We consider a gas of non-interacting spinless fermions in a rotating optical lattice and calculate the density profile of the gas in an external confinement potential. The density profile exhibits distinct plateaus, which correspond to gaps in the single particle spectrum known as the Hofstadter butterfly. The plateaus result from insulating behavior whenever the Fermi energy lies within a gap. We discuss the necessary conditions to realize the Hofstadter insulator in a cold atom setup and show how the quantized Hall conductance can be measured from density profiles using the Středa formula.Recently, some fundamental models of many-particle quantum systems have been experimentally realized using trapped ultra-cold fermions. Some of these experiments, such as transport in optical lattices [1], has demonstrated well known effects with improved precision, while others, such as spin imbalanced superfluidity [2,3], have provided access to previously unexplored regimes. With constantly improving experimental control over ultra-cold systems, it is expected that many other fundamental ideas can be tested in laboratory for the first time.A basic problem in quantum mechanics is the dynamics of a charged particle moving in a periodic potential under a magnetic field. The single particle spectrum depends sensitively on the ratio of the flux through a unit cell of the lattice to flux quantum. For a tight binding lattice, a single band splits into narrow magnetic bands, forming a self-similar energy spectrum known as the Hofstadter butterfly [4]. The gaps in the Hofstadter spectrum form continuous regions for a finite range of flux. For a system of non-interacting fermions, it was shown by Thouless et. al. that whenever the Fermi energy lies in one of these gaps, the Hall conductance of the system is quantized [5]. This quantization is topological in nature, and the quantized Hall conductance is determined uniquely by the magnetic translation symmetry [6]. This Hofstadter insulating phase is a topological insulator that can be characterized by the first Chern number.Despite its mathematical elegance, the Hofstadter insulator can hardly be achieved in solid state systems because the magnetic field needs to be thousands of Tesla in order to create a magnetic flux which can be comparable to one flux quantum per unit cell [9]. While in some experiments super-lattice structures have been used to study the splitting of Landau levels under a periodic potential, the tight-binding regime (strong magnetic field limit) has never been experimentally realized [10]. The main purpose of this Letter is to propose an alternative way to achieve and to experimentally study this topological insulator by using ultra-cold Fermi gases in a rotating optical lattice. We discuss (i) the conditions to realize the Hofstadter insulator in rotating optical lattices, and (ii) the manifestation of this insulator in real space density profile and the method to detect the Hall conductance in a cold atom setup.The Hamiltonian for a particle in a rotating latti...

We consider the superfluid-insulator transition for cold bosons under an effective magnetic field. We investigate how the applied magnetic field affects the Mott transition within mean-field theory and find that the critical hopping strength (t U)c increases with the applied field. The increase in the critical hopping follows the bandwidth of the Hofstadter butterfly at the given value of the magnetic field. We also calculate the magnetization and superfluid density within mean-field theory. © 2007 The American Physical Society

We consider the Bose-Hubbard model in a two dimensional rotating optical lattice and investigate the consequences of the effective magnetic field created by rotation. Using a Gutzwiller type variational wavefunction, we find an analytical expression for the Mott insulator(MI)-Superfluid(SF) transition boundary in terms of the maximum eigenvalue of the Hofstadter butterfly. The dependence of phase boundary on the effective magnetic field is complex, reflecting the self-similar properties of the single particle energy spectrum. Finally, we argue that fractional quantum Hall phases exist close to the MI-SF transition boundaries, including MI states with particle densities greater than one.

We discuss how a three level system can be used to change the frequency dependent magnetic permeability of an atomic gas to be significantly different from one. We derive the conditions for such a scheme to be successful and briefly discuss the resulting macroscopic electrodynamics. We find that it may be possible to obtain left handed electrodynamics for an atomic gas using three atomic levels.

The formula ͓Eq. ͑19͔͒ for the induced dipole moment of the atom contains a sign mistake in the exponent. After correcting the error it should read ͑1͒As the expression includes the complex conjugate, there is a term matched to the probe field and the conclusions in the paper are not affected.In the sixth paragraph, it is stated that two media related as ⑀ 2 ͑͒ =−⑀ 1 ͑͒ and 2 ͑͒ =− 1 ͑͒ would have the same indices of refraction. This is incorrect. These relations give two media with the same wave impedance but opposite indices of refraction n 2 =−n 1 .