Computational Quantum Mechanics in Phase Space
A/Prof. Si-Hong Shao
School of Mathematical Sciences, Peking University

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 2D phase space. Few results have been reported for higher dimensional simulations. My group has made substantial progress in both aspects. 
We presented a first attempt to obtain a deterministic method to solve the Wigner equation in 4-D phase space by exploiting its intriguing mathematical structure, and succeeded to detail the quantum dynamics of a Helium-like system and the quantum interference fringes in the double-slit experiment. Our algorithm combines an efficient conservative semi-Lagrangian scheme in the temporal-spatial space with an accurate spectral element method in the momentum space. On the other hand, we built the bridge between the Wigner equation and a stochastic particle method in a rigorous manner and proposed a SPA (Stationary Phase Approximation) + SPADE (Sequential-clustering Particle Annihilation via Discrepancy Estimation) strategy to overcome the sign problem where the curse of dimensionality which causes the unattainable exponential wall is translated into the NP-hard problems that may have approximate solutions. The performance of 6-D simulations demonstrates the accuracy and the efficiency of our stochastic particle methods.
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. 

[1] S. Shao, Y. Xiong, SPADE: Sequential-clustering particle annihilation via discrepancy estimation, arXiv:2005.05129, 2020.
[2] Y. Xiong, S. Shao, Overcoming the numerical sign problem in Wigner dynamics via particle annihilation, arXiv:2008.05161, 2020.
[3] S. Shao, Y. Xiong, Branching random walk solutions to the Wigner equation, SIAM Journal on Numerical Analysis 58 (2020) 2589.
[4] S. Shao, Y. Xiong, A branching random walk method for many-body Wigner quantum dynamics, Numerical Mathematics: Theory, Methods and Applications 12 (2019) 21.
[5] Z. Chen, S. Shao, W. Cai, A high order efficient numerical method for 4-D Wigner equation of quantum double-slit interferences, Journal of Computational Physics 396 (2019) 54. 
[6] Y. Xiong, S. Shao, The Wigner branching random walk: Efficient implementation and performance evaluation, Communications in Computational Physics 25 (2019) 871.
[7] Z. Chen, Y. Xiong, S. Shao, Numerical methods for the Wigner equation with unbounded potential, Journal of Scientific Computing 79 (2019) 345.
[8] Y. Xiong, Z. Chen, and S. Shao, An advective-spectral-mixed method for time-dependent many-body Wigner simulations, SIAM Journal on Scientific Computing 38 (2016) B491.
[9] S. Shao and J. M. Sellier, Comparison of deterministic and stochastic methods for time-dependent Wigner simulations, Journal of Computational Physics 300 (2015) 167.
[10] S. Shao, T. Lu, and W. Cai, Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Communications in Computational Physics 9 (2011) 711.

About the Speaker


2021-03-31 10:30 AM
Room: A203 Meeting Room
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