  We design faster-than-adiabatic state transfers (switching of quantum
numbers) in time-dependent coupled-oscillator Hamiltonians. The manipulation to
drive the process is found using a two-dimensional invariant recently proposed
in S. Simsek and F. Mintert, Quantum 5 (2021) 409, and involves both rotation
and transient scaling of the principal axes of the potential in a Cartesian
representation. Importantly, this invariant is degenerate except for the
subspace spanned by its ground state. Such degeneracy, in general, allows for
infidelities of the final states with respect to ideal target eigenstates.
However, the value of a single control parameter can be chosen so that the
state switching is perfect for arbitrary (not necessarily known) initial
eigenstates. Additional 2D linear invariants are used to find easily the
parameter values needed and to provide generic expressions for the final states
and final energies. In particular we find time-dependent transformations of a
two-dimensional harmonic trap for a particle (such as an ion or neutral atom)
so that the final trap is rotated with respect to the initial one, and
eigenstates of the initial trap are converted into rotated replicas at final
time, in some chosen time and rotation angle.
