Causality and phase transitions inPT-symmetric optical systems

Abstract

We discuss phase transitions in - symmetric optical systems. We show that due to frequency dispersion of the dielectric permittivity, an optical system can have - symmetry at isolated frequency points only. An assumption of the existence of a - symmetric system in a continuous frequency interval violates the causality principle. Therefore, the ideal symmetrybreaking transition cannot be observed by simply varying the frequency.In the last decade, there has been rising interest in optics of artificial materials. Among such materials, the systems with balanced loss and gain regions have attracted particular attention [1][2][3]. The concept of these systems stems from the idea of the extension of quantum mechanics to non-Hermitian Hamiltonians possessing - symmetry [4,5].- symmetrical systems are invariant with respect to the simultaneous spatial inversion and time inversion In optics, - symmetry is usually studied in the frequency domain by considering solutions of the scalar Helmholtz equation for the z-component of the electric field E:

Instantaneous modulations in time-varying complex optical potentials

“…We should mention that our choice of the permittivity suggests that the causality principle is not fulfilled, i.e., the parameters of the permittivity do not follow conventional dispersion constraints, as represented by the Kramers-Kronig relations. Given the known inconsistencies with the Kramers-Kronig relations for  -symmetric systems [60][61][62][63] and other artificial metamaterials with complex-valued permittivity [64][65][66][67][68], analogous relations must be constructed for various types of  -symmetric time-varying dielectric permittivities, relaxing the strict assumptions made for causality.…”

PT-symmetric metasurfaces: wave manipulation and sensing using singular points

“…Due to the fact that the conductivity of graphene and the metasurface admittance are dispersive, the condition for PT symmetry cannot be fulfilled over a finite frequency interval, but only at a specific frequency point , 0 w as a result of causality and Kramers-Kronig relations [64]. Figure 7(c) presents the eigenvalues of S as a function of frequency for this PT system, showing that when the operating frequency shifts away from the EP, the degenerate eigenvalues may quickly split into complex ones, accompanied by dramatic changes in scattering properties ( figure 7(a)).…”

Broadband quasi-PT-symmetry sustained by inhomogeneous broadening of the spectral line

“…The fundamental importance of this last constraint, stems from the causality principle, usually expressed through the Kramers-Kronnig relations. Indeed, the latter can be satisfied only for isolated frequencies [5]. That is why, the major development of the PT -symmetric optics is achieved, so far, within the paraxial approximation proposed in [4], rather than with optical pulses, as initially suggested in [2].…”