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arXiv:1012.4475 · Physical Review A

Robust PT -symmetric chain and properties of its Hermitian counterpart

Yogesh N. Joglekar, Avadh Saxena

Published May 4, 2011• 56 citations
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Abstract

We study the properties of a parity- and time-reversal- ($\mathcal{P}\mathcal{T}$) symmetric tight-binding chain of size $N$ with position-dependent hopping amplitude. In contrast to the fragile $\mathcal{P}\mathcal{T}$-symmetric phase of a chain with constant hopping and imaginary impurity potentials, we show that, under very general conditions, our model is always in the $\mathcal{P}\mathcal{T}$-symmetric phase. We numerically obtain the energy spectrum and the density of states of such a chain, and show that they are widely tunable. By studying the size dependence of inverse participation ratios, we show that although the chain is not translationally invariant, most of its eigenstates are extended. Our results indicate that tight-binding models with non-Hermitian, $\mathcal{P}\mathcal{T}$-symmetric hopping have a robust $\mathcal{P}\mathcal{T}$-symmetric phase and rich dynamics which may be explored in coupled waveguides.

DOI
10.1103/physreva.83.050101

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We study the properties of a parity- and time-reversal- (PT\mathcal{P}\mathcal{T}PT) symmetric tight-binding chain of size NNN with position-dependent hopping amplitude. In contrast to the fragile PT\mathcal{P}\mathcal{T}PT-symmetric phase of a chain with constant hopping and imaginary impurity potentials, we show that, under very general conditions, our model is always in the PT\mathcal{P}\mathcal{T}PT-symmetric phase. We numerically obtain the energy spectrum and the density of states of such a chain, and show that they are widely tunable. By studying the size dependence of inverse participation ratios, we show that although the chain is not translationally invariant, most of its eigenstates are extended. Our results indicate that tight-binding models with non-Hermitian, PT\mathcal{P}\mathcal{T}PT-symmetric hopping have a robust PT\mathcal{P}\mathcal{T}PT-symmetric phase and rich dynamics which may be explored in coupled waveguides.
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.
We study the non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an NNN-site tight-binding chain with two conjugated imaginary potentials $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ at two end sites, which Hamiltonian has parity-time symmetry (PT\mathcal{P}\mathcal{T}PT symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases: an unbroken symmetry phase with a purely real energy spectrum in the region $\ensuremath{\gamma}\ensuremath{\prec}{\ensuremath{\gamma}}_{c}$ and a spontaneously broken symmetry phase with $N\ensuremath{-}2$ real and two imaginary eigenvalues in the region $\ensuremath{\gamma}\ensuremath{\succ}{\ensuremath{\gamma}}_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of ${\ensuremath{\gamma}}_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.
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.)
We investigate the robustness of the parity- and time-reversal (PT\mathcal{P}\mathcal{T}PT) symmetric phase in an NNN-site lattice with a position-dependent, parity-symmetric hopping function and a pair of imaginary, PT\mathcal{P}\mathcal{T}PT-symmetric impurities. We find that the ``fragile'' PT\mathcal{P}\mathcal{T}PT-symmetric phase in these lattices is stronger than its counterpart in a lattice with constant hopping. With an open system in mind, we explore the degrees of broken PT\mathcal{P}\mathcal{T}PT symmetry and their signatures in single-particle wave-packet evolution. We predict that, when the PT\mathcal{P}\mathcal{T}PT-symmetric impurities are closest to each other, the time evolution of a wave packet in an even-NNN lattice is remarkably different from that in an odd-NNN lattice. Our results suggest that PT\mathcal{P}\mathcal{T}PT symmetry breaking in such lattices is accompanied by rich, hitherto unanticipated, phenomena.
The occurrence of parity-time reversal (PT)(\mathcal{P}\mathcal{T})(PT) symmetry breaking is discussed in a non-Hermitian spin chain. The Hermiticity of the model is broken by the presence of an alternating, imaginary, transverse magnetic field. A full real spectrum, which occurs if and only if all the eigenvectors are PT\mathcal{P}\mathcal{T}PT symmetric, can appear only in presence of dimerization, i.e., only if the hopping amplitudes between nearest-neighbor spins assume alternate values along the chain. In order to make a connection between such system and the Hermitian world, we study the critical magnetic properties of the model and look for the conditions that would allow to observe the same phase diagram in the absence of the imaginary field. Such procedure amounts to renormalizing the spin-spin coupling amplitudes.
We explore an alternative way of finding the link between a PT\mathcal{P}\mathcal{T}PT non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of the scattering problem for an imaginary potential and its time-reversal process, it is shown that any real-energy eigenstate of a PT\mathcal{P}\mathcal{T}PT tight-binding lattice with on-site imaginary potentials shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain. It indicates that the PT\mathcal{P}\mathcal{T}PT eigenstate of a PT\mathcal{P}\mathcal{T}PT non-Hermitian Hamiltonian has a connection to the resonance transmission state of the extended Hermitian Hamiltonian.