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We study analytically and numerically the properties of phonon modes in anion quantum computer. The ion chain is placed in a harmonic trap with anadditional periodic potential which dimensionless amplitude $K$ determinesthree main phases available for quantum computations: at zero $K$ we have thecase of Cirac-Zoller quantum computer, below a certain critical amplitude$K<K_c$ the ions are in the Kolmogorov-Arnold-Moser (KAM) phase, withdelocalized phonon modes and free chain sliding, and above the criticalamplitude $K>K_c$ ions are in the pinned Aubry phase with a finite frequencygap protecting quantum gates from temperature and other external fluctuations.For the Aubry phase, in contrast to the Cirac-Zoller and KAM phases, the phonongap remains independent of the number of ions placed in the trap keeping afixed ion density around the trap center. We show that in the Aubry phase thephonon modes are much better localized comparing to the Cirac-Zoller and KAMcases. Thus in the Aubry phase the recoil pulses lead to local oscillations ofions while in other two phases they spread rapidly over the whole ion chainsmaking them rather sensible to external fluctuations. We argue that theproperties of localized phonon modes and phonon gap in the Aubry phase provideadvantages for the ion quantum computations in this phase with a large numberof ions.

Properties of phonon modes of ion trap quantum computer in the Aubry phase