Redfield Equation Research Articles

Non-canonical distribution and non-equilibrium transport beyond weak system-bath coupling regime: A polaron transformation approach

The concept of polaron, emerged from condense matter physics, describes the dynamical interaction of moving particle with its surrounding bosonic modes. This concept has been developed into a useful method to treat open quantum systems with a complete range of system-bath coupling strength. Especially, the polaron transformation approach shows its validity in the intermediate coupling regime, in which the Redfield equation or Fermi’s golden rule will fail. In the polaron frame, the equilibrium distribution carried out by perturbative expansion presents a deviation from the canonical distribution, which is beyond the usual weak coupling assumption in thermodynamics. A polaron transformed Redfield equation (PTRE) not only reproduces the dissipative quantum dynamics but also provides an accurate and efficient way to calculate the non-equilibrium steady states. Applications of the PTRE approach to problems such as exciton diffusion, heat transport and light-harvesting energy transfer are presented.

Linear and Nonlinear Spectra in Photosynthetic Light Harvesting Complexes: Benchmark Tests of Modified Redfield Method

We employ the numerically exact hierarchical equations of motion (HEOM) method to perform benchmark tests for the popular modified Redfield method in calculating linear and nonlinear spectroscopic signals of molecular aggregates in photosynthetic light harvesting complexes. It is currently well known that the perturbative and Markovian approximations involved in the modified Redfield equation may give inappropriate description of the excitation energy transfer processes in the intermediate coupling regime. An interesting topic is thus to test the validity of the modified Redfield method in calculating various types of spectroscopic signals. By using model dimers with different sets of parameters and a model of the Fenna-Matthews-Olson complex, we calculate and compare the absorption, emission, and 2D spectra using the modified Redfield and HEOM methods. It is found that results from the modified Redfield method agree well with the HEOM ones in a wide range of parameter regimes. The comparison also helps to understand the quantum beating signals in the 2D spectra of the photosynthetic light harvesting complexes.

Quantum heat transport of a two-qubit system: Interplay between system-bath coherence and qubit-qubit coherence.

We consider a system consisting of two interacting qubits that are individually coupled to separate heat baths at different temperatures. The quantum effects in heat transport are investigated in a numerically rigorous manner with a hierarchial equations of motion (HEOM) approach for non-perturbative and non-Markovian system-bath coupling cases under non-equilibrium steady-state conditions. For a weak interqubit interaction, the total system is regarded as two individually thermostatted systems, whereas for a strong interqubit interaction, the two-qubit system is regarded as a single system coupled to two baths. The roles of quantum coherence (or entanglement) between the two qubits (q-q coherence) and between the qubit and bath (q-b coherence) are studied through the heat current calculated for various strengths of the system-bath coupling and interqubit coupling for high and low temperatures. The same current is also studied using the time convolutionless (TCL) Redfield equation and using an expression derived from the Fermi golden rule (FGR). We find that the HEOM results exhibit turnover behavior of the heat current as a function of the system-bath coupling strength for all values of the interqubit coupling strength, while the results obtained with the TCL and FGR approaches do not exhibit such behavior, because they do not possess the capability of treating the q-b and q-q coherences. The maximum current is obtained in the case that the q-q coherence and q-b coherence are balanced in such a manner that coherence of the entire heat transport process is realized. We also find that the heat current does not follow Fourier's law when the temperature difference is very large, due to the non-perturbative system-bath interactions.

Coherent quantum transport in disordered systems: A unified polaron treatment of hopping and band-like transport.

Quantum transport in disordered systems is studied using a polaron-based master equation. The polaron approach is capable of bridging the results from the coherent band-like transport regime governed by the Redfield equation to incoherent hopping transport in the classical regime. A non-monotonic dependence of the diffusion coefficient is observed both as a function of temperature and system-phonon coupling strength. In the band-like transport regime, the diffusion coefficient is shown to be linearly proportional to the system-phonon coupling strength and vanishes at zero coupling due to Anderson localization. In the opposite classical hopping regime, we correctly recover the dynamics described by the Fermi's Golden Rule and establish that the scaling of the diffusion coefficient depends on the phonon bath relaxation time. In both the hopping and band-like transport regimes, it is demonstrated that at low temperature, the zero-point fluctuations of the bath lead to non-zero transport rates and hence a finite diffusion constant. Application to rubrene and other organic semiconductor materials shows a good agreement with experimental mobility data.

Surface hopping outperforms secular Redfield theory when reorganization energies range from small to moderate (and nuclei are classical)

We evaluate the accuracy of Tully's surface hopping algorithm for the spin-boson model in the limit of small to moderate reorganization energy. We calculate transition rates between diabatic surfaces in the exciton basis and compare against exact results from the hierarchical equations of motion; we also compare against approximate rates from the secular Redfield equation and Ehrenfest dynamics. We show that decoherence-corrected surface hopping performs very well in this regime, agreeing with secular Redfield theory for very weak system-bath coupling and outperforming secular Redfield theory for moderate system-bath coupling. Surface hopping can also be extended beyond the Markovian limits of standard Redfield theory. Given previous work [B. R. Landry and J. E. Subotnik, J. Chem. Phys. 137, 22A513 (2012)] that establishes the accuracy of decoherence-corrected surface-hopping in the Marcus regime, this work suggests that surface hopping may well have a very wide range of applicability.

Coherent quantum transport in disordered systems: II. Temperature dependence of carrier diffusion coefficients from the time-dependent wavepacket diffusion method

The time-dependent wavepacket diffusion method for carrier quantum dynamics (Zhong and Zhao 2013 J. Chem. Phys. 138 014111), a truncated version of the stochastic Schrödinger equation/wavefunction approach that approximately satisfies the detailed balance principle and scales well with the size of the system, is applied to investigate the carrier transport in one-dimensional systems including both the static and dynamic disorders on site energies. The predicted diffusion coefficients with respect to temperature successfully bridge from band-like to hopping-type transport. As demonstrated in paper I (Moix et al 2013 New J. Phys. 15 085010), the static disorder tends to localize the carrier, whereas the dynamic disorder induces carrier dynamics. For the weak dynamic disorder, the diffusion coefficients are temperature-independent (band-like property) at low temperatures, which is consistent with the prediction from the Redfield equation, and a linear dependence of the coefficient on temperature (hopping-type property) only appears at high temperatures. In the intermediate regime of dynamic disorder, the transition from band-like to hopping-type transport can be easily observed at relatively low temperatures as the static disorder increases. When the dynamic disorder becomes strong, the carrier motion can follow the hopping-type mechanism even without static disorder. Furthermore, it is found that the memory time of dynamic disorder is an important factor in controlling the transition from the band-like to hopping-type motions.

Nonequilibrium quantum phase transitions in the XY model: comparison of unitary time evolution and reduced density operator approaches

We study nonequilibrium quantum phase transitions in the XY spin 1/2 chain using the algebra. We show that the well-known quantum phase transition at a magnetic field of h = 1 also persists in the nonequilibrium setting as long as one of the reservoirs is set to absolute zero temperature. In addition, we find nonequilibrium phase transitions associated with an imaginary part of the correlation matrix for any two different reservoir temperatures at h = 1 and , where γ is the anisotropy and h the magnetic field strength. In particular, two nonequilibrium quantum phase transitions coexist at h = 1. In addition, we study the quantum mutual information in all regimes and find a logarithmic correction of the area law in the nonequilibrium steady state independent of the system parameters. We use these nonequilibrium phase transitions to test the utility of two models of a reduced density operator, namely the Lindblad mesoreservoir and the modified Redfield equation. We show that the nonequilibrium quantum phase transition at h = 1, related to the divergence of magnetic susceptibility, is recovered in the mesoreservoir approach, whereas it is not recovered using the Redfield master equation formalism. However, none of the reduced density operator approaches could recover all the transitions observed by the algebra. We also study the thermalization properties of the mesoreservoir approach.

Electron dynamics in charged wet TiO2 anatase (001) surface functionalised by ruthenium ions

The surface of charged wet TiO2 anatase (001) functionalised by ruthenium ion at ambient temperatures is studied by computational modelling. Response of this model to photoexcitations at ambient temperatures is explored with the Redfield density matrix equation of motion on the basis of Kohn–Sham orbitals. The parameters of the Redfield equation are on-the-fly non-adiabatic couplings for electronic degrees of freedom obtained along the ab initio molecular dynamics nuclear trajectories. The main results in this study are the following: (1) optical properties of the doped models such as light absorption intensity and transition energies can be tuned by modifying total charge; (2) electron and hole relaxation rates depend on the initial excitation; and (3) in the doped model, excitations of lower energy provide quicker relaxation. Results of computational modelling would benefit understanding of the mechanism of electron transfer processes on the surface of ruthenium-doped TiO2.

Quantification of memory effects in the spin-boson model

Employing a recently proposed measure for quantum non-Markovianity, we carry out a systematic study of the size of memory effects in the spin-boson model for a large region of temperature and frequency cutoff parameters. The dynamics of the open system is described utilizing a second-order time-convolutionless master equation without the Markov or rotating wave approximations. While the dynamics is found to be strongly non-Markovian for low temperatures and cutoffs, in general, we observe a special regime favoring Markovian behavior. This effect is explained as resulting from a resonance between the system's transition frequency and the frequencies of the dominant environmental modes. We further demonstrate that the corresponding Redfield equation is capable of reproducing the characteristic features of the non-Markovian quantum behavior of the model.

Heat transport in quantum spin chains: Relevance of integrability

We investigate heat transport in various quantum spin chains using the Redfield equation. We find that anomalous heat transport is linked not to the integrability of the Hamiltonian, but to whether it can be mapped to a model of noninteracting fermions. Our results also suggest how seemingly anomalous transport may occur at low temperatures in a much wider class of models.

On the influence of nonlocal molecular vibrations and charge transfer on the spectra of aggregated push–pull chromophores

Through their fluorescence spectrum, aggregates of push-pull chromophores are good reporters of their microenvironment temperature and polarity. The understanding of the fluorescence and charge-separation dynamics in arrays composed of this type of species is consequently of considerable interest. In this article, we study the effect of charge fluctuations induced by molecular nonlocal vibrations on the electronic coupling between a pair of linear push-pull chromophores, for side-to-side or head-to-tail orientations, using a valence-bond charge-transfer (VB-CT) model and the Redfield equation. The results show that the exciton-vibrational dynamics along the bond length alternation coordinate can significantly modify the inter-molecular electronic coupling, which determines the fluorescence spectral band redshift due to aggregation. Numerical results for the electronic and exciton-vibrational contributions to the Coulombic coupling between two of these chromophores are obtained using experimentally based parameters for polyene linker species. The exciton-vibrational contribution is significant relative to the electronic contribution at room temperature in some ranges of the energy gap between the VB and CT states, and it is more important for the side-to-side than for the head-to-tail configuration. Our calculations also show that, even without including solvation effects, the spectral band associated with an S(0) → S(1) transition is redshifted with increasing temperature.

Efficient energy transfer in light-harvesting systems, I: optimal temperature, reorganization energy and spatial–temporal correlations

Understanding the mechanisms of efficient and robust energy transfer in light-harvesting systems provides new insights for the optimal design of artificial systems. In this paper, we use the Fenna–Matthews–Olson (FMO) protein complex and phycocyanin 645 (PC 645) to explore the general dependence on physical parameters that help maximize the efficiency and maintain its stability. With the Haken–Strobl model, the maximal energy transfer efficiency (ETE) is achieved under an intermediate optimal value of dephasing rate. To avoid the infinite temperature assumption in the Haken–Strobl model and the failure of the Redfield equation in predicting the Forster rate behavior, we use the generalized Bloch–Redfield (GBR) equation approach to correctly describe dissipative exciton dynamics, and we find that maximal ETE can be achieved under various physical conditions, including temperature, reorganization energy and spatial–temporal correlations in noise. We also identify regimes of reorganization energy where the ETE changes monotonically with temperature or spatial correlation and therefore cannot be optimized with respect to these two variables.

Non-equilibrium stationary states from the equation of motion of open systems

Non-equilibrium stationary states are solved from the equation of motion of open systems via a Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY)-like hierarchy. As an example, we demonstrate this approach for the Redfield equation, which is derived from the whole dynamic equation of motion of a central system coupled to two baths through the projector technique. Generalization to other equations of motion is straightforward. For non-interacting central systems, the first equation out of the hierarchy is closed, whereas for interacting systems it is coupled to higher-order equations in the hierarchy. In the case of interacting systems, two systematic approximations, in the form of perturbation theories, are proposed to truncate and solve the hierarchy. A non-equilibrium Wick's theorem is proved to provide a basis for the perturbation theories. As a test of reliability of the proposed methods, we apply them to small systems, where it is also possible to apply other exact direct methods. Consistent results were found.

Self-Consistent Quantum Master Equation Approach to Molecular Transport

We propose a self-consistent generalized quantum master equation (GQME) to describe electron transport through molecular junctions. In a previous study [M.Esposito and M.Galperin. Phys. Rev. B 79, 205303 (2009)], we derived a time-nonlocal GQME to cure the lack of broadening effects in Redfield theory. To do so, the free evolution used in the Born-Markov approximation to close the Redfield equation was replaced by a standard Redfield evolution. In the present paper, we propose a backward Redfield evolution leading to a time-local GQME which allows for a self-consistent procedure of the GQME generator. This approach is approximate but properly reproduces the nonequilibrium steady state density matrix and the currents of an exactly solvable model. The approach is less accurate for higher moments such as the noise.

μSR study on the spin singlet state in the S = 1/2 cluster magnet GaNb4S8

GaNb4S8 is a S = 1/2 Nb4-cluster magnet exhibiting a structural phase transition at Ts = 32 K. We have investigated the magnetic ground state of the compound by means of the zero-field (ZF) and the longitudinal-field (LF) μSR techniques. In the ZF-μSR spectra, no muon spin precession was observed below Ts, revealing an absence of magnetic long-range orders. Instead, an stretched exponential relaxation was observed, which remains under the LF up to 3950 G. The field dependence of the relaxation rate obeys the Redfield's equation, indicating that a dynamic component of electronic spin fluctuations is dominant below Ts. These μSR results can be understand as sporadic dynamics of spin fluctuations in a spin singlet sea and support the spin singlet state of GaNb4S8 due to the formation of Nb8 octamer from two Nb4 tetramers.

Exact solution of Markovian master equations for quadratic Fermi systems: thermal baths, open XY spin chains and non-equilibrium phase transition

We generalize the method of third quantization to a unified exact treatment of Redfield and Lindblad master equations for open quadratic systems of n fermions in terms of diagonalization of a 4n×4n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how one can compute all the physically relevant quantities, such as non-equilibrium expectation values of local observables, various entropies or information measures, or time evolution and properties of relaxation. We also discuss how to exactly treat explicitly time-dependent problems. The general formalism is then applied to study a thermally driven open XY spin 1/2 chain. We find that the recently proposed non-equilibrium quantum phase transition in the open XY chain survives the thermal driving within the Redfield model. In particular, the phase of long-range magnetic correlations can be characterized by hypersensitivity of the non-equilibrium steady state to external (bath or bulk) parameters. Studying the heat transport, we find negative differential thermal conductance for sufficiently strong thermal driving as well as non-monotonic dependence of the heat current on the strength of the bath coupling.

A new form of the redfield equation for dissipative systems related to the matrix of correlation functions

In the Redfield theory framework, we consider the problem of the vibrational dynamics in dissipative systems. We decompose the Hamiltonian of interaction of the observed system with a thermal bath into terms that are products of system transition operators and bath transition operators. Using the decomposition, we construct a correlation function matrix containing all the information about the interaction of the system with the bath and obtain a new operator form of the Redfield equation. We consider the procedure for factoring the interaction operator and constructing the correlation function matrix. Using the diagonalization of the obtained matrix, we give correlated dissipation operators, whose introduction simplifies the form of the Redfield equations. We show that for several problems in which fundamental transition frequencies can be chosen, this procedure significantly reduces the dimensionality of the dissipative dynamics problem.

Unified treatment of quantum coherent and incoherent hopping dynamics in electronic energy transfer: Reduced hierarchy equation approach

A new quantum dynamic equation for excitation energy transfer is developed which can describe quantum coherent wavelike motion and incoherent hopping in a unified manner. The developed equation reduces to the conventional Redfield theory and Forster theory in their respective limits of validity. In the regime of coherent wavelike motion, the equation predicts several times longer lifetime of electronic coherence between chromophores than does the conventional Redfield equation. Furthermore, we show quantum coherent motion can be observed even when reorganization energy is large in comparison to intersite electronic coupling (the Forster incoherent regime). In the region of small reorganization energy, slow fluctuation sustains longer-lived coherent oscillation, whereas the Markov approximation in the Redfield framework causes infinitely fast fluctuation and then collapses the quantum coherence. In the region of large reorganization energy, sluggish dissipation of reorganization energy increases the time electronic excitation stays above an energy barrier separating chromophores and thus prolongs delocalization over the chromophores.

On the adequacy of the Redfield equation and related approaches to the study of quantum dynamics in electronic energy transfer

The observation of long-lived electronic coherence in photosynthetic excitation energy transfer (EET) by Engel et al. [Nature (London) 446, 782 (2007)] raises questions about the role of the protein environment in protecting this coherence and the significance of the quantum coherence in light harvesting efficiency. In this paper we explore the applicability of the Redfield equation in its full form, in the secular approximation and with neglect of the imaginary part of the relaxation terms for the study of these phenomena. We find that none of the methods can give a reliable picture of the role of the environment in photosynthetic EET. In particular the popular secular approximation (or the corresponding Lindblad equation) produces anomalous behavior in the incoherent transfer region leading to overestimation of the contribution of environment-assisted transfer. The full Redfield expression on the other hand produces environment-independent dynamics in the large reorganization energy region. A companion paper presents an improved approach, which corrects these deficiencies [A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130, 234111 (2009)].

Decoherence of a Double Quantum Dot Charge Qubit Approached with Redfield Equation

In this paper, we investigate the decoherence time of a double quantum dot (DQD) charge qubit in three kinds of baths through solving dynamics of the qubit. The dynamics of the qubit is investigated with Redfield master equation. It is shown that the decoherence time of the qubit in Ohmic bath has the same order of magnitude as the experiments reported. When the environment is modeled with the supra-Ohmic bath the decoherence time of the qubit is shorter than the experimental result. And when modeled with the sub-Ohmic bath the decoherence time of the qubit is longer than the experimental result.