The phenomenon of PTsymmetric breaking in classical systems with balanced loss and gain is associated with a sharp transition from a purely real to complex eigenvalue spectrum of the underlying dynamical matrix. Over the past years this phenomenon has been extensively studied, for example, using coupled optical modes, where however, the system is always in a large amplitude classical state. In this thesis we study for the first time the effect of PTsymmetry breaking in the quantum regime where the effects of nonlinearities and intrinsic quantum noise become important. In the first part we analyze the stationary states of two coupled harmonic oscillators with engineered loss and gain. By applying different numerical techniques to solve the corresponding master equation for this system we observe an unconventional transition from a highnoise symmetric state to a paritybroken lasing state with strongly reduced fluctuations. Moreover, we show that the transition point strongly depends on the quantumness of the system. In the second part we develop numerical techniques for the simulation of extended PTsymmetric spin chains, which we use to demonstrate a crossover from a symmetric to a symmetry broken phase also for finite dimensional quantum systems.
