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Realization of the Harper Hamiltonian with Ultracold Atoms in Optical Lattices

Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody Burton, Wolfgang Ketterle

Published Jan 1, 2014• 0 citations
Publisher

Abstract

We experimentally realized the Harper Hamiltonian with charge neutral, ultracold atoms in optical lattices using laser-assisted tunneling and a potential energy gradient. The energy spectrum of this Hamiltonian is the fractal Hofstadter butterfly.
DOI
10.1364/cleo_qels.2014.ff2d.1

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The first sentence of the abstract should read as ''We experimentally implement the Harper Hamiltonian for neutral particles in optical lattices using laser-assisted tunneling and a potential energy gradient provided by gravity or magnetic field gradients.'' The paper has been corrected as of 28 October 2013.

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We experimentally implement the Harper Hamiltonian for neutral particles in optical lattices using laser-assisted tunneling and a potential energy gradient provided by gravity or magnetic field gradients. This Hamiltonian describes the motion of charged particles in strong magnetic fields. Laser-assisted tunneling processes are characterized by studying the expansion of the atoms in the lattice. The band structure of this Hamiltonian should display Hofstadter's butterfly. For fermions, this scheme should realize the quantum Hall effect and chiral edge states.
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.
We demonstrate that all of the salient features of the Harper-Hofstadter model can be implemented with ultracold atoms trapped in a bichromatic ring-shaped lattice. Using realistic sinusoidal lattice potentials rather than assume the idealized tight-binding picture, we determine the optimal conditions necessary to realize the critical point where the spectrum becomes fractal, and we identify the nature and cause of the departures from the discrete model predictions. We also show that even with a commensurate ring with a few lattice sites, the Aubry-Andr\'e localization transition can be realized. Localized states that behave like edge states with energies that reside in the band gaps can be generated by introducing a surprisingly small local perturbation within the ring. Spectrum oscillation arising from complex coupling can be implemented by uniform rotation of the ring, but with certain significant differences that are explained.