Irreversible quantum mechanics in nonlocal nonlinear optics

We review one of our recent theoretical and experimental developments on the simulation of fundamental physical models by nonlocal nonlinear optics. Specifically, we consider dispersive shock waves which dominate wave-breaking phenomena in Hamiltonian systems. In the absence of loss, these highly irregular and disordered waves are potentially reversible. However, no experimental evidence has been given about the possibility of inverting their dynamics and turn them into a regular wave-front. Nevertheless, the opposite scenario, i.e., a smooth wave generating turbulent dynamics is well studied and observed in experiments. Here we introduce a new theoretical formulation for the dynamics in a highly nonlocal and defocusing medium described by the nonlinear Schroedinger equation, showing that a defocusing medium realizes the optical analog of a reversed harmonic oscillator in the highly nonlocal regime. It has been shown [A. R. Bohm, R. Scurek, and S. Wikramasekara, "Resonances, Gamow Vectors and Time Asymmetric Quantum Theory," (1999), arXiv:nucl-th/9902076v1 [nucl-th]; A. R. Bohm, J. Math. Phys. 22 (1981); D. Chruscinski, Open Sys. Information Dyn. 9, 207221 (2002), arXiv:math-ph/0206009v1 [math-ph]] that this theoretical model is the paradigm for irreveribility and can describe dissipative systems where no time-symmetry holds.Our theory is based on the concept of Gamow vectors, that are commonly adopted in the Rigged Hilbert Space formulation of irreversible quantum mechanics. This nonlinear vectors are a power dependent generalizations of the counter-intuitive and hereto elusive exponentially decaying states in Hamiltonian systems. We show that nonlinear Gamow vectors play a fundamental role in nonlinear Schroedinger models: they may be used as a generalized basis for describing the dynamics of the shock waves, and affect the degree of irreversibility of wave-breaking phenomena. Gamow vectors allow to analytically calculate the amount of breaking of time-reversal with a quantitative agreement with numerical solutions.We also report on the experimental observation of these states, characterized by a quantized decay rate, generated by an optical photothermal nonlinearity [S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, Phys. Rev. A 87, 053811 (2013)].