Hierarchical equations of motion

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 $$ \rho(t)$$ 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


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

Hierarchical equations of motion
HEOMs are developed to describe the time evolution of the density matrix $$ \rho(t)$$ for an open quantum system. It is a non-perturbative, non-Markovian approach to propagating in time a quantum state. Motivated by the path integral formalism presented by Feynman and Vernon, Tanimura derive the HEOM from a combination of statistical and quantum dynamical techniques. Using a two level spin-boson system Hamiltonian


 * $$ \hat{H} = \hat{H}_A(\hat{a}^{+},\hat{a}^{-}) + V(\hat{a}^{+},\hat{a}^{-})\sum_{j}c_j\hat{x}_j + \sum_{j}\big[ {\ \hat{p}^2\over{2}} + \frac{1}{2}\hat{x}_{j}^{2}  \big] $$

Characterising the bath phonons by the spectral density $$ J(\omega) = \sum\nolimits_j c_j^{2}\delta(\omega - \omega_j)$$

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 $$ J(\omega) = \hbar\eta\gamma^2\omega/\pi(\gamma^2 + \omega^2) $$ and taking the time derivative of the path integral form density matrix the equation and writing it in hierarchal form yields


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

where $$ \Theta $$ destroys system excitation and hence can be referred to as the relaxation operator.


 * $$ \hat{\Theta} = -\frac{\eta\gamma}{\beta} \big( \hat{V}^{\times} - i \frac{\beta\hbar\gamma}{2} \hat{V}^{\circ }\big) $$

The second term in $$\hat{\Theta} $$ is the temperature correction term with the inverse temperature $$ \beta = 1/k_B T$$ and the "Hyper-operator" notation is introduced.


 * $$ \hat{A}^{\times} \hat{\rho} = \hat{A}\hat{\rho} - \hat{\rho} \hat{A}$$


 * $$ \hat{A}^{\circ} \hat{\rho} = \hat{A}\hat{\rho} + \hat{\rho} \hat{A}$$

As with the Kubo's stochastic Liouville equation in hierarchal form, the counter $$ n $$ can go up to infinity which is a problem numerically, however Tanimura and Kubo provide a method by which the infinite hierarchy can be truncated to a finite set of $$ N $$ differential equations where $$ N $$ is determined by some constraint sensitive to the characteristics of the system i.e. frequency, amplitude of fluctuations, bath coupling etc. The "Terminator" defines the depth of the hierarchy. A simple relation to eliminate the $$ \hat{\rho}_{n+1}$$ term is found. $$\ \hat{\rho}_{N+1} = - \hat{\Theta} \hat{\rho}_N/ \hbar\gamma$$. With this terminator the hierarchy is closed at the depth $$ N $$ of the hierarchy by the final term:


 * $$ \frac{\partial}{\partial t}{\hat{\rho}}_N = -(\frac{i}{\hbar}\hat{H}^{\times}_A + N\gamma) \hat{\rho}_N - {i\over \gamma\hbar^2}\hat{V}^{\times}\hat{\Theta}\hat{\rho}_{N} + {iN\over\hbar}\hat{\Theta}\hat{\rho}_{N-1}$$.

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.

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Arbitrary spectral density and low temperature correction
It was pointed out by Dattani et al. in 2012 that the HEOM method can be employed as long as the bath correlation function is written as a sum of exponentials, and therefore an arbitrarily complicated spectral distribution function $$J(\omega)$$ can be fitted to any of the forms listed in Table 1 of whose bath response function can analytically be written as a sum of exponentials, and then the HEOM can be applied for that spectral density at arbitrary temperature. In a subsequent paper, it was suggested that the bath response function be fitted directly to a sum of exponentials rather than fitting the spectral density to one of the forms in Table 1 of and then calculating the bath response function as a sum of exponentials analytically.

In the Drude case, by modifying the correlation function that describes the noise correlation function strongly non-Markovian and non-perturbative system-bath interactions can be dealt with. The equations of motion in this case can be written in the form

$$

\begin{align} \frac{\partial}{\partial t}{\hat{\rho}}_{n, j_1,..,j_K} = &- \bigg[ {i\over \hbar}\hat{H}_A^{\times} + n\gamma + \sum_{k=1}^{K}(j_k\nu_k - {1\over{\nu_k\hbar^2}}\hat{V}^{\times}\hat{\Theta}_k) + \hat{\Gamma}_0 \bigg]{\hat{\rho}}_{n, j_1,..,j_K} - {i\over\hbar}\hat{V}^{\times}\bigg[ \hat{\rho}_{(n+1), j_1,..,j_K} + \sum_{k=1}^{K}\hat{\rho}_{n, j_1,..,(j_k+1),..,j_K} \bigg] \\ &-{in\over\hbar}\hat{\Theta}_0\hat{\rho}_{(n-1), j_1..j_K} - \sum_{k=1}^{K}{ij_k\over\hbar}\hat{\Theta}_k\hat{\rho}_{n, j_1,..,(j_k-1),..,j_K} \end{align}

$$

In this equation, only $$ {\hat{\rho}}_{0, 0,...,0}$$ contains all order of system bath interactions with other elements $$ {\hat{\rho}}_{n, j_1...j_K}$$ being auxiliary terms, moving deeper into the hierarchy, the order of interactions decreases, which is contrary to usual perturbative treatments of such systems. $$ \hat{\Theta}_k = c_k\hat{V}^{\times} $$ where $$c_k $$ is a constant determined in the correlation function.

$$

\hat{\Gamma}_0 \equiv {\eta\over{\beta\hbar^2}}\big( 1 - {\beta\over\gamma n}c_0 \big)\hat{V}^{\times}\hat{V}^{\times}

$$

This $$ \hat{\Gamma}_0 $$ term arises from the Matsubara cut-off term introduced to the correlation function and thus holds information about the memory of the noise.

Below is the terminator for the HEOM

$$

\frac{\partial}{\partial t}{\hat{\rho}}_{n, j_1,...,j_K} \simeq - \big( {i\over\hbar}\hat{H}_A^{\times} -

\sum_{k=1}^{K}{1\over{\nu_k\hbar^2}}\hat{V}^{\times}\hat{\Theta}_k + \hat{\Gamma}_0 \big) {\hat{\rho}}_{n, j_1,...,j_K}

$$

Performing a Wigner transformation on this HEOM, the quantum Fokker-Planck equation with low temperature correction terms emerges.

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Computational cost
When the open quantum system is represented by $$M$$ levels and $$M$$ baths with each bath response function represented by $$K$$ exponentials, a hierarchy with $$\mathcal{N}$$ layers will contain:



\frac{\left(MK + \mathcal{N}\right)!}{\left(MK\right)!\mathcal{N}!} $$

matrices, each with $$M^2$$ complex-valued (containing both real and imaginary parts) elements. Therefore, the limiting factor in HEOM calculations is the amount of RAM required, since if one copy of each matrix is stored, the total RAM required would be:



16M^2\frac{\left(MK + \mathcal{N}\right)!}{\left(MK\right)!\mathcal{N}!} $$

bytes (assuming double-precision).

Implementations
The HEOM method is implemented in a number of freely available codes. A number of these are at the website of Yoshitaka Tanimura including a version for GPUs which used improvements introduced by David Wilkins and Nike Dattani. The nanoHUB version provides a very flexible implementation. An open source parallel CPU implementation is available from the Schulten group.