The study of nonlinear propagation of light at a quantum level may offer a complete a scenario to understand the action of nonlinearity at a low photon number regime can unveil unknown and totally new optical effects.
The first phenomenon we decided to investigate is the propagation of quantum optical solitons. In fact, though classical solitons (CSs) are very well studied, QSs have a much richer dynamics that may provide many new solutions for quantum computing and cryptography.
For this reason, we studied the fundamental properties of the soliton solution of the quantum nonlinear Schrödinger equation (QNLSE), first analytically and then numerically. We got the QNSE analytical solutions both by Bethe-ansatz and by quantum inverse scattering, whereas we obtained the numerical solutions by using phase-space methods, such as the Glauber- Sudarshan P-representation and the positive P-representation. This way we transformed operator equations in stochastic differential equations, solvable by a stochastic Runge-Kutta algorithm. The most important difference between the QSs and the CSs proved both by theory and by simulations, is given by the breaking of the shape-invariance during the propagation. Indeed, the numerical simulations show the quantum wave-packet spreading and the QS emission spectrum.
Controlling quantum nonlinear optical processes, as the generation of highly nonclassical states and broadband spectra, is a major challenge in optics. Specifically, controlling quantum solitons is a problem which is very broad in scope; one can control the number of generated solitons or their parameters as amplitude or speed. We focused our attention on the process of quantum spreading of solitons and on the increase of spectral emission, which can bring to supercontinuum generation. By combining stochastic partial differential equations and the so-called Chopped Random Basis Optimization (CRAB), we looked for the best control function to limit the quantum spreading. This result opens up a number of further possibilities for controlling quantum dynamics.