A mathematical simulation of intramolecular proton phototransfer

A system of equations for the matrix elements of the density operator of a seven-level model molecule interacting with a light pulse was solved numerically to determine the time dependences of the populations of molecule states at various radiation pulse parameters and parameters characterizing radiative and nonradiative spontaneous molecule transitions and reversible transitions between some of its states. The results were used to characterize the photoisomerization of molecules between states with different positions of the proton of the intramolecular H-bond (the keto and enol forms). Examples of oscillating molecular state population modulation in isomer-isomer tunnel proton shifts are given. Changes in the development of photoionization in time as molecular parameters and radiation pulse width and intensity changed were considered. An analysis of the results obtained is an example of the use of mathematical simulation of intramolecular dynamics for increasing the effectiveness of using spectral-time data in the determination of the mechanism of proton phototransfer in molecules with intramolecular H-bonds     

Mathematical simulation of phototransfer of the proton of a hydrogen bond under irradiation of a molecule by two light pulses

The time-dependences of the populations of molecular states are determined by the numerical solution of a set of equations for the matrix elements of a density operator for the five-level model of a molecule with different values of parameters of two nonoverlapping in time irradiation pulses and of the constants governing the rates of induced radiative transitions of the molecule, as well as the radiative and nonradiative decays of molecular states. It is proposed to use the results obtained as reference points in interpreting experimental data on the spectro-temporal characteristics of secondary emission upon photoisomerization of molecules with an intramolecular hydrogen bond under irradiation by two light pulses and in determining the mechanism of phototransfer of the hydrogen-bond proton occurring in this process.     

Numerical simulation of the intramolecular dynamics of photoisomerization

The results of mixed quantum-classical and quantum-mechanical numerical calculations of the intramolecular dynamics of photoisomerization under conditions similar to ordinary natural conditions, i.e., for irradiation of the molecule by a light pulse not shorter than the lifetime of the resonant excited electronic state of the molecule and with an intensity comparable to that of solar light at the Earth’s surface, are presented. It was concluded that the dynamics of such photoisomerization should be modeled using quantum-mechanical methods. The simplest approach to modeling the photoisomerization of a molecule with two isomeric forms can be based on the density matrix formalism for describing the interaction of a light pulse with a three-level system of Λ configuration.     

Quantum-classical simulation of the photoisomerization during the transformation of ultrashort light pulses by a molecule

The dynamics of photoisomerization of a model molecule during its transformation of ultrashort (with a duration much shorter than the lifetime of the resonant excited electronic state) light pulses is simulated numerically. The two-level electronic subsystem of the molecule is described using the quantum theory, while the nuclear subsystem (taking into account the two isomeric states of the molecule) and the radiation field are described using the classical theory. The ranges of the carrier frequency, the peak intensity, and the durations of nπ sinusoidal pulses (n = 1–10) irradiation with which results in the photoisomerization of molecules of the type under study (for example, cyanine dyes) are determined from the analysis of solutions to self-consistent equations that describe the motion of the “isomerization oscillator” and the time evolution of the population amplitude of the resonant electronic state of the molecule. Each of these non-overlapping ranges corresponds to a particular value of n. Bifurcation values of the above parameters of the light pulse are boundaries of these ranges.     

Numerical simulation of intramolecular dynamics during dual fluorescence

The dynamics of transformation of a light pulse by a five-level model molecule whose secondary emission spectrum can contain two fluorescence bands is simulated. The system of equations that determine the time behavior of the matrix elements of the statistical operator of the molecule interacting with the light pulse is numerically solved. From this solution, the time dependences of the populations of the molecular states are determined for different values of the parameters of the irradiation pulse, which is described in terms of the classical theory, and of the parameters that characterize the rates of radiative and nonradiative spontaneous transitions of the molecule. Based on particular examples of the choice of these parameters, it is demonstrated that the mechanism by which dual fluorescence occurs in molecules with intramolecular hydrogen bonds can be efficiently established from the numerically simulated intramolecular dynamics.     

Collective electronically excited states of a uniform chain of atoms

Electronically excited states of finite uniform chains of atoms were considered taking into account the influence of the continuous energy spectrum. Traditional quantum-chemical methods for calculating two-electron transitions between neighboring chain atoms were combined with the asymptotic theory of interactions between excited atoms and neutral particles and the mathematical apparatus of the theory of multiple scattering for taking into account intercenter transitions in an ensemble of interacting centers. Recurrence equations for describing energy zones containing symmetrical and antisymmetric excited state levels of chains with an arbitrary length were obtained. Depending on system parameters, different modes of the distribution of the electron density of collective excited states were possible. At a certain ratio between level shifts and exchange integral values, excited states with a uniform electron density distribution over all chain nodes could form for certain solutions. This was a fortuitous circumstance caused by the influence of the continuous spectrum. Such states appeared at small principal quantum number n values, they were similar to one-electron excitations of the type of Frenkel excitons, when an electron was localized near its Coulomb center. These conditions were rapidly disturbed as n increased, and one-electron excitations of a linear molecule were formed in the system (that is, limiting excitations of the type of Wannier-Mott excitons did not form).     

Nonresonant interaction of femtosecond laser pulse with centrosymmetric molecules

We have derived a system of equations that describes the evolution of the density matrix of a centrosymmetric molecule interacting with a single nonresonant femtosecond laser pulse. The dynamics of the ground electronic state is expressed in terms of the effective Hamiltonian and the coherences between the ground an excited electronic states are functionals of the ground state density matrix. Using the time-dependent perturbation theory, we have calculated the energy deposited in the molecule as a result of rotational stimulated Raman scattering. The effective absorption coefficient is found to be proportional to the fourth power of the pulse amplitude and has a resonance-like dependence with respect to the pulse duration.     

Quantum theory of open systems based on stochastic differential equations of generalized Langevin (non-Wiener) type

It is shown that the effective Hamiltonian representation, as it is formulated in author??s papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are ??locked?? inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.     

Mathematical modeling of the population dynamics of “dark” states of a three-level system

Mathematical modeling of the population dynamics is performed for states of a three-level system (atom) with a V-type configuration transforming a light pulse. It is assumed that the excited eigenstates of the atom are degenerate and coupled by coherent interaction, one of the states being radiating (radiative), while the other state is nonradiating (??dark??). The population dynamics of atomic states is described on the basis of numerical solutions of equations for the matrix elements of the density operator. The dependence of the efficiency of population of the atomic dark state from the values of the parameters of an irradiation pulse and from the ratio of the period of population oscillations of excited atomic states (caused by their coherent interaction) to the lifetime of the atomic radiating state is determined. Typical examples of the time dependence of the population of states of the atom considered are presented for the cases of irradiation by a short (as compared to the lifetime of the radiating state) sinusoidal light pulse and by a long rectangular light pulse with the resonance carrier frequency.     

Study of low-lying collective states using a Microscopic Anharmonic Vibrator Approach

Anharmonic features of the low-lying collective states in the ruthenium isotopes have been investigated systematically by using the Microscopic Anharmonic Vibrator Approach (MAVA). MAVA is based on a realistic microscopic G-matrix Hamiltonian, only slightly renormalized in the adopted large realistic single-particle spaces. This Hamiltonian is used to derive equations of motion for the mixing of one-and two-phonon degrees of freedom starting from collective phonons of the quasiparticle random-phase approximation. Interesting applications to analysis of double-beta-decay rates to excited collective states are foreseen. Presented by J. Suhonen at the Workshop on calculation of double-beta-decay matrix elements (MEDEX’05), Corfu, Greece, September 26–29, 2005.     

Nonclassical effects in a highly nonlinear generalized homogeneous Dicke model

An intensity dependent nonlinear coupling model of N two-level atoms (generalized Dicke model) interacting dispersively with a bimodal cavity field via two-photon transitions is investigated in a scenario where the rotating wave approximation is assumed. The model becomes homogeneous in the sense that the spin transition frequency is the same for all atoms and the coupling constants emerging from the collective interactions of the atomic system with the cavity field depend only on the particular radiation field mode. This allows us to represent the Dicke Hamiltonian entirely in terms of the total angular momentum J. It is assumed that, initially, the atomic system and the field are in a disentangled state where the field modes are in Glauber coherent states and the atomic system is a superposition of states |JM〉 (Dicke states). The model is numerically tested against simulations of normal squeezing variance of the field, squeezing factors based on the Heisenberg uncertainty principle, along with the statistical properties of the light leading to the possible production of nonclassical effects, such as degree of second-order coherence in the modes, degree of intermode correlation, as well as violation of the Cauchy–Schwartz inequality. Analytical expression of the total density operator matrix elements at t>0 shows the present nonlinear model to be strongly entangled, which is reflected in the time evolution of the linear entropy, where the superposition states are reduced to statistical mixtures. Thus, the present generalized Dicke model does not preserve the modulus of the Bloch vector. The computations, performed in the weak coupling and strong field limits, were conducted via second-order Dyson perturbative expansion of the time evolution operator matrix elements for the totality of the angular momentum states of the atomic system.     

A time convolution less density matrix approach to the nonlinear optical response of a coupled system-bath complex

Time convolution less density matrix theory (TCL) is a powerful and well established tool to investigate strong system-bath coupling for linear optical spectra. We show that TCL equations can be generalised to the nonlinear optical response up to a chosen order in the optical field. This goal is achieved via an time convolution less perturbation scheme for the reduced density matrices of the electronic system. In our approach, the most important results are the inclusion of a electron-phonon coupling non-diagonal in the electronic states and memory effects of the bath: First, the considered model system is introduced. Second, the time evolution of the statistical operator is expanded with respect to the external optical field. This expansion is the starting point to explain how a TCL theory can treat the response up to in a certain order in the external field. Third, new TCL equations, including bath memory effects, are derived and the problem of information loss in the reduced density matrix is analysed. For this purpose, new dimensions are added to the reduced statistical operator to compensate lack of information in comparison with the full statistical operator. The theory is benchmarked with a two level system and applied to a three level system including non-diagonal phonon coupling. In our analysis of pump-probe experiments, the bath memory is influenced by the system state occupied between pump and probe pulse. In particular, the memory of the bath influences the dephasing process of electronic coherences developing during the time interval between pump and probe pulses.     

Variance squeezing and information entropy squeezing via Bloch coherent states in two-level nonlinear spin models

The nonclassical squeezing effect emerging from a nonlinear coupling model (generalized Jaynes–Cummings model) of a two-level atom interacting resonantly with a bimodal cavity field via two-photon transitions is investigated in the rotating wave approximation. Various Bloch coherent initial states (rotated states) for the atomic system are assumed, i.e., (i) ground state, (ii) excited state, and (iii) linear superposition of both states. Initially, the atomic system and the field are in a disentangled state, where the field modes are in Glauber coherent states via Poisson distribution. The model is numerically tested against simulations of time evolution of the based Heisenberg uncertainty relation variance and Shannon information entropy squeezing factors. The quantum state purity is computed for the three possible initial states and used as a criterion to get information about the entanglement of the components of the system. Analytical expression of the total density operator matrix elements at t > 0 shows, in fact, the present nonlinear model to be strongly entangled, where each of the definite initial Bloch coherent states is reduced to statistical mixtures. Thus, the present model does not preserve the modulus of the Bloch vector.     

On electromagnetic properties of rotating nuclei in crpa

An analysis of quasiparticle correlations with special emphasis on transition matrix elements have been done for a self-consistent cranking model. It is pointed out that a second order of the boson representation of a transition operator leads to a signature dependence of transition probabilities between excited one-phonon states in even-even nuclei. Moreover, it brings about the new contribution to an expectation value of electromagnetic moments in the yrast line states of deformed nuclei.     

Alternative generalized master equation approach to the dc phonon-assisted hopping

As the momentum operator has no diagonal elements between localized states, the hopping conduction theory should be formulated in terms of the linear response of the site-off-diagonal elements of the single-electron density matrix to an external field. A theory of this kind, starting from generalized master equations and yielding the dc phonon-assisted hopping conductivity and thermopower is formulated. This is an alternative to the usual approach treating the current conduction via a time-derivative of the electric dipole momentum.Stimulating discussions with Dr. B. Velický turning the present author's attention to his own old ideas about the hopping conduction via off-diagonal elements of the single-electron density matrix are appreciated.     

Effect of the vibrational pattern of out-of-plane vibrational modes on vibronically induced spin-orbit coupling between ππ* states involved in nonradiative intersystem crossing transitions

To describe nonradiative intersystem crossing between excited ππ* states, the matrix elements of the operator of spin-orbit interaction are calculated. Adiabatic electronic functions that depend on out-of-plane normal vibrational coordinates are used as a zero approximation. Using dibenzofuran and tetrachloro-substituted dioxin as examples and taking into account all the out-of-plane vibrational modes, the effects of the vibrational patterns of these modes on the matrix elements under consideration, as well as of the spin-orbit coupling in carbon, oxygen, and chlorine atoms of individual atomic groups of the molecule, are discussed.     

Electron spin polarization in a rotating triplet

The triplet model of electron spin polarization in fluid media is evaluated. The model consists of an initial singlet molecule rotating in a static, externally applied magnetic field. Intersystem crossing into different zero-field states is represented by a rate matrix diagonal in the molecular frame, and this matrix is expressed as an effective spin operator. The triplet rotates, and the motion affects the polarization in the laboratory frame, and also causes spin relaxation in the triplet manifold. The triplet is chemically quenched, and the polarization appears in the doublet fragments. The model is treated in a density matrix formalism and on the basis of anisotropic rotational diffusion of the triplet molecule. Explicit expressions are obtained in terms of the molecular parameters, the various rate constants, and the rotational correlation time.