Coherence turned on by incoherent light

One of the most pertinent problems in the debate on non-trivial quantum effects in biology concerns natural photosynthesis. Since sunlight is composed of thermal photons, it was argued to be unable to induce quantum coherence in matter, and that quantum mechanics is therefore irrelevant for the dynamical processes following photoabsorption. Our present analysis of a toy ``molecular aggregate"-- composed of two dipole-dipole interacting two-level atoms treated as an open quantum system -- however shows that incoherent excitations indeed can trigger coherent dynamics that persist: We demonstrate that collective decay processes induced by the dipole-dipole interaction create coherent intermolecular transport -- regardless of the coherence properties of the incoming radiation. Our analysis shows that the steady state coherence is mediated by the population imbalance between the molecules and, therefore, {\it increases} with the energy difference between the two-level atoms. Our results establish the importance of collective decay processes in the study of ultrafast photophysics, and especially their potential role to generate stationary coherence in incoherently driven quantum transport.


INTRODUCTION
A detailed understanding of the microscopic processes which underlie natural photosynthesis represents an important and intriguing source of inspiration for technologies which seek to efficiently capture, transform, and store solar energy [1,2]. One of the most important open questions in this research area is whether quantum interference effects could play a role in solar light harvesting, and possibly be the source for the high efficiency of the energy conversion. That transient quantum coherence can prevail in such complex structures, at ambient temperatures, can by now be regarded a solidly established experimental fact [3,4], and has also been reported in the charge separation process in organic solar cells [5,6].
It however has been argued [7] that the evidence provided by the above experiments is inconclusive, because the conditions under which quantum effects were experimentally observed in certain light harvesting complexes (LHC) differ from conditions in vivo. Indeed, laboratory experiments rely on photon echo spectroscopy [8], where the energy transfer is induced by a series of ultrashort coherent laser pulses. In contrast, sunlight (driving the natural process) can be described as continuous wave (or stationary) thermal (incoherent) radiation [9]. Thus, it is a priori crucial to distinguish the coherence observed in photon echoes [10] from coherence which may arise in non-equilibrium open system quantum dynamics -as we will outline below.
Moreover, it is argued in [7,11,12] that the coupling of a quantum system to a thermal radiation bath rapidly leads to a formation of a stationary state in the system that does not exhibit any dynamical coherences. This apparently contradicts the point of view that it is some form of intrinsic quantum coherence, which can be triggered by any photoabsorption event, regardless of the source of photons, that underlies the efficient energy transport [13][14][15]. Such intrinsic coherence could stem from different scenarios: system-bath correlation [16], ratchetlike effects [17], resonant coupling to vibrational modes [10,[18][19][20][21], or to a single light mode [22], as well as Fanotype resonances with the electromagnetic continuum [23].
Here we offer a microscopic quantum optical theory to resolve this longstanding controversy. Specifically, we establish that nonstationary steady state coherence can indeed emerge in an incoherently driven molecular complex, under realistic assumptions on the incident wave lengths and molecular separations. To begin with, we recall that the primary process of photosynthesis is the absorption of a single photon by a chlorophyll molecule, whereby the molecule undergoes a transition from the ground to the excited electronic state [2]. The photoabsorption event can then launch energy transfer towards the reaction center, where it triggers a charge separation cascade with almost unit efficiency [24]. This transfer process from the initial absorption event to the charge separation has has a finite duration, of the order of a picosecond [25], and it is during this process that transient electronic coherences have been observed [3,4,26]. After the excited state energy is transformed, the molecule resets in its ground electronic state and is able to absorb the next photon. We will show that, when averaging over many such single photon absorption and transfer cycles, one ends up with a master equation type ensemble description which exhibits non-vanishing coherence in the non-equilibrium steady state.
Since any LHC can be thought of as a photodetector designed by nature let us start by drawing an analogy between the operation of an LHC and that of a man-made broadband single-photon detector used in quantum optics [27,28]. Such a detector's sensitivity is non-vanishing over a frequency window of finite width ∆ω. When a photon with frequency ω within the sensitivity range hits the detector, it excites a wavepacket which depends on the photon's carrier frequency and bandwidth, and may in turn produce a click -a transient electric responsewhose minimal duration is bounded by the inverse of ∆ω. Should quantum coherent processes accompany the detector response, they must be completely independent of the light source. Consequently, an absorption of a photon from not only a coherent laser source but also from an incoherent thermal source may be accompanied by transient coherent evolution -for natural and artificial photon detectors alike -provided that it is intrinsic to the detection process.
We model an LHC as a "molecular aggregate" which consists of two effective two-level atoms [29] -which we shall refer to as molecules in the following -that are located in each other's near field and embedded into a common electromagnetic bath. Thereby, we abstract ourselves from the details of the structure and energy spectra of a real photosynthetic complex [2]. Nonetheless, our dimer model is able to describe two absorption bands associated with the widths of the electronic excited state, as well as the dipole-dipole interaction between the molecules. We study the interaction of this system with an external incoherent field which emulates the sunlight, and show that coherent evolution is induced by photo-absorption, and survives even in the nonequilibrium steady state of the incoherently driven system, as a reflection of the transient coherences induced on the level of single photon absorption and transport processes.

MODEL
Our model is presented in Fig. 1. It consists of two molecules embedded, at a distance r 12 , into a common radiation bath and interacting with an external incoherent radiation field in the optical frequency range. We assume that the molecules have distinct dipole transition matrix elements d 1 = d 2 between their electronic ground and excited states |g k and |e k , k = 1, 2, respectively, and that their optical transition frequencies ω 1 and ω 2 are detuned with ∆ = ω 1 − ω 2 ω 1 , ω 2 . Furthermore, we can ignore the ambient thermal photons (since ω/k B ∼ 6000 K at optical frequencies) and therefore 1 2 Two molecules at a distance r12 and with transition frequencies ω1 and ω2, detuned by ∆ = ω1 −ω2, are embedded into a common electromagnetic bath. The bath induces radiative decay of the molecules (γ k is the decay rate of atom k), as well as a dipole-dipole interaction with complex coupling constant T12 [see Eq. (2)]. An external, incoherent thermal source stimulates absorption and emission processes. The molecular line widths γ1, γ2 determine the dimer's sensitivity range ∆ω and its response time to the incident incoherent radiation (see Fig. 2).
assume the relevant modes of the radiation reservoir in the vacuum state. This bath induces spontaneous decay of the individual molecules with rates γ k , as well as their dipole-dipole interaction with complex coupling strength T 12 . Additionally, the coupling to other, e.g. vibrational, degrees of freedom may cause further dissipation [13,30], which is not considered in this work. As for the external incoherent field, we assume that its energy density is a slowly varying function around the transition frequency. The external field generates absorption and stimulated emission processes at the rate γ k N (ω k ) [31], where N (ω k ) is the average number of the (incoherent) source photons at the transition frequency, which is defined by the source temperature.
Using standard quantum optical methods [32,33], one can trace out the bath degrees of freedom, to arrive at the master equation governing the evolution of the "aggregate" density matrix ρ, in the basis of the uncoupled individual molecules' energy eigenstates {|g 1 , g 2 , |e 1 , g 2 , |g 1 , e 2 , |e 1 , e 2 }, and in the frame rotating at the frequency ω 0 = (ω 1 + ω 2 )/2, and for the molecules initially in their ground states, the master equation readṡ In this equation, the atomic (de-)excitation operators are given as σ k − = |g k e k |, σ k + = |e k g k |, the atomic decay rates explicitly read γ k = d 2 k ω 3 k /6π 0 c 3 , and Γ ≡ Γ(ω 0 r 12 /c), Ω ≡ Ω(ω 0 r 12 /c) are the real and imaginary parts of the retarded dipole-dipole interaction strength T 12 = Γ + iΩ, which in particular describes collective effects such as super-radiance [33][34][35] [36]. The physical meaning of the real and imaginary parts of T 12 can be unambiguously identified from the structure of the master equation (1): Terms proportional to iΩ describe oscillatory, reversible, non-radiative excitation exchange between both molecules. Terms proportional to Γ represent (collective) radiative decay processes, following a non-radiative excitation exchange between the molecules. Accordingly, Γ and Ω are associated with life time and energy shift of the (entangled, Dicke) eigenstates [37] (|e 1 , g 2 ± |e 2 , g 1 / √ 2), |e 1 , e 2 of the dipole-coupled molecular dimer, respectively [33,38,39]. Thus, the retarded dipole-dipole interaction coupling describes collective effects, and is in particular important for understanding super-radiance and entanglement in the Dicke model [38,40]. Explicitly, they are given as [33,34,41], where ξ ≡ ω 0 r 12 /c is the effective intermolecular distance [42], andd k andr 12 are unit vectors directed along the kth molecular dipole and along the vector connecting the molecules, respectively. Note that the far-field terms in Eqs. (2a,2b) (i.e., the terms decreasing as ξ −1 for ξ 1) describe retardation effects proper that are associated with the exchange of real photons [34]. These effects start playing a role at r 12 ≥ 10 nm [43], though they are deemed unimportant at inter-molecular distances of less than 10 nm (i.e. ξ < 0.1).
One of the key processes in the theory of photosynthesis is resonance energy transfer [29]. This transfer is effective between molecules whose transition frequencies are close to each other (hence, the name of the process) and is characterized by the rate proportional to |Γ + iΩ| 2 [34]. In the non-retarded limit ξ 1, Γ(ξ) is dominated by Ω(ξ). It is therefore standard to neglect Γ(ξ), and to retain only the non-retarded contributions of Ω(ξ) [17,29,44,45]. In this limit, Ω(ξ) → V dd / , with V dd the static dipole-dipole interaction energy [33,35]: This wide-spread approximation however neglects that also Γ(ξ) does remain finite as r 12 → 0, with Γ(ξ) → √ γ 1 γ 2d1 ·d 2 , and Eq. (3) is thus imprecise at small distances. As we show below, a consequence of using the approximate expression (3) is that a collective coherent effect -the stationary excitation current in the dipoleinteracting system -is erroneously predicted to vanish. Furthermore, when Γ is kept while γ 1 , γ 2 are neglected then Eq. (1) is not more diagonal in the exciton basis (i.e., the eigenbasis of the Hamiltonian part of Eq. (1) [29]). Therefore, the presence of Γ shall create coherence both in the site and in the exciton basis. Let us therefore inspect the time-dependent expecta- tion value Im { σ 2 + (t)σ 1 − (t) } ≡ Im { e 1 , g 2 | (t)|g 1 , e 2 } as the quantifier of the electronic inter-site coherence of our "molecular aggregate" under incoherent driving. Upon multiplication by 2Ω this yields the excitation current (in short, current), which is proportional to the probability per unit time for an excitation transfer from molecule 1 to molecule 2 (see SI). For simplicity and without loss of essential physics, we study the temporal behavior of this coherence-induced current under the assumptions that both dipoles point in the same direction and that the excited state of molecule 1 has slightly larger natural linewidth than that of molecule 2: γ 2 = 0.9999γ 1 . In a fully symmetric system, where γ 1 = γ 2 and ∆ = 0, the expectation value of the excitation current vanishes for all times, Im { σ 2 Fig. 2(a)-(d), where we vary either the mean photon number N (ω 1 ) ≈ N (ω 2 ) = N (ω 0 ) (since ∆ ω 0 ), or the detuning ∆, or the orientation f = (d 1 ·r 12 ), or the effective distance ξ, while keeping the three remaining parameters fixed. It is evident that a non-vanishing current is a generic feature of the intramolecular excitation transfer that follows the photo-absorption process by the molecular aggregate prepared in its ground (reset) state at t = 0. On transient time scales, 2Ω Im { σ 2 + (t)σ 1 − (t) } exhibits decaying oscillations which correspond to the excitation bouncing back and forth between the molecules. The frequency and the decay rate of these oscillations are given by [∆ 2 + 4Ω 2 ] 1/2 and a constant ∼ γ 1 (which can be associated with the dimer's sensitivity range, γ 1 ∼ ∆ω), respectively. On top of the decaying oscillations, there is a slow relaxation, at a rate ∼ N (ω 0 )γ 1 , to the non-equilibrium steady state coherence, analytically given by where N ≡ N (ω 0 ) and R is given in (19). Equation (4) shows that the stationary excitation current only emerges for a non-vanishing collective decay rate Γ, giving rise to to the irreversibility of the excitation exchange process. [46] As a result, the stationary populations of the excited levels of the two molecules become unequal, which is crucial for the emergence of the stationary current. It is also evident from Eq. (4) that, if γ = γ 1 = γ 2 , then the population imbalance and, hence, the current in the steady state increase with ∆. Furthermore, using Eqs. (4) and (18), we can establish an identity for the special case γ = γ 1 = γ 2 (while ∆ = 0): Here, κ = 2γ(1 + 2N ) and σ k + σ k − = e k | k |e k , with k = Tr l ( ) (k, l = 1, 2, k = l) the reduced density matrix of molecule k. This result can be interpreted as an energy balance relation for our dipole-dipole coupled system: The left hand side of (5) yields the number of photons that is transferred per unit time from molecule 1 to molecule 2; the right hand side yields the difference between the total number of photons that are emitted, or absorbed, per unit time, by molecule 2 and 1, due to spontaneous and stimulated emission.
Equation (5) is reminiscent of a relation well known in single atom resonance fluorescence [27]: There, quantum coherence between the atomic ground and excited states arises due to the presence of the laser field, characterized by the Rabi frequency. The quantity describing the atomic coherence, Im { σ + }, is coupled to the atomic excited state population, such that a single atom energy balance relation similar to (5) holds. In our present case of two molecules, quantum coherence between the molecular dipoles arises due to the dipole-dipole interaction, playing the role of the Rabi frequency. The directed excitation current leads to an imbalance of the molecular excited state populations: the molecule with larger transition frequency ω 1 > ω 2 has smaller excited state population because part of it is coherently transferred to the molecule with smaller transition frequency ω 2 .
The directionality of the average steady-state current (of the order of 10 −5 excitation transfer events per second, for intermolecular distances in the range 1 to 5 nm and at the resonant wavelength of 460 nm), which does not exhibit any oscillations, is a signature of the following cycle: For ∆ > 0 (see Fig. 1) thermal photons are absorbed mainly by molecule 1 and, after the excitation transfer to molecule 2, they are emitted therefrom. This behavior is consistent with our intuitive picture of an individual incident of an excitation transfer process in our "molecular aggregate": Initially (re-)set in its ground state, the molecular aggregate absorbs a photon at t = 0, followed by an oscillation of the excitation current. The latter fades away on a time scale γ −1 , until the photon is emitted by molecule 2 and the dimer is reset to its ground state ... until the next photon absorption occurs. The smaller the mean number N of incident thermal photons, the less frequent are the random photo-absorption events that trigger such excitation transfer cycles, and the larger are the fluctuations of the excitation current. Averaging the evolution over many such elementary cycles leads to the time evolution depicted in Fig. 2, finally settling in the non-equilibrium steady state. This scenario of directional cycling may be violated at very short inter-molecular separations, when the difference between the natural line widths of the molecules multiplied by the dipole-dipole interaction strength exceeds the term proportional to the detuning in the numerator in Eq. (4). In that case the directions of the transient and stationary currents are opposite (see SI), which means that in the long time limit, a photon is emitted by the same molecule that absorbed it. CONCLUSION We have studied the dynamics of the electronic coherence of a toy "molecular aggregate" composed of two closely located two-level atoms coupled to a vacuum reservoir and excited by an incoherent field. We have shown that the transient behavior of this quantity exhibits coherent oscillations which are indicative of the excitation exchange between the dimer's constituents. The frequency and decay rate of these oscillations are defined by the inter-molecular dipole-dipole interaction strength and by the local relaxation rates of the individual molecular sites' excitations. Furthermore, we have established the emergence of stationary coherence in the non-equilibrium steady state of the aggregate, giving rise to a stationary current, as a consequence of retarded dipole interaction-induced collective decay processes which prevail at small inter-molecular distances. Although retardation does not play a functional role at such distances, in the non-retarded limit (3) typically employed in the literature [29], these collective decay processes are neglected. As a result, the incoherent excitation instantaneously creates an incoherent mixture of eigenstates, and steady state coherence is absent. Thus, in contrast to the results of [7], our results establish a realistic scenario where intermolecular electronic steady state coherence can be triggered by the absorption of photons coming from an incoherent source, mediating transient population oscillations which relax into a coherent and directed flux of excitations in the steady state.
The latter quantity can be inferred from solutions of the following equation of motion: where κ i = 2γ i {1 + 2N (ω 0 )}, and T = Γ + iΩ, with Γ ≡ Γ(ξ), Ω ≡ Ω(ξ) given by (2a) and (2b), respectively [47]. We assume that at time t = 0 both molecules are in their ground states, hence the vector of initial conditions is The formal time dependent solution of Eq. (9) reads For arbitrary times, the temporal behavior of x(t) can be studied numerically. Several examples of the transient and asymptotic behavior are given in Fig. 2. Here we present another example (see Fig. 3), which complements Fig. 2(d), where we illustrate how the transient oscillations develop into steady values for different effective distances ξ. In particular, Fig. 3 shows that, for ξ < 0.05, the directions of the excitation current in the transient and steady regimes are opposite.
Let us now address the steady state limit where analytical solutions x(∞) = −A −1 L are readily available.
First, let us consider the steady state solutions for Γ = 0. In this case, the entries of the vector x(∞) read: where N ≡ N (ω 0 ). The above solutions indicate equal population distributions of both molecules and the absence of intermolecular electronic coherence. In contrast, for Γ = 0, we obtain σ 1 z = σ 2 z , and a non-trivial two-molecule coherence in the two-level system. Below we present the explicit expressions for two quantities: the excitation current, Im { σ 2 + σ 1 − }, and the difference between the excited state populations of the two molecules, σ 2 + σ 2 − − σ 1 + σ 1 − = ( σ 2 z − σ 1 z )/2: with Direct inspection of Eqs. (17) and (18), for γ = γ 1 = γ 2 , yields the energy balance relation (5).