Coherent optimal control of photosynthetic molecules

We demonstrate theoretically that open-loop quantum optimal control techniques can provide efficient tools for the verification of various quantum coherent transport mechanisms in natural and artificial light-harvesting complexes under realistic experimental conditions. To assess the feasibility of possible biocontrol experiments, we introduce the main settings and derive optimally shaped and robust laser pulses that allow for the faithful preparation of specified initial states (such as localized excitation or coherent superposition, i.e., propagating and nonpropagating states) of the photosystem and probe efficiently the subsequent dynamics. With these tools, different transport pathways can be discriminated, which should facilitate the elucidation of genuine quantum dynamical features of photosystems and therefore enhance our understanding of the role that coherent processes may play in actual biological complexes.

Figure 1

FMO energy-level structure where $|i\rangle$ denotes a single excitation in site $i$. The states $|\pm \rangle$ are the symmetric and antisymmetric superpositions of the states $|1\rangle$ and $|2\rangle$. The green (light gray) and red (dark gray) bubbles identify the fast and slow transport path, as detailed in Ref. [10] where the interplay between different transport pathways in FMO dynamics was discussed at length. The main effect of the inclusion of dephasing noise is the opening of an incoherent relaxation channel from level $|-\rangle$ to level $|+\rangle$ (red wiggled line) and therefore the effective suppression of the coherent oscillation between level $|-\rangle$ and sites 6-7-4 that dominates the coherent dynamics and is responsible for the very slow transport once the sink population has reached $50\%$ (initial population held in site $|+\rangle$). The proposed quantum control strategies efficiently probe this dynamical model.

Figure 2

Error probability ${\varepsilon }_{\alpha }$ vs dephasing rate $\gamma$ (in units of ps${}^{-1}$) for different initial states (i.e., $\alpha =B,D$). Inset: Transfer efficiency vs dephasing rate $\gamma$ (uniform for all of the sites) at $t=10\mathrm{ps}$, when one excitation is initially in site 1.

Figure 3

Optimal errors ${\epsilon }_{B,D}$ vs the pulse amplitude rescaling ($R$) in the preparation of the initial states $B$ and $D$, for $\gamma \sim 1{\mathrm{ps}}^{-1}$.

Figure 4

Probability distribution of the quantity ${\varepsilon }_B$ (top) and ${\varepsilon }_D$ (bottom) for the preparation of the state bright $B$ (top) and the state dark $D$ (bottom), in the presence of dephasing with rate $\gamma \sim 1{\mathrm{ps}}^{-1}$, by using the optimal and a standard Gaussian laser pulse. Since in the laboratory one has a sample of FMO complexes in random orientations, we plot these probability distribution functions (PDFs) for two extreme cases in which we add $1\%$ and $100\%$ of random disorder to the two angles defining the orientation of the FMO complex. We considered a sample of ${10}^4$ FMO complexes to get enough statistics. Inset: Amplitude of the optimal pulse, optimized to prepare the state $B$ (top) and $D$ (bottom) in $250\mathrm{fs}$. Notice that the laser-pulse amplitude is always shown in units of D${}^{-1}$ cm${}^{-1}\sim 6\times {10}^6$ V/m. Finally, the optimal values are (top) $\Delta \theta =2.5$, $\Delta \phi =7.6$, ${\omega }_l=121.76$ cm${}^{-1}$, ${\varepsilon }_B=0.20$, without averaging, and (bottom) $\Delta \theta =3.09$, $\Delta \phi =3.76$, ${\omega }_l=504.46$ cm${}^{-1}$, ${\varepsilon }_D=0.09$, without averaging. Interestingly enough, by considering a simpler linear shape (dashed line), we find a quite similar error, i.e., ${\varepsilon }_B=0.24$ and ${\varepsilon }_D=0.10$, by showing the “robustness” of the optimal pulse to prepare the initial state.

Figure 5

Probability distribution of the quantity ${\varepsilon }_B$ (top) and ${\varepsilon }_D$ (bottom), for the preparation of the states $B$ and $D$, respectively, in the case of $100\%$ orientation disorder, in the presence and absence of inhomogeneous broadening (set to 50 cm${}^{-1}$).

Figure 6

Transfer efficiency as a function of time (ps) for different initial states. We consider an idealized preparation of the states $|-\rangle$ and $|B\rangle$, i.e., ${\varepsilon }_{\alpha }=0$ (dashed lines), and a more realistic scenario where the states $|D\rangle$ and $|B\rangle$ are prepared applying the optimally shaped laser pulses, in the absence (dotted lines) and presence (full lines) of dephasing noise, $\gamma \sim 1{\mathrm{ps}}^{-1}$.

Figure 7

Probability distribution of RC transfer efficiency after 2 ps, in the presence of local dephasing with rate $\gamma \sim 1{\mathrm{ps}}^{-1}$, for the optimal preparations of the state $D$ and the state $B$, with $100\%$ of random disorder in the orientation of ${10}^4$ FMO complexes. The two distributions are distinguishable with less than $5\%$ error.

Figure 8

Modulus of the ensemble average of the elements of the FMO density matrix in the site basis (over ${10}^4$ samples), after applying the optimally shaped laser pulse to prepare the states $B$ (top) and $D$ (bottom), for $\gamma \sim 1{\mathrm{ps}}^{-1}$. We introduce also orientation disorder, set to $1\%$ (left) and $100\%$ (right), and we get the following errors: ${\varepsilon }_B=0.21$ (top left), ${\varepsilon }_B=0.75$ (top right), ${\varepsilon }_D=0.09$ (bottom left), ${\varepsilon }_D=0.53$ (bottom right).

Figure 9

Modulus of the ensemble average of the elements of the FMO density matrix in the site basis (over ${10}^4$ samples), after applying a standard Gaussian pulse, for $\gamma \sim 1{\mathrm{ps}}^{-1}$. We consider also the presence of orientation disorder, which is set to $1\%$ (left) and $100\%$ (right), respectively.

Figure 10

Modulus of the ensemble average of the elements of the FMO density matrix in the site basis (over ${10}^4$ samples), after applying the optimally shaped laser pulse to prepare the states $B$ (left) and $D$ (right), respectively, for $\gamma \sim 1{\mathrm{ps}}^{-1}$, and orientation disorder set to $10\%$ in both cases, considering that some orientation will be obtained in the experiment. The errors are ${\varepsilon }_B=0.44$ and ${\varepsilon }_D=0.10$, respectively.

Figure 11

Probability distribution of the quantity ${\varepsilon }_D$ for the preparation of the state $D$, for $\gamma \sim 1{\mathrm{ps}}^{-1}$, by using the optimal pulses obtained by considering a single FMO complex, by averaging on a small sample of 20 (almost isotropic) FMO orientations pointing to the 20 vertices of a dodecahedron, and on a sample of 21 orientations inside a cone with an opening angle of $0.1\pi$. Hence, one considers the probability distribution of ${\varepsilon }_D$ when these optimal pulses are applied to a sample of ${10}^4$ FMO complexes in completely random orientations in the first two cases and, for the last case, randomly oriented inside that cone. The dashed lines represent the corresponding averaged values, i.e., 0.53 (single orientation), 0.46 (dodecahedron), and 0.29 (cone). The corresponding distribution widths are also plotted.

Figure 12

Top: Behavior of $\Delta$ [defined in Eq. (8)], in units of cm${}^{-1}$ ($\sim$0.1 meV) as a function of the angles $\Delta \theta$ and $\Delta \phi$ defining the orientation of the FMO complex, in the case of ${\omega }_l=-1000$ cm${}^{-1}$ and $E_0=70$ D${}^{-1}$ cm${}^{-1}\sim 42\times {10}^7$ V/m. The difference between the maximum and the minimum value is comparable to the thermal energy, and the maximum is obtained for $\Delta \theta \sim 1.75$ and $\Delta \phi \sim 2$, which seems to be the preferred orientation when the sample is subjected to this constant laser field. Bottom: Maximum value of the Rabi frequencies ${\vec{\mu }}_i·\vec{e}{E}_0$ vs $\Delta \theta$ and $\Delta \phi$. Notice that the detuning values are much larger than the Rabi frequencies.

Figure 13

Top: Boltzmann-Gibbs (BG) probability distribution in orienting the FMOs in the sample, as a function of the angles $\Delta \theta$ and $\Delta \phi$, in the case of ${\omega }_l=-1000$ cm${}^{-1}$ and $E_0=70$ D${}^{-1}$ cm${}^{-1}$. The dashed area represents the region in which we will sum up all of the probabilities in the case, for instance, of a $20\%$ cone opening angle, i.e., $0.2\pi$. Notice, however, that a slightly tilted strip would get a higher probability for the same width and so further improves our results. Bottom: Sample population ratio of FMOs oriented within a cone with some opening angle ($20\%$, i.e., $0.2\pi$, and $50\%$, i.e., $0.5\pi$), as a function of temperature (K), with ${\omega }_l=-1000$ cm${}^{-1}$, and $E_0=70$ D${}^{-1}$ cm${}^{-1}$, according to the BG distribution.

Figure 14

Probability distribution of the quantity ${\varepsilon }_P$, when the FMO is initially prepared with all the population in site 3 and then subjected a laser probe pulse for a time interval of 125 fs to detect the site-3 population, in the presence of dephasing with rate $\gamma \sim 1{\mathrm{ps}}^{-1}$. In particular, we consider the case of a Gaussian pulse on resonance with the site 3, i.e., ${\omega }_l=0$ cm${}^{-1}$, and polarized along some optimal orientation (with regards to a single FMO), and the case of an optimal pulse obtained by averaging the error function ${\varepsilon }_P$ over 21 directions inside a cone with an opening angle of $0.1\pi$, whose optimal carrier frequency is ${\omega }_l=12.63$ cm${}^{-1}$. To get the probability distribution, we apply both pulses to a sample of ${10}^4$ samples randomly oriented inside that cone. Inset: Amplitude (in units of D${}^{-1}$ cm${}^{-1}\sim 6\times {10}^6$ V/m) of both (Gaussian and optimal) probe pulses.