D. Segal

The text below was written during two visits that Daniel Segal made at Universit{é} Paris 13. Danny stayed at Laboratoire de Physique des Lasers the summers of 2008 and 2009 to participate in the exploration of a novel lead in the field of ultra-high resolution spectroscopy. Our idea was to probe trapped molecular ions using Quantum Logic Spectroscopy (QLS) in order to advance our understanding of a variety of fundamental processes in nature. At that time, QLS, a ground-breaking spectroscopic technique, had only been demonstrated with atomic ions. Our ultimategoals were new approaches to the observation of parity violation in chiral molecules and tests of time variations of the fundamental constants. This text is the original research proposal written eight years ago. We have added a series of notes to revisit it in the light of what has been since realized in the field.

Since Hund's work on the ammonia molecule, the double well potential has formed a key paradigm in physics. Its importance is further underlined by the central role it plays in the Landau theory of phase transitions. Recently, the study of entanglement properties of many-body systems has added a new angle to the study of quantum phase transitions of discrete and continuous degrees of freedom, i.e., spin and harmonic chains. Here we show that control of the radial degree of freedom of trapped ion chains allows for the simulation of linear and non-linear Klein-Gordon fields on a lattice, in which the parameters of the lattice, the non-linearity and mass can be controlled at will. The system may be driven through a phase transition creating a double well potential between different configurations of the ion crystal. The dynamics of the system are controllable, local properties are measurable and tunnelling in the double well potential would be observable.