Manipulating atoms in an optical lattice: Fractional fermion number and its optical quantum measurement

TL;DR

This paper analyzes a scheme to engineer fractional fermion numbers in a 1D optical lattice of fermionic atoms, demonstrating how topological defects induce fractionalization and how optical measurements can detect these effects.

Contribution

It provides a detailed analytical and numerical study of fractional fermion number in optical lattices with topological defects, linking atomic Hamiltonians to relativistic Dirac models and proposing optical detection methods.

Findings

Fractional fermion number arises at topological defects in engineered optical lattices.

Optical scattering can detect fractionalization through atom counting statistics.

The atomic Hamiltonian is shown to be equivalent to a relativistic Dirac Hamiltonian in the low-energy limit.

Abstract

We provide a detailed analysis of our previously proposed scheme [Phys. Rev. Lett. 88, 180401, (2002)] to engineer the profile of the hopping amplitudes for atomic gases in a 1D optical lattice so that the particle number becomes fractional. We consider a constructed system of a dilute two-species gas of fermionic atoms where the two components are coupled via a coherent electromagnetic field with a topologically nontrivial phase profile. We show both analytically and numerically how the resulting atomic Hamiltonian in a prepared dimerized optical lattice with a defect in the pattern of alternating hopping amplitudes exhibits a fractional fermion number. In particular, in the low-energy limit we demonstrate the equivalence of the atomic Hamiltonian to a relativistic Dirac Hamiltonian describing fractionalization in quantum field theory. Expanding on our earlier argument [Phys. Rev. Lett. 91, 150404 (2003)] we show how the fractional eigenvalues of the particle number operator can be detected via light scattering. In particular, we show how scattering of far-off resonant light can convey information about the counting statistics of the atoms in an optical lattice, including state-selective atom density profiles and atom number fluctuations. Optical detection could provide a truly quantum mechanical measurement of the particle number fractionalization in a dilute atomic gas.