Pauli Master Equation Research Articles

Towards a microscopic understanding of the phonon bottleneck

The problem of the phonon bottleneck in the relaxation of two-level systems (spins) to a narrow group of resonant phonons via emission-absorption processes is investigated from first principles. It is shown that the kinetic approach based on the Pauli master equation is invalid because of the narrow distribution of the phonons exchanging their energy with the spins. This results in a long-memory effect that can be best taken into account by introducing an additional dynamical variable corresponding to the nondiagonal matrix elements responsible for spin-phonon correlation. The resulting system of dynamical equations describes the phonon-bottleneck plateau in the spin excitation, as well as a gap in the spin-phonon spectrum, for any finite concentration of spins. On the other hand, it does not accurately render the line shape of emitted phonons and still needs improving.

Temperature-Dependent Relaxation of Excitons in Tubular Molecular Aggregates: Fluorescence Decay and Stokes Shift

We report temperature-dependent steady-state and time-resolved fluorescence studies to probe the exciton dynamics in double-wall tubular J-aggregates formed by self-assembly of the dye 3,3'-bis(3-sulfopropyl)-5,5',6,6'-tetrachloro-1,1'-dioctylbenzimidacarbocyanine. We focus on the lowest energy fluorescence band, originating from the inner cylindrical wall. At low temperatures, the experiments reveal a nonexponential decay of the fluorescence, with a typical time scale that depends on the emission wavelength. At these temperatures we also find a dynamic Stokes shift of the fluorescence spectrum and its nonmonotonic dependence on temperature under steady-state conditions. All these data indicate that below about 20 K the excitons in the lowest fluorescence band do not reach thermal equilibrium before emission occurs, while above about 60 K thermalization on this time scale is complete. By comparing the two lowest fluorescence bands, we also find indications for fast energy transfer from the outer to the inner wall. We show that the Frenkel exciton model with diagonal disorder, which previously has been proposed to explain the absorption and linear dichroism spectra of these aggregates, yields a quantitative explanation to the observed dynamics. To this end, we extend the model to account for weak phonon-induced scattering of the localized exciton states; the spectral dynamics are then described by solving a Pauli master equation for the exciton populations.

Fewest Switches Adiabatic Surface Hopping As Applied to Vibrational Energy Relaxation

In this contribution quantum/classical surface hopping methodology is applied to vibrational energy relaxation of a quantum oscillator in a classical heat bath. The model of a linearly damped (harmonic) oscillator is chosen which can be mapped onto the Brownian motion (Caldeira-Leggett) Hamiltonian. In the simulations Tully's fewest switches surface hopping scheme is adopted with inclusion of dephasing in the adiabatic basis using a simple decoherence algorithm. The results are compared to the predictions of a Redfield-type quantum master equation modeling using the classical heat bath force correlation function as input. Thereby a link is established between both types of quantum/classical approaches. Viewed from the latter perspective, surface hopping with dephasing may be interpreted as "on-the-fly" stochastic realization of a quantum/classical Pauli master equation.

Temperature, charge carrier density, and electric field dependence of mobilities in disordered conjugated polymers: simulation results

We present the results of simulation studies of the dependence on temperature, charge carrier density, and electric field of the mobility μ in disordered conjugated polymers. The disorder is modeled by a Gaussian density of states (DOS) with width σ. We base our analysis on an exact numerical solution of the Pauli Master equation. The recently experimentally determined carrier density dependence, ranging from densities typically found in light-emitting diodes (LEDs) up to those found in field-effect transistors (FETs), is fully reproduced. At low temperatures T, we find deviations from the generally accepted μ∼exp[−const.(σ/ kT) 2] behavior. Our calculations show that the electric field dependence does not play a prominent role at the driving voltages of polymer devices.

Excitation Migration along Oligophenylenevinylene-Based Chiral Stacks: Delocalization Effects on Transport Dynamics

Atomistic models based on quantum-chemical calculations are combined with time-resolved spectroscopic investigations to explore the migration of electronic excitations along oligophenylenevinylene-based chiral stacks. It is found that the usual Pauli master equation (PME) approach relying on uncoherent transport between individual chromophores underestimates the excitation diffusion dynamics, monitored here by the time decay of the transient polarization anisotropy. A better agreement to experiment is achieved when accounting for excitation delocalization among acceptor molecules, as implemented in a modified version of the PME model. The same models are applied to study light harvesting and trapping in guest-host systems built from oligomers of different lengths.

Dynamics of a Fermi system with resonant dissipation and dynamical detailed balance

The dissipative dynamics of a system of Fermions is described in the framework of a resonance model—the quantum master equation describes two-body correlations of the system with the environment particles. This equation, with microscopic coefficients depending on the exactly known two-body potential between the system and the environment particles, is discussed in comparison with other master equations, obtained on axiomatic grounds, or derived from a coupling with an environment of harmonic oscillators without altering the quantum conditions. The asymptotic solution is in accordance with the detailed balance principle, and with other generally accepted conditions satisfied during the whole time-evolution: Pauli master equations for the diagonal elements of the density matrix, and damped Bloch–Feynman equations for the non-diagonal ones, that we call dynamical detailed balance. For a harmonic oscillator coupled with the electromagnetic field through dipole interaction, a master equation with transition operators between successive levels is obtained. As an application, the decay width of a quantum logic gate is calculated.

Current in Open Quantum Systems

We show that a dissipative current component is present in the dynamics generated by a Liouville-master equation, in addition to the usual component associated with Hamiltonian evolution. The dissipative component originates from coarse graining in time, implicit in a master equation, and needs to be included to preserve current continuity. We derive an explicit expression for the dissipative current in the context of the Markov approximation. Finally, we illustrate our approach with a simple numerical example, in which a quantum particle is coupled to a harmonic phonon bath and dissipation is described by the Pauli master equation.

A 1D coupled Schrödinger drift-diffusion model including collisions

We consider a one-dimensional coupled stationary Schrödinger drift-diffusion model for quantum semiconductor device simulations. The device domain is decomposed into a part with large quantum effects (quantum zone) and a part where quantum effects are negligible (classical zone). We give boundary conditions at the classic–quantum interface which are current preserving. Collisions within the quantum zone are introduced via a Pauli master equation. To illustrate the validity we apply the model to three resonant tunneling diodes.

Coulomb effects in tunneling through a quantum dot stack

Tunneling through two vertically coupled quantum dots is studied by means of a Pauli master equation model. The observation of double peaks in the current-voltage characteristic in a recent experiment is analyzed in terms of the tunnel coupling between the quantum dots and the coupling to the contacts. Different regimes for the emitter chemical potential indicating different peak scenarios in the tunneling current are discussed in detail. We show by comparison with a density matrix approach that the interplay of coherent and incoherent effects in the stationary current can be fully described by this approach.

Low-temperature dynamics of weakly localized Frenkel excitons in disordered linear chains.

We calculate the temperature dependence of the fluorescence Stokes shift and the fluorescence decay time in linear Frenkel exciton systems resulting from the thermal redistribution of exciton population over the band states. The following factors, relevant to common experimental conditions, are accounted for in our kinetic model: (weak) localization of the exciton states by static disorder, coupling of the localized excitons to vibrations in the host medium, a possible nonequilibrium of the subsystem of localized Frenkel excitons on the time scale of the emission process, and different excitation conditions (resonant or nonresonant). A Pauli master equation, with microscopically calculated transition rates, is used to describe the redistribution of the exciton population over the manifold of localized exciton states. We find a counterintuitive nonmonotonic temperature dependence of the Stokes shift. In addition, we show that depending on experimental conditions, the observed fluorescence decay time may be determined by vibration-induced intraband relaxation, rather than radiative relaxation to the ground state. The model considered has relevance to a wide variety of materials, such as linear molecular aggregates, conjugated polymers, and polysilanes.

Intraband relaxation and temperature dependence of the fluorescence decay time of one-dimensional Frenkel excitons: The Pauli master equation approach

In molecular J-aggregates one often observes an increase of the fluorescence decay time when increasing the temperature from 0 K. This phenomenon is usually attributed to the thermal population of the dark Frenkel exciton states that lie above the superradiant bottom state of the exciton band. In this paper, we study this effect for a homogeneous one-dimensional aggregate in a host medium and we model the scattering between different exciton states as arising from their coupling to the host vibrations. A Pauli master equation is used to describe the redistribution of excitons over the band. The rates entering this equation are calculated within the framework of first-order perturbation theory, assuming a linear on-site interaction between excitons and acoustic phonons. Solving the master equation numerically for aggregates of up to 100 molecules, we calculate the temperature dependence of the fluorescence kinetics in general and the decay time scale in particular. The proper definition of the fluorescence decay time is discussed in detail. We demonstrate that, even at a quantum yield of unity, the possibility to directly interpret fluorescence experiments in terms of a simple radiative time scale depends crucially on the initial excitation conditions in combination with the competition between spontaneous emission and intraband phonon-assisted relaxation.

DISORDER-INDUCED RELAXATION OF FRENKEL EXCITONS IN MOLECULAR AGGREGATES

A model, based on the intraband scattering of excitons in one-dimensional J-aggregates is proposed to describe the lengthening of the experimentally observed radiative lifetime with temperature. According to this mechanism, the exciton-phonon scattering transfers the oscillator strength fromt he lowest k≅0 optically allowed state to the other states within the excitonic band. A Pauli master equation, in which the hopping rates are calculated, is used to describe the thermalization of the excitonic band. Assuming a fast relaxation mechanism, the temperature dependence of the exciton radiative lifetime is simulated for various chain lengths.

Disorder-induced temperature dependence of the radiative lifetime of Frenkel excitons in molecular aggregates

We calculate the temperature dependence of the fluorescence decay of a 1D localized Frenkel exciton resulting from the thermal redistribution of population over the localized states. The Pauli master equation is used to describe the exciton hopping.

FROM THE VON-NEUMANN EQUATION TO THE QUANTUM BOLTZMANN EQUATION IN A DETERMINISTIC FRAMEWORK

In this paper, we investigate the rigorous convergence of the Density Matrix Equation (or Quantum Liouville Equation) towards the Quantum Boltzmann Equation (or Pauli Master Equation). We start from the Density Matrix Equation posed on a cubic box of size L with periodic boundary conditions, describing the quantum motion of a particle in the box subject to an external potential V. The physics motivates the introduction of a damping term acting on the off-diagonal part of the density matrix, with a characteristic damping time α−1. Then, the convergence can be proved by letting successivelyL tend to infinity and α to zero. The proof relies heavily on a lemma which allows to control some oscillatory integrals posed in large dimensional spaces. The present paper improves a previous announcement [CD].

The electron state occupation and photoluminescence spectra from nanoscale Si particles

The occupancy of higher electron energy levels under optical excitation is obtained by solving the Pauli master equation. The shift in peak positions with reducing average particle sizes L 0 for the PL spectra from nanoscale Si particles and luminescence centers in the oxide layer is different. The critical scale is estimated under a definite excitation. The critical scale is not a fixed size at which the PL intensity from Si and LCs equal to each other. On the contrary, it decreases for stronger excitation.

Picosecond time-resolved fluorescence studies on excitation energy transfer in a histidine 117 mutant of the D2 protein of photosystem II in Synechocystis 6803.

The role of the peripheral reaction center chlorophyll a molecule associated with His117 of the D2 polypeptide in photosystem II was investigated in Synechocystis sp. PCC 6803 using a combination of steady state, pump-probe, and picosecond time-resolved fluorescence spectroscopy. Data were obtained from intact cells and isolated thylakoid membranes of a control mutant and a D2-H117T mutant, both of which lacked photosystem I. Excitation energy transfer and trapping were investigated by analyzing the data with a kinetic model that used an exact numerical solution of the Pauli master equation, taking into account available photosystem II spectral and structural information. The results of our kinetic analysis revealed the observed difference in excited-state dynamics between the H117T mutant and the control to be consistent with a retardation of the rate of excitation energy transfer from the peripheral chlorophyll of D2 (Chl at His117) to the electron-transfer pigments and an increase of the rate constant for charge recombination in the H117T mutant. The kinetic model was able to account for the experimentally observed changes in absorption cross section and fluorescence decay kinetics between the control and mutant by invoking changes in only these two rate constants. The results rule out quenching of excitation by a chlorophyll cation radical as a mechanism responsible for the lower efficiency of excitation energy utilization in the H117T mutant. Our work also demonstrates the importance of the chlorophyll associated with His117 of the D2 protein for excitation energy transfer to the PSII electron-transfer pigments and for the effective stabilization of the primary radical pair.

Energy Transfer among CP29 Chlorophylls: Calculated Förster Rates and Experimental Transient Absorption at Room Temperature

The energy transfer rates between chlorophylls in the light harvesting complex CP29 of higher plants at room temperature were calculated ab initio according to the Förster mechanism (Förster T. 1948, Ann. Physik. 2:55–67). Recently, the transition moment orientation of CP29 chlorophylls was determined by differential linear dichroism and absorption spectroscopy of wild-type versus mutant proteins in which single chromophores were missing (Simonetto R., Crimi M., Sandonà D., Croce R., Cinque G., Breton J., and Bassi R. 1999. Biochemistry. 38:12974–12983). In this way the Q y transition energy and chlorophyll a/b affinity of each binding site was obtained and their characteristics supported by reconstruction of steady-state linear dichroism and absorption spectra at room temperature. In this study, the spectral form of individual chlorophyll a and b ligands within the protein environment was experimentally determined, and their extinction coefficients were also used to evaluate the absolute overlap integral between donors and acceptors employing the Stepanov relation for both the emission spectrum and the Stokes shift. This information was used to calculate the time-dependent excitation redistribution among CP29 chlorophylls on solving numerically the Pauli master equation of the complex: transient absorption measurements in the (sub)picosecond time scale were simulated and compared to pump-and-probe experimental data in the Q y region on the native CP29 at room temperature upon selective excitation of chlorophylls b at 640 or 650 nm. The kinetic model indicates a bidirectional excitation transfer over all CP29 chlorophylls a species, which is particularly rapid between the pure sites A1-A2 and A4-A5. Chlorophylls b in mixed sites act mostly as energy donors for chlorophylls a, whereas site B5 shows high and bidirectional coupling independent of the pigment hosted.

The Dynamics of Electronic Energy Transfer in Novel Multiporphyrin Functionalized Dendrimers: A Time-Resolved Fluorescence Anisotropy Study

The dynamics of electronic energy transfer (EET) for a series of spherical porphyrin arrays based on different generations of poly(propylene imine) dendrimers have been investigated using time-resolved fluorescence anisotropy measurements (TRAMS) in a glass environment. The first, third, and fifth generation dendrimers consisting of 4, 16, and 64 porphyrin chromophores, respectively, are investigated in this study. We observe a depolarization of the fluorescence in all three dendrimers as compared to the monoporphyrin model compound, indicating that EET takes place between the chromophores within the dendrimers. The experimental TRAMS results were compared to computationally simulated data obtained from the Pauli master equation. For the first generation dendrimer, we find the rate of energy transfer is well described by Forster theory. Anomalous behavior is observed in the third generation dendrimer where the limiting anisotropy value suggests that energy transfer is confined to only the porphyrins conta...

Quantum-mechanical tunneling in associative neural networks

We investigate the quantum-mechanical tunneling between the “patterns" of the, so-called, associative neural networks. Being the relatively stable minima of the “configuration-energy" space of the networks, the “patterns" represent the macroscopically distinguishable states of the neural nets. Therefore, the tunneling represents a macroscopic quantum effect, but with some special characteristics. Particularly, we investigate the tunneling between the minima of approximately equal depth, thus requiring no energy exchange. If there are at least a few such minima, the tunneling represents a sort of the “random walk" process, which implies the quantum fluctuations in the system, and therefore “malfunctioning" in the information processing of the nets. Due to the finite number of the minima, the “random walk" reduces to a dynamics modeled by the, so-called, Pauli master equation. With some plausible assumptions, the set(s) of the Pauli master equations can be analytically solved. This way comes the main result of this paper: the quantum fluctuations due to the quantum-mechanical tunneling can be “minimized" if the “pattern"-formation is such that there are mutually “distant" groups of the “patterns", thus providing the “zone" structure of the “pattern" formation. This qualitative result can be considered as a basis of the efficient deterministic functioning of the associative neural nets.

Localization length and intraband scattering of excitons in linear aggregates

A theoretical model to describe the intraband scattering of excitons in linear aggregates of finite size which exhibit strong intermolecular interactions is presented. From the calculation of the aggregate eigenstates, the localization length of excitons is evaluated for various configurations featuring physical situations like trapping, edge effects, inclusion of diagonal and/or orientational disorders. The intraband scattering is studied by considering the exciton–phonon stochastic coupling induced by the thermal bath. This coupling creates local dynamical fluctuations in the site energies which are characterized by their amplitude ( Δ) and their correlation time ( τ c). Expressions of scattering rates are provided and used in a Pauli master equation to calculate the time dependence of the eigenstates populations after initial excitation of the quasi exciton-band. It is shown that the time evolution of the lowest state population as well as the Stokes shift strongly depend on τ c. Comparison of the theoretical results to time-resolved experiments performed on triaryl pyrylium salts allows us to interpret the observed Stokes shift and to derive an average value of the exciton–phonon correlation time.