Optical simulation of Majorana physics

We show a procedure to classically simulate the Majorana equation in $1+1$ dimensions via two one-dimensional photonic crystals. We use a decomposition of the Majorana equation into two Dirac equations and propose an approach that uses a bichromatic refractive index distribution and nearest-neighbor couplings of the type found in Glauber-Fock lattices. This allows us to escape the restriction of staying near the Brillouin zone imposed by the classical simulation of Dirac dynamics with bichromatic lattices. Furthermore, it is possible to simulate the evolution of Gaussian wave packets under the Majorana-Dirac equation with light impinging only into the first waveguide of our bichromatic-Glauber-Fock lattice.

Figure 1

Time evolution of the modulus squared of an initial Gaussian wave packet, ${\Psi }_0\left(q\right)=e^{-q^2/2}/{\pi }^{1/4}$, simulated by the propagation of light impinging the first waveguide of an array of photonic waveguides described by the matrix $H_{+,+}$ or $H_{-,-}$ with the following effective masses: (a) $m=0.1/\sqrt{2}$, (b) $m=1/\sqrt{2}$, and (c) $m=10/\sqrt{2}$.

Figure 2

Evolution of the center of mass for an initial Gaussian wave packet, ${\Psi }_0\left(q\right)=e^{-q^2/2}/{\pi }^{1/4}$, under Dirac dynamics recovered from the propagation of an initial beam impinging the first waveguide in an array of photonic waveguides described by the matrix $H_{+,+}$ or $H_{-,-}$ related to effective-mass parameters described in Fig. 1: (a) $m=0.1/\sqrt{2}$, (b) $m=1/\sqrt{2}$, and (c) $m=10/\sqrt{2}$.