Decoherence as a signature of an excited-state quantum phase transition

4 pages, 4 figures.-- PACS nrs.: 03.65.Yz; 64.70.Tg; 05.70.Fh. We analyze the decoherence induced on a single qubit by the interaction with a two-level boson system with critical internal dynamics. We explore how the decoherence process is affected by the presence of quantum phase transitions in the environment. We conclude that the dynamics of the qubit changes dramatically when the environment passes through a continuous excited state quantum phase transition. If the system-environment coupling energy equals the energy at which the environment has a critical behavior, the decoherence induced on the qubit is maximal and the fidelity tends to zero with finite size scaling obeying a power law. This work has been partially supported by the Spanish Ministerio de Educación y Ciencia and by the European Regional Development Fund (FEDER) under Projects No. FIS2005-01105, No. FIS2006-12783-C03-01, No. FPA2006-13807-C02-02, and No. FPA2007-63074, by Comunidad de Madrid and CSIC under Project No. 200650M012, and by Junta de Analucía under Projects No. FQM160, No.FQM318, No. P05-FQM437, and No. P07-FQM-02962. A.R.was supported by the Spanish program "Juan de la Cierva", and P. P-F. by a grant from the Plan Propio of the University of Sevilla.