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Communications in Theoretical Physics

Light-Induced Hofstadter's Butterfly Spectrum in Optical Lattices

Jing-Min Hou

Published Mar 1, 2009• 1 citation
Publisher

Abstract

We propose a scheme to create an effective magnetic field, which can be perceived by cold neutral atoms in a two-dimensional optical lattice, with a laser field with a space-dependent phase and a conventional laser field acting on Λ-type three-level atoms. When the dimensionless parameter α, being the ratio of flux through a lattice cell to one flux quantum, is rational, the energy spectrum shows a fractal band structure, which is so-called Hofstadter's butterfly.

DOI
10.1088/0253-6102/51/3/11

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We propose a scheme to create an effective magnetic field, which can be perceived by cold neutral atoms in a two-dimensional optical lattice, with a laser field with a space-dependent phase and a conventional laser field acting on Λ-type three-level atoms. When the dimensionless parameter α, being the ratio of flux through a lattice cell to one flux quantum, is rational, the energy spectrum shows a fractal band structure, which is so-called Hofstadter's butterfly.
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 that an artificial electromagnetic field can be engineered in the context of cold fermionic atoms that are coupled to a cavity mode via two-photon processes in a two-dimensional optical lattice. There is a standing-wave pump laser inducing the electric effect in one spatial direction, and a second running-wave laser beam generates the magnetic flux perpendicular to the lattice plane. In the static scenario, the bulk spectrum resembles the fractal structure of the Hofstadter butterfly and the edge mode spectrum indicates the occurrence of the quantum Hall phase. The Keldysh formulism is utilized to capture the time evolution. The back action between atoms and cavity field gives a picture of the time-dependent non-equilibrium dynamics. We find that the spontaneous emergence of the artificial electromagnetic field stimulates the Hall current, and the superradiant cavity field emerges without pump threshold. Moreover, the magnetic flux not only modify the bulk topology, but also drives a series of dynamical phase transitions.
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.
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.