atom-cavity system has been realized recently by placing a bosonic quantum gas into an optical high-finesse resonator subjected to a perpendicular off-resonant pump beam [10][11][12][13][14]. Above a critical pump strength, the occupation of the cavity mode is stabilized and the bosonic atoms organize into a checkerboard density pattern [12]. Many different proposals have been put forward to realize the self-organization of more complex quantum phases [13] reaching from the Mott-insulator [15] over fermionic phases [16][17][18][19][20] and disordered structures [21][22][23][24] to phases with spin-orbit coupling [25][26][27].In this work, we engineer a direct coupling mechanism of the cavity photons to the tunneling of atoms in an optical lattice. This is achieved using a Raman transition induced by the combination of a transverse pump beam and the cavity field (Fig. 1). The frequencies of the pump beam and the cavity mode are chosen such that the energy transfer is close to the energy offset between neigh- boring lattice sites. The resulting process represents an effective tunneling of the atoms. Due to the running-wave nature of the pump beam, a phase ϕ can be imprinted onto the tunneling of the atoms around a plaquette in the presence of cavity photons. We demonstrate that due t...

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confidence: 80%

Section: Bogolioubov Transformationmentioning

confidence: 99%

Section: Bogolioubov Transformationmentioning

confidence: 99%

mentioning

confidence: 99%

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Ultracold Fermions in a Cavity-Induced Artificial Magnetic Field

et al. 2016

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“…However, while the existence of such edge states is linked to the non-trivial global topological properties of bulk bands, a direct measurement of the Berry curvature itself has so far not been realized. In ultracold atoms, instead, there has been much progress in this direction, for example, with experiments exploiting the effects of Berry curvature on particle transport to extract, in 2D systems, the Chern number 19 and topological phase diagrams 20 , and, in 1D systems, both the quantized 'Thouless pumping' of atoms in a filled band 23,24 and the un-quantized 'geometrical pumping' of a Bose-Einstein condensate 25 .…”

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confidence: 99%

Experimental measurement of the Berry curvature from anomalous transport

et al. 2017

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The geometric properties of energy bands underlie fascinating phenomena in many systems, including solid-state, ultracold gases and photonics. The local geometric characteristics such as the Berry curvature 1 can be related to global topological invariants such as those classifying the quantum Hall states or topological insulators. Regardless of the band topology, however, any non-zero Berry curvature can have important consequences, such as in the semi-classical evolution of a coherent wavepacket. Here, we experimentally demonstrate that the wavepacket dynamics can be used to directly map out the Berry curvature. To this end, we use optical pulses in two coupled fibre loops to study the discrete time evolution of a wavepacket in a one-dimensional geometric 'charge' pump, where the Berry curvature leads to an anomalous displacement of the wavepacket. This is both the first direct observation of Berry curvature e ects in an optical system, and a proof-ofprinciple demonstration that wavepacket dynamics can serve as a high-resolution tool for mapping out geometric properties.The Berry curvature is a geometrical property of an energy band, which plays a key role in many physical phenomena as it encodes how eigenstates evolve as a local function of parameters 1 . In a twodimensional (2D) quantum Hall system, for example, the integral of the Berry curvature over the 2D Brillouin zone determines the Chern number: a global topological invariant that underlies the quantization of Hall transport for a 2D filled energy band 2 . As first explained by Thouless 3 , there can be an analogous topological quantization of particle transport in a 1D band insulator, when the lattice potential is 'pumped' , that is, is slowly and periodically modulated in time. Also in this case, the geometrical and topological properties are defined for an effective 2D parameter space, but now one spanned by the 1D Bloch momentum and the external periodic pumping parameter.A local non-zero Berry curvature can have striking physical effects in both 2D systems and 1D pumps, regardless of whether the global topological Chern number is non-trivial. In the simplest case, a semi-classical wavepacket moves as a coherent object governed by classical equations of motion with an additional 'anomalous' Hall velocity due to the geometrical Berry curvature at its centre-ofmass, as an external force is applied or as the control parameter is pumped 1,4 . As highlighted further below, this anomalous transport can be understood physically as the Berry curvature acting like a magnetic field in parameter space [5][6][7] .In recent years, there have been many landmark experiments to engineer and study geometrical and topological energy bands in ultracold gases and photonics . In photonics, for example, topological edge states have been studied in a wide variety of (effectively) 1D set-ups, such as quantum walks 14 and pumping in optical quasicrystals 13,29,30 , as well as in 2D quantum Hall-like systems of photonic crystals 9-11 , propagating waveguides 12 and silico...

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Solitons in One‐Dimensional Lattices with a Flat Band

2017

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We investigate the spectral properties of a quasi-one-dimensional lattice in two possible dimerisation configurations. Both configurations are characterised by the same lattice topology and the identical spectra containing a flat band at zero energy. We find that, one of the dimerised configuration has similar symmetry to a one-dimensional chain proposed by Su-Schrieffer-Heeger for studying solitons in conjugated polymers. Whereas, the other dimerised configuration only shows non-trivial topological properties in the presence of chiral-symmetry breaking adiabatic pumping.

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Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms

Cited by 999 publications

References 37 publications

Ultracold Fermions in a Cavity-Induced Artificial Magnetic Field

Ultracold Fermions in a Cavity-Induced Artificial Magnetic Field

Experimental measurement of the Berry curvature from anomalous transport

Solitons in One‐Dimensional Lattices with a Flat Band

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