PT-symmetry and supersymmetry: interconnection of broken and unbroken phases

Abstract

The broken and unbroken phases of P T and supersymmetry in optical systems are explored for a complex refractive index profile in the form of a Scarf potential, under the framework of supersymmetric quantum mechanics. The transition from unbroken to the broken phases of P T -symmetry, with the merger of eigenfunctions near the exceptional point is found to arise from two distinct realizations of the potential, originating from the underlying supersymmetry. Interestingly, in P T -symmetric phase, spontaneous breaking of supersymmetry occurs in a parametric domain, possessing non-trivial shape invariances, under reparametrization to yield the corresponding energy spectra. One also observes a parametric bifurcation behaviour in this domain. Unlike the real Scraf potential, in P T -symmetric phase, a connection between complex isospecrtal superpotentials and modified Korteweg-de Vries equation occurs, only with certain restrictive parametric conditions. In the broken P T -symmetry phase, supersymmetry is found to be intact in the entire parameter domain yielding the complex energy spectra, with zero-width resonance occurring at integral values of a potential parameter.

PT-symmetric KdV Solutions and Their Algebraic Extension with Zero-Width Resonances

“…Given a potential solves the KdV system, the corresponding superpotential satisfies the modified KdV (mKdV) equations 33,45 as the respective solutions to these two equations are related through the Miura transformation: u = v 2 ± v x 46, 47 . Moreover, since there are two distinct mKdV equations with solutions connected as v → iv, there is another class of KdV solutions with functional form: u = −v 2 ± iv x 46, 48 .…”

$$\mathscr{P}\mathscr{T}$$-symmetric KdV solutions and their algebraic extension with zero-width resonances

“…Given a Schrödinger potential that is a solution to the KdV system, the corresponding superpotential satisfies the modified KdV (mKdV) equation 37 , 49 . This is because the respective solutions to these two equations are related through the Miura transformation: 50 , 51 .…”

Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential

“…In the literature, Scarf II potential has recently attracted much attention, in general not by itself, but for its complexification as a particular simple model to display analytically some properties of complex potentials (see for instance [4][5][6][7][8][9]).…”

“…In general, complex potentials (non-Hermitian Hamiltonians) emerges frequently in optical systems [17,18]. For example, the Scarf-II potential, or its complexifications, is used as the refractive index profile in [9] and the broken and unbroken PT and SUSY potentials in optical systems related with Scarf II potential were investigated in [8,9]. For complex potentials, PT symmetry may assure real eigenvalues, so it is an important important concept.…”