Generalized Master Equation Research Articles

Generalized master equation for a two-level atom in a strong field and tailored reservoirs

Generalized master equation for a two-level atom in a strong field and tailored reservoirs

Generalized master equation for a two-level atom in a strong field and tailored reservoirs

Generalized master equation for a two-level atom in a strong field and tailored reservoirs

Generalized master equation for a two-level atom in a strong field and tailored reservoirs

The generalized master equation for a two-level atom driven by a strong classical field and damped into a ‘tailored’ reservoir with a non-flat density of modes is derived under the Born-Markov approximation. To derive the master equation the dressing transformation on the atomic operators is performed first and next the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The modifications introduced by a strong field and/or by the reservoir with the non-flat density of modes lead to non-standard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs. The optical Bloch equations based on this generalized master equation are obtained and solved for the steady state. The solutions are discussed from the point of view of both bare and dressed atoms. Analytical formulas for the fluorescence and probe absorption spectra are obtained and illustrated graphically.

Non-homogeneous random walks, generalised master equations, fractional Fokker-Planck equations, and the generalised Kramers-Moyal expansion

A generalised random walk scheme for random walks in an arbitrary external potential field is investigated. From this concept which accounts for the symmetry breaking of homogeneity through the external field, a generalised master equation is constructed. For long-tailed transfer distance or waiting time distributions we show that this generalised master equation is the genesis of apparently different fractional Fokker-Planck equations discussed in literature. On this basis, we introduce a generalisation of the Kramers-Moyal expansion for broad jump length distributions that combines multiples of both ordinary and fractional spatial derivatives. However, it is shown that the nature of the drift term is not changed through the existence of anomalous transport statistics, and thus to first order, an external potential Φ(x) feeds back on the probability density function W through the classical term ∝∞ ∂/∂ xФ´(x)W(x, t), i.e., even for Levy flights, there exists a linear infinitesimal generator that accounts for the response to an external field.

A new formula of exciton coherence length in linear array of pigments

By solving the generalized master equation with a memory function properly expressed by using the time-correlation function of the exciton-vibration interaction, we derived an empirical formula for the exciton coherence length Ncoh in the linear array of pigments as a function of exciton coupling U and the decay factor for excitation transfer g as follows: Ncoh = 1.40 + 1.32 (U/g ) Adopting the intermediate coupling theory of exciton in dimer (Kimura et al., J. Phys. Chem. B 104 (2000) 9276), we can express the decay factor g by using the reorganization energy l and energy gap D G due to the excitation transfer between neighboring pigments. Applying this general formula to the B850 ring of LH2 of photosynthetic purple bacteria (U = 350 cm-1, l = 80 cm-1 and so g = 140 cm-1), we obtain Ncoh = 4.7. When we evaluate g by considering the Gaussian inhomogeneous broadening (HWHM 200 cm-1), we obtain Ncoh = 3.2. Namely, the exciton domain extends only 3-4 BCl units among the 18 BCl units of B850 in LH2.

Exact solution to the general non-radiative energy transfer master equations in crystalline materials

The microscopic general non-radiative energy transfer master equations are solved for the case of co-doped crystalline materials. With the calculated exact solution, the macroscopic dopant emissions are obtained. In particular, it is shown that our solution unifies the predictions for the donor emission reported by other models for the cases of direct energy transfer and migration among donors. In particular, we refer to the Forster–Dexter, D.L. Huber, Yokota–Tanimoto, and A.I. Burshtein models. Further, our modeling includes the back-transfer and migration among acceptor processes.

Dynamical principles in biological processes: a model of charge migration in proteins and DNA.

The generalized master equations (GMEs) that contain multiple time scales have been derived quantum mechanically. The GME method has then been applied to a model of charge migration in proteins that invokes the hole hopping between local amino acid sites driven by the torsional motions of the floppy backbones. This model is then applied to analyze the experimental results for sequence-dependent long-range hole transport in DNA reported by Meggers et al. [Meggers, E., Michel-Beyerle, M. E., & Giese, B. (1998) J. Am. Chem. Soc. 120, 12950-12955]. The model has also been applied to analyze the experimental results of femtosecond dynamics of DNA-mediated electron transfer reported by Zewail and co-workers [Wan, C., Fiebig, T., Kelley, S. O., Treadway, C. R., Barton, J. K. & Zewail, A. H. (1999) Proc. Natl. Acad. Sci. USA 96, 6014-6019]. The initial events in the dynamics of protein folding have begun to attract attention. The GME obtained in this paper will be applicable to this problem.

Two-level atom-field interaction: Exact master equations for non-Markovian dynamics, decoherence, and relaxation

We perform a first- principles derivation of the general master equation to study the non-Markovian dynamics of a two-level atom (2LA) interacting with an electromagnetic field (EMF). We use the influence functional method which can incorporate the full backreaction of the field on the atom, while adopting Grassmannian variables for the 2LA and the coherent state representation for the EMF. We find exact master equations for the cases of a free quantum field and a cavity field in the vacuum. In response to the search for mechanisms to preserve maximal coherence in quantum computations in ion trap prototypes, we apply these equations to analyse the decoherence of a 2LA in an EMF, and fine that decoherence time is close to relaxation time. This is at variance to the claims by authors who studied the same system but used a different coupling model. We explain the source of difference and argue that, contrary to common belief, the EMF when resonantly coupled to an atom does not decohere it as efficiently as a bath does on a quantum Brownian particle. The master-equations for non-Markovian dynamics derived here is expected to be useful for exploring new regimes of 2LA-EMF interaction, which is becoming physically important experimentally.

Generalised Master Equation for the Spin-Boson Hamiltonian by the Projection Operator Technique

Starting from the von Neuman equation of quantum theoretical dynamics, by using the projection operator technique, we write the generalised master equation (GME) of the reduced statistical operator. By using this technique we find a series expansion of the memory kernel which allows, in principle, a treatment of the higher orders of approximation. By keeping only the first term of this expansion, which corresponds to the second order of the perturbation treatment, GME is applied, in the Markov approximation, to the spin boson Hamiltonian to describe the dynamics of the relevant pseudo-spin subsystem. In the secular and rotating wave approximations this dynamics is found to be described by the Bloch equations.

Random walks in one-dimensional environments with feedback-coupling

Random walks in one-dimensional environments with an additional dynamical feedback-coupling is analyzed numerically. The feedback introduced via a generalized master equation is controlled by a memory kernel of strength λ the explicit form of which is motivated by arguments used in mode-coupling theories. Introducing several realizations of the feedback mechanism within the simulations we obtain for a negative memory term, λ <, superdiffusion in the long time limit while a positive memory leads to localization of the particle. The numerical simulations are in agreement with recent predictions based on renormalization group techniques. A slight modification of the model including an exponentially decaying memory term and some possible applications for glasses and supercooled liquids are suggested. The relation to the true self-avoiding is discussed.

Incoherent Zener tunneling and its application to molecular magnets

We generalize the Landau-Zener theory of coherent tunneling transitions by taking thermal relaxation into account. The evaluation of a new generalized master equation containing a dynamic tunneling rate that includes the interaction between the relevant system and its environment leads to an incoherent Zener transition probability with an exponent that is twice as large as the one of the coherent Zener probability in the limit T -> 0. We apply our results to molecular clusters, in particular to recent measurements of the tunneling transition of spins in Fe8 crystals performed by Wernsdorfer and Sessoli [Science 284, 133 (1999)].

Spin tunneling and phonon-assisted relaxation inMn12-acetate

We present a comprehensive theory of the magnetization relaxation in a Mn12-acetate crystal in the high-temperature regime (T>1 K), which is based on phonon-assisted spin tunneling induced by quartic magnetic anisotropy and weak transverse magnetic fields. The overall relaxation rate as function of the longitudinal magnetic field is calculated and shown to agree well with experimental data including all resonance peaks measured so far. The Lorentzian shape of the resonances, which we obtain via a generalized master equation that includes spin tunneling, is also in good agreement with recent data. We derive a general formula for the tunnel splitting energy of these resonances. We show that fourth-order diagonal terms in the Hamiltonian lead to satellite peaks. A derivation of the effective linewidth of a resonance peak is given and shown to agree well with experimental data. In addition, previously unknown spin-phonon coupling constants are calculated explicitly. The values obtained for these constants and for the sound velocity are also in good agreement with recent data. We show that the spin relaxation in Mn12-acetate takes place via several transition paths of comparable weight. These transition paths are expressed in terms of intermediate relaxation times, which are calculated and which can be tested experimentally.

Purity and decoherence in the theory of a damped harmonic oscillator.

For the generalized master equations derived by Karrlein and Grabert for the microscopic model of a damped harmonic oscillator, the conditions for purity of states are written, in particular for different initial conditions and different types of damping, including Ohmic, Drude, and weak coupling cases, and the Agarwal and Weidlich-Haake models. It is shown that the states which remain pure are the squeezed states with variances that are constant in time. For pure states, generalized nonlinear Schrödinger-type equations corresponding to these master equations are also obtained. Then the condition for purity of states of a damped harmonic oscillator is considered in the framework of Lindblad theory for open quantum systems. For a special choice of the environment coefficients, correlated coherent states with constant variances and covariance are shown to be the only states which remain pure all the time during the evolution of the considered system. In Karrlein-Grabert and Lindblad models, as well as in the particular models considered, expressions for the rate of entropy production are written, and it is shown that state which preserve their purity in time are also states which minimize entropy production and, therefore, are the most stable state under evolution in the presence of the environment, and play an important role in the description of decoherence phenomenon.

Two-photon absorption in strong fields

We present a theoretical formalism to describe simultaneous absorption of two single-mode photons by an atom. The appropriate interaction Hamiltonian, derived by a unitary transformation, is used to obtain a very general master equation for the photon density operator that includes the effects of Stark shifts. The special case in which the atomic levels are in resonance with the field, i.e., zero detuning, is considered. The master equation for the photon probability predicts a linear dependence of absorption on intensity, rather than the usual quadratic dependence, and results in a slower rate of absorption as a function of time. Coherence in the presence of two-photon absorption is also demonstrated.

Neuronal Oscillatory Network with Stored Patterns**The project supported by National Basic Research Project “Nonlinear Science”

We consider a neuronal oscillatory network in which the interactions among neurons are constructed by Hebb rule with finite stored patterns. A general master equation is obtained by dividing the lattice into sublattices. In some simple cases, the dynamics of the system is explicitly solved.

Excitation energy transfer mechanisms in allophycocyanin (I)

The mechanisms of excitation energy transfer within allophycocyanin monomer with the theory of generalized master equation (GME) and the technique of time-resobed fluorescence anisotropic spectroscopy are studied. In the case of known information of its structure and spectra, the theory applied is based on the assumption that the coupling interaction between two chromophores is fairly weak. The theory correctly predicts the experimentlly observed rate for excitation energy transfer in allophycocyanin monomer. Based on the results, the energy transfer mechanism can be described as Forster and these processes cannot take place from the high vibrational levels of donor to acceptor.

A thermal bath induced new resonance on the linear and nonlinear spectra of a two-level system

Within the scope of generalized master equation we have shown the effect of bath with finite band width on the linear absorption and on the resonance fluorescence and absorption spectra of a driven two-state system. It is found that depending on the band width and temperature of the bath a reversible dynamics may set in even in the case of linear absorption induced by weak field. At higher temperature the bath induces new resonance on the spectral profiles.

Variable-range hopping: Role of Coulomb interactions

The effect of Coulomb interactions on hopping conduction in the variable-range hopping regime is analyzed within a linear-response formalism. Here the conductivity and the dielectric function are related to the density-density response function for which a generalized master equation (GME) can be derived using the Mori-Zwanzig projector formalism. The GME can be thought of as a random resistor network with frequency-dependent internode conductances, whose values can be determined from a function related to the current-current correlator at the two nodes. We evaluate the internode conductances using a diagrammatic perturbation formalism. For a single electron hop with all the other charges frozen, we obtain hop rates correct to all orders in Coulomb interaction. This gives us a finite temperature generalization of existing results for the interacting system. We then incorporate relaxation effects that accompany electron hops, using a dynamical model of the Coulomb gap. We argue that the parameter that governs the local relaxation is related to the conductivity itself. These internode conductances are then used to calculate the dc conductivity of the network by effective- medium approximation. We show that a crossover from Efros-Shklovskii's ${T}^{1/2}$ behavior to Mott's ${T}^{1/4}$ behavior occurs due to the relaxation effects, as the temperature is increased. At low temperatures the relaxation is slow so that electrons hop in a frozen charge background and thereby sense the Coulomb gap. This gives the ${T}^{1/2}$ behavior. At higher temperatures the relaxation gets faster and the Coulomb gap is alleviated leading to Mott's behavior.

RELAXATION IN TWO-LEVEL LONG-RANGE-MEMORY SYSTEMS

Recently, new rigorous results concerning integrals of exact memory functions in the time-convolution Generalized Master Equations (GME) for state occupation probabilities, governing relaxation of open quantum systems, have been obtained. They include that a) time integrals of exact memories \(\int_0^{ + \infty } \user1{w} ij(t){\text{d}}t = 0\) and b) memories wij(t) have tails unobtainable by perturbational arguments which cause that \(\int_0^{ + \infty } {\user1{t} \cdot \user1{w}} ij(t){\text{d}}t\) does not exist or is infinite. For a two-level system, a simple model for such memories is considered and solved. It is concluded that GME may yield that with increasing time, the system unphysically more and more deviates from equilibrium, indicating thus instability of the equilibrium distribution. Thus, in contrast to, e.g., the famous Boltzmann equation, the mathematical structure of GME alone does not guarantee the stability of the equilibrium state.

Stochastic Wave Function Approach to Generalized Master Equations

A generalization of the stochastic wave function method is presented that allows the unraveling of arbitrary linear quantum master equations that are not necessarily in Lindblad form and, moreover, the explicit treatment of memory effects by employing the time-convolutionless projection operator technique. The crucial point of this construction is the description of the open system in a doubled Hilbert space, which has already been successfully used for the computation of multitime correlation functions.