# Electronic-vibrational resonance does not alter steady-state transport in
natural light-harvesting systems

Leonardo F. Calderón<EMAIL_ADDRESS>Chemical Physics Theory
Group, Department of Chemistry, and Center for Quantum Information and Quantum
Control, University of Toronto, Toronto, Ontario M5S 3H6, Canada  Chern Chuang
<EMAIL_ADDRESS>Chemical Physics Theory Group, Department of
Chemistry, and Center for Quantum Information and Quantum Control, University
of Toronto, Toronto, Ontario M5S 3H6, Canada  Paul Brumer
<EMAIL_ADDRESS>Chemical Physics Theory Group, Department of
Chemistry, and Center for Quantum Information and Quantum Control, University
of Toronto, Toronto, Ontario M5S 3H6, Canada

###### Abstract

Oscillations in time-dependent 2D electronic spectra appear as evidence of
quantum coherence in light-harvesting systems related to electronic-
vibrational resonant interactions. Nature, however, takes place in a non-
equilibrium steady-state, so the relevance of these arguments to the natural
process is unclear. Here we examine the role of intramolecular vibrations in
the non-equilibrium steady-state of photosynthetic dimers in the natural
scenario of incoherent light excitation. It is found that vibrations resonant
with the energy difference between exciton states do not increase the quantum
yield nor the imaginary part of the intersite coherence that is relevant for
transport compared with non-resonant vibrations in the natural non-equilibrium
steady state. That is, the vibration-electronic resonance interaction does not
alter energy transport under natural incoherent-light excitation conditions.

Grupo de Física Computacional en Materia Condensada, Escuela de Física,
Facultad de Ciencias, Universidad Industrial de Santander, Cra 27 calle 9,
Bucaramanga, Colombia.

Ultrafast spectroscopy has provided crucial information about the specific
vibronic structure of biological light-harvesting systems and their
interaction with the surrounding environment 1, 2, 3, 4, 5, 6. The role of
intramolecular vibrations in energy transfer has been examined in coherent
pulsed laser excitation studies, where it has been argued that the electronic-
vibrational resonant interaction can be responsible for energy transfer
enhancements, the character of quantum coherence associated with nonlinear
spectroscopic signals, and for non-trivial quantum behaviour of collective
pigment motions 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28. However, these laboratory-designed/controlled
conditions differ fundamentally from nature 29, 30, 31, 32, 33, 34, 35, 36,
where the light-harvesting systems are continuously illuminated by incoherent
natural radiation (sunlight), leading to a non-equilibrium steady-state (NESS)
37, 35, 36, 38, 39. The key issue addressed in this letter is the extent to
which vibration-electronic resonance affects electronic energy transport in
the NESS.

We consider a prototype photosynthetic dimer to evaluate the impact of the
electronic-vibrational resonance and vibronic coupling under natural
incoherent light excitation. Specifically, we analyze a model PEB dimer in the
cryptophyte algae PE545 antenna protein 40, 41, 42. Each chromophore
($\mathrm{PEB_{50/61C}}$ and $\mathrm{PEB_{50/61D}}$) is modelled as a two-
level system coupled to a quantized high-energy intramolecular vibrational
mode and is excitonically coupled with one another. We solve for the steady-
state of this vibronic dimer (system) using an open quantum system approach.
We model the sunlight as a blackbody radiation bath, the protein/solvent
environment as a thermal phonon bath, and consider other processes such as
exciton recombination and exciton harvesting (trapping at the reaction
center), acting on nanosecond and picosecond time-scales, respectively, via
effective Lindbladians 43, 37, 39, 38, 44.

In this Letter, we analyze the different non-unitary contributions from the
environment on the NESS and its relation to the vibronic effects. First, the
electronic-vibrational resonance pattern is analyzed by computing the quantum
yield for different Huang-Rhys factors, intramolecular vibrational
frequencies, as well as the system-phonon bath coupling strengths. The quantum
yield is contrasted with the imaginary part of the intersite coherence, which
dictates the transport between the chromophores. We then compute the dynamics
assuming a coherent excitation initial condition to look for similarities in
the vibronic resonance pattern concerning steady-state results with different
system-phonon bath coupling strengths. In doing so, we demonstrate that _the
vibration-electronic resonance interaction does not alter the NESS energy
transport under natural incoherent-light excitation conditions._

Model.—Light-induced energy transport in molecular aggregates depends on the
interaction between the electronic and nuclear degrees of freedom, the latter
being associated with intermolecular and intramolecular vibrations 45, 15. As
is done here, the intermolecular vibrations are usually treated as a low-
frequency continuous phonon bath described by a spectral density, and the
intramolecular vibrations as discrete underdamped high-frequency vibrations 8,
15. In what follows, we denote the coupling and resonance interaction between
electronic excitations and intramolecular vibrations as vibronic coupling and
vibronic resonance, respectively.

Consider a prototype photosynthetic vibronic dimer system, immersed within a
protein-solvent environment and excited by incoherent sunlight. For the
$i^{\mathrm{th}}$ chromophore (site/molecule) only the electronic ground state
$|g_{i}\rangle$ and the electronic first excited state
$|\varepsilon_{i}\rangle$ are taken into account (two-level system
approximation), and a single intramolecular vibrational mode of frequency
$\Omega_{i}$ per chromophore is considered. The electronic creation
(annihilation) operators
$\hat{\varphi}_{i}^{+}\left(\hat{\varphi}_{i}^{-}\right)$ are defined by
$|\varepsilon_{i}\rangle=\hat{\varphi}_{i}^{+}|g_{i}\rangle,\,|g_{i}\rangle=\hat{\varphi}_{i}^{-}|\varepsilon_{i}\rangle$.
The intramolecular vibrational creation (annihilation) operators
$\hat{c}_{i}^{\dagger}\left(\hat{c}_{i}\right)$ then define the intramolecular
vibrational excited state
$|\chi_{i,\nu}\rangle=\left(\hat{c}_{i}^{\dagger}\right)^{\nu}/\sqrt{\nu!}|0_{i}\rangle$,
where $\nu$ is the vibrational quantum number, and $|0_{i}\rangle$ stands for
the vibrational ground state. The Hamiltonian of the $i^{\mathrm{th}}$
chromophore reads

$\hat{H}_{i}=\varepsilon_{i}\hat{\varphi}_{i}^{+}\hat{\varphi}_{i}^{-}+\hbar\mathcal{G}_{i}\hat{\varphi}_{i}^{+}\hat{\varphi}_{i}^{-}(\hat{c}_{i}^{\dagger}+\hat{c}_{i})+\hbar{}_{i}\Omega_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i},$
(1)

where $\mathcal{G}_{i}=\Omega_{i}\sqrt{S_{i}}$ is the vibronic coupling at
frequency $\Omega_{i}$, and $S_{i}$ is the dimensionless Huang-Rhys factor
that characterizes the electron-vibration coupling.

The global system-bath Hamiltonian reads
$\hat{H}=\hat{H}_{\mathrm{Sys}}+\hat{H}_{\mathrm{Sys-B}}+\hat{H}_{\mathrm{B}}$.
Consider then a vibronic dimer, which is the system of interest from an open
quantum system perspective, composed of two interacting chromophores $A$
(acceptor) and $D$ (donor), where the intersite interaction is assumed to be
dipolar and represented by the electronic coupling $V_{AD}$
($V_{AA}=V_{DD}=0$). The vibronic dimer Hamiltonian is given by

$\hat{H}_{\mathrm{Sys}}=\sum_{i}^{A,D}\varepsilon_{i}\hat{\varphi}_{i}^{+}\hat{\varphi}_{i}^{-}+\sum_{i\neq
j}^{A,D}V_{ij}\hat{\varphi}_{i}^{+}\hat{\varphi}_{j}^{-}+\sum_{i}^{A,D}\hbar\mathcal{G}_{i}\hat{\varphi}_{i}^{+}\hat{\varphi}_{i}^{-}(\hat{c}_{i}^{\dagger}+\hat{c}_{i})+\sum_{i}^{A,D}\hbar\Omega_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}.$
(2)

The intramolecular vibrational degrees of freedom are treated explicitly in
the vibronic dimer model 8, 15, while we discuss an electronic dimer model
without the former in the Supporting Information. Several discrete
intramolecular vibrations are reported for cryptophyte algae 40, 41. Here, we
performed the simulations considering different frequencies and Huang-Rhys
factors to get an overall picture of the impact of the intramolecular
vibrations on the NESS under incoherent light illumination. The remaining
intermolecular vibrations are considered as parts of a thermal phonon bath
described by a spectral density (see below).

We consider a model of the two phycoerythrobilin $\mathrm{PEB_{50/61C}}$
(acceptor) and $\mathrm{PEB_{50/61D}}$ (donor) chromophores (PEB dimer) from
the protein-antenna phycoerythrin 545 (PE545) in cryptophyte algae. There is a
large energy gap between the monomer electronic excited states $\Delta
E_{\varepsilon}=1042\,\mathrm{cm}^{-1}$ and a small excitonic coupling
$V_{AD}=92\,\mathrm{cm}^{-1}$ between the PEB dimer. Exciton states are highly
localized over a specific chromophore, and the exciton energy gap is $\Delta
E_{e}=1058\,\mathrm{cm}^{-1}$. The transition dipole moments are
$12.17\,\mathrm{D}$ ($\mathrm{PEB_{50/61C}}$) and $11.87\,\mathrm{D}$
($\mathrm{PEB_{50/61D}}$). We assume that both chromophores have the same
intramolecular vibrational frequencies and vibronic couplings, i.e.,
$\Omega=\Omega_{A}=\Omega_{D}$ and
$\mathcal{G}=\mathcal{G}_{A}=\mathcal{G}_{D}$. The first three levels of
quantized intramolecular vibrations are considered in the simulations below.

Interaction with the environment.—The system-bath and bath Hamiltonians read

$\hat{H}_{\mathrm{Sys-B}}=\sum_{i}^{A,D}\sum_{l}\hbar
g_{il}^{(\mathrm{e})}\hat{\varphi}_{i}^{+}\hat{\varphi}_{i}^{-}\left(\hat{b}_{l}^{(i)}+{\hat{b}_{l}}^{(i){\dagger}}\right)+\sum_{i}^{A,D}\sum_{m}\hbar
g_{im}^{(\mathrm{v})}(\hat{c}_{i}^{\dagger}+\hat{c}_{i})\left(\hat{b}_{m}^{(i)}+{\hat{b}_{m}}^{(i){\dagger}}\right)-\sum_{i}^{A,D}\hat{\boldsymbol{\mu}}_{i}\cdot\hat{\mathbf{E}}(t),$
(3)
$\hat{H}_{\mathrm{B}}=\sum_{i}^{A,D}\sum_{l}\hbar\omega_{l}^{(i)}\hat{b}_{l}^{(i){\dagger}}\hat{b}_{l}^{(i)}+\sum_{i}^{A,D}\sum_{m}\hbar\omega_{m}^{(i)}\hat{b}_{m}^{(i){\dagger}}\hat{b}_{m}^{(i)}+\sum_{\mathbf{k},s}\hbar
ck\hat{a}_{\mathbf{k},s}^{{\dagger}}\hat{a}_{\mathbf{k},s}.$ (4)

Here, $g_{il}^{(\mathrm{e})}$ ($g_{im}^{(\mathrm{v})}$) represents the
coupling between the electronic (intramolecular vibrational) degrees of
freedom of the $i^{\mathrm{th}}$ chromophore and the $l^{\mathrm{th}}$
($m^{\mathrm{th}}$) phonon mode, where
$\hat{b}_{l}^{(i){\dagger}},\hat{b}_{m}^{(i){\dagger}}$
$\big{(}\hat{b}_{l}^{(i)},\hat{b}_{m}^{(i)}\big{)}$ are the creation
(annihilation) operators of the phonon modes of frequencies
$\omega_{l}^{(i)},\omega_{m}^{(i)}$ that couple to the electronic and
intramolecular vibrational degrees of freedom of the $i^{\mathrm{th}}$
chromophore. The electric field of the radiation 46 is given by
$\hat{\mathbf{E}}(t)=\hat{\mathbf{E}}^{(+)}(t)+\hat{\mathbf{E}}^{(-)}(t)$,
with
$\hat{\mathbf{E}}^{(+)}(t)=\mathrm{i}\sum_{\mathbf{k},s}\left(\frac{\hbar\omega}{2\epsilon_{0}V}\right)^{1/2}\hat{a}_{\mathbf{k},s}(\varepsilon_{\mathbf{k},s})\mathrm{e}^{-\mathrm{i}\omega
t}$ and
$\hat{\mathbf{E}}^{(-)}(t)=\left[\hat{\mathbf{E}}^{(+)}(t)\right]^{\dagger}$,
with $\hat{a}_{\mathbf{k},s}^{{\dagger}}$ ($\hat{a}_{\mathbf{k},s}$) being the
creation (annihilation) operator for the $\mathbf{k}^{\mathrm{th}}$ radiation
field mode in the $s^{\mathrm{th}}$ polarization state, and
$\hat{\boldsymbol{\mu}}_{i}$ is the dipole operator for the $i^{\mathrm{th}}$
chromophore.

The open quantum system dynamics of the density operator $\hat{\rho}$ for the
vibronic dimer system described in Eq. 2 is given by the master equation

$\frac{\partial}{\partial
t}\hat{\rho}=-\frac{\mathrm{i}}{\hbar}\big{[}\hat{H}_{\mathrm{Sys}},\hat{\rho}\big{]}+\mathscr{L}_{\mathrm{RB}}\left[\hat{\rho}\right]+\mathscr{L}_{\mathrm{PB}}\left[\hat{\rho}\right]+\mathscr{L}_{\mathrm{rec}}\left[\hat{\rho}\right]+\mathscr{L}_{\mathrm{trap}}\left[\hat{\rho}\right].$
(5)

The first term on the right-hand side describes the unitary evolution of the
dimer system. The Liouvillian operator for the incoherent radiation
environment (blackbody radiation bath denoted as RB) and the protein/solvent
environment (phonon baths denoted as PB) read
$\left(\mathscr{L}_{\mathrm{\mathrm{RB,PB}}}\left[\hat{\rho}\right]\right)_{ij}=-\sum_{kl}R_{ij,kl}^{\mathrm{RB,PB}}\rho_{kl}(t)$,
where we use the standard non-secular Redfield approach for thermal baths
comprised of harmonic modes 47, 45

$R_{ij,kl}^{\mathrm{\mathrm{RB,PB}}}=\delta_{ik}\sum_{m}\Gamma_{jm,ml}^{\mathrm{\mathrm{RB,PB}}}(\omega_{lm})+\delta_{jl}\sum_{m}\Gamma_{im,mk}^{\mathrm{\mathrm{RB,PB}}}(\omega_{km})-\Gamma_{ki,jl}^{\mathrm{\mathrm{RB,PB}}}(\omega_{lj})-\Gamma_{lj,ik}^{\mathrm{\mathrm{RB,PB}}}(\omega_{ki}).$
(6)

The damping matrix elements

$\Gamma_{ij,kl}^{\mathrm{RB,PB}}(\omega)=\sum_{u,v}\int_{0}^{\infty}\mathrm{d}t\mathrm{e}^{\mathrm{i}\omega
t}C_{u,v}^{\mathrm{RB,PB}}(t)K_{u,ij}^{\mathrm{RB,PB}}K_{v,kl}^{\mathrm{RB,PB}},$
(7)

determine the time span for correlations. Here, $K_{u,ij}^{\mathrm{RB,PB}}$
denote the observables of the vibronic dimer system that are coupled to the
radiation and phonon baths modes (see Eq. 3), and

$C_{u,v}^{\mathrm{RB,PB}}(t)=\int_{0}^{\infty}\mathrm{d}\omega\,\omega^{2}J_{i}^{\mathrm{\mathrm{RB,PB}}}(\omega)\bigg{[}\coth\left(\frac{\hbar\omega\beta}{2}\right)\cos(\omega
t)-\mathrm{i}\sin(\omega t)\bigg{]},$ (8)

the correlation functions of the radiation and phonon baths. Formally, the
radiation bath is described by a super-Ohmic spectral density with cubic-
frequency dependence which generates long-lasting coherent dynamics provided
by the lack of pure dephasing dynamics and the strong dependence of the
decoherence rate on the system level spacing 34

$\omega^{2}J_{j}^{\mathrm{\mathrm{RB}}}(\omega)=\frac{2\hbar\omega^{3}}{3(4\epsilon_{0}\pi^{2}c^{3})}.$
(9)

The phonon baths introduce dissipation and decoherence into the electronic
$(\mathrm{e})$ and intramolecular vibrational $(\mathrm{v})$ degrees of
freedom of the system Hamiltonian (Eq. 2). Its effects are encoded in the
Drude-Lorentz spectral density

$\omega^{2}J_{j}^{\mathrm{\mathrm{PB}}}(\omega)=\frac{2\gamma_{j}^{(\mathrm{e},\mathrm{v})}\lambda_{j}^{(\mathrm{e},\mathrm{v})}\omega}{\hbar(\omega^{2}+\gamma_{j}^{(\mathrm{e},\mathrm{v})2})},$
(10)

where $\gamma^{(\mathrm{e},\mathrm{v})}$ represents the cutoff frequency and
$\lambda^{(\mathrm{e},\mathrm{v})}$ the reorganization energies. The same
spectral densities are assumed on each chromophore, with
$\gamma_{A,D}^{(\mathrm{e},\mathrm{v})}=100\,\mathrm{cm}^{-1}$ and different
values for $\lambda_{A,D}^{(\mathrm{e},\mathrm{v})}$ below. The temperature
assumed for the phonon bath it is $T^{\mathrm{PB}}=300\,\mathrm{K}$, and
$T^{\mathrm{RB}}=5600\,\mathrm{K}$ for the blackbody radiation bath. The
transition dipole moments of the chromophores are considered constants in time
and parallel to the incoherent light electric field.

The exciton recombination and the trapping at the reaction center processes
are considered localized processes on each chromophore 43, 37, 39, 38, 44 and
modelled through the effective Lindbladians

$\mathscr{L}_{\mathrm{\mathrm{rec}}}\left[\hat{\rho}\right]=\tau_{\mathrm{rec}}^{-1}\sum_{i}^{A,D}\left(|g_{i}\rangle\langle\varepsilon_{i}|\hat{\rho}|\varepsilon_{i}\rangle\langle
g_{i}|-\frac{1}{2}\left[|\varepsilon_{i}\rangle\langle\varepsilon_{i}|,\hat{\rho}\right]_{+}\right),$
(11)
$\mathscr{L}_{\mathrm{\mathrm{trap}}}\left[\hat{\rho}\right]=\tau_{\mathrm{trap}}^{-1}\left(|\mathrm{RC}\rangle\langle\varepsilon_{A}|\hat{\rho}|\varepsilon_{A}\rangle\langle\mathrm{RC}|-\frac{1}{2}\left[|\varepsilon_{A}\rangle\langle\varepsilon_{A}|,\hat{\rho}\right]_{+}\right),$
(12)

where $\left[\mathcal{\hat{O}}_{1},\mathcal{\hat{O}}_{2}\right]_{+}$
represents the anticommutator between operators $\mathcal{\hat{O}}_{1}$ and
$\mathcal{\hat{O}}_{2}$. The recombination mechanism (Eq. 11) accounts for the
electronic excitation decay to the ground state due to nonradiative processes,
and is determined by the recombination time $\tau_{\mathrm{rec}}$, which is
assumed the same for both chromophores. The trapping process (Eq. 12) dictates
the electronic excitation harvesting at the reaction center from the acceptor
chromophore on a time scale determined by the trapping time
$\tau_{\mathrm{\mathrm{trap}}}$. Recombination and trapping times are expected
to be on the order of nanoseconds and picoseconds, respectively 48, 41, 43.

Energy transport.—We assume that only the donor chromophore $D$ interacts with
the incoherent light and the acceptor chromophore $A$ has a lower electronic
excitation energy, i.e., $\varepsilon_{D}>\varepsilon_{A}$. To analyze the
transport between chromophores, note that the Hamiltonian of the vibronic
dimer (Eq. 2) in the electronic-vibrational localized state basis (single-
excitation manifold)
$\\{|g_{j}^{\beta}\varepsilon_{i}^{\alpha}\rangle=|\chi_{j,\beta}^{(g)},g_{j},\chi_{i,\alpha}^{(\varepsilon)},\varepsilon_{i}\rangle\\}$
is given by

$\displaystyle\hat{H}_{\mathrm{Sys}}=$ $\displaystyle\sum_{i\neq
j}^{A,D}\sum_{\alpha,\beta}\varepsilon_{i}|g_{j}^{\beta}\varepsilon_{i}^{\alpha}\rangle\langle\varepsilon_{i}^{\alpha}g_{j}^{\beta}|+\sum_{i\neq
j}^{A,D}\sum_{\alpha,\beta}V_{ij}|g_{j}^{\beta}\varepsilon_{i}^{\alpha}\rangle\langle\varepsilon_{j}^{\alpha}g_{i}^{\beta}|$
$\displaystyle+\sum_{i\neq
j}^{A,D}\sum_{\alpha,\nu,\mu}\hbar\mathcal{G}_{i}\left(\delta_{\nu,\mu+1}\sqrt{\mu+1}+\delta_{\nu,\mu-1}\sqrt{\mu}\right)|g_{j}^{\alpha}\varepsilon_{i}^{\nu}\rangle\langle\varepsilon_{i}^{\mu}g_{j}^{\alpha}|+\sum_{i\neq
j}^{A,D}\sum_{\alpha,\beta}\alpha\,\hbar\Omega_{i}|g_{j}^{\beta}\varepsilon_{i}^{\alpha}\rangle\langle\varepsilon_{i}^{\alpha}g_{j}^{\beta}|.$
(13)

The first term of the right-hand site of Eq. Electronic-vibrational resonance
does not alter steady-state transport in natural light-harvesting systems
accounts for the electronic site energies, the second stands for the excitonic
coupling, the third is the vibronic coupling, and the last is the vibrational
energy.

The change in the population of the state
$|g_{D}^{\mu}\varepsilon_{A}^{\nu}\rangle$ (see Eqs. 5 and Electronic-
vibrational resonance does not alter steady-state transport in natural light-
harvesting systems) is equal to

$\displaystyle\langle
g_{D}^{\mu}\varepsilon_{A}^{\nu}|\dot{\hat{\rho}}|g_{D}^{\mu}\varepsilon_{A}^{\nu}\rangle=$
$\displaystyle\,2V_{AD}\imaginary\left[\rho_{DA}^{\nu,\mu;\nu,\mu}\right]+2\hbar\mathcal{G}_{A}\left(\sqrt{\nu}\imaginary\left[\rho_{AA}^{\nu-1,\mu;\nu,\mu}\right]+\sqrt{\nu+1}\imaginary\left[\rho_{AA}^{\nu+1,\mu;\nu,\mu}\right]\right)$
$\displaystyle-\tau_{\mathrm{rec}}^{-1}\rho_{AA}^{\nu,\mu;\nu,\mu}-\tau_{\mathrm{trap}}^{-1}\rho_{AA}^{\nu,\mu;\nu,\mu}.$
(14)

The unitary contributions to the change in the population of the acceptor
state $|g_{D}^{\mu}\varepsilon_{A}^{\nu}\rangle$ originate from the electronic
interaction ($V_{AD}$) with the donor state
$|g_{A}^{\mu}\varepsilon_{D}^{\nu}\rangle$ through the imaginary part of the
coherence density matrix element $\rho_{DA}^{\nu,\mu;\nu,\mu}$, and from the
vibronic interaction $\mathcal{G}_{A}$ with acceptor states with adjacent
vibrational numbers $\nu-1$ and $\nu+1$. The non-unitary contributions arise
from recombination and trapping. It should be noted that the phonon baths do
not contribute directly to the change of the population of the state
$|g_{D}^{\mu}\varepsilon_{A}^{\nu}\rangle$ due to the pure dephasing
interaction in the system-bath Hamiltonian 39, 49 (see Eq. 3).

By tracing over all the vibrational states, the change in the population at
the acceptor chromophore reads

$\displaystyle\frac{\differential P_{A}}{\differential
t}=\sum_{\nu,\mu}\langle
g_{D}^{\mu}\varepsilon_{A}^{\nu}|\dot{\hat{\rho}}|g_{D}^{\mu}\varepsilon_{A}^{\nu}\rangle=2V_{AD}\imaginary\left[\rho_{DA}\right]-(\tau_{\mathrm{rec}}^{-1}+\tau_{\mathrm{trap}}^{-1})\rho_{AA},$
(15)

where
$\imaginary\left[\rho_{DA}\right]=\sum_{\nu,\mu}\imaginary\left[\rho_{DA}^{\nu,\mu;\nu,\mu}\right]$
and $\rho_{AA}=\sum_{\nu,\mu}\rho_{AA}^{\nu,\mu;\nu,\mu}$, which are the
intersite coherence and acceptor population of the electronic subsystem. The
second term on the right-hand side of Eq. Electronic-vibrational resonance
does not alter steady-state transport in natural light-harvesting systems
disappears since contributions from different vibrational levels on the
acceptor add up to zero. From Eq. 15 it can be inferred that the flux from the
donor chromophore is $F_{DA}=2V_{AD}\imaginary\left[\rho_{DA}\right]$. The
above allows us to connect with the case of transport in an electronic dimer
without intramolecular vibrational modes 39, i.e. without the effects of
vibronic resonance.

Quantum yield: Energy transfer efficiency.—We consider a definition for the
quantum yield $\eta$ based on currents38, 44 (transfer rates)

$\eta=\frac{\mathcal{J}_{\mathrm{trap}}}{\mathcal{J}_{\mathrm{abs}}},$ (16)

where the
$\mathcal{J}_{\mathrm{trap}}=\sum_{i}\langle\psi_{i}^{(e)}|\mathscr{L}_{\mathrm{\mathrm{trap}}}\left[\hat{\rho}\right]|\psi_{i}^{(e)}\rangle$
is the current of trapping at the reaction center, and
$\mathcal{J}_{\mathrm{abs}}=\sum_{i}\langle\psi_{i}^{(e)}|\mathscr{L}_{\mathrm{\mathrm{RB}}}^{(abs)}\left[\hat{\rho}\right]|\psi_{i}^{(e)}\rangle$
is the incoherent light absorption current. Note that
$\mathcal{J}_{\mathrm{trap}}\neq\tau_{\mathrm{\mathrm{trap}}}^{-1}$ (see Eq.
12). The states ${|\psi_{i}^{(e)}\rangle}$ are (single-excitation manifold)
eigenstates of the vibronic dimer Hamiltonian,
$\hat{H}_{\mathrm{Sys}}|\psi_{i}^{(e)}\rangle=E_{i}|\psi_{i}^{(e)}\rangle$.
While both currents $\mathcal{J}_{\mathrm{trap}}$ and
$\mathcal{J}_{\mathrm{abs}}$ contribute to the determination of the quantum
yield, we find that $\mathcal{J}_{\mathrm{abs}}$ is essentially constant. This
suggests that the changes in the quantum yield are mainly dependent on the
trapping current $\mathcal{J_{\mathrm{trap}}}$.

In the steady-state regime, the following continuity equation must be
satisfied

$\frac{\mathcal{J}_{\mathrm{emi}}+\mathcal{J}_{\mathrm{rec}}+\mathcal{J}_{\mathrm{trap}}}{\mathcal{J}_{\mathrm{abs}}}=1.$
(17)

Here,
$\mathcal{J}_{\mathrm{emi}}=\sum_{i}\langle\psi_{i}^{(e)}|\mathscr{L}_{\mathrm{\mathrm{RB}}}^{(emi)}\left[\hat{\rho}\right]|\psi_{i}^{(e)}\rangle$
and
$\mathcal{J}_{\mathrm{rec}}=\sum_{i}\langle\psi_{i}^{(e)}|\mathscr{L}_{\mathrm{\mathrm{rec}}}\left[\hat{\rho}\right]|\psi_{i}^{(e)}\rangle$
denote the incoherent light emission and recombination currents, respectively.
The incoherent light currents $\mathcal{J}_{\mathrm{abs}}$ and
$\mathcal{J}_{\mathrm{emi}}$ are related to the blackbody radiation bath
occupation number
$(\mathrm{e}^{\hbar\omega/k_{\mathrm{B}}T^{\mathrm{RB}}}-1)^{-1},\>\omega\geq
0$ (absorption) and
$(\mathrm{e}^{\hbar\omega/k_{\mathrm{B}}T^{\mathrm{RB}}}-1)^{-1}+1,\>\omega<0$
(emission) 38. From Eq. 17, it can be concluded that the quantum yield
$\eta=1$ when $\mathcal{J}_{\mathrm{emi}}=\mathcal{J}_{\mathrm{rec}}=0$.
Although, the current definitions discussed above have been described for the
single-excitation eigenstate manifold, their definitions are basis-invariant
44, i.e., the sum $\sum_{i}\langle\psi_{i}^{(e)}|\dots|\psi_{i}^{(e)}\rangle$
represents the trace over the system of interest.

Quantum yield under an electronic-vibrational resonance interaction.—

Figure 1:  Quantum yield (top panels) and the imaginary part of the intersite
coherence (bottom panels) as a function of the Huang-Rhys factor $S$ and the
intramolecular vibrational frequency $\Omega$ considering different
reorganization energies $\lambda^{(\mathrm{e},\mathrm{v})}$ (system–phonon
bath coupling strength). Typical values for recombination
$\tau_{\mathrm{rec}}=1\,\mathrm{ns}$ and trapping
$\tau_{\mathrm{trap}}=10\,\mathrm{ps}$ times are assumed.

Given the recent interest in the potential quantum effects and energy transfer
enhancements driven by the electronic-vibrational resonance interaction 24,
25, 19, we first compute the steady-state
$\frac{\partial\rho_{\mathrm{SS}}}{\partial t}=0$ in Eq. 5 for different
values of the intramolecular vibrational frequency $\Omega$ close to the
exciton energy gap $\Delta E_{e}=1058\;\mathrm{cm}^{-1}$ (vibronic resonance)
and Huang-Rhys factors $S$ (vibronic coupling strengths) reported for
cryptophyte algae 40, 41. To quantify a possible effect due to the vibronic
resonance, Figure 1 shows the quantum yield (top panels) as a function of the
intramolecular vibrational frequency $\Omega$ and the Huang-Rhys factor $S$
for selected values of the system-phonon bath coupling strength
$\lambda^{(\mathrm{e},\mathrm{v})}$. We assume typical values for
recombination $\tau_{\mathrm{rec}}=1\,\mathrm{ns}$ and trapping
$\tau_{\mathrm{trap}}=10\,\mathrm{ps}$ times (See Figure S1 presented in the
Supporting Information where a trapping time
$\tau_{\mathrm{trap}}=1\,\mathrm{ps}$ is considered). A vertical dashed line
in Figure 1 indicates the on-resonance intramolecular vibrational frequency
$\Omega=1058\,\mathrm{cm}^{-1}$, and the two black points mark the frequencies
$\Omega=813\,\mathrm{cm}^{-1}$ (off-resonance) and
$\Omega=1111\,\mathrm{cm}^{-1}$ (quasi-resonance) for the Huang-Rhys factor
$S=0.0578$ reported in previous works 41, 10.

In the case of very weak coupling to the phonon baths
$(\lambda^{(\mathrm{e})},\lambda^{(\mathrm{v})})=(1,1)\,\mathrm{cm}^{-1}$,
Figure 1 shows a significant change in the quantum yield and the imaginary
part of the intersite coherence in the region close to the resonance frequency
$\Omega=1058\,\mathrm{cm}^{-1}$ and for a half of the resonance frequency
$\Omega=529\,\mathrm{cm}^{-1}$ for different values of the Huang-Rhys factor.
The formation of a resonance pattern is observed in this case. Hence, the NESS
allows for the vibronic resonance effect. However, these values for the
reorganization energies are far smaller than those estimated for the real
biophysical systems, which are in an intermediate coupling regime with the
phonon bath, i.e., $\lambda^{(\mathrm{e})}\sim V_{AD}\sim\mathcal{G}_{A/D}$.

As can be seen in Figure 1, as the values of system-phonon bath couplings
increase, the pattern of the electronic-vibrational resonance observed for
$(\lambda^{(\mathrm{e})},\lambda^{(\mathrm{v})})=(1,1)\,\mathrm{cm}^{-1}$
fades away. In fact, for realistic values of the reorganization energies,
e.g.,
$(\lambda^{(\mathrm{e})},\lambda^{(\mathrm{v})})=(100,10)\,\mathrm{cm}^{-1}$,
as expected for photosynthetic complexes such as the protein-antenna
phycoerythrin 545, the changes in the quantum yield due to the vibronic
resonance are insignificant (see also Figure 3). Therefore, in the case of
realistic coupling strengths to the phonon environment in the NESS the
vibronic resonance effects disappear. Comparing the upper and the bottom
panels in Figure 1 it can be concluded that the changes in the quantum yield
have the same trend as the imaginary part of the intersite coherence, as
pointed out above in the analysis of energy transport. In other words, the
current from the donor chromophore is proportional to the imaginary part of
the intersite coherence $F_{DA}=2V_{AD}\imaginary\left[\rho_{DA}\right]$.

Figure 2:  Quantum yield as function of the intramolecular vibrational
frequency for specific values of the Huang-Rhys factor $S=0.001$ (left)
$S=0.01$ (middle) $S=0.1$ (right) and different values of the reorganization
energies $\lambda^{(\mathrm{e},\mathrm{v})}$ (color coding is shown on the
top). $\tau_{\mathrm{rec}}=1\,\mathrm{ns}$ and
$\tau_{\mathrm{trap}}=10\,\mathrm{ps}$.

This physical picture is corroborated in Figure 2, which displays the quantum
yield $\eta$ as a function of intramolecular vibrational frequency $\Omega$
for the Huang-Rhys factors $S=\\{0.001,\,0.01,\,0.1\\}$ and different values
of the reorganization energies (color coded). For all the Huang-Rhys factors
analyzed, vibronic resonance signatures (peaks) lead to significant changes in
the quantum yield only for weak system-phonon bath couplings. For the expected
physical scenario where
$(\lambda^{(\mathrm{e})},\lambda^{(\mathrm{v})})=(100,10)\,\mathrm{cm}^{-1}$
(red lines), there is no effect on the quantum yield, neither with frequency
changes nor changes in the Huang-Rhys factor.

Figure 3 shows detailed information on the quantum yield as a function of the
intramolecular vibrational frequency (left panel) and the Huang-Rhys factor
(right panel) for the case
$(\lambda^{(\mathrm{e})},\lambda^{(\mathrm{v})})=(100,10)\,\mathrm{cm}^{-1}$
(corresponding to the top right panel in Fig. 1 and red lines in Fig. 2). Even
though small spikes appear close to the resonance region, the changes in the
quantum yield are marginal for changes in the intramolecular vibrational
frequency or the Huang-Rhys factor (note the vertical scale). Analogous
results are obtained for different recombination and trapping times where
quantum yields are far less than one. _Therefore, when all non-unitary
contributions in Eq. 5 are considered, i.e., radiation and phonon baths,
exciton recombination and trapping at the reaction center, and the physical
parameters are within the regime expected for the natural function, there is
no significant changes in the quantum yield resulting from vibronic resonance
in the NESS_ (We present additional NESS results supporting the above by using
the hierarchical equations of motion (HEOM) method in Figure S2 of the
Supporting Information).

Figure 3:  Quantum yield as a function of the intramolecular vibrational
frequency $\Omega$ (left) and the Huang-Rhys factor $S$ (right) for the case
$\lambda^{(\mathrm{e})}=100\,\mathrm{cm}^{-1}$ and
$\lambda^{(\mathrm{v})}=10\,\mathrm{cm}^{-1}$ (see the top right panel in Fig.
1). Color coding is shown on the top of each figure.
$\tau_{\mathrm{rec}}=1\,\mathrm{ns}$ and
$\tau_{\mathrm{trap}}=10\,\mathrm{ps}$.

Non-unitary contributions in the establishment of the NESS.—Below we examine
individual terms (recombination, trapping and phonon bath coupling) in an
effort to identify the particular feature (or features) that allow the
electronic-vibrational resonance effect. Although the natural scenario
corresponds to the situation discussed above, where the exciton population is
harvested at the reaction center, to quantify a possible effect due to the
vibronic resonance in the case of no trapping at the reaction center, we
consider the population in the acceptor chromophore $\rho_{AA}$.

Figure 4:  NESS population of the acceptor chromophore as a function of the
intramolecular vibrational frequency $\Omega$ and the Huang-Rhys factor $S$
when the coupling strengths to the phonon baths are zero
($\lambda^{(\mathrm{e})}=0$ and $\lambda^{(\mathrm{v})}=0$), and there is no
trapping at the reaction center. Left panel: $\tau_{\mathrm{rec}}=\infty$
(Incoherent light only). Right panel $\tau_{\mathrm{rec}}=1\,\mathrm{ns}$
(Incoherent light + recombination only).

First, consider the case where the system only interacts with the incoherent
light and reaches a canonical thermal equilibrium at long times
$\hat{\rho}^{(\mathrm{eq,\,RB})}=\mathrm{e^{-\beta^{\mathrm{RB}}\hat{H}_{\mathrm{Sys}}}}/\mathrm{Tr\,\mathrm{e^{-\beta^{\mathrm{RB}}\hat{H}_{\mathrm{Sys}}}}}$,
where $\beta^{\mathrm{RB}}=1/(k_{\mathrm{B}}T^{\mathrm{RB}})$. Due to the high
temperature of the blackbody radiation bath
$T^{\mathrm{RB}}=5600\,\mathrm{K}$, higher energy eigenstates get a larger
population (e.g., compared with the case when the phonon baths are turned on
($T^{\mathrm{PB}}=300\,\mathrm{K}$). Thus, as expected for the vibronic dimer
examined here, Figure 4 (left panel) shows that the population of the acceptor
chromophore increases for higher intramolecular vibrational frequencies and
Huang-Rhys factors. However, the population of the acceptor chromophore does
not peak when the frequency of the intramolecular vibrational mode is in
resonance with the exciton energy gap, as seen in the left panel of Figure 4.
That is, the distinctive resonance pattern observed in Figure 1 for the case
of very low reorganization energies is nonexistent when the system thermalizes
with the incoherent photon bath only or, more generally, any singular thermal
bath. Hence, the equilibrium state reached under the effect of the incoherent
radiation bath is not sensitive to the vibronic resonance effect.

However, if the system interacts with the incoherent light and is also
subjected to recombination, the corresponding NESS allows for the vibronic
resonance effect by facilitating the transport to the ground vibrational level
in the electronically excited state of the acceptor chromophore, as seen in
the right panel of Figure 4. That is, the population of the ground vibrational
level in the electronically excited state of the acceptor increases in the
resonance, leading to a deviation from the thermal population
$\hat{\rho}^{(\mathrm{eq,\,RB})}$. Meanwhile, the acceptor population
increases when the energy frequency of the intramolecular vibrational mode is
in resonance with the exciton energy gap, indicating that there is a vibronic
resonance effect. Furthermore, the population in the acceptor chromophore is
larger with increasing Huang-Rhys factor. On the other hand, for very low
Huang-Rhys factors ($S\leq 0.001$), the vibronic resonance effect is barely
noticeable since the vibronic couplings are too small. Similar observations
can be made if the system interacts with the incoherent light and is subjected
to recombination and trapping (see Figure S3 in the Supporting Information).
Formally, when $\tau_{\mathrm{rec}}\to\infty$ and
$\tau_{\mathrm{trap}}\to\infty$, then $\mathcal{J}_{\mathrm{rec}}\to 0$ and
$\mathcal{J}_{\mathrm{trap}}\to 0$; the NESS corresponds to an equilibrium
thermal state $\rho_{\mathrm{SS}}\to\rho^{(\mathrm{eq,\,RB})}$, and therefore
$\mathrm{Im}[\rho_{AD}]\to 0$. Therefore, _when the reached NESS is close to
an equilibrium state, the signature pattern of the electronic-vibrational
resonance does not arise._

When the effects of incoherent light, recombination, and phonon baths are
considered, there is no change in the acceptor NESS population associated with
the electronic-vibrational resonance interaction. Specifically, _it is the
coupling to the phonon baths that suppresses vibronic resonance effects for
realistic reorganization energies_ (e.g.,
$(\lambda^{(\mathrm{e})},\lambda^{(\mathrm{v})})=(100,10)\,\mathrm{cm}^{-1}$).
In particular, the interaction with the phonon baths is stronger than the
interaction with incoherent light 34, 50, so that the resonance-induced
effects observed in the case of ‘incoherent light + recombination only’ owing
to the vibronic coupling are overshadowed by the dissipative effect of the
intermolecular vibrational degrees of freedom associated with the low-
frequency phonon baths.

Time-dependent results: Coherent initial excitation.—

Figure 5:  Population of the acceptor chromophore, population of the reaction
center, and the imaginary part of the intersite coherence as a function of
time for different values of the intramolecular vibrational frequency $\Omega$
(note the color coding). Two scenarios for the system-phonon bath coupling are
considered: small $\lambda^{(\mathrm{e})}=10\,\mathrm{cm}^{-1}$ (top panels)
and intermediate $\lambda^{(\mathrm{e})}=100\,\mathrm{cm}^{-1}$ (bottom
panels). For all cases, the vibronic coupling is the same
$\mathcal{G}=250\,\mathrm{cm}^{-1}$. $\tau_{\mathrm{rec}}=1\,\mathrm{ns}$ and
$\tau_{\mathrm{trap}}=1\,\mathrm{ps}$.

To assess the similarities regarding the vibronic resonance between NESS
results discussed above and time-dependent results related to coherent light
excitation discussed in the literature, Figure 5 shows the population
(acceptor and reaction center) and intersite coherence (imaginary part)
dynamics for times up to one picosecond, assuming an initial coherent
excitation condition $\rho_{D}(t=0)=1$, i.e., the entire population initially
in the donor. The top and bottom panels in Figure 5 display the dynamics using
the HEOM method (see Supporting Information) for weak and intermediate system-
phonon bath coupling $\lambda^{(\mathrm{e})}=10\,\mathrm{cm}^{-1}$
($\lambda^{(\mathrm{e})}=100\,\mathrm{cm}^{-1}$) for three intramolecular
vibrational frequencies: $\Omega=1058\,\mathrm{cm}^{-1}$ (on-resonance) and
$\Omega=\\{700,\,1400\\}\,\mathrm{cm}^{-1}$ (off-resonance). The same vibronic
coupling $\mathcal{G}=250\,\mathrm{cm}^{-1}$ is considered in all the cases,
and typical values for recombination $\tau_{\mathrm{rec}}=1\,\mathrm{ns}$ and
trapping $\tau_{\mathrm{trap}}=1\,\mathrm{ps}$ times are assumed.

For weak system-phonon bath coupling, the time-dependent results evidence an
enhancement in the acceptor and the reaction center population, as well as in
the imaginary part of the intersite coherence related to the flux between
chromophores for the on-resonance vibrational frequency compared to the off-
resonance frequencies (see top panels in Figure 5). However, for intermediate
system-phonon bath coupling, the increase in the acceptor and the reaction
center population, as well as in the imaginary part of the intersite
coherence, is not guaranteed for the vibrational frequency on-resonance with
the exciton energy gap (see red curves in bottom panels of Figure 5). In fact,
higher values in the acceptor and the reaction center population, as well as
in $\imaginary\left[\rho_{DA}\right]$, are obtained for the off-resonance
frequency $\Omega=700\,\mathrm{cm}^{-1}$ (see blue curves in bottom panels of
Figure 5). This is in contrast to prior work 17 which did not examine the
lower off-resonance frequencies (here $\Omega=700\,\mathrm{cm}^{-1}$) and
concluded that a significant resonance effect exists.

We observe then that for weak system-phonon bath coupling the time-dependent
results evidence an electronic-vibrational resonance effect, as observed in
the case of the NESS results discussed above. For realistic values of the
reorganization energy, that is, in the intermediate system-phonon bath
coupling regime, the results differ from the NESS case, in that time-dependent
results could lead to significantly different population and coherence
dynamics as a function of the intramolecular vibrational frequency. However,
it is possible to obtain higher values for the acceptor and the reaction
center population for off-resonance vibrational frequencies as compared to the
on-resonance vibrational frequency. These observations merit further study,
ongoing in our laboratory.

Conclusions.—Natural incoherent light-induced processes in photosynthetic
light-harvesting systems occur in a non-equilibrium steady-state (NESS). Here,
the extent to which electronic-vibrational resonance associated with high-
energy intramolecular vibrations affects the steady-state energy transport in
the PEB dimer was analyzed using a prototype vibronic dimer, i.e., two two-
level systems, each coupled to an intramolecular vibrational mode. The results
indicate that in the NESS for the PEB dimer, intramolecular vibrations with
frequency resonant with the energy difference between exciton states do not
alter the quantum yield nor the imaginary part of intersite coherence relevant
for transport compared to non-resonant vibrations. The results clearly
motivate studies of other relevant biological models.

Individual non-unitary contributions (incoherent light, recombination,
trapping and phonon coupling) were analyzed to determine the physical feature
that destroys the electronic-vibrational resonance effect. We conclude that in
the natural scenario of incoherent light excitation and under realistic
system-phonon bath couplings, the steady-state reached is dominated mainly by
the effect of the phonon baths, and the impact of the vibronic resonance is
not evident. Moreover, under initial coherent excitation, as prepared by
pulsed lasers and commonly assumed in other studies about vibronic effects in
photosynthetic light-harvesting systems, and considering an intermediate
system-phonon bath coupling regime, it is possible to obtain higher values for
the acceptor population and the imaginary part of the intersite coherence for
off-resonance vibrational frequencies as compared to the on-resonance
vibrational frequency. These PEB results imply that energy transport is not
dramatically affected by electronic-vibrational resonance.

This work was supported by the U.S. Air Force Office of Scientific Research
(AFOSR) under Contract No. FA9550-20-1-0354.

Electronic dimer model; Quantum yield analysis considering a trapping time of
one picosecond; presentation of NESS results using the HEOM method; Figure S3:
quantum yield for the case of incoherent light + recombination \+ trapping
only.

## References

  * Renger et al. 2001 Renger, T.; May, V.; Kühn, O. Ultrafast excitation energy transfer dynamics in photosynthetic pigment–protein complexes. _Phys. Rep._ 2001, _343_ , 137–254
  * Jonas 2003 Jonas, D. M. Two-Dimensional Femtosecond Spectroscopy. _Annu. Rev. Phys. Chem._ 2003, _54_ , 425–463
  * Brixner et al. 2005 Brixner, T.; Stenger, J.; Vaswani, H. M.; Cho, M.; Blankenship, R. E.; Fleming, G. R. Two-dimensional spectroscopy of electronic couplings in photosynthesis. _Nature_ 2005, _434_ , 625–628
  * Collini et al. 2010 Collini, E.; Wong, C. Y.; Wilk, K. E.; Curmi, P. M. G.; Brumer, P.; Scholes, G. D. Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature. _Nature_ 2010, _463_ , 644–647
  * Jang and Mennucci 2018 Jang, S. J.; Mennucci, B. Delocalized excitons in natural light-harvesting complexes. _Rev. Mod. Phys._ 2018, _90_ , 035003
  * Mančal 2020 Mančal, T. A decade with quantum coherence: How our past became classical and the future turned quantum. _Chemical Physics_ 2020, _532_ , 110663
  * Womick and Moran 2011 Womick, J. M.; Moran, A. M. Vibronic enhancement of exciton sizes and energy transport in photosynthetic complexes. _J. Phys. Chem. B_ 2011, _115_ , 1347–1356
  * Butkus et al. 2012 Butkus, V.; Valkunas, L.; Abramavicius, D. Molecular vibrations-induced quantum beats in two-dimensional electronic spectroscopy. _J. Chem. Phys._ 2012, _137_ , 044513
  * Christensson et al. 2012 Christensson, N.; Kauffmann, H. F.; Pullerits, T.; Mančal, T. Origin of Long-Lived Coherences in Light-Harvesting Complexes. _J. Phys. Chem. B_ 2012, _116_ , 7449–7454
  * Kolli et al. 2012 Kolli, A.; O’Reilly, E. J.; Scholes, G. D.; Olaya-Castro, A. The fundamental role of quantized vibrations in coherent light harvesting by cryptophyte algae. _J. Chem. Phys._ 2012, _137_ , 174109
  * Chenu et al. 2013 Chenu, A.; Christensson, N.; Kauffmann, H. F.; Mančal, T. Enhancement of vibronic and ground-state vibrational coherences in 2D spectra of photosynthetic complexes. _Sci. Rep._ 2013, _3_ , 2029
  * Tiwari et al. 2013 Tiwari, V.; Peters, W. K.; Jonas, D. M. Electronic resonance with anticorrelated pigment vibrations drives photosynthetic energy transfer outside the adiabatic framework. _Proc. Natl. Acad. Sci. USA_ 2013, _110_ , 1203–1208
  * Chin et al. 2013 Chin, A. W.; Prior, J.; Rosenbach, R.; Caycedo-Soler, F.; Huelga, S. F.; Plenio, M. B. The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigment-protein complexes. _Nat. Phys._ 2013, _9_ , 113–118
  * Fuller et al. 2014 Fuller, F. D.; Pan, J.; Gelzinis, A.; Butkus, V.; Senlik, S. S.; Wilcox, D. E.; Yocum, C. F.; Valkunas, L.; Abramavicius, D.; Ogilvie, J. P. Vibronic coherence in oxygenic photosynthesis. _Nat. Chem._ 2014, _6_ , 706–711
  * Butkus et al. 2014 Butkus, V.; Valkunas, L.; Abramavicius, D. Vibronic phenomena and exciton-vibrational interference in two-dimensional spectra of molecular aggregates. _J. Chem. Phys._ 2014, _140_ , 034306
  * Halpin et al. 2014 Halpin, A.; Johnson, P. J.; Tempelaar, R.; Murphy, R. S.; Knoester, J.; Jansen, T. L.; Miller, R. D. Two-dimensional spectroscopy of a molecular dimer unveils the effects of vibronic coupling on exciton coherences. _Nat. Chem._ 2014, _6_ , 196–201
  * O’Reilly and Olaya-Castro 2014 O’Reilly, E. J.; Olaya-Castro, A. Non-classicality of the molecular vibrations assisting exciton energy transfer at room temperature. _Nat. Commun._ 2014, _5_ , 3012
  * Romero et al. 2014 Romero, E.; Augulis, R.; Novoderezhkin, V. I.; Ferretti, M.; Thieme, J.; Zigmantas, D.; Van Grondelle, R. Quantum coherence in photosynthesis for efficient solar energy conversion. _Nat. Phys._ 2014, _10_ , 676
  * Dijkstra et al. 2015 Dijkstra, A. G.; Wang, C.; Cao, J.; Fleming, G. R. Coherent Exciton Dynamics in the Presence of Underdamped Vibrations. _J. Phys. Chem. Lett._ 2015, _6_ , 627–632
  * Novelli et al. 2015 Novelli, F.; Nazir, A.; Richards, G. H.; Roozbeh, A.; Wilk, K. E.; Curmi, P. M.; Davis, J. A. Vibronic resonances facilitate excited-state coherence in light-harvesting proteins at room temperature. _J. Phys. Chem. Lett._ 2015, _6_ , 4573–4580
  * Schröter et al. 2015 Schröter, M.; Ivanov, S.; Schulze, J.; Polyutov, S.; Yan, Y.; Pullerits, T.; Kühn, O. Exciton–vibrational coupling in the dynamics and spectroscopy of Frenkel excitons in molecular aggregates. _Phys. Rep._ 2015, _567_ , 1–78
  * Malý et al. 2016 Malý, P.; Somsen, O. J.; Novoderezhkin, V. I.; Mančal, T.; Van Grondelle, R. The Role of Resonant Vibrations in Electronic Energy Transfer. _ChemPhysChem_ 2016, _17_ , 1356–1368
  * Dean et al. 2016 Dean, J. C.; Mirkovic, T.; Toa, Z. S.; Oblinsky, D. G.; Scholes, G. D. Vibronic enhancement of algae light harvesting. _Chem_ 2016, _1_ , 858–872
  * Yeh et al. 2019 Yeh, S.-H.; Hoehn, R. D.; Allodi, M. A.; Engel, G. S.; Kais, S. Elucidation of near-resonance vibronic coherence lifetimes by nonadiabatic electronic-vibrational state character mixing. _Proc. Natl. Acad. Sci. U.S.A._ 2019, _116_ , 18263–18268
  * Cao et al. 2020 Cao, J.; Cogdell, R. J.; Coker, D. F.; Duan, H.-G.; Hauer, J.; Kleinekathöfer, U.; Jansen, T. L.; Mančal, T.; Miller, R. D.; Ogilvie, J. P. et al. Quantum biology revisited. _Sci. Adv._ 2020, _6_ , eaaz4888
  * Arsenault et al. 2020 Arsenault, E. A.; Yoneda, Y.; Iwai, M.; Niyogi, K. K.; Fleming, G. R. Vibronic mixing enables ultrafast energy flow in light-harvesting complex II. _Nat. Commun._ 2020, _11_ , 1–8
  * Higgins et al. 2021 Higgins, J. S.; Lloyd, L. T.; Sohail, S. H.; Allodi, M. A.; Otto, J. P.; Saer, R. G.; Wood, R. E.; Massey, S. C.; Ting, P.-C.; Blankenship, R. E. et al. Photosynthesis tunes quantum-mechanical mixing of electronic and vibrational states to steer exciton energy transfer. _Proc. Natl. Acad. Sci. U.S.A._ 2021, _118_ , e2018240118
  * Policht et al. 2022 Policht, V. R.; Niedringhaus, A.; Willow, R.; Laible, P. D.; Bocian, D. F.; Kirmaier, C.; Holten, D.; Mančal, T.; Ogilvie, J. P. Hidden vibronic and excitonic structure and vibronic coherence transfer in the bacterial reaction center. _Sci. Adv._ 2022, _8_ , eabk0953
  * Jiang and Brumer 1991 Jiang, X.-P.; Brumer, P. Creation and dynamics of molecular states prepared with coherent vs partially coherent pulsed light. _J. Chem. Phys._ 1991, _94_ , 5833–5843
  * Mančal and Valkunas 2010 Mančal, T.; Valkunas, L. Exciton dynamics in photosynthetic complexes: excitation by coherent and incoherent light. _New J. Phys._ 2010, _12_ , 065044
  * Brumer and Shapiro 2012 Brumer, P.; Shapiro, M. Molecular response in one-photon absorption via natural thermal light vs. pulsed laser excitation. _Proc. Natl. Acad. Sci. U.S.A._ 2012, _109_ , 19575–19578
  * Pachón and Brumer 2012 Pachón, L. A.; Brumer, P. Computational methodologies and physical insights into electronic energy transfer in photosynthetic light-harvesting complexes. _Phys. Chem. Chem. Phys._ 2012, _14_ , 10094–10108
  * Kassal et al. 2013 Kassal, I.; Yuen-Zhou, J.; Rahimi-Keshari, S. Does coherence enhance transport in photosynthesis? _J. Phys. Chem. Lett._ 2013, _4_ , 362–367
  * Pachón et al. 2017 Pachón, L. A.; Botero, J. D.; Brumer, P. W. Open system perspective on incoherent excitation of light harvesting systems. _J. Phys. B_ 2017, _50_ , 184003
  * Brumer 2018 Brumer, P. Shedding (Incoherent) Light on Quantum Effects in Light-Induced Biological Processes. _J. Phys. Chem. Lett._ 2018, _9_ , 2946–2955
  * Dodin and Brumer 2021 Dodin, A.; Brumer, P. Noise-induced coherence in molecular processes. _J. Phys. B_ 2021, _54_ , 223001
  * Tscherbul and Brumer 2018 Tscherbul, T. V.; Brumer, P. Non-equilibrium stationary coherences in photosynthetic energy transfer under weak-field incoherent illumination. _J. Chem. Phys._ 2018, _148_ , 124114
  * Chuang and Brumer 2020 Chuang, C.; Brumer, P. LH1–RC light-harvesting photocycle under realistic light–matter conditions. _J. Chem. Phys._ 2020, _152_ , 154101
  * Jung and Brumer 2020 Jung, K. A.; Brumer, P. Energy transfer under natural incoherent light: Effects of asymmetry on efficiency. _J. Chem. Phys._ 2020, _153_ , 114102
  * Doust et al. 2004 Doust, A. B.; Marai, C. N.; Harrop, S. J.; Wilk, K. E.; Curmi, P. M.; Scholes, G. D. Developing a structure–function model for the cryptophyte phycoerythrin 545 using ultrahigh resolution crystallography and ultrafast laser spectroscopy. _J. Mol. Biol._ 2004, _344_ , 135–153
  * Novoderezhkin et al. 2010 Novoderezhkin, V. I.; Doust, A. B.; Curutchet, C.; Scholes, G. D.; Van Grondelle, R. Excitation dynamics in phycoerythrin 545: modeling of steady-state spectra and transient absorption with modified Redfield theory. _Biophys. J._ 2010, _99_ , 344–352
  * Blankenship 2014 Blankenship, R. E. _Molecular mechanisms of photosynthesis_ ; WILEY, 2014
  * León-Montiel et al. 2014 León-Montiel, R. d. J.; Kassal, I.; Torres, J. P. Importance of excitation and trapping conditions in photosynthetic environment-assisted energy transport. _J. Phys. Chem. B_ 2014, _118_ , 10588–10594
  * Janković and Mančal 2020 Janković, V.; Mančal, T. Nonequilibrium steady-state picture of incoherent light-induced excitation harvesting. _J Chem. Phys._ 2020, _153_ , 244110
  * May and Kühn 2011 May, V.; Kühn, O. _Charge and Energy Transfer Dynamics in Molecular Systems_ ; John Wiley & Sons, Ltd, 2011
  * Mandel and Wolf 1995 Mandel, L.; Wolf, E. _Optical coherence and quantum optics_ ; Cambridge University Press, 1995
  * Nitzan 2006 Nitzan, A. _Chemical Dynamics in Condensed Phases_ ; Oxford University Press, 2006
  * Mohseni et al. 2008 Mohseni, M.; Rebentrost, P.; Lloyd, S.; Aspuru-Guzik, A. Environment-assisted quantum walks in photosynthetic energy transfer. _J. Chem. Phys._ 2008, _129_ , 11B603
  * Roden and Whaley 2016 Roden, J. J.; Whaley, K. B. Probability-current analysis of energy transport in open quantum systems. _Phys. Rev. E_ 2016, _93_ , 012128
  * Pachón and Brumer 2013 Pachón, L. A.; Brumer, P. Incoherent excitation of thermally equilibrated open quantum systems. _Phys. Rev. A_ 2013, _87_ , 022106