We investigate the parity- and time-reversal ($\mathcal{PT}$-) symmetry breaking in lattice models in the presence of long-ranged, non-Hermitian, $\mathcal{PT}$-symmetric potentials that remain finite or become divergent in the continuum limit. By scaling analysis of the fragile $\mathcal{PT}$ threshold for an open finite lattice, we show that continuum loss-gain potentials ${V}_{\ensuremath{\alpha}}(x)\ensuremath{\propto}i{|x|}^{\ensuremath{\alpha}}\mathrm{sgn}(x)$ have a positive $\mathcal{PT}$-breaking threshold for $\ensuremath{\alpha}>\ensuremath{-}2$, and a zero threshold for $\ensuremath{\alpha}\ensuremath{\le}\ensuremath{-}2$. When $\ensuremath{\alpha}<0$ localized states with complex (conjugate) energies in the continuum energy band occur at higher loss-gain strengths. We investigate the signatures of $\mathcal{PT}$-symmetry breaking in coupled waveguides, and show that the emergence of localized states dramatically shortens the relevant time scale in the $\mathcal{PT}$-symmetry broken region.
We investigate the parity- and time-reversal (PT-) symmetry breaking in lattice models in the presence of long-ranged, non-Hermitian, PT-symmetric potentials that remain finite or become divergent in the continuum limit. By scaling analysis of the fragile PT threshold for an open finite lattice, we show that continuum loss-gain potentials V\ensuremathα(x)\ensuremath∝i∣x∣\ensuremathαsgn(x) have a positive PT-breaking threshold for \ensuremathα>\ensuremath−2, and a zero threshold for \ensuremathα\ensuremath≤\ensuremath−2. When \ensuremathα<0 localized states with complex (conjugate) energies in the continuum energy band occur at higher loss-gain strengths. We investigate the signatures of PT-symmetry breaking in coupled waveguides, and show that the emergence of localized states dramatically shortens the relevant time scale in the PT-symmetry broken region.
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].
The study of electromagnetic waves in engineering optical materials presents opportunities to manipulate light transmission through various optical media such as fiber optic cables. Recent experiments in this field have explored materials that have balanced gain and absorption that occur at different locations. We can model such a system as a chain with nearest neighbor hopping, and balanced, parity and time reversal symmetric (PT-symmetric) impurities, and it is represented by a non-Hermitian Hamiltonian matrix. The emergence of complex energy eigenvalues for such a matrix corresponds to the PT-symmetry breaking. We numerically and analytically investigate the eigenvalues of such a non-Hermitian Hamiltonian for an open chain with N sites. We find that the critical impurity strength, when the PT-symmetry breaks, is determined by the hopping amplitude between the impurities t_b, and hopping amplitude outside the impurities t_o in a non-trivial way. As a consequence, we show that the PT-breaking in such an open chain can be dramatically tuned by changing the hopping amplitude. Our results suggest that small changes in such systems, created in optical waveguides, can lead to significant changes for wave propagation through them (Joglekar, Y. N.,& Barnett, J. L., (2011). Origin of maximal symmetry breaking in even PT-symmetric lattices, Physical Review A, 84, 024103.)
Parity-time (PT) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a PT-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the PT breaking threshold. Here, we obtain the PT threshold for a one-dimensional, finite Kitaev chain---a prototype for a p-wave superconductor---in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. In addition to a robust, nonlocal threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band structure of the Hermitian Kitaev model. In particular, for an even chain with zero on-site potential, we find a re-entrant PT-symmetric phase bounded by second-order EP contours. Our numerical results are supplemented by analytical calculations for small system sizes.
Thresholdless dynamical instability in transverse PT-symmetric scattering systems: The hidden role of bound states in the continuum
The stationary scattering properties of transverse parity-time (PT) symmetric systems have been extensively studied, yet their dynamical stability, a prerequisite for any stationary description, remains largely unexplored. Here we uncover a thresholdless dynamical instability in such systems, driven by symmetry-protected bound states in the continuum (BICs). Without gain and loss, the up-down mirror symmetry of the transverse geometry generically protects a BIC, which manifests as the coalescence of an S-matrix pole and zero on the real axis of the complex wave-number plane. A Hermitian symmetry-breaking perturbation shifts the poles into the lower half-plane, converting the BIC into a resonance with a Fermi-golden-rule decay width. An anti-Hermitian PT-symmetric perturbation instead reverses the sign of the second-order energy shift, driving the poles into the upper half-plane and producing a time-growing bound state with ImE=γ2ΓV/2+O(γ3), where γ is the gain-loss strength and ΓV is the golden-rule coupling of the BIC to the continuum. The instability therefore sets in at arbitrarily small γ whenever ΓV>0. We confirm this mechanism in a three-site side-coupled model that is unstable despite possessing a unitary scattering matrix, and in a four-site rhombic model where transverse and longitudinal gain-loss placements yield vanishing and finite thresholds, respectively. These results establish a microscopic stability criterion and a design principle for stable PT-symmetric scattering devices.
We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for non-hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of T-breaking, and T is broken in a generic manner (without preserving other discrete symmetries), then the standard PT-symmetry breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However the realness of the entire spectrum can be preserved over a finite interval if additional discrete symmetries X are imposed when T is broken, if X decouple all degenerate modes. When this is true only for a subset of the degenerate spectrum, there can be a partial PT transition in which this subset remains real over a finite interval of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different T-breaking perturbations of a uniform dielectric disk and sphere, and a group theoretical analysis is given in the disk case. Finally, we show that multimode coupling is capable of restoring the T-symmetric phase at finite T-breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.