Looking for Hofstadter’s Butterfly in Cold Atoms

Abstract

We demonstrate the experimental implementation of an optical lattice that allows for the generation of large homogeneous and tunable artificial magnetic fields with ultracold atoms. Using laser-assisted tunneling in a tilted optical potential, we engineer spatially dependent complex tunneling amplitudes. Thereby, atoms hopping in the lattice accumulate a phase shift equivalent to the Aharonov-Bohm phase of charged particles in a magnetic field. We determine the local distribution of fluxes through the observation of cyclotron orbits of the atoms on lattice plaquettes, showing that the system is described by the Hofstadter model. Furthermore, we show that for two atomic spin states with opposite magnetic moments, our system naturally realizes the time-reversal-symmetric Hamiltonian underlying the quantum spin Hall effect; i.e., two different spin components experience opposite directions of the magnetic field. DOI: 10.1103/PhysRevLett.111.185301 PACS numbers: 67.85.Àd, 03.65.Vf, 03.75.Lm, 73.20.Àr Ultracold atoms in optical lattices constitute a unique experimental setting to study condensed matter Hamiltonians in a clean and well-controlled environment [1], even in regimes not accessible to typical condensed matter systems [2]. Especially intriguing is their promising potential to realize and probe topological phases of matter, for example, by utilizing the newly developed quantum optical high-resolution detection and manipulation techniques [3,4]. One compelling possibility in this direction is the quantum simulation of electrons moving in a periodic potential exposed to a large magnetic field, described by the Hofstadter-Harper Hamiltonian [5,6]. For a filled band of fermions, this model realizes the paradigmatic example of a topological insulator that breaks time-reversal symmetry-the quantum Hall insulator. Moreover, the atomic realization of time-reversal-symmetric topological insulators based on the quantum spin Hall effect [7] promises new insights for spintronic applications.The direct quantum simulation of orbital magnetism in ultracold quantum gases is, however, hindered by the charge neutrality of atoms, which prevents them from experiencing a Lorentz force. Overcoming this limitation through the engineering of synthetic gauge potentials is currently a major topic in cold-atom research. Artificial magnetic fields were first accomplished using the Coriolis force in a rotating atomic gas [8,9] and later by inducing Berry's phases through the application of Raman lasers [10,11]. Recently, staggered magnetic fields in optical lattices were achieved using laser-induced tunneling in superlattice potentials [12] or through dynamical shaking [13]. In one dimension, tunable gauge fields have been implemented in an effective ''Zeeman lattice' ' [14] and using periodic driving [15]. Furthermore, the free-space spin Hall effect was observed using Raman dressing [16]. Despite intense research efforts, 2D optical lattices featuring topological many-body phases have so far been beyond the reach of experiments.I...

Quasi-periodic Wannier–Stark ladders from driven atomic Bloch oscillations

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Spectral functions of a time-periodically driven Falicov-Kimball model: Real-space Floquet dynamical mean-field theory study

“…It is therefore of experimental relevance to choose the special driving amplitude satisfying A = 1.435. To observe a Hofstadter butterfly, the temperature should be smaller than the gaps between subbands [42]. Additional peaks for a nonoptimal choice of A would imply even smaller gaps and increase the experimental challenge.…”

Realization of Hofstadter's butterfly and a one-way edge mode in a polaritonic system

“…For continuous systems, examples include the generation of magnetic field for cold atoms gases using rotation [6] and laser illumination [5,7]. For lattice systems, different microwave [8], optical [9][10][11][12][13], and cold atom [14][15][16] setups were considered recently, and recipes for artificial gauge field generations were proposed. Up to date, Hofstadter's butterfly was demonstrated using engineered superlattice structures with microwave resonators [17], bilayer graphene [18,19], optical ring microresonators [20], and cold atom lattices [14,[21][22][23].…”