Annals of Physics 341 (2014) 1–11

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Annals of Physics

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Environment assisted energy transfer indimer system

Salman Khan a,∗, M. Ibrahim b, M.K. Khan b

a Department of Physics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad, Pakistanb Department of Physics, Quaid-i-Azam University, Islamabad, Pakistan

h i g h l i g h t s

• The dynamics of energy transfer between the levels of a dimer are studied.• Coupling of collective as well as individual environments are considered.• The environments are in spin star configurations.• The environment assists the energy transfer between the levels.• For correlated multilocal environments, the transition probability is almost 100%.

a r t i c l e i n f o

Article history:Received 29 May 2013Accepted 13 November 2013Available online 22 November 2013

Keywords:EntanglementDecoherenceQuantum transport

a b s t r a c t

The influence of collective and multilocal environments on theenergy transfer between the levels of a dimer is studied. Thedynamics of energy transfer are investigated by consideringcoupling of collective environment with the levels of the dimerin the presence of both two individuals and mutually correlatedmultilocal environments. It is shown that every way of couplingwe consider assists, though differently, the probability of transitionbetween the levels of dimer. The probability of transition isstrongly enhanced when the two local environments are mutuallycorrelated.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

The control of the loss of quantum correlations in simple systems under the influence ofexternal noise is of great importance for practical application of quantum computation and for the

∗ Corresponding author.E-mail addresses: sksafi@comsats.edu.pk, safiqau@gmail.com (S. Khan).

0003-4916/$ – see front matter© 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.aop.2013.11.009

2 S. Khan et al. / Annals of Physics 341 (2014) 1–11

development of quantum information processing devices [1–6]. The quantum coherence betweendifferent components of a system may last long enough for totally isolated quantum systems. Forexample, entanglement between components of a composite systemmay survive against the decayingparameter (time) up to a considerable value. However, completely isolating realistic physical systemsfrom their environment is quite impractical. It is now a well established fact that the unavoidableinteraction of a quantum system with its environment rapidly destroys the quantum coherence andhence limits the practical realization of quantum computers, information processing and storage ofinformation [7]. On the other hand, there are some recent studies based on spectroscopic observationsthat predict long-lived quantum coherence in biological species, such as the photosynthetic light-harvesting complexes of a species of green sulfur bacteria [8,9], a species of purple bacteria[10] and two species of cryptophyte algae [11]. It has also been suggested that the persistentpresence of quantum entanglement speeds up the electronic excitation transfer in photosynthesis[12].

Recently, the effect of quantum coherence on electronic energy transfer in a simple model of anaggregate of monomers that interact through dipole–dipole forces was investigated in Ref. [13]. Itis shown that in such systems quantum effects bring nothing new than what the classical effectspresent. On the other hand, counter-intuitively, it has also been investigated that a dissipativequantum network subject to dephasing can exhibit an enhanced capacity for transmission of classicalinformation [14]. Excitons are created when Chromophoric complexes absorb light and are thentransported to a reaction center where it is used to trigger more processes to bind its energy ina chemical form. The enhancement of quantum transport due to environmental influences in suchsystems has also been reported [15–18]. The dynamic of energy transfer in a dimer system in thepresence of two local spin environments is studied in Ref. [19], where each level of the dimer systeminteracts with its own local environment and it is shown that the existence of the local environmentscauses a drastic increase in the transition probability. The role of entanglement in the enhancementof quantum transport has also been studied [20–22].

In most cases, mathematical modeling of a system that represents a real physical situation cannotbe solved in a straightforward, analytical way. However, extension of the simplest existing modelstowards a realistic physical system is a common practice. In this paper we exercise such a practiceand extend the work of [19] by considering the effects of a spin environment (spin 1/2) that actscollectively on the dimer system, already coupled to two multilocal environments. We show that inthe presence of collective environment, themaximumof the probability of transition occurs when thecoupling between the system and the environment becomes strong. Moreover, at room temperatureand in the presence of inter-environment coupling, the probability of transition can be increased to ashigh as 0.996. However, simultaneous coupling of all the environments with the levels of the dimerresults in the considerable reduction of the probability of transition (see Fig. 4). Most generally, thecollective environment could interact with the multilocal environments thereby indirectly effectingthe dynamics of energy transfer of the dimer system and could directly interact with the levels of thedimer as shown in Fig. 1. Nevertheless, for mathematical ease, we will limit our findings to the casein which the collective environment is coupled only to the two levels of the dimer and will set thecoupling parameters between the collective and multilocal environments equal to zero. We will referto the collective andmultilocal environments jointly as a global environment. Let Ei (i = 1, 2) stand forthe energies of the two levels |i⟩ of the dimer and J for the amplitude of transition between the levels,then, the Hamiltonian of an isolated dimer is given by Hd = E1|1⟩⟨1| + E2|2⟩⟨2| + J(|1⟩⟨2| + |2⟩⟨1|).For the dimer initially in state |1⟩, the maximum of the probability of transition [P1→2(t)] from state|1⟩ to state |2⟩ is given by Max[P1→2(t)] = J2/(J2 + E2), where E = (E2 − E1)/2 [19]. This means thatthe excitation transfer takes place with certainty when the two levels are at the same energy and thatthe probability of transition of excitation is always less than 1 otherwise. For a dimer interacting withan environment, however, this may not always be true.

In this paper, we investigate the effect on the transfer of energy between the levels of a dimer inthe presence of a collective environment and twomultilocal environments. In Section 2, we introducethe total Hamiltonian of ourmodel. In Section 3, we develop themathematicalmachinery for studyingthe dynamic of energy transfer and present the discussion of our results. In Section 4, we summarizeour paper with some concluding remarks.

S. Khan et al. / Annals of Physics 341 (2014) 1–11 3

Fig. 1. Scheme of the dimer coupled to multilocal and collective environments, where each environment consists ofindependent spins in spins star configurations. The two levels with energies E1 = E2 have amplitudes of transition probabilityJ . Each level |1⟩ and |2⟩ is coupled with its local and collective environments with coupling constants γM1, γM2 and γc . Thedashed double arrow indicates the possibility of correlations between the multilocal baths with strength q.

2. Dimer with decoherence

The total Hamiltonian of the dimer coupled to a global environment can be written as

H = Hd + HM1 + HM2 + Hc + HdM1 + HdM2 + Hdc, (1)

where subscripts d, Mi, and c stand for dimer, multilocal environments and collective environment,respectively. The subscripts dMi and dc represent the interaction part of the total Hamiltonianbetween the levels of dimer and the respective environments. If we consider the multilocal andcollective environments to consist of independent spins of spin 1/2 particles in the spin starconfiguration [23], then, the Hamiltonians of the multilocal environments and of the collectiveenvironment can be written as

Hn = αn

Nnk=1

σ k,nz

2, (2)

4 S. Khan et al. / Annals of Physics 341 (2014) 1–11

where n = {Mi, c}, σz is the Pauli phase flip matrix and Nn is the number of independent spins in thecorresponding bath. If γMi and γc represent the coupling strengths of the multilocal and collectiveenvironments, respectively, with the levels of the dimer, then the interaction parts of the totalHamiltonian can be written as

Hdn =

Nnk=1

γn|i⟩⟨i|σ k,nz

2. (3)

The total Hamiltonian can be rewritten in a more convenient way in terms of the collective spinoperator Szn , as follows [19]

H = J(|1⟩⟨2| + |2⟩⟨1|) + αcSzc +

2i=1

(Ei|i⟩⟨i| + αMiSzMi + γMi|i⟩⟨i|SzMi + γc |i⟩⟨i|Szc ), (4)

where

Szn =

Nnk=1

σ k,nz

2. (5)

The projectors |i⟩⟨j| of the dimer system in terms of Pauli matrices can be expressed as follows

|1⟩⟨1| =I2 − σz

2, |2⟩⟨2| =

I2 + σz

2,

|2⟩⟨1| = σ+, |1⟩⟨2| = σ−, (6)where I2 is the 2 × 2 identity matrix, σ±

= σx ± iσy are the spin raising and lowering operators and

the bases are chosen such that |1⟩ =

01

. Using Eq. (6) and the fact that γc |i⟩⟨i|Szc = γc(|1⟩⟨1|Szc1 +

|2⟩⟨2|Szc2), the total Hamiltonian becomes

H =

E12

+E22

+γM1

2SzM1 +

γM2

2SzM2 +

γc

2(Szc1 + Szc2)

I2

+

E2 − E1

2−

γM1SzM1 − γM2SzM2

2+

γc

2(Szc2 − Szc1)

σz

+ Jσx + αM1SzM1 + αM2SzM2 + αcSzc , (7)where σx and σy are Pauli spin operators and Szci are the collective spin operators of the spins ofcollective bath that interact with the corresponding levels of the dimer.

3. Transition analysis

In this section, we present our results for the transition probability between the levels of the dimerunder different environmental conditions. In the first case, we consider the environments at zerotemperature. Under such constraint, the spins of the global environment are all in the ground state.The initial state of the global environment is a pure state and is given by

|ΨB(0)⟩ =

NM1

2, −

NM1

2

⊗

NM2

2, −

NM2

2

⊗

Nc

2, −

Nc

2

, (8)

where Nn is the number of spins in each environment and |j,m⟩ are the eigenvectors of the angularmomentumoperator S2 and Sz with eigenvalues j(j+1) andm, respectively. The state of the compositesystem at any time t can be written in the form

|Ψtotal(t)⟩ =

2i=1

Ci(t)|i⟩ ⊗ |ΨB(0)⟩

=

C2(t)C1(t)

⊗ |ΨB(0)⟩ (9)

S. Khan et al. / Annals of Physics 341 (2014) 1–11 5

where Ci(t) are the probability amplitudes for dimer to be in either of the two states.We consider thatinitially (t = 0) the excitation is in level |1⟩ of the dimer, then, C1(0) = 1 and C2(0) = 0. Moreover,we assume that N ′

c spins of the collective bath are coupled with level |1⟩ and Nc −N ′c with level |2⟩ of

the dimer. Then, the solution of time dependent Schrödinger equation

H|Ψtotal(t)⟩ = ih∂|Ψtotal(t)⟩

∂t,

with Hamiltonian in its matrix form given by

H =

E2 − E1

2−

γM1SzM1 − γM2SzM22

+

γc (Szc2 − Szc1)

2J

J −

E2 − E1

2−

γM1SzM1 − γM2SzM22

−

γc (Szc2 − Szc1)

2

,

(10)

results in

i

∂C2(t)∂t

∂C1(t)∂t

=

E2 − E12

−γM2NM2 − γM1NM1

4

+

γc

4(2N ′

c − Nc)

C2(t) + JC1(t)

JC2(t) −

E2 − E1

2−

γM2NM2 − γM1NM1

4+

γc

4(2N ′

c − Nc)

C1(t)

. (11)

Note that in Eq. (10) we have ignored the first part of the total Hamiltonian of Eq. (7), because itcommutes with all the rest terms in the Hamiltonian. Its contribution is to add a global phase thatleaves the dynamics of probabilities of transition unaffected. In deriving Eq. (11), we have taken theinner product with ⟨ΨB(0)| and have used the fact that

Szi |ΨB(0)⟩ = −Ni

2|ΨB(0)⟩, ⟨ΨB(0)||ΨB(0)⟩ = 1.

It is straightforward to solve Eq. (11) for the probability amplitudes, which results in

C1(t) = cos

t

J2 +

∆ +

γc

4(2N ′

c − Nc)2

+ i∆ +

γc4 (2N ′

c − Nc)J2 +

∆ +

γc4 (2N ′

c − Nc)2 sin

t

J2 +

∆ +

γc

4(2N ′

c − Nc)2

, (12)

C2(t) = iJ

J2 +∆ +

γc4 (2N ′

c − Nc)2

× sin

t

J2 +

∆ +

γc

4(2N ′

c − Nc)2

exp−

i(E2 − E1)t2

, (13)

where ∆ =E2−E1

2 −γ2NM2−γ1NM1

4 . The probability of transition P1→2(t) from level |1⟩ to level |2⟩ ofthe dimer is given by

P1→2(t) = |C2(t)|2 =J2

J2 + (∆ +γc4 (2N ′

c − Nc))2

× sin2

t

J2 +

∆ +

γc

4(2N ′

c − Nc)2

. (14)

It can be seen that switching off the coupling of the dimer with collective environment, Eq. (14)reduces to the result of Ref. [19]. To further investigate the dynamics of the probability of transitionin the present set up, we first switch off the coupling of multilocal environments (γ1 = γ2 = 0)with the levels of dimer and consider only the effect of collective environment. The dynamics of the

6 S. Khan et al. / Annals of Physics 341 (2014) 1–11

Fig. 2. (Color online) The transition probability is plotted against the coupling parameter γc (ps−1)with collective environmentand time t (ps). The coupling parameters of multilocal baths in Fig. 2(a) are γM1 = γM2 = 0 and in Fig. 2(b) they areγM1 = 10 ps−1 , γM2 = 2 ps−1 . The other parameters for both figures are set to, NM1 = 10, NM2 = 20, N ′

c = 4, N ′= 20,

E2 − E1 = 20 ps−1 , J = 10 ps−1 .

probability of transition for this case is shown in Fig. 2(a). It is seen that the difference in energy,which reduces the probability of transition between the levels, is compensated by the couplingwith collective environment. That is, the collective environment assists to increase the probabilityof transition between the levels. Fig. 2(b) shows the dynamics of the probability of transition whenthe coupling with multilocal environments is also present. One can see that not only the maximumprobability of transition is achievable but also the maximum of the probability of transition shifts tothe region of strong correlation between the environment and the dimer system. The inset in eachfigure shows the contour map of the corresponding surface plot.

Next, we consider a more realistic physical situation of baths at nonzero temperature andinvestigate the influence on transition between the levels of dimer when coupled to such a globalenvironment. The initial density matrix for the global environment being at nonzero temperature isgiven by the canonical distribution

ρB(0) = Πn1Zn

exp[−βnαnSzn], (15)

where Zn represents the partition function of the corresponding bath and is given by [24,25]

Zn =

Nn/2jn=0

jnmn=−jn

ν(Nn, jn) ⟨jn,mn| exp[−βnαnSzn] |jn,mn⟩ . (16)

The parameter βn in the above equations is the inverse temperature and ν(Nn, jn) stands for thedegeneracy of the spin bath [24,25], which is given by

ν(Nn, jn) =

Nn

Nn2 − jn

−

Nn

Nn2 − jn − 1

. (17)

The final density matrix of the composite system becomes

ρ(t) = U(t)ρ(0)U(t)Ď, (18)

where U(t) = exp(−iHt) is the time evolution operator of the composite system and ρ(0) =

ρd(0) ⊗ ρB(0) is the initial density matrix of the composite system. The final density matrix ρd(t)of the dimer system is obtained by tracing out over the environment space. Using the commutivity ofinteraction Hamiltonian and the total Hamiltonian, it can be shown that the transition probability inthis case becomes

P1→2(t) = TrB⟨2|U(t) |1⟩ ρB(0) ⟨1|UĎ(t) |2⟩

. (19)

S. Khan et al. / Annals of Physics 341 (2014) 1–11 7

Employing the trace operation leads to the following relation

P1→2(t) =

Nn/2jn=0

jnmn=−jn

ν(NM1, jM1)ν(NM2, jM2)ν(Nc, jc)C2(t)n

⟨jn,mn|

×

n

1Zn

e−βαn Szn

⟨2| U(t) |1⟩ ⟨1| UĎ(t) |2⟩ C∗

2 (t)n

|jn,mn⟩ , (20)

where we have used

Nn/2jn=0

=

NM1/2jM1=0

NM2/2jM2=0

Nc/2jc=0

,

jnmn=−jn

=

jM1mM1=−jM1

jM2mM2=−jM2

jcmc=−jc

, (21)

and n

|jn,mn⟩ = |jM1,mM1⟩ ⊗ |jM2,mM2⟩ ⊗ |jc,mc⟩ . (22)

Using the fact that

exp[−βnαnSzn] |jn,mn⟩ = exp[−βnαnmn] |jn,mn⟩ , (23)

the above equation can be reduced to the following form

P1→2(t) =1

ZM1ZM2Zc

Nn/2jn=0

jnmn=−jn

ν(NM1, jM1)ν(NM2, jM2)ν(Nc, jc)

× exp

−

i

βMiαMimMi + βcαcmc

J2

J2 +∆mMi −

γc2 (2m′

c − mc)2

× sin2

t

J2 +

∆mMi −

γc

2(2m′

c − mc)2

, (24)

where ∆mMi =E2−E1

2 +γM2mM2−γM1mM1

2 and m′c is the net spin quantum number of spins in the

collective environment interacting with level |1⟩ of the dimer. We confirm that in the absenceof collective environment, Eq. (24) reduces to the result of Ref. [19]. The probability of transitionwhen only the collective environment is coupled with the levels of the dimer is plotted in Fig. 3(a),(b) for two different temperatures of the environment. In both cases, the environment assists toincrease the probability of transition. The peaks happen at different sets of values of the plottingparameters, such that the coupling parameter γc is higher for delayed time t . At room temperature, themaximum of the probability of transition spreads over a larger range of the plotting parameters andat lower temperature the peaks get sharp and thus are defined by a narrow range of the parameters.In the range of large values of γc , the peaks, at room temperature, rapidly get flat, however, theprobability of transition does not go abruptly to zero. In either case, the environment assists thetransition probability, however, its maximum value is smaller at room temperature than its valueat lower temperature. As before, for more clarity, we include the contour map in the inset of eachfigure.

In Fig. 4(a), we show the behavior of the probability of transition between the levels of the dimerwhen it is coupled to the global environment with γM1 = 0. In this case, the peaks occur at strongercoupling between the environment and the system and are sharp enough. Themaximum is not as highas in the absence of multilocal environments. The situation when all the environments are coupledwith the levels of the dimer is shown in Fig. 4(b). One can see that the maximum is strongly reducedin this case. Moreover, if the temperature of all the baths are set to 77 K, the probability of transitiongoes to zero for the chosen values of other parameters. The inset in each figure is the contour map ofthe data of the corresponding surface plot. It is important to note that all the peaks that correspond to

8 S. Khan et al. / Annals of Physics 341 (2014) 1–11

a b

Fig. 3. (Color online) The transition probability is plotted against the coupling parameter γc (ps−1)with collective environmentand time t (ps). The temperature of the baths in Fig. 2(a) is 77 K, and in Fig. 2(b) it is 300 K. The other parameters for both figuresare set to, N ′

c = 4, N ′= 20, E2 − E1 = 20 ps−1 , J = 10 ps−1 , αc = 250 ps−1 .

a b

Fig. 4. (Color online) The transition probability is plotted against the coupling parameter γc (ps−1)with collective environmentand time t (ps). (a) The coupling of level |1⟩ with its local environment is switched off. The other parameters for both figuresare set to, NM2 = J = 10 ps−1 , α2 = αc = 300 ps−1 , γM1 = γM2 = 2 ps−1 , N ′

c = 4, N ′= 20, E2 − E1 = 20 ps−1 , J = 10,

T2 = Tc = 77 K. (b) All the environments are on. The additional parameters are set toα1 = 300 ps−1 , γM1 = 2 ps−1 , T1 = 300 K,NM1 = 2.

the increased transition rate in all of the above cases occur on time scales of the order of a fewhundredfemtosecond, which corresponds to experimental observation [8,9] and theoretical predictions[15].

It is quite impractical to completely abandon inter-environments interactions, therefore, wealso consider the effects of such interactions on the probability of transition. However, includingall possible interactions between the environments will, mathematically, make the solution of theproblem very complicated. To avoid such mathematical difficulties, we consider the case in whichonly the multilocal environments are coupled with each other through an Ising like interaction andhave no interaction with the collective environment. Under this condition the Hamiltonian HM of themultilocal environments takes the form [19]

HM = aM1SzM1 + aM2SzM2 + qSzM1SzM2, (25)

S. Khan et al. / Annals of Physics 341 (2014) 1–11 9

where q is the coupling strength between the multilocal environments. For nonzero temperature, theinitial density matrix of the environment can be written as

ρB(0) =1

ZcZMexp

−

n

βn(αnSzn) + qβMSzM1SzM2

, (26)

where ZM is the partition function of the correlated multilocal environments and is given by

ZM =

NMi/2jMi=0

jimMi=−ji

ν(NM1, jM1)ν(NM2, jM2) ⟨jM1,mM1| exp[−βMαM1SzM1] |jM1,mM1⟩

× ⟨jM2,mM2| e−βMαM2SzM2 |jM2,mM2⟩

× ⟨jM1,mM1| ⟨jM2,mM2| e−βMqSzM1SzM2 |jM1,mM1⟩ |jM2,mM2⟩ . (27)

Note that in Eq. (27), we have taken βM = βM1 = βM2. The final density matrix ρ(t) can be obtained,as before, by using Eq. (18) with ρB(0) given by Eq. (26). It is straightforward to find the probability oftransition by using Eq. (19) with U(t) = exp(−iHM t), where HM is the Hamiltonian given by Eq. (25).After carrying out the trace operation, the probability of transition simplifies to the following form

P1→2(t) =

Nn/2jn=0

jnmn=−jn

ν(NM1, jM1)ν(NM2, jM2)ν(Nc, jc)C2 (t)

×1ZM

⟨jM1,mM1| e−βMαM1SzM1 |jM1,mM1⟩

× ⟨jM2,mM2| e−βMαM2SzM2 |jM2,mM2⟩ ⟨jM1,mM1|

× ⟨jM2,mM2| e−βMqSzM1SzM2 |jM1,mM1⟩ |jM2,mM2⟩

×1Zc

⟨jc,mc | e−βcαcSzc |jc,mc⟩ C∗

2 (t) ⟨2| U(t) |1⟩ ⟨1| UĎ(t) |2⟩ . (28)

With the help of Eq. (23) we can rewrite Eq. (28) in the following form

P1→2(t) =1

ZcZM

Nn/2jn=0

jnmn=−jn

ν(NM1, jM1)ν(NM2, jM2)ν(Nc, jc)

× exp

−βM

i

αMimMi + qmM1mM2

− βcαcmc

×J2

J2 +∆mMi −

γc2 (2m′

c − mc)2 sin2

t

J2 +

∆mMi −

γc

2(2m′

c − mc)2

, (29)

where use of the following relation is made

|C2 (t)|2 =J2

J2 +∆mMi −

γc2 (2m′

c − mc)2 sin2

t

J2 +

∆mMi −

γc

2(2m′

c − mc)2

.

In Eq. (29) all the terms havemeanings unchanged. The influence of correlation between the two localenvironments on the probability of transition in the presence of collective environment is shown inFig. 5 for two different temperatures of the baths. Fig. 5(a) shows the situation when all the baths areat 77 K and Fig. 5(b) represents the case for 300 K of the baths. For lower temperature of the baths(77 K), the probability of transition goes as high as 0.996 and at 300 K temperature of the baths itdrops to 0.97, which is still higher than when the collective environment is absent [19]. At low baths’temperature, the probability of transition oscillates and the amplitude damps as the coupling strengthincreases. Whereas at high baths’ temperature, the oscillatory behavior is not persistent and dies outsoon. The behavior can more clearly be observed in the corresponding contour map given in the insetof each figure.

10 S. Khan et al. / Annals of Physics 341 (2014) 1–11

a b

Fig. 5. (Color online) The transition probability is plotted against the coupling parameter γc (ps−1)with collective environmentand the coupling parameter q (ps−1) between themultilocal environments. The coupling of level |1⟩with its local environmentis switched off. The other parameters for both figures are set to, NM1 = 22, NM2 = Nc = 20, J = 10 ps−1 , t = 0.155 ps αM1 =

αM2 = αc = 250 ps−1 , γM1 = γM2 = 0.8 ps−1 , N ′c = 6, N ′

= 20, E2 − E1 = 20 ps−1 .

4. Summary

In summary,wehave investigated the energy transfer between the levels of a dimer in the presenceof different spin environments. We have considered the simplest, among the many possible, ways ofinteractions between the environment and the levels of the dimer. Generally saying, it is shown thatin the presence of collective environment with biologically applicable parameters ranges and timescales, both ways of coupling, the correlated and uncorrelated multilocal environment, assists thetransfer of energy in the dimer system. In particular, the increase in probability of transition stronglydepends on the temperature of the environment as well as on the mutual coupling between theenvironments. It is shown that at higher temperature, stronger coupling between the levels of thedimer and the environment is needed for maximizing the probability of transition. In the absenceof inter-environment coupling, the global environment, although assists the transport of energy,however, the maximum of transition of probability is considerably reduced. The quantum transportis considerably enhanced when the inter-environment coupling is present.

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