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hierarchicalequationsofmotion

hierarchical equations of motion

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HIERARCHICAL EQUATIONS OF MOTION - The Hierarchical equations of motion (HEOM) technique derived by Yoshitaka Tanimura and Ryogo Kubo in 1989, and generalized to arbitrary temperature ...

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Hierarchical equations of motion might refer to
The Hierarchical equations of motion (HEOM) technique derived by Yoshitaka Tanimura and Ryogo Kubo in 1989, and generalized to arbitrary temperature and arbitrary spectral densities by Nike Dattani in 2012, is a non-perturbative approach developed to study the evolution of a density matrix * * * * ρ * ( * t * ) * * * {\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 * * * * * * ∂ * * ∂ * t * * * * * * * * ρ * ^ * * * * * n * * * = * − * i * ( * * * * * H * ^ * * * * * A * * * + * n * γ * ) * * * * * ρ * ^ * * * * * n * * * − * * * 1 * ℏ * * * * * * * V * ^ * * * * * × * * * * * * * ρ * ^ * * * * * n * + * 1 * * * + * * * * i * n * * ℏ * * * * * * Θ * ^ * * * * * * * * ρ * ^ * * * * * n * − * 1 * * * * * {\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}}

Hierarchical equations of motion might refer to

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* {\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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