Exact solution for the interaction of two decaying quantized fields.

We show that the Markovian dynamics of two coupled harmonic oscillators may be analyzed using a Schrödinger equation and an effective non-Hermitian Hamiltonian. This may be achieved by a non-unitary transformation that involves superoperators; such transformation enables the removal of quantum jump superoperators, which allows us to rewrite the Lindblad master equation in terms of a von Neumann-like equation with an effective non-Hermitian Hamiltonian. This may be generalized to an arbitrary number of interacting fields. Finally, by applying an extra non-unitary transformation, we may diagonalize the effective non-Hermitian Hamiltonian to obtain the evolution of any input state in a fully quantum domain.

Kapitza-Dirac photonic lattices.

We show that the Kapitza-Dirac effect may be modeled by classical light propagation in photonic lattices having a square power law for the refraction index coefficient. The dynamics is shown to be fully soluble because both systems share the same time-independent Schrödinger equation: the angular Mathieu equation. We examine the trajectories of classical light propagating in such structures.

Mathematical and diffractive modeling of self-healing.

To analyze the self-healing of a partially obstructed optical beam, we represent it by two orthogonal field components. The first component is an exact copy of the unobstructed beam, attenuated by a factor that is computed by a simple formula. The second component represents a pure distortion field, due to its orthogonality respect to the first one. This approach provides a natural measure of the beam damage, due to the obstruction, and the degree of self-healing, during propagation of the obstructed beam. As interesting results, derived in our approach, we obtain that the self-healing reaches a limit degree at the far field propagation domain, and that certain relatively small phase obstructions may produce a total damage on the beam. The theory is illustrated considering a Gaussian beam, distorted by different amplitude and phase obstructions. In the case of a soft Gaussian obstruction we obtain simple formulas for the far field limit values of the beam damage and the self-healing degree.

Generalized revival and splitting of an arbitrary optical field in GRIN media.

Assuming a non-paraxial propagation operator, we study the propagation of an electromagnetic field with an arbitrary initial condition in a quadratic GRIN medium. We show analytically that at certain specific periodic distances, the propagated field is given by the fractional Fourier transform of a superposition of the initial field and of a reflected version of it. We also prove that for particular wavelengths, there is a revival and a splitting of the initial field. We apply this results, first to an initial field given by a Bessel function and show that it splits into two generalized Bessel functions, and second, to an Airy function. In both cases our results are compared with the numerical ones.

Revival and splitting of a Gaussian beam in gradient index media.

The short range revival of an arbitrary monochromatic optical field, which propagates in a quadratic GRIN rod, is a well-known effect that is established assuming the first-order approximation of the propagation operator. We discuss the revival and multiple splitting of an off-axis Gaussian beam propagating to relatively long distances in a quadratic GRIN medium. These effects are obtained assuming the second-order approximation of the propagation operator in this medium.

Optical realization of quantum Kerr medium dynamics.

We use the propagation of a conveniently shaped Gaussian beam in a GRIN media to mimic a quantum cavity filled with a Kerr medium. This is attained by introducing a second-order correction to the paraxial propagation of the beam. An additional result is that a Gaussian beam propagating in GRIN media may split into two Gaussian beams, corresponding to the generation of superposition of coherent states (Schrödinger cat states) in the cavity filled with the Kerr medium.

Ermakov-Lewis symmetry in photonic lattices.

We present a class of waveguide arrays that is the classical analog of a quantum harmonic oscillator, where the mass and frequency depend on the propagation distance. In these photonic lattices, refractive indices and second-neighbor couplings define the mass and frequency of the analog quantum oscillator, while first-neighbor couplings are a free parameter to adjust the model. The quantum model conserves the Ermakov-Lewis invariant, thus the photonic crystal also possesses this symmetry.

Jacobi photonic lattices and their SUSY partners.

We present a classical analog of quantum optical deformed oscillators in arrays of waveguides. The normal modes of these one-dimensional photonic crystals are given in terms of Jacobi polynomials. We show that it is possible to attack the problem via factorization by exploiting the corresponding quantum optical model. This allows us to provide an unbroken supersymmetric partner of the proposed Jacobi lattices. Thanks to the underlying SU(1, 1) group symmetry of the lattices, we present the analytic propagators and impulse functions for these one-dimensional photonic crystals.

A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices: reply to comment.

We regret that such a misleading comment [Opt. Express (2013)] has been made to our paper. First Lo states in his abstract that "However, the nonlinear Rabi model has already been rigorously proven to be undefined" to later recoil and use the contradictory statement "(. . . ) regarding the BS model with the counter-rotating terms (. . . ) Lo and his co-authors have proven that the model is well defined only if the coupling stregth g is smaller than a critical value gc = ω/4". While Lo focuses on the validity of the quantum optics Hamiltonians and gives a misleading assesment of our manuscript, the focus of our paper is the method to map such a set of Hamiltonians from quantum optics to photonic lattices. Our method is valid for the given class of Hamiltonians and, indeed, precaution must be exerted on the paramater ranges where those Hamiltonians are valid and where their classical simulation is feasible. These parameter ranges have to be specified in for each particular case studied. Furthermore, we gave as example the Buck-Sukumar model including counter-rotating terms which is a valid Hamiltonian for some coupling parameters.

A classical simulation of nonlinear Jaynes-Cummings and Rabi models in photonic lattices.

The interaction of a two-level atom with a single-mode quantized field is one of the simplest models in quantum optics. Under the rotating wave approximation, it is known as the Jaynes-Cummings model and without it as the Rabi model. Real-world realizations of the Jaynes-Cummings model include cavity, ion trap and circuit quantum electrodynamics. The Rabi model can be realized in circuit quantum electrodynamics. As soon as nonlinear couplings are introduced, feasible experimental realizations in quantum systems are drastically reduced. We propose a set of two photonic lattices that classically simulates the interaction of a single two-level system with a quantized field under field nonlinearities and nonlinear couplings as long as the quantum optics model conserves parity. We describe how to reconstruct the mean value of quantum optics measurements, such as photon number and atomic energy excitation, from the intensity and from the field, such as von Neumann entropy and fidelity, at the output of the photonic lattices. We discuss how typical initial states involving coherent or displaced Fock fields can be engineered from recently discussed Glauber-Fock lattices. As an example, the Buck-Sukumar model, where the coupling depends on the intensity of the field, is classically simulated for separable and entangled initial states.

Optical realization of quantum-mechanical invariants.

We study the problem of paraxial propagation in two grade-index media by using invariant techniques that allow a continuous solution of the problem. By using the well-known fact that this problem is analogous to the time-dependent harmonic oscillator in quantum mechanics, known methods there may be imported producing on the one hand a solution to the propagation problem and on the other hand a realization of a quantum-mechanical invariant.

Optical realization of a quantum beam splitter.

We show how the quantum process of splitting light may be modeled in classical optics. A second result is the possibility to engineer specific forms of a classical field.

Unifying distribution functions: some lesser known distributions.

We show that there is a way to unify distribution functions that describe simultaneously a classical signal in space and (spatial) frequency and position and momentum for a quantum system. Probably the most well known of them is the Wigner distribution function. We show how to unify functions of the Cohen class, Rihaczek's complex energy function, and Husimi and Glauber-Sudarshan distribution functions. We do this by showing how they may be obtained from ordered forms of creation and annihilation operators and by obtaining them in terms of expectation values in different eigenbases.