The article information
 Zhihao Gong, Zhoufei Tang, Jianshu Cao, Jianlan Wu
 龚志浩, 唐舟飞, 曹建树, 吴建澜
 Optimal Initialization of a Quantum System for an Efficient Coherent Energy Transfer^{†}
 关于量子体系的高效相干能量转移的最优初始化的研究
 Chinese Journal of Chemical Physics, 2018, 31(4): 421432
 化学物理学报, 2018, 31(4): 421432
 http://dx.doi.org/10.1063/16740068/31/cjcp1804068

Article history
 Received on: April 14, 2018
 Accepted on: June 21, 2018
b. Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
Optimizing system and environmentrelated parameters is a fundamental question in quantum dynamics and thermodynamics, appearing in contexts of energy and charge transfer [1], quantum heat engine [2, 3], optimal control [4, 5], and many other problems. For example, the transfer efficiency of electronic excitation energy
in biological photosynthetic protein complexes is maximized (
The environment surrounding a quantum system induces quantum dissipation [23, 24], which plays an interesting role of adjusting quantum transport processes such as energy transfer. In the strong dissipation limit, the conventional hopping kinetics predicts a small transfer rate and a low efficiency. In the opposite limit of a longlasting quantum coherence, a delocalized eigenstate (exciton) leads to an instantaneous longrange transfer, but energy oscillates within the quantum system before being irreversibly absorbed by an outer energy sink. The transfer efficiency stays at a low level. As the dissipation strength is gradually applied to disrupt coherence of the quantum system, the transfer efficiency is significantly enhanced until reaching a maximum value at an intermediate dissipation strength. The phenomenon that the efficiency increases with the dissipation strength is termed with different names such as the noiseenhanced energy transfer (NEET) and environmentassisted quantum transport (ENAQT) [622].
Taking a biased twosite system as a simple example, we can interpret the NEET using the Förster resonance energy transfer (FRET) theory [25], where the energy transfer rate (more accurately the time integration of the rate kernel) is proportional to the spectral overlap between donor emission and acceptor absorption. The environmental noise broadens two lineshapes, which subsequently increases the spectral overlap and the transfer rate. A similar phenomenon is observed in the classical Kramer's theory, where the friction accelerates the diffusion in the energy space and also leads to the increase of the reaction rate in the weak damping regime [26]. For a general multisite quantum network, the NEET has been interpreted with the concepts of the invariant subspace [9] and the trappingfree subspace [15]. In the eigen basis representation, the trappingfree subspace consists of excitons free of the irreversible trapping process (orthogonal to the trapping operator) and the assistance of the environmental noise can break this orthogonality for the enhancement of efficiency. In our previous study [15], the trappingfree subspace is demonstrated in a highlysymmetric dendrimer system, and a more comprehensive construction is required.
As a contradiction, the time integration of the rate kernel for an unbiased twosite system approaches the infinity in the complete coherent limit and the transfer efficiency is maximized accordingly. Another example is a homogeneous onedimensional (1D) chain, which is the simplest polymer model [27]. The coherent transfer efficiency can also be maximized by a ballistic quantum motion. It may seem that such systems disobey the NEET behavior. Instead, we take a different angle to think that the maximized coherent transfer efficiency is induced by a finely tuned initial system state. In other words, the NEET is a universal behavior of a quantum network with an irreversible population depletion but the optimization on the system initialization can suppress the NEET behavior, which is also universal. In this paper, we will perform a mathematical analysis to reveal this conceptual point and verify it in various model systems.
In this work, we briefly review
the theoretical modelling of a quantum dynamic network with an irreversible energy
trapping process. The trapping time is approximated
by its partial value within the exciton population subspace. Following
a rational polynomial expression, the partial trapping time is analyzed
to derive the constraints of the NEET as well as
the optimal initial system states to suppress the
For an open quantum system with an irreversible population depletion process,
e.g. the trapping process, the time evolution of the system reduced density matrix (RDM)
$ \begin{eqnarray} \dot{\rho}_{\rm{S}}(t) = {\cal L}_{\rm{S}}\rho_{\rm{S}}(t) {\cal L}_{\rm{t}}[\rho_{\rm{S}}(t)]{\cal L}_{\rm{d}}\left[\rho_{\rm{S}}(t)\right] \label{eq_01} \end{eqnarray} $  (1) 
The three superoperators,
For simplicity, a nonHermitian Hamiltonian
$ \begin{eqnarray} {\cal L}^{\rm{L}}_{{\rm{t}}; mn, m' n'} = \delta_{m', m}\delta_{n', n}\frac{k_{{\rm{t}};m}+k_{{\rm{t}}; n}}{2} \label{eq_02} \end{eqnarray} $  (2) 
The trapping rates can be similarly assigned to delocalized excitons (eigenstates) if necessary [19].
The dissipation of a quantum system induced by an interaction between the system and the surrounding environment
is reflected by the phenomena of population redistribution and decoherence [23, 24].
In a microscopic description, we introduce the total Hamiltonian,
In the simplest case, the environment is modelled as a classical white noise. The quantum dissipation is described by the HakenStroblReineker (HSR) model as [30, 31]
$ \begin{eqnarray} {\cal L}^{\rm{L}}_{{\rm{d}}; mn, m' n'} = (1\delta_{m, n})\delta_{m', m}\delta_{n', n}\Gamma \label{eq_03} \end{eqnarray} $  (3) 
where
$ \begin{align} & {{\mathcal{L}}_{\text{d}}}[\rho _{\text{S}}^{(\text{I})}(t)]\to \int_{0}^{t}{\text{d}}\tau {\mathcal{P}}{{\mathcal{L}}_{\text{SB}}}(t){{\mathcal{T}}_{+}}\left[ {{\text{e}}^{\int_{\tau }^{t}{\text{d}}{\tau }'{\mathcal{Q}}{{\mathcal{L}}_{\text{SB}}}({\tau }')}} \right]\cdot \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathcal{Q}}{{\mathcal{L}}_{\text{SB}}}(\tau ){\mathcal{P}}\rho _{\text{tot}}^{(\text{I})}(\tau ) \\ \end{align} $  (4) 
where
$ \begin{align} & {{\mathcal{L}}_{\text{d}}}[\rho _{\text{S}}^{(\text{I})}(t)]\approx \int_{0}^{t}{\text{d}}\tau \text{T}{{\text{r}}_{\text{B}}}\{{{\mathcal{L}}_{\text{SB}}}(t){{\mathcal{L}}_{\text{SB}}}(\tau )\rho _{\text{S}}^{(\text{I})}(\tau )\rho _{\text{B}}^{\text{eq}}\} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\int_{0}^{t}{\text{d}}\tau \text{T}{{\text{r}}_{\text{B}}}\{[{{H}_{\text{SB}}}(t),[{{H}_{\text{SB}}}(\tau ),\rho _{\text{S}}^{(\text{I})}(\tau )\rho _{\text{B}}^{\text{eq}}]]\} \\ \end{align} $  (5) 
which is a good approximation in the limit of weak dissipation. Furthermore, the BornMarkov approximation and the random phase approximation are applied. In the Schrödinger picture and the eigen basis representation, Eq.(5) is simplified to be
$ \begin{eqnarray} {\cal L}_{\rm{d}}[\rho^{\rm{E}}_{\rm{S}}(t)] &\approx& \sum\limits_{i, j} R_{ii, jj}\rho^{\rm{E}}_{{\rm{S}}; jj}(t)+\sum\limits_{i\neq j} R_{ij, ij} \rho^{\rm{E}}_{{\rm{S}}; ij}(t)\quad \label{eq_06} \end{eqnarray} $  (6) 
where
$ \begin{eqnarray} \dot{\boldsymbol\sigma}(t) =  {\cal W} \boldsymbol\sigma(t) \label{eq_07} \end{eqnarray} $  (7) 
where the transition rate matrix
$ \begin{eqnarray} {\cal L}_{\rm{d}}[\rho_{\rm{S}}(t)] =\int_0^t \textrm{d}\tau {\cal L}_{\rm{d}}(t\tau)\rho_{\rm{S}}(\tau) \label{eq_08} \end{eqnarray} $  (8) 
The dissipation kernel
$ \begin{eqnarray} {\cal L}_{\rm{d}}(t) = \mathit{\rm{LT}}^{1}\bigg[{\cal W}_{0, 1}\cdot\nonumber\\ &&\frac{1}{z+{\cal W}_{1, 1}+{\cal W}_{1, 2}\frac{1}{z+{\cal W}_{2, 2}+\cdots}{\cal W}_{2, 1} }{\cal W}_{1, 0} \bigg] \label{eq_09} \end{eqnarray} $  (9) 
In general, the equation of motion in Eq.(1) can be
formally solved in the Laplace
$ \begin{eqnarray} \tilde{\rho}_{\rm{S}}(z) = \left[z+{\cal L}_{{\rm{s}}}+{\cal L}_{\rm{t}}+\tilde{{\cal L}}_{\rm{d}}(z)\right]^{1}\rho_{\rm{S}}(0) \label{eq_10} \end{eqnarray} $  (10) 
where
For an irreversible quantum dynamic network, a key quantity is the trapping time
$ \begin{align} & \langle t\rangle =\text{T}{{\text{r}}_{\text{S}}}\{{{{\tilde{\rho }}}_{\text{S}}}(z=0)\} \\ & \ \ \ \ =\text{T}{{\text{r}}_{\text{S}}}\left\{ {{\left[ {{\mathcal{L}}_{\text{S}}}+{{\mathcal{L}}_{\text{t}}}+{{\widetilde{\mathcal{L}}}_{\text{d}}}(z=0) \right]}^{1}}{{\rho }_{\text{S}}}(0) \right\} \\ \end{align} $  (11) 
Since
$ \begin{eqnarray} q\approx \frac{1}{1+k_\mathit{\rm{decay}} \langle t\rangle} \label{eq_12} \end{eqnarray} $  (12) 
In a real system,
In the HSR model [30, 31], the dissipation strength from the classical white noise is purely determined by the dephasing rate
In the weak dissipation limit (
$ \begin{eqnarray} \ell^{\rm{E}}_{{\rm{d}}; ij, kl} \approx \ell^{\rm{E}}_{{\rm{d}}; ii, kk}\delta_{i, j}\delta_{k, l}+\ell^{\rm{E}}_{{\rm{d}}; ij(\neq i), ij}\delta_{k, i}\delta_{l, j} \label{eq_13} \end{eqnarray} $  (13) 
in the eigen basis representation. Eq.(13) is obtained from the secular Redfield equation in Eq.(6).
In the Liouville space, we partition the RDM into the exciton population and coherence subspaces [56], given by
$ \begin{eqnarray} {\cal L}_{\rm{S}}^{\rm{E}} \hspace{0.12cm}&=&\hspace{0.12cm} \left( {\begin{array}{*{20}{c}} 0&0\\ 0&{\cal L}_{\rm{S;C}}^{\rm{E}} \end{array}} \right)\nonumber \\ {\cal L}_{\rm{d}}^{\rm{E}} \hspace{0.12cm}&=&\hspace{0.12cm}\left( {\begin{array}{*{20}{c}} \ell_{\rm{d;P}}^{\rm{E}}&0\\ 0&\ell_{\rm{d;C}}^{\rm{E}} \end{array}} \right)\\ {\cal L}_{\rm{t}}^{\rm{E}} \hspace{0.12cm}&=&\hspace{0.12cm} \left( {\begin{array}{*{20}{c}} {\cal L}_{\rm{t;P}}^{\rm{E}} &{\cal L}_{\rm{t;PC}}^{\rm{E}}\\ {\cal L}_{\rm{t;CP}}^{\rm{E}}&\ell_{\rm{t;C}}^{\rm{E}} \end{array}} \right)\nonumber \end{eqnarray} $  (14) 
In general, the trapping process (
$ \begin{eqnarray} \langle t\rangle^{\rm{E}}_{\rm{P}} = {\rm{Tr}}_{\rm{S}}\left\{\left[\Gamma\ell^{\rm{E}}_{{\rm{d}}; {\rm{P}}} + {\cal L}^{\rm{E}}_{{\rm{t}}; {\rm{P}}} \right]^{1} \rho^{\rm{E}}_{\rm{P}}(0) \right\} \label{eq_15} \end{eqnarray} $  (15) 
where the matrices in the trace
For a finite
$ \begin{eqnarray} \langle t \rangle^{\rm{E}}_{\rm{P}} &=& \frac{\displaystyle\sum\limits_{k=0}^{N1}a_k \Gamma^{k}}{\displaystyle\sum\limits_{k=0}^{N1}b_k \Gamma^{k}}= t_0+\sum\limits_{k=1}^{N1}\frac{t_k}{\Gamma+\Gamma_k} \label{eq_16} \end{eqnarray} $  (16) 
where all the four parameter sets,
$ \begin{eqnarray} \sum\limits_{k=0}^{N1}b_k \Gamma^{k}=0 \label{eq_17} \end{eqnarray} $  (17) 
To include a weak contribution of the exciton coherence, we add a linear
$ \begin{eqnarray} \langle t\rangle\approx \langle t\rangle^{\rm{E}}_{\rm{P}}+\delta t_0+f_{\mathit{\rm{hop}}}\Gamma \label{eq_18} \end{eqnarray} $  (18) 
where the two positive parameters,
$ \begin{eqnarray} \langle t \rangle &\approx& \left(t_0+\delta t_0+\sum\limits_{k=1}^{N1} \frac{t_k}{\Gamma_k}\right) +\nonumber\\ &&\left(f_{\mathit{\rm{hop}}}\sum\limits_{k=1}^{N1}\frac{t_k}{\Gamma^2_k}\right) \Gamma +O(\Gamma^2) \label{eq_19} \end{eqnarray} $  (19) 
As a result, the condition
$ \begin{eqnarray} \sum\limits_{k=1}^{N1}t_k/\Gamma^2_k>f_{\mathit{\rm{hop}}} \label{eq_20} \end{eqnarray} $  (20) 
is a general requirement of the NEET that
However, one or more zero roots in Eq.(17) dramatically change
the
$ \begin{eqnarray} \langle t \rangle^{\rm{E}}_{\rm{P}} = \left(t_0+\sum\limits_{k=2}^{N1}\frac{t_k}{\Gamma_k}\right)+ \frac{t_1}{\Gamma}+O(\Gamma) \label{eq_21} \end{eqnarray} $  (21) 
where
$ \begin{eqnarray} b_0 \propto \mathit{\rm{Det}}\left[{\cal L}^{\rm{E}}_{{\rm{t}}; {\rm{P}}}\right] = 0 \label{eq_22} \end{eqnarray} $  (22) 
which is our definition of the rigorous trappingfree subspace
The rigorous solution of
$ \begin{eqnarray} b_0\ll b_{N1} \Gamma_\textrm{c}^{N1} \label{eq_23} \end{eqnarray} $  (23) 
where
$ \begin{eqnarray} \begin{array}{l} \left\langle t \right\rangle _{\rm{P}}^{\rm{E}} \approx \left[{{t_0} + \displaystyle\sum\limits_{k = {N_{\rm{t}}} + 1}^{N1} {\frac{{{t_k}}}{{{\Gamma _k}}}} } \right] + \displaystyle\sum\limits_{k = 1}^{{N_{\rm{t}}}} {\frac{{{t_k}}}{{\Gamma + {\Gamma _k}}}} \\ \hspace{3cm}\textrm{for}\ \Gamma\ll\Gamma_{k(>N_{\rm{t}})};\\ \left\langle t \right\rangle _{\rm{P}}^{\rm{E}} \approx \left[{{t_0} + \displaystyle\sum\limits_{k = {N_{\rm{t}}} + 1}^{N1} {\frac{{{t_k}}}{{{\Gamma _k}}}} } \right] + \displaystyle\frac{{\displaystyle\sum\limits_{k = 1}^{{N_{\rm{t}}}} {{t_k}} }}{\Gamma }\\ \hspace{3cm}\textrm{for}\ \Gamma_{k(<N_{\rm{t}})}\ll\Gamma\ll\Gamma_{k(>N_{\rm{t}})} \end{array} \label{eq_24} \end{eqnarray} $  (24) 
To achieve the approximate
In Section Ⅲ.A, we formulate the exact and approximate
$ \begin{eqnarray} a_0:a_1:\cdots:a_{N1} = b_0:b_1:\cdots:b_{N1} \label{eq_25} \end{eqnarray} $  (25) 
from which all the
Next we provide a mathematical procedure of optimizing the initial system state.
In the subspace of exciton population, the partial reduced dissipation superoperator
$ \begin{eqnarray} {\cal D}={\cal V}^{1}\ell^{\rm{E}}_{{\rm{d}}; {\rm{P}}}{\cal V} = \left( {\begin{array}{*{20}{c}} 0&{}&{}\\ {}&{{\Lambda _2}}&{}\\ {}&{}& \ddots \end{array}} \right) \label{eq_26} \end{eqnarray} $  (26) 
where
$ \begin{eqnarray} \langle t\rangle^{\rm{E}}_{\rm{P}} = \mathit{\rm{Tr}}_{\rm{S}}\left\{\mathcal V(\Gamma{\cal D}+{\cal T})^{1}\varrho(0)\right\} \label{eq_27} \end{eqnarray} $  (27) 
After tedious but straightforward steps, we obtain
$ \begin{eqnarray} \mathit{\rm{Det}}\left[{\cal T}\right] \ll \prod\limits_{i=2}^{N} \Lambda_{i} [{\cal T}]_{11} \Gamma_c^{N1} \label{eq_28} \end{eqnarray} $  (28) 
which implies an upper limit of the trapping rates.
With respect to the zero eigen rate of
$ \begin{align} & {{[\varrho (0)]}_{1}}:{{[\varrho (0)]}_{2}}:\cdots :{{[\varrho (0)]}_{N}}= \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{[T]}_{11}}:{{[T]}_{21}}:\cdots :{{[T]}_{N1}} \\ \end{align} $  (29) 
Since the initial exciton coherence varies freely, an infinite possibility of the initial system states (either pure or mixed) can satisfy Eq.(29).
If the reduced dissipation superoperator
In Section Ⅲ, we have derived the requirements of the rigorous and approximate
The first example is a biased twosite system (see FIG. 1(a)).
The system Hamiltonian is defined in the local basis as
The analytical expression of the trapping time for this model was shown previously [11]. Here we apply the theoretical procedure developed in Section Ⅲ to analyze the NEET and determine the optimal initial system states. In the exciton population subspace, the two relevant superoperators are explicitly written as
$ \begin{eqnarray} \begin{array}{l} \ell^{\rm{E}}_{{\rm{d}}; {\rm{P}}} = \displaystyle\frac{\sin^22\theta }{2}\left( {\begin{array}{*{20}{c}} 1&{{\rm{  }}1}\\ {{\rm{  }}1}&1 \end{array}} \right)\\ {\cal L}^{\rm{E}}_{{\rm{t}}; {\rm{P}}} = k_{\rm{t}}\left( {\begin{array}{*{20}{c}} {\sin {\theta ^2}}&0\\ \theta &{\cos {\theta ^2}} \end{array}} \right)\\ \end{array} \label{eq_30} \end{eqnarray} $  (30) 
with
$ \begin{eqnarray} \langle t\rangle = t_0+ \frac{t_1}{\Gamma+k_{\rm{t}}/2} +\delta t_0 + f_{\mathit{\rm{hop}}} \Gamma \label{eq_31} \end{eqnarray} $  (31) 
where the parameters,
To suppress the
$ \begin{eqnarray} \rho^{\rm{L}}_{{\rm{S}}; 11}(0)+\frac{2J}{\Delta}\mathit{\rm{Re}} \{\rho^{\rm{L}}_{{\rm{S}}; 12}(0)\}=0 \label{eq_32} \end{eqnarray} $  (32) 
in the local basis representation. A trivial solution of Eq.(32) is
The second example is a symmetric threesite branching system (see FIG. 1(b)).
The system Hamiltonian is given by
The explicit expression of the trapping time was also provided previously [11]. Following the procedure in Section Ⅲ, we obtain dissipation and trapping superoperators,
$ \begin{eqnarray} \begin{array}{l} \ell^{\rm{E}}_{{\rm{d}}; {\rm{P}}}= \displaystyle\frac{1}{8} \left( {\begin{array}{*{20}{c}} 4&{  2}&{  2}\\ {  2}&5&{  3}\\ {  2}&{  3}&5 \end{array}} \right)\\ [2ex] {\cal L}^{\rm{E}}_{{\rm{t}}; {\rm{P}}} = \displaystyle\frac{k_{\rm{t}}}{2}\left( {\begin{array}{*{20}{c}} 0&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right)\\ \end{array} \label{eq_33} \end{eqnarray} $  (33) 
in the exciton population subspace.
The first exciton state,
As discussed in Section Ⅲ.B, a straightforward way to optimize
the coherent energy transfer is to avoid the initial population at the trappingfree subspace, i.e.,
$ \begin{eqnarray} \rho^{\rm{L}}_{11}(0)+\rho^{\rm{L}}_{33}(0) = 2 \mathit{\rm{Re}}\{\rho^{\rm{L}}_{13}(0) \} \label{eq_34} \end{eqnarray} $  (34) 
in the local basis representation. Eq.(34) is identical to
the condition of zero population in the trappingfree subspace.
Here we design an optimal initial system state,
The third example is a homogeneous 1D
$ \begin{eqnarray} H^{\rm{L}}_{\rm{S}} = \sum\limits_{n=1}^{N1} J (n\rangle\langle n+1+n+1\rangle\langle n) \label{eq_35} \end{eqnarray} $  (35) 
with the nearest neighboring interaction. The dissipation is simulated by the HSR model, while the trapping process is defined by an irreversible rate
Following the procedure in Section Ⅲ.B, we numerically calculate Eq.(28)
and determine an upper trapping rate limit,
To extract the approximate
$ \begin{eqnarray} \langle t\rangle^{\rm{E}}_{\rm{P}} \approx \frac{N}{k_\textrm{t}}+\frac{t_1}{\Gamma+\Gamma_1}+\frac{t_2}{\Gamma+\Gamma_2} \label{eq_36} \end{eqnarray} $  (36) 
over a broad range of
$ \begin{eqnarray} \langle t\rangle^{\rm{E}}_{\rm{P}} \approx \frac{N}{k_{\rm{t}}}+\frac{t_1+t_2}{\Gamma} \label{eq_37} \end{eqnarray} $  (37) 
which is confirmed for the result of
Next we calculate the optimal initial system state of the coherent energy transfer. Without
tedious details, Eq.(29) is transformed into an
$ \begin{eqnarray} \sum\limits_{i=2}^{N1}x^k_{i}\rho^{\rm{L}}_{i}(0)+\sum\limits_{i=1}^{N2}\sum\limits_{j=1}^{(Ni)/2} 2 x^k_{i, i+2j}\mathit{\rm{Re}}\rho^{\rm{L}}_{i, i+2j}(0)=0\quad \label{eq_38} \end{eqnarray} $  (38) 
with
$ \begin{eqnarray} \rho^{\rm{L}}_{\rm{S}}(0) &=& \frac{1}{2(N+1)}\bigg[31\rangle\langle 1+3N\rangle\langle N+2\sum\limits_{i=2}^{N1}i\rangle\langle i \nonumber\\ &&\sum\limits_{i=1}^{N2}\left(i\rangle\langle i+2+i+2\rangle\langle i \right) \bigg] \label{eq_39} \end{eqnarray} $  (39) 
As shown in FIG. 5, the total trapping time
By comparing the results of the nonoptimal and optimal initial system states in FIGs. 4 and 5, we observe that the trapping time of
Our final example is the 8chromophore FennaMatthewsOlson (FMO) protein complex (see FIG. 1(d)), which is an important lightharvesting system in green sulfur bacteria [60, 61]. The effective Hamiltonian of an FMO monomer is taken from Refs. [13, 62]. The influence of the bosonic bath is simulated by a Debye spectral density,
$ \begin{eqnarray} J(\omega) = \frac{2\lambda}{\pi} \frac{\omega\omega_\textrm{D}}{\omega^2+\omega^2_\textrm{D}} \label{eq_40} \end{eqnarray} $  (40) 
where
For the nonsymmetric FMO system, the upper limit of the trapping rate for the NEET is estimated by Eq.(28) as
As a demonstration, we consider a natural initial condition of
$ \begin{eqnarray} \langle t\rangle^{\rm{E}}_{\rm{P}} \approx t_0 + \frac{t_1}{\lambda+\lambda_1}+\frac{t_2}{\lambda+\lambda_2}+\frac{t_3}{\lambda+\lambda_3} \label{eq_41} \end{eqnarray} $  (41) 
can reliably describe the partial trapping time over a broad range of the dissipation strength
(
$ \begin{eqnarray} \langle t\rangle^{\rm{E}}_{\rm{P}} \sim \left(t_0+\frac{t_3}{\lambda_3}\right) +\frac{t_1+t_2}{\lambda} \label{eq_42} \end{eqnarray} $  (42) 
is extracted in the weak dissipation regime of 0.01 cm
On the opposite side, the optimal initial system state in Eq.(29) is determined by the secular Redfield equation, from which the efficiency of the coherent energy transfer (
$ \begin{eqnarray*} \rho^{\rm{L}}_{\rm{S}}(0) = \left( {\begin{array}{*{20}{c}} 0.0081 & 0.0146 &0.0444 &0.0127 &0.0020 &0.0013 &0.0015 &0.0010 \\ 0.0146 & 0.0303 &0.1252 &0.0394 &0.0058 &0.0042 &0.0059 &0.0018 \\ 0.0444 &0.1252 & 0.8378 & 0.2675 & 0.0392 & 0.0259 & 0.0420 & 0.0045 \\ 0.0127 &0.0394 & 0.2675 & 0.1065 & 0.0184 & 0.0061 & 0.0242 & 0.0016 \\ 0.0020 &0.0058 & 0.0392 & 0.0184 & 0.0043 &0.0009 & 0.0055 & 0.0002 \\ 0.0013 &0.0042 & 0.0259 & 0.0061 &0.0009 & 0.0037 &0.0011 & 0.0002 \\ 0.0015 &0.0059 & 0.0420 & 0.0242 & 0.0055 &0.0011 & 0.0091 & 0.0003 \\ 0.0010 &0.0018 & 0.0045 & 0.0016 & 0.0002 & 0.0002 & 0.0003 & 0.0001 \end{array}} \right) \end{eqnarray*} $  (43) 
The trapping time
In this paper, we extend our previous studies of efficiency optimization [12, 14, 15]
to probing the mechanism of the noiseenhanced energy transfer (NEET) and engineering the
initial system state to maximize the efficiency of coherent energy transfer from a conceptual point of view.
In the weak dissipation limit, the trapping time
Our theoretical predictions of the
The studies in this paper confirm that the NEET is a universal behavior in the energy transfer network, which can be applied to other irreversible quantum dynamic systems such as a quantum heat engine. Here we focus on the dephasing rate and reorganization energy, but the concept of the dissipation strength can be extended to other system and bathrelated parameters such as temperature. Our calculation of the optimal initial system states shows that quantum (sitesite) coherence to accelerate energy transfer must be tuned finely to be compatible with dissipation and trapping. Although such a fine tuning is difficult to be achieved in natural systems, it is experimentally accessible in preciselycontrolled artificial quantum devices, which will be an interesting problem to be explored in the future [65].
Ⅵ. ACKNOWLEDGEMENTSThe work reported was supported by the National Natural Science Foundation of China (No.21573195) and the Ministry of Science and Technology of China (MOST2014CB921203).
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b. 麻省理工学院化学系, 马萨诸塞州, 剑桥 02139