Research Seminar in Mathematics - Phase-space representation for fermionic Quantum Dynamics

Speaker

François Rousse, Örebro University.

Abstract

Phase-space representations for bosonic Quantum Dynamics were introduced in Quantum Optics (i.e. for photons) in the 60’s. Some more practical computational maturity was achieved first in the 80-90’s through the development of the positive-P representation and the Truncated Wigner Approximation (TWA), and through new numerical implementation of the corresponding stochastic equations.  Then experimental progress in the field of Bose-Einstein condensates motivated applications of phase-space representations for (massive) bosonic particles, such as cold dilute atoms around 2000.

The more general understanding of matter also requires investigations of fermionic particles and of electronic structures, i.e. there is a need to develop simulation methods for fermions.  Phase-space methods maps the Hamiltonians to multidimensional Partial Differential Equations (PDE) for a quasi-probability density representing the Quantum Dynamics in an overcomplete basis. The PDEs are then mapped to stochastic differential equations (SDE) for stochastic sampling of physical observables. This last step is a standard procedure but allows for different types of so-called gauge degrees of freedoms, that can be explored to increase the practical numerical performance of the methods.  

We have explored a new phase-space representation for fermions, the fermionic Truncated Wigner Approximation (fTWA), first presented in 2017. fTWA is an approximative method which we have shown have major advantages over the standard mean-field method, in capturing higher order correlations of the quantum states. The first fTWA method have been explored on a large hexagonal 2D lattice, resembling the electronic structure of graphene. While the latter exact numerical-matrix-equation-based simulation have only been carried out on minimalistic 1D systems so far. 

In addition we have worked on extending the useful simulation time for another exact phase-space representation for fermions called the Gaussian phase-space representation (GPSR). We have done this by building in the mentioned gauge freedom into numerical matrices, obeying a constrained matrix equation.

Welcome!