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

We study analytically and numerically the properties of phonon modes in an ion quantum computer. The ion chain is placed in a harmonic trap with an additional periodic potential which dimensionless amplitude $K$ determines three main phases available for quantum computations: at zero $K$ we have the case of Cirac-Zoller quantum computer, below a certain critical amplitude $K<K_c$ the ions are in the Kolmogorov-Arnold-Moser (KAM) phase, with delocalized phonon modes and free chain sliding, and above the critical amplitude $K>K_c$ ions are in the pinned Aubry phase with a finite frequency gap protecting quantum gates from temperature and other external fluctuations. For the Aubry phase, in contrast to the Cirac-Zoller and KAM phases, the phonon gap remains independent of the number of ions placed in the trap keeping a fixed ion density around the trap center. We show that in the Aubry phase the phonon modes are much better localized comparing to the Cirac-Zoller and KAM cases. Thus in the Aubry phase the recoil pulses lead to local oscillations of ions while in other two phases they spread rapidly over the whole ion chains making them rather sensible to external fluctuations. We argue that the properties of localized phonon modes and phonon gap in the Aubry phase provide advantages for the ion quantum computations in this phase with a large number of ions.