Incoherent amplification cannot compensate for nonclassicality loss in PT-symmetric systems

Abstract

We show that the loss of nonclassicality (including quantum entanglement) cannot be compensated by the (incoherent) amplification of PT -symmetric systems. We address this problem by manipulating the quantum fluctuating forces in the Heisenberg-Langevin approach. Specifically, we analyze the dynamics of two nonlinearly coupled oscillator modes in a PT -symmetric system. An analytical solution allows us to separate the contribution of reservoir fluctuations from the evolution of quantum statistical properties of the modes. In general, as reservoir fluctuations act constantly, the complete loss of nonclassicality and entanglement is observed for long times. To elucidate the role of reservoir fluctuations in a long-time evolution of nonclassicality and entanglement, we consider and compare the predictions from two alternative models in which no fatal long-time detrimental effects on the nonclassicality and entanglement are observed. This is so as, in the first semiclassical model, no reservoir fluctuations are considered at all. This, however, violates the fluctuation-dissipation theorem. The second, more elaborated, model obeys the fluctuation-dissipation relations as it partly involves reservoir fluctuations. However, to prevent from the above long-time detrimental effects, the reservoir fluctuations have to be endowed with the nonphysical properties of a sink model. In both models, additional incorporation of the omitted reservoir fluctuations results in their physically consistent behavior. This behavior, however, predicts the gradual loss of the nonclassicality and entanglement. Thus the effects of reservoir fluctuations related to damping cannot be compensated by those related to amplification. This qualitatively differs from the influence of damping and amplification to a direct coherent dynamics of PT -symmetric systems in which their mutual interference results in a periodic behavior allowing for nonclassicality and entanglement at arbitrary times.