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arXiv:1210.6374 · Physical Review A

Incoherent excitation of thermally equilibrated open quantum systems

Leonardo A. Pachón, Paul W. Brumer

Published Feb 7, 2013• 53 citations
arXivHTMLPDF

Abstract

Under natural conditions, excitation of biological molecules, which display nonunitary open system dynamics, occurs via incoherent processes such as temperature changes or irradiation by sunlight or moonlight. The dynamics of such processes is explored analytically in a non-Markovian generic model. Specifically, a system S in equilibrium with a thermal bath TB is subjected to an external incoherent perturbation BB (such as sunlight) or another thermal bath TB${}^{\ensuremath{'}}$, which induces time evolution in ($\text{S}+\text{TB}$). Particular focus is on (i) the extent to which the resultant dynamics is coherent, and (ii) the role of ``stationary coherences'' established in the ($\text{S}+\text{TB}$) equilibration in response to the second incoherent perturbation. Results for systems with parameters analogous to those in light-harvesting molecules in photosynthesis show that the resultant dynamical behavior is incoherent beyond a very short response to the turn on of the perturbation.

DOI
10.1103/physreva.87.022106

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Under natural conditions, excitation of biological molecules, which display nonunitary open system dynamics, occurs via incoherent processes such as temperature changes or irradiation by sunlight or moonlight. The dynamics of such processes is explored analytically in a non-Markovian generic model. Specifically, a system S in equilibrium with a thermal bath TB is subjected to an external incoherent perturbation BB (such as sunlight) or another thermal bath TB\ensuremath′{}^{\ensuremath{'}}\ensuremath′, which induces time evolution in (S+TB\text{S}+\text{TB}S+TB). Particular focus is on (i) the extent to which the resultant dynamics is coherent, and (ii) the role of ``stationary coherences'' established in the (S+TB\text{S}+\text{TB}S+TB) equilibration in response to the second incoherent perturbation. Results for systems with parameters analogous to those in light-harvesting molecules in photosynthesis show that the resultant dynamical behavior is incoherent beyond a very short response to the turn on of the perturbation.
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 persistent, coherent dynamics in both the site and the exciton bases: we demonstrate that collective decay processes induced by the dipole–dipole interactions 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, 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.
Closed-form analytic solutions to non-secular Bloch-Redfield master equations for quantum dynamics of a V-type system driven by weak coupling to a thermal bath, relevant to light harvesting processes, are obtained and discussed. We focus on noise-induced Fano coherences among the excited states induced by incoherent driving of the V-system initially in the ground state. For suddenly turned-on incoherent driving, the time evolution of the coherences is determined by the damping parameter ζ=12(γ1+γ2)/Δp, where γi are the radiative decay rates of the excited levels i = 1, 2, and Δp=Δ(2)+(1-p(2))γ1γ2 depends on the excited-state level splitting Δ > 0 and the angle between the transition dipole moments in the energy basis. The coherences oscillate as a function of time in the underdamped limit (ζ ≫ 1), approach a long-lived quasi-steady state in the overdamped limit (ζ ≪ 1), and display an intermediate behavior at critical damping (ζ = 1). The sudden incoherent turn-on is shown to generate a mixture of excited eigenstates |e1〉 and |e2〉 and their in-phase coherent superposition |ϕ+〉=1r1+r2(r1|e1〉+r2|e2〉), which is remarkably long-lived in the overdamped limit (where r1 and r2 are the incoherent pumping rates). Formation of this coherent superposition enhances the decay rate from the excited states to the ground state. In the strongly asymmetric V-system where the coupling strengths between the ground state and the excited states differ significantly, additional asymptotic quasistationary coherences are identified, which arise due to slow equilibration of one of the excited states. Finally, we demonstrate that noise-induced Fano coherences are maximized with respect to populations when r1 = r2 and the transition dipole moments are fully aligned.
Dynamics and coherences in retinal isomerization are investigated in a standard two-mode two-state model irradiated by natural incoherent light using the Markovian partial-secular Bloch-Redfield formalism. The two-mode two-state model is a minimal model of retinal that considers vibronic states on a ground and excited electronic manifold coupled to two continuous Ohmic harmonic baths. All light-induced coherent oscillations are shown to disappear as the turn-on time becomes realistically slow. Rather, an interplay between incoherent-light induced coherences and environmentally induced coherences is exposed as the system approaches a nonequilibrium steady state. The dynamics of the system reveal stable steady state coherences under realistic conditions, producing a small but robust transient enhancement of quantum yield.
We develop a perturbative technique for solving Markovian quantum dissipative dynamics, with the perturbation parameter being a small gap in the eigenspectrum. As an example, we apply the technique and straightforwardly obtain analytically the dynamics of a three-level system with quasidegenerate excited states, where quantum coherences persist for very long times, proportional to the inverse of the energy splitting squared. We then show how to bypass this long-lived coherent dynamics and accelerate the relaxation to thermal equilibration in a hyper-exponential manner, a Markovian quantum-assisted Mpemba-like effect. This hyperacceleration of the equilibration process manifests if the initial state is carefully prepared, such that its coherences precisely store the amount of population relaxing from the initial condition to the equilibrium state. Our analytical method for solving quantum dissipative dynamics readily provides equilibration timescales, and as such it reveals how coherent and incoherent effects interlace in the dynamics. It further advises on how to accelerate relaxation processes, which is desirable when long-lived quantum coherences stagnate dynamics.
Molecular excitation with incoherent light is examined using realistic turn-on time scales, and results are compared to those obtained via commonly used sudden turn-on, or pulses. Two significant results are obtained. First, in contrast to prior studies involving sudden turn-on, realistic turn-on is shown to lead to stationary coherences for natural turn-on time scales. Second, the time to reach the final stationary mixed state, known to result from incoherent excitation, is shown to depend directly on the inverse of the molecular energy level spacings, in both sudden and realistic turn-on cases. The S0 → S2/S1 internal conversion process in pyrazine is used as an example throughout. Implications for studies of natural light harvesting systems are noted.
We present a detailed study of the non-Markovian two-state system dynamics for the regime of incoherent quantum tunneling. Using perturbation theory in the system tunneling amplitude $\ensuremath{\Delta}$, and in the limit of strong system-bath coupling, we determine the short-time evolution of the reduced density matrix and thereby find a general equation of motion for the non-Markovian evolution at longer times. We relate the nonlocality in time due to the non-Markovian effects with the environment characteristic response time. In addition, we study the incoherent evolution of a system with a double-well potential, where each well consists of several quantized energy levels. We determine the crossover temperature to a regime where many energy levels in the wells participate in the tunneling process, and observe that the required temperature can be much smaller than the one associated with the system plasma frequency. We also discuss experimental implications of our theoretical analysis.