Single-photon absorption by single photosynthetic light-harvesting complexes

As described in [31, 32], a single-photon pulse of a coherent state with average photon number of one can be represented by a multi-mode wavepacket whose form is controlled by the temporal pulse profile $\xi (t)$. We parameterize this as $\alpha (t)\equiv \alpha \xi (t)$ for complex α and square-normalized pulse profile $\xi (t)$. The average number of photons contained in the pulse is equal to

$$\begin{eqnarray}&&\langle n\rangle =\int | \alpha (t){| }^{2}{\rm{d}}{t}=| \alpha {| }^{2},\end{eqnarray} \tag{ 9 }$$

since $\xi (t)$ is already squared-normalized. For a single pulse we may take α to be real, without loss of generality. Note that this average photon number for a pulsed coherent state does not have the same meaning as the thermodynamic average photon number per mode $\overline{n}$ of a thermal radiation field (see below). We shall employ a Gaussian pulse profile,

$$\begin{eqnarray}&&\xi (t)\equiv {\left(\displaystyle \frac{{{\rm{\Omega }}}^{2}}{2\pi }\right)}^{\tfrac{1}{4}}\exp \left[-\displaystyle \frac{{{\rm{\Omega }}}^{2}}{4}{t}^{2}\right]\exp (-{\rm{i}}{\omega }_{0}t),\end{eqnarray} \tag{ 10 }$$

with pulse frequency bandwidth parameter Ω and carrier frequency ${\omega }_{0}$. For this pulse form, the full width at half maximum (FWHM) of the pulse amplitude in the frequency domain is ${\rm{\Delta }}\omega =2\sqrt{{\rm{ln}}2}{\rm{\Omega }}$ in angular frequency units and ${\rm{\Delta }}\nu =(\sqrt{{\rm{ln}}2}/\pi ){\rm{\Omega }}={\rm{\Omega }}/3.77$ in linear frequency units. Other pulse forms have been studied in [32] (see also section 5). The value of Ω also defines the coherence time ${\tau }_{{\rm{coh}}}$ of the radiation field. Using the simple definition ${\tau }_{{\rm{coh}}}=1/{\rm{\Delta }}\nu$ for this [33] gives ${\tau }_{{\rm{coh}}}=(\pi /\sqrt{{\rm{ln}}2}){\rm{\Omega }}=3.77/{\rm{\Omega }}$, which is related to the pulse width by ${\tau }_{{\rm{coh}}}=4.53/{\rm{\Delta }}t$, with ${\rm{\Delta }}t$ the FWHM of the pulse equation (10) in the time domain. In the numerical calculations we start the pulse at time ${t}_{0}-3\sigma$ and end it at ${t}_{0}+3\sigma$, where $\sigma =\sqrt{2}/{\rm{\Omega }}$. The pulse duration for the numerical calculations is thus $6\sqrt{2}/{\rm{\Omega }}$.

The quantum optical master equation for a dimeric chromophore system interacting with such a pulse can be obtained from the QSDE for a system interacting with a quantum optical field [34]. As shown explicitly in [26], this allows generation of quantum master equations for general forms of the optical field. For an optical field in the form of a coherent state and a single chromophore represented by its ground and first excited electronic state, the resulting system master equation is equivalent to the well-known master equation for a two-level system interacting with a coherent state drive [35]. Generalizing this to a dimeric chromophore system yields the excitonic density matrix evolution equation

$$\begin{eqnarray}\dot{\rho } & = & -\displaystyle \frac{{\rm{i}}}{{\hslash }}[{H}_{s},\rho ]-{\rm{i}}\displaystyle \sum _{k=1}^{2}[{\rm{i}}{\alpha }^{* }(t){L}_{k}-{\rm{i}}\alpha (t){L}_{k}^{\dagger },\rho ]\\ & & +\displaystyle \sum _{k=1}^{2}\left({L}_{k}\rho {L}_{k}^{\dagger }-\displaystyle \frac{1}{2}\{{L}_{k}^{\dagger }{L}_{k},\rho \}\right),\end{eqnarray} \tag{ 11 }$$

where L_k are single-chromophore lowering operators given by

$$\begin{eqnarray}&&{L}_{k}\equiv \sqrt{{{\rm{\Gamma }}}_{k}}{\sigma }_{k},\end{eqnarray} \tag{ 12 }$$

with ${{\rm{\Gamma }}}_{k}$ the Weisskopf–Wigner spontaneous emission decay rate of the kth chromophore,

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{k}=\displaystyle \frac{1}{4\pi {\epsilon }_{0}}\displaystyle \frac{4{\omega }_{k}^{3}{\mu }_{k}^{2}}{3{\hslash }{c}^{3}}.\end{eqnarray} \tag{ 13 }$$

Here ${\omega }_{k}$ and ${\mu }_{k}$ are the energy and transition dipole moment of the kth chromophore, respectively. The terms in equation (11) involving α(t) correspond to coherent excitation and de-excitation by the coherent pulse, while those in the last term correspond to spontaneous emission.

We shall occasionally refer also to the chromophore oscillator strength [36],

$$\begin{eqnarray}&&{f}_{k}={{\rm{\Gamma }}}_{k}/3{{\rm{\Gamma }}}_{{\rm{cl}}},\end{eqnarray} \tag{ 14 }$$

with

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{\rm{cl}}}={\omega }_{k}^{2}{e}^{2}/(6\pi /{\epsilon }_{0}{m}_{e}{c}^{3})\end{eqnarray} \tag{ 15 }$$

defined as the dimensionless ratio of the spontaneous emission decay rate to that of a classical oscillator with the corresponding frequency ${\omega }_{k}$, which is more commonly used in molecular spectroscopy.

For ease of numerical simulation, it is more convenient to transform equation (11) to the interaction picture, in which case $\alpha (t)$ is replaced by the slowly varying envelope of the oscillation, $\tilde{\alpha }(t)$:

$$\begin{eqnarray}{\dot{\rho }}^{r} & = & -\displaystyle \frac{{\rm{i}}}{{\hslash }}[{H}_{\delta },{\rho }^{r}]-{\rm{i}}\displaystyle \sum _{k=1}^{2}[{\rm{i}}{\tilde{\alpha }}^{* }(t){L}_{k}-{\rm{i}}\tilde{\alpha }(t){L}_{k}^{\dagger },{\rho }^{r}]\\ & & +\displaystyle \sum _{k=1}^{2}\left({L}_{k}{\rho }^{r}{L}_{k}^{\dagger }-\displaystyle \frac{1}{2}\{{L}_{k}^{\dagger }{L}_{k},{\rho }^{r}\}\right),\end{eqnarray} \tag{ 16 }$$

with

$$\begin{eqnarray}&&\tilde{\alpha }(t)=\alpha {\left(\displaystyle \frac{{{\rm{\Omega }}}^{2}}{2\pi }\right)}^{\tfrac{1}{4}}\exp \left[-\displaystyle \frac{{{\rm{\Omega }}}^{2}}{4}{t}^{2}\right].\end{eqnarray} \tag{ 17 }$$

The standard description for interaction of a system of two-level chromophores with a thermal radiation field is obtained with the well-known reservoir theory of quantum optics [28] in which the radiation field is treated as a large Markovian reservoir. For a dimeric chromophore system, this yields the following Lindblad form for evolution of the system density matrix:

$$\begin{eqnarray}\dot{\rho } & = & -\displaystyle \frac{{\rm{i}}}{{\hslash }}[{H}_{s},\rho ]+\displaystyle \sum _{k=1}^{2}\overline{n}({\omega }_{k})\left[{L}_{k}^{\dagger }\rho {L}_{k}-\displaystyle \frac{1}{2}\{{L}_{k}{L}_{k}^{\dagger },\rho \}\right]\\ & & +\displaystyle \sum _{k=1}^{2}(\overline{n}({\omega }_{k})+1)\left[{L}_{k}\rho {L}_{k}^{\dagger }-\displaystyle \frac{1}{2}\{{L}_{k}^{\dagger }{L}_{k},\rho \}\right],\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}&&\overline{n}(\omega )=\displaystyle \frac{1}{\exp (\beta {\hslash }\omega )-1}\end{eqnarray} \tag{ 19 }$$

is the thermal average photon number per mode for energy $E={\hslash }\omega$, and inverse temperature $\beta \equiv 1/{k}_{B}T$.

This description represents the quantum driving of the system by an incoherent radiation field. In equation (18), the terms in the square brackets multiplied by $\overline{n}$ represent stimulated absorption, while those multiplied by $(\overline{n}+1)$ represent stimulated and spontaneous emission. These two terms provide incoherent driving of the system density matrix, in contrast to the unitary drive provided by the first term $-\tfrac{{\rm{i}}}{{\hslash }}[{H}_{s},\rho ]$. The latter contains the excitonic couplings between chromophores, which play a key role in the time evolution of off-diagonal elements of the system density matrix for systems larger than a monomer. We note that inclusion of retardation effects can result in additional small coherent light-induced coupling terms between chromophores [18].

Applying the rotating frame transformation (7) to the thermal equation (18), results in an equation of the same form but with the coherent driving term replaced by $-\tfrac{{\rm{i}}}{{\hslash }}[{H}_{\delta },{\rho }^{r}]$ and ρ replaced by ${\rho }^{r}$, i.e.,

$$\begin{eqnarray}{\dot{\rho }}^{r} & = & -\displaystyle \frac{{\rm{i}}}{{\hslash }}[{H}_{\delta },{\rho }^{r}]+\displaystyle \sum _{k=1}^{2}\overline{n}({\omega }_{k})\left[{L}_{k}^{\dagger }{\rho }^{r}{L}_{k}-\displaystyle \frac{1}{2}\{{L}_{k}{L}_{k}^{\dagger },{\rho }^{r}\}\right]\\ & & +\displaystyle \sum _{k=1}^{2}(\overline{n}({\omega }_{k})+1)\left[{L}_{k}{\rho }^{r}{L}_{k}^{\dagger }-\displaystyle \frac{1}{2}\{{L}_{k}^{\dagger }{L}_{k},{\rho }^{r}\}\right],\end{eqnarray} \tag{ 20 }$$

To see this, note that in going to the rotating frame the lowering operators will be transformed as ${L}_{k}\,\to {U}_{r}{L}_{k}{U}_{r}^{\dagger }\,={{\rm{e}}}^{-\tfrac{{\rm{i}}}{{\hslash }}{E}^{{\prime} }t}{L}_{k}$. Since each term in equation (18) contains both L_k and its adjoint ${L}_{k}^{\dagger }$, the time dependent phases cancel out.

We emphasize here that such a thermal radiation bath is characterized by thermodynamic average photon numbers $\overline{n}({\omega }_{k})$ that are derived from the temperature of the radiation source. Unlike a coherent pulse, a thermal bath is not characterized by finite amount of energy and thus it is not possible to directly compare the energy content of a single coherent pulse with that of a thermal radiation bath. In fact, the model for a thermal bath model is implicitly assumed to be of infinite size, and therefore of unlimited energy.

Figure 2 shows the time-dependent excitation probability P_e for excitation of the Chla by a coherent state with an average of one photon, i.e., $\langle n\rangle =1$. In this calculation the Gaussian pulse form has carrier frequency ${\omega }_{0}$ equal to the monomer bare excited state energy (15287 cm⁻¹) and the bandwidth parameter Ω is set to $4.41\times {10}^{13}\,\mathrm{Hz}$ (1471 cm⁻¹) to ensure that the pulse amplitude covers the FWHM of a typical Chla absorption in solution (${\rm{\Delta }}\nu \sim 390$ cm⁻¹ [20]). The insert of figure 2 shows that the 10–90 rise time⁴ for excitation of the chromophore from its ground state, ${t}_{{\rm{rise}}}=62.5$ fs, occurs on the timescale of the pulse duration, and that this is followed by a much slower decay due to spontaneous emission on a timescale of ${{\rm{\Gamma }}}^{-1}\sim 56$ ns for this Chla.

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One of the most marked features of the Chla excitation by a single-photon coherent pulse in figure 2 is the very low value of the excitation probability P_e, which is of order 10⁻⁶ for both the bare chromophore and the chromophore coupled to its vibrational bath. Further analysis based on studying the variation of P_e^max as Ω is varied reveals two distinct reasons for this reduction, as well as a difference between the excitation probabilities of the bare and phonon-bath coupled chromophore. For this value of ${\rm{\Omega }}=4.41\times {10}^{13}\,\mathrm{Hz}$ and larger, the value of the excitation probability with and without the phonon bath are quite similar, with both considerably reduced below unity to a value of order 10⁻⁶. Yet the excitation probabilities show quite different dependences on the coherent pulse bandwidth parameter Ω when the chromophore is coupled to the phonon bath. Figure 3 summarizes the maximal values of excitation probability P_e^max in both situations, for values of Ω varying from 10⁷ to 10¹⁵ Hz.

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For the bare Chla monomer without the phonon bath (blue symbols in figure 3), the value of ${P}_{e}^{\max }$ is seen to depend strongly on the order of magnitude of Ω. In this case, P_e^max increases from $\sim {10}^{-7}$ to ${P}_{e}^{{\rm{\max }}}=0.48$ as Ω is decreased from $\sim {10}^{15}\,\mathrm{Hz}$ to ${{\rm{\Omega }}}_{{\rm{opt}}}=4.30\times {10}^{7}\,\mathrm{Hz}$, consistent with the maximal excitation probability for excitation of an isolated two-level system by a pulsed single-photon Gaussian coherent state in vacuum with optimal bandwidth parameter ${{\rm{\Omega }}}_{{\rm{opt}}}=2.4{\rm{\Gamma }}$ [32]. This is due to the increasing contribution of resonant absorption as Ω decreases to ${{\rm{\Omega }}}_{{\rm{opt}}}$.

In contrast, when the chromophore is coupled to the phonon bath (red symbols in figure 3), P_e^max shows a plateau at the value ${P}_{e}^{{\rm{\max }}}\sim {10}^{-6}$ over a broad range of Ω values. The bare chromophore absorption (blue symbols) approaches the absorption calculated with phonon coupling at the Ω value corresponding to the Chla absorption (${\rm{\Omega }}=4.41\times {10}^{13}$ Hz) and the two values of P_e^max are equal for larger Ω values. As Ω increases beyond this value and approaches ${{\rm{\Omega }}}_{{\rm{sun}}}$, both values of P_e^max then decrease, continuing the smooth fall with Ω seen for the bare chromophore absorption for ${\rm{\Omega }}\gt {10}^{8}\,\mathrm{Hz}$.

This behavior of P_e^max reflects the role of dephasing fluctuations induced by the thermal phonon bath at T = 300 K, which substantially lower the excitation probability, substantially reducing any gain due to greater resonant absorption at smaller values of Ω. The resulting phonon-induced reduction is similar to the differential transmission factors ${\rm{\Delta }}T/T\sim {10}^{-6}$ seen in single-molecule attenuation measurements at ambient temperatures in solution [58, 60], from which empirical absorption cross-sections may be extracted (see below). In contrast, at very high bandwidths the dephasing plays a secondary role to the dominant effect of off-resonance of the chromophore transition energy that results in the fall off of P_e^max. The fall off at very low bandwidths is due to the weaker photon density of significantly longer pulses, which effectively reduces the molecule-field coupling [32].

Carrying out the coherent pulse calculations with bandwidth parameter ${\rm{\Omega }}={{\rm{\Omega }}}_{{\rm{sun}}}$ (section 3.1) results in excitation evolution profiles that are qualitatively similar to those shown in figure 2, differing only in a shorter pulse time duration that results in a 10–90 rise time of ${t}_{{\rm{rise}}}=2.44$ fs.

The broad range of Ω in figure 3 corresponds to a very broad range of optical coherence time ${\tau }_{{\rm{coh}}}=1/{\rm{\Delta }}\nu$ (section 2.2.1). The relevant values of ${\tau }_{{\rm{coh}}}$ for the Chla chromophore are 87.7 ns at ${{\rm{\Omega }}}_{{\rm{opt}}}=4.30\times {10}^{7}\,\mathrm{Hz}$, 85.6 fs at ${\rm{\Omega }}=4.41\times {10}^{13}\,\mathrm{Hz}$, and 3.11 fs at ${{\rm{\Omega }}}_{{\rm{sun}}}=1.21\times {10}^{15}\,\mathrm{Hz}$. Comparing this timescale with that of the inverse exciton–phonon and phonon–phonon couplings (γ and λ, respectively), it is evident that when the pulse bandwidth is selected to cover the Chla absorption spectrum (${\rm{\Omega }}=4.41\times {10}^{13}$ Hz), the optical coherence time is of the same order of magnitude as the timescale for phonon relaxation and exciton–phonon energy transfer. This coincidence is intriguing in that it mirrors the coincidence between the energetic scales of exciton–exciton, exciton–phonon, phonon–phonon couplings and energetic disorder characterizing natural photosynthetic light-harvesting systems (see section 1). This suggests that this coincidence of timescales between the optical coherence of single photons absorbed by chlorophylls in light-harvesting complexes and the critical timescales of excitonic energy transfer in these systems may constitute a key factor in overcoming the limitations imposed by the ultraweak chromophore–light coupling under illumination by sunlight (see section 1), as well as playing a role in ensuring the robustness of optimal excitation under sunlight conditions [55, 56]. Further evidence for this derives from additional calculations carried out with randomization of the optical phase at discrete time intervals during the single-photon pulse, which showed that the excitation probability decreases as the time duration of the coherent intervals decreases and hence as the effective pulse bandwidth increases beyond its optimal value. For example, when the optical phase is randomized every 2 fs, the value of P_e^max decreases by two orders of magnitude relative to the value shown in figure 2 and also becomes independent of the presence or absence of phonon coupling.

With the coherent pulse QSDE approach we may also readily study the intensity dependence of absorption outside the single-photon regime by varying the average photon number $\langle n\rangle =| \alpha {| }^{2}$ (equation (9)). This dependence is shown in figure 4 for excitation of Chla by the single-photon coherent pulse with the same bandwidth parameter ${\rm{\Omega }}=4.41\times {10}^{13}\,\mathrm{Hz}$, including (red line) and not including (blue line) the chromophore coupling to the phonon bath. The linear behavior seen for both situations shows that for coherent fields with up to ten photons at these large bandwidths where the excitation is primarily non-resonant, excitation of the chromophore coupled to the coherent photon field with or without a phonon bath is consistent with a linear response description of absorption of chromophores in pigment–protein complexes, in agreement with conventional analyses of molecular absorption for bulk samples. Note that even for $\langle n\rangle =10$, the excitation probability is still exceedingly low, of order 10⁻⁵. Calculations for larger $\langle n\rangle$ values (not displayed) show that for this pulse bandwidth the linear behavior continues to $\langle n\rangle \sim {10}^{4}$, after which the excitation starts to saturate, reaching a plateau at a value near unity for $\langle n\rangle \sim {10}^{13}$, followed by a decrease at even larger $\langle n\rangle$.

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Treating the multi-mode single-photon pulse as a 'broadened single-mode pulse' at the carrier frequency ${\omega }_{0}$ further allows us to use these direct calculations of single-photon absorption to estimate a full single-photon absorption spectrum as a function of ${\omega }_{0}$ for the Chla monomer coupled to the phonon bath without making the usual Franck–Condon assumption of a vertical electronic transition in which the nuclear coordinates of the bath are artificially assumed to be stationary. This is achieved by scanning the value of ${\omega }_{0}$ over the frequency range relevant to absorption of the chromophore and recording the maximum absorption probability P_e^max at each value, resulting in the absorption spectrum ${P}_{e}^{{\rm{\max }}}({\omega }_{0})$. Figure 5 shows this absorption spectrum calculated using two different values of the pulse bandwidth parameter Ω, and compares these direct single-photon absorption spectra with the linear response prediction calculated using the approach described in section 2.4, which relies on the Franck–Condon approximation, i.e., assuming that the electronic excitation is instantaneous. The linear response spectrum is shown as the solid green line in figure 5 and has a FWHM of 267 cm${}^{-1}=8.00\times {10}^{12}$ Hz. The dotted red line presents direct single-photon calculations with bandwidth parameter ${{\rm{\Omega }}}_{1}=4.41\times {10}^{13}$ Hz corresponding to a FWHM equal to the experimental value of 390 cm⁻¹ for Chla in solution [20], and the dashed red line presents direct calculations with bandwidth parameter ${{\rm{\Omega }}}_{2}=3.02\times {10}^{13}\,\mathrm{Hz}$, corresponding to a FWHM covering the linear response spectrum. The peak values and FWHM of all three spectra are listed in table 3.

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Table 3.  Peak values and FWHM of the spectra in figure 5. See caption of figure 5 for definitions of ${{\rm{\Omega }}}_{1}$ and ${{\rm{\Omega }}}_{2}$.

	Peak	FHWM
Bare chromophore energy ${E}_{1}^{(0)}$	$15287\ {\mathrm{cm}}^{-1}$	N/A
QSDE (${{\rm{\Omega }}}_{1}$)	$15292\ {\mathrm{cm}}^{-1}$	$393\ {\mathrm{cm}}^{-1}$
QSDE (${{\rm{\Omega }}}_{2}$)	$15294\ {\mathrm{cm}}^{-1}$	$334\ {\mathrm{cm}}^{-1}$
Linear response	$15314\ {\mathrm{cm}}^{-1}$	$267\ {\mathrm{cm}}^{-1}$

The direct single-photon absorption spectrum calculation is seen to be quite robust to small variations in the bandwidth parameter, as long as this is consistent with the target spectral width. For all three spectra, both the two direct QSDE spectra and the linear response spectrum, the values in table 3 show that the peak position is blue shifted from the bare excitation energy but by a smaller extent than the reorganization energy, i.e., the peak position is red-shifted from the 'site energy' of the Chla in the pigment–protein environment, which is conventionally expressed as ${E}_{s}={E}_{1}^{(0)}+\lambda$. This is the usual situation with linear response calculations for chromophores in light harvesting situations at finite temperatures. On decreasing the temperature (not shown here), the spectra show an increasing blue shift from the bare excitation energy, i.e., the red shift from the 'site energy' decreases, and the width of the spectra decreases.

While there is good overall agreement between the direct single-photon absorption spectrum and the linear response spectrum, as one might expect given the linear dependence of excitation probability on intensity seen in figure 4, there are nevertheless small systematic differences that reflect the lack of any Franck–Condon assumption of vertical transitions in the direct calculation. At T = 0 when there is no thermal population of excited vibrations in the ground electronic state, the Franck–Condon principle predicts a peak at E_s, but as temperature increases, the growth of thermal excitations of vibrations in the ground electronic state modifies the spectral density ${J}_{s}[\omega ]$ [39] and results in a shift of the absorption peak to the red of E_s as well as increased absorption component to the red of ${E}_{1}^{(0)}$. However the differences between the direct and linear response calculations in the wings of the absorption go beyond this Franck–Condon analysis. In the QSDE continuous time description of coupled radiation–chromophore–phonon dynamics we now have a timescale for photon absorption which is finite on the timescale of relaxation dynamics of phonons, as evidenced by the finite rise times of P_e in the direct calculations of figure 2 (10–90 rise times of 62.5 fs and 89.3 fs for the blue and red lines, respectively). This results in greater time overlap between the phonon and photon dynamics and enables partial vibrational relaxation to occur during the timescale of photon absorption, modifying the dynamical effect of the finite temperature spectral density.

We now address the absolute magnitudes of the excitation probabilities obtained from these direct calculations with the single-photon QSDE approach. We note first that the conventional linear response evaluation of the absorption spectrum yields only relative magnitudes of absorption and not absolute absorption intensities or cross-sections. Absolute absorption intensities are usually quantified by molecular extinction coefficients or absorption cross-sections, measured in attenuation experiments relying on the Beer–Lambert law [57]. Recent progress in single-molecule spectroscopy has allowed detection of absorption by single molecules using several techniques [54, 58–60]. Of particular note is the extraction of direct estimates for single-molecule cross-sections from modulation free transient absorption spectroscopy [60]. Assuming similar focusing for a single-photon pulse as in [58, 60] leads to an estimated absorption cross-section of ${{\sigma }}_{{a}{b}{s}}\sim 1-10\,\mathring{\rm{A}}^{2}$, consistent with typical values for Chla extracted from bulk extinction measurements [20, 65].

Such an estimate for the absorption cross-section follows the empirical approach of direct absorption experiments. We can also make a more fundamental analysis that connects our excitation probability P_e to values of ${\sigma }_{{abs}}$ derived from bulk absorption experiments, by a simple estimate based on estimation of the density of geometric modes at each value of the carrier frequency, together with a physically relevant mode volume. Recall that P_e is derived under idealized optical conditions with a single geometric mode (distinct from the multiple frequency modes present in the pulse). Thus, given a density of geometric modes per unit frequency in vacuum (at a given carrier frequency ${\omega }_{0}$ and for a single polarization), $n({\omega }_{0})\equiv 4\pi {\nu }^{2}/{c}^{3}$, with $\nu ={\omega }_{0}/2\pi$, and a physically relevant mode volume V_m, the probability of excitation by one or more of the possible modes is one minus the probability of not being excited by any modes. This is given by the combinatorial factor

$$\begin{eqnarray}&&P=1-{(1-{P}_{e})}^{n({\omega }_{0}){V}_{m}{\rm{d}}{\omega }_{0}},\end{eqnarray} \tag{ 30 }$$

where the differential ${\rm{d}}{\omega }_{0}$ accounts for the continuous nature of the carrier frequency. For ${P}_{e}\ll 1$, this reduces to ${P}_{e}n({\omega }_{0}){V}{\rm{d}}{\omega }_{0}$, so that we may then estimate the total absorption probability under conditions of irradiation within a mode volume V_m as

$$\begin{eqnarray}&&{P}_{\mathrm{tot}}={V}_{m}\int {P}_{e}({\omega }_{0})n({\omega }_{0}){\rm{d}}{\omega }_{0}.\end{eqnarray} \tag{ 31 }$$

We may take P_e^max as an estimate of P_e, noting that the excited state population at times longer than the pulse duration is not relevant to the absorption. The relevant mode volume V_m should be the volume in which the irradiating field is coherent. While this is not easy to estimate for chromophores in solution, for natural photosynthetic systems we may estimate this 'coherence volume' [33, 61] by taking the coherence volume of sunlight ${V}_{\text{coh}}$ to be given by the product of the Cittert–Zernike coherence area $A{}_{{\rm{c}}{\rm{o}}{\rm{h}}}\sim 3\times {10}^{-3}$ mm² [62] and the coherence length $L{}_{{\rm{c}}{\rm{o}}{\rm{h}}}\sim c/{\rm{\Delta }}\nu$ [63, 64], where ${\rm{\Delta }}\nu$ is the relevant bandwidth. For the Chla monomer, using the bandwidth of the absorption spectrum, which we have argued above is the relevant bandwidth for absorption by the chromophore, results in an integrated absorption probability of ∼0.147, in good agreement with values estimated from bulk absorption cross-section values [20, 65].

We can further estimate the photon flux within the spectral bandwidth that is incident on the Chla monomer under full sunlight using solar spectral irradiance data [66], which yields photon flux ${I}\sim 122\,{\mu }\text{mol}\,{{\rm{m}}}^{-2}\,{{\rm{s}}}^{-1}$, equivalent to $\sim$73.7 photons per second incident on a Chla molecule (size ∼100 Ų [20]). Combining this with the absorption probability value of 0.147 results in an estimate of 0.09 s for the average time required for a single Chla to absorb the full energy equivalent of a single photon. Scaling by the number of Chla per reaction center in the PSII complex ($\sim 270\mbox{--}300$) and a factor of 3 for random polarization, yields an estimated average time of ∼0.1ms for absorption of the full energy equivalent of a single photon by PSII, under the simplest possible assumption of uncorrelated absorptions and no cooperativity, i.e., no exciton delocalization (this is of order 3–4 chromophores in PSII [67]). Since the timescale for excitonic energy transport to the reaction center is considerably faster, of order ns [53], this estimated photon energy absorption timescale is smaller than, and hence consistent with the measured turnover rate of 200–300 second (i.e., ∼3.3–5 ms per electron) for PSII in its active state [68, 69]. Note that a smaller coherence volume will increase the estimated average time for absorption of the full energy equivalent, e.g., a reduction in V_coh by a factor of ten increases the time to ∼1 ms, which is still consistent with the measured turnover rate for PSII.

Figure 6 shows the time-dependent absorption probability for the Chla monomer under irradiation by the thermal field of equation (18) with a thermal average photon number equal to unity and the field switched on and off at arbitrary times (on at 10 ns and off at 200 ns in this instance). It is evident that coupling of the excited monomer state to phonons has no effect on the monomer absorption probability. This results from use of the idealized form of equation (18), which implicitly contains a single-mode description of the photons at a single frequency. Consequently the chlorophyll is coupled to the radiation field with the same rate regardless of the energetic fluctuations of the excited state resulting from the phonon-induced dephasing. The initial rise of the excited state population, its steady-state value, and the subsequent decay after the radiation field is switched off, can all be obtained by analysis of the dynamics in the absence of the phonon coupling, equation (20). For the monomer there is no coherent drive in the rotating frame and the population dynamics given by the diagonal density matrix elements decouple from the off-diagonal matrix elements. The off-diagonal elements (ground-excited state coherences) remain zero at all times, while the excited state population satisfies the following master equation for general $\overline{n}$:

$$\begin{eqnarray}&&{\dot{\rho }}_{e}={\rm{\Gamma }}\overline{n}{\rho }_{g}-{\rm{\Gamma }}(\overline{n}+1){\rho }_{e}.\end{eqnarray} \tag{ 32 }$$

For $\overline{n}=1$ and initial condition of ground state Chla, i.e., ${[{\rho }^{r}]}_{{jk}}\,=1,j=k=1$, ${[{\rho }^{r}]}_{{jk}}=0$ otherwise, the explicit solution is

$$\begin{eqnarray}&&{\rho }_{{\rm{e}}}=\displaystyle \frac{N}{3}(1-{{\rm{e}}}^{-3{\rm{\Gamma }}t}),\end{eqnarray} \tag{ 33 }$$

which shows an exponential rise of the excited state population P_e with rate $3{\rm{\Gamma }}$, yielding a 10–90 rise time of $\sim 2/3{\rm{\Gamma }}$. For the Chla monomer this yields a 10–90 rise time of 40.9 ns, in excellent agreement with the value ${t}_{10-90}=40.9$ ns extracted numerically from figure 6. The condition ${\dot{\rho }}^{r}=0$ further yields a steady-state value of P_e = 0.333 in excellent agreement with the plateau value in figure 6. After the field is switched off, the excited state population decays by spontaneous emission.

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The excitation probabilities with a thermal radiation field are most similar to those resulting from irradiation with a coherent pulse when the bandwidth of the pulse is very small, corresponding to a long pulse that may in the extreme case be regarded as quasi-adiabatic on the timescale of the phonon motions. Figure 7 shows the dependence of P_e^max on the mean photon number $\langle n\rangle$ for coherent excitation with optimal bandwidth parameter ${{\rm{\Omega }}}_{{\rm{opt}}}=2.4{\rm{\Gamma }}$ (blue line—no phonons, red line—with phonons) and compares this with the corresponding dependence of P_e^max on the thermal average photon number $\overline{n}$ (green line—with and without phonons). For coherent excitation with the optimal bandwidth and coupling to phonons, the excitation is again primarily off-resonant as a result of the dephasing effect of the phonons, and we find linear behavior as in figure 4, consistent with linear response. However when the chromophore is now coherently excited without coupling to phonons, the resonant contribution dominates and P_e^max reaches its saturation value after $\langle n\rangle \sim 8$. This saturation from coherent driving is now more similar to the saturation obtained from classical driving of population with the thermal radiation field (green line), which is independent of the coupling to phonons, as noted above, although the underlying kinetics are quite different.

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Figure 8 shows the linear absorption spectrum obtained for this heterodimer using the linear response formalism described in section 2.4. The absorptions of the two chromophores overlap to make a single broad asymmetric absorption peak. Superimposed we show the frequency profile of a multi-mode single-photon Gaussian pulse centered at the maximum of the absorption, with a bandwidth parameter ${\rm{\Omega }}=6.11\times {10}^{13}\,\,\mathrm{Hz}$, chosen to cover the dimer absorption band⁵ .

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Figure 9(a) shows the excitation probability P_e as a function of time for single-photon coherent excitation of the two excitonic states of a heterodimer representing the a603 and a602 Chla chromophores in LHCII of PSII. The excitation is made here with a multi-mode single-photon Gaussian pulse centered at the average of the bare excitation energies of the two chromophores. The result in figure 9 is obtained with the bandwidth parameter ${\rm{\Omega }}=6.11\times {10}^{13}\,\mathrm{Hz}$ which excites both chromophores (see footnote 5). The probabilities of exciting the two excitonic eigenstates are denoted here by $| + \rangle$ and $| - \rangle$, with $| + \rangle$ the higher energy state and $| - \rangle$ the lower energy state.

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Analysis of the dependence of the dimer spectra on bandwidth parameter Ω (not shown here) reveals that the behavior with coupling to the phonon bath is very similar to that seen for the monomer spectra in figure 3, namely a plateau value of ${P}_{e}^{{\rm{\max }}}\sim 5\times {10}^{-6}$ over the range ${\rm{\Omega }}\sim {10}^{7}$–10¹⁷ Hz, with a fall off for ${\rm{\Omega }}\lt {10}^{8}\,\mathrm{Hz}$ and for ${\rm{\Omega }}\gt {10}^{14}\,\mathrm{Hz}$. In the absence of coupling to the bath, the dimer dependence on Ω resembles the monomer dependence for ${\rm{\Omega }}\gt {10}^{13}\,\mathrm{Hz}$, but drops off sharply as Ω decreases below this value, in contrast to the increase in P_e^max that is seen in figure 3 as Ω is decreased to the optimal vacuum value and the monomer absorption becomes increasingly resonant. In both cases, the behavior at high Ω values results from the increasingly dominant role of the off-resonant nature of absorption, just as for the monomer. However when the FHWM of the pulse approaches and then decreases below the energy separation of the two bare chromophore excitonic energies (i.e. Ω approaches and then decreases below ∼1.70 × 10¹³ Hz), the dimer absorption now becomes increasingly off-resonant again. In the absence of phonons, this results in a sharp drop in P_e^max, while the dephasing induced by the phonons will still broaden the absorptive regime of each chromophore and hence that of the full dimer.

In the absence of coupling to the phonon bath, the rise time of the excitation (47.2 fs) and the long timescale (ns) decrease of excitation after the pulse has passed that is due to spontaneous emission (with average rate ∼57 ns for these two Chla), are both similar to that seen for the monomer in figure 2.

The dimer behavior at sub ps timescales shows several interesting additional features when the chromophores are coupled to the bath (red lines). The two inserts show that just after the turnover there are two additional timescales. One is a fast timescale of tens of fs over which weak oscillations corresponding to excitonic coherence are seen (lower insert). These oscillations become more pronounced when the bandwidth parameter Ω is further increased and the pulse length shortens (see below). The second is a slower timescale of ∼3 ps that corresponds to the time to reach the quasi-equilibrium excitonic distribution in the presence of phonon coupling. Note that without coupling to phonons, the higher energy excitonic state, $| +\rangle$, which has a larger transition dipole moment, shows a larger excitation probability at all times. When the coupling to phonons is added, the population of $| +\rangle$ is still larger at short (fs) times but the dissipative effect of the non-Markovian coupling causes relaxation to the lower $| -\rangle$ state, resulting in a crossover of the excitation probabilities at approximately t_c + 315 fs, after which the populations approach their steady-state values on a ps timescale.

As noted above, when the bandwidth parameter Ω is increased, e.g., to the value of ${{\rm{\Omega }}}_{{\rm{sun}}}$ (section 3.1), with a correspondingly shorter pulse duration, the fs timescale excitonic oscillations induced by coupling to the phonons are more pronounced. Conversely, for significantly longer pulses, i.e., pulses of duration 50 ps or greater that correspond to bandwidths less than $\sim {10}^{11}\,\mathrm{Hz}$, the exciton populations rise smoothly to their steady-state values with a very early crossover of the populations so that the lower energy state $| -\rangle$ always appears larger than that of the higher energy state $| +\rangle$ on the timescale resolution of figure 9(b), and do not show any apparent oscillatory relaxation dynamics (red lines in figure 9(b)). This opposite situation can be understood as an inversion of the conventional hierarchy of timescales for photon absorption from short pulses, resulting instead in a fast, and in the extreme case, an 'instantaneous' phonon relaxation during slow photon absorption dynamics. In figure 9(b) the electric field is changing on a much slower timescale than that of the phonon relaxation and the exciton relaxation dynamics therefore occurs on a much faster timescale than this. The result is a very slow rate of interaction of the pulse with the chlorophyll complex, at each point of which excitonic relaxation may be regarded as essentially instantaneous. Note that in the absence of coupling to phonons (blue lines in figure 9(b)), the exciton populations now show oscillatory dynamics during the pulse. They also do not saturate at a steady-state value, decaying instead with the pulse. These features can also be rationalized in terms of the slow rate of change of the electric field relative to the excitonic dynamics: since the monomers have different transition amplitudes $\sqrt{{{\rm{\Gamma }}}_{i}}$, the state reached by the slowly changing field is a superposition of excitonic states with a characteristic oscillation time given by the inverse of twice the energy difference between the excitons, which is now much faster than the rate of change of the electric field. Such oscillatory behavior at twice the excitonic frequency difference also results from a perturbative semiclassical analysis for interaction of the dimer with an oscillating field of constant amplitude, which is consistent with the picture of fast excitonic dynamics following an adiabatically changing electric field. Under these conditions the subsequent decay of the pulse results in de-excitation of the excited states, the dynamics of which essentially follow the pulse adiabatically throughout its duration.

This range of behavior can be summarized by the change in the relationship between the optical coherence time ${\tau }_{{\rm{coh}}}$ and the characteristic timescales of exciton and vibrational dynamics, ${\tau }_{{\rm{exc}}}=1/J,{\tau }_{{\rm{vib}}}=1/\gamma$. For the pulse excitation conditions used in figure 9, ${\tau }_{{\rm{coh}}}=44.2$ fs and we have ${\tau }_{{\rm{coh}}}\lt {\tau }_{{\rm{vib}}},{\tau }_{{\rm{exc}}}$. In this case, during the coherent excitation by the pulse the excitonic and bath degrees of freedom do not have time to relax and we are therefore exciting superpositions of excitonic and vibrational states that show subsequent relaxation dynamics. For the pulse durations longer than 1 ps and corresponding coherence times (recall ${\tau }_{{\rm{coh}}}=4.53{\rm{\Delta }}t$), the timescales now satisfy ${\tau }_{{\rm{coh}}}\gg {\tau }_{{\rm{vib}}},{\tau }_{{\rm{exc}}}$ which ensures rapid equilibration of both excitonic and vibrational degrees of freedom during the optical excitation.

Figure 10 shows the dimer excitation probabilities obtained with a thermal radiation field, equation (18), where the thermal occupation numbers are now slightly different for the a603 and a602 chromophores ($\overline{n}=1.00$ and 1.01, respectively) due to their different bare excited state energies ${E}_{k}^{(0)}$. As in the monomer calculation (figure 6), the field is switched on at 10 ns and off at 200 ns. As in the comparison for the monomer absorption above, we employ a thermal average photon number of one here. The overall behavior of both excitation probabilities (figure 10(a), plotted here in the excitonic basis) is similar to that of the monomer (figure 6), showing a rise on ns timescales, with 10–90 rise times ∼30 ns, to a steady-state saturation value. In the absence of phonons (blue lines), both excitonic states are near equally populated, as expected from equation (18) since for this dimer both the stimulated absorption rates ${{\rm{\Gamma }}}_{k}$ (table 1) and the values of $\overline{n}$ (see above) are very similar for the two monomers. However, when the phonon coupling is present (red lines) we now see that the lower energy excitonic state $| -\rangle$ has higher steady-state excitation P_e, and that the excitonic probabilities show a crossover of P_e for the $| +\rangle$ and $| -\rangle$ states at short times. For the a603–a602 Chla dimer, this crossover occurs at t₀ + 78.5 fs (figure 10(b)) after which the $| -\rangle$ state rises to a larger steady-state value on account of the relaxation between the excited states that is induced by the non-Markovian dephasing of the chromophore excited states. This relaxation is now fast on the ns timescale of light absorption from the thermal radiation field so that the crossover between the populations of $| +\rangle$ and $| -\rangle$ occurs relatively early on in the initial rise of the populations.

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For this heterodimer the time evolution of the excitonic populations is mirrored by a similar exponential rise to a saturation value of off-diagonal excitonic density matrix coherences ${\rho }_{+-}$ (figure 10(c)–(d)). Such finite off-diagonal excitonic coherences are the result of the non-equivalence of the monomer energies (and hence of their stimulated absorption rates ${{\rm{\Gamma }}}_{k}$) and are absent for a homodimer as well as for uncoupled (J = 0) chromophores. The ground-excited state coherences are in all cases zero, as for the monomer in section 4.1.3.

Just as for the monomer under thermal irradiation (section 4.1.3), these features may be understood by detailed analysis of the dynamics of diagonal and off-diagonal terms in equation (20). For a dimeric system, this analysis is now complicated by the presence of the coherent drive term $-\tfrac{{\rm{i}}}{{\hslash }}[{H}_{\delta },{\rho }^{r}]$. In the site basis this term now couples the diagonal population terms to the off-diagonal coherences. Thus a constant drive by incoherent thermal radiation will provide a coherent drive of the excited state coherences via the excitonic coupling between chromophores.

Detailed analysis reveals that the rise times of both populations and coherences are similar to those of the monomer, consistent with the initial absorption being essentially defined by the rates ${{\rm{\Gamma }}}_{i}$ for each excitonic state $| i\rangle$, analogous to the analysis in section 4.1.3. However, the additional coherent drive term introduces several interesting and distinct features into the dynamics. Explicit analysis of the coupled equations of motion for the dimer density matrix elements, equation (20), shows that the excited state coherence ${\rho }_{12}$ between excited states of monomers 1 and 2, and hence also the excitonic coherence ${\rho }_{+-}$, is initially driven by the coherent term $-\frac{{i}}{{\hslash }}{J}({{\rho }}_{22}-{{\rho }}_{11})$, with subsequent additional contributions from stimulated absorption and stimulated plus spontaneous emission terms proportional to ${\rho }_{12}$, once the value of this coherence has risen from its initial zero value. This role of the coherent drive is manifested by the slight concavity evident in the initial growth of $\text{Im}({\rho }_{+-})$ (see insert of figure 10(d)). For a heterodimer, the incoherent driving ensures different excited state populations of the two chromophores, so that provided there is some non-zero excitonic coupling J, the coherent drive term will then generate non-zero excited state coherences. These are shown in the exciton basis in figures 10(c)–(d) and in the site basis for the case of no phonon coupling in figure 10(e). In contrast, for a homodimer the site populations ${\rho }_{11}$ and ${\rho }_{22}$ will always be equal, and then no excited state coherence is generated by absorption from the thermal radiation field, regardless of both the value of J and of whether or not there is coupling to phonons. We note that this analysis applies to the current model of a single light-harvesting complex, i.e., without inhomogeneous broadening effects, and uncorrelated phonon coupling at each chromophore site. Relaxing either of these constraints can potentially give rise to homodimer coherences, as discussed further below.

The calculations without phonon coupling reveal a further interesting feature of these incoherently initiated coherences, namely the presence of oscillations on fs timescales, shown in figure 10(d) for times up to 8 ps. These oscillations decay over a ns timescale of order ${({{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2})}^{-1}\sim 28$ ns. In the absence of phonon coupling we can analytically solve for the steady-state solutions of all density matrix elements in equation (20). This yields a steady-state solution with small but finite values of the coherences. Specifically, the magnitude of the imaginary parts of ${{\rho }}_{-+}$ and ${\rho }_{+-}$ are $\pm \sim {10}^{-10}$ and that of the real parts is $\sim {10}^{-16}$ in both cases. The steady-state populations are in agreement with the numerical simulation plateau values that are established on ns timescales (figure 10(a)). To gain a physical understanding of these fast oscillatory coherences, it is useful to analyze them in the site basis, which can be more directly related to a probability current analysis [70]. Figure 10(e) shows the behavior of ${\rho }_{12}$ and ${\rho }_{21}$ over the same ps timescale. It is evident that the imaginary parts of ${\rho }_{12}$ and ${\rho }_{21}$ are of opposite sign but equal magnitude. The unitary contribution to the excited state probability current within the dimer, which is constituted of the sum of the imaginary components of the coherence, is therefore zero. This eventually results in a zero overall probability current, as required in a closed system [70].

In the presence of exciton–phonon coupling, the coherence oscillations are suppressed and die out near the relaxation crossover point. The suppression increases with the coupling to phonons and is not visible for the value of λ used in figure 10 (table 1). Nevertheless it is evident from figure 10(c) that the magnitude of the steady-state values of the coherence are now significantly larger than the corresponding values in the case of no coupling to phonons. Thus the phonon bath is assisting in maintaining the coherence. It is possible that this is due to the non-Markovian nature of the coupling, but further analysis is required to characterize the mechanism of this enhancement. It should be noted that coupling to phonons that are correlated can further enhance the coherence, due to suppression of the relative fluctuations in the excited state energies [4, 71]. Another factor that can provide additional enhancement of the coherences in ensemble systems is inhomogeneous broadening, which can lead to accidental degeneracies that result in long-lived coherences in photon echo experiments [72, 73].

The above analysis shows that a continuous drive by an incoherent radiation reservoir will generate excitonic coherences in a hetero-chromophore system as a result of the combination of excitonic coupling J and different excitation rates ${{\rm{\Gamma }}}_{i}$ (or equivalently, different oscillator strengths f_i), both in absence and presence of coupling to a phonon bath. Light-harvesting systems thus constitute a natural example of an open quantum system that can generate steady-state coherences from incoherent driving (see also [74]). The steady-state coherences seen here cannot be derived from a classical Einstein rate equation analysis of the excited state populations, which will also predict incorrect population dynamics, since the coherences also enter into the equations of motion for the diagonal terms of equation (20). We further note that the excitonic coherences generated here by interaction of a dimeric chromophore system from a thermal radiation field are distinct from those observed in [15, 18]. Reference [18] reports transient excited state coherences due to additional retarded light-induced inter-chromophore coupling terms in the Hamiltonian, while [15] reports small oscillatory excited state coherences for an uncoupled dimer (J = 0) that are due to a collective coherent (superradiant) interaction of the dimer with the field. In contrast, no collective coherent interaction of the field is assumed in the thermal radiation reservoir analysis employed here.