Hierarchical equations of motion

The Hierarchical equations of motion (HEOM) technique derived by R. Kubo and Y. Tanimura 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

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

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