Effects of an exceptional point on the dynamics of a single particle in a time-dependent harmonic trap

Abstract

The time evolution of a single particle in a harmonic trap with time-dependent frequency $\ensuremath{\omega}(t)$ has been well studied. Nevertheless, here we show that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter $\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)$ fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at $\ensuremath{\mu}=2$. At this transition point, the time evolution has an 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. In the present case, we show that the order of the EP depends on the set of observables used to describe the dynamics. 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.