The hierarchical equations of motion (HEOM) technique derived by Yoshitaka Tanimura and Ryogo Kubo in 1989, is a non-perturbative approach developed to study the evolution of a density matrix
\rho(t)
The hierarchical equation of motion for a system in a harmonic Markovian bath is
\partial | |
\partialt |
{\hat{\rho}}n=-(
i | |
\hbar |
x | |
\hat{H} | |
A |
+n\gamma)\hat{\rho}n-{i\over\hbar}\hat{V} x \hat{\rho}n+1+{in\over\hbar}\hat{\Theta}\hat{\rho}n-1
HEOMs are developed to describe the time evolution of the density matrix
\rho(t)
\hat{H}=
+ | |
\hat{H} | |
A(\hat{a} |
,\hat{a}-)+V(\hat{a}+,\hat{a}-)\sumjcj\hat{x}j+\sumj[{ \hat{p}2\over{2}}+
1 | |
2 |
2 | |
\hat{x} | |
j |
]
Characterising the bath phonons by the spectral density
J(\omega)=\sum\nolimitsj
2 | |
c | |
j |
\delta(\omega-\omegaj)
By writing the density matrix in path integral notation and making use of Feynman–Vernon influence functional, all the bath coordinates in the interaction terms can be grouped into this influence functional which in some specific cases can be calculated in closed form. Assuming a high temperature heat bath with the Drude spectral distribution
J(\omega)=\hbarη\gamma2\omega/\pi(\gamma2+\omega2)
\partial | |
\partialt |
{\hat{\rho}}n=-(
i | |
\hbar |
x | |
\hat{H} | |
A |
+n\gamma)\hat{\rho}n-{i\over\hbar}\hat{V} x \hat{\rho}n+1+{in\over\hbar}\hat{\Theta}\hat{\rho}n-1
where
\Theta
\hat{\Theta}=-
η\gamma | |
\beta |
(\hat{V} x -i
\beta\hbar\gamma | |
2 |
\hat{V}\circ)
The second term in
\hat{\Theta}
\beta=1/kBT
\hat{A} x \hat{\rho}=\hat{A}\hat{\rho}-\hat{\rho}\hat{A}
\hat{A}\circ\hat{\rho}=\hat{A}\hat{\rho}+\hat{\rho}\hat{A}
As with the Kubo's stochastic Liouville equation in hierarchal form, the counter
n
N
N
\hat{\rho}n+1
\hat{\rho}N+1=-\hat{\Theta}\hat{\rho}N/\hbar\gamma
N
\partial | |
\partialt |
{\hat{\rho}}N=-(
i | |
\hbar |
x | |
\hat{H} | |
A |
+N\gamma)\hat{\rho}N-{i\over\gamma\hbar2}\hat{V} x \hat{\Theta}\hat{\rho}N+{iN\over\hbar}\hat{\Theta}\hat{\rho}N-1
The statistical nature of the HEOM approach allows information about the bath noise and system response to be encoded into the equation of motion doctoring the infinite energy problem of Kubo's SLE by introducing the relaxation operator ensuring a return to equilibrium.
When the open quantum system is represented by
M
M
K
l{N}
\left(MK+l{N | |
\right)!}{\left(MK\right)!l{N}!} |
matrices, each with
M2
| ||||
16M |
bytes (assuming double-precision).
The HEOM method is implemented in a number of freely available codes. A number of these are at the website of Yoshitaka Tanimura[1] including a version for GPUs which used improvements introduced by David Wilkins and Nike Dattani.[2] The nanoHUB version provides a very flexible implementation.[3] An open source parallel CPU implementation is available from the Schulten group.[4]