New confining optical media generated by Darboux transformations

Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

“…Additional applications may include the propagation of electromagnetic signals in waveguides, where the Helmholtz equation is formally paired with the Schrodinger one [86][87][88], and the self-focusing is relevant [82][83][84][85]. Finally, the approach can be extended to study supersymmetric structures in quantum mechanics [89] with time-dependent potentials [16,17] Using (9) and (A-1), the Schrödinger equation of the stationary oscillator (3) becomes a partial differential equation of the desired form S def .…”

“…is well known in the literature and finds many application in physics [22-25, 30, 57, 64, 65, 82-85, 90, 91]. It arises quite naturally in the studies of parametric oscillators [22][23][24][25]30], in the description of structured light in varying media [82][83][84][85], and in the study of non-Hermitian Hamiltonians with real spectrum [57,64,65]. The key to solve (C-1) is to consider the homogeneous linear equation q + Ω 2 (t) q = 0, (C-2) which coincides with the equation of motion for a classical parametric oscillator.…”

Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

“…Besides, they have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics [63][64][65][66][67][68][69]. Recent applications include propagation of optical beams in parabolic media [171][172][173] and Kerr media [174][175][176][177] as well, studies of the geometry of the Riccati equation [178] and the fourth-order Schrödinger equation with the energy spectrum of the Pöschl-Teller system [179]. Further discussion on the subject can be found in the Schuch's book on a nonlinear perspective to quantum theory [180].…”

“…However, it may be also useful in the study of multilevel quantum systems [272]. Of course, the studies on the propagation of optical beams in parabolic [171][172][173] and Kerr [174][175][176][177] media include a refractive index with special properties that can be expressed as a concrete parametrization of the quantum states of light. On the other hand, quantum field theory in curved spacetime also permits classical analogies [260], so that Hawking radiation can be studied in nonlinear Kerr media (including the analysis of the vacuum state for a star collapsing to a black hole which leads to the controversial effect named after Unruh [273][274][275][276]).…”

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

“…Supersymmetric quantum mechanics (Susy-QM) is presently a very robust formulation embracing a very wide set of applications [5,13,14], including optics [15][16][17][18][19][20][21][22][23][24][25], where supersymmetry may be interpreted as describing two light beams of different colors that form a standing periodic interference pattern along the waveguide axis [15]. In the Helmholtz regime, the supersymmetric partners can be constructed to display parabolic [16] or sech-like index profile [17][18][19][20]. They also show identical coefficients of transmission and reflection for any angle of incidence [5,18,19,21,22], which may render them perfectly indistinguishable to an external observer [23].…”