Decoherence and Quantum Interference in a Four ... - ACS Publications

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Decoherence and Quantum Interference in a Four-Site Model System: Mechanisms and Turnovers Mahdi Zarea,* Daniel Powell, Nicolas Renaud, Michael R. Wasielewski, and Mark A. Ratner Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois, United States ABSTRACT: We study quantum interference effects in a four-level system which can be used as a minimal model to understand such behavior in systems from synthetic molecular structures to the photosystem-1 reaction center. The effects of environmental decoherence and relaxation on the electron transfer rate are investigated for several types of decoherence processes. The rate as a function of decoherence amplitude shows Kramers turnover, as expected. However, various decoherence processes affect the quantum interference differently. It is shown that when the bridge sites are not dephased the superexchange transfer is enhanced by constructive quantum interference. Dephasing on bridge sites opens a (classical) diffusive channel for fast electron transfer, which can dominate the superexchange current and reduce the constructive quantum interference.

1. INTRODUCTION X-ray crystallographic studies have shown that reaction center (RC) proteins of photosystem I (PS1) have two sets of electron transport “bridge” cofactors arrayed in a parallel pseudo C2symmetric arrangement (Figure 1). The electron transfers from the primary donor (P700), which is a dimer of chlorophyll molecules, through either intermediate chlorophyll (A0A or A0B), to the corresponding phylloquinones (A1A or A1B) to the common bridging iron−sulfur cluster (FX), and ultimately to the terminal iron−sulfur acceptors (FA and FB). In PS1 reaction centers (PS1 RCs), the electron transfer cofactors are chemically identical, forming two nearly symmetric pathways.1−4 The symmetry of the pathways suggests the possibility that the PS1 RC protein may function as a sort of interferometer in which electrons traversing the two pathways are in a quantum superposition state. There is evidence that in PS1 both paths engage in electron transport5−13 as opposed, for example, to the photosynthetic reaction center.14,15 Moreover, it is possible to design and synthesize dual pathway molecules which mimic the PS1 RC, as shown in Figure 1.16 In this work, we present a minimal model study of quantum coherent electron transfer (ET) from the primary donor to the acceptor. It is shown that, with proper choice of the initial state, constructive quantum interference enhances the transfer rate in the superexchange regime. However, the environment causes decoherence that might make quantum interference effects unobservable. We will show that a small amount of damping increases the rate constant while too much damping reduces it, a signature of classic and quantum Kramers turnover.17−19 The rate constant can be affected in many different ways. For example, we show that, when one or both of the bridge sites is coupled to the bath, decoherence reduces the amount of effective coupling and thus the interference; this adds a new © 2013 American Chemical Society

pathway for ET which has no quantum interference. This pathway can be faster than coherent superexchange, and therefore, it can make interference effects unobservable. One may naively expect that any kind of decoherence process that reduces the quantum correlation between two bridge sites must reduce the quantum interference. We show that this happens, for example, when only the acceptor is coupled to the bath, where the sole mechanism for ET is superexchange. The paper is organized as follows. A simple minimal quantum model for the isolated PS1 RC is presented in section II. Different decoherence processes and their effects on ET rates in a PS1 model system are studied in section III using an effective Liouvillian method. The system is studied in a wider range of parameters and in the presence of more involved relaxation processes, using the steady state method, in section IV. Conclusion and discussion follow. Appendix A explains the effective Liouvillian and the steady state methods applied to a simple two level system. The superexchange and diffusive transport mechanisms are introduced in Appendix B.

2. THE MODEL The exact values of the local energies and the coupling strengths between different segments of the PS1 system remain uncertain.20−22 However, there is a general consent that the energy difference between the primary dimer P700 and the chlorophyll molecules A0A and A0B is small, of the order of 10 meV. The energy of phylloquinones A1A and A1B is around 450 meV lower, while the iron−sulfur cluster (FX) energy is 250 meV lower than phylloquinones. The typical coupling constant between donor and chlorophyll molecules is not more than 5 Received: October 17, 2012 Revised: December 5, 2012 Published: January 3, 2013 1010

dx.doi.org/10.1021/jp3102942 | J. Phys. Chem. B 2013, 117, 1010−1020

The Journal of Physical Chemistry B

Article

Figure 1. The PS1 RC (left) and its artificial analogue (right) have two nearly symmetric paths.1

Throughout this work, the typical energy gap between different segments is assumed to be larger than the coupling terms in accordance with the realistic data. The Hamiltonian of the 4LS system is given by H: = H0 + HV where

meV, while the coupling between chlorophyll molecules and phylloquinones is 0.1 meV.23 The cluster FX is coupled to phylloquinones very weakly, and this mixing is very hard to measure experimentally. To study the two-path quantum interference, we model both PS1 and its artificial analogues by a minimal four-level system (4LS) with local sites d, a, c1, and c2 representing the donor, acceptor, and two bridges (or connectors) respectively (Figure 2). This is a reasonable model for the artificial molecule. For

H0 = ,dd†d + ,aa†a + ,c1c1†c1 + ,c2c 2†c 2 HV = (Vdc1d†c1 + Vdc2d†c 2 + Vac1a†c1 + Vac2a†c 2) + h.c. (1)

where h.c. stands for hermitian conjugate. We set ,a = 0 as the reference energy. The bridge site energies are assumed to be equal and are relabeled by ,c1 = ,c2 = ω throughout this paper. For the PS1 RC, we take Vdc1 ≈ Vdc2 ≈ Vac1 ≈ Vac2 = V ≪ ω, and therefore the energy eigenstates are approximately described by local energy sites.29 In the Hamiltonian of eq 1, the donor and acceptor are not coupled to each other directly but through the bridge sites. Using the standard Green function technique, the effective coupling (at the energy ε) between the donor−acceptor pair is given by28

Figure 2. PS1 RC modeled as a four-level (left figure) or a five-level (right figure) system. The labeling shows the local energy of each site.

eff ( ε) = V da

PS1, the FX cluster and the phylloquinones are represented by the acceptor and two bridges, respectively. The dimer P700 and chlorophyll molecules (which have almost the same energies) are considered as a single donor d in this simplest model. In a slightly more realistic model of PS1, these components can be differentiated.23 Even the primary donor compound consists of two coupled molecules (chlorophyll dimer). The detailed analysis of these models is more involved and will be presented elsewhere.23 However, we note that this model with two primary donors can be mapped to two 4LSs introducing bonding/antibonding donors d± = (d1 ± d2)/√2 (Figure 2). Here, d+ couples to both bridge sites with (V1 + V2)/√2. On the other hand, d− couples to c1 with (V1 − V2)/√2 and to c2 with (V2 − V1)/√2, i.e., with an opposite sign, which causes destructive quantum interference.24−27 The 4LS model serves as the simplest two-pathway prototype system in which quantum interference affects the ET. Nevertheless, some general features of the 4LS also hold for more sophisticated models.

Vdc1Vc1a ε − ,c1

+

Vdc2Vc2a ε − ,c2

≈

2V 2 ε − ,c1

(2)

where Vdc1 ≈ Vdc2 ≈ Vac1 ≈ Vac2 = V ≪ ω. The effective coupling is twice the effective coupling for a single-path d−c1−a model. On the other hand, if one of the four coupling constants has a negative sign, the total effective coupling is zero, resulting in destructive interference.24−26 This form for the effective matrix elements, first used in a molecular context by McConnell, is referred to as superexchange.30,31 The symmetric 4LS can also be mapped to a single d−c+−a chain model.32,33 Here, c± = (c1 ± c2)/√2 are bonding/antibonding bridge sites and c− is decoupled from the donor and acceptor. Both above approches indicate that the effective coupling between donor and acceptor in the two-path model is twice the corresponding effective coupling in the one-path model. For an isolated 4LS and in the absence of any decoherence, the electron population density oscillates between local states. In the presence of environmental dephasing, the quantum 1011



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In eq 3, ρ is the N × N density matrix (for an N-level system). We go to the super-Liouville space where the density matrix ρ is represented by the vector r ⃗ which has 0, 1, ..., N2 − 1 components. As an example, the 2 × 2 density matrix of a 2LS is written as a four-component vector (see Appendix A). Equation 3 is then written as

correlation between different sites is reduced.34 We will show that, as long as the bridge sites are not dephased, the ET from the donor to the acceptor happens via the effective coupling (eq 2) between them. This is the superexchange transfer regime in which the bridge sites are not populated but used as mediators for ET. In this case, since the effective coupling of a two-path model (eq 2) is twice that for the single path, the Fermi golden rules imply that the rate constant increases by a factor of 4. When bridge sites are dephased, the diffusive (classical) mechanism also contributes to the transport, i.e., electrons hop to the bridge sites and from there to the acceptor site. In the extreme case when all correlations are absent, the ET from the donor to the acceptor is just diffusive, and the donor−acceptor rate constant for the two-path model is simply the sum of the rate constants from each path. In general, both of these mechanisms can contribute to ET.

i∂t r ⃗ = 3 r ⃗

where 3 is the N2 × N2 super-Liouvillian matrix containing both the unitary and the dissipative dynamics of the system. The eigenvectors of 3 are represented by rn⃗ , where n = 0, ..., N2 − 1. The corresponding eigenvalue is called λn. In the eigenbasis of 3 , the master equation is written as the sum of n = 0, 1, ..., N2 − 1 independent equations ∂t rn⃗ = −λn rn⃗

3. SUPEREXCHANGE AND DIFFUSIVE TRANSPORT REGIME IN 4LS Below we study in detail a few types of decoherence processes in a 4LS. We show in which situations the classical transfer dominates and when, in contrast, the two-path quantum interference dominates the ET. As usual, we include the bath in the overall Hamiltonian: H = H: + H) + H:) with the terms representing the system, the bath, and their interaction Hamiltonian, respectively. The coupling of the system to the bath is assumed to be linear H:) ∝ FB, where the dimensionless operator F (or B) is an operator from the system : (or bath ) ). There are two variations of the coupling Hamiltonian, depending on which system operator F is used. In one, exemplified by F = a†d, the bath phonons help transfer electrons between the two levels, contributing directly to the population relaxation of the two sites. This type of system−bath interaction is included briefly in the next section and in more detail elsewhere.23 In the second model, the system operator F is diagonal in local basis, e.g., F = a†a. This model reflects the effect of a boson bath on the states of the system that are associated with distinctly different polarizations of the environment in the different system states. The local states of the system couple to different bath modes causing fluctuations in local energies, e.g., ,a → ,a + δ ,a with ⟨δ ,a⟩ = 0 but ⟨δ ,a 2⟩ ≠ 0. As a result, every off-diagonal element of density matrix related to this site decays, e.g., ∂tρda = −κρda/2 + ..., where κ is the dephasing parameter proportional to the bath relaxation time. In contrast to the first kind, this kind of decoherence does not cause the direct relaxation of diagonal density matrices. However, since local states are coupled by V (i.e., they are not exact eigenstates) their lifetimes are changed. To derive the master equation, we use the Lindblad formalism35 rather than the Redfield equations.36 The Lindblad master equation is given by37−39 (ℏ = 1) ⎛ ⎞ 1 1 ∂tρ = i[ρ , H:] + κ ⎜F†ρF − FF†ρ − ρFF†⎟ ⎝ ⎠ 2 2

(4)

(5)

The eigenvector rn⃗ is a linear combination of the density matrix elements. The eigenvalue λn is a complex number whose imaginary (respectively real) part is the oscillation frequency (respectively the decaying rate constant) of rn⃗ . We assume that initially all electron density is on the donor site. The Lindblad formalism conserves the total density. Therefore, always one of the characteristic equations (eq 5) which we label by n = 0 corresponds to the total density, i.e., λ0 = 0 and r0⃗ = ρtotal = ∑i ρii. On the other hand, in all cases studied here, the donor density ρdd appears only in one of the remaining rn⃗ . We represent this particular eigenstate by r1⃗ and its eigenvalue by Γ ≡ λ1. Approximate methods can be used to solve eq 5. The effective Liouvillian method and the steady state method are two examples which are briefly explained in Appendix A. For example, application of the effective Liouvillian method to a two-level system (2LS) coupled to the bath by the Lindblad operator F = a†a results in ∂t(ρdd − ρaa ) = −Γ(ρdd − ρaa )

(6)

with Γ=

8κVda 2 κ 2 + 4(,d − ,a)2

(7)

As shown in Appendix A, Γ is also the inverse lifetime of the donor site. As will be made clear later, the above rate constant reveals some general features of other models like 4LS. We note that, as a function of dephasing parameter κ, the above rate constant shows the Kramers turnover, i.e., the rate first increases linearly with κ and then decreases as 1/κ. Before we proceed, it is useful to list the various notations we have used throughout this work: • The energy gap between the acceptor and the bridge sites is called ω. The energy gap between donor and acceptor is ,d − ,a and is a variable parameter. ω appears in the superexchange rate constant because the effective donor−acceptor coupling depends on ω. It also appears in the diffusive rate constant (which depends on the energy gap between bridge sites and the acceptor). • The dephasing parameter of ρda, ρad is always called κ, e.g., ∂tρda = −κρda/2 + .... This parameter appears only in the superexchange rate constant.

(3)

The dephasing parameter κ measures the strength of system− bath coupling.37 The Lindblad master equation has the advantage that the density matrix remains positive. However, the choice of the F operator is not specified in this formalism. Here, we select different F operators to study the effect of only dephasing (not relaxation) on quantum interference. 1012



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• γ is the dephasing parameter of the bridge sites, e.g., ∂tρcia = −γρcia/2 + .... It appears solely in the diffusive rate constant. • For cases considered here with small coupling V, the donor density appears in the total density r0⃗ with λ0 = 0 and one of the remaining equations in eq 5 called r1⃗ with the corresponding eigenvalue λ1 = Γ. 3.1. Superexchange Regime of 4LS. We start from the phenomenological situation where the only dephasing parameter in the system is κ, the dephasing of ρad, ρda. More dephasing processes will be added gradually. Consider first the situation where the donor and acceptor are degenerate ,a = ,d = 0 but the bridge sites are higher in energy by ,c1 = ,c2 = ω. The effective coupling (eq 2) between donor and acceptor at (ε = 0) is 2

Vda = Veff ≈ 2V /ω

Figure 3. The population densities of local sites in 4LS in the superexchange regime. Initially, all electrons are on the donor site. Bridge sites are very weakly populated. The inset shows that ρda and ρad are subjected to dephasing with rate κ.

that there would be diffusive transport between the acceptor and bridge sites. However, for the donor, still the only mechanism for ET is superexchange. That is because the quantum correlation between donor and bridge sites is still preserved and so the donor population cannot diffuse to the bridge sites. This is evident in Figure 4. Here the acceptor is first populated via superexchange transfer from the donor. After that, the bridge sites are populated by diffusive transport. More precisely, this can be proved by calculating the rate constant. For small coupling V, one of the eigenstates of the characteristic equation (eq 5) is r1⃗ = (2ρdd − ρaa − ρc+c+)/√6 corresponding to

(8)

Under these assumptions, the bridge sites act only as mediators which couple the donor to the acceptor. The system behaves effectively as a 2LS whose approximate donor− acceptor transfer rate constant is given by eq 7, with Vda = Veff given by the above relation. That results in Γ ≈ 32V 4 /κω 2

(9)

which is the rate constant of ρdd − ρaa. This is 4 times the superexchange rate constant of a single path d−c−a system (see eq 35 in Appendix B): the rate constant for a two-path system is not simply the sum of the rates from each path as expected classically. The enhancement is due to the constructive quantum interference between two paths. It can be easily shown that if the energy of one of the bridge sites is raised with respect to the other site by δ > 0 the effective coupling drops to Veff ≈ V2/ω + V2/(ω + δ), which reduces the superexchange rate. Increasing the energy difference δ will eventually force all electrons to transfer through the lower energy bridge site. In artificial molecules which mimic the PS1 system,16 when a substitutional compound is attached to one bridge site, its energy will be altered and thus the rate constant changes. Therefore, these molecules might be used as sensitive chemical sensors.23 If one of the four couplings has a negative sign, the effective couplings of two paths cancel each other, resulting in destructive quantum interference. In this case, increasing the energy of one of the bridge sites reduces its absolute effective coupling and so superexchange transfer is recovered. Inelastic scattering with energy exchanges between one bridge and the environment is also shown to recover the ET in systems with destructive interference.25,26 Figure 3 shows the time evolution of local densities. The coupling constant V is selected here to be relatively large to stress the presence of fast oscillations in local densities. These oscillations are not captured in the approximate rate constant (eq 9) which only shows the decaying (real) part of the densities. 3.2. Four-Level System with the Acceptor Coupled to the Bath. Now we consider the case when the acceptor site is coupled to the bath via H:) ∝ a†aB. As before, the corresponding dephasing parameter is called κ and it is the decaying rate of ρad as well as ρac1, ρac2 (and their complex conjugates), as shown in Figure 4. From the figure, one expects

Γ ≈ 24V 4 /(κω 2)

(10)

where c± = (c1 ± c2)/√2 are the bonding/antibonding bridge operators. The donor density ρdd appears only here and in total density r0⃗ . Therefore, it can be shown that eq 10 is also the inverse lifetime of the donor site whose time evolution is approximately given by ρdd (t ) ≈ (2e−Γt + 1)/3

(11)

The rate constant (eq 10) clearly corresponds to the superexchange rate. Interestingly, its value has reduced as compared with eq 9, where there is no diffusion between any local state pairs. Therefore, although the diffusive transport is only between the acceptor and bridge sites, it does affect the value of superexchange transfer from the donor. Nevertheless, the rate constant of eq 10 has the signature of complete constructive quantum interference; had we calculated the same rate constant for a single-path system, the rate would be Γ1path = 6V4/(κω2), 4 times smaller. Therefore, for destructive interference, the dephasing of the acceptor site will recover the ET. Figure 4 also shows another interesting feature of bridge site relaxation. One might expect that at t → ∞ the electron density will be distributed evenly on local sites (note that so far we have not included any thermal relaxation in our model). However, in Figure 4, at t → ∞ the density of each bridge site is half of the acceptor density. That is because the correlation between two bridge sites is preserved and remains nonzero. Had we plotted ρc1c2, ρc2c1, we would see that ρc1c2 = ρc2c1 = ρc1c1 = ρc2c2. In fact, only the density of c+ (and not the individual bridge densities) enters in the characteristic eigenstate r1⃗ ∝ 2ρdd − ρaa − ρc+c+. We explore this point further introducing a phenomenological dephasing parameter η which causes the correlation between two bridge sites to decay, i.e., ∂tρc1c2 = −ηρc1c2/2 + .... At 1013



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Figure 4. When a is coupled to the bath, ρac1, ρac2, ρad, and their complex conjugates decay with the rate κ/2. The donor population is transferred by superexchange to the acceptor, and from there, it diffuses to the bridge sites. It should be noted that at t → ∞ the correlation between two bridge sites is preserved, ρc1c2 = ρc2c1 = ρcici (not shown in figure).

large values of η, the two bridges are not correlated anymore. A careful analysis of the master equation in this case reveals that the rate constant reduces to Γ ≈ 64V 4 /3κω 2

acceptor the donor is coupled to the bath modes H:) ∼ d†dB. As before, both superexchange and diffusion contribute to ET. The donor population is transferred by fast diffusion to the bridge sites (Figure 7). On the other hand, since ρdci is not dephased, the acceptor is mainly populated by slow superexchange transfer from the donor site.

(12)

As compared to eq 10, due to the reduction of correlation between two paths, the rate constant has dropped by a factor of 8/9, as shown in Figure 5. Γ (in eq 12) is the eigenvalue of r1⃗ =

Figure 7. When d is coupled to the bath, ρdc1, ρdc2, ρad, and their complex conjugates decay with the rate κ/2. The donor population is transferred by diffusion to the bridge. The acceptor is slowly populated by superexchange.

Figure 5. The rate constant Γ as a function of the acceptor dephasing parameter κ. When quantum correlation between two bridges is reduced ρc1c2 → 0, the rate constant drops from 24V4/(κω2) (red line) to 64V4/(3κω2) (black dashed line). All parameters are in arbitrary energy (inverse time) units.

3.4. The Acceptor and One of the Bridge Sites Are Coupled to the Bath. Now we assume that besides the acceptor one of the bridge sites (c2) is also coupled to the bath by H:) ∝ c 2†c 2B. Off-diagonal matrix elements ρc2c1, ρc2a, ρc2d, and their complex conjugates decay with the rate γ/2. The situation is schematically depicted in Figure 8. From this figure, it is expected that, besides the superexchange, the donor population can reach the acceptor via a diffusive channel through d−c2−a. The total rate constant is given approximately by

(3ρdd − ρaa − ρc1c1 − ρc2c2)/(12)1/2 in which, instead of the density of the bonding bridge site c+, the individual densities of bridge sites are present. Figure 6 shows that as expected in this case the bridge site populations approach the donor and acceptor population at t → ∞. The donor density is ρdd (t ) ≈ (3e−Γt + 1)/4

(13)

3.3. The Donor Is Coupled to the Bath. This situation is very similar to the previous case except that now instead of the

Γ = Γsup + Γdif

(14)

Here the diffusive rate constant Γdif is given by eq 34 in Appendix B. Dephasing of the bridge site c2 reduces the superexchange rate constant Γsup in two ways; it reduces the effective donor−acceptor coupling and it reduces the correlation between the two bridge sites (which, as seen in the previous case, suppresses the superexchange). Figure 9 shows the behavior of the rate constant as a function of acceptor (respectively bridge site) dephasing parameter κ (respectively γ). At a given κ, the rate constant first increases by γ due to the opening of a diffusive channel for transport. At very small system−bath coupling, where γ < V2/κ, the total rate constant eq 14 is

Figure 6. The time evolution of local densities when correlation between two bridge sites is reduced ρc1c2 → 0. 1014



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Figure 8. When acceptor a and bridge c2 are coupled to the bath, the donor population is reduced by both superexchange and diffusive ET. Whenever the diffusive transport dominates the dephased bridge, c2 is populated before the acceptor. The dotted black line is the donor density time evolution if the superexchange contribution is not considered.

Figure 9. Dephasing of bridge c2 opens a new (diffusive) channel for transport which first enhances the total rate but eventually becomes overdamped (left figure). Dephasing of the bridge site also reduces the superexchange rate constant from Γ ≈ 24V4/κω2 (red line) to a quarter of it, Γ ≈ 6V4/κω2 (black line), by removing the constructive quantum interference between two paths (right figure). All parameters are in arbitrary energy units.

Γ ≈ 24

V4 V2 + κω 2 γω 2

superexchange transfer from the c1 bridge is not affected and thus eq 16 is valid for this case too. 3.5. Acceptor and Both Bridge Sites Are Dephased. When both bridge sites are dephased, there are two diffusive channels, one from each path d−ci−a. Their total contributions to the rate constant is the (classical) sum. Adding the superexchange contribution and at the limit γ, γ′ ≪ ω the total rate constant is

(small system−bath coupling) (15)

and approximately corresponds to rn⃗ ≈ (ρdd − ρaa). The first term (the superexchange contribution) has the signature of constructive quantum interference. The second term is the diffusive rate constant in its linear regime. At the opposite limit γ ≫ ω, the total rate constant eq 14 is 4

Γ≈6

Γ ≈ 24

2

V V +4 2 γ κω

(large system−bath coupling)

(16)

V 2(γ + γ ′) V4 + κω 2 ω2

(small system−bath coupling)

As compared to eq 15, the diffusive contribution is in the overdamped regime. Moreover, the superexchange rate constant has dropped by a factor of 4. That is because the path d−c2−a does not contribute to the effective donor− acceptor coupling anymore. At this limit, all the transport is done through the bridge c1. Therefore, even without changing the bridge average energy, dephasing has suppressed the transport through bridge c2. Figure 8 also shows that the bath-coupled bridge site is populated first (by diffusive transport from the donor site). The acceptor and then the second bridge site are populated later. When the dephasing parameters κ and γ are of the same order, since the effective coupling V2/ω is much smaller than V, the diffusive transport dominates, Appendix B. If we neglect the superexchange contribution in the time evolution of donor site, we get the dotted line in Figure 8 which still represents the actual donor density well. When one coupling constant is negative, the destructive quantum interference between two paths cancels the superexchange contribution in eq 15. In the overdamped regime, the

(17)

As compared to the case where only one bridge site is dephased (eq 15), we have only the extra diffusive contribution from the second bridge. However for γ, γ′ ≫ ω the rate constant is ⎛1 1⎞ Γ ≈ 4V 2⎜ + ⎟ γ′ ⎠ ⎝γ

(large system−bath coupling) (18)

As compared to eq 16, the superexchange contribution is totally suppressed. Only the sum of diffusive rate constants from the two paths survives. The case of destructive interference is closely analogous except that the superexchange term in eq 17 is zero.

4. STEADY STATE RESULTS OF FOUR-LEVEL SYSTEM So far, it was assumed that the donor and the acceptor are in resonance ,d = ,a . Although this condition might be realized for the bioinspired compounds which mimic PS1, it does not represent the PS1 system. Below, this condition is relaxed and 1015



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(blue dashed line) and at ,d = 2,ci = 2ω (black dotted line). The qualitative features of the graphs are the samethey show Kramers turnover. Next, each graph is discussed in more detail. 4.1.1. The McConnell Superexchange Regime. When ,d = ,a and for κ ≪ V, the rate constant is Γss ≈ κ. For κ ≫ V, it follows the McConnell superexchange relation,30 i.e.,

we show that the above results change quantitatively but not qualitatively. Moreover, so far, the bath was assumed to couple to the local states densities, e.g., F = a†a, etc. In this section, we include processes which cause direct population relaxation of states. The bath is now coupled to system operators like F = a†d, which cause transitions between local states. To solve the master equation, the steady state approximation, which is easier to implement, is used in this section. A short introduction to this method is given in Appendix A. 4.1. The Transfer Rate Constant Dependence on the Donor Energy. Figure 10 shows the steady state rate constant

Γss ≈ 16V 4 /(κω 2)

(20)

Γ1path ss

For a one-path system, one gets ≈ 4V /(κω ), which indicates the presence of quantum interference in a two-path system. 4.1.2. Donor-Bridge Resonance Regime. Here, ,d = ,c1. In the linear regime when κ ≪ ω, the rate constant is 4

2

Γss ≈ 2κV 2/ω 2

(21)

which shows slower increase with V compared to the McConnell regime. However, for large κ, the rate constant exceeds the superexchange rate constant. For κ ≫ ω, the rate constant follows

Γss ≈ 8V 2/κ

(22)

which is the sum of the rate constants from two paths, i.e., the classical rate constant. 4.1.3. The Suppressed Transfer Regime. Upon increasing the donor energy over the bridge energy, the rate constant drops rapidly. For example, at ,d = 2,ci = 2ω, the linear regime κ ≪ ω is given by Γss ≈ κV 4 /ω4

Figure 10. The steady state transfer rate constant as a function of the donor energy. Here ,a = 0 and ,ci = ω and all parameters are in arbitrary energy units.

(23)

For κ ≫ ω,

Γss ≈ 16V 4 /κω 2 of the 4LS as a function of donor energy ,d . The exact expression for the steady state rate constant Γss is

(24)

which is again the superexchange rate constant with quantum interference signature. 4.1.4. Discussion. The behavior of steady state transfer rate constant for different values of donor energy is summarized in Table 1.

(κ 2/4 + (,d − ,a)2 )((,d − ,ci)2 + 2V 2) 1 1 = + κ Γss 4κV 4 (19)

Table 1

The first peak in Figure 10 with high transfer rate constant corresponds to the donor and acceptor being in resonance, i.e., ,d = ,a = 0. The second peak appears when the donor and bridges have the same energy ,d = ,c1 = ω. Figure 11 shows the steady state transfer rate constant as a function of κ. The donor energy is selected to be donor− acceptor resonance (red solid line), donor−bridge resonance

local energies

small κ

large κ

,d = ,a = 0, ,ci = ω

κ

16V4/κω2

,d = ,ci = ω , ,a = 0

2κV2/ω2

8V2/κ

,d = 2,ci = 2ω , ,a = 0

κV4/ω4

16V4/κω2

As mentioned before, except for the ,d = ,ci case where Γss is the sum of the rate constants from the two paths (the classical rate), for all other cases, |,d − ,ci| ≫ V , the quantum interference effect is present and enhances the rate constant. This can be explained by a simple argument. For the Hamiltonian of eq 1, the donor and acceptor are coupled to the bonding operator c+ = (c1 + c2)/√2 with coupling constant √2V, but they are decoupled from c− = (c1 − c2)/√2. Within the steady state method, the ρdc+, ρc+d preserve their coherence. Therefore, consider the eigenbasis of the isolated pair d, c+

Figure 11. The steady state transfer rate constant as a function of κ for different values of the donor energy: solid red line, ,d = ,a ; dashed blue line, ,d = ,c1; dotted black line, ,d = − ,a . All parameters are in arbitrary energy units.

d ̃ = cos θd + sin θc+ c+̃ = cos θc+ − sin θd 1016

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Figure 12. The left figure shows a process in which electrons can jump to the bridge site by emitting phonons and from there to the acceptor. The total rate constant increases for small γ but drops in the overdamped regime.

(and the bridge sites) and so the overall rate constant is reduced. The total rate constant (in the absence or the presence of γ) is reduced by orders of magnitude as compared to section III where the donor and the acceptor are degenerate. Nevertheless, constructive quantum interference still enhances the rate constant. For completeness, we consider an example of pure dephasing processes. These are processes in which [H: , H:)] = 0; i.e., the system and the bath do not exchange energy. The bath only causes dephasing of the system eigenstates. This happens when the bath couples to the exact eigenstate of the system. Note that H: includes the coupling constants (V’s) as well, so local states are not exact eigenstates. A trivial example is to set F ∼ H in H:) ∝ FB with the corresponding dephasing parameter γ. Figure 13 shows the resulting effect on the rate constant. In

with θ=

1 2 2V V 2 tan−1 ≈ 2 ,d − ,ci ,d − ,ci

The operator d̃ has a large overlap with the local donor state d and is coupled to the acceptor a by Vad ̃ ≈ Vc+a sin θ =

2V 2 ,d − ,ci

(26)

The master equation at steady state is derived using the Lindblad operator F = a†d = a†(cos θd̃ + sin θc̃) ≈ a†d̃ + θa†c+̃ . Calculating the rate constant of the d̃, a pair with the coupling (eq 26) between them results in Γss̃ ≈ κ

16V 4 /(,d − ,ci)2 κ 2 + 4(,d − ,a)2

(27)

which corresponds to the steady state rate constant (eq 19) for |,d − ,ci| ≫ V . In the PS1 RC, the primary dimer P700 site is higher in energy than the sites for phylloquinones and iron−sulfur clusters. Within the steady state method, its dynamics is described by eq 27. In the suppressed transfer regime where κ ≫ ,d − ,a this equation reduces to Γss ≈ 16V 4 /κ(,d − ,ci)2 which is the superexchange rate. Therefore, the signature of quantum interference is expected to be observed in donor−acceptor ET. 4.2. Other Relaxation Processes in the Steady State Regime. In this section, the effects of other relaxation and dephasing processes in the steady state transfer rate constant are investigated. Two particular examples are considered here: As explained in Appendix A, the steady state conditions are equivalent to the situation where a donor electron emits phonons and relaxes to the acceptor (with the corresponding dephasing parameter κ). A natural extension of the model is to include processes in which the electron first relaxes to the bridge sites and from there to the acceptor site by emitting more phonons. For simplicity, we assume that this process happens through one of the bridge sites, i.e., c2, and that the decoherence parameter for both transitions from the donor to c2 and from c2 to the acceptor is γ. More precisely, the system− bath coupling is H:) ∝ F1·B + F2·B, where F1 = c†2d and F2 = a†c2. Figure 12 shows the rate constant as a function of γ. The rate constant is first enhanced due to the opening of an extra diffusive channel for ET through bridge c2. When γ is very large, the diffusive channel becomes overdamped and the rate drops. Note that here the donor energy is higher than the acceptor

Figure 13. The steady state rate constant for the pure dephasing processes. All parameters are in arbitrary energy units.

contrast to all cases studied so far, the steady state rate does not show Kramers turnover. Instead, at large γ, it saturates at constant κ.

5. CONCLUSION The PS1 RC is a nearly symmetric structure that provides two similar pathways for ET from the donor to the acceptor. We have studied the decoherence in a simple four-level model, that can be understood as a minimal model for PS1. More sophisticated models of PS1 can be mapped to a 4LS. The model shows behaviors that occur in any complicated system capable of quantum interference. In principle, the constructive quantum interference of the two paths enhances the electronic transport from the donor to the acceptor. Coupling to the bath causes decoherence and relaxation that might destroy the interference. It is already known that inelastic interaction with the bath reduces the destructive quantum interference.25,26 Elastic interaction, in general, is also known to revive the donor−acceptor ET for destructive interference models.24 1017



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leading to the fluctuation in local energies. Using the Lindblad formalism and in the Liouville space, the master equation reads

Here, we have discerned between various types of elastic processes. Each different dephasing process has its own signature and can open diffusive ET channels, reduce the superexchange transfer of each path and/or reduce (increase) the constructive (destructive) two-path quantum interference. We show that the quantum interference affects the rate constant in the superexchange regime. For example, when only the acceptor is coupled to the bath, the ET from the donor to the acceptor is coherent and there is complete constructive interference between two paths. On the other hand, when one or both of the bridge sites are coupled to the bath, the donor population can also reach the acceptor by diffusive transport through bridge sites. Even a small amount of dephasing in bridge sites is enough for the diffusive transport to dominate. Although the superexchange transfer does not vanish, its contribution is often much smaller than the diffusive transport which makes quantum interference effects hard to observe. To study the behavior of the model over a wider range of parameters, we use the steady state method. It is shown that when the donor and acceptor are degenerate the superexchange transfer is maximum (as expected). The other peak of the rate constant is at donor−bridge degeneracy. However, for this peak, the rate constant is the sum of the rates from each path, i.e., the classical rate. In the PS1 RC, the donor energy is higher than all other sites. For this system, although the transfer rate constant is very small, quantum interference effects are present. One could also consider more complicated decoherence processes using the steady state method. This includes in particular processes in which higher energy electrons can emit phonons and relax to lower local energy states. If during these processes the ET occurs directly from the donor to the acceptor, quantum interference effects affect the rate. However, indirect transfer (through bridge sites) is classical. Another example is where the bath causes only the dephasing of exact energy eigenstates, and there is no energy exchange between system and bath. Pure dephasing does not show the Kramers turnover; as the coupling to the bath increases, the rate constant increases monotonically until it saturates at a constant value. It might be possible to design a nanoscale Mach−Zehnder interferometer which mimics the PS1 RC. Since any kind of structural asymmetry can interrupt the quantum interference, this system might be used as a sensor with potential singlemolecule sensitivity. Our theoretical analysis of the interference in a four-level system suggests that the key point for the device to function as a quantum interferometer is to reduce the coupling between the bridge sites and the environment.

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3= ⎛0 ⎜ ⎜0 ⎜ ⎜ Vad ⎜ ⎝− Vad

0 − Vad Vad

⎞ ⎟ Vad − Vad ⎟ ⎟ ,d − ,a − iκ /2 0 ⎟ ⎟ 0 − ,d + ,a − iκ /2 ⎠ Vad

0

− Vad

(28)

which is written in the (ρdd, ρaa, ρad, ρda) basis. For simplicity, instead of writing the whole Liouville matrix 3 , we represent the system schematically as in Figure 14 (see the explanation in the figure caption).

Figure 14. The schematic representation of the situation where bath modes couple to the local system levels. The wavy lines between local levels d and a indicate that the energy difference between them is fluctuating. Parameters ± (,d − ,a) − iκ /2 indicate that two offdiagonal density matrix elements ρad and ρda decay with the rate κ/2 and oscillate with frequencies ± (,d − ,a), respectively, e.g, i∂tρad = (,d − ,a − iκ /2)ρad .

We use two approximate methods in this paper: the effective Liouvillian method and the steady state approximation. A.1. The Effective Liouvillian Method. The Liouvillian space can be partitioned and projected to the smaller space 1 or 2. Assuming that 1 is the space of interest, the effective Liouvillian in this space 3eff is given by 11 12 22 21 311 eff (ε) = 3 + 3 . 0 (ε)3

(29)

22

w h e r e . (ε) i s t h e s u p e r G r e e n f u n c t i o n 22 .22 − iδ) of space 2. 0 (ε) = 1/(ε − . For a 2LS, take space 1 to be ρ1 = (ρdd , ρaa ) and space 2 as ρad, ρba. Since the coupling Vda is assumed to be the smallest parameter of the model, it is justified to approximate the exact result (eq 29) by setting ε = 0. The effective master equation for the space 1 is then ⎛ ρdd ⎞ ⎛−1 1 ⎞⎛ ρdd ⎞ 4Vda 2κ ⎜ ⎟⎜ i∂t⎜ ⎟ = i 2 ⎟ κ + 4(,d − ,a)2 ⎝1 −1⎠⎝ ρaa ⎠ ⎝ ρaa ⎠

APPENDIX

(30)

The dynamics of ρdd − ρaa is given in eqs 6 and 7. Given that the total density ρdd + ρaa is a constant and all electrons are initially on the donor site, one finds that ρdd(t) = (e−Γt + 1)/2, which indicates why Γ in eq 7 also is a measure of the inverse lifetime of the donor density. Note that the donor density ρdd acquires the lifetime Γ indirectly through the coupling to the decaying off-diagonal terms ρad, ρda. A.2. The Steady State Solution. In the steady state approach,40,41 a constant current J is added to the donor site ∂tρdd → ∂tρdd + J. The acceptor density is assumed to decay with an additional rate constant κ, i.e., ∂tρaa → ∂tρaa − κρaa. For every off-diagonal term ρai, we replace ∂tρai → ∂tρai − κρaa/2

A. The Transfer Rate Constant and Donor Lifetime

Let us start from a simple two-level system (2LS) in which the donor and acceptor are separated by the energy gap ,d − ,a and are coupled to each other by Vad. The Hamiltonian is H: = (,d − ,a)σ 3/2 + Vadσ 1 with σi, i = 1, 2, 3, being Pauli matrices, and the index : refers to the system. The 2LS is used as a prototype example to show how decoherence affects the electron transfer rate; many general features of the rate constant in 2LS are also present in more elaborate models. The system and bath are coupled to each other by H:) ∝ a†aB. Here, the bath bosons couple to the density of local charges, 1018



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density matrix elements ρdc, ρcd, ρac, and ρca decay with inverse lifetime γ/2, e.g., ∂tρdc = −γρdc/2 + .... For the d−c and c−a pairs, one expects that ∂t(ρdd − ρcc) ∝ −Γ(ρdd − ρcc) and ∂t(ρcc − ρaa) ∝ −Γ(ρcc − ρaa), respectively. Therefore, it is reasonable to think that the donor density relaxes to the bridge site and from there it goes to the acceptor. This is the classical diffusion or thermal hopping transfer mechanism. A careful analysis of the master equation shows that ∂t(ρdd − ρaa) ≈ −Γdif(ρdd − ρaa), where the diffusive rate constant Γdif is

and the same for ρia. The conservation of total population implies that J = κρaa. Applying these boundary conditions, the stationary solution of the master equation ∂tρ = 0 is found. The transfer rate constant is then defined by40,41 Γss ≡ κρaa /ρdd

(31)

As an example, for a 2LS with the energy gap ,d − ,a and the coupling Vda, the Liouvillian will be 3= ⎛0 ⎜ ⎜0 ⎜ ⎜ Vad ⎜ ⎝− Vad

κ −κ − Vad Vad

⎞ ⎟ Vad − Vad ⎟ ⎟ ,d − ,a − iκ /2 0 ⎟ ⎟ 0 − ,d + ,a − iκ /2 ⎠ Vad

Γdif =

− Vad

(diffusive rate constant)

(34)

At equilibrium, all local sites are populated including the bridge site c. The second mechanism for ET is superexchange. Consider the situation in which the bridge site is isolated from the bath but ρda, ρad are decaying with the rate κ/2 (Figure 16). The

(32)

Solving for stationary solutions of the master equation, the steady state rate constant is Γss =

4V 2γ γ + 4ω 2 2

4Vad 2κ 4Vad 2 + κ 2 + 4(,d − ,a)2

(33)

It should be noted that this result is different from eq 7 but both share the general feature of Kramers turnover. The reason is that the steady state conditions are different from the conditions which lead to eq 7. This difference is also reflected in the above Liouvillian equation (eq 32) as compared to eq 28 by the extra elements κ in the first two rows. It is interesting to note that eq 32 can be derived in another way by using a Lindblad operator F = a†d with the corresponding dephasing parameter called κ. It can be shown that this is valid for the four-level system too. The system−bath interaction H:) ∝ FB with the above F describes the dilute phonon limit in which electrons in the higher level d emit phonons and relax to the lower level a. Therefore, the steady state method provides insight to the dynamics of the system when this kind of process is considered.

Figure 16. Superexchange ET. Bridge sites are not populated.

presence of the bridge site introduces the effective coupling Veff ≈ V2/ω between donor and acceptor. The system behaves effectively like a 2LS, with ∂t(ρdd − ρaa) ≈ −Γsup(ρdd − ρaa). For κ ≫ V, the rate constant Γsup is approximately given by eq 7 with Veff replacing V

B. Diffusive (Hopping) and Superexchange Transfer

When there is a bridge site between the donor and the acceptor, there are two mechanisms for electron transfer between them. For simplicity, in this section, we assume that ,d ≈ ,a = 0. In the first mechanism, the d−c−a chain behaves like two catenated 2LS. This happens, for example, when the bridge site is coupled to phonons by H:) ∝ c †cB (Figure 15). Then,

Γsup ≈ 8V 4 /κω 2

(superexchange rate constant)

(35)

This is the superexchange mechanism in which the bridge site is not populated but used as a virtual state for ET.40,41 Note that the superexchange rate constant Γsup ∝ Veff2 ∝ V4/ω2 while the diffusive rate constant (eq 34) is Γdif ∝ V2. Therefore, when the donor/acceptor dephasing parameter κ is comparable to the bridge dephasing parameter γ, the diffusive transport dominates the superexchange transfer.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the National Science Foundation under grant number CHE-1112258 (M.R.W.) and by QUBEDARPA SPONSOR: N66001-10-1-4066/P00001.

Figure 15. Diffusive or hopping ET. Bridge sites are populated. 1019



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(38) (39) (40) (41) M. A.

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