Correlation functions

The bath correlation functions $C_{ab}(t)$ determines how the environment fluctuations affect the system through the couplings $F_a$. The bath correlation functions are given as

$$C_{ab}(t) = \left< u_a(t)u_b(0)\right>_B = {\rm tr} _B\left\... ...rm tr} _B\left\{\exp\left(-\beta H_B\right)\right\}} \right\}.$$ (10)

Here $u_a(t)$ evolve according to the interaction representation with respect to $H_{SB}$ and $\beta=1/T$ is the inverse temperature. The correlation function tells us how a perturbation of the environment caused by the coupling $F_a$ affects the system at a later time through the coupling $F_b$. The correlation functions are usually specified through the Fourier-Laplace transform of the spectral density $J_a(\omega)$ as

$$C_{ab}(t) = \frac{1}{\pi} \int_0^{\infty} d\omega J_{ab}\left(\omega\right) \frac{e^{-i \omega t}}{1-e^{-\beta\hbar\omega}},$$ (11)

where,

$$J_{ab}(\omega) = \sum_\xi \frac{\pi}{2} \sum_{\xi} \frac{c_{a\xi} c_{b\xi}}{m_\xi \omega_\xi} \delta(\omega-\omega_\xi).$$ (12)

The HEOM arise by assuming a form of bath correlation functions given by

$$C_{ab}(t) = \sum_{k=0}^K c_{abk} e^{- \left(\nu_{abk}+i\Omega_{abk}\right) t}.$$ (13)

The huge computational expense of such arbitrary correlation functions (especially with large M and K) restricts us to forms with $C_{ab}(t) = \delta_{ab} C_a(t)$ and $\Omega_{abk}=0$. This gives the bath correlation functions as

$$C_a(t) = \sum_{k=0}^\infty c_{ak} e^{- \nu_{ak}t },$$ (14)

which corresponds to spectral densities of the Drude form given by

$$J_a(\omega) = 2 \lambda_a\frac{ \omega\gamma_a } {\omega^2 + \gamma_a^2}.$$ (15)

The Drude spectral density, and employing a Matsubara expansion of $1/({1-\exp({-\beta\hbar\omega}}))$ is Eq. 11, results in correlation function coefficients

$$c_{a0} = {\gamma_a \lambda_a}\left[ \cot (\beta \hbar \gamma_a/2) - i\right]$$ (16)

$$c_{ak\ge1} = \frac{4 \lambda_a \gamma_a}{\beta \hbar} \frac{\nu_{ak}}{\nu_{ak}^2-\gamma_a^2}.$$ (17)

and damping constants $\nu_{a0} = \gamma_a$, $\nu_{ak\ge1} = 2\pi k/\beta\hbar$. The infinite number of Matsubara terms $\nu_{ak}$ in Eq. 14 is truncated to a finite $K$ where for all $k>K$, $\nu_{ak}\exp(-\nu_{ak}t)\approx \delta(t)$.

The parameters needed to model the system are summarized in Table 1.

Table 1: Parameters defining model of quantum system

Parameter	Meaning	Units
$N$	Number of states ${\left\vert i \right>}$
$E_i$	Energy level of state ${\left\vert i \right>}$	[Energy]
$V_{ij}$	Interaction between states ${\left\vert i \right>}$ and ${\left\vert j \right>}$	[Energy]
$M$	Number of system-environment coupling terms
$f_{ai}$	Elements of $F_a$ specifying system-environment coupling
$\lambda_a$	Strength of coupling $F_a$	[Energy]
$\gamma_a$	Response frequency of environment from $F_a$ coupling	[time]$^{-1}$
$K$	Number of Matsubara terms to include
$L_T$	Hierarchy truncation level