Abstract
Nearunity energy transfer efficiency has been widely observed in natural photosynthetic complexes. This phenomenon has attracted broad interest from different fields, such as physics, biology, chemistry, and material science, as it may offer valuable insights into efficient solarenergy harvesting. Recently, quantum coherent effects have been discovered in photosynthetic light harvesting, and their potential role on energy transfer has seen the heated debate. Here, we perform an experimental quantum simulation of photosynthetic energy transfer using nuclear magnetic resonance (NMR). We show that an Nchromophore photosynthetic complex, with arbitrary structure and bath spectral density, can be effectively simulated by a system with log_{2} N qubits. The computational cost of simulating such a system with a theoretical tool, like the hierarchical equation of motion, which is exponential in N, can be potentially reduced to requiring a just polynomial number of qubits N using NMR quantum simulation. The benefits of performing such quantum simulation in NMR are even greater when the spectral density is complex, as in natural photosynthetic complexes. These findings may shed light on quantum coherence in energy transfer and help to provide design principles for efficient artificial light harvesting.
Introduction
Efficient exciton energy transfer (EET) is crucial in photosynthesis and solar cells,^{1,2} especially when the systems are large,^{3} e.g., as in Photosystem I (PSI) and Photosystem II (PSII), which have hundreds of chromophores. A comprehensive knowledge of the quantum dynamics of such systems would be of potential importance to the study on EET.^{4} Much effort has been made to reveal the effects of quantum coherence on efficient energy transfer.^{4,5,6,7,8,9,10,11} In order to try to mimic EET, a Frenkelexciton Hamiltonian is required and this can be studied with quantum chemistry approaches,^{1} e.g., fitting experimental spectra, or calculations by density functional theory. Because photosynthetic pigment–protein complexes are intrinsically open quantum systems, with systembath couplings comparable to the intrasystem couplings, it is difficult to faithfully mimic the exact quantum dynamics of EET. Among the methods for describing EET,^{1,11,12,13} the hierarchical equation of motion (HEOM) yields a numerically exact solution at the cost of considerable computing time.^{11,14} For the widelystudied Fenna–Matthews–Olson (FMO) complex,^{5,6,11,13,15,16} a reliable result could be produced by 16,170 coupled differential equations at low temperatures, based on the HEOM^{11,14} with 4 layers. Moreover, the HEOM encounters difficulties when the system has nontrivial spectral density, making it often difficult to verify the results. On the other hand, due to heterogeneity, e.g., static disorder and conformational change, it is difficult to experimentally verify the theoretical predictions in natural photosynthetic systems. Because quantum simulations with N qubits can powerfully mimic the quantum dynamics of 2^{N} states by virtue of quantum mechanics,^{17,18,19} the quantum simulation of photosynthetic energy transfer with an arbitrary Hamiltonian by nuclear magnetic resonance (NMR) provides an intriguing approach to verify theoretical predictions. Recently, a newlydeveloped technique which could simulate the effect of a bath with an arbitrary spectral density by a set of classical pulses has been successfully realized with ion traps^{20} and NMR.^{21} Therefore, photosynthetic lightharvesting with an arbitrary Hamiltonian and spectral density, describing a structured environment, can be experimentally simulated by NMR.
Results
NMR is an excellent platform for quantum simulation since it is easy to operate and it can have long coherence times.^{22,23} In this paper, NMR is utilized to simulate the quantum coherent dynamics in photosynthetic light harvesting. As a prototype, a tetramer including four chlorophylls,^{13} as schematically illustrated in Fig. 1a, is employed in the NMR quantum simulation. In a previous investigation,^{13} the tetramer model was exploited to study the clustered geometry utilizing exciton delocalization and energy matching to accelerate the energy transfer. The EET in photosynthesis is described by the Frenkelexciton Hamiltonian
where \(\left i \right\rangle\) (i = 1, 2, 3, 4) is the state with a single excitation at site i and other sites at the ground state, ε_{i} is the site energy of \(\left i \right\rangle\), J_{ij} is the excitonic interaction between sites i and j. In our quantum simulations, we adopt the site energies^{13} ε_{1} = 13,000 cm^{−1}, ε_{2} = 12,900 cm^{−1}, ε_{3} = 12,300 cm^{−1}, and ε_{4} = 12,200 cm^{−1}, and the couplings J_{12} = J_{34} = 126 cm^{−1}, J_{13} = J_{24} = 16 cm^{−1}, J_{23} = 132 cm^{−1}, and J_{14} = 5 cm^{−1}, which are typical parameters in photosynthetic systems.
For a photosynthetic complex with N chlorophylls, there are only N singleexcitation states involved in the energy transfer. Therefore, only log_{2} N qubits are required to realize the quantum simulation. To mimic the energy transfer described by the above Hamiltonian (1), two qubits are necessary for the quantum simulation. In this case, the photosynthetic singleexcitation state \(\left i \right\rangle\) is encoded as a twoqubit product state, i.e., \(\left {00} \right\rangle\), \(\left {01} \right\rangle\), \(\left {10} \right\rangle\), and \(\left {11} \right\rangle\). By a straightforward calculation, the Frenkelexciton Hamiltonian can be mapped to the NMR Hamiltonian as
where \(\sigma _j^u\) (j = 1, 2, u = x, y, z) is the Pauli operator for qubit j, numerically \(\varepsilon _j^\prime = \pi \varepsilon _j{\mathrm{/}}10\) and \(J_{ij}^\prime = \pi J_{ij}{\mathrm{/}}10\), but the dimension cm^{−1} should be replaced by kHz. In other words, all realistic parameters have been scaled down in energy by a factor of 1 cm^{−1}/(π/10 kHz) = 3 × 10^{8}/π.
The excitonic coupling J_{ij} between sites i and j makes the exciton energy hop in both directions, i.e., \(\left i \right\rangle\) ⇆ \(\left j \right\rangle\). Apart from this, the systembath couplings will facilitate the energy flow towards an energy trap, where the captured photon energy is converted into chemical energy.^{2,3} The interaction between the system and bath in photosynthetic complexes can be described by
where \(a_{ik}^\dagger\) is the creation operator for the kth phonon mode of chlorophyll i with coupling strength g_{ik}. H_{SB} will induce puredephasing on the ith chromophore when it is in the excited state. Generally, the systembath couplings are given by the spectral density
which we assume identical for all chromophores. For typical photosynthetic complexes, the systembath couplings are of the same order as the intrasystem couplings. In order to mimic the effects of noise, we utilize the bathengineering technique which has been successfully implemented in ion traps and NMR. The implementation of a puredephasing Hamiltonian
relies on generating stochastic errors by performing phase modulations on a constantamplitude carrier, i.e.,
where Ω_{0} is the constant amplitude of a magnetic field with driving frequency ω_{μ}, ψ_{j} is a random number, ω_{j} = jω_{0} with ω_{0} and ω_{J} = Jω_{0} being base and cutoff frequencies, respectively, and α is a global scaling factor. The powerspectral density \(S_z(\omega ) \equiv {\int} {\kern 1pt} d\tau \left\langle {\dot \phi _N(t + \tau )\dot \phi _N(t)} \right\rangle e^{i\omega \tau }\) is the Fourier transform of the autocorrelation function of \(\dot \phi _N(t)\),
By appropriately choosing F(j) and the cutoff J, an arbitrary powerspectral density of the bath can be realized, e.g., white, Ohmic, or Drude–Lorentz spectral densities. For the details, please refer to the Supplementary Information.
As illustrated in Fig. 1b, the nuclear spins of the carbon atom and hydrogen atom in a chloroform molecule are chosen to encode the two qubits with the Hamiltonian written as
where ω_{1} = 3206.5 Hz, ω_{2} = 7787.9 Hz are the chemical shifts of the two spins, and J = 215.1 Hz (ref. ^{24}). Therefore, in order to simulate the quantum dynamics of energy transfer, the entire evolution is decomposed into repetitive identical cycles. After each cycle, there is a small difference between the exact evolution and the simulated one. However, because there are tens of thousands of cycles before completing the energy transfer, the accumulated error might be significant so that the simulation becomes unreliable. To avoid this problem, we utilize the gradient ascent pulse engineering (GRAPE) algorithm,^{25} which has been successfully applied to a number of quantum simulations in NMR,^{23,26,27} to mimic the quantum dynamics of light harvesting.
Before studying the population dynamics, we shall explore the characteristics of the artificially injected noise. Generally, the noise is characterized by Ramsey interference measurements, as shown in Fig. 2. When there is no noise applied to the system, we would expect coherent evolution without damping. However, as the artificial noise is applied, the oscillations become damped. The envelope of the damped oscillations is fully determined by the noise strength. Interestingly, for the parameters employed in this work for a broadfrequency background bath, the NMR experimental results exactly fit the theoretical prediction by HEOM. This suggests that our technique for injecting artificial noise can, in this limit, faithfully reproduce the effect of the protein environment on natural photosynthetic complexes.
In Fig. 3a, the quantum coherent energy transfer for the above Hamiltonian H_{NMR} + H_{PDN}, using an initial condition where an excitation is localized on the first chromophore, is simulated in NMR. In the shorttime regime, cf. Fig. 3b, there are Rabilike oscillations of coherent energy transfer between the two levels with the highest energies, because there is a strong coupling between these two levels. Furthermore, the oscillation quickly damps as the energy transfer is irreversible due to the puredephasing noise. After an exponential decay process, the populations on all levels reach thermal equilibrium. Noticeably, there are small oscillations for the two lowestenergy levels as a result of their strong coupling. For each point in the quantum simulation, the data is averaged over M random realizations. For a given realization, the system undergoes a coherent evolution by applying a timedependent magnetic field with fluctuating phases, as shown in Eqs. (6) and (7). However, for an ensemble of random realizations, since each realization experiences a different phase at a given time, the ensemble average manifests itself as a single realization undergoing a puredephasing noise. In this regard, the deviation of the NMR simulation from that predicted by the HEOM decreases as the number of realizations in the ensemble increases. In Fig. 4, we show the effect on the simulation error of the number of realizations in each ensemble. Both results, by NMR and by HEOM, converge as M increases, such that the difference between them can hardly be noticed when M ~ 500. This effect would be more remarkable if M were increased further, as confirmed by our theoretical simulations in the Supplementary Information. In conclusion, the dephasing Hamiltonian (3) in photosynthesis is effectively mimicked by a classical timefluctuating magnetic field ((6) and (7)).
It has been suggested in the literature^{6,28,29} that the longlasting electronic coherence observed in experiments might be attributed to strong coupling to vibrational modes. Therefore, it is interesting to consider whether it is possible to simulate some features of underdamped vibrational modes in the NMR platform. In photosynthesis, the coupling to a vibrational mode can be described by \(\mathop {\sum}\nolimits_i {\kern 1pt} g_{iv}\left i \right\rangle \left\langle i \right\left( {a_{iv}^\dagger + a_{iv}} \right)\). In order to simulate the underdamped vibrational mode in our protocol, we can supplement an additional term, i.e., αF(v) sin(ω_{v}t) into Eq. (7). In Fig. 5, we show the numerical simulation for the above 4level system including such a “semiclassical” vibrational mode. Because we set the vibrational mode to be closelyresonant with the two lower exciton states, there is significantly prolonged coherence in the two lower energy states in the shorttime regime, as compared to Fig. 3. Since there is no random phase associated with this modulation, and quantum properties of the vibrational mode are neglected, each realization will evolve uniformly in Fig. 5. Therefore, the longlasting coherence is driven by the nearlyresonant driving. Going beyond a semiclassical approach, and accurately capturing the quantum features of this mode, requires the use of ancillary degrees of freedom, as discussed in the conclusions.
Discussion
Note that the proofofconcept NMR experiment presented in this work goes beyond reproducing the HEOM results. To faithfully simulate the EET dynamics on a large photosynthetic system, using the HEOM would be computationally unaffordable. In addition, the HEOM^{11,14} is known to encounter difficulties when the spectral density is not in a simple Drude–Lorentz form. As shown in the Supplementary Information, the computational cost of HEOM scales exponentially with the system size and the number of exponentials in the bath correlation function.^{11,14} Our NMR simulations do not have such shortcomings, as it scales polynomially with respect to the system size.^{30,31} Moreover, the quantum simulation algorithm also scales more favorably in terms of the number of exponentials in the bath correlation function. Notice that our simulation proposal is designed for a Gaussian bath. Interestingly, a recentlydeveloped stochastic Liouville equation approach^{32} can effectively handle opensystem dynamics for spin and fermion bath. In the future it might be beneficial to generalize our approach to handle nonGaussian baths. With our current approach, in principle, a photosynthetic system with ~100 sites (e.g., the PSI complex) requires a 7qubit quantum simulator. This means that the coherent EET dynamics of a full functional biological unit for photosynthesis (e.g., the PSII supercomplex that contains ~300 sites) can be faithfully simulated by a 9qubit NMR quantum computer. Recently, the quantum control on 12 qubits has been successfully demonstrated on an NMR system,^{31} where the shortest \(T_2^ \ast\) is 150 ms. In natural photosynthetic complexes, the energy transfer is completed within 50 ps. For the scaling factor in our proposal, it corresponds to 7.5 ms, which is smaller than \(T_2^ \ast\) by one order of magnitude. Therefore, it is reasonable to simulate the coherent energy transfer within the intrinsic decoherence time of NMR systems. Thus, we believe that NMR quantum simulation has the potential to help clarify the mysteries of light harvesting in natural photosynthesis. In this regard, other approaches, which do not take advantage of this scaling provided by encoding multiple sites into a single qubit,^{33} currently lack the required size to simulate such large photosynthetic systems, and thus may also benefit from the approach we develop here.
On the other hand, in natural photosynthetic complexes, the energy is transferred from the outer antenna to the reaction center. In principle, the excitation depletion at the reaction center could be simulated in the NMR platform by coupling auxiliary spins to the energy sink site to remove the population, which is similar to introducing an amplitude damping error in the simulations.^{34} In order to prevent the back transfer of energy, there is a large energy gap between the lowest eigenstate of the outer antenna and the reaction center. Thus, the transfer rate from the lowestenergy state to the reaction center is much smaller than the transfer rate within the outer antenna. It is reasonable to simulate the energy transfer without the reaction center.^{35} In our theoretical exploration,^{13} we investigate the geometrical effects on the energy transfer. In another exploration^{36}, it was shown that tuning the phases of the intermolecular couplings can further optimize the energytransfer efficiency. In future work, it might be interesting to investigate the effect of the couplings’ phases on the NMR quantum simulation.
In this paper, the photosynthetic energy transfer is experimentally simulated using NMR. As a prototype, a twoqubit NMR system is utilized to demonstrate both: coherent oscillations in the shorttime regime and steadystate thermalization for long times. By using the GRAPE technique, an arbitrary photosynthetic system can be faithfully mapped to the NMR system. Moreover, the effect of the environment on EET can be effectively mimicked by a set of welldesigned pulses, which act as a classical puredephasing noise. The quantum simulation of photosynthetic energy transfer in NMR would facilitate the investigation of quantum coherent effects on the EET and provide clearer design principles for artificial lightharvesting devices together with structured baths.^{13,37}
It was experimentally verified in circuit QED^{33} that a structured bath can optimize the energy transfer in EET.^{37} Vibrationally assisted energy transfer was demonstrated in a trappedion quantum simulator.^{38} In both cases, the physical setups therein showed their potential to verify the design principles in optimal photosynthetic light harvesting. However, here we showed that we can use NMR and bathengineering techniques to efficiently mimic exactly photosynthetic light harvesting in large systems with arbitrary system structure. On the other hand, with our current approach, the simulation may not capture quantum features of the system–environment interaction that arises at stronger couplings and lower temperatures. Capturing such correlations by encoding them in ancillary qubits^{39} is a potentially rich avenue for future explorations.
Methods
Physical parameters
The Hamiltonian H_{NMR} implemented in our NMR simulations is the same as the photosynthetic Hamiltonian H_{EET}, but with the unit cm^{−1} divided by 3 × 10^{4}/π. The diagonal terms are
The interlevel couplings are: the nearest neighboring couplings
the nextnearestneighboring couplings
and the coupling between the two ends
The temperature of the photosynthetic energy transfer and NMR experiment are respectively T_{EET} = 3 × 10^{4} K and T_{NMR} = 5 × 10^{−5} K. The reorganization energy of the bath is λ_{EET} = 0.2 cm^{−1}. The cutoff frequency of the bath is γ_{EET} = 900 cm^{−1}.
Both the Hamiltonian and bath parameters for NMR experiment are those for the photosynthetic energy transfer which are scaled down by a factor of 3 × 10^{8}/π.
Initialization
Starting from the thermalequilibrium state, we prepare a pseudopure state^{40,41} using the spatialaverage technique.^{41}
Measurement
The goal is to acquire probability distributions of four states \(\left {00} \right\rangle\), \(\left {01} \right\rangle\), \(\left {10} \right\rangle\), and \(\left {11} \right\rangle\) after the pulse sequences that we designed are implemented on the twoqubit NMR system, namely, four diagonal values of the final density matrix. The density matrices of the output states are reconstructed completely via quantum state tomography.^{42} Therefore, the density matrix of the system can be estimated from ensemble averages of a set of observables. For the onequbit system, the observable set is {σ_{i}} (i = 0, 1, 2, 3). Here, σ_{0} = I, σ_{1} = σ_{x}, σ_{2} = σ_{y}, σ_{3} = σ_{z}. For the twoqubit system, the observable set is {σ_{i} ⊗ σ_{j}} (i, j = 0, 1, 2, 3). In our experiments, the complete density matrix tomography is not necessary. All we need is to perform two experiments in which the readingout pulses exp(−iπσ_{y}/4) ⊗ I and I ⊗ exp(−iπσ_{y}/4) are respectively implemented on the final states of ^{1}H and ^{13}C. The symbols in Fig. 3 are obtained by averaging over M = 150 random realizations.
Ramsey spectroscopy
In our experiments, we perform Ramsey spectroscopy for a single qubit subject to Drude–Lorentz noise. In particular, the experiment is divided into three steps:

(1)
Preparing the state \(\left 0 \right\rangle\) and rotating it to the xy plane by a π/2 pulse along the xaxis.

(2)
Free evolution while adding the Drude–Lorentz noise.

(3)
Applying a reverse π/2 pulse and measuring the population of the state \(\left 0 \right\rangle\).
We observe that the decay time constant of the population of the state \(\left 0 \right\rangle\) is significantly reduced in the presence of the Drude–Lorentz noise, as shown in Fig. 2. Ramsey fringes are a fit to a cosine with a simple exponential decay envelope. The result by the HEOM theory (blue curve in Fig. 2) is consistent with that in experiments (red dots in Fig. 2). In other words, we demonstrate that χ(t) = Re[g(t)].
Vibrational mode
In photosynthetic energy transfer, the coupling to a vibrational mode is described by \(\mathop {\sum}\nolimits_i {\kern 1pt} g_{iv}\left i \right\rangle \left\langle i \right\left( {a_{iv}^\dagger + a_{iv}} \right)\). In order to simulate the underdamped vibrational mode, we can supplement an additional term into Eq. (7), i.e., αF(v) sin(\(\omega _{{\mathrm{NMR}}}^vt\)). Here we choose \(\omega _{{\mathrm{NMR}}}^v\) = 2π × 12.5 kHz and \(\alpha F(v) = \sqrt {0.1} \times \omega _{{\mathrm{NMR}}}^v\). Since there is no random phase associated with this modulation, i.e., ψ_{v} = 0, each realization will be driven uniformly. Because we set the vibrational mode to be closelyresonant with the two lower exciton states, i.e.,
and the Huang–Rhys factor (g_{iv}/\(\omega _{{\mathrm{EET}}}^v\))^{2} = 0.1, the longlived coherence observed in experiments can be effectively simulated, as shown in Fig. 5.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank XingLong Zhen and KeRen Li for discussions. G.L.L. was supported by National Natural Science Foundation of China under Grant Nos. 11175094 and 91221205, and the National Basic Research Program of China under Grant No. 2015CB921001. F.G.D. was supported by the National Natural Science Foundation of China under Grant Nos. 11474026 and 11674033. F.N. was supported in part by the MURI Center for Dynamic MagnetoOptics via the Air Force Office of Scientific Research (AFOSR) (FA95501410040), Army Research Office (ARO) (Grant No. 73315PH), Asian Office of Aerospace Research and Development (AOARD) (Grant No. FA23861814045), Japan Science and Technology Agency (JST) (the ImPACT program and CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (JSPSRFBR Grant No. 175250023), RIKENAIST Challenge Research Fund, and the John Templeton Foundation. Q.A. was supported by the National Natural Science Foundation of China under Grant No. 11505007. This work was supported in part by the Beijing Advanced Innovation Center for Future Chip (ICFC).
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All work was carried out under the supervision of G.L.L., F.G.D., Y.C.C., and F.N. B.X.W. and T.X. performed the experiments. Y.C.C., B.X.W., Q.A., M.J.T., and N.L. analyzed the experimental data. M.J.T., Q.A., and N.L. wrote the HEOM simulation code and performed the numerical simulation. Y.C.C. and Q.A. contributed the theory. All authors contributed to writing the manuscript.
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Wang, BX., Tao, MJ., Ai, Q. et al. Efficient quantum simulation of photosynthetic light harvesting. npj Quantum Inf 4, 52 (2018). https://doi.org/10.1038/s4153401801022
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DOI: https://doi.org/10.1038/s4153401801022
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