We provide an experimental framework where periodically driven PT -symmetric systems can be investigated. The set-up, consisting of two UHF oscillators coupled by a time-dependent capacitance, demonstrates a cascade of PT -symmetric broken domains bounded by exceptional point degeneracies. These domains are analyzed and understood using an equivalent Floquet frequency lattice with local PT -symmetry. Management of these PT -phase transition domains is achieved through the amplitude and frequency of the drive.PACS numbers: 42.25.Bs, 11.30.Er Introduction -Non-Hermitian Hamiltonians H which commute with the joint parity-time (PT ) symmetry might have real spectrum when some parameter γ, that controls the degree of non-hermiticity, is below a critical value γ PT [1]. In this parameter domain, termed exact PT -phase the eigenfunctions of H are also eigenfunctions of the PT -symmetric operator. In the opposite limit, coined the broken PT -phase, the spectrum consists (partially or completely) of pairs of complex conjugate eigenvalues while the eigenfunctions cease to be eigenfunctions of the PT operator. The transition point γ = γ PT shows all the characteristic features of an exceptional point (EP) singularity where both eigenfunctions and eigenvalues coalesce.Although originally the interest on PT -symmetric systems was driven by a mathematical curiosity [1], during the last five years the field has blossomed and many applications in areas of physics, ranging from optics [2-18], matter waves [19,20] and magnonics [21,22] [4, 9, 10, 12-14, 17, 18, 24-26]. Importantly, the existence of the PT phase transition and specifically of the EP singularity played a prominent role in many of these studies, and subsequent technological applications.Though the exploitation of PT -symmetric systems has been prolific, most of the attention has been devoted to static (i.e. time-independent) potentials. Recently, however, a parallel activity associated with time-dependent PT -symmetric systems has started to attract increasing attention [29][30][31][32][33][34][35][36][37][38][39]. The excitement for this line of research stems from two reasons: the first one is fundamental and it is associated with the expectation that new pathways in the PT -arena can lead to new exciting phenomena. This expectation is further supported by the fact that the investigation of time-dependent Hermitian counterparts led to a plethora of novel phenomena-examples include Rabi oscillations [40], Autler-Townes splitting [41], dynamical localization [42], dynamical Anderson localization [43], and coherent destruction of tunneling [44,45] (for a review see [46]). The second reason is technological and it is associated with the possibility to use driving schemes as a flexible experimental knob to realize new forms of reconfigurable synthetic matter [47,48]. Specif-

Using a pair of coupled LRC cavities we experimentally demonstrate that instabilities and amplification action can be tamed by a spatially inhomogenous gain. Specifically we observe the counter-intuitive phenomenon of stabilization of the system even when the overall gain provided is increased. This behavior is directly related to lasing death via asymmetric pumping, recently proposed in [M. Liertzer et al., Phys. Rev. Lett. 108, 173901 (2012)]. The stability analysis of other simple systems reveals the universal nature of the lasing death phenomenon.

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The effect of prebunching of the electron beam and tapering of the wiggler amplitude on the harmonic upconversion in free-electron laser amplifier is studied in three dimensions. A set of coupled nonlinear first-order differential equations that describe the three-dimensional simulation of the system is solved numerically. This set of equation describes self-consistently the longitudinal spatial dependence of radiation waists, curvatures, and amplitudes together with the evaluation of the electron beam. The analysis is related to extreme ultraviolet and x-ray emission. In addition to uniform beam, prebunched electron beam has also been studied. The effect of sinusoidal distribution of entry times for the electron beam on the evolution of radiation is compared with uniform distribution. It is shown that prebunching reduces the saturation length substantially. For efficiency enhancement, the wiggler is set to decrease linearly when the radiation of the third harmonic saturates. The optimum starting point and the slope of tapering of the amplitude of the wiggler are found by a successive run of the code. It was found that tapering can increase the saturated power of the third harmonic considerably.

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