Delocalization and Quantum Entanglement in Physical Systems - The

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Delocalization and Quantum Entanglement in Physical Systems Rajesh Dutta, and Biman Bagchi J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00382 • Publication Date (Web): 04 Apr 2019 Downloaded from http://pubs.acs.org on April 4, 2019

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The Journal of Physical Chemistry Letters

Delocalization and Quantum Entanglement in Physical Systems Rajesh Dutta1 and Biman Bagchi1,* 1SSCU,

Indian Institute of Science, Bangalore 560012, India.

*Email: [email protected], [email protected]

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Abstract Quantum coherence and entanglement in an extended interacting system where energy levels are non-degenerate and coupled to a dissipative environment is a common occurrence in nature, like in photosynthetic reaction systems and conjugated polymers. Temperature dependence of quantum coherence in trimer complex (first three subunits of Fenna-Matthews-Olson (FMO) complex) is studied using a temperature dependent Quantum Stochastic Liouville equation. In the non-Markovian limit, the lowering of temperature induces long-lasting quantum coherence which in turn leads to delocalization whose length grows in the non-Markovian limit. Measure of entanglement and coherence length determines the nature of dynamic localization.

TOC Graphic

KEYWORDS: Excitation energy transfer, Non-vanishing quantum entanglement, Dynamic localization,

Coherence

length,

coherence

in

equilibrium

and

excited

bath

states.


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The existence of long lived quantum coherence in noisy environments has attracted the attention of scientists in the area of quantum computation and information. In theoretical descriptions, quantum coherence is described by the off-diagonal elements of density matrix which arise from the superposition of the states. Our main concern is the role of quantum coherence for the excitation energy transfer (EET) in noisy but spatially correlated environments. Here we study long lived quantum coherence in terms within a consistent dynamical disorder model. By definition, when two particles are entangled they continue to share information no matter how far apart they become in space as well as time. The field started from the well-known EPR paradox developed by Einstein and co-workers1. They concluded that the quantum theory is incomplete and should be extended with local hidden variable and termed the non-classical behavior as “spooky action at a distance”. At the similar time, Schrodinger2 described the behavior as “the characteristics trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought”. Later the development of Bell theorem3 rules out the thought of local hidden variable. The violation of Bell’s inequality due to quantum entanglement that can’t be represented by any version of the classical picture of physics opened a new door to the experimental verification of quantum physics. In recent times, observation of coherent energy transfer in photosynthetic4-8 and conjugated polymer9,10 complex has motivated recent studies to investigate electronic energy transfer from the perspective of quantum entanglement. Entanglement is a quantum correlation between different parts of a quantum system. If a quantum state does not satisfy the separability criteria, it is said to be entangled. For photosynthetic systems the sites can be addressed individually then one can speak about site-entanglement, i.e., quantum correlations across distinguishable locations. Quantum superpositions of the states provide non-trivial quantum features such as coherence and 3 ACS Paragon Plus Environment


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entanglement. Quantum coherence is related to the interference effects whereas entanglement is subjected to non-classical correlations between the subsystems. Presence of coherence is necessary but not the sufficient condition for entanglement. Since all the off-diagonal elements of the density matrix are involved, quantum coherence and entanglement are related. The experiments to extract the elements of density matrix is not well established and yet under development. However if one assumes the single excitation approximation, coherence and entanglement are equivalent. In this scenario, quantum entanglement is the quantum delocalized states which is the superposition of the wave packets and could be spectroscopically detectable via the calculation of energy eigen states of the system. In this work we calculate quantum entanglement in single exciton manifold in the site basis which refers the non-local quantum correlations between the electronic states of the spatially correlated chromophores of FMO complex.11 Despite recent advances in EET in photosynthetic complex and conjugated polymer, the role of quantum coherence and entanglement in presence of dynamic disorder in EET process remains as an unsettled issue. In this work, we consider dimer correlated bath and trimer uncorrelated or independent bath for first three sub units of FMO complex. For the correlated bath case, all the fluctuations are spatially correlated all the times. For uncorrelated bath, site energy and coupling fluctuations remain independent all the times. Correlated bath case mimics the excitation transfer at low temperature where bath is correlated over large distance. However, the completely correlated bath model may not be applicable for real photosynthetic complexes. We calculate quantum coherence and entanglement to observe their contribution in EET process. We explore the role of fluctuation strength and fluctuation rate in the propagation of coherence and entanglement. Temperature independent study may not capture the non-vanishing 4 ACS Paragon Plus Environment

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coherences and entanglement in the long time limit. We compare the coherences and entanglement at different temperatures with temperature independent results. To observe dynamic localization we calculate coherence length. The schematic picture of our model is provided as

Figure 1. Conjugated polymer and photosynthetic complexes can be modeled as a series of two level systems.

Coherence is defined for correlated bath case as follows

Coherence  k  m l

(1)

where  is the reduced density matrix (Eq. 9) and suffix “m” can take 0 and 1 value for two level Poisson bath. For Gaussian bath, however, it can take values from 0 to .12 To quantify quantum



entanglement we calculate concurrence.13

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For bipartite entanglement between two sites,

concurrence in single exciton manifold is defined for correlated bath as12,

Cklm  2 k  m l

(2)

where, concurrence is the absolute value of coherences in site representation.  is the reduced density matrix. In order to obtain the reduced density matrix σ, we employ the well-known Haken-StroblReineker-Silbey14-16 exciton Hamiltonian,

H tot  H S  H B  H int

(3)

where system (exciton) Hamiltonian is defined as

H S   Ek k k

where Ek is

the

k   J kl k

energy

l

k ,l k l

of

. an

exciton

(4) localized

at

site

k

and

J kl

is

the

time-independent off-diagonal interaction between excitations at site k and l. We assume that bath is a collection of harmonic oscillators

 p 2j 1  HB    m j 2j x 2j    2 j  2m j 

(5)

If system-bath interaction Hamiltonian is assumed to be given as H int  VX where, V consists of system part and X is a collective bath variable, X   c j x j , then one can use Feynman-Vernon j

influence functional17 to eliminate X from the total Hamiltonian to write it as


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H tot  H S  V (t )

(6)

Feynman-Vernon allows for a procedure to obtain V(t). When bath is a collection of harmonic oscillators and the coupling is linear, V(t) can be approximated as a Gaussian random variable. In the next step, a joint probability distribution is defined in system and bath variables as follows P   , V, t        (t )    V  V(t )  .

(7)

where, P is the joint probability and V is the random energy variable. The equation of motion of P

is

given

by

Kubo’s

quantum

stochastic

Liouville

equation18,19

i   P   ,V , t    H tot ,   P  V P t h 

(QSLE) (8)

where V is a stochastic diffusion operator. For Gaussian bath it is a Fokker-Planck operator. (Supporting information SI) We next define a reduced QSLE by averaging the density matrix as,

 (t )   d   P   , V, t 

(9)

By recombination of Eq. (8) and (9) and followed by integration by parts, the QSLE for the full density matrix reduces to

i     H (t ),     V t h

(10)

The temperature corrected QSLE was derived by Tanimura and Kubo20 using dynamical approach. The equation of motion (EOM) in reduced density matrix can be given as,



  V,t   i     i b    o    H (t ) x  b V   V   V   V,t  t V  V  2  V    h

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(11)

The non-trivial nature of Eq. (11) precludes an easy analytical solution but one can follow Kubo's method of expansion of the RDM in the eigen-states bm of the bath operator  V with  m as the expansion coefficient,     m bm which leads to a hierarchical equation of motion as, m

 m i iV VB o   H exx  m  V x  m 1   m 1   V  m 1  mB m t h h 2

where,  

o 1 , anti-commutator O f k BT

(12)

 Of  fO and commutatror O x f  Of  fO . B is the

eigen value of stochastic diffusion operator and gives the decay rate whereas V designates fluctuation strength. For a two level Poisson bath stochastic diffusion operator is a 22 matrix with eigen value 0 and –B. For Gaussian bath stochastic diffusion operator is Fokker-Planck operator with eigen values –mB.21-23 (SI) Now one needs to calculate coherence length to observe the dynamical localization process in terms of population and coherences. Coherence length is defined for trimer system with uncorrelated bath as24 2

 N      kmnpl  k ,l m, n, p  LC   2 N

N



(13)

 kmnpl

k ,l m, n, p

where, k and l is the site number with maximum value N. As we consider trimer complex with uncorrelated bath, there are three independent indices m, n and p indicating the three bath states. 8 ACS Paragon Plus Environment

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In the high temperature limit, populations of all the sites are equal and there is no non-vanishing coherence in long time limit. Consequently in the high temperature limit coherence length attains the value 1. However in the opposite limit i.e. in the absence of bath fluctuations coherence length attains the limiting value N. Recent interest in EET process and role of quantum coherence in photosynthetic complex was prompted by the pioneering experiment of 2D Fourier transform electronic spectroscopy, Fleming and co-workers4 observed long lived coherent EET in photosynthetic complexes. Colliniand Scholes9,10 observed coherent nature of intra-chain EET in conjugated polymer even at room temperature. Later Ishizaki and Fleming5,6 investigated the role of environment fluctuation in energy transport in FMO complex. Another eminent approach is based on polaron transfer technique developed by Jang and co-workers25 to investigate the EET dynamics. Aspuru-Guzik and co-workers26 explored environment assisted quantum transport in real photosynthetic complex. Cao, Silbey and co-workers27,28 computed the efficiency of exciton transfer in case of dimer model system and population relaxation for FMO complex. Earlier we also investigated several aspects of latter’s quantum dynamics.29-31 Recently Plenio and co-workers32 studied entanglement for photo-synthetic FMO complex through the entanglement measure, logarithmic negativity, using Lindblad equation. Fassioli and Olaya-Castro33 investigated the relationship between the quantum yield of a light-harvesting complex and the distribution of entanglement. Fleming and co-workers34,35 observed nonvanishing entanglement in photosynthetic complex (LHCII and FMO complex) even at very long time and also mentioned that in addition with bipartite entanglement a small amount of multipartite entanglement might be present there.



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We now proceed to explicit numerical calculation of the hetero trimer. Here we use system parameters from FMO Hamiltonian (SI). We have numerically solved the coupled EOM using Runge-Kutta fourth order method. Coherence can be represented for uncorrelated bath model as follows,

Coherence (k , l )  k 

N

 ai

l

(14)

i 1

where, k and l are the site number, N is the total number of sites. The indices a1 , a2 , a3 ,....aN describes independent environment around each chromophore. In this case we consider trimer complex and consequently we have three indices a1.a2.a3. To describe excitation transfer we consider Gaussian bath for trimer complex by taking four bath states. Our investigations showed that there could be small changes in the amplitude of oscillation after 4th bath states but the nature of coherences will remain the same The expression of coherence in EBS and ExBS for dimer correlated bath model with no site energy heterogeneity for temperature independent case is obtained analytically and is given by,   C  Bod  J 1  0 2  sin 2 t exp   t  h  2  2 

2iVodC h

1 1 2 

B 

C 2 od

 16

VodC 

2

C Bod

B 

C 2 od

V   16

C 2 od

sinh

B 

C 2 od

V   16

C 2 od

h2

2

1 t  cosh 2

B 

C 2 od

C  Bod  J cos 2 t exp   t  sinh h  2 

B  C od

V   16

C 2 od

h2

2

t

(15)

h2


C 2 od

h2

2

h2

2

V   16

   t   

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Here, VodC and BodC denote the off-diagonal fluctuation strength and rate of fluctuation for correlated bath case. 1  0 2 indicates coherence in equilibrium bath states and

1 1 2

designates

coherence in excited bath states between site 1 and 2 respectively. After initial coherent or oscillatory dynamics at high value of inverse of bath correlation time and at long time limit coherence vanishes and we obtain classical incoherent hopping motion. We obtained the expression for coherence analytically from the temperature corrected EOM as follows,





C VC 1  1 2   od 1  e  Bod t  2k BT

2iVodC h

B 

C 2 od

 16

V 

C 2 od

 BC J cos 2 t exp   od h  2

 t  sinh 

B 

C 2 od

V   16

C 2 od

h2

2

t

h2

(16) There is no effect of temperature in the propagation of quantum coherence in equilibrium bath states. Temperature correction enters through the real part of coherence in excited bath states (ExBS) only. We could also calculate coherence analytically for the dimer system with only off-diagonal dynamic disorder. Numerically, we evaluate coherences from both temperature dependent and independent QSLE for uncorrelated bath model. At high temperature limit, these two merge into each other. We plot absolute value of coherences in strong and weak coupling limit by varying the diagonal fluctuation strength and bath correlation time for uncorrelated bath case as VdUC and

B 

UC 1 d

. Here C0 denotes absolute value of coherence in equilibrium bath states and C1represents

absolute value of coherence in excited bath states. 11 ACS Paragon Plus Environment


0.5

0 C13

(Temp. Indep)

0

C12 (200K)

0.3 0.2

0 C13

(200K)

0 C12

(300K)

0 C13

(300K)

0.1 0 0

0.5

1 1.5 Time (ps)

2

1

C12 (Temp. Indep)

(b)

Coherence

Coherence

0.08

0

C12 (Temp. Indep)

(a)

0.4

1

C13 (Temp. Indep)

0.06

1

C12 (200K) 1

0.04

C13 (200K)

0.02

C13 (300K)

1

C12 (300K) 1

0 0

2.5

0.5

1 1.5 Time (ps)

2

2.5

Figure2.(a)-(b) depict absolute value of coherence in equilibrium bath states and excited bath states





UC 1

for trimer system at fluctuation strength VdUC = 50 cm-1 and bath correlation time Bd

= 10 fs.

Solid line indicates results obtained from temperature independent calculation. Dashed line designates absolute value of coherence at 200K. Dotted line indicates absolute value of coherence at 300K.0 and 1 indicate equilibrium and excited bath respectively.

In Figure 2 the temperature dependency of absolute values of coherences is plotted indifferent regimes. In the non-Markovian limit the decay of coherence is slow and the dynamics is oscillatory. With decrease in temperature the decay further slows down, for both coherence in EBS and ExBS. However, the effect of temperature is more prominent for coherence between nearest neighboring sites than non-neighbors.

0.3 0.2

0.2

0

C12 (Temp. Indep.) 0

C13 (Temp. Indep.) 0

C12 (200K) 0

C13 (200K) 0

0.1 0 0

C12 (300K) 0

C13 (300K)

0.5 1 Time (ps)

1.5

1

C12 (Temp. Indep)

(b)

1

C13 (Temp. Indep)

0.15 Coherence

(a) (a)

Coherence

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1

C12 (200K) 1

C13 (200K)

0.1

1

C12 (300K) 1

C13 (300K)

0.05 0 0

ACS Paragon Plus Environment

0.5

1 Time (ps)

12

1.5

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Figure3.(a)-(b) depict absolute value of coherence in equilibrium bath states and excited bath states





UC 1

for trimer system at fluctuation strength VdUC = 350 cm-1 and bath correlation time Bd

= 10 fs.

Solid line indicates results obtained from temperature independent calculation. Dashed line designates absolute value of coherence at 200K. Dotted line indicates absolute value of coherence at 300K.0 and 1 indicate equilibrium and excited bath respectively. At high temperature limit temperature independent and dependent case show similar behavior.

In Figure 3, we plot the absolute value of coherence in the weak coupling-Markovian regime. In this case, we observe pronounced temperature effect in coherence between nearest neighbor sites. In weak coupling-Markovian limit we observe non-oscillatory decay of coherence. We also observe that with increase in the number of EBS the contribution of coherences towards the EET dynamics decreases. Figure 2 (a) and (b) show that coherence in equilibrium bath state (EBS) is more prominent than that of coherence in excited bath state (ExBS). However, in case of Figure 3, the contributions towards the excitation transfer dynamics are more or less the same. The coherence in ExBS directly governs by the value of bath correlation time. In Figure 2, the bath correlation time is small and consequently the coherence in ExBS contain negligible contribution. However, the coherence in EBS is mainly dictated by the value of fluctuation strength. When fluctuation strength is large, contribution of coherence in EBS towards the dynamics decreases (Figure 3). However, the contribution of coherence in ExBS increases as the bath correlation time increases. As a result coherence in both EBS and ExBS show similar behaviour in weak coupling limit. This can be explained qualitatively using Eq. (15) and Eq. (16) as well.



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Coherence in the eigen basis behaves differently from that of coherence in the localized site basis. In the eigen basis the oscillation in coherence is less and the latter decays more rapidly than that of coherence in site basis. In the long time limit coherence in eigen basis vanishes. However, the coherence in site basis shows finite value. (SI) Increase in temperature helps in transition from coherent to incoherent state and vice versa. Temperature acts as an effective decay term in the excitonic equation of motion. Lowering temperature helps in preserving phase relation between excitonic states for long time. In long time limit when oscillation disappears, low temperature helps in slow decay which essentially indicates localization of the energies on corresponding site. Coherences between non-local sites create interference between pathways of energy transfer. Non-local coherences open up new channels for energy transfer and facilitate energy transfer dynamics. For our model system the non-local coherence leads to energy transfer from site 1 to site 3 by avoiding the barrier i.e. site 2. When fluctuation strength (V) and fluctuation rate (B) are large and the ratio V2/B is greater than 2J, the oscillation vanishes, in this limit for uncorrelated bath case, coherence in EBS decays much faster than coherence in equilibrium bath state. However, in intermediate limit that is appropriate for photosynthetic and conjugated systems, coherence in EBS and coherence in equilibrium bath states play equally important roles. Temperature independent QSLE provides an equal population of all sites in long time limit which indicate the vanishing of quantum entanglement in long time limit. However, temperature dependent QSLE shows that unequal population leads to non-vanishing quantum entanglement at long time limit. 14 ACS Paragon Plus Environment

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Although the quantum entanglement in the single-exciton manifold is conceptually the same as the quantum delocalization caused by the excitonic coupling, investigations with tools quantifying the entanglement provide us with some insights into quantum delocalized states found in photosynthetic EET problems. Eigen vectors for trimer system

 1  0.0761  0.1152  0.9913  2  0.8851  0.4512  0.123  3  0.461  0.8852  0.0673

(17)

The EET process occurs in the presence of fluctuating environment. Now to observe the dynamical localization process one can not use the definition of inverse participation ratio in terms of coefficient of eigen states because the dynamical interaction between the system and environment changes the delocalized states.

3.5

3 2.5

Temp. Indep.

(a)

3

300K

Coherence Length

Coherence Length

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


200K

2 1.5 1 0.5 0 0

Temp. Indep. 300K 200K

(b)

2.5 2 1.5 1 0.5

0.5

1 1.5 Time (ps)

2

2.5

0 0

0.5



UC

Figure 4.depictcoherence length for trimer system at (a) VdUC = 50 cm-1 and Bd 350 cm-1 and

B 

UC 1 = d

1 1.5 Time (ps) 1



2

= 10 fs. (b) VdUC =

10 fs. Blue line indicates results obtained from temperature independent


2.5


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calculation. Green, Red and Blue line designates coherence length at 200K, 300K and for temperature independent case respectively.

Figure 4 describes the dynamic localization-delocalization in strong coupling non-Markovian and weak coupling Markovian limit. For both the cases with decreasing temperature the phase relation between the states increase which essentially leads to the delocalization. At the complete delocalization limit the expected value of coherence length is 3 for trimer system. However, in the opposite limit when coherence in the long time limit attains zero value, the expected coherence length should be 1. At the short time limit and strong coupling non-Markovian limit the coherence length exceeds 2.5 which indicate the coherent transport and complete delocalization. In case of weak coupling Markovian limit initially the coherence length surpasses 2 for temperature dependent case. From both the figures it is clear that even at long time the value of coherence length is greater than unity for temperature dependent case. This behavior signifies finite coherence even in the long time limit. For temperature independent case in the long time limit coherence vanishes and all the sites acquire equal population at steady state. As a result the coherence length attains unity. With lowering the time independent off-diagonal coupling or intersite coupling we obtain the unit value of coherence length which essentially signifies the localization of excitation energy. Below we summarize the main features of our study (1) In the non-Markovian limit we observe coherent dynamics and transition from coherent to incoherent dynamics while going from non-Markovian to Markovian limit. We analytically show for correlated bath model coherence is propagated through ExBS which only contains non-vanishing term even in long time limit in absence of site energy heterogeneity (SEH).


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(2) In strong coupling and non-Markovian limit coherence in EBS is more dominant than coherence in ExBS. However, time scale of decay is more or less the same. In the Markovian limit non-oscillatory dynamics corresponds to incoherent EET dynamics. (3) Non-local coherences between non-neighboring sites lead to creation of new pathways of energy transfer. Non-local coherence helps to overcome the energy barrier thus facilitating energy transfer dynamics. Double of the absolute coherence is known as concurrence which is an entanglement measure for single exciton manifold. (4) Non-local coherences show weak dependency of temperature. Temperature dependent study shows non-vanishing quantum entanglement even in long time limit. No such non-vanishing entanglement is observed in case of temperature independent case. Initially the rapid increase of concurrence is due to a quick delocalization of excitation. (5) For both non-Markovian and Markovian limit and temperature dependent case coherence length overcome the value 2.5 which indicates complete delocalization in the short time limit and dynamics is coherent. As finite amount of coherence survives in the long time limit for temperature dependent case, there exists delocalization at least in the nearest neighbor level. However, for the temperature independent case coherence vanishes in the long time and all the sites acquire equal population at equilibrium. This indicates the localization of excitation and the EET process occurs via completely hopping or incoherent mechanism. Briggs and Eisfeld36 studied excitation transfer in FMO complex using quantum and classical dynamics. They observed that in realistic coupling regime classical and quantum dynamics provide more or less similar results. However, in the strong coupling limit classical and quantum dynamics are quite different. The study of Briggs and Eisfeld was carried out in the absence of the environment. The dynamics are substantially different in the presence of fluctuating environment. 17 ACS Paragon Plus Environment


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Finally, we observe that this non-perturbative hierarchical EOM can capture the quantum entanglement for correlated and uncorrelated environments in strong, weak and intermediate coupling limit. The nature of entanglement and the coherence length provide us information about dynamical localization process of EET dynamics. In single exciton manifold one can measure the entanglement by the calculation of energy eigen states of the system via non-linear femtosecond spectroscopic techniques. We have chosen the parameters such that we can explore all the limits. One can obtain functional form of bath spectral density by fitting the experimental data37. Now using the function one get the relaxation kernel for bath. From the relaxation kernel one can obtain fluctuation strength and bath correlation time from pre-factor and time constant respectively. One can also obtain the parameters from simulation38 by using the same technique.

ACKNOWLEDGMENTS BB thanks Department of Science and Technology (DST, India) and Sir J. C. Bose fellowship for providing partial financial support. RD thanks Mr. Saumyak Mukherjee for the help in making figure.

Supporting Information Available Temperature independent and dependent quantum stochastic Liouville equation and diffusion operator for Poisson and Gaussian stochastic process and a table for parameters for the system Hamiltonian and coherence in eigen basis are plotted.

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Delocalization and Quantum Entanglement in Physical Systems - The

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