  Quantum phase transitions (QPTs) are usually associated with many-body
systems with large degrees of freedom approaching the thermodynamic limit. In
such systems, the many-body ground state shows abrupt changes at zero
temperature when the control parameter of the Hamiltonian is scanned across a
quantum critical point. Recently it has been realized that a QPT can also occur
in a simple system composed of only a two-level atom and a single-mode bosonic
field, described by the quantum Rabi model (QRM). Here we report the first
experimental demonstration of a QPT in the QRM using a single trapped ion. We
measure the average spin-up state population of the ion and the average phonon
number in its spatial oscillation mode as two order parameters and observe the
clear evidences of the phase transition via slow quench of the coupling between
the ion and its spatial motion. An experimental probe of the phase transitions
in a fundamental quantum optics model without imposing the thermodynamic limit
opens up a new window for the controlled study of QPTs and quantum critical
phenomena.
