A high order efficient numerical method for 4-D Wigner equation of quantum double-slit interferences

Abstract

We propose a high order numerical method for computing time dependent 4-D Wigner equation with unbounded potentials and study a canonical quantum double-slit interference problem. To address the difficulties of 4-D phase space computations and higher derivatives from the Moyal expansion of the nonlocal pseudo-differential operator for unbounded potentials, an operator splitting technique is adopted to decompose the 4-D Wigner equation into two sub-equations, which can be computed either analytically or numerically with high efficiency. The first sub-equation contains only a linear convection term in (x,t)-space and can be solved with an upwinding characteristic method, while the second involves the pseudo-differential term and can be approximated by a plane wave expansion in k-space. By exploiting properties of Fourier transformations, the expansion coefficients for the second sub-equation have explicit forms and the resulting scheme is shown to be unconditionally stable for any high order derivatives in the Moyal expansion, ensuring the feasibility of 4-D Wigner numerical simulations for quantum double-slit interferences. Numerical experiments demonstrate the spectral convergence in (x,k)-space and provide highly accurate information on the number, position, and intensity of the interference fringes for different types of slits, quantum particle masses, and initial states (pure and mixed).