  The time evolution of a single particle in a harmonic trap with time
dependent frequency omega(t) is well studied. Nevertheless here we show that,
when the harmonic trap is opened (or closed) as function of time while keeping
the adiabatic parameter mu = [d omega(t)/dt]/omega(t)^2 fixed, a sharp
transition from an oscillatory to a monotonic exponential dynamics occurs at mu
= 2. At this transition point the time evolution has a third-order exceptional
point (EP) at all instants. This situation, where an EP of a time-dependent
Hermitian Hamiltonian is obtained at any given time, is very different from
other known cases. Our finding is relevant to the dynamics of a single ion in a
magnetic, optical, or rf trap, and of diluted gases of ultracold atoms in
optical traps.
