Hierarchical equations of motion

The hierarchical equations of motion (HEOM) technique derived by Yoshitaka Tanimura and Ryogo Kubo in 1989,^[1] is a non-perturbative approach developed to study the evolution of a density matrix ${\displaystyle \rho (t)}$ of quantum dissipative systems. The method can treat system-bath interaction non-perturbatively as well as non-Markovian noise correlation times without the hindrance of the typical assumptions that conventional Redfield (master) equations suffer from such as the Born, Markovian and rotating-wave approximations. HEOM is applicable even at low temperatures where quantum effects are not negligible.

The hierarchical equation of motion for a system in a harmonic Markovian bath is^[2]

${\displaystyle {\frac {\partial }{\partial t}}{\hat {\rho }}_{n}=-({\frac {i}{\hbar }}{\hat {H}}_{A}^{\times }+n\gamma ){\hat {\rho }}_{n}-{i \over \hbar }{\hat {V}}^{\times }{\hat {\rho }}_{n+1}+{in \over \hbar }{\hat {\Theta }}{\hat {\rho }}_{n-1}}$