  Quantum simulation provides quantum systems under study with analogous
controllable quantum systems and has wide applications from condensed-matter
physics to high energy physics and to cosmology. The quantum system of a
homogeneous and isotropic field in the Friedmann-Robertson-Walker universe can
be simulated by a charge in an electrically modulated ion trap. The quantum
states of these time-dependent oscillators are constructed by quantum
invariants. Further, we propose simulation of quantum
Friedmann-Robertson-Walker universe with a minimal massive scalar field by a
charged scalar field in a homogeneous, time-dependent, magnetic field in
quantum electrodynamics and investigate the Cauchy problem of how the wave
functions evolve.
