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arXiv:1104.1666 · Physical Review A

Degrees and signatures of broken PT symmetry in nonuniform lattices

Derek D. Scott, Yogesh N. Joglekar

Published May 23, 2011• 57 citations
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Abstract

We investigate the robustness of the parity- and time-reversal ($\mathcal{P}\mathcal{T}$) symmetric phase in an $N$-site lattice with a position-dependent, parity-symmetric hopping function and a pair of imaginary, $\mathcal{P}\mathcal{T}$-symmetric impurities. We find that the ``fragile'' $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$ symmetry and their signatures in single-particle wave-packet evolution. We predict that, when the $\mathcal{P}\mathcal{T}$-symmetric impurities are closest to each other, the time evolution of a wave packet in an even-$N$ lattice is remarkably different from that in an odd-$N$ lattice. Our results suggest that $\mathcal{P}\mathcal{T}$ symmetry breaking in such lattices is accompanied by rich, hitherto unanticipated, phenomena.

DOI
10.1103/physreva.83.050102

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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.
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 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.
PT\mathcal {PT}PT-symmetry breaking and maximal chirality in a nonuniform PT\mathcal {PT}PT symmetric ring
We study the properties of an N -site tight-binding ring with parity and time-reversal ( ) symmetric, Hermitian, site-dependent tunneling and a pair of non-Hermitian, -symmetric, loss and gain impurities ±iγ. The properties of such lattices with open boundary conditions have been intensely explored over the past 2 years. We numerically investigate the -symmetric phase in a ring with a position-dependent tunneling function t α ( k ) = [ k ( N − k )] α/2 that, in an open lattice, leads to a strengthened -symmetric phase, and study the evolution of the -symmetric phase from the open chain to a ring. We show that, generally, periodic boundary conditions weaken the -symmetric phase, although for experimentally relevant lattice sizes N ∼ 50, it remains easily accessible. We show that the chirality, quantified by the (magnitude of the) average transverse momentum of a wave packet, shows a maximum at the -symmetric threshold. Our results show that although the wavepacket intensity increases monotonically across the -breaking threshold, the average momentum decays monotonically on both sides of the threshold. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
By investigating a parity- and time-reversal- (PT\mathcal{P}\mathcal{T}PT-) symmetric, NNN-site lattice with impurities $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ and hopping amplitudes t0(tb){t}_{0}\phantom{\rule{0.28em}{0ex}}({t}_{b})t0​(tb​) for regions outside (between) the impurity locations, we probe the interimpurity-distance dependence of the critical impurity strength and the origin of maximal PT\mathcal{P}\mathcal{T}PT-symmetry breaking that occurs when the impurities are nearest neighbors. Through a simple and exact derivation, we prove that the critical impurity strength is equal to the hopping amplitude between the impurities, ${\ensuremath{\gamma}}_{c}={t}_{b}$, and the simultaneous emergence of NNN complex eigenvalues is a robust feature of any PT\mathcal{P}\mathcal{T}PT-symmetric hopping profile. Our results show that the threshold strength ${\ensuremath{\gamma}}_{c}$ can be widely tuned by a small change in the global profile of the lattice and thus have experimental implications.
We investigate the properties of an NNN-site tight-binding lattice with periodic boundary conditions (PBCs) in the presence of a pair of gain and loss impurities $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ and two tunneling amplitudes, t0{t}_{0}t0​ and tb{t}_{b}tb​, that are constant along the two paths that connect them. We show that the parity and time-reversal (PT\mathcal{P}\mathcal{T}PT)-symmetric phase of the lattice with PBCs are robust, insensitive to the distance between the impurities, and that the critical impurity strength for PT\mathcal{P}\mathcal{T}PT-symmetry breaking is given by ${\ensuremath{\gamma}}_{\mathrm{PT}}=|{t}_{0}\ensuremath{-}{t}_{b}|$. We study the time-evolution of a typical wave packet, initially localized on a single site, across the PT\mathcal{P}\mathcal{T}PT-symmetric phase boundary. We find that it acquires nonzero chirality for $\ensuremath{\gamma}\ensuremath{\ne}0$, and the associated momentum reaches a universal maximum value at the threshold, $\ensuremath{\gamma}={\ensuremath{\gamma}}_{\mathrm{PT}}$, irrespective of the initial location of the wave packet and the lattice parameters. Our results imply that PT\mathcal{P}\mathcal{T}PT-symmetry breaking on a lattice with PBCs has consequences that have no counterpart in open chains.
We investigate the single-particle time evolution and two-particle quantum correlations in a one-dimensional NNN-site lattice with a site-dependent nearest-neighbor tunneling function ${t}_{\ensuremath{\alpha}}(k)={t}_{0}[k(N\ensuremath{-}k)]{}^{\ensuremath{\alpha}/2}$. Since the bandwidth and the energy-level spacings for such a lattice both depend upon $\ensuremath{\alpha}$, we show that the observable properties of a wave packet, such as its spread and the relative phases of its constituents, vary dramatically as $\ensuremath{\alpha}$ is varied from positive to negative values. We also find that the quantum correlations are exquisitely sensitive to the form of the tunneling function. Our results suggest that arrays of waveguides with position-dependent evanescent couplings will show rich dynamics with no counterpart in present-day, traditional systems.