Strengthened PT-symmetry with

Carl M. Bender

Miloslav Znojil

𝒫𝒯 ( = parity times time-reversal) symmetry of complex Hamiltonians with real spectra is usually interpreted as a weaker mathematical substitute for Hermiticity. Perhaps an equally important role is played by the related strengthened analyticity assumptions. In a constructive illustration we complexify a few potentials solvable only in s -wave. Then we continue their domain from the semi-axis to the whole axis and obtain new exactly solvable models. Their energies turn out to be real as expected. The new one-dimensional spectra themselves differ quite significantly from their s -wave predecessors.

Miloslav Znojil

Qinghai Wang, Song-zhi Chia, Jie-hong Zhang

The Hilbert space in -symmetric quantum mechanics is formulated as a linear vector space with a dynamic inner product. The most general -symmetric matrix Hamiltonians are constructed for the 2 × 2 and 3 × 3 cases. In the former case, the -symmetric Hamiltonian represents the most general matrix Hamiltonian with a real spectrum. In both cases, Hermitian matrices are shown to be special cases of -symmetric matrices. This finding confirms and strengthens the early belief that the -symmetric quantum mechanics is a generalization of the conventional Hermitian quantum mechanics.

Derek D. Scott

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.