  Quantum vacuum fluctuations are a direct manifestation of Heisenberg's
uncertainty principle. The dynamical Casimir effect allows for the observation
of these vacuum fluctuations by turning them into real, observable photons.
However, the observation of this effect in a cavity QED experiment would
require the rapid variation of the length of a cavity with relativistic
velocities, a daunting challenge. Here, we propose a quantum simulation of the
dynamical Casimir effect using an ion chain confined in a segmented ion trap.
We derive a discrete model that enables us to map the dynamics of the multimode
radiation field inside a variable-length cavity to radial phonons of the ion
crystal. We perform a numerical study comparing the ion-chain quantum
simulation under realistic experimental parameters to an ideal Fabry-Perot
cavity, demonstrating the viability of the mapping. The proposed quantum
simulator, therefore, allows for probing the photon (respectively phonon)
production caused by the dynamical Casimir effect on the single photon level.
