Revisiting the Optical PT-Symmetric Dimer

Abstract

Abstract:Optics has proved a fertile ground for the experimental simulation of quantum mechanics. Most recently, optical realizations of P T -symmetric quantum mechanics have been shown, both theoretically and experimentally, opening the door to international efforts aiming at the design of practical optical devices exploiting this symmetry. Here, we focus on the optical P T -symmetric dimer, a two-waveguide coupler where the materials show symmetric effective gain and loss, and provide a review of the linear and nonlinear optical realizations from a symmetry-based point of view. We go beyond a simple review of the literature and show that the dimer is just the smallest of a class of planar N-waveguide couplers that are the optical realization of the Lorentz group in 2 + 1 dimensions. Furthermore, we provide a formulation to describe light propagation through waveguide couplers described by non-Hermitian mode coupling matrices based on a non-Hermitian generalization of the Ehrenfest theorem.

Photon Propagation through Linearly Active Dimers

“…Note that the spontaneous generation is asymptotically equal in both waveguides when the dimer is at the Kato exceptional point for any configuration with linear gain, Figure 2b,e,h. In addition, in the broken symmetry regime, the asymptotic value for the renormalized spontaneous generation is the same for the different configurations including linear gain, Figure 2c,f,i, and converges to the same value than the classical P T -symmetric dimer [1,18], Figure 2. Instantaneously renormalized spontaneous generation,ñ (00) j (ζ), along different realizations of the effective P T -symmetric dimer.…”

“…The coupling matrix exponential can be easily calculated following an approach similar to that used in the classical P T -symmetric dimer [1],…”

“…Ref. [1] and references therein from the initial description of linear loses in directional couplers with a more complex non-Hermitian symmetry [2], the quantum-like description of classical planar waveguides [3], to all the contemporaneous work derived from the seminal introduction of optical P T -symmetric structures [4]. It is well known that the propagation of electromagnetic field through a linearly active two-waveguide coupler can be described by the classical P T -symmetric dimer,…”

$${\mathscr{PT}}$$ -symmetry from Lindblad dynamics in a linearized optomechanical system

“…In such a simulation with classical light, the complex potentials in the PT -symmetric Hamiltonian of a quantum system translate into complex refractive media that represent localized amplification or absorption. These parity-time symmetric structures are described by a Schrödinger-like differential equation, where the renormalized paraxial propagation mimics quantum dynamics of a non-relativistic particle in the presence of complex optical potentials [2][3][4]. A key feature of the PT -symmetric Hamiltonian is that at small gain-loss strength, its spectrum remains purely real, its linearly independent eigenfunctions are no longer orthogonal, but continue to remain simultaneous eigenfunctions of the combined PT operator.…”

Nonclassical light at exceptional points of a quantum PT -symmetric two-mode system

“…Two considered optical modes, one damped and the other amplified, interact via the linear coupling (for scrutinizing the results, see Ref. [72]). A Kerr-type nonlinearity, typical for physical PT -symmetric systems and originating, e.g., in the gain saturation, is assumed in both modes [33,34].…”