$\mathcal{PT}$ -Symmetric Periodic Optical Potentials

Abstract

In quantum theory, any Hamiltonian describing a physical system is mathematically represented by a self-adjoint linear operator to ensure the reality of the associated observables. In an attempt to extend quantum mechanics into the complex domain, it was realized few years ago that certain non-Hermitian parity-time (PT ) symmetric Hamiltonians can exhibit an entirely real spectrum. Much of the reported progress has been remained theoretical, and therefore hasn't led to a viable experimental proposal for which non Hermitian quantum effects could be observed in laboratory experiments. Quite recently however, it was suggested that the concept of PT -symmetry could be physically realized within the framework of classical optics. This proposal has, in turn, stimulated extensive investigations and research studies related to PT -symmetric Optics and paved the way for the first experimental observation of PT -symmetry breaking in any physical system. In this paper, we present recent results regarding PT -symmetric Optics.

Stability analysis for solitons inPT-symmetric optical lattices

“…For this system, we first show analytically that when the strength of the gain-loss component (the imaginary part of V (x, y)) in the PT lattice rises above a certain threshold (phase-transition point), an infinite number of linear Bloch bands turn complex simultaneously. This simultaneous bifurcation of an infinite number of complex eigenvalues at the phase transition point has never been reported before for any PT-symmetric potentials to our best knowledge [1,9]. Second, we show that while stable families of solitons can exist in PT lattices (below the phase transition point), increasing the gain-loss component has an overall destabilizing effect on soliton propagation.…”

“…These phenomena may also be studied in a nonlinear * Corresponding author, email: jyang@math.uvm.edu context by considering the existence of localized modes called solitons [8,9]. When a system contains gain and loss, solitons generally exist only at special values of the propagation constant [10].…”

PT symmetry via electromagnetically induced transparency

“…It is just this point that provides us the possibility to realize a PT symmetric potential in our system by using the periodic external laser fields. (2). If the incoherent pumping is absent, the probe field has only absorption but no gain and hence not possible to realize PT symmetry.…”

“…In recent years, a lot of efforts have been made on a class of non-Hermitian Hamiltonian with parity-time (PT) symmetry, which in a definite range of system parameters may have an entirely real spectrum [1,2]. PT symmetry requires that the real (imaginary) part of the complex potential in the Hamiltonian is an even (odd) function of space, i.e.…”

Solitary Waves of a <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}$</tex-math> </inline-formula><inline-formula> <tex-math notation="LaTeX">$\mathcal {T}$</tex-math> </inline-formula>-Symmetric Nonlinear Dirac Equation

“…The study of open systems bearing gain and loss (especially so in a balanced form) is a topic that has emerged over the past two decades as a significant theme of study [1]- [3]. While the realm of PT -symmetry introduced by Bender and collaborators was originally intended as an alternative to the standard Hermitian quantum mechanics, its most canonical realizations (beyond the considerable mathematical analysis of the theme in its own right at the level of operators and spectral theory in mathematical physics) emerged elsewhere in physics.…”