Quantum gravity simulation by nonparaxial nonlinear optics

We show that an analog of the physics at the Planck scale can be found in the propagation of tightly focused laser beams. Various equations that occur in generalized quantum mechanics are formally identical to those describing the nonlinear nonlocal propagation of nonparaxial laser beams. The analysis includes a generalized uncertainty principle and shows that the nonlinear focusing of a light beam with dimensions comparable to the wavelength corresponds to the spontaneous excitation of the so-called maximally localized states. The approach, driven by the ideas of the quantum gravity physics, allows one to predict the existence of self-trapped subwavelength solitary waves for both focusing and defocusing nonlinearities, and opens the way to laboratory simulations of phenomena that have been considered to be inaccessible.

Figure 1

(a) Profiles of the ground state $\Phi \left(y\right)$ vs the dimensionless generalized momentum $y$ for three values of the nonlinearity $\varepsilon$; (b) energy $\epsilon$ vs $\varepsilon$.

Figure 2

(a) SW profiles scaled to unitary norm for $\chi =1$, calculated in the cardinal basis after Eq. (19) and the numerical solution of Eq. (24), for $\mathcal{N}/\lambda =0.075$ (circles), $\mathcal{N}/\lambda =0.134$ (triangles), $\mathcal{N}/\lambda =3.00$ (squares); (b) $\epsilon$ and $\Delta X$ when increasing the strength of nonlinearity; (c) as in (a) for $\chi =-1$ and $\mathcal{N}/\lambda =0.85$ (circles), $\mathcal{N}/\lambda =1$ (triangles), $\mathcal{N}/\lambda =3$ (squares); (d) as in (b) for $\chi =-1$.