Noise-assisted quantum electron transfer in photosynthetic complexes

Abstract

Electron transfer (ET) between primary electron donor and acceptor is modeled in the photosynthetic complexes. Our model includes (i) two discrete energy levels associated with donor and acceptor, which are directly interacting and (ii) two continuum manifolds of electron energy levels (“sinks”), each interacting with the donor and acceptor. We also introduce external (classical) noise which acts on both donor and acceptor. We derive a closed system of integro-differential equations which describes the non-Markovian quantum dynamics of the ET. A region of parameters is found in which the ET dynamics can be simplified, and described by coupled ordinary differential equations. Using these simplified equations, both cases of sharp and flat redox potentials are analyzed. We analytically and numerically obtain the characteristic parameters that optimize the ET rates and efficiency in this system. In particular, we demonstrate that even for flat redox potential a simultaneous influence of sink and noise can significantly increase the efficiency of the ET. We discuss a relation between our approach and the Marcus theory of ET.

Appendices

Appendix A: Non-Hermitian effective Hamiltonian

We consider the time-dependent Hamiltonian of a $N$-level system coupled with independent sinks through each level:

$$\begin{aligned} {\fancyscript{H}}(t)&= \sum ^N_{n=1} \epsilon _n(t)|n\rangle \langle n| + \sum _{m \ne n} \beta _{nm}(t)|n\rangle \langle m| \nonumber \\&+ \sum ^{N}_{n=1}\sum ^{N_n}_{i_n=1} \left( E_{i_n}|i_n\rangle \langle i_n | +V_{ni_n}|n\rangle \langle i_n | + V_{i_n n}|i_n\rangle \langle n |\right) , \end{aligned}$$

(67)

where $m,n = 1,2, \ldots ,N$. We assume that the sinks are sufficiently dense, so that one can perform an integration instead of a summation. Then we have,

$$\begin{aligned} {\fancyscript{H}}(t)&= \sum _{n}\epsilon _n(t)|n\rangle \langle n| + \sum _{m \ne n} \beta _{mn}(t)|n\rangle \langle m| \nonumber \\ \nonumber \\&+\sum _{n} \left( \int \alpha _n(E)|n\rangle \langle E|g_n(E) dE + \mathrm h.c.\right) +\sum _{n}\int E|E\rangle \langle E|g_n(E)dE\nonumber \\ \end{aligned}$$

(68)

where $g_n(E)$ is the density of states, and $V_{ni_n}\rightarrow \alpha _n(E)$.

With the state vector written as

$$\begin{aligned} |\psi \rangle = \sum _{n}\left( c_n(t) |n\rangle + \int c_{nE}(t)|E\rangle g_n(E)dE \right) , \end{aligned}$$

(69)

the Schrödinger equation,

$$\begin{aligned} i\frac{\partial |\psi (t) \rangle }{\partial t} = {{\fancyscript{H}}} |\psi (t) \rangle , \end{aligned}$$

(70)

takes the form

$$\begin{aligned} i\dot{c}_n(t)&= E(t) c_n(t) + \sum _{m \ne n} \beta _{nm}(t) c_m(t) + \int \limits ^\infty _{0}\alpha _n^*(E) c_{nE}(t)g_n(E) dE \end{aligned}$$

(71)

$$\begin{aligned} i\dot{c}_{nE}(t)&= E c_E(t) + \alpha _n(E) c_n(t). \end{aligned}$$

(72)

In order to eliminate the continuum amplitudes from the equations for the discrete states, we first apply the Laplace transformation:

$$\begin{aligned} c_n(t)&= \int \limits _0^\infty e^{-st} c_n(s) ds, \end{aligned}$$

(73)

$$\begin{aligned} c_{nE}(t)&= \int \limits _0^\infty e^{-st} c_{nE}(s) ds. \end{aligned}$$

(74)

Then, from Eq. (72) we obtain

$$\begin{aligned} (s + iE)c_{nE}(s) = - i \alpha _n(E) c_{n}(s). \end{aligned}$$

(75)

This yields $c_E(s) = - i \alpha _n(E) c_2(s)/(s + iE)$. Inserting this expression for $c_E(s)$ into Eq. (71), we obtain the following system of integro-differential equations, describing the non-Markovian dynamics of the TLS,

$$\begin{aligned} i\dot{c}_n(t)= E c_n(t) + \sum \limits _{n \ne m} \beta _{nm}(t) c_m(t) -i \int \limits ^{\infty }_{0} c_{n}(s) e^{-st} ds\int \frac{|\alpha _n(E)|^2g_n(E) \, dE _n}{s + iE}. \nonumber \\ \end{aligned}$$

(76)

To proceed further, we change the variable in the last integral, $s \rightarrow -E^{\prime }$, so that

$$\begin{aligned} \int \frac{|\alpha _n(E)|^2g_n(E) \, dE }{s + iE} \rightarrow -i \int \frac{|\alpha _n(E)|^2g_n(E) \, dE _n}{E- E^{\prime }} \end{aligned}$$

(77)

The next step is to use the identity

$$\begin{aligned} \frac{1}{x-x^\prime + i0} = {\fancyscript{P}} \left\{ \frac{1}{x-x^\prime }\right\} - i\pi \delta (x-x^\prime ), \end{aligned}$$

(78)

where ${\fancyscript{P}}$ = Principal value. This yields

$$\begin{aligned} \int \frac{|\alpha _n(E)|^2g_n(E) \, dE }{E- E^{\prime }}= \varDelta (E^{\prime }) - \frac{i}{2}\varGamma _n(E^{\prime }), \end{aligned}$$

(79)

where

$$\begin{aligned} \varDelta (E^{\prime })&= {\fancyscript{P}} \int \frac{|\alpha _n(E)|^2g_n(E) \, dE }{E- E^{\prime }}, \end{aligned}$$

(80)

$$\begin{aligned} \varGamma _n(E^{\prime })&= 2\pi \int {|\alpha _n(E)|^2 }g_n(E){\delta (E- E^{\prime })}\, dE. \end{aligned}$$

(81)

Now using the Weisskopf–Wigner pole approximation, we evaluate the integrals as follows [25, 26, 47]:

$$\begin{aligned} \varDelta (E^{\prime }) \approx \varDelta (\epsilon _n)&= {\fancyscript{P}} \int \frac{|\alpha _n(E)|^2 g_n(E) dE }{E - \epsilon _n}, \end{aligned}$$

(82)

$$\begin{aligned} \varGamma _n(E^{\prime })\approx \varGamma _n(\epsilon _n)&= 2\pi \int {|\alpha _n(E)|^2 }g_n(E){\delta (E -\epsilon _n)}\, dE = 2\pi g_n(\epsilon _n)|\alpha _n(\epsilon _n)|^2. \nonumber \\ \end{aligned}$$

(83)

The Weisskopf–Wigner pole approximation basically corresponds to the assumption that the coupling constant to the continuum is a smoothly varying function of the energy, e.g. the continuum is treated as a single discrete level.

Inserting (82) into Eq. (71), we obtain

$$\begin{aligned} i\dot{c}_n(t)= \varepsilon _n(t) c_n(t) + \sum _{m \ne n} \beta _{nm}(t) c_m(t) - \frac{i\varGamma _n}{2} c_n(t), \end{aligned}$$

(84)

where $\varGamma _n = \varGamma _n(E_n)$ and $\varepsilon (t) = \epsilon _n(t) -\varDelta (E_n)$.

Writing $|\psi _N\rangle = \sum _{n}c_n(t) |n\rangle$, we find that the dynamics of the $N$-level system interacting with the continuum is described by the Schrödinger equation,

$$\begin{aligned} i\frac{\partial |\psi _N(t) \rangle }{\partial t} =\tilde{{\fancyscript{H}}} |\psi _N(t) \rangle , \end{aligned}$$

(85)

where $\tilde{{\fancyscript{H}}}= {{\fancyscript{H}}}- i {\fancyscript{W}}$ is the effective non-Hermitian Hamiltonian,

$$\begin{aligned} {{\fancyscript{H}}} = \sum _{n}\varepsilon _n|n\rangle \langle n| +\sum _{m \ne n} \beta _{mn}(t) |m\rangle \langle n| \end{aligned}$$

(86)

being the dressed Hamiltonian, and

$$\begin{aligned} W = \frac{1}{2}\sum _{n}\varGamma _n|n\rangle \langle n|. \end{aligned}$$

(87)

Equivalently, the dynamics of this system can be described by the Liouville equation,

$$\begin{aligned} i \dot{ \rho } =\tilde{{\fancyscript{H}}}\rho - \rho {\tilde{{\fancyscript{H}}}}^\dagger = [{\fancyscript{H}},\rho ] - i\{{\fancyscript{W}},\rho \}, \end{aligned}$$

(88)

where $\rho$ is the density matrix projected on the intrinsic states, and $\{{\fancyscript{W}},\rho \}= {\fancyscript{W}}\rho +\rho {\fancyscript{W}}$.

In particular case of the two-level system considered in this paper, the effective non-Hermitian Hamiltonian takes the form

$$\begin{aligned} \tilde{{\fancyscript{H}} } = \frac{1}{2} \left( \begin{array}{cc} 2\varepsilon _1- i\varGamma _1 &{} V \\ V&{} 2\varepsilon _2- i\varGamma _2 \end{array}\right) . \end{aligned}$$

(89)

Comments. The results of this section can be obtained using the standard Feshbach projection method [26, 32–35].

Appendix B: Equation of motion for the average density matrix

In this Appendix, we derive from the Liouville equation, $i\dot{ \rho } = [\tilde{{\fancyscript{H}}},\rho ] - i \{{\fancyscript{W}},\rho \}$, the equation of motion for the average density matrix. We will use the interaction representation. Considering the off-diagonal elements as perturbations, so that $\tilde{{\fancyscript{H}}}={{\fancyscript{H}}}_0 + V(t)- i{\fancyscript{W}}$, where

$$\begin{aligned} {{\fancyscript{H}}}_0&= \sum _{n} \varepsilon _n |n\rangle \langle n | + \sum _{n} \lambda _{nn} (t) |n\rangle \langle n |, \end{aligned}$$

(90)

$$\begin{aligned} V(t)&= \sum _{m \ne n} ( V_{mn} +\lambda _{mn}(t))|m\rangle \langle n |, \end{aligned}$$

(91)

$$\begin{aligned} {\fancyscript{W}}&= \frac{\varGamma _1}{2} |1\rangle \langle 1|+ \frac{\varGamma _2}{2}|2\rangle \langle 2|, \end{aligned}$$

(92)

we obtain the following equations of motion:

$$\begin{aligned} {\dot{\tilde{\rho }}}_{11}&= i({\tilde{\rho }}_{12}{\tilde{V}}_{21}- {\tilde{V}}_{12} {\tilde{\rho }}_{21})- \varGamma _1 {\tilde{\rho }}_{11}, \end{aligned}$$

(93)

$$\begin{aligned} {\dot{\tilde{\rho }}}_{22}&= i({\tilde{\rho }}_{21}{\tilde{V}}_{12}- {\tilde{V}}_{21} {\tilde{\rho }}_{12})-\varGamma _2 {\tilde{\rho }}_{22}, \end{aligned}$$

(94)

$$\begin{aligned} {\dot{\tilde{\rho }}}_{12}&= i{\tilde{V}}_{12}({\tilde{\rho }}_{11}- {\tilde{\rho }}_{22})- \varGamma {\tilde{\rho }}_{12}, \end{aligned}$$

(95)

$$\begin{aligned} {\dot{\tilde{\rho }}}_{21}&= i{\tilde{V}}_{21}({\tilde{\rho }}_{11}- {\tilde{\rho }}_{22}) - \varGamma {\tilde{\rho }}_{21}, \end{aligned}$$

(96)

where $\varGamma =(\varGamma _1 + \varGamma _2)/2$,

$$\begin{aligned} \tilde{\rho }= T\left( e^{i\int \limits _0^t H_0(\tau ) d \tau }\right) \rho T\left( e^{-i\int \limits _0^t H_0(\tau ) d\tau }\right) , \end{aligned}$$

(97)

and

$$\begin{aligned} \tilde{V}= T\left( e^{i\int \limits _0^t H_0(\tau ) d \tau }\right) V T\left( e^{-i\int \limits _0^t H_0(\tau ) d\tau }\right) . \end{aligned}$$

(98)

Using Eqs. (93)–(96), we obtain

$$\begin{aligned} {\tilde{\rho }}_{11}(t)&= { {\tilde{\rho }}}_{11}(0) + i\int \limits _0^t e^{-\varGamma _1 (t- t^{\prime })}({\tilde{\rho }}_{12}(t^{\prime }){\tilde{V}}_{21}(t^{\prime })- {\tilde{V}}_{12}(t^{\prime }) {\tilde{\rho }}_{21}(t^{\prime }))dt^{\prime }, \end{aligned}$$

(99)

$$\begin{aligned} {\tilde{\rho }}_{22}(t)&= {\tilde{\rho }}_{22}(0)+ i\int \limits _0^t e^{-\varGamma _2 (t- t^{\prime })}({\tilde{\rho }}_{21}(t^{\prime }){\tilde{V}}_{12}(t^{\prime })- {\tilde{V}}_{21}(t^{\prime }) {\tilde{\rho }}_{12}(t^{\prime })) dt^{\prime }, \end{aligned}$$

(100)

$$\begin{aligned} {\tilde{\rho }}_{12}(t)&= {\tilde{\rho }}_{12}(0) + i \int \limits _0^t e^{-\varGamma (t- t^{\prime })}{\tilde{V}}_{12}(t^{\prime }) ({\tilde{\rho }}_{11}(t^{\prime })- {\tilde{\rho }}_{22}(t^{\prime }))dt^{\prime }, \end{aligned}$$

(101)

$$\begin{aligned} {\tilde{\rho }}_{21}(t)&= {\tilde{\rho }}_{21}(0) + i \int \limits _0^t e^{-\varGamma (t- t^{\prime })}{\tilde{V}}_{21}(t^{\prime })({\tilde{\rho }}_{11}(t^{\prime })- {\tilde{\rho }}_{22}(t^{\prime }))dt^{\prime }. \end{aligned}$$

(102)

We assume that initially ${\tilde{\rho }}_{12}(0)={\tilde{\rho }}_{21}(0)=0$. Now, inserting (99)–(102) into Eqs. (93)–(96), and taking into account that ${\tilde{\rho }}_{11} = \rho _{11}$ and ${\tilde{\rho }}_{22} = \rho _{22}$, we obtain the following system of integro-differential equations,

$$\begin{aligned} {\dot{\rho }}_{11}(t)&= - \int \limits _0^t e^{-\varGamma (t- t^{\prime })}\left( {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime })+ {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\right) \left( { \rho }_{11}(t^{\prime }) -{ \rho }_{22}(t^{\prime })\right) dt^{\prime } \nonumber \\&- \varGamma _1 {\rho }_{11}(t), \end{aligned}$$

(103)

$$\begin{aligned} {\dot{\rho }}_{22}(t)&= \int \limits _0^te^{-\varGamma (t- t^{\prime })}\left( {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime })+ {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\right) \left( { \rho }_{11}(t^{\prime }) -{\rho }_{22}(t^{\prime })\right) dt^{\prime } \nonumber \\&- \varGamma _2 {\rho }_{22}(t), \end{aligned}$$

(104)

$$\begin{aligned} \dot{\tilde{\rho }}_{12}(t)&= - \int \limits _0^t\left( e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\right) \left( {\tilde{V}}_{21}(t^{\prime }){\tilde{\rho }}_{12}(t^{\prime })- {\tilde{V}}_{12}(t^{\prime }){\tilde{\rho }}_{21}(t^{\prime })\right) {\tilde{V}}_{12}(t)dt^{\prime } \nonumber \\&- \varGamma {\rho }_{12}(t) + i{\tilde{V}}_{12}(t)({ \rho }_{11}(0)- { \rho }_{22}(0)), \end{aligned}$$

(105)

$$\begin{aligned} \dot{\tilde{\rho }}_{21}(t)&= - \int \limits _0^t\left( e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\right) \left( {\tilde{V}}_{21}(t^{\prime }){\tilde{\rho }}_{12}(t^{\prime })- {\tilde{V}}_{12}(t^{\prime }){\tilde{\rho }}_{21}(t^{\prime })\right) {\tilde{V}}_{21}(t)dt^{\prime } \nonumber \\&- \varGamma {\rho }_{21}(t) + i{\tilde{V}}_{21}(t)({ \rho }_{11}(0)- {\rho }_{22}(0)). \end{aligned}$$

(106)

For the average components of the density matrix this yields

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{11}(t)\rangle&= - \int \limits _0^te^{-\varGamma (t- t^{\prime })}\Big \langle \left( {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime })+ {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\right) \nonumber \\&\times \left( { \rho }_{11}(t^{\prime }) -{ \rho }_{22}(t^{\prime })\right) \Big \rangle dt^{\prime }-\varGamma _1 \langle {\rho }_{11}(t)\rangle , \end{aligned}$$

(107)

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{22}(t)\rangle&= \int \limits _0^te^{-\varGamma (t- t^{\prime })}\Big \langle \left( {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime })+ {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\right) \left( { \rho }_{11}(t^{\prime }) -{\rho }_{22}(t^{\prime })\right) \Big \rangle dt^{\prime } \nonumber \\&- \varGamma _2 \langle {\rho }_{22}(t)\rangle , \end{aligned}$$

(108)

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{12}(t)\rangle&= - \int \limits _0^t\left( e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\right) \nonumber \\&\times \Big \langle \left( {\tilde{V}}_{21}(t^{\prime }){\tilde{\rho }}_{12}(t^{\prime })- {\tilde{V}}_{12}(t^{\prime }){\tilde{\rho }}_{21}(t^{\prime })\right) {\tilde{V}}_{12}(t)\Big \rangle dt^{\prime } - \varGamma \langle {\rho }_{12}(t)\rangle \nonumber \\&+ i\langle {\tilde{V}}_{12}(t)\rangle ({ \rho }_{11}(0)- {\rho }_{22}(0)), \end{aligned}$$

(109)

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{21}(t)\rangle&= - \int \limits _0^t+\left( e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\right) \nonumber \\&\times \Big \langle \left( {\tilde{V}}_{21}(t^{\prime }){\tilde{\rho }}_{12}(t^{\prime })- {\tilde{V}}_{12}(t^{\prime }){\tilde{\rho }}_{21}(t^{\prime })\right) {\tilde{V}}_{21}(t)\Big \rangle dt^{\prime } - \varGamma \langle {\rho }_{21}(t)\rangle \nonumber \\&+ i\langle {\tilde{V}}_{21}(t)\rangle ({\rho }_{11}(0)-{\rho }_{22}(0)), \end{aligned}$$

(110)

where the average $\langle \; \rangle$ is taken over the random process describing noise.

In the spin-fluctuator model of noise with the number of fluctuators, ${\fancyscript{N}} \gg 1$, one has the following relations for the splitting of correlations [43],

$$\begin{aligned}&\left\langle \left( {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime })+ {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\right) \left( {\tilde{\rho }}_{11}(t^{\prime }) -{\tilde{\rho }}_{22}(t^{\prime })\right) \right\rangle \nonumber \\&\quad = \left( \left\langle {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime }) \right\rangle + \left\langle {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\right\rangle \right) \left( \left\langle {\tilde{\rho }}_{11}(t^{\prime })\right\rangle -\left\langle {\tilde{\rho }}_{22}(t^{\prime })\right\rangle \right) , \end{aligned}$$

(111)

and so on. Employing (111), we obtain the following system of integro-differential equations for the average components of the density matrix,

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{11}(t)\rangle&= -\int \limits _0^t e^{-\varGamma (t- t^{\prime })}\big (\big \langle {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime }) \big \rangle \nonumber \\&+ \big \langle {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\big \rangle \big )\big (\big \langle {\rho }_{11}(t^{\prime })\big \rangle -\big \langle {\rho }_{22}(t^{\prime })\big \rangle \big )dt^{\prime }- \varGamma _1 \langle { \rho }_{11}(t)\rangle , \end{aligned}$$

(112)

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{22}(t)\rangle&= \int \limits _0^t e^{-\varGamma (t- t^{\prime })}\big (\big \langle {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime }) \big \rangle \nonumber \\&+ \big \langle {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\big \rangle \big )\big (\big \langle {\rho }_{11}(t^{\prime })\big \rangle -\big \langle {\rho }_{22}(t^{\prime })\big \rangle \big )dt^{\prime } - \varGamma _2 \langle { \rho }_{22}(t)\rangle , \end{aligned}$$

(113)

$$\begin{aligned} \frac{d}{dt}\langle {\tilde{\rho }}_{12}(t)\rangle&= i\langle {\tilde{V}}_{12}(t)\rangle ({ \rho }_{11}(0)- { \rho }_{22}(0))\nonumber \\&- \int \limits _0^t \Big (e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\Big ) \langle {\tilde{V}}_{12}(t) {\tilde{V}}_{21}(t^{\prime })\rangle \langle {\tilde{\rho }}_{12}(t^{\prime })\rangle dt^{\prime } \nonumber \\&+ \int \limits _0^t \Big (e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\Big ) \langle {\tilde{V}}_{12}(t) {\tilde{V}}_{12}(t^{\prime })\rangle \langle {\tilde{\rho }}_{21}(t^{\prime })\rangle dt^{\prime } - \varGamma \langle {\tilde{\rho }}_{12}(t)\rangle , \nonumber \\\end{aligned}$$

(114)

$$\begin{aligned} \frac{d}{dt}\langle {\tilde{\rho }}_{21}(t)\rangle&= i\langle {\tilde{V}}_{21}(t)\rangle ({ \rho }_{11}(0)- { \rho }_{22}(0)) \nonumber \\&- \int \limits _0^t \Big (e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\Big ) \langle {\tilde{V}}_{21}(t) {\tilde{V}}_{21}(t^{\prime })\rangle \langle {\tilde{\rho }}_{12}(t^{\prime })\rangle dt^{\prime } \nonumber \\&+ \int \limits _0^t \Big (e^{-\varGamma _1(t- t^{\prime })}+ e^{-\varGamma _2(t- t^{\prime })}\Big )\langle {\tilde{V}}_{21}(t) {\tilde{V}}_{12}(t^{\prime })\rangle \langle {\tilde{\rho }}_{21}(t^{\prime })\rangle dt^{\prime } - \varGamma \langle {\tilde{\rho }}_{21}(t)\rangle . \nonumber \\ \end{aligned}$$

(115)

Using these relations, we obtain the following system of integro-differential equations for the diagonal components of the density matrix,

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{11}(t)\rangle&= - \int \limits _0^t { K}(t,t^{\prime })\left( \left\langle {\rho }_{11}(t^{\prime })\right\rangle -\left\langle {\rho }_{22}(t^{\prime })\right\rangle \right) dt^{\prime } - \varGamma _1 \langle { \rho }_{11}(t)\rangle , \end{aligned}$$

(116)

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{22}(t)\rangle&= \int \limits _0^t { K}(t,t^{\prime })\left( \left\langle {\rho }_{11}(t^{\prime })\right\rangle -\left\langle {\rho }_{22}(t^{\prime })\right\rangle \right) dt^{\prime } - \varGamma _2 \langle { \rho }_{22}(t)\rangle , \end{aligned}$$

(117)

where the kernel is given by

$$\begin{aligned} K(t,t^{\prime }) = e^{-\varGamma (t- t^{\prime })}\left( \left\langle {\tilde{V}}_{21}(t){\tilde{V}}_{12}(t^{\prime }) \right\rangle + \left\langle {\tilde{V}}_{21}(t^{\prime }){\tilde{V}}_{12}(t)\right\rangle \right) . \end{aligned}$$

(118)

For the diagonal noise, so that $\lambda _{mn}=0 \, (m\ne n)$, the kernel can be recast as

$$\begin{aligned} K(t -t^{\prime }) =|V_{12}|^2e^{-\varGamma (t- t^{\prime })}\left( e^{i\varepsilon _{12}(t-t^{\prime })} \left\langle e^{-i\kappa (t-t^{\prime })}\right\rangle + e^{-i\varepsilon _{12}(t-t^{\prime })} \left\langle e^{i\kappa (t-t^{\prime })}\right\rangle \right) ,\qquad \end{aligned}$$

(119)

where $\varepsilon _{12} = \varepsilon _{1} - \varepsilon _{2},\,\kappa (t-t^{\prime }) = D\int _0^{t-t^{\prime }}\xi (\tau )d\tau$ and $D =|g_1-g_2|$.

Applying the cumulant expansion, the generating functional can be recast in terms of the line shape function and correlation function

$$\begin{aligned} \left\langle e^{i\kappa (t-t^{\prime })} \right\rangle = e^{i\lambda _0(t-t^{\prime })-\langle \kappa ^2(t-t^{\prime })\rangle /2}, \end{aligned}$$

(120)

where $\lambda _0 = D\langle \xi (0)\rangle$ and

$$\begin{aligned} \langle \kappa ^2(t-t^{\prime })\rangle =2 D^2\int \limits ^{t-t^{\prime }}_0 d\tau ^{\prime }\int \limits _{0}^{\tau ^{\prime }}d\tau ^{\prime \prime }\chi (\tau ^{\prime } -\tau ^{\prime \prime }). \end{aligned}$$

(121)

Employing Eqs. (120)–(121), we obtain

$$\begin{aligned} K(t \!-\!t^{\prime }) =\frac{V^2}{2} \cos (\varepsilon (t-t^{\prime }))\exp \left( -\varGamma (t-t^{\prime })- D^2\int \limits ^{t-t^{\prime }}_0 d\tau ^{\prime }\int \limits _{0}^{\tau ^{\prime }}d\tau ^{\prime \prime }\chi (\tau ^{\prime } \!-\!\tau ^{\prime \prime }) \right) , \nonumber \\ \end{aligned}$$

(122)

where $\varepsilon =\varepsilon _{12} -\lambda _0$.

Using the results obtained in Sec. III, one can show that for $V< D\sigma$ the system of integro-differential Eqs. (116)–(117) can be approximated by the following system of ordinary differential equations:

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{11}\rangle&= - {\mathfrak{R }}(t)\left( \left\langle {\rho }_{11}\right\rangle -\left\langle {\rho }_{22}\right\rangle \right) - \varGamma _1 \langle {\rho }_{11}\rangle , \end{aligned}$$

(123)

$$\begin{aligned} \frac{d}{dt}{\langle {\rho }}_{22}\rangle&= \,{\mathfrak{R }}(t)\left( \left\langle {\rho }_{11}\right\rangle -\left\langle {\rho }_{22}\right\rangle \right) - \varGamma _2 \langle {\rho }_{22}\rangle , \end{aligned}$$

(124)

where ${\mathfrak{R }}(t)= \int _{0}^{t} \tau K(\tau ) d\tau$. Performing the integration we obtain

$$\begin{aligned} {\mathfrak{R }}(t)&= \frac{\sqrt{\pi }q}{4p}\exp \left( \frac{q^2}{4p^2}\right) \left( \mathrm{erf}\left( \frac{q}{2p} +pt\right) - \mathrm{erf}\left( \frac{q}{2p}\right) \right) \nonumber \\&+\frac{\sqrt{\pi }\bar{q}}{4p} \exp \left( \frac{{\bar{q}}^2}{4p^2}\right) \left( \mathrm{erf}\left( \frac{\bar{q}}{2p}+pt\right) - \mathrm{erf}\left( \frac{\bar{q}}{2p}\right) \right) , \end{aligned}$$

(125)

where $p=D\sigma /\sqrt{2},\,q=\varGamma + i\varepsilon ,\,\bar{q}=\varGamma - i\varepsilon$, and $\mathrm{erf}(z)$ is the error function [44].