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Letter Cite This: J. Phys. Chem. Lett. 2019, 10, 2037−2043

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Delocalization and Quantum Entanglement in Physical Systems Rajesh Dutta and Biman Bagchi* SSCU, Indian Institute of Science, Bangalore 560012, India

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S Supporting Information *

ABSTRACT: Quantum coherence and entanglement in an extended interacting system where energy levels are nondegenerate and coupled to a dissipative environment is a common occurrence in nature, like in photosynthetic reaction systems and conjugated polymers. The temperature dependence of quantum coherence in a trimer complex (first three subunits of the Fenna−Matthews−Olson complex) is studied using a temperaturedependent quantum stochastic Liouville equation. In the non-Markovian limit, the lowering of temperature induces long-lasting quantum coherence that, in turn, leads to delocalization, whose length grows. The entanglement and coherence length determine the nature of the dynamic localization.

T

across distinguishable locations. Quantum superpositions of the states provide nontrivial quantum features such as coherence and entanglement. Quantum coherence is related to the interference effects, whereas entanglement is subjected to nonclassical correlations between the subsystems. The presence of coherence is necessary but is not a sufficient condition for entanglement. Because all of the off-diagonal elements of the density matrix are involved, quantum coherence and entanglement are related. The experiments to extract the elements of the density matrix are not well established and are still under development. However, if one assumes the single excitation approximation, then coherence and entanglement are equivalent. In this scenario, quantum entanglement is the quantum delocalized states that is the superposition of the wave packets and could be spectroscopically detectable via the calculation of energy eigenstates of the system. In this work, we calculate quantum entanglement in the single-exciton manifold in the site basis, which refers to the nonlocal quantum correlations between the electronic states of the spatially correlated chromophores of the Fenna− Matthews−Olson (FMO) complex.11 Despite recent advances in EET in the photosynthetic complex and conjugated polymer, the role of quantum coherence and entanglement in the presence of dynamic disorder in the EET process remains an unsettled issue. In this work, we consider dimer correlated bath and trimer uncorrelated or independent bath for the first three subunits of the FMO complex. For the correlated bath case, all of the fluctuations are spatially correlated at all times. For the uncorrelated bath, site energy and coupling fluctuations remain independent at all

he 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 the density matrix that arise from the superposition of the states. Our main concern is the role of quantum coherence in the excitation energy transfer (EET) in noisy but spatially correlated environments. Here we study long-lived quantum coherence in terms of 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 Einstein−Podolsky−Rosen (EPR) paradox developed by Einstein and coworkers.1 They concluded that the quantum theory is incomplete and should be extended with local hidden variables, and they termed the nonclassical behavior as “spooky action at a distance”. At the similar time, Schrödinger2 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 the Bell theorem3 ruled out the thought of a local hidden variable. The violation of Bell’s inequality due to quantum entanglement that cannot 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, the observation of coherent energy transfer in photosynthetic4−8 and conjugated polymer9,10 complexes has motivated 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, then it is said to be entangled. For photosynthetic systems, the sites can be addressed individually; then, one can speak about site entanglement, that is, quantum correlations © 2019 American Chemical Society

Received: February 10, 2019 Accepted: April 4, 2019 Published: April 4, 2019 2037

DOI: 10.1021/acs.jpclett.9b00382 J. Phys. Chem. Lett. 2019, 10, 2037−2043

Letter

The Journal of Physical Chemistry Letters

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

where Ek is the energy of an exciton localized at site k and Jkl is the time-independent off-diagonal interaction between excitations at sites k and l. We assume that the bath is a collection of harmonic oscillators

times. The correlated bath case mimics the excitation transfer at low temperature, where the bath is correlated over a large distance. However, the completely correlated bath model may not be applicable to real photosynthetic complexes. We calculate quantum coherence and entanglement to observe their contribution in the EET process. We explore the role of the fluctuation strength and the fluctuation rate in the propagation of coherence and entanglement. The temperatureindependent study may not capture the nonvanishing 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 the coherence length. The schematic picture of our model is provided as Figure 1. Coherence is defined for the correlated bath case as follows coherence = ⟨k|σm|l⟩

ij p2 yz j j z 1 HB = ∑ jjjj + mjωj2xj2zzzz j 2mj z 2 j k {

If the system−bath interaction Hamiltonian is assumed to be given as Hint = −VX, where V consists of the system part and X is a collective bath variable, X = ∑j cjxj , then one can use the Feynman−Vernon influence functional17 to eliminate X from the total Hamiltonian to write it as Htot = HS + V (t )

where σ is the reduced density matrix (RDM) (eq 9) and the suffix “m” can take a 0 or 1 value for the two-level Poisson bath. For the Gaussian bath, however, it can take values from 0 to ∞.12 To quantify quantum entanglement, we calculate concurrence.13 For bipartite entanglement between two sites, the concurrence in the single-exciton manifold is defined for the correlated bath as12

P(ρ , V , t ) = ⟨δ(ρ − ρ(t ))δ(V − V (t ))⟩

where concurrence is the absolute value of coherences in the site representation. σ is the RDM. To obtain the RDM, σ, we employ the well-known Haken− Strobl−Reineker−Silbey14−16 exciton Hamiltonian

∂ i ∂ P(ρ , V , t ) = [Htot , ρ]P + ΓV P ∂t ℏ ∂ρ

∑ Ek|k⟩⟨k| + ∑ Jkl |k⟩⟨l| k

k ,l k≠l

(8)

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

(3)

where the system (exciton) Hamiltonian is defined as HS =

(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 equation (QSLE)18,19

(2)

Htot = HS + HB + Hint

(6)

The Feynman−Vernon influence functional allows for a procedure to obtain V(t). When the 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

(1)

Cklm = 2|⟨k|σm|l⟩|

(5)

σ (t ) =

(4) 2038

∫ dρ ρP(ρ , V , t )

(9) DOI: 10.1021/acs.jpclett.9b00382 J. Phys. Chem. Lett. 2019, 10, 2037−2043

Letter

The Journal of Physical Chemistry Letters

by Jang and coworkers25 to investigate the EET dynamics. Aspuru-Guzik and coworkers26 explored environment-assisted quantum transport in a real photosynthetic complex. Cao, Silbey, and coworkers27,28 computed the efficiency of exciton transfer in the case of the dimer model system and population relaxation for the FMO complex. Previously, we also investigated several aspects of the latter’s quantum dynamics.29−31 Recently Plenio and coworkers32 studied entanglement for the photosynthetic FMO complex through the entanglement measure, logarithmic negativity, using the 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 coworkers34,35 observed nonvanishing entanglement in a photosynthetic complex (light-harvesting complex II (LHCII) and FMO complex) even at very long times and also mentioned that, in addition, with bipartite entanglement, a small amount of multipartite entanglement might be present there. We now proceed to explicit numerical calculation of the heterotrimer. Here we use system parameters from the FMO Hamiltonian (SI). We have numerically solved the coupled EOM using the Runge−Kutta fourth-order method. Coherence can be represented for the uncorrelated bath model as follows

By recombination of eqs 8 and 9, followed by integration by parts, the QSLE for the full density matrix reduces to ∂σ i = − [H(t ), σ ] + ΓV σ ∂t ℏ

(10)

The temperature-corrected QSLE was derived by Tanimura and Kubo20 using a dynamical approach. The equation of motion (EOM) in the RDM can be given as Ä ∂σ(V , t ) ÅÅÅÅ i ∂ i ∂ yz zz = ÅÅ− H(t )x − B jjjV + Å ∂t ∂V k ∂V { ÅÇ ℏ É iβB ij ∂ yz oÑÑÑÑ jjV + zzV ÑÑσ(V , t ) − ∂V { ÑÑÖ 2 k (11) The nontrivial nature of eq 11 precludes an easy analytical solution, but one can follow Kubo’s method of expansion of the RDM in the eigenstates |bm⟩ of the bath operator ΓV with σm as the expansion coefficient, σ = ∑m σm|bm⟩, which leads to a hierarchical equation of motion as ∂σm i iV = − Hexx σm − V x(σm + 1 + σm − 1) ∂t ℏ ℏ βVB o V σm − 1 − mBσm − 2

where β =

1 , kBT x

(12)

coherence (k , l) = ⟨k|σ ∏N ai|l⟩

anticommutator Oof = Of + f O, and

commutator O f = Of − f O. B is the eigenvalue of stochastic diffusion operator and gives the decay rate, whereas V designates the fluctuation strength. For a two-level Poisson bath, the stochastic diffusion operator is a 2 × 2 matrix with eigenvalues 0 and −B. For the Gaussian bath the stochastic diffusion operator is the Fokker−Planck operator with eigenvalues −mB (SI).21−23 Now, one needs to calculate the coherence length to observe the dynamical localization process in terms of population and coherences. The coherence length is defined for the timer system with an uncorrelated bath as24 LC

(∑ =

N k ,l

where k and l are the site numbers and N is the total number of sites. The indices a1, a2, a3, ..., aN describe the independent environment around each chromophore. In this case, we consider the trimer complex; consequently, we have three indices a1, a2, and a3. To describe excitation transfer, we consider the Gaussian bath for the trimer complex by taking four bath states. Our investigations showed that there could be small changes in the amplitude of the oscillation after four bath states, but the nature of coherences will remain the same The expression of coherence in the equilibrium bath state (EBS) and the excited bath state (ExBS) for the dimer correlated bath model with no site energy heterogeneity for the temperature-independent case is obtained analytically and is given by

2

∑m , n , p |σkmnpl|

)

N

N ∑k , l ∑m , n , p |σkmnpl|2

(14)

i=1

ÄÅ ÅÅ Å C Å C Bod J jij Bod zyzÅÅÅÅ z j t zzÅÅ ⟨1|σ0|2⟩ = sin 2 t expjj− Å ℏ (V C )2 C 2 k 2 {ÅÅÅÅ 2 (Bod ) − 16 od2 ÅÅÇ ℏ

(13)

where k and l are the site numbers with a maximum value N. As we consider the trimer complex with an uncorrelated bath, there are three independent indices m, n, and p indicating the three bath states. In the high-temperature limit, the populations of all of the sites are equal, and there is no nonvanishing coherence in the long time limit. Consequently, in the high-temperature limit, the coherence length attains the value 1; however, in the opposite limit, that is, in the absence of bath fluctuations, the coherence length attains the limiting value N. Recent interest in the EET process and the role of quantum coherence in the photosynthetic complex was prompted by the pioneering experiment of 2D Fourier transform electronic spectroscopy, in which Fleming and coworkers4 observed longlived coherent EET in photosynthetic complexes. Collini and Scholes9,10 observed the coherent nature of intrachain EET in conjugated polymers even at room temperature. Later, Ishizaki and Fleming5,6 investigated the role of environment fluctuation in energy transport in the FMO complex. Another eminent approach is based on the polaron transfer technique developed

sinh

C 2 (Bod ) − 16

C 2 (Vod ) 2

ℏ

2 C 2iVod ℏ

⟨1|σ1|2⟩ =

C 2 (Bod ) −

sinh

(V C )2 16 od2

C 2 (Bod )

ℏ

− 16 2

1 t + cosh 2

C 2 (Bod ) − 16

ij BC yz J cos 2 t expjjjj− od t zzzz ℏ k 2 {

2

C 2 (Vod ) 2

ℏ

ÑÉÑ ÑÑ ÑÑ Ñ t ÑÑÑÑ ÑÑ ÑÑ ÑÑ ÑÖ

C 2 (Vod )

ℏ2

t

(15)

Here and denote the off-diagonal fluctuation strength and rate of fluctuation for the correlated bath case. ⟨1|σ0|2⟩ indicates the coherence in the EBSs, and ⟨1|σ1|2⟩ designates coherence in the ExBSs between sites 1 and 2, respectively. After the initial coherent or oscillatory dynamics at a high value VCod

2039

BCod

DOI: 10.1021/acs.jpclett.9b00382 J. Phys. Chem. Lett. 2019, 10, 2037−2043

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The Journal of Physical Chemistry Letters

Figure 2. Absolute value of coherence in (a) equilibrium bath states and (b) excited bath states for the trimer system at fluctuation strength VUC d = −1 = 10 fs. The solid line indicates the results obtained from temperature-independent calculations. The 50 cm−1 and bath correlation time (BUC d ) dashed line designates the absolute value of coherence at 200 K. The dotted line indicates the absolute value of coherence at 300 K. 0 and 1 indicate equilibrium and excited bath, respectively.

Figure 3. Absolute value of coherence in (a) equilibrium bath states and (b) excited bath states for the trimer system at fluctuation strength VUC d = −1 = 10 fs. The solid line indicates results obtained from temperature-independent calculations. The 350 cm−1 and bath correlation time (BUC d ) dashed line designates the absolute value of coherence at 200 K. The dotted line indicates the absolute value of coherence at 300 K. 0 and 1 indicate equilibrium and excited bath, respectively. At the high-temperature limit, temperature-independent and -dependent cases show similar behavior.

dynamics is oscillatory. With a decrease in temperature, the decay further slows down for coherence in both the EBS and the ExBS. However, the effect of temperature is more prominent for coherence between nearest-neighbor sites than non-neighbors. In Figure 3, we plot the absolute value of coherence in the weak-coupling Markovian regime. In this case, we observe a pronounced temperature effect on coherence between nearestneighbor sites. In the weak-coupling Markovian limit, we observe the nonoscillatory decay of coherence. We also observe that with an increase in the number of EBSs, the contribution of coherences toward the EET dynamics decreases. Figure 2a,b shows that coherence in the EBS is more prominent than that in the ExBS. However, in the case of Figure 3, the contributions toward the excitation-transfer dynamics are more or less the same. The coherence in the ExBS is directly governed by the value of bath correlation time. In Figure 2, the bath correlation time is small; consequently, the coherence in the ExBS contains a negligible contribution. However, the coherence in the EBS is mainly dictated by the value of fluctuation strength. When the fluctuation strength is large, the contribution of coherence in the EBS toward the dynamics decreases (Figure 3). However, the contribution of coherence in the ExBS increases as the bath correlation time increases. As a result, coherence in both the EBS and the ExBS shows similar behavior in the weak coupling limit. This can be explained qualitatively using eqs 15 and 16 as well. Coherence in the eigenbasis behaves differently from coherence in the localized site basis. In the eigenbasis, there is less oscillation in coherence, and the coherence decays more rapidly than that in the site basis. In the long time limit,

of inverse of bath correlation time and at long time limit coherence vanishes, we obtain a classical incoherent hopping motion. We obtained the expression for coherence analytically from the temperature-corrected EOM as follows C VC ⟨1|σ1|2⟩ = − od (1 − e−Bod t ) + 2kBT

C 2iVod ℏ

C 2 (Bod ) − 16

ij BC yz J cos 2 t expjjjj− od t zzzz sinh ℏ k 2 {

C 2 (V od )

ℏ2

C 2 (Bod ) − 16

2

C 2 (V od )

ℏ2

t

(16)

There is no effect of temperature on the propagation of quantum coherence in EBSs. The temperature correction only enters through the real part of coherence in ExBSs. 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 QSLEs for the uncorrelated bath model. At the high-temperature limit, these two merge into each other. We plot the absolute value of coherences in strong and weak coupling limits by varying the diagonal fluctuation strength and bath correlation time for the uncorrelated bath case as VUC d and −1 (BUC d ) . Here C° denotes the absolute value of coherence in EBSs, and C1 represents the absolute value of coherence in ExBSs. In Figure 2, the temperature dependency of absolute values of coherences is plotted in different regimes. In the nonMarkovian limit, the decay of coherence is slow, and the 2040

DOI: 10.1021/acs.jpclett.9b00382 J. Phys. Chem. Lett. 2019, 10, 2037−2043

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The Journal of Physical Chemistry Letters

−1 −1 −1 −1 Figure 4. Coherence length for the trimer system at (a) VUC and (BUC = 10 fs and (b) VUC and (BUC = 10 fs. The blue d = 50 cm d ) d = 350 cm d ) line indicates results obtained from the temperature-independent calculation. Green, red, and blue lines designate coherence lengths at 200 and 300 K and the temperature-independent case, respectively.

Figure 4 describes the dynamic localization−delocalization in strong-coupling non-Markovian and weak-coupling Markovian limits. For both of the cases with decreasing temperature, the phase relation between the states increases, which essentially leads to the delocalization. At the complete delocalization limit, the expected value of coherence length is 3 for the 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 and strong coupling non-Markovian limit, the coherence length exceeds 2.5, which indicates the coherent transport and complete delocalization. In the case of the weak-coupling Markovian limit, initially the coherence length surpasses 2 for the temperature-dependent case. From both panels of Figure 4, it is clear that even at long time the value of coherence length is greater than unity for the temperature-dependent case. This behavior signifies finite coherence even in the long-time limit. For the temperature-independent case in the long-time limit, coherence vanishes, and all of the sites acquire an equal population at the 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 a transition from coherent to incoherent dynamics while going from the non-Markovian to the Markovian limit. We analytically show for the correlated bath model that coherence is propagated through the ExBS, which contains only the nonvanishing term, even in long-time limit in the absence of site energy heterogeneity (SEH). (2) In the strong-coupling and non-Markovian limit, coherence in the EBS is more dominant than coherence in the ExBS. However, the time scale of decay is more or less the same. In the Markovian limit, nonoscillatory dynamics corresponds to incoherent EET dynamics. (3) Nonlocal coherences between non-neighboring sites lead to the creation of new pathways of energy transfer. Nonlocal coherence helps to overcome the energy barrier, thus facilitating energy-transfer dynamics. Double the absolute coherence is known as concurrence, which is an entanglement measure for the single-exciton manifold. (4) Nonlocal coherences show a weak dependency on temperature. The temperature-dependent study shows nonvanishing quantum entanglement even in the long-time limit. No such nonvanishing entanglement is observed in the

coherence in the eigenbasis vanishes; however, the coherence in the site basis shows a finite value (SI). An increase in temperature helps in the transition from coherent to incoherent state and vice versa. Temperature acts as an effective decay term in the excitonic equation of motion. Lowering the temperature helps to preserve the phase relation between excitonic states for a long period of time. In the long time limit, when oscillation disappears, low temperature helps in slow decay, which essentially indicates the localization of the energies on a corresponding site. Coherences between nonlocal sites create interference between pathways of energy transfer. Nonlocal coherences open up new channels for energy transfer and facilitate energytransfer dynamics. For our model system, the nonlocal coherence leads to energy transfer from site 1 to site 3 by avoiding the barrier, that is, 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 the uncorrelated bath case, coherence in ExBS decays much faster than coherence in EBS. However, in the intermediate limit that is appropriate for photosynthetic and conjugated systems, coherence in the EBS and coherence in the ExBS play equally important roles. The temperature-independent QSLE provides an equal population of all sites in the long-time limit, which indicates the vanishing of quantum entanglement in the long-time limit. However, the temperature-dependent QSLE shows that an unequal population leads to nonvanishing quantum entanglement at the long-time limit. 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. Eigenvectors for the trimer system ψ1 = −0.076ϕ1 + 0.115ϕ2 + 0.991ϕ3 ψ2 = 0.885ϕ1 + 0.451ϕ2 + 0.12ϕ3 ψ3 = −0.46ϕ1 + 0.885ϕ2 + 0.067ϕ3

(17)

The EET process occurs in the presence of a fluctuating environment. Now, to observe the dynamical localization process, one cannot use the definition of the inverse participation ratio in terms of the coefficient of eigenstates because the dynamical interaction between the system and the environment changes the delocalized states. 2041

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The Journal of Physical Chemistry Letters temperature-independent case. Initially, the rapid increase in concurrence is due to a quick delocalization of excitation. (5) For both non-Markovian and Markovian limits and the temperature dependent case, the coherence length overcomes the value 2.5, which indicates complete delocalization in the short-time limit, and dynamics is coherent. Because a finite amount of coherence survives in the long-time limit for the temperature-dependent case, there exists delocalization at least at the nearest-neighbor level. However, for the temperatureindependent case, coherence vanishes in the long-time limit, and all of the sites acquire an equal population at equilibrium. This indicates that the localization of excitation and the EET process occurs via completely hopping or an incoherent mechanism. Briggs and Eisfeld36 studied excitation transfer in the FMO complex using quantum and classical dynamics. They observed that in the 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 a fluctuating environment. Finally, we observe that this nonperturbative hierarchical EOM can capture the quantum entanglement for correlated and uncorrelated environments in strong, weak,and intermediate coupling limits. The nature of entanglement and the coherence length provide us information about the dynamical localization process of EET dynamics. In the single-exciton manifold, one can measure the entanglement by the calculation of energy eigenstates of the system via nonlinear femtosecond spectroscopic techniques. We have chosen the parameters such that we can explore all of the limits. One can obtain the functional form of the bath spectral density by fitting the experimental data.37 Now, using the function, one get the relaxation kernel for the bath. From the relaxation kernel, one can obtain the fluctuation strength and the bath correlation time from the prefactor and the time constant, respectively. One can also obtain the parameters from the simulation38 by using the same technique.

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ACKNOWLEDGMENTS

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REFERENCES

B.B. acknowledges the Department of Science and Technology (DST, India) and Sir J. C. Bose fellowship for providing partial financial support. R.D. thanks Mr. Saumyak Mukherjee for the help in making the figures.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b00382. Temperature-independent and -dependent quantum stochastic Liouville equations; diffusion operators for Poisson and Gaussian stochastic processes; a table of parameters for the system Hamiltonian; and plots of absolute value of coherence in the eigenbasis are plotted (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. ORCID

Biman Bagchi: 0000-0002-7146-5994 Notes

The authors declare no competing financial interest. 2042

DOI: 10.1021/acs.jpclett.9b00382 J. Phys. Chem. Lett. 2019, 10, 2037−2043

Letter

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DOI: 10.1021/acs.jpclett.9b00382 J. Phys. Chem. Lett. 2019, 10, 2037−2043