Quantum quenches of ion Coulomb crystals across structural instabilities II: thermal effects

We theoretically analyze the efficiency of a protocol for creating mesoscopic superpositions of ion chains, described in [Phys. Rev. A 84, 063821 (2011)], as a function of the temperature of the crystal. The protocol makes use of spin-dependent forces, so that a coherent superposition of the electronic states of one ion evolves into an entangled state between the chain's internal and external degrees of freedom. Ion Coulomb crystals are well isolated from the external environment, and should therefore experience a coherent, unitary evolution, which follows the quench and generates structural Schr\"odinger cat-like states. The temperature of the chain, however, introduces a statistical uncertainty in the final state. We characterize the quantum state of the crystal by means of the visibility of Ramsey interferometry performed on one ion of the chain, and determine its decay as a function of the crystal's initial temperature. This analysis allows one to determine the conditions on the chain's initial state in order to efficiently perform the protocol.