Rivas Vargas, ÁngelLuis Aina, Alfredo2023-06-202023-06-2020121367-263010.1088/1367-2630/14/9/093052https://hdl.handle.net/20.500.14352/44589© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. We are grateful to Professors A Smerzi, L Maccone, M Tsang and M J W Hall for useful discussions. We acknowledge financial support from the project QUITEMAD S2009-ESP-1594 of the Consejeria de Educacion de la Comunidad de Madrid. AR acknowledges support from MICINN FIS2009-10061. AL acknowledges support from project no. FIS2008-01267 of the Spanish Direccion General de Investigacion del Ministerio de Ciencia e Innovacion.We provide evidence that the uncertainty in detection of small and deterministic phase-shift deviations from a working point can be lower than the Heisenberg bound, for a fixed finite mean number of photons. We achieve that by exploiting nonlinearity of estimators and coherence with the vacuum.engAtribución 3.0 Españahttps://creativecommons.org/licenses/by/3.0/es/journal articlehttp://dx.doi.org/10.1088/1367-2630/14/9/093052http://iopscience.iop.org/open access535QuantumLikelihoodLimitStatisticsÓptica (Física)2209.19 Óptica Física