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We investigate the normal mode spectrum of a trapped ion chain at thesymmetry-breaking linear to zigzag transition and at finite temperatures. Forthis purpose we modulate the amplitude of the Doppler cooling laser in order toexcite and measure mode oscillations. The expected mode softening at thecritical point, a signature of the second-order transition, is not observed.Numerical simulations show that this is mainly due to the finite temperature ofthe chain. Inspection of the trajectories suggest that the thermal shifts ofthe normal-mode spectrum can be understood by the ions collectively jumpingbetween the two ground state configurations of the symmetry broken phase. Wedevelop an effective analytical model, which allows us to reproduce thelow-frequency spectrum as a function of the temperature and close to thetransition point. In this model the frequency shift of the soft mode is due tothe anharmonic coupling with the high frequency modes of the spectrum, actingas an averaged effective thermal environment. Our study could prove importantfor implementing ground-state laser cooling close to the critical point.

Finite temperature spectrum at the symmetry-breaking linear-zigzag transition