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We provide a detailed analysis of our previously proposed scheme [Phys. Rev.Lett. 88, 180401, (2002)] to engineer the profile of the hopping amplitudes foratomic gases in a 1D optical lattice so that the particle number becomesfractional. We consider a constructed system of a dilute two-species gas offermionic atoms where the two components are coupled via a coherentelectromagnetic field with a topologically nontrivial phase profile. We showboth analytically and numerically how the resulting atomic Hamiltonian in aprepared dimerized optical lattice with a defect in the pattern of alternatinghopping amplitudes exhibits a fractional fermion number. In particular, in thelow-energy limit we demonstrate the equivalence of the atomic Hamiltonian to arelativistic Dirac Hamiltonian describing fractionalization in quantum fieldtheory. Expanding on our earlier argument [Phys. Rev. Lett. 91, 150404 (2003)]we show how the fractional eigenvalues of the particle number operator can bedetected via light scattering. In particular, we show how scattering of far-offresonant light can convey information about the counting statistics of theatoms in an optical lattice, including state-selective atom density profilesand atom number fluctuations. Optical detection could provide a truly quantummechanical measurement of the particle number fractionalization in a diluteatomic gas.

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