Second-order imaginary differential operator for effective absorption in the numerical solution of the time-dependent Schrödinger equation

We propose a method to implement absorbing layers for the numerical solution of the time-dependent Schr\"odinger equation (TDSE) in the open domain. The method is based on introducing an imaginary second-order differential operator which effectively absorbs waves propagating both toward the boundary and away from it. The survival probability is 0.01 for de Broglie wavelength equal to the layer width and decreases exponentially as the wavelength decreases. The proposed method can be used in both length and velocity gauges for the electric field when solving strong-field physics problems. We also propose the propagator for the absorbing operator for convenient use in the Fourier split-step method of the TDSE solution. We demonstrate the high accuracy of using the proposed method in simulations of the high-order harmonics generation in atomic hydrogen by femtosecond laser pulses and calculations of photoelectron momentum distributions using projections of the absorbed wave packets onto Volkov states.