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arXiv:quant-ph/0304038 · New Journal of Physics

Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms

Dieter Jaksch, P. Zoller

Published May 30, 2003• 777 citations
arXivHTMLPDF

Abstract

We investigate the dynamics of neutral atoms in a 2D optical lattice which traps two distinct internal states of the atoms in different columns. Two Raman lasers are used to coherently transfer atoms from one internal state to the other, thereby causing hopping between the different columns. By adjusting the laser parameters appropriately we can induce a non-vanishing phase of particles moving along a closed path on the lattice. This phase is proportional to the enclosed area and we thus simulate a magnetic flux through the lattice. This set-up is described by a Hamiltonian identical to the one for electrons on a lattice subject to a magnetic field and thus allows us to study this equivalent situation under very well defined controllable conditions. We consider the limiting case of huge magnetic fields—which is not experimentally accessible for electrons in metals—where a fractal band structure, the Hofstadter butterfly, characterizes the system.

DOI
10.1088/1367-2630/5/1/356

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We investigate the dynamics of neutral atoms in a 2D optical lattice which traps two distinct internal states of the atoms in different columns. Two Raman lasers are used to coherently transfer atoms from one internal state to the other, thereby causing hopping between the different columns. By adjusting the laser parameters appropriately we can induce a non-vanishing phase of particles moving along a closed path on the lattice. This phase is proportional to the enclosed area and we thus simulate a magnetic flux through the lattice. This set-up is described by a Hamiltonian identical to the one for electrons on a lattice subject to a magnetic field and thus allows us to study this equivalent situation under very well defined controllable conditions. We consider the limiting case of huge magnetic fields—which is not experimentally accessible for electrons in metals—where a fractal band structure, the Hofstadter butterfly, characterizes the system.
We propose a simple setup for introducing an artificial magnetic field for neutral atoms in 2D optical lattices. This setup is based on the phenomenon of photon-assistent tunneling and involves a low-frequency periodic driving of the optical lattice. We also address the problem of detecting this effective magnetic field. In particular, we discuss interference patterns of the atomic wave function and study the center of mass wave-packet dynamics, which shows some features of cyclotron dynamics of a classical charged particle. Introduction. – A major motivation of current research with cold atoms in optical lattices is the prospect of simulating the solid state physics. However, to have a full access to phenomena of the solid state physics this system (which can be considered as an artificial crystal) should be accomplished by artificial electric and magnetic fields. In present days experiments the former case of external electric field is routinely mimicked by accelerating the optical lattice [1–3], or by using the gravitational force [4–6] or a combination of gravitational and levitational forces [7]. The case of magnetic field is more difficult for laboratory realization because it requires a setup, where atomic wave function acquires a finite phase when the atom tunnels along a closed path on the lattice. This setup was suggested in the seminal paper [8], where the authors use two independent 2D optical lattices for two different internal atomic states, which are coupled by additional Raman lasers. The Rabi transition between internal states induces hopping of the atom between nearest lattice sites, where the required phase accumulation is achieved by using a special geometry for the Raman beams. This idea was developed further in Ref. [9, 10]. It was shown that one can also introduce an exotic (unphysical) field and non-Abelian gauge potentials by using the Raman-laser technique. For atoms in a harmonic trap (no lattice) the Raman scheme with 3 spin states of the F = 1 electronic ground state of Rb atom was realized in the recent experiment [11]. In this setup the magnitude of the effective magnetic field is defined by the gradient of the real magnetic field which splits the F = 1 level into Zeeman sublevels. In the paper the authors put forward a conjecture that this scheme should work in the presence of an optical lattice as well. We note that the cited experiment also illuminated a drawback of the Raman-laser based techniques. Namely, the spontaneous emission from the Raman beams removes atom from the trap, causing the population to decay within approx. 2 seconds.
Abstract Topological many-body phases of matter exhibit remarkable electronic properties and ultracold atoms in optical lattices constitute promising candidates to study them in a well-controlled environment. In two-dimensional (2D) electron gases topological phases may emerge in the presence of strong magnetic fields. This situation is not directly applicable to cold atoms because they are charge neutral. Therefore, novel experimental techniques have been developed to engineer novel lattice systems, whose Hamiltonian is formally equivalent to the one of charged particles in magnetic fields. In this Tutorial, we introduce a paradigmatic topological lattice model, the Hofstadter model, and explain how it can be implemented with ultracold atoms using laser-assisted tunneling. The technique is based on imprinting phases on the tunneling matrix elements using additional laser beams. These phases are reminiscent of Aharonov–Bohm phases and can be interpreted as a magnetic flux piercing the lattice unit cell. We present experimental results on the cyclotron-like dynamics of neutral atoms in isolated four-site square plaquettes and discuss the first measurement of a 2D topological invariant, the Chern number, in artificially generated Hofstadter bands. The work presented in this Tutorial was one of the four shortlisted finalists of the 2016 DPG SAMOP dissertation prize.
Energy bands of electrons in a square lattice potential threaded by a uniform magnetic field exhibit a fractal structure known as the Hofstadter butterfly. Here we study a Fermi gas in a 2D optical lattice within a linear cavity with a tilt along the cavity axis. The hopping along the cavity axis is only induced by resonant Raman scattering of transverse pump light into a standing-wave-cavity mode. Choosing a suitable pump geometry allows us to realize the Hofstadter-Harper model with a cavity-induced dynamical synthetic magnetic field, which appears at the onset of the superradiant phase transition. The dynamical nature of this cavity-induced synthetic magnetic field arises from the delicate interplay between collective superradiant scattering and the underlying fractal band structure. Using a sixth-order expansion of the free energy as a function of the order parameter and by numerical simulations, we show that at low magnetic fluxes the superradiant ordering phase transition is first order, while it becomes second order for higher flux. The dynamic nature of the magnetic field induces a nontrivial deformation of the Hofstadter butterfly in the superradiant phase. At strong pump far above the self-ordering threshold, we recover the Hofstadter butterfly one would obtain in a static magnetic field.
We describe a simple technique for generating a cold-atom lattice pierced by a uniform magnetic field. Our method is to extend a one-dimensional optical lattice into the "dimension" provided by the internal atomic degrees of freedom, yielding a synthetic two-dimensional lattice. Suitable laser coupling between these internal states leads to a uniform magnetic flux within the two-dimensional lattice. We show that this setup reproduces the main features of magnetic lattice systems, such as the fractal Hofstadter-butterfly spectrum and the chiral edge states of the associated Chern insulating phases.
We devise experimental schemes that are able to mimic uniform and staggered magnetic fluxes acting on ultracold two-electron atoms, such as ytterbium atoms, propagating in a honeycomb lattice. The atoms are first trapped into two independent state-selective triangular lattices and then further exposed to a suitable configuration of resonant Raman laser beams. These beams induce hops between the two triangular lattices and make atoms move in a honeycomb lattice. Atoms traveling around each unit cell of this honeycomb lattice pick up a nonzero phase. In the uniform case, the artificial magnetic flux sustained by each cell can reach about two flux quanta, thereby realizing a cold-atom analog of the Harper model with its notorious Hofstadter's butterfly structure. Different condensed-matter phenomena such as the relativistic integer and fractional quantum Hall effects, as observed in graphene samples, could be targeted with this scheme.