Zitterbewegung in a Kerr medium

The evolution of a quantum particle propagating in a nonlinear medium and under the action of electric and magnetic fields is studied. For this purpose, the integrable unification of the Dirac and Manakov equations is derived. The integrability conditions of the system and the corresponding representation of the zero curvature of the system are found. We have applied an approach based on the Riemann-Hilbert problem to derive soliton solutions of the unified model. The study of the solution shows that the action of the electric and magnetic fields leads to the Zitterbewegung in the nonlinear interaction mode. It was also found that the collision of two solitons in the presence of an electric field can lead to the rectification of the particle trajectory and the disappearance of its Zitterbewegung.

Figure 1

Image of the average position of a soliton calculated by the formula (57) for $h=H=0$ in dimensionless units. The black dots, red triangles, and blue squares correspond to the values $g=0.05,G=-4,g=0.5,G=-4$, and $g=0.5,G=-1$, respectively. Soliton parameters are ${\eta }_1=1,{\xi }_1=0.4,{\alpha }_1=1,\text{and}{\beta }_1=1.$

Figure 2

Rectification of the soliton form due to collision for $h=H=0,g=0.25,\text{and}G=-1$. The dependence of the absolute values of the polarization component $\left|p\right|$ on the dimensionless variables $x$ and $t$ is shown. Soliton parameters are ${\eta }_1=1,{\eta }_2=1,{\xi }_1=-0.4,{\xi }_2=0.4,{\alpha }_1=0,{\alpha }_2=-2,{\beta }_1=1,\text{and}{\beta }_2=1$.