Comparison between length and velocity gauges in quantum simulations of high-order harmonic generation

We solve the time-dependent Schrodinger equation for atomic hydrogen in an intense field using spherical coordinates with a radial grid and a spherical harmonic basis for the angular part. We present the high-order harmonic spectra based on three different forms, the dipole, dipole velocity, and acceleration forms, and two gauges, the length and velocity gauges. The relationships among the harmonic phases obtained from the Fourier transform of the three forms are discussed in detail. Although quantum mechanics is gauge invariant and the length and velocity gauges should give identical results, the two gauges present different computation efficiencies, which reflects the different behavior in terms of characteristics of the physical couplings acting in the two gauges. In order to obtain convergence, more angular momentum states are required in the length gauge, while more grid points are required in the velocity gauge. At lower laser intensity, the calculation in the length gauge is faster than that in the velocity gauge, while at high laser intensity, the calculation in the velocity gauge is more efficient. The velocity gauge is also expected to be more efficient in higher-dimensional calculations.