Shedding (Incoherent) Light on Quantum Effects in Light-Induced

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Perspective

Shedding (Incoherent) Light on Quantum Effects in Light-Induced Biological Processes Paul Brumer J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b00874 • Publication Date (Web): 15 May 2018 Downloaded from http://pubs.acs.org on May 16, 2018

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The Journal of Physical Chemistry Letters

Shedding (Incoherent) Light on Quantum Effects in Light-Induced Biological Processes Paul Brumer1 1

Chemical Physics Theory Group, Department of Chemistry,

and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario, M5S 3H6, Canada (Dated: May 1, 2018)

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The Journal of Physical Chemistry Letters 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

ABSTRACT: Light-induced processes that occur in nature, such as photosynthesis and photoisomerization in the first steps in vision, are often studied in the laboratory using coherent pulsed laser sources, which induce time-dependent coherent wavepacket molecule dynamics. Nature, however, uses stationary incoherent thermal radiation, such as sunlight, leading to a totally different molecular response, the time-independent steady state. It is vital to appreciate this difference in order to assess the role of quantum coherence effects in biological systems. Developments in this area are discussed in detail.

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The interaction of light with matter is ubiquitous in nature and central to the function of many natural processes.1 Sample issues, of longstanding physical chemistry interest, are bacterial or plant photosynthesis,2 and isomerization in the first steps in vision.3,4 Challenging questions that have emerged5,6 from pulsed laser experiments7–12 have motivated vigorous discussions as to the role of quantum mechanics and "non-trivial" quantum effects (such as interference, entanglement, non-locality, etc.13–16 ) in natural nanoscale biological systems. In addition, information gleaned from these systems are stimulating new directions in lightbased technologies such as photocells,17,18 and recent results have provided new insights into the way in which these systems should be considered computationally and conceptually.19 There is, however, a significant issue regarding the relationship between the results of the primary laboratory methods of studying these processes (via coherent pulsed laser excitation) and natural scenarios (that occur due to excitation with stationary incoherent radiative sources, such as sunlight). These two methods of molecular excitation lead to dramatically different molecular responses, although this difference is often unappreciated. The central issue is readily stated. Modern studies of the interaction of light with matter utilize coherent pulsed laser sources, with ever decreasing pulse durations (nsec to psec to fsec to asec). This has lead to a trend in the approach used to understand these processes. For example, a recent Faraday discussion20 provides an overview of the current focus of experimental, computational and theoretical studies of the excitation of matter by incident light. Pulsed laser intensity, coherence and time duration serve as the underlying themes of the papers presented and of the associated discussions. Indeed, these topics, and particularly coherent wave packet dynamics,21 dominate the approach invoked to understand the interaction of light with matter. However, this approach is not directly relevant to natural light-induced processes, where excitation occurs with natural stationary incoherent radiation (such as sunlight, noise, etc.) As discussed in detail in this perspective, stationary incoherent radiation has a profoundly different effect on molecules than does pulsed coherent laser radiation.22–27 Indeed, tools developed to understand excitation with pulsed coherent sources often lead to misunderstandings as to the characteristics of natural processes. This is particularly true for the issue of the rates of processes as well as coherences, and the time evolution of the processes themselves. This Perspective provides an overview of developments in understanding natural light ACS Paragon 3Plus Environment


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(or noise) induced processes and tools appropriate for their analysis. The first part of this Perspective summarizes features of natural light induced processes and the differences between these features and results emerging from laboratory laser studies. The second part provides sample computational results supporting this view, as well as a discussion of developments in the tools that are emerging to study such processes. We allude below, as sample problems of interest, to light absorption in the components of photosynthetic lightharvesting systems,10,11 and the isomerization of retinal in the first steps in vision.8,28,29 Examples are shown schematically in Figs. 1 and 2. The light absorption in both cases is the first step in a complex sophisticated biological molecular machinery.

Fig. 1: Eight bilin chromophores embedded in their protein environment. They absorb light at the various indicated wavelengths in the Light Harvesting Antenna Phycocyanin 645. From Ref. 30 Note first that the interaction between natural incoherent radiation (e.g. sunlight or noise) and the absorbing molecules is sufficiently weak to allow a first order perturbation theory treatment using a classical electric field-dipole interaction. That is, the Hamiltonian is given by H = H M − µ · E(t).

(1)

where H M is the Hamiltonian of the molecule with H M |αi = Eα |αi = ~ωα |αi, µ the dipole operator, and E(t) is the electric field. A straightforward application22,32 of perturbation theory gives the excited state after excitation from the ground state |gi (where spontaneous emission and other relaxation mechanisms are neglected here, but included below): ACS Paragon 4Plus Environment

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Fig. 2: Schematic representation of cis-trans isomerization of retinal upon absorption of light, the first step in animal vision. This structure is embedded in a complex protein environment that is not shown. From Ref. 31

ραβ (t) ≡ hα|ρ(t)|βi µα µ∗β −iωαβ t = 2 e ~

Z

t iωαg τ2

Z

t

dτ2 e −∞

dτ1 e−iωβg τ1

(2)

−∞

×hE(τ1 )∗ E(τ2 )i , with ωαβ = ωα − ωβ and µγ = hγ|µ|gi. Hence, natural light-induced biological processes are described by ραβ (t) which, in turn, reflect hE(τ1 )∗ E(τ2 )i. Here we have utilized a density matrix description of the state ρ of the molecule, which is required if we are to treat a variety of light sources. That is, a formalism based on wavefunctions and the associated Schrödinger equation is appropriate only if we possess maximal information on both the system and on any incident perturbation. (If any uncertainty is present, it arises solely from quantum limitations.) However, a density matrix description33 is required if either the system or the perturbation possesses additional uncertainties. This is indeed the case for natural incoherent radiation which is described by an ensemble, as discussed below. Note that the ensemble description of the incident light necessitates an ensemble description of the system as well. In accord with Eq. (2) the key relevant characteristic of the incident radiation is, for these processes, encapsulated in the degree of first order coherence g (1) , defined by the normalized ACS Paragon 5Plus Environment


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first order correlation function:

g (1) = hE ∗ (τ2 )E(τ1 )i/|E ∗ (τ2 )E(τ1 )|.

(3)

Note that the process is linear and described by g (1) so that a treatment involving quantum light is unnecessary; use of classical light is equally valid34 . Above, the density matrix ρ is written in the basis of energy eigenstates |αi of the system Hamiltonian H M . Hence ραα (t) denotes the populations of energy eigenstate α and ραβ (t), with α 6= β, indicate coherences between energy eigenstates |αi and |βi. Particular attention should be paid to the latter, which are of two types. The first, where the coherences are time dependent, encapsulate the time evolution of the system. The second, also discussed below, indicate time-independent (stationary) coherences, associated with coherences between energy eigenstates when the system is either at equilibrium or in a steady state. Note, however, that “coherences" here do not refer to the spatial spread of a wavefunction over sites in the system; an alternative, sometimes used, terminology.35 Below, differences between natural light induced processes and laboratory coherent laser experiments are emphasized. Specifically, we note the following qualitative differences. (a) The nature of the light: pulsed laser excitation experiments, such as pump-dump studies, or 2D photon-echo utilize well defined pulsed coherent laser light E(t) leading to an analytic time-dependent perturbation incident on the molecule. Such fully coherent sources are characterized in first order36 by g (1) = 1 . In sharp contrast, natural light is incoherent, i.e., an ensemble of E(t) fields entirely described by its statistical properties. In the case of stationary thermal blackbody radiation the first order correlation function is given by37 , with τ = τ1 − τ2 , ~ hE (τ2 )E(τ1 )i = 2 3 6π 0 c ∗

Z

∞

dω ω 3 n(ω)e−iωτ ,

(4)

0

with n(ω) = (e~ω/(kB T ) −1)−1 being the average photon number at temperature T and kB the Boltzmann constant. This essentially corresponds to blackbody radiation with a coherence time of a few fsec. Note that the above blackbody correlation function assumes a fully stationary thermal source, i.e., with no consideration of the time associated with turning on the light, an important issue discussed later below. ACS Paragon 6Plus Environment

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The fact that excitation with weak natural weak incoherent light is described entirely as an ensemble characterized by hE ∗ (τ2 )E(τ1 )i statistics has deep formal, computational, and conceptual consequences. Specifically, there are a myriad of possible, equally acceptable, realizations that can be used as the basis to describe the ensemble. Examples range from a collection of photons (number states)36 to uniformly distributed coherent states32 to phase randomized CW fields.37 None of these realizations of the incident incoherent light is more "real" than any other. Indeed, the only way that a specific realization can be regarded as "real" is if the radiation is incident on a detector that measures that specific property. Further analysis is currently ongoing to disentangle the measurement process in several processes in nature.38 However, it is important to note that the natural molecular-weak light induced process is insensitive to whatever realization mechanism that may ultimately be established, being sensitive, rather, to the ραβ (t) in Eq. (2). That is, the system is responsive to hE ∗ (τ2 )E(τ1 )i which is an ensemble average. Several additional differences distinguish the radiation associated with typical pulsed laser sources and natural incident light. Principal amongst these are time scales. That is, constantly improving pulsed laser technology yields light pulses with pulse durations on the order of those of rotation, vibration, and electronic motion (e.g., nsec to asec.) Hence, excitation with these light sources can produce dynamics on these ever faster time scales. By contrast, natural incident incoherent light is on "forever" from the molecular viewpoint, e.g., minutes in the case of photosynthesis and seconds (between blinks) in the case of vision. Transient turn-on time effects, as shown below, are generally irrelevant. (b) The molecular state subsequent to excitation. As can be readily appreciated and, in accord with Eq. (2), excitation with pulsed coherent laser light or natural incoherent light sources constitute excitation with different perturbations to the system and, hence, will generate different responses, with major consequences. Two are readily noted. (b1) Time Evolution and Coherence: Pulsed coherent laser excitation induces transient coherent molecular dynamics, manifest formally in time-dependent molecular coherences ρα,β , often measured as time-dependent oscillations in the optical polarization.10–12 In a totally isolated molecular system these coherences will survive indefinitely barring spontaneous emission. In an open system, i.e., a system coupled to an environment, they will decay due to interactions with the surroundings, providing a measure of the strength and nature of the interaction of the system with its environment. By contrast, in the case of natural ACS Paragon 7Plus Environment


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incoherent excitation the molecule sees a time-independent (stationary) perturbation that leads to a stationary process, i.e., there is no dynamics.22,24 As noted below, there may be coherences in the system, but they are time-independent. Two comments make this qualitatively clear. First, since the incident perturbative light is stationary (as is any coupling to the environment), i.e., devoid of time dependence, then so too is the state of the entire system, molecule included. Second, the stationary radiation field is a blackbody with which the system comes to steady state. Such systems, as is well known, display no average time dependence. This difference implies a re-evaluation of the implications of coherent pulsed laser experiments, both historical8,9 and recent,10–12 that display time evolving coherences in photosynthetic components such as FMO or PC64510,11 , or visual components like retinal isomerization in vision.8,9,12 In particular, these experiments, which display time-dependent coherences and their decay over time, provide important information on the nature of the chromophore and its coupling to its environment. This information is crucial for the determination of the Hamiltonian of such systems, necessary for modeling and understanding the result of any excitation. However, the coherences observed in these experiments are induced by the laser pulses, and will not be observed in nature. Finally, we note that a system in contact with incident incoherent radiation, and possibly coupled to an external environment, is best treated using a Master Equation approach.39,40 Given the system Hamiltonian H M we can write the time evolution of the system density matrix ρs as Lρs = dρs /dt

(5)

where L is the Liouville operator that incorporates all effects. That is, L includes the time evolution of the system in the absence of any environment plus effects due to the coupling to any material environment, and coupling to the radiation field. At longer times the system reaches either an equilibrium state or a steady state,41–43 characterized by solutions to the equation Lρs = dρs /dt = 0.

(6)

Solutions to Eq. (6) would occur if the system is, for example, coupled to incoherent radiation, or coupled to both incoherent radiation and a material environment. (b2) Rates. Although the term "rate of a process" is often stated as if it is unique, ACS Paragon 8Plus Environment

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a rate is, in fact, dependent on the circumstances by which the system is prepared and how it is measured.44,45 It should therefore not be surprising that the time scales measured with transient pulsed laser excitation can be significantly different from rates under natural conditions. For example, the rate obtained in a pulsed laser experiment may well depend upon the duration of the laser pulse: faster pulses can induce faster light-induced dynamics. Hence, rates measured in this way are not those that occur in nature when excitation is with incoherent light and the process occurs in the steady state. To see this formally, consider any of the radiationless transition processes of interest, such as internal conversion, intersystem crossing, electronic energy transfer between sites, unimolecular decay, photodissociation, as well as others that correspond to transitions between one part of Hilbert space and another. These are describable by a Hamiltonian of the form: H M = H1 + H2 + H12 , (i)

(7)

(i)

(i)

where Hi |j i = j |j i. Typically, pulsed laser studies are concerned with the time scales P (1) for evolving from an initial state |φi = j cj |j i within H1 , prepared by laser excitation ¯ jg )h(1) |µ|gi reflect the from a ground state |gi, to states within H2 . The coefficients cj ≈ E(ω j

¯ jg ) is the amplitude of the incident laser at the tranlaser-molecule interaction, where E(ω (1)

sition frequency ωjg between levels |j i and |gi. Expanding |φi in terms of the eigenstates |αi of H M gives the time dependence of |φi as: |φ(t)i =

X

(1)

cj |αihα|j ie−iEα t/~ ,

(8)

αj (2)

with the probability P2 (t) of having moved to states |k i in H2 at time t given by: P2 (t) =

X

(2)

| hk | φ(t)i |2 =

k

X X X (2) (1) | cj hk | αihα | j ie−iEα t/~ |2 k

j

(9)

α

Qualitatively21,44 , the time dependence of P2 (t) is dictated by the product of the energy P (2) (1) widths of two terms, the molecular property D(E) = α hk | αihα | j i and the state preparation coefficents cj . Specifically, the wider in energy the D(E), the faster the possible P2 (t) dynamics. (Here, E and the width of D(E) are implicitly defined by the range of (2)

(1)

nonzero overlaps hk | αihα | j i.) However, this maximal width (shown schematically as a black curve in Fig. 3) is modulated by the cj coefficients, which reflect characteristics of the incident laser light. An incident long-duration pulse produces a narrow cj energy ACS Paragon 9Plus Environment


width (e.g., the green curve in Fig. 3) and hence slower than the fastest possible dynamics. Shortening the laser duration (e.g., the red curve in Fig. 3) produces faster dynamics. Hence, for example, a nsec laser tends to induce dynamics on the nsec timescale whereas, for the same molecular system, a fsec laser may be able to induce fsec timescale dynamics. Ultimately, as the incident pulse gets faster, the pulse bandwidth exceeds the width of black D(E) curve, resulting in the fastest rate for the process.

Fig. 3: Schematic representation of contributions to pulsed laser excitation of a molecule with characteristic molecular profile D(E) in black and laser induced profiles in green and red, as discussed in the text.

Hence, the observed coherence and associated time scales produced by the coherent laser pulse,21 are distinct from those in natural processes where excitation with natural incoherent light results in a mixture of stationary eigenstates of H M , and associated steady-state rates. For example, studies26,46 of this kind show, for the case of incoherent excitation with natural light, that the slow rate of light absorption can be a rate-determining step. Thus, for example, natural isomerization of retinal shows a steady state rate, discussed below, of ACS Paragon10 Plus Environment

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(a)

D

J

(b)

A

ee+

Δ=2J

ΓRC Γ1

r

γ

Γ2

RC

RC

g

μ1

μ2

Fig. 4: A minimal photosynthetic dimer model in the site basis (a) and the energy eigenstate basis (b). The donor (D) and acceptor (A) chromophores are modeled by two-level systems coupled by the Coulomb interaction J. Only the donor is assumed to be excited by incoherent light. The extended V-type system consists of the eigenstates of the symmetric dimer in the single-excitation subspace (|gi and |e± i) coupled to the reaction center (RC). From Ref. 47 .

greater than a msec, rather than the pulsed laser < 200 fsec rate.8,9,12

Model Computations: A great deal of the overview discussed above is supported by exact solutions to minimal model problems that contain much of the physics of interest. Consider, for example, the systems shown in Fig. 4 which demonstrates a model two-site Donor (D)Acceptor (A) system. Two bases are shown, the site basis, with a ground and excited state on each site, and the same system transformed to the eigenstate basis. As is common, the latter neglects the joint excited state of both the donor and acceptor, whose population is assumed negligibly small due to the weakness of the exciting light. This system clearly contains many of the essential features of interest in light harvesting, including the continuous light-induced excitation of the donor state by the incoherent light and associated stimulated emission from the excited states back to the ground state (yellow arrows), the relaxation of excited state populations via environmental couplings with rates Γi (yellow arrow) and spontaneous emission rate γ (red arrow) and excited state decoherence, as well as deposition of the energy into the reaction center (RC). This model serves as motivation for studies of the three level “V-system” discussed below. ACS Paragon11 Plus Environment


e1

e2

Δ γ2

γ1 ω0 r1

r2

g

Fig. 5: Schematic representation of a V-type system. ∆ is the excited state splitting, γi is the radiative line-width, and ri is the incoherent pumping rate of excited state |ei i. From Ref. 48

Consider the case where there is no coupling to the reaction center, shown in Fig. 5 as a three level system in the energy eigenstate basis. (The notation in this figure is standard for the V-level excitation without transport to RC.) Upon irradiation the system, starting in the ground state, is excited to the higher lying states and the system establishes an equilibrium with the ongoing incident incoherent radiation field. Two regions are of interest, the initial transient region associated with the initial excitation, and the longer time steady-state result. Many of the initial studies of this type assumed the sudden turn-on of the radiation field, e.g., Ref. 49. However, even elementary perturbation theory indicates that the sudden turnon of any perturbation creates time-dependent coherences. Hence these studies were aimed at understanding the subsequent relaxation to the steady state associated with the system interacting with the ongoing radiation field, which functions as a thermal bath. However, and crucially, natural light-induced processes occur with a finite turn-on time which is extremely long on molecular time scales.48,50 For example, typical turn-on times for photosynthesis can be seconds, and a "blink of the eye", which constitutes turn-on for the visual light-initiated dynamics (discussed below), is ≈ 0.1 sec. Consider then Figs. 6 and 7 which show (using γ1 = γ2 = γ and r1 = r2 = r corresponding to sunlight, i.e. 5800o K) the two dramatically different types of coherences computed48 in these V-level studies, and the effect of turn-on time scales on these coherences. Specifically, ACS Paragon12 Plus Environment

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Fig. 6 shows results for the "small molecule regime", where the ratio of level spacing between excited levels |e1 i and |e2 i to the radiative line width is large, i.e. ∆/γ 1. In this regime the coherences are seen to display familiar characteristic oscillations that decay on a time scale dictated by the radiative lifetime γ. Three sets of panels are shown in Fig. 6 corresponding to different ranges of τr , a measure of the turn-on time of the radiation, τ∆ = 1/∆, the characteristic time for the dynamics of the molecule, and τγ = 1/γ, the radiative lifetime. As the turn on time exceeds the characteristic time for the molecular dynamics (Panels B and C) the magnitude of the excited state population is unchanged, but the coherences diminish by orders of magnitude. (Note the different ordinate scales in the three lower figures.) For example, for the case of τr = 100 τγ (Panel C), the coherences ρe1 ,e2 are 1000 times smaller than in the case of the faster turn on (τr = 0.024 τ∆ ) shown in Panel A. It is significant to note, however, that even Panel C corresponds to rapid turn-on relative to turn-on times of natural processes. That is, with a typical τγ = 1 nsec, the turn-on time here is τr = 10−7 sec. However, typical turn-on times in nature are greater than 10−3 sec, where, following the trend in Fig. 6, coherences would be entirely negligible. Figure 7 shows the analogous results in the domain associated with “large molecules" where the spacing between excited state energy levels that are excited is small compared to the radiative linewidth γ. Turn-on times τr are compared to τs = 2γ/∆2 , a measure of the time scale of the dynamics, and τγ , the radiative lifetime. Here, the coherences are seen not to oscillate and can live longer than τγ , where smaller ∆ gives longer coherence lifetimes. However, as in Fig. 6, coherences diminish by orders of magnitude as the turnon time lengthens compared to the time scale of the molecular dynamics. Once again then, coherences will be entirely negligible on the time scales of natural turn-on times in biological systems. These results strongly support the view that the coherences experimentally observed in pulsed laser experiments on FMO, PC645, retinal, etc. are a result of the use of coherent pulsed lasers, and that these coherences are not expected to occur in nature (or man-made photocells) where turn-on times are long on the time scales of molecule dynamics, and the light is incoherent. Rate Computations. In addition to issues related to the coherences discussed above, “rates” obtained from pulsed laser studies and from steady state processes differ substantially. ACS Paragon13 Plus Environment


A. τr ≪ τ∆ ≪ τγi

0.08

0.02

0.04 0.02

−4

10

−2

0

10

10

0

2

10

−4

−2

0

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1

ρe1 ,e2 (t)

−2

3

−4

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t

2

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−1

0

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x 10

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x 10

0.5

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x 10

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−6

0.04 0.02

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−3

4

C. τ∆ ≪ τγi ≪ τr

0.06

ρei ,ei (t)

0.04

0

0.08

0.06

ρei ,ei (t)

ρei ,ei (t)

0.06

B. τ∆ ≪ τr ≪ τγi

ρe1 ,e2 (t)

0.08

ρe1 ,e2 (t)

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1 0

−4

10

−2

0

10

10

2

10

t

−1

−4

10

−2

0

10

10

2

10

t

Fig. 6: Evolution of populations (upper panels) and coherences (lower panels) of an “underdamped" V-system (i.e., where ∆/γ 1). Here γ1 = 1.0 = γ2 = γ and ∆ = 24.0. Three different turn on regimes are shown here. Panels A show the ultrafast turn-on of the field with τr = 0.024τ∆ while Panels B and C show the intermediate (τr = 24τ∆ ) and slow (τr = 100τγ ) turn-on regimes respectively. Note the difference in ordinate scales for the coherence plots. Solid red lines indicate the real part of the coherence ρR e1 e2 with the imaginary part ρIe1 e2 indicated by the dashed blue line. From Ref. 48. Indeed, many computational physical chemistry tools designed to determine timescales of radiationless transition processes are built to treat ultrafast (e.g., psec and faster) processes induced by pulsed laser excitation. As such, they are not appropriate for the longer time processes associated with establishment and maintenance of the steady state under incoherent excitation. For such processes the most general route would be to extract the rate from the solution to the steady state equation [Eq. (6)] Lρs = 0. We have recently developed such a general formalism46 and applied it to a number of systems. As noted above and for cases studied, the results implied that the steady-state rate for natural processes excited by incoherent solar can be limited by the rate of absorption of the weak incident light, and is hence markedly slow. However, the general structure is clear from an earlier,26 less formal, procedure that we had applied to a model of retinal photoisomerization, the process shown in Fig. 2. That is, if the probability of producing the final trans state in cis-trans phoACS Paragon14 Plus Environment

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A.τr ≪

τγ 2

0.08

B. 2γ ≪ τr ≪ τs

0.08

0.06

ρei ,ei (t)

ρei ,ei (t)

0.06 0.04 0.02 0

τ

τ

≪ τs

0.04 0.02

0

10

3

10

0

6

10

C. 2γ ≪ τs ≪ τr

0.06

ρei ,ei (t)

0.08

0.04 0.02

0

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x 10

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ρe1 ,e2 (t)

0.04

ρe1 ,e2 (t)

ρe1 ,e2 (t)

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


3

10

6

10

t

0

0

10

3

10

6

10

t

Fig. 7: Evolution of populations (upper panels) and coherences (lower panels) of a V-system in the “overdamped" region (i.e., where ∆/γ 1). Here γ1 = 1.0 = γ2 and ∆ = 0.001. Three different, turn-on regimes are shown. Panels A show the ultrafast turn-on of the field with τr = 10−3 τγ while Panels B and C show the intermediate (τr = 100τγ = 5 × 10−5 τs ) and slow (τr = 20τs ) turn-on regimes, respectively. Note the difference in ordinate scales for the different coherence plots. From Ref. 48.

toisomerization is found to grow linearly in time, then the rate of product production can be found by noting that the linear growth continues in time and that the rate is given by the linear slope. This was found to be the case for retinal isomerization in a minimal 1-D model,26 and verified in a 2-D model. Sample results, with further details provided in Ref. 26, are shown in Fig. 8 for a minimal model of retinal isomerization as a function of the luminescence L of the source (bottom panel) and of coupling to an external environment (top panel) whose strength is defined by η. The straight line and associated slope provide a direct means of determining the rate of the process. Timescales were found to be in excess of a msec, in stark contrast to the fsec rates seen in laser induced isomerization. This is a clear example of the interplay of the results of pulsed laser experiments and steady state calculations. The former provides insight into the system Hamiltonian and its ACS Paragon15 Plus Environment


Ptrans(t) x10

18

2

(L = 0.03 Cd/m )

coupling to the environment, which informs the steady state calculation. 1.0

(a)

η = 50 η = 25 η = 12.5

0.8 0.6 0.4 0.2

18

(η = 25)

1.0

Ptrans(t) x10

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

(b)

2

L = 0.060 Cd/m 2 L = 0.030 Cd/m 2 L = 0.015 Cd/m

0.8 0.6 0.4 0.2 0.0 0

1

2

3 4 time (ps)

5

6

7

FIG. 8: a) Time dependence of Ptrans (the probability of forming trans retinal by photoisomerization of cis-retinal) for three η values with incoherent light luminescence L = 0.03 Cd·m−2 . Here, η defines the strength of the coupling of the retinal model to its environment. b) Time dependence of Ptrans for various values of L, with system–environment coupling η = 25. In all cases there is a deviation from strictly linear behavior at the early times that corresponds to timescales of isomerization dynamics due to the sudden turn-on of the radiation in this computation. From Ref. 26. Steady State Coherences. The coherences discussed above arise from the interaction between the system and a thermal “bath" of incoherent radiation. They are transient or irrelevant coherences because of the nature of the coupling between the molecular system and the radiative bath, as discussed in Ref. 51. Alternative couplings, which can arise between the molecular system and a phonon environment (such as a molecule coupled to a protein environment) can allow for stationary coherences which are substantial and that do survive in time. Indeed, although light-induced coherences are irrelevant in nature, these “stationary coherences" due to system-bath interactions have been proposed52 to occur in both equilibrium systems and in steady-state systems showing transport, e.g. between two baths, and may well be biologically relevant. In addition, the presence of stationary ACS Paragon16 Plus Environment

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coherences in either equilibrium53 or steady state47,54–56 presents an interesting possibility to observe and sculpt dynamics through the system-bath interaction. To gain insight into the magnitude of such steady state coherences consider57 a simple analytically tractable Bloch-Redfield model that is an extension of that in Fig. 5. Here, a three level V-System with ground state |gi and excited states |e1 i and |e2 i, interacts with an incoherent radiation and, in addition, with a non-radiative phonon bath. Energy is transferred from the former to the latter. As above, the ground and excited state manifolds are separated by a transition frequency ω0 while the two excited states are separated by an energy ∆ = ~ω12 where ω12 is the transition frequency between |e1 i and |e2 i. The radiative bath drives absorption and stimulated emission between |gi and |ei i with incoherent pumping rate ri and drives spontaneous emission at a rate γi . Similarly, a non-radiative bath (not shown in Fig. 5) drives absorption and stimulated emissions between these states with pumping rate Ri and spontaneous emission rate Γi . In many cases of interest, non-radiative relaxation processes occur on timescales much faster than fluorescence; therefore it is reasonable to assume Γi γi . Further, the phonon bath in these systems is far too low of a temperature to drive excitations between the ground and excited states and so ri Ri . giving the weak pumping limit ri γi Γi . in PSBR Theory. Overall, this gives a V-system model pumped by the photon bath with rates ri while spontaneous relaxation is primarily driven by the non-radiative bath with relaxation rates Γi . Hence, the model is similar to that in Fig. 5 but with γi replaced by Γi . The magnitude of the real and imaginary parts of the steady state coherences can then be analytically obtained57 as (for radiative pumping rates r1 = r2 = r) lim

t→∞

ρR e1 ,e2 (t)

√ √ √ Γ1 Γ2 ( Γ1 − Γ2 )2 √ √ = Γ1 + Γ2 ( Γ1 − Γ2 )2 + 2∆

lim ρIe1 ,e2 (t) = −

t→∞

∆ r+

1 (Γ1 2

+ Γ2 )

lim ρR 1,2 (t)

t→∞

(10a)

(10b)

Hence, the system establishes a nonzero stationary coherence for Γ1 6= Γ2 . Interestingly, it vanishes for equal relaxation nonzero rates Γ1 = Γ2 . This same calculation can be repeated for arbitrary values of r1 and r2 . The resultant expression is extremely unwieldy, but it can be shown57 that steady-state coherences vanish when r1 /Γ1 = r2 /Γ2 . In essence, it is the competition between the two baths, the radiative and non-radiative, with “incompatible” rates that here leads to the steady-state coherences. That is, these steady-state coherences ACS Paragon17 Plus Environment


are manifestations of a non-equilibrium steady state arising due to the system behaving as a transport medium between two baths. It is of great interest, in future work, to ascertain the extent to which these coherences participate in biological processes and the extent to which they manifest quantum features.56,58 Work of this kind is ongoing in our research group. Computational and Experimental Tools: Finally, we note that successful computational studies of such natural radiationless transition processes requires the time-dependent excitation of a molecule with steady-state sources for both adiabatic and non-adiabatic scenarios.19 Since the molecular system of interest is coupled to an incoherent excitation source, and possibly to an external environment as well, master equation treatments39,40 provide the preferred approach. To this end, it is advantageous to note a number of recent relevant developments. These include a partial secular master equation for electronic excitation,59 non-secular completely positive master equations for model systems, and a highly efficient approach to computing steady state rates.46 However, despite these advances, a surprising number of basic issues require further attention. These issues arise because treating the quantum master equation usually requires introducing a number of approximations. These can include the Markovian approximation, where the dynamics of the environment is assumed to evolve much faster than the system, the secular approximation, where the evolution of populations and coherences are decoupled from one-another, and the second-order approximation, where the master equation is truncated at second order in the system-environment interaction. For example, a direct experimental test of the regimes of validity of the secular approximation has only recently been proposed,54 and questions regarding the range of applicability of the Markov approximation and the second-order approximation require further examination,60–62 as do issues of excitation and final state description.63 Similarly, issues have arisen related to “global" vs. “local" master equations, essentially a question of the proper basis in which to set up and deal with master equations for various systems.64–69 A focus on the steady state also calls for new experimental directions. That is, experimental studies on light-induced processes relevant to biological systems have, as noted above, focused on pulsed laser excitation. Such studies yield insight into the nature of the system Hamiltonian and its coupling to the environment. However, as shown above, experimental studies involving excitation with weak incoherent sources would provide information that directly reflects processes as they occur in nature. It is the case that a number of studies ACS Paragon18 Plus Environment

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on light-harvesting systems have been carried out with incoherent sources. However these experiments have focused on effects due to nonlinear, rather than the required linear, excitation. To this end we have proposed an experimental approach using available pulsed laser equipment whose measured signal would correspond to result of incoherent excitation. Specifically,32 one irradiates the system with a sequence of pulses, each with a spectrum analogous to that of solar radiation, and assembles the collective output signal. In doing so, the detector averages over pulse times and yields the incoherent excitation result. The computed results were shown to be exactly that of the (sudden turn-on of) incoherent excitation and we look forward to the experimental implementation of this and related protocols. In summary, pulsed laser experiments provide important information on system and system-environment properties. However, the light-induced coherences that have been observed in pulsed laser experiments on photosynthesis and on the first photoisomerization step in retinal will not appear in nature where incoherent light is turned on slowly and yields a long-time steady-state process. Alternative coherences relating to the coupling of the system and surroundings may, however, reveal interesting quantum effects. Work on the latter is in progress in our laboratory. ACKNOWLEDGMENTS: It is a pleasure to thank former members of my research group who have made major contributions to this work, including Amro Dodin, Prof. Timur Tscherbul and Prof. Leonardo Pachon. Working with them has been a very rewarding experience. I also thank Susan Arbuckle for technical and administrative support. She is always keen to ensure the best of our output. This work was supported by the US AFOSR through Grants number FA9550-17-1-0310 and FA9550-13-1-0005.

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Tscherbul, T. V.; Brumer, P. Quantum Coherence Effects in Natural Light-Induced Processes: cis-trans Photoisomerization of Model Retinal Under Incoherent Excitation. Phys. Chem. Chem. Phys. 2015, 17, 30904.

30

Marin, A.; Doust, A. B.; Scholes, G. D.; et al. Flow of Excitation Energy in the Cryptophyte Light-Harvesting Antenna Phycocyanin 645. Biophys. J. 2011, 101, 1004-1013.

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Chenu, A.; Brumer, P. Transform-Limited-Pulse Representation of Excitation with Natural Incoherent Light. J. Chem. Phys. 2016, 144, 044103. ACS Paragon21 Plus Environment


33

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Axelrod, S.; Brumer, P. Manuscript in preparation.

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Tscherbul, T. V.; Brumer, P. Long-lived Quasistationary Coherences in V-type System Driven by Incoherent Light. Phys. Rev. Lett. 2014, 113, 113601.

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Grinev, T.; Brumer, P. Realistic vs Sudden Turn-On of Natural Incoherent Light: Coherences and Dynamics in Molecular Excitation and Internal Conversion. J. Chem. Phys. 2015, 143, ACS Paragon22 Plus Environment

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54

Dodin, A.; Tscherbul, T.; Alicki, R.; Vutha, A.; Brumer, P. Secular versus Nonsecular Redfield Dynamics and Fano Coherences in Incoherent Excitation: An Experimental Proposal. Phys. Rev. A 2018, 97, 013421.

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ACS Paragon Plus Environment

coherent/ incoherent light

molecular dynamics/ steady state


Paul Brumer—Brief biography

Professor Paul Brumer is a Distinguished University Professor and the Roel Buck Professor of Chemical Physics in the Department of Chemistry, University of Toronto. He received his B.Sc. in Chemistry at Brooklyn College, his Ph.D. in Chemical Physics at Harvard University, and was subsequently a postdoc at the Weizmann Institute of Science and at the Harvard College Observatory. His work, mainly on coherence, decoherence and incoherence in light-induced quantum dynamics has been recognized through numerous awards, including the Palladium Medal of the Chemical Institute of Canada, the Noranda Award of the Canadian Society for Chemistry, and the Killam Memorial Prize for Natural Sciences. He is a Fellow of the American Physical Society, the Chemical Institute of Canada, and the Royal Society of Canada.


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1411x1058mm (72 x 72 DPI)


Shedding (Incoherent) Light on Quantum Effects in Light-Induced

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