The Wigner function has provided an equivalent and convenient way to render quantum mechanics in phase space. It allows one to express macroscopically measurable quantities, such as currents and heat fluxes, in statistical forms as usually does in classical statistical mechanics, thereby facilitating its applications in nanoelectronics, quantum optics and etc. Distinct from the Schrödinger equation, the most appealing feature of the Wigner equation, which governs the dynamics of the Wigner function, is that it shares many analogies to the classical mechanism and simply reduces to the classical counterpart when the reduced Planck constant vanishes. Despite the theoretical advantages, numerical resolutions for the Wigner equation is notoriously difficult and remains one of the most challenging problems in computational physics, mainly because of the high dimensionality and nonlocal pseudo-differential operator. On one hand, the commonly used finite difference methods fail to capture the highly oscillatory structure accurately. On the other hand, all existing stochastic algorithms, including the affinity-based Wigner Monte Carlo and signed particle Wigner Monte Carlo methods, have been confined to at most 4D phase space. Few results have been reported for higher dimensional simulations. My group has made substantial progress in both aspects.

(1) We completed the design and implementation of a highly accurate numerical scheme for the Wigner quantum dynamics in 4D phase space. Our algorithm combines an efficient conservative semi-Lagrangian scheme in the temporal-spatial space with an accurate spectral element method in the momentum space. This accurate Wigner solver has been successfully applied into the investigation of quantum tunneling in double well and quantum double slit interference. Moreover, the Wigner function for a one-dimensional Helium-like system was clearly shown for the first time.

(2) We explored the inherent relation between the Wigner equation and a stochastic branching random walk model. With an auxiliary function, we can cast the Wigner equation into a renewal-type integral equation and prove that its solution is equivalent to the first moment of a stochastic branching random walk. In order to realize an efficient, reliable and integrated particle-based scheme to capture complicated quantum features in phase space, we utilized the probabilistic interpretation of the Wigner equation, efficient Monte Carlo strategies and non-parameter density estimation techniques.

It should be noted that all proposed numerical schemes fully exploit the mathematical structure of the Wigner equation. Our target is an efficient simulator for analyzing some fundamental issues in many-body quantum mechanics, such as the nuclear quantum effect and dynamical correlation.

References:

[1] Z. Chen, Y. Xiong and S. Shao, Numerical methods for the Wigner equation with unbounded potential, arXiv:1712.03019, 2017.

[2] Y. Xiong and S. Shao, Efficient implementation and performance evaluation of the Wigner branching random walk, arXiv:1709.02121, 2017.

[3] S. Shao and Y. Xiong, A computable branching random walk for the many-body Wigner quantum dynamics, arXiv:1603.00159, 2016.

[4] Y. Xiong, Z. Chen, and S. Shao, An advective-spectral-mixed method for time-dependent many-body Wigner simulations, SIAM J. Sci. Comput. 38 (2016) B491-B520.

[5] S. Shao and J. M. Sellier, Comparison of deterministic and stochastic methods for time-dependent Wigner simulations, J. Comput. Phys. 300 (2015) 167-185.

[6] S. Shao, T. Lu, and W. Cai, Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Commun. Comput. Phys. 9 (2011) 711-739.