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arXiv:cond-mat/0604600 · Physical Review Letters

High-Field Fractional Quantum Hall Effect in Optical Lattices

Rebecca N Palmer, Dieter Jaksch

Published May 12, 2006• 152 citations
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Abstract

We consider interacting bosonic atoms in an optical lattice subject to a large simulated magnetic field. We develop a model similar to a bilayer fractional quantum Hall system valid near simple rational numbers of magnetic flux quanta per lattice cell. Then we calculate its ground state, magnetic lengths, fractional fillings, and find unexpected sign changes in the Hall current. Finally we study methods for detecting these novel features via shot noise and Hall current measurements.

DOI
10.1103/physrevlett.96.180407

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We explore the behavior of interacting bosonic atoms in an optical lattice subject to a large artificial magnetic field. We extend earlier investigations of this system where the number of magnetic flux quanta per unit cell $\ensuremath{\alpha}$ is close to a simple rational number [R. N. Palmer and D. Jaksch, Phys. Rev. Lett. 96, 180407 (2006)]. Interesting topological states such as the Laughlin and Read-Rezayi states can occur even if the atoms experience a weak trapping potential in one direction. An explicit numerical calculation near $\ensuremath{\alpha}=1/2$ shows that the system exhibits a striped vortex lattice phase of one species, which is analogous to the behavior of a two-species system for small $\ensuremath{\alpha}$. We also investigate methods to probe the encountered states. These include spatial correlation functions and the measurement of noise correlations in time-of-flight expanded atomic clouds. Characteristic differences arise which allow for an identification of the respective quantum Hall states. We furthermore discuss that a counterintuitive flow of the Hall current occurs for certain values of $\ensuremath{\alpha}$.
We analyze a recently proposed method to create fractional quantum Hall (FQH) states of atoms confined in optical lattices [A. S\o{}rensen et al., Phys. Rev. Lett. 94, 086803 (2005)]. Extending the previous work, we investigate conditions under which the FQH effect can be achieved for bosons on a lattice with an effective magnetic field and finite on-site interaction. Furthermore, we characterize the ground state in such systems by calculating Chern numbers which can provide direct signatures of topological order and explore regimes where the characterization in terms of wave-function overlap fails. We also discuss various issues which are relevant for the practical realization of such FQH states with ultracold atoms in an optical lattice, including the presence of a long-range dipole interaction which can improve the energy gap and stabilize the ground state. We also investigate a detection technique based on Bragg spectroscopy to probe these systems in an experimental realization.
We consider the purity-based entanglement detection scheme introduced in [C. Moura Alves and D. Jaksch, Phys. Rev. Lett. 93, 110501 (2004)]. We describe how it could be implemented in an optical lattice using two-atom loss, and prove that in this form it detects all pure entangled states even without any spatial resolution. We then prove that correcting for certain reasonable types of experimental error is possible, and practical for error rates up to the order of one over the number of lattice sites considered. Limited spatial resolution similarly becomes a significant improvement over no spatial resolution only at nearly single site level. We also show how to use this process for state parameter estimation and collapse-revival evidence of entanglement, for which it remains useful even when the error rate is too high to permit unambiguous entanglement detection. We also consider an optical lattice bosonic analogue of the fractional quantum Hall (FQH) effect. This system can reach high “magnetic fields” very difficult to attain in the solid state FQH system, where the discrete nature of the lattice becomes important. Near simple rational numbers l/n of flux quanta per lattice cell, we find that the single particle states become nearly periodic with period n lattice sites, and have an n fold degeneracy which leads to FQH states resembling those of n-internal-state particles. Standard time of flight expansion would reveal this periodicity and be able to distinguish FQH states from vortex lattice or Mott insulator states. Shot noise correlation would provide further information on the nature of the FQH states.
We consider the purity-based entanglement detection scheme introduced in [C. Moura Alves and D. Jaksch, Phys. Rev. Lett. 93, 110501 (2004)]. We describe how it could be implemented in an optical lattice using two-atom loss, and prove that in this form it detects all pure entangled states even without any spatial resolution. We then prove that correcting for certain reasonable types of experimental error is possible, and practical for error rates up to the order of one over the number of lattice sites considered. Limited spatial resolution similarly becomes a significant improvement over no spatial resolution only at nearly single site level. We also show how to use this process for state parameter estimation and collapse-revival evidence of entanglement, for which it remains useful even when the error rate is too high to permit unambiguous entanglement detection. We also consider an optical lattice bosonic analogue of the fractional quantum Hall (FQH) effect. This system can reach high “magnetic fields” very difficult to attain in the solid state FQH system, where the discrete nature of the lattice becomes important. Near simple rational numbers l/n of flux quanta per lattice cell, we find that the single particle states become nearly periodic with period n lattice sites, and have an n fold degeneracy which leads to FQH states resembling those of n-internal-state particles. Standard time of flight expansion would reveal this periodicity and be able to distinguish FQH states from vortex lattice or Mott insulator states. Shot noise correlation would provide further information on the nature of the FQH states.
In this research, we study the bosonic fractional quantum Hall (FQH) states in a system of ultracold bosons in a two-dimensional optical lattice in the presence of a synthetic magnetic field, described by the bosonic Harper–Hofstadter Hamiltonian. We use the cluster Gutzwiller mean-field and exact diagonalization techniques in our work. We obtain incompressible states as ground states at various filling factors similar to those of the FQH states. We focus in particular on the ν = 1/2 FQH state, and it is characterized by the two-point correlation function and the many-body Chern number. We further investigate the effect of dipolar interaction on the ν = 1/2 FQH state. We find that the dipolar interaction stabilizes the FQH state against the competing superfluid state.
We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low-energy physics is mapped to a multispecies continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin. We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wave functions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wave functions can be destabilized when umklapp processes become strong.