Enhanced quantum efficiency of light-harvesting in a biomolecular quantum “steam engine”

In 1824, the French military engineer Nicolas Léonard Sadi Carnot laid the grounds (1) for the understanding of the basic principles of any heat engine that converts heat into useful work. Now, Dorfman et al. (2) exploit these insights to explore the reaction center of a photoactive biomolecule as a highly efficient biological, but quantum mechanical, heat engine driven by the thermal radiation of the sun.

In his lifetime, Carnot recognized the possibilities of the steam engine, which, however, could only be improved empirically because a fundamental understanding of the physical principles of a heat engine was lacking. Carnot extracted empirical but universal principles by which heat is transformed into useful mechanical work; at the same time, he avoided speculative assumptions about the microscopic constituents of the involved materials (3). Carnot imagined an engine that takes in heat from a hot reservoir at temperature T_h, releases heat to a cold reservoir at temperature T_c, and converts the difference in energy to mechanical work. He determined the maximum efficiency (the ratio of the extracted work and the invested heat) to be η = 1 − T_c/T_h. Carnot understood that these principles apply to “every imaginable heat engine” that relies on a “periodic repetition” (1). The Carnot cycle as an idealized process is still an invaluable conceptual tool used today to understand the function of classic heat and cooling engines. The Carnot efficiency sets the limit of the maximally possible efficiency of a heat engine based on classic (equilibrium) thermodynamics.

When Carnot’s principles are transferred to the realm of quantum mechanics (4), surprises (5) are possible. In 2003, Scully et al. showed (4) that quantum coherence can fundamentally alter the operation of a heat engine. The efficiency of a quantum heat engine, such as a maser (6) or a photocell (7), is also limited by the Carnot efficiency because of the principle of detailed balance between the emission and absorption processes, which lead Einstein in the early days of quantum mechanics to the concept of stimulated emission (8). In thermal equilibrium, the decay of an atomic electron in the excited state to the ground state by emitting a photon is as probable as the promotion of an electron to the excited state by absorption of a photon. A Carnot cycle results when photons from a thermal background radiation at temperature T_h excite the atomic electrons and a load uses this energy, thereby deexciting the electrons back to the ground state, the energy of which determines the reservoir temperature T_c. Quantum coherence may break detailed balance (4): When the ground state is replaced by two quantum states of almost equal energy, which are prepared in a quantum-coherent superposition, their relative phase determines the relative population of the two low-energy states. Such a phase-coherent three-level system in general does not fulfill detailed balance between the lower two states and the upper state. By adjusting the phase with an external microwave field, the classic Carnot efficiency can be exceeded.

Another example of broken detailed balance is the concept of lasing without inversion (9): When a coherent superposition of the quantum system is prepared by a weak probe radiation, the destructive quantum interference of the transition pathways can erase absorption but stimulated emission is retained. Thus, radiation can be amplified without inverting the population, but solely by exciting low-frequency coherences.

In 2010, Scully (10) exploited quantum coherence between states of a quantum dot between positively and negatively doped semiconductors. In a minimal model of a p-n–junction photocell, incoming radiation excites an electron from the valence to the conduction band. This electronic band structure provides a “built-in” electric field in the depletion layer, which separates the electron and the hole, thus generating an electric potential difference. In the steady state, this “Carnot voltage” depends on the ratio of the lower temperature of the surroundings and the higher temperature of the solar radiation. However, a major problem that reduces conversion efficiency is that the electron-hole pair can radiatively recombine before it is separated. The losses can be reduced when the single quantum state at higher energy is replaced by two quantum states separated in energy by a small Δε. An alternating resonant electric field may generate a quantum-coherent superposition of the two states, thereby performing work. By adjusting their relative phase, recombination for both states can be minimized, yielding an increase of the generated voltage by Δε.

A different scheme to improve efficiency has been proposed by Nozik (11) and involves the generation of multiple bound electron-hole pairs (excitons) in small semiconductor quantum-dot nanocrystals when a single photon is absorbed and more than one exciton is created. When the photon energy is at least twice the band gap energy, the excess kinetic energy of the electrons and holes can create further excitons, thereby increasing the overall conversion efficiency.

As an alternative to the coherent radiation, quantum coherence can also be induced by randomly fluctuating forces from outside the quantum system. These forces typically originate from an incoherent environment that hosts the system. Incoherent lattice fluctuations (phonons) in a host crystal in condensed matter devices, fluctuating molecular vibrational “background” modes of large biomolecules, fluctuating polar solvent molecules in a physiological liquid, or a bath of photons comprising the thermal light of the sun are typical sources of environmental fluctuations, which rattle the energy levels of the two quantum states. Thus, without external coherent radiation, the efficiency can still be increased (12). This noise-induced generation of quantum coherence (known as the Fano effect) is at the heart of the “biological quantum heat engine” of Dorfman et al. (2). These authors transfer this concept to the light-harvesting in photoactive biomolecules. There, the excitation energy is transiently stored in a bound electron-hole pair after the harvest of a solar or a laser photon. The energy is transferred to the reaction center, which is the molecular subunit where the energy is used to separate charges in a chemical reaction. It has been a longstanding question as to what determines the transfer efficiency being close to 99% and whether the nature of the transfer is the reason. These first transfer steps can occur in the form of a quantum-coherent wave-like process along the molecule or via a sequence of uncorrelated hoppings. Ultrafast optical spectroscopy allows us to study these rapid-transfer processes on very short time scales. Signatures of electronic quantum coherence have been found as early as in 1997 (13). More recent experiments on photosynthetic antenna proteins (14–16) and reaction center proteins (17) have reported quantum-coherent time evolution over surprisingly long time scales after excitation with coherent laser pulses. Theoretical investigations (18, 19) indicate for the Fenna–Matthews–Olson complex that a combination of electronic and molecular vibrational (i.e., vibronic) degrees-of-freedom are at the origin of the long-lived coherent quantum evolution, while the quantum-beating signals in the photosynthetic reaction center of Rhodobacter sphaeroides are attributed (17) to purely electronic intermolecular coherences.

Dorfman et al. (2) propose a concept in the form of a four-level quantum heat engine composed of a donor special pair, D₁ and D₂, and an acceptor A molecule (Fig. 1). The global ground state b describes the configuration when all the molecules are in their lowest energy state. Initially, a cooperative absorption of an energy quantum (either provided by the solar photon directly or fed in by a molecular “wire,” which connects the antenna with the reaction center) promotes the donor special pair D₁ and D₂ to a common excitation of their higher energy states a₁ and a₂ (b →a₁ and b →a₂). Both of those states are only separated in energy by the small Davydov splitting. Their neighborhood in energy ensures that the states can be in a quantum-coherent superposition state. The excitation is still localized in the donor. In a first step, the excitation of states a₁ and a₂ are then transferred to the lower lying excited state α of the acceptor A (a₁ →α and a₂ →α). The state α is a charge-separated state with the electron in A and the hole in the donor part. The excess energy is emitted as phonons. In a second step, the acceptor relaxes from its excited state α to its ground state β (α →β) via an effective “load” connecting both states. The excess energy can produce useful work in form of an electric current. To complete the cycle, a final transfer in step 3 takes the electron back to the global ground state b with the emission of a phonon and the cycle can start again (step 4).

Fig. 1.

Scheme of a biomolecular quantum Carnot heat engine realized by a donor (D)–acceptor (A) excitation energy transfer system. “b” denotes the molecular ground-state configuration, and a₁ and a₂ refer to the two excitation states separated by the Davydow splitting. α is the excited state of the acceptor and β its ground state. The inner scheme shows the classic Carnot cycle.

This quantum Carnot cycle will suffer from unavoidable losses by stimulated or spontaneous emission of thermal photons, which can bring the whole configuration back to its ground state b before contributing to the work. The key observation is that when a quantum-coherent phase relation, which is sufficiently long-lived, can be induced between a₁ and a₂, the losses can be minimized by destructive interference of the loss transitions such that the energy can be released to the state α, thereby contributing to the work.

The crucial coherence between a₁ and a₂ can be generated by noise-induced cross couplings between the two states (12). This process works best in the overdamped regime where environmental fluctuations strongly joggle the quantum levels and produce a large energetic overlap. The level splitting is less than the noise-induced level widths, and an enhancement of the output peak power by 27% is obtained in the simple toy model, assuming somewhat realistic parameters. The gain is still 18% in the intermediate regime between over- and underdamped relaxation: It is counterintuitive that it requires strong environmental fluctuations to asymptotically generate steady-state oscillations of the coherences, while they vanish in the underdamped regime. The fluctuations are used as a “source of coherence” for the biological quantum-heat engine in a constructive manner. Finally, the Carnot efficiency is recovered for an open circuit when the load is removed and the system remains at equilibrium. It is also important to emphasize that the cycle does not rely on coherent laser irradiation but works with natural thermal light (20).

The principle insights are gained on the basis of a generic minimal model. How (and whether) this concept is realized by nature remains to be shown (e.g., by identifying the relevant molecular states in particular natural biomolecules or by including more realistic sources of fluctuations). Similarly, artificial photoactive proteins based on organic compounds could be designed to test the concept. In the long run, this approach might have important technological applications in form of cheap organic solar cells with an increased light-harvesting efficiency (21, 22).