Redfield Equation Research Articles

Nonperturbative non-Markovian quantum master equation: Validity and limitation to calculate nonlinear response functions

Based on the influence functional formalism, we have derived a nonperturbative equation of motion for a reduced system coupled to a harmonic bath with colored noise in which the system–bath coupling operator does not necessarily commute with the system Hamiltonian. The resultant expression coincides with the time-convolutionless quantum master equation derived from the second-order perturbative approximation, which is also equivalent to a generalized Redfield equation. This agreement occurs because, in the nonperturbative case, the relaxation operators arise from the higher-order system–bath interaction that can be incorporated into the reduced density matrix as the influence operator; while the second-order interaction remains as a relaxation operator in the equation of motion. While the equation describes the exact dynamics of the density matrix beyond weak system–bath interactions, it does not have the capability to calculate nonlinear response functions appropriately. This is because the equation cannot describe memory effects which straddle the external system interactions due to the reduced description of the bath. To illustrate this point, we have calculated the third-order two-dimensional (2D) spectra for a two-level system from the present approach and the hierarchically coupled equations approach that can handle quantal system–bath coherence thanks to its hierarchical formalism. The numerical demonstration clearly indicates the lack of the system–bath correlation in the present formalism as fast dephasing profiles of the 2D spectra.

Role of quantum coherence and energetic disorder in exciton transport in polymer films

The cross over from coherent to incoherent exciton transport in disordered polymer films is studied by computationally solving a modified form of the Redfield equation for the exciton density matrix. This theory models quantum-mechanical (ballistic) and incoherent (diffusive) transport as limiting cases. It also reproduces F\orster transport for certain parameter regimes. Using model parameters appropriate to polymer thin films it is shown that short-time quantum-mechanical coherence increases the exciton diffusion length. It also causes rapid initial energy relaxation and larger linewidths in agreement with experiment. The route to equilibrium is, however, more questionable, as the equilibrium populations of the model do not satisfy the Boltzmann distributions over the site energies. The Redfield equation for the dimer is solved exactly to provide insight into the numerical results.

The spherical tensor formalism applied to relaxation in magnetic resonance

Spherical tensor operators are used to simplify the relaxation terms of the Redfield equation. A method for treating relaxation is developed using trace-normalized spherical tensor operators. The Redfield equation is recast as a set of coupled linear differential equations for the expectation values of a complete set of spherical tensor operators. A single spin-1/2 particle is treated, and the correspondence of the equation of motion with the Bloch equation is demonstrated. The advantage of the spherical tensor approach is that both the coherent and relaxation terms of the equation of motion are written as matrices that operate on a common vector consisting of the expectation values of spin variables. The process used to construct the equation of motion helps to organize the various contributions to spin relaxation and can be applied uniformly regardless of spin system size. As an example, the relaxation of a two-spin system by the chemical shift anisotropy and dipolar Hamiltonians is developed using the spherical tensor formalism. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 270–290, 2006

Biogeochemistry: now and into the future

Biogeochemistry can be defined as the mutual interactions (two-way) between the biology and chemistry of the Earth system, and as such is clearly an important component of the broader discipline of geobiology. It is a well-developed field, having a dedicated journal and textbook, and many hundreds of publications appearing in the scientific literature each year. Perhaps the best example of biogeochemistry is the balance between photosynthesis and respiration, autotrophy and heterotrophy, on Earth, the so-called Redfield equation. The modern practice of biogeochemical research has been focused on present-day processes such as those in the Redfield equation, but new tools and approaches are beginning to explore the biogeochemistry of the ancient Earth; this is the challenge for biogeochemists in the future.

Vibrational coherence transfer characterized with Fourier-transform 2D IR spectroscopy.

Two-dimensional infrared (2D IR) spectroscopy of the symmetric and asymmetric C[Triple Bond]O stretching vibrations of Rh(CO)(2)acac in hexane has been used to investigate vibrational coherence transfer, dephasing, and population relaxation in a multilevel vibrational system. The transfer of coherence between close-lying vibrational frequencies results in extra relaxation-induced peaks in the 2D IR spectrum, whose amplitude depends on the coherence transfer rate. Coherence transfer arises from the mutual interaction of the bright CO stretches with dark states, which in this case reflects the mutual d-pi(*) back bonding of the Rh center to both the terminal carbonyls and the acetylacenonate ligand. For 2D IR relaxation experiments with variable waiting times, coherent dynamics lead to the modulation of peak amplitudes, while incoherent population relaxation and exchange results in the growth of the relaxation-induced peaks. We have modeled the data by propagating the density matrix with the Redfield equation, incorporating all vibrational relaxation processes during all three experimental time periods and including excitation reorientation effects arising from relaxation. Coherence and population transfer time scales from the symmetric to the asymmetric stretch were found to be 350 fs and 3 ps, respectively. We also discuss a diagrammatic approach to incorporating all vibrational relaxation processes into the nonlinear response function, and show how coherence transfer influences the analysis of structural variables from 2D IR spectroscopy.

Quantum master equation for a system influencing its environment.

A perturbative quantum master equation is derived for a system interacting with its environment, which is more general than the ones derived before. Our master equation takes into account the effect of the energy exchanges between the system and the environment and the conservation of energy in the finite total system. This master equation describes relaxation mechanisms in isolated nanoscopic quantum systems. In its most general form, this equation is non-Markovian and a Markovian version of it rules the long-time relaxation. We show that our equation reduces to the Redfield equation in the limit where the energy of the system does not affect the density of state of its environment. This master equation and the Redfield one are applied to a spin-environment model defined in terms of random matrices and compared with the solutions of the exact von Neumann equation. The comparison proves the necessity to allow energy exchange between the subsystem and the environment in order to correctly describe the relaxation in an isolated nanoscopic total system.

Weak-field laser control of systems experiencing dissipation and decoherence. An application to the enhancement of the coherent fluorescence of the B850 ring in light harvesting complexes

A previously developed approach for the computation of optimal laser pulses in the weak-field limit is extended to systems undergoing dephasing and dissipation and thus requiring a density matrix treatment. The starting point is a multicolor gaussian pulse which can be shaped by modifying its individual amplitudes and phases. The method is grounded on the generation of random pulses to form a linearly-independent basis set from which the optimal pulse is determined by linear algebra manipulations. As an interesting example we show that the coherent component of the fluorescence (free induction decay, FID) from the B850 ring present in the light-harvesting 2 (LH2) complex of the purple bacterium Rhodopseudomonas acidophila can be significantly enhanced. The calculations have been performed using a Redfield equation built up from a model Frenkel Hamiltonian taken from the literature with a phenomenological dephasing time for electronic coherences.

Influence of phonons on exciton transfer dynamics: comparison of the Redfield, Förster, and modified Redfield equations

The effect of the reorganization of phonons on exciton transfer dynamics is examined. Starting from a general quantum master equation, we obtain population transfer rates for the Redfield, the Förster, and the modified Redfield equations and compare the predictions of these three models over various ranges of parameters. It is shown that the validity of the traditional Redfield equation is justified by a broad spectrum of phonons rather than a small magnitude of electron–phonon coupling strength as is usually implied. The modified Redfield equation suggested by Zhang et al. [J. Chem. Phys. 108 (1998) 7763] is shown to have a wide range of applicability and to reduce numerically to the traditional Redfield equation and the Förster equation in the respective limiting cases.

Influence of phonons on exciton transfer dynamics: comparison of the Redfield, Förster, and modified Redfield equations

The effect of the reorganization of phonons on exciton transfer dynamics is examined. Starting from a general quantum master equation, we obtain population transfer rates for the Redfield, the Förster, and the modified Redfield equations and compare the predictions of these three models over various ranges of parameters. It is shown that the validity of the traditional Redfield equation is justified by a broad spectrum of phonons rather than a small magnitude of electron–phonon coupling strength as is usually implied. The modified Redfield equation suggested by Zhang et al. [J. Chem. Phys. 108 (1998) 7763] is shown to have a wide range of applicability and to reduce numerically to the traditional Redfield equation and the Förster equation in the respective limiting cases.

Symplectic integrators for the multilevel Redfield equation

We develop a method for solving the Redfield equation using symplectic integrators (SI). We show how the Redfield equation for a reduced density matrix of a multi-level system interacting with a thermal bath can be rewritten in a symplectic form so that numerical algorithms used in classical mechanics can readily be applied. The symplectic structure of the Redfield equation is established for a superoperator which is complex, non-Hermitian, and explicitly time-dependent in some cases. The efficiency of several SI schemes are studied and compared for a model dissipative system: electron transfer coupled to a vibrational mode and a thermal bath.

Relaxation equation for muonium-like systems with electron spin more than one half

A phenomenological derivation of the Wangsness–Bloch–Redfield equation for the density matrix of muonium-like systems is given in this paper. In the case of the isotropic relaxation, it is shown that the corresponding relaxation term is described by only two independent constants. A typical form of the density matrix of the isotropic hyperfine interaction is given, and it is shown that the density matrix is decomposed into diagonal blocks. Relaxation rates corresponding to the different blocks can differ by orders of magnitude.

An adiabatic picture for electron transfer in mixed-valence systems

An adiabatic picture is proposed to analyze short-time electronic coherence and dephasing in mixed-valence systems, which cannot be described by the Redfield equation or the golden-rule rate theory. In this picture, electronic coherence arises from Rabi oscillations between two adiabatic surfaces and decays as a result of inhomogeneous distributions and thermal fluctuations of Rabi frequencies. The initial preparation and wavepacket dynamics modulate Rabi oscillations and thus influence electronic coherence. Comparisons between exact numerical results and the prediction from the proposed theory verify the accuracy of the adiabatic approximation in describing the short-time electronic dynamics.

Slippage of initial conditions for the Redfield master equation

For a slow open quantum subsystem weakly coupled to a fast thermal bath, we derive the general form of the slippage to be applied to the initial conditions of the Redfield master equation. This slippage is given by a superoperator which describes the non-Markovian dynamics of the subsystem during the short-time relaxation of the thermal bath. We verify in an example that the Redfield equation preserves positivity after the slippage superoperator has been applied to the initial density matrix of the subsystem. For δ-correlated baths, the Redfield master equation reduces to the Lindblad master equation and the slippage of initial conditions vanishes consistently.

Short-time Chebyshev propagator for the Liouville–von Neumann equation

A Chebyshev interpolation scheme is proposed for the short-time Liouville–von Neumann propagator. For each propagation step, a small number of Chebyshev polynomials is used to construct the propagator. The method involves only matrix-vector multiplication and is memory efficient since the three-term Chebyshev recursion needs only two vectors stored. It is also numerically stable since neither matrix diagonalization nor inversion is involved. The short Chebyshev recursion ensures that the divergence due to the complex eigenvalues of the Liouville superoperator is kept under control. Numerical tests carried out for the Redfield equation of a one-dimensional dissipative harmonic system demonstrate that the short-time Chebyshev propagator is accurate and significantly more efficient than the commonly used fourth-order Runge–Kutta scheme.

Finite dimensional systems with random external fields and neutrino propagation in fluctuating media

We develop the general formalism for the study of neutrino propagation in presence of stochastic media. This formalism allows the systematic derivation of evolution equations for averaged quantities as survival probabilities and higher order distribution moments. The formalism applies equally to any finite dimensional Schroedinger equation in presence of a stochastic external force. New integro-differential equations valid for finite correlated processes are obtained for the first time. For the particular case of exponentially correlated processes a second order ordinary equation is obtained. As a consequence, the Redfield equation valid for Gaussian delta-correlated noise is rederived in a simple way. The formalism, together with the quantum correlation theorem is applied to the computation of higher moments and correlation functions of practical interest in forthcoming high precision neutrino experiments. It is shown that equal and not equal time correlators follow similar differential equations.

Reduced density matrix description of gas–solid interactions: Scattering, trapping, and desorption

A formalism is developed for modeling the interaction between a particle scattering from a surface and the thermal vibrations of the lattice, using the reduced density matrix. A short time propagation algorithm is constructed, with the dissipative terms derived directly from the full particle–bath Hamiltonian. The resulting Bloch–Redfield equation is used to study the scattering of He, Ne, and Ar from model Cu and Ru surfaces, over a wide range of energy and temperature. Trapping and desorption are also examined, and comparisons are made with experimental sticking data. The method is shown to give reasonable results and to accurately describe the relaxation to equilibrium over long times.

Classical-quantum correspondence in the Redfield equation and its solutions

In a recent paper we showed the equivalence, under certain well-characterized assumptions, of Redfield’s equations for the density operator in the energy representation with the Gaussian phase space ansatz for the Wigner function of Yan and Mukamel. The equivalence shows that the solutions of Redfield’s equations respect a striking degree of classical-quantum correspondence. Here we use this equivalence to derive analytic expressions for the density matrix of the harmonic oscillator in the energy representation without making the almost ubiquitous secular approximation. From the elements of the density matrix in the energy representation we derive analytic expressions for Γ1n(1/T1n) and Γ2nm(1/T2nm), i.e., population and phase relaxation rates for individual matrix elements in the energy representation. Our results show that Γ1n(t)=Γ1(t) is independent of n; this is contrary to the widely held belief that Γ1n is proportional to n. We also derive the simple result that Γ2nm(t)=|n−m|Γ1(t)/2, a generalization of the two-level system result Γ2=Γ1/2. We show that Γ1(t) is the classical rate of energy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Γ1(t) is directly proportional to the classical friction, γ. An additional element of classical-quantum correspondence concerns the time rate of change of the phase of the off diagonal elements of the density matrix, ωnm, a quantity which has received little attention previously. We find that ωnm is time-dependent, and equal to |n−m|Ω(t), where Ω(t) is the rate of change of phase space angle in the classical damped harmonic oscillator. Finally, expressions for a collective Γ1(t) and Γ2(t) are derived, and shown to satisfy the relationship Γ2=Γ1/2. This familiar result, when applied to these collective rate constants, is seen to have a simple geometrical interpretation in phase space.

Multilevel Redfield Treatment of Bridge-Mediated Long-Range Electron Transfer: A Mechanism for Anomalous Distance Dependence

The density matrix formalism is used to consider long-range electron transfer through a delocalized bridge system that is dissipatively coupled to a thermal bath. Under conditions when the bridge is weakly populated, a distance-independent transfer mechanism arises that eventually dominates the exponentially distance-dependent nonadiabatic tunneling process typically observed. This model offers a possible explanation for the anomalously rapid long-range photoelectron transfer recently observed by Barton and Turro et al, for donor and acceptor species intercalated into a DNA double helix. Predicted electron-transfer rates are obtained by numerically solving the Redfield equation for the density matrix of the full donor-bridge-acceptor system in the presence of stochastic fluctuations induced by the bath. 43 refs., 17 figs., 4 tabs.

Neutrino oscillations in noisy media.

We develop the Redfield equation for \ensuremath{\delta}-correlated Gaussian noise and apply it to the case of two neutrino flavor or spin precession in the presence of a noisy matter density or magnetic field, respectively. The criteria under which physical fluctuations can be well approximated by the \ensuremath{\delta}-correlated Gaussian noise for the above cases are examined. Current limits on the possible neutrino magnetic moment and solar magnetic field suggest that a reasonably noisy solar magnetic field would not appreciably affect the solar electron-neutrino flux. However, if the solar electron density has fluctuations of a few percent of the local density and a small enough correlation length, the MSW effect is suppressed for a range of parameters.

Solution of the Redfield equation for the dissipative quantum dynamics of multilevel systems

We present a new method for solving the Redfield equation, which describes the evolution of the reduced density matrix of a multilevel quantum-mechanical system interacting with a thermal bath. The method is based on a new decomposition of the Redfield relaxation tensor that makes possible its direct application to the density matrix without explicit construction of the full tensor. In the resulting expressions, only ordinary matrices are involved and so any quantum system whose Hamiltonian can be diagonalized can be treated with the full Redfield theory. To efficiently solve the equation of motion for the density matrix, we introduce a generalization of the short-iterative-Lanczos propagator. Together, these contributions allow the complete Redfield theory to be applied to significantly larger systems than was previously possible. Several model calculations are presented to illustrate the methodology, including one example with 172 quantum states.