Mutually coupled modes of a pair of active LRC circuits, one with amplification and another with an equivalent amount of attenuation, provide an experimental realization of a wide class of systems where gain/loss mechanisms break the Hermiticity while preserving parity-time PT symmetry. For a value γPT of the gain/loss strength parameter the eigen-frequencies undergo a spontaneous phase transition from real to complex values, while the normal modes coalesce acquiring a definite chirality. The consequences of the phase-transition in the spatiotemporal energy evolution are also presented. PACS numbers: 11.30.Er, 03.65.Vf Parity (P) and time -reversal (T ) symmetries, as well as their breaking, belong to the most basic notions in physics. Recently there has been much interest in systems which do not obey P and T -symmetries separately but do exhibit a combined PT -symmetry. Examples of such PT -symmetric systems range from quantum field theories and mathematical physics [1-3] to atomic [4], solid state [5,6] and classical optics [7][8][9][10][11][12][13][14][15]. A PTsymmetric system can be described by a phenomenological "Hamiltonian" H. Such Hamiltonians may have a real energy spectrum, although in general are nonHermitian. Furthermore, as some parameter γ that controls the degree of non-Hermiticity of H changes, a spontaneous PT symmetry breaking occurs. The transition point γ = γ PT show the characteristic behaviour of an exceptional point (EP) where both eigenvalues and eigenvectors coallesce (for experimental studies of EP singularities of lossy systems see Ref.[16]). For γ > γ PT , the eigenfunctions of H cease to be eigenfunctions of the PToperator, despite the fact that H and the PT -operator commute [1]. This happens because the PT -operator is anti-linear, and thus the eigenstates of H may or may not be eigenstates of PT . As a consequence, in the broken PT -symmetric phase the spectrum becomes partially or completely complex. The other limit where both H and PT share the same set of eigenvectors, corresponds to the so-called exact PT -symmetric phase in which the spectrum is real. This result led Bender and colleagues to propose an extension of quantum mechanics based on nonHermitian but PT -symmetric operators [1,2]. The class of non-Hermitian systems with real spectrum has been extended by Mostafazadeh in order to include Hamiltonians with generalized PT (antilinear) symmetries [17].While these ideas are still debatable, it was recently suggested that optics can provide a particularly fertile ground where PT -related concepts can be realized [7] and experimentally investigated [8,9]. In this framework, PT symmetry demands that the complex refractive index obeys the condition n( r) = n * (− r). PTsynthetic materials can exhibit several intriguing features. These include among others, power oscillations and non-reciprocity of light propagation [7,9,11], absorption enhanced transmission [8], and unidirectional invisibility [15]. In the nonlinear domain, such nonreciprocal effects can be used to realiz...

We show both theoretically and experimentally that a pair of inductively coupled active LRC circuits (dimer), one with amplification and another with an equivalent amount of attenuation, display all the features which characterize a wide class of non-Hermitian systems which commute with the joint parity-time PT operator: typical normal modes, temporal evolution, and scattering processes. Utilizing a Liouvilian formulation, we can define an underlying PT -symmetric Hamiltonian, which provides important insight for understanding the behavior of the system. When the PT -dimer is coupled to transmission lines, the resulting scattering signal reveals novel features which reflect the PT -symmetry of the scattering target. Specifically we show that the device can show two different behaviors simultaneously, an amplifier or an absorber, depending on the direction and phase relation of the interrogating waves. Having an exact theory, and due to its relative experimental simplicity, PT -symmetric electronics offers new insights into the properties of PT -symmetric systems which are at the forefront of the research in mathematical physics and related fields.

The beat time τ fpt associated with the energy transfer between two coupled oscillators is dictated by the bandwidth theorem which sets a lower bound τ fpt ∼ 1/δω. We show, both experimentally and theoretically, that two coupled active LRC electrical oscillators with parity-time (PT ) symmetry, bypass the lower bound imposed by the bandwidth theorem, reducing the beat time to zero while retaining a real valued spectrum and fixed eigenfrequency difference δω. Our results foster new design strategies which lead to (stable) pseudo-unitary wave evolution, and may allow for ultrafast computation, telecommunication, and signal processing.One of the fundamental principles of wave physics is the Bandwidth Theorem [1] which in quantum mechanics takes the form of the celebrated energy-time Heisenberg Uncertainty relation [2]. A direct consequence of this principle is the fact that the time for evolution between two orthogonal states τ fpt (first passage time) is bounded by τ fpt ∼ 1/δω [3,4]. A basic example where this lower bound can be exhibited is the beat time associated with the energy transfer between two coupled oscillators. Even though the validity of the uncertainty principle is undoubted, it has recently been suggested that possible extensions of quantum mechanics invoking non-Hermitian PT −symmetric Hamiltonians [5] can generate arbitrarily fast state evolution referred to as brachistochrone dynamics [6, 7,[9][10][11]. The main characteristic of this class of Hamiltonians H [6, 8-10] is that they commute with an anti-linear operator PT , where the time-reversal operator T is the anti-linear operator of (generalized) complex conjugation and P is a (generalized) parity operator [5]. Examples of such PT -symmetric systems range from quantum field theories to solid state physics and classical optics [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Due to the anti-linear nature of the PT operator, the eigenstates of H may or may not be eigenstates of PT . In the former case, all the eigenvalues of H are strictly real and the PT -symmetry is said to be exact. Otherwise the symmetry is said to be spontaneously broken. In many physical realizations, the transition from the exact to the broken PT -symmetric phase is due to the presence of various gain/loss mechanisms that are controlled by some parameter γ of H.At the same time, the brachistochrone evolution has a long standing history and is significant both in theory and in application. It was one of the earliest problems posed in the calculus of variations, and in the framework of classical mechanics it dictates "the curve down which a particle sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time" [35]. The quantum mechanical brachistochrone problem has recently been revived in the emerging fields of quantum computation and sig-nal processing where one studies the possibility to use dynamical protocols in solving computational problems and enhancing signal transport respectively....

We investigate experimentally parity-time (PT ) symmetric scattering using LRC circuits in an inductively coupled PT -symmetric pair connected to transmission line leads. In the single-lead case, the PT -symmetric circuit acts as a simple dual device -an amplifier or an absorber depending on the orientation of the lead. When a second lead is attached, the system exhibits unidirectional transparency for some characteristic frequencies. This non-reciprocal behavior is a consequence of generalized (non-unitary) conservation relations satisfied by the scattering matrix.

First of all, it would be impossible to overstate the thanks I owe to Fred, whose mentorship for the past four years has been beyond formative. Thanks for teaching me almost everything I know! Many thanks also go to Tsampikos, for making this entire project and thesis possible, for always providing direction and relevance to our research, and for many exciting discussions. I owe Gideon and Nick immensely, without them I literally couldn't have finished the octoboard project. Their hard work on the Axon interfacing was amazing, and deeply appreciated. Thanks to Hamid and Zin for making collaborating so much fun, and I hope our collaborations extend to the future. Sam, it's been a great time down in the lab, glad to share it with you. Sorry in advance if you have to make all those octoboard connectors. And to the rest of my physics peers, the physics grad students, my teachers, and all the physics faculty: thanks for making my time in the department fun and enlightening. Thanks to Andy, Anna, Max, AK, FMR, Ty, and everyone else who spent time with me in the underworld, making the writing process more fun and considerably slower. Finally, thanks to my family for being the best, and to NF for the same reason. i References 90