Hierarchical theory of quantum dissipation: Partial fraction decomposition scheme

We implement the partial fraction decomposition scheme proposed by Croy and Saalmann [A. Croy, U. Saalmann, Phys. Rev. B 80 (2009) 073102] to the construction of the hierarchical equations of motion for bosonic quantum dissipation systems [R.X. Xu, Y.J. Yan, Phys. Rev. E 75 (2007) 031107]. The expansion of Bose-Einstein function in this scheme shows similar properties as it applies for Fermi function. The performance of the resulting quantum dissipation theory is exemplified with spin-boson systems. The new theoretical construction performs much better (about an order of magnitude faster) than the best available conventional formalism based on Matsubara spectral decomposition scheme.