The appearance of so-called exceptional points in the complex spectra of non-Hermitian systems is often associated with phenomena that contradict our physical intuition. One example of particular interest is the state-exchange process predicted for an adiabatic encircling of an exceptional point. In this work we analyse this and related processes for the generic system of two coupled oscillator modes with loss or gain. We identify a characteristic system evolution consisting of periods of quasi-stationarity interrupted by abrupt non-adiabatic transitions, and we present a qualitative and quantitative description of this switching behaviour by connecting the problem to the phenomenon of stability loss delay. This approach makes accurate predictions for the breakdown of the adiabatic theorem as well as the occurrence of chiral behavior observed previously in this context, and provides a general framework to model and understand quasi-adiabatic dynamical effects in non-Hermitian systems.

Using amplitude equations, we show that groups of identical nano-mechanical resonators, interacting with a common mode of a cavity microwave field, synchronize to form a single mechanical mode which couples to the cavity with a strength dependent on the square sum of the individual mechanical-microwave couplings. Classically this system is dominated by periodic behaviour which, when analyzed using amplitude equations, can be shown to exhibit multi-stability. In contrast groups of sufficiently dissimilar nano-mechanical oscillators may lose synchronization and oscillate out of phase at significantly higher amplitudes. Further the method by which synchronization is lost resembles that for large amplitude forcing which is not of the Kuramoto form.

%e present a solution for the dynamics of an anharmonic oscillator coupled to a zerotemperature heat bath. Comparison of observable properties in a classical and quantum description uses true joint phase-space probability densities. The time evolution of the density in the quantum case is rapidly "reduced" to that given in the classical description. The rate of reduction is proportional to the product of the damping rate and the oscillator's initial energy. Quite rapidly, typical quantum recurrence effects are destroyed and the classical "whorl" structure restored. %e point out the close similarity with rapid destruction of quantum coherence through dissipation. In a quantum description, it is well known that the possible joint probability densities which arise are restricted to a subset of those occurring in a classical description. In a previous paper' one of us investigated the quantum and classical dynamics of joint phase-space probability When the system is described classically, an initial Gaussian joint density displaced from the origin develops into a "whorl" 2; contours of the initial density undergo a rotational shear and as time proceeds spiral out from the origin. However, when the same system is described quantum mechanically the density undergoes a more complicated evolution; "interference" fringes develop, the initial state recurs at a fixed period, and no whorl develops. This more complicated behavior is manifested in the density evolution equation by the appearance of second-order derivatives with complex coefficients. Similar behavior for a related anharmonic model has recently been reported by Takahashi and Saito. 3 In this Letter we consider the effect of dissipation on the evolution of phase-space densities for a particular anharmonic oscillator model. We show that quite apart from the overall contraction of phase space, the effect of dissipation is to destroy the interference (3) terms, prevent a recurrence of the initial state and to restore the classical whorl structure. The destruction of interference effects becomes much more rapid as the average energy of the initial state increases (i.e. , when the initial density is concentrated at a large radius). This dependence on the initial energy is similar to the decay of off-diagonal coherence in the harmonic oscillator.~We are thus lead to interpret the interference terms as a manifestation of quantum coherences, between parts of the density wrapped on neighboring tori.We incorporate dissipation in our model by coupling the anharmonic oscillator to a reservoir of oscillators which we assume to be at zero temperature. The reservoir is then eliminated and a Markovian master equation for the oscillator-density operator in the interaction picture is obtained. A unique phase-space density is associated with the density operator by means of a bounded positive map from the state space of density operators to the classical state space of probability densities on phase space. This map is then used to transform the evolution equation for the dens...

The simplest model of three coupled Bose-Einstein condensates is investigated using a group theoretical method. The stationary solutions are determined using the SU͑3͒ group under the mean-field approximation. This semiclassical analysis, using system symmetries, shows a transition in the dynamics of the system from self trapping to delocalization at a critical value for the coupling between the condensates. The global dynamics are investigated by examination of the stable points, and our analysis shows that the structure of the stable points depends on the ratio of the condensate coupling to the particle-particle interaction, and undergoes bifurcations as this ratio is varied. This semiclassical model is compared to a full quantum treatment, which also displays a dynamical transition. The quantum case has collapse and revival sequences superimposed on the semiclassical dynamics, reflecting the underlying discreteness of the spectrum. Nonzero circular current states are also demonstrated as one of the higher-dimensional effects displayed in this system.