  We study a nonlinear oscillator, which is parametrically driven at a
frequency close to twice its eigenfrequency. By judiciously choosing the
frequency detuning and linearly increasing the driving amplitude, one can
prepare any even quasienergy state starting from the oscillator ground state.
Such state preparation is effectively adiabatic. We find the Wigner
distribution of the prepared states. For a different choice of the frequency
detuning, the adiabaticity breaks down, which allows one to prepare on demand a
superposition of quasienergy states using Landau-Zener-type transitions. We
find the characteristic spectrum of the transient radiation emitted by the
oscillator after it has been prepared in a given quasienergy state.
