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We propose a method to factor numbers based on the quantum dynamics of two interacting bosonic atoms where the single-particle energy spectrum depends logarithmically on the quantum number. We show that two atoms initially prepared in the ground state are preferentially excited by a time-dependent interaction into a two-particle energy state characterized by the factors. Hence, a measurement of the energy of one of the two atoms yields the factors. The number to be factored is encoded in the frequency of a sinusoidally modulated interaction. We also discuss the influence of off-resonant transitions and the limitation of the number to be factored imposed by experimental conditions.

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We study the shape and the energy of 2m phase solitons, i.e. , vacancies or interstitials of the electronic crystal formed by the density wave (DW), and the distortion of the DW induced by impurities. We show that at low temperatures nonlinear screening by electronic excitations strongly modifies the characteristic spatial and energy scales of phase distortions in DW conductors, resulting in the formation of metallic regions at the soliton center and around the impurities. PACS numbers: 71.45.Lr, 72.15.v, 72.20.Pa, 75.30.Fv Quasi-one-dimensional conductors are known to exhibit the Peierls transition to a semiconducting state which contains a mobile electronic crystal called density wave (DW). Whereas the main properties of DW conductors at higher temperatures are well understood [1,2], they demonstrate many intriguing properties at low temperatures which are still not explained. Even in small electrical fields, below the threshold field for the sliding of the DW, there are many puzzling experimental results.For example, the activation energy of the temperaturedependent conductivity along the chains decreases with temperature decreasing, while the activation energy for the conductivity in the perpendicular direction remains unchanged [3,4]. At lower temperatures a behavior typical for hopping conductivity caused by impurity-induced local states inside the energy gap was observed [5]. In the same temperature region where the activation energy diminishes, an abrupt drop of thermopower was observed; moreover, in some compounds, e.g. , TaS3 [6] and (NbSe4)tol3I [7], the thermopower even changes its sign, a transition from electron to hole conductivity being excluded because no changes in the Hall constant in this temperature range are observed (see, e.g. , [8,9]). Another puzzling problem is the existence of low-energy excitations with characteristic energies well below the Peierls gap, demonstrated in measurements of specific heat [10] and of the dielectric function [11,12] at low temperatures. Excitations are important to understand the lowtemperature, probably glassy, DW behavior [13],but their origin is still not completely clear. One-particle electronic excitations are known to be important at higher temperatures, determining, e.g. , the characteristic length scale (the stiffness constant) [14,15] and the damping coefficient [16,17]. It was suggested [5,18 -20] that at low temperatures, where the excitation of quasiparticles above the gap is negligible, solitonlike distortions of the DW phase contribute to the conductivity. We found that due to nonlinear coupling of those distortions to the electrical as well as K Rv dp2 eg d + 47r dx2 hv dp"i 2 dx) (2) Here v is the Fermi velocity, K is the inverse screening length in the metallic state, J "describes the interchain interaction, and e~gives a contribution to the dielectric constant due to the opening of the gap 5 below the DW transition.p, "(x) is the local shift of the chemical potential of the nth chain from midgap position. It is related to 4"by the equil...