Semiconductor Bloch equations in Wannier gauge with well-behaved dephasing

The semiconductor Bloch equations (SBEs) with a dephasing operator for the microscopic polarizations are a well established approach to simulate high-harmonic spectra in solids. We discuss the impact of the dephasing operator on the stability of the numerical integration of the SBEs in the Wannier gauge. It is shown that the commonly used phenomenological approach to apply dephasing is ill-defined in the presence of band crossings and leads to artifacts in the carrier distribution. They are caused by rapid changes of the dephasing operator matrix elements in the Wannier gauge, which render the convergence of the simulation in the stationary basis infeasible. In the comoving basis, also called Houston basis, these rapid changes can be resolved, but only at the cost of a largely increased computation time. As a remedy, we propose a modification of the dephasing operator with reduced magnitude in energetically close subspaces. This approach removes the artifacts in the carrier distribution and significantly speeds up the calculations, while affecting the high-harmonic spectrum only marginally. To foster further development, we provide our parallelized source code.