Non-Markovian theories based on a decomposition of the spectral density

Emergence of Boltzmann Subspaces in Open Quantum Systems Far from Equilibrium

Single molecule junctions are important examples of complex out-of-equilibrium many-body quantum systems. We identify a nontrivial clustering of steady state populations into distinctive subspaces with Boltzmann-like statistics, which persist far from equilibrium. Such Boltzmann subspaces significantly reduce the information needed to describe the steady state, enabling modeling of high-dimensional systems that are otherwise beyond the reach of current computations. The emergence of Boltzmann subspaces is demonstrated analytically and numerically for fermionic transport systems of increasing complexity.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Bath-Induced Correlations Enhance Thermometry Precision at Low Temperatures

We study the role of bath-induced correlations in temperature estimation of cold bosonic baths. Our protocol includes multiple probes, that are not interacting, nor are they initially correlated to each other. They interact with a bosonic sample and reach a nonthermal steady state, which is measured to estimate the temperature of the sample. It is well known that in the steady state such noninteracting probes may get correlated to each other and even entangled. Nonetheless, the impact of these correlations in metrology has not been deeply investigated yet. Here, we examine their role for thermometry of cold bosonic gases and show that, although being classical, bath-induced correlations can lead to significant enhancement of precision for thermometry. The improvement is especially important at low temperatures, where attaining high precision thermometry is particularly demanding. The proposed thermometry scheme does not require any precise dynamical control of the probes and tuning the parameters and is robust to noise in initial preparation, as it is built upon the steady state generated by the natural dissipative dynamics of the system. Therefore, our results put forward new possibilities in thermometry at low temperatures, of relevance, for instance, in cold gases and Bose-Einstein condensates.

Ultrafast Charge Transfer and Relaxation at a Donor-Acceptor Interface

The efficiency of charge separation in organic photovoltaic materials is crucially determined by the underlying dynamics of the charge transfer (CT) excitons and their dissociation into free electrons and holes. To unravel the main principles of the underlying mechanism on a molecular level, we construct a toy model of electronically coupled donors interacting with a manifold of CT exciton states. In particular, we set up a ladder of CT site energies to model the exciton dissociation. To mimic the complexity of the exciton dynamics at the donor-acceptor interface, the electronic CT manifold is designed to include two vibrational modes that are vibronically coupled to the excitons. We examine the impact of the electronic and vibrational coherences and the structure of the vibronic manifold on the transfer efficiency and charge recombination. Optimal configurations of the vibronic CT manifold are revealed. In particular, the rate of charge recombination can be minimized when the transient dynamics are carefully explored. Such a toy model can be used as a guide for the design of organic materials for efficient photovoltaic devices.

Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors

We develop new methods to efficiently propagate the hierarchical equations of motion (HEOM) by using the Tucker and hierarchical Tucker (HT) tensors to represent the reduced density operator and auxiliary density operators. We first show that by employing the split operator method, the specific structure of the HEOM allows a simple propagation scheme using the Tucker tensor. When the number of effective modes in the HEOM increases and the Tucker representation becomes intractable, the split operator method is extended to the binary tree structure of the HT representation. It is found that to update the binary tree nodes related to a specific effective mode, we only need to propagate a short matrix product state constructed from these nodes. Numerical results show that by further employing the mode combination technique commonly used in the multi-configuration time-dependent Hartree approaches, the binary tree representation can be applied to study excitation energy transfer dynamics in a fairly large system including over 10⁴ effective modes. The new methods may thus provide a promising tool in simulating quantum dynamics in condensed phases.

Controllable dynamics of a dissipative two-level system

We propose a strategy to modulate the decoherence dynamics of a two-level system, which interacts with a dissipative bosonic environment, by introducing an ancillary degree of freedom. It is revealed that the decay rate of the two-level system can be significantly suppressed under suitable steers of the assisted degree of freedom. Our result provides an alternative way to fight against decoherence and realize a controllable quantum dissipative dynamics.

Does electronic coherence enhance anticorrelated pigment vibrations under realistic conditions?

The light-harvesting efficiency of a photoactive molecular complex is largely determined by the properties of its electronic quantum states. Those, in turn, are influenced by molecular vibrational states of the nuclear degrees of freedom. Here, we reexamine two recently formulated concepts that a coherent vibronic coupling between molecular states would either extend the electronic coherence lifetime or enhance the amplitude of the anticorrelated vibrational mode at longer times. For this, we study a vibronically coupled dimer and calculate the nonlinear two-dimensional (2D) electronic spectra that directly reveal electronic coherence. The time scale of electronic coherence is initially extracted by measuring the antidiagonal bandwidth of the central peak in the 2D spectrum at zero waiting time. Based on the residual analysis, we identify small-amplitude long-lived oscillations in the cross-peaks, which, however, are solely due to groundstate vibrational coherence, regardless of having resonant or off-resonant conditions. Our studies neither show an enhancement of the electronic quantum coherence nor an enhancement of the anticorrelated vibrational mode by the vibronic coupling under ambient conditions.

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures

The time evolution in open quantum systems, such as a molecular aggregate in contact with a thermal bath, still poses a complex and challenging problem. The influence of the thermal noise can be treated using a plethora of schemes, several of which decompose the corresponding correlation functions in terms of weighted sums of exponential functions. One such scheme is based on the hierarchical equations of motion (HEOM), which is built using only certain forms of bath correlation functions. In the case where the environment is described by a complex spectral density or is at a very low temperature, approaches utilizing the exponential decomposition become very inefficient. Here, we utilize an alternative decomposition scheme for the bath correlation function based on Chebyshev polynomials and Bessel functions to derive a HEOM approach up to an arbitrary order in the environmental coupling. These hierarchical equations are similar in structure to the popular exponential HEOM scheme, but are formulated using the derivatives of the Bessel functions. The proposed scheme is tested up to the fourth order in perturbation theory for a two-level system and compared to benchmark calculations for the case of zero-temperature quantum Ohmic and super-Ohmic noise. Furthermore, the benefits and shortcomings of the present Chebyshev-based hierarchical equations are discussed.

Ultrafast Energy Transfer in Excitonically Coupled Molecules Induced by a Nonlocal Peierls Phonon

Molecular vibration can influence exciton transfer via either a local (intramolecular) Holstein or a nonlocal (intermolecular) Peierls mode. We show that a strong vibronic coupling to a nonlocal mode dramatically speeds up the transfer by opening an additional transfer channel. This Peierls channel is rooted in the formation of a conical intersection of the excitonic potential energy surfaces. For increasing Peierls coupling, the electronically coherent transfer for weak coupling turns into an incoherent transfer of a localized exciton through the intersection for strong coupling. The interpretation in terms of a conical intersection intuitively explains recent experiments of ultrafast energy transfer in photosynthetic and photovoltaic molecular systems.

Origin of poor doping efficiency in solution processed organic semiconductors

Doping is an extremely important process where intentional insertion of impurities in semiconductors controls their electronic properties. In organic semiconductors, one of the convenient, but inefficient, ways of doping is the spin casting of a precursor mixture of components in solution, followed by solvent evaporation. Active control over this process holds the key to significant improvements over current poor doping efficiencies. Yet, an optimized control can only come from a detailed understanding of electronic interactions responsible for the low doping efficiencies. Here, we use two-dimensional nonlinear optical spectroscopy to examine these interactions in the course of the doping process by probing the solution mixture of doped organic semiconductors. A dopant accepts an electron from the semiconductor and the two ions form a duplex of interacting charges known as ion-pair complexes. Well-resolved off-diagonal peaks in the two-dimensional spectra clearly demonstrate the electronic connectivity among the ions in solution. This electronic interaction represents a well resolved electrostatically bound state, as opposed to a random distribution of ions. We developed a theoretical model to recover the experimental data, which reveals an unexpectedly strong electronic coupling of ∼250 cm^-1 with an intermolecular distance of ∼4.5 Å between ions in solution, which is approximately the expected distance in processed films. The fact that this relationship persists from solution to the processed film gives direct evidence that Coulomb interactions are retained from the precursor solution to the processed films. This memory effect renders the charge carriers equally bound also in the film and, hence, results in poor doping efficiencies. This new insight will help pave the way towards rational tailoring of the electronic interactions to improve doping efficiencies in processed organic semiconductor thin films.

Primary Charge Separation in the Photosystem II Reaction Center Revealed by a Global Analysis of the Two-dimensional Electronic Spectra

The transfer of electronic charge in the reaction center of Photosystem II is one of the key building blocks of the conversion of sunlight energy into chemical energy within the cascade of the photosynthetic reactions. Since the charge transfer dynamics is mixed with the energy transfer dynamics, an effective tool for the direct resolution of charge separation in the reaction center is still missing. Here, we use experimental two-dimensional optical photon echo spectroscopy in combination with the theoretical calculation to resolve its signature. A global fitting analysis allows us to clearly and directly identify a decay pathway associated to the primary charge separation. In particular, it can be distinguished from regular energy transfer and occurs on a time scale of 1.5 ps under ambient conditions. This technique provides a general tool to identify charge separation signatures from the energy transport in two-dimensional optical spectroscopy.

Chebyshev Expansion Applied to Dissipative Quantum Systems

To determine the dynamics of a molecular aggregate under the influence of a strongly time-dependent perturbation within a dissipative environment is still, in general, a challenge. The time-dependent perturbation might be, for example, due to external fields or explicitly treated fluctuations within the environment. Methods to calculate the dynamics in these cases do exist though some of these approaches assume that the corresponding correlation functions can be written as a weighted sum of exponentials. One such theory is the hierarchical equations of motion approach. If the environment, however, is described by a complex spectral density or if its temperature is low, these approaches become very inefficient. Therefore, we propose a scheme based on a Chebyshev decomposition of the bath correlation functions and detail the respective quantum master equations within second-order perturbation theory in the environmental coupling. Similar approaches have recently been proposed for systems coupled to Fermionic reservoirs. The proposed scheme is tested for a simple two-level system and compared to existing results. Furthermore, the advantages and disadvantages of the present Chebyshev approach are discussed.

Quantum effects in ultrafast electron transfers within cryptochromes

Cryptochromes and photolyases are flavoproteins that may undergo ultrafast charge separation upon electronic excitation of their flavin cofactors.

Influence of Coherent Tunneling and Incoherent Hopping on the Charge Transfer Mechanism in Linear Donor-Bridge-Acceptor Systems

The mechanism of charge transfer has been observed to change from tunneling to hopping with increasing numbers of DNA base pairs in polynucleotides and with the length of molecular wires. The aim of this paper is to investigate this transition by examining the population dynamics using a tight-binding Hamiltonian with model parameters to describe a linear donor-bridge-acceptor (D-B-A) system. The model includes a primary vibration and an electron-vibration coupling at each site. A further coupling of the primary vibration with a secondary phonon bath allows the system to dissipate energy to the environment and reach a steady state. We apply the quantum master equation (QME) approach, based on second-order perturbation theory in a quantum dissipative system, to examine the dynamical processes involved in charge-transfer and follow the population transfer rate at the acceptor, ka, to shed light on the transition from tunneling to hopping. With a small tunneling parameter, V, the on-site population tends to localize and form polarons, and the hopping mechanism dominates the transfer process. With increasing V, the population tends to be delocalized and the tunneling mechanism dominates. The competition between incoherent hopping and coherent tunneling governs the mechanism of charge transfer. By varying V and the total number of sites, we also examine the onset of the transition from tunneling to hopping with increasing length.

Electron transfer within a reaction path model calibrated by constrained DFT calculations: application to mixed-valence organic compounds

The quantum dynamics of electron transfer in mixed-valence organic compounds is investigated using a reaction path model calibrated by constrained density functional theory (cDFT). Constrained DFT is used to define diabatic states relevant for describing the electron transfer, to obtain equilibrium structures for each of these states and to estimate the electronic coupling between them. The harmonic analysis at the diabatic minima yields normal modes forming the dissipative bath coupled to the electronic states. In order to decrease the system-bath coupling, an effective one dimensional vibronic Hamiltonian is constructed by partitioning the modes into a linear reaction path which connects both equilibrium positions and a set of secondary vibrational modes, coupled to this reaction coordinate. Using this vibronic model Hamiltonian, dissipative quantum dynamics is carried out using Redfield theory, based on a spectral density which is determined from the cDFT results. In a first benchmark case, the model is applied to a series of mixed-valence organic compounds formed by two 1,4-dimethoxy-3-methylphenylene fragments linked by an increasing number of phenylene bridges. This allows us to examine the coherent electron transfer in extreme situations leading to a ground adiabatic state with or without a barrier and therefore to the trapping of the charge or to an easy delocalization.

Two-Dimensional Electronic Spectroscopy of Light-Harvesting Complex II at Ambient Temperature: A Joint Experimental and Theoretical Study

We have performed broad-band two-dimensional (2D) electronic spectroscopy of light-harvesting complex II (LHCII) at ambient temperature. We found that electronic dephasing occurs within ∼60 fs and inhomogeneous broadening is approximately 120 cm(-1). A three-dimensional global fit analysis allows us to identify several time scales in the dynamics of the 2D spectra ranging from 100 fs to ∼10 ps and to uncover the energy-transfer pathways in LHCII. In particular, the energy transfer between the chlorophyll b and chlorophyll a pools occurs within ∼1.1 ps. Retrieved 2D decay-associated spectra also uncover the spectral positions of corresponding diagonal peaks in the 2D spectra. Residuals in the decay traces exhibit periodic modulations with different oscillation periods. However, only one of them can be associated with the excitonic cross-peaks in the 2D spectrum, while the remaining ones are presumably of vibrational origin. For the interpretation of the spectroscopic data, we have applied a refined exciton model for LHCII. It reproduces the linear absorption, circular dichroism, and 2D spectra at different waiting times. Several components of the energy transport are revealed from theoretical simulations that agree well with the experimental observations.

Using non-Markovian measures to evaluate quantum master equations for photosynthesis

When dealing with system-reservoir interactions in an open quantum system, such as a photosynthetic light-harvesting complex, approximations are usually made to obtain the dynamics of the system. One question immediately arises: how good are these approximations, and in what ways can we evaluate them? Here, we propose to use entanglement and a measure of non-Markovianity as benchmarks for the deviation of approximate methods from exact results. We apply two frequently-used perturbative but non-Markovian approximations to a photosynthetic dimer model and compare their results with that of the numerically-exact hierarchy equation of motion (HEOM). This enables us to explore both entanglement and non-Markovianity measures as means to reveal how the approximations either overestimate or underestimate memory effects and quantum coherence. In addition, we show that both the approximate and exact results suggest that non-Markonivity can, counter-intuitively, increase with temperature, and with the coupling to the environment.

Optically induced transport through semiconductor-based molecular electronics

A tight binding model is used to investigate photoinduced tunneling current through a molecular bridge coupled to two semiconductor electrodes. A quantum master equation is developed within a non-Markovian theory based on second-order perturbation theory with respect to the molecule-semiconductor electrode coupling. The spectral functions are generated using a one dimensional alternating bond model, and the coupling between the molecule and the electrodes is expressed through a corresponding correlation function. Since the molecular bridge orbitals are inside the bandgap between the conduction and valence bands, charge carrier tunneling is inhibited in the dark. Subject to the dipole interaction with the laser field, virtual molecular states are generated via the absorption and emission of photons, and new tunneling channels open. Interesting phenomena arising from memory are noted. Such a phenomenon could serve as a switch.

Using the Chebychev expansion in quantum transport calculations

Irradiation by laser pulses and a fluctuating surrounding liquid environment can, for example, lead to time-dependent effects in the transport through molecular junctions. From the theoretical point of view, time-dependent theories of quantum transport are still challenging. In one of these existing transport theories, the energy-dependent coupling between molecule and leads is decomposed into Lorentzian functions. This trick has successfully been combined with quantum master approaches, hierarchical formalisms, and non-equilibrium Green's functions. The drawback of this approach is, however, its serious limitation to certain forms of the molecule-lead coupling and to higher temperatures. Tian and Chen [J. Chem. Phys. 137, 204114 (2012)] recently employed a Chebychev expansion to circumvent some of these latter problems. Here, we report on a similar approach also based on the Chebychev expansion but leading to a different set of coupled differential equations using the fact that a derivative of a zeroth-order Bessel function can again be given in terms of Bessel functions. Test calculations show the excellent numerical accuracy and stability of the presented formalism. The time span for which this Chebychev expansion scheme is valid without any restrictions on the form of the spectral density or temperature can be determined a priori.

Reduced quantum dynamics with arbitrary bath spectral densities: hierarchical equations of motion based on several different bath decomposition schemes

We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.

Relaxation dynamics in quantum dissipative systems: the microscopic effect of intramolecular vibrational energy redistribution

We investigate the effect of inter-mode coupling on the vibrational relaxation dynamics of molecules in weak dissipative environments. The simulations are performed within the reduced density matrix formalism in the Markovian regime, assuming a Lindblad form for the system-bath interaction. The prototypical two-dimensional model system representing two CO molecules approaching a Cu(100) surface is adapted from an ab initio potential, while the diatom-diatom vibrational coupling strength is systematically varied. In the weak system-bath coupling limit and at low temperatures, only first order non-adiabatic uni-modal coupling terms contribute to surface-mediated vibrational relaxation. Since dissipative dynamics is non-unitary, the choice of representation will affect the evolution of the reduced density matrix. Two alternative representations for computing the relaxation rates and the associated operators are thus compared: the fully coupled spectral basis, and a factorizable ansatz. The former is well-established and serves as a benchmark for the solution of Liouville-von Neumann equation. In the latter, a contracted grid basis of potential-optimized discrete variable representation is tailored to incorporate most of the inter-mode coupling, while the Lindblad operators are represented as tensor products of one-dimensional operators, for consistency. This procedure results in a marked reduction of the grid size and in a much more advantageous scaling of the computational cost with respect to the increase of the dimensionality of the system. The factorizable method is found to provide an accurate description of the dissipative quantum dynamics of the model system, specifically of the time evolution of the state populations and of the probability density distribution of the molecular wave packet. The influence of intra-molecular vibrational energy redistribution appears to be properly taken into account by the new model on the whole range of coupling strengths. It demontrates that most of the mode mixing during relaxation is due to the potential part of the Hamiltonian and not to the coupling among relaxation operators.

Open Quantum Dynamics Calculations with the Hierarchy Equations of Motion on Parallel Computers

Calculating the evolution of an open quantum system, i.e., a system in contact with a thermal environment, has presented a theoretical and computational challenge for many years. With the advent of supercomputers containing large amounts of memory and many processors, the computational challenge posed by the previously intractable theoretical models can now be addressed. The hierarchy equations of motion present one such model and offer a powerful method that remained under-utilized so far due to its considerable computational expense. By exploiting concurrent processing on parallel computers the hierarchy equations of motion can be applied to biological-scale systems. Herein we introduce the quantum dynamics software PHI, that solves the hierarchical equations of motion. We describe the integrator employed by PHI and demonstrate PHI's scaling and efficiency running on large parallel computers by applying the software to the calculation of inter-complex excitation transfer between the light harvesting complexes 1 and 2 of purple photosynthetic bacteria, a 50 pigment system.

How Quantum Coherence Assists Photosynthetic Light Harvesting

This perspective examines how hundreds of pigment molecules in purple bacteria cooperate through quantum coherence to achieve remarkable light harvesting efficiency. Quantum coherent sharing of excitation, which modifies excited state energy levels and combines transition dipole moments, enables rapid transfer of excitation over large distances. Purple bacteria exploit the resulting excitation transfer to engage many antenna proteins in light harvesting, thereby increasing the rate of photon absorption and energy conversion. We highlight here how quantum coherence comes about and plays a key role in the photosynthetic apparatus of purple bacteria.

Non-Markovian finite-temperature two-time correlation functions of system operators: beyond the quantum regression theorem

An extremely useful evolution equation that allows systematically calculating the two-time correlation functions (CF's) of system operators for non-Markovian open (dissipative) quantum systems is derived. The derivation is based on perturbative quantum master equation approach, so non-Markovian open quantum system models that are not exactly solvable can use our derived evolution equation to easily obtain their two-time CF's of system operators, valid to second order in the system-environment interaction. Since the form and nature of the Hamiltonian are not specified in our derived evolution equation, our evolution equation is applicable for bosonic and/or fermionic environments and can be applied to a wide range of system-environment models with any factorized (separable) system-environment initial states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equation is valid for both L = L(†) and L ≠ L(†) cases, in contrast to those evolution equations valid only for L = L(†) case in the literature. The derived equation that generalizes the quantum regression theorem (QRT) to the non-Markovian case will have broad applications in many different branches of physics. We then give conditions on which the QRT holds in the weak system-environment coupling case and apply the derived evolution equation to a problem of a two-level system (atom) coupled to the finite-temperature bosonic environment (electromagnetic fields) with L ≠ L(†).

High-Performance Solution of Hierarchical Equations of Motion for Studying Energy Transfer in Light-Harvesting Complexes

Excitonic models of light-harvesting complexes, where the vibrational degrees of freedom are treated as a bath, are commonly used to describe the motion of the electronic excitation through a molecule. Recent experiments point toward the possibility of memory effects in this process and require one to consider time nonlocal propagation techniques. The hierarchical equations of motion (HEOM) were proposed by Ishizaki and Fleming to describe the site-dependent reorganization dynamics of protein environments ( J. Chem. Phys. 2009 , 130 , 234111 ), which plays a significant role in photosynthetic electronic energy transfer. HEOM are often used as a reference for other approximate methods but have been implemented only for small systems due to their adverse computational scaling with the system size. Here, we show that HEOM are also solvable for larger systems, since the underlying algorithm is ideally suited for the usage of graphics processing units (GPU). The tremendous reduction in computational time due to the GPU allows us to perform a systematic study of the energy-transfer efficiency in the Fenna-Matthews-Olson (FMO) light-harvesting complex at physiological temperature under full consideration of memory effects. We find that approximative methods differ qualitatively and quantitatively from the HEOM results and discuss the importance of finite temperature to achieving high energy-transfer efficiencies.