Molecular Heat Engines: Quantum Coherence Effects

Recent developments in nanoscale experimental techniques made it possible to utilize single molecule junctions as devices for electronics and energy transfer with quantum coherence playing an important role in their thermoelectric characteristics. Theoretical studies on the efficiency of nanoscale devices usually employ rate (Pauli) equations, which do not account for quantum coherence. Therefore, the question whether quantum coherence could improve the efficiency of a molecular device cannot be fully addressed within such considerations. Here, we employ a nonequilibrium Green function approach to study the effects of quantum coherence and dephasing on the thermoelectric performance of molecular heat engines. Within a generic bichromophoric donor-bridge-acceptor junction model, we show that quantum coherence may increase efficiency compared to quasi-classical (rate equation) predictions and that pure dephasing and dissipation destroy this effect.


I. INTRODUCTION
Single molecules are used as building blocks of molecular devices for electronics, biosensors, nanoscale motors, controllable chemical reactivity and energy transfer [1][2][3] . The development of nano-fabrication led to tremendous progress in the ability to detect and manipulate molecules on surfaces and in junctions [4][5][6][7][8][9] . Due to the small size of nanodevices, their characterization is necessarily quantum, and interference is expected to be crucial in the response of molecular electronic devices. Experimental measurements have demonstrated quantum coherence effects on the transport 10-12 and optical response 13,14 of molecular junctions. The importance of quantum coherence in energy transfer was demonstrated in experimental studies of the initial stages of photosynthesis [15][16][17][18] .
Understanding and controlling the combined motion of charges and excitations (energy) is crucial for the development of new materials and state-of-the-art guiding principles for building efficient energy conversion and storage devices. Historically, this research has been focused on thermoelectric properties in bulk materials 19 . With the development of nanofabrication techniques, the study of thermoelectric properties at the nanoscale attracted a lot of attention experimentally [20][21][22][23][24][25][26][27][28] and theoretically [29][30][31][32][33][34][35][36][37][38][39][40] . The small size of these nanodevices gives rise to new physical phenomena (such as quantum coherence) that are not present at the macroscopic level, and which promise to improve the performance of energy conversion.
These studies are concerned with characterizing charge and energy fluxes in the system.
A closely related set of works utilized the thermodynamic approach to determine the efficiency of photoelectric devices [41][42][43][44][45] . With rate (Pauli) equations employed in the thermodynamic description of such devices, quantum coherence could not be fully taken into account (see below). The effect of coherence on the thermodynamics of quantum heat engines consisting of n-level systems coupled to thermal bath(s) was considered in a number of publications 46-50 at the Lindblad-Redfield level of theory. These considerations were either restricted to closed systems or disregarded charge transfer between the system and the baths (the latter is inherent in molecular devices). Thus, naturally also in thermodynamic studies, the question of how quantum coherence affects a molecular device performance can be raised.
We note in passing that the distinction between populations and coherences (diagonal and off-diagonal elements of the density matrix) is basis dependent. For example, transforming the local basis to the eigenbasis of the system accounts for coherences of the local basis simply by converting them into populations in the eigenbasis. For an isolated system, local basis coherences are taken into account exactly as a result of such a transformation. In the presence of baths, for the consideration to be complete one has to account also for bathinduced coherences between the eigenstates of the system. The latter cannot be treated properly at the Lindblad-Redfield quantum master equation (QME) level of theory (one has to go to at least the fourth order in system-bath coupling to account for the coherences 51 ), while approximate consideration may lead to qualitative failures (see, e.g., [52][53][54] ). In addition, Lindblad-Redfield QME does not allow to model the gradual transition from coherent to incoherent transport. The latter is the focus of our study.
Here, we consider a generic donor-bridge-acceptor (DBA) molecular system, which is coupled to fermionic baths and is driven against the bias applied by solar photons. This setup is a simple model for a continuous steady-state heat engine 55-57 , whose thermoelectric efficiency was previously considered in 41,42,44 with the effects of quantum coherence disregarded. The latter were shown to play an important role in the charge and energy transport in similar models of DBA molecular junctions 58,59 .
We utilize nonequilibirum Green function (NEGF) methodology 60 , which is capable of accounting for quantum coherence in an open nonequilibrium system, to study the effects of quantum coherence on the average efficiency of photoelectric molecular devices. For simplicity, our consideration is restricted to a non-interacting (electron-electron and electronvibration interactions are disregarded) molecular system, although intra-system interactions in principle can be taken into account within many-body flavors of the methodology 61-65 .
We show that quantum coherence may lead to an increase and a decrease in the efficiency of the device and study the transition to a quasi-classical regime by destroying coherence with pure dephasing or dissipation. The former is achieved by employing a Büttiker probe 66 , and the latter is induced by increasing the strength of the system-contacts coupling. We note in passing that although a general formulation of quantum thermodynamics for currentcarrying junctions has not yet been established 67,68 , the formulation is clear as long as the junction operates in a steady-state 69,70 . The latter is the situation considered here.
The paper is structured as follows: In Section II, we introduce the model and discuss the technical details of the simulations. We present the results of the numerical simulations and compare them with previously published (coherence-free) studies in Section III. We draw conclusions in Section IV.

II. MODEL
We consider a molecular junction comprised of a DBA molecular complex coupled to metallic contacts L and R (see Figure 1). The contacts are equilibrium reservoirs of free charge carriers maintained at the same temperature T . The junction is biased so that the electrochemical potential of contact L is lower than that of R, µ L < µ R . The donor and the acceptor are modeled as two-level (highest occupied molecular orbital-lowest unoccupied molecular orbital, HOMO-LUMO) systems. Following 44 , we assume that the HOMO of the acceptor is always populated (i.e., does not participate in transport), and thus can be disregarded. The bridge provides super-exchange coupling between the donor and the acceptor and is accounted for by effective electron hopping matrix element t. The donor is subjected to solar radiation which is modeled by coupling to a thermal bath S of high (solar) temperature T S (T S ≫ T ). Electron transfer is allowed between the LUMOs of the donor and the acceptor; thus, solar radiation drives the electronic flux against the applied bias (heat engine). Bridge-induced coupling between the LUMOs (levels 2 and 3 in Figure   1) is the cause of intra-molecular quantum coherence. For an isolated two-level system, such coherence leads to permanent Rabi oscillations in the electron population on both levels. In our model, these oscillations are damped by coupling to baths (solar radiation and contacts) with the damping rate depending on the strength of the couplings.

The Hamiltonian of the model iŝ
where theĤ M andĤ B Hamiltonians represent the molecule (M), contacts (L and R) and thermal bath (S), respectively.V B couples the molecule to the baths (contacts and Sun radiation). Explicit expressions arê Here,d † m (d m ) andĉ † k (ĉ k ) create (annihilate) the electron at the molecular level m or contact state k, respectively.D † =d † 2d 1 is the donor excitation operator.â † α (â α ) creates (annihilates) excitation quanta in the thermal bath. We previously discussed a similar model 58,59 in a study of the quantum coherence effects on electron transport in DBA junctions. Einax et al. 42,44 utilized the model in consideration of a molecular heat engine within a hopping transport regime. Here, we elucidate the effects of intra-molecular quantum coherence (Rabi oscillation between the LUMOs, levels 2 and 3; see Figure 1) on the thermodynamic performance of the engine.
The response properties of the junction can be conveniently expressed in terms of NEGFs.
In particular, we use single-and two-particle electron Green functions defined on the Keldysh contour as (here and below e = = k B = 1) Here, m 1 and m 2 indicate the molecular levels, T c is the contour ordering operator and τ 1 and τ 2 are the contour variables. The single-particle Green function is obtained by solving the Dyson equation where G (0) m 1 m 2 (τ 1 , τ 2 ) is the single-particle Green function in the absence of coupling to baths and is the total electron self-energy due to coupling to the baths. In our consideration, we use second-order (in system-baths couplings) diagrammatic expansion. The procedure is self-consistent, because single-electron self-energy due to coupling to the radiation field Σ S m 1 m 2 (τ 1 , τ 2 ) depends on the single-electron Green function G m 1 m 2 (τ 1 , τ 2 ), while the latter is defined by the self-energy. Below, we utilize level populations, at subsequent iteration steps to judge the convergence of the procedure. Here, G < is a lesser projection of the single-particle Green function. Explicit expressions of the self-energies are given in Appendix A.
To evaluate the two-particle Green function G, we employ an approximation which disregards multi-photon processes in the evaluation of the heat flux (see below). This approximation was employed in earlier studies 71 , and for the parameters of the simulations (strength of coupling to the radiation field), the approximation is reasonable.
Below, we calculate the particle flux at the interface with the right contact, I R , and the energy (heat) flux at the interface with the solar bath S, J S . In terms of these fluxes, the average thermodynamic efficiency of the molecular heat engine is defined as where P ≡ (µ R − µ L )I R is the power of the engine. We are interested in the efficiency at the maximum power η max . Within NEGF, fluxes I R and J S are defined as the rates of the change in the electronic population in R and the energy in S, respectively: They can be expressed exactly in terms of the Green functions and self-energies. At steady state within the NEGF, fluxes I R and J S are (see Appendix B for the derivation) Here, Tr[. . .] is the trace over the molecular levels, Σ R < (E) and Σ R > (E) are defined in Equations (A4) and (A5), Π < (ω) and Π > (ω) are given in Equations (A11) and (A12).

III. RESULTS
We present the results of the simulations for the DBA heat engine model (see Figure 1).
Unless stated otherwise, the parameters of the simulation are ambient temperature T =  detuning of the molecular system LUMOs (see Figure 2b). Note that this channel control is to some extent similar to consideration of 72 with the role of the resonant driving field there played by the gate potential in our consideration. Note also that channel control by level detuning does not affect the overall thermodynamic formulation, because (for static levels) no external thermodynamic forces are acting on the system.
While maximum power (see Figure 2c) is insensitive to the change in the dominant scattering channel, the thermodynamic efficiency at maximum power shows a non-monotonic behavior due to the sudden change in the bias at which the device performs at maximum power when the dominant channel is switched. We stress that this behavior is possible only due to the presence of quantum coherence in the molecular system and that when coherence is destroyed the efficiency at maximum power attains its monotonic dependence on level detuning in agreement with the classical study of the system 42 .
To destroy system coherence, we employ Büttiker probes coupled to LUMOs of the donor and the acceptor. Büttiker probes are widely used in quantum transport literature. They are modeled as an additional bath (represented by additional self-energy), whose role is to induce dephasing in the system. The probes should destroy the phase at the same time not allowing for either electron or energy exchange between the system and the probe.
Here, δ is the dephasing parameter. This self-energy can be derived considering the electronphonon interaction in the second order of the diagrammatic expansion 60 in the limit of low phonon frequency 59 . Such physical insight shows that neither electron nor energy flux can be induced between the system and the probe. Thus, the probe can destroy only quantum coherence in the molecular system. Figure 3 demonstrates the effect of coherence destruction with Büttiker probes. The dephasing parameter employed in the simulation is δ = 0.02 eV 2 . Introducing the probe has two effects on the device's performance. First, by destroying coherence the two transport channel situation in the purely coherent case changes to a single transport channel situation in the purely classical (hopping) case. The dephasing parameter employed in the simulation partially destroys system coherence; therefore, the second channel is less prominent (compare panels (a) and (b) in Figures 2 and 3), although the impact of coherence on efficiency is still pronounced (compare the dotted lines in Figure 3c). Note that destroying coherence increases the efficiency for V g < 0.6 V, while decreasing it for V g ≥ 0.6 V. This indicates the prevalence (in the purely coherent case) of destructive and  two effects on the device's performance. First, by destroying coherence the two transport channel situation in the purely coherent case changes to a single transport channel situation in the purely classical (hopping) case. The dephasing parameter employed in the simulation partially destroys system coherence; therefore, the second channel is less prominent (compare (a) and (b) panels in Figures 2 and 3), although the impact of coherence on efficiency is still pronounced (compare the dotted lines in the bottom panel in Figure 3). Note that destroying coherence increases the efficiency for V g < 0.6 V, while decreasing it for V g ≥ 0.6 V.
This indicates the prevalence (in the purely coherent case) of destructive and constructive interference in the two regions. Second, destroying coherence leads to a slight increase in the maximum power (compare the dashed lines in the bottom panel in Figure 3). The effect is due to the disruption by the Büttiker probe of the Rabi oscillations between the donor and acceptor LUMOs: The electron that previously spent time in the system now escapes faster in the right contact, which results in an increase in the current.
Another way to destroy system coherence is by increasing dissipation due to coupling to contacts (increase escape rate parameters Γ L and Γ R ). We use Γ L = Γ R = 0.05 eV in the simulations. Similar to the Bütiiker probe case, here the destruction of coherence results in a transition from two-channel to single-channel transport (and from quantum to classical behavior). The difference between the two cases is that while the former is pure decoherence, which results in a transition from coherent transport to hopping, the latter effectively eliminates the acceptor LUMO by incorporating it into the right contact. Thus, the extreme of the Bütiiker probe is the classical model of 42 while strong acceptor-contact coupling results in the classical consideration of 41 . Figure 4 shows that for the parameters of the simulation coherence is completely destroyed, which results in a monotonic behavior of efficiency and an essential increase in the maximum power.

IV. CONCLUSIONS
We study thermoelectric properties of a bi-chromophoric DBA junction model driven by solar radiation of the donor complex. In particular, utilizing NEGF formalism, we calculate the efficiency at the maximum power and elucidate the role of intra-molecular quantum coherence in the efficiency. Coherence in the junction is controlled by effective donor-acceptor coupling and by detuning of molecular levels at the two nodes of the molecular complex. We show that quantum coherence results in non-monotonic behavior of efficiency at maximum power vs. level detuning. The observed sudden jump indicates a switch between dominant scattering eigenchannel in our two-site model (see Figure 1). Although the non-monotonic behavior of the current-voltage characteristic in junctions is well-known, thus far nobody has discussed this effect with respect to efficiency. We stress that this behavior is possible only due to the presence of quantum coherence in the molecular system and that when coherence is destroyed, efficiency at maximum power attains its monotonic dependence on level detuning in agreement with previously performed classical studies of the system. To show the quantum-classical transition, we destroy quantum coherence in the system by either employing a Büttiker probe or increas-ing molecule-contacts coupling. In both cases, destruction of coherence implies a transition from two transport channels to one transport channel and from quantum to classical con- Expressions for self-energies Equation (5) can be derived following the standard diagrammatic perturbation theory formulated on the Keldysh contour 60,73,74 . The resulting expression for the electron self-energies due to coupling to contacts L and R is exact is the Green function of the free electron in state k. For Equation (2), V mk = V ℓ for m = 1 and V mk = V r for m = 3. At steady state and within the wide-band approximation, the projections of (A1) after performing the Fourier transform are where K = L, R, superscript − (+) indicates the forward (backward) branches of the Keldysh contour, f K (E) is the Fermi-Dirac distribution, and is the dissipation matrix due to coupling to contacts. In the wide-band approximation, the matrix is energy-independent. For Equation (2), Γ L m 1 m 2 = δ m 1 ,m 2 ,1 Γ L and Γ R m 1 m 2 = δ m 1 ,m 2 ,3 Γ R , where Γ L and Γ R are the electron escape rates into contacts L and R, respectively. Note Σ K <(>) ≡ Σ K −+(+−) .
The expression for the self-energy due to coupling to the thermal bath is derived within the second order of the diagrammatic perturbation theory where is the self-energy due to the coupling of the molecular excitations to the thermal bath. The N(E) is the Bose-Einstein distribution, and is the energy dissipation rate.
where G is the two-particle Green function defined in Equation (3) and is the Green function of the free photon of the radiation field. Substituting Equation (B4) into Equation (B2) and using (here, θ(x) is the Heaviside step function) leads to Finally, taking into account that for the free photon in mode α F <(>) α (t, t 1 ) ∼ e −iωα(t−t 1 ) , and thus, ω α F <(>) α (t, t 1 ) ≡ i∂ t F <(>) α (t, t 1 ), we get where Π(τ 1 , τ 2 ) is defined in Equation (A8). At steady-state, the correlation functions depend only on the time difference; therefore, performing the Fourier transform in Equation (B9) leads to Equation (11).