Hierarchy equations of motion

The exponential terms in Eq. 14 lead to a hierarchy of matrices $\rho_{\bf n}$(t), called auxiliary density matrices (ADMs), to take into account the non-Markovian evolution of $\rho(t)$. The ADMs are indexed by a vector ${\bf n}=\left(n_{10},\cdots,n_{1K},\cdots,n_{M0},\cdots,n_{MK}\right)$ and coupled through the index operators ${\bf n}_{ak}^{\pm} = \left(n_{10},\cdots,n_{ak}\pm 1,\cdots,n_{MK}\right)$. The hierarchy equations of motion are

$$\dot {\rho}_{\bf n} = -\frac i \hbar \left[ H_S, \rho_{\bf n}\right]-\sum_{a=1}^M\sum_{k=0}^K n_{ak}\nu_{ak} \rho_{\bf n}$$ (18)

$$-\sum_{a=1}^M\left(\frac{2\lambda_a}{\beta\hbar^2\gamma_a}-\sum_{... ...{c_{ak}}{\hbar\nu_{ak}}\right)\left[ F_a,\left[ F_a, \rho_{\bf n}\right]\right]$$ (19)

$$-i\sum_{a=1}^M\left[ F_a,\sum_{k=0}^K \rho_{{\bf n}_{ak}^+}\right... ...t(c_{ak} F_a \rho_{{\bf n}_{ak}^-} - \rho_{{\bf n}_{ak}^-} F_a c_{ak}^*\right).$$ (20)

More simply written as

$$\dot {\rho}_{\bf n} = \mathcal{L}_{eff}\;{\rho}_{\bf n} + \s... ...} + \sum_{a}\sum_{k}\mathcal{N}_{ak}\;{\rho}_{{\bf n}_{ak}^-},$$ (21)

where $\mathcal{L}_{eff},\mathcal{P}_{ak}$ and $\mathcal{N}_{ak}$ are Liouville space operators $\mathcal{X}$, such that $\mathcal{X} \bullet = X\bullet - \bullet X^\dag$ for the corresponding Hilbert space operator $X$. Any ADM with a negative in the index vector ${\bf n}$ is set to 0.

The ADMs need to be truncated to a finite number. Each ADM can be assign to a level $L = \sum_{a=1}^M\sum_{k=0}^K n_{ak}$. There are two ways to truncate the hierarchy of ADMs to some level $L_T$: time non-local (TNL) truncation and time local (TL) truncation. In case of TNL truncation, all ADMs with $L \ge L_T$ is set to zero. In case of TL truncation, all ADMs with $L=L_T-1$ the Markovian approximation is assumed, such that

$$\sum_{k=0}^K\hat \rho_{{\bf n}_{ak}^+} \approx -i \left( \ha... ...^K(t) \rho_{\bf n} - \rho_{\bf n} {\hat Q_a^K(t)}^\dag\right),$$ (22)

where

$$\hat Q_a^K(t) = \int_0^t \left(\sum_{k=0}^K c_{ak} \exp\left... ...ight)\hat F_a \exp\left(\frac{i} {\hbar}H_S\,\tau\right)d\tau.$$ (23)