K. Lendi

Influence of a magnetic field on delayed fluorescence of aromatic hydrocarbons in solution: II. A theoretical approach

Abstract The calculations are based on the following model: Triplet pair states are formed by diffusion controlled encounters of molecules in the triple state. The pair state is characterized by JQ, the energetic splitting of its singlet and quintet component, by the rate constants λS, λT and λQ for the transitions to singlet, triplet and quintet product states as well as by k1, the rate constant for redissociation into non-interacting molecular triplets. The zero field splittings of the molecular triplets are taken from ESR measurements. The theoretical treatment is based on a generalized Liouville equation for the triplet pair density matrix and some of the mathematical techniques are explained in detail. In fitting the theoretical calculations to experimental results for excimers (Part I) it is seen that (a) JQ must be smaller than 1010 s− (angular frequency), (b) k−1 is proportional to the absolute temperature divided by the viscosity of the solvent, (c) λQ cannot be put equal to zero. The model suggests: (a) The pair state has to be considered as a rather loose complex of two molecules in the triplet state. (b) Excimers are predominantly formed from pairs having neear sandwich structure. (c) In the molecules investigated (pyrene, 1,2-benzanthracene, phenanthrene, 3,4-benzpyrene) a quintet state exists, the energy of which is lower than twice the triplet energy. The magnetic field effects on the monomer delayed fluorescence seem to arise from randomly oriented triplet pairs.

On quantum beats in polyatomic molecules

Abstract Very recently, the first true intramolecular quantum beats have experimentally been observed in the fluorescence of a polyatomic molecule whereas general theoretical predictions have already been made many years ago. In this paper, particular attention is payed to specific singlet—triplet systems where quantum beats as due to intersystem crossing may manifest themselves not only in the fluorescence but also in the phosphorescence. The theoretical analysis is based on a very simple, exactly solvable model which accounts for the relevant key features such as multiexponential decay and superimposed quantum oscillations and permits to discuss frequency and relative beat amplitudes, i.e. the fraction of the oscillatory component within the total luminescence intensity as well as the new phenomenon of a phase shift between fluorescence and phosphorescence beats as a function of the parameters which describe the intramolecular mechanism. One may hope that these considerations provide some guideline in the experimental search for suitable molecules which might exhibit quantum beats. This field is a challenge for future subpicosecond techniques.

Entropy studies for the damped and the undamped Jaynes-Cummings model

On the basis of a representation in terms of photon-number states we derive an analytically solvable set of ordinary differential equations for the matrix elements of the density operator belonging to the Jaynes-Cummings model. We allow for atomic detuning, spontaneous emission, and cavity damping, but we do not take into account the presence of thermal photons. The exact results are employed to perform a careful investigation of the evolution in time of atomic inversion and von Neumann entropy. A factorization of the initial density operator is assumed, with the privileged field mode being in a coherent state. We invoke the mathematical notion of maximum variation of a function to construct a measure for entropy fluctuations. In the undamped case the measure is found to increase during the first few revivals of Rabi oscillations. Hence, the influence of the surroundings on the atom does not decrease monotonically from time zero onwards. A further non-Markovian feature of the dynamics is given by the strong dependence of our measure on the initial atomic state, even for times at which damping brings about irreversible decay. For weak damping and high initial energy density the atomic evolution exhibits a crossover between quasireversible revival dynamics and irreversible Markovian decay. During this stage the state of maximum entropy acts as an attractor for the trajectories in atomic phase space. Subsequently, all trajectories follow a unique route to the atomic ground state, for which the off-diagonals of the atomic density matrix equal zero. From our entropy studies one learns what kind of difficulties must be overcome in establishing formulae for entropy production, the use of which is not limited to semigroup-induced dynamics.

Superoperator formalism in the perturbation theory of density operators obeying generalized liouville equations

Abstract The superoperator formalism is a very convenient tool in the perturbation theory for density - or any operators obeying equations of the general (stochastic) Liouville type. The new concept of a superoperator norm is introduced for the discussion of convergence of perturbation series as well as for the error estimates of finite-order approximations. Explicit formulas for the diagonal elements of the density matrix up to second order are given. The results hold for the stationary state of a system or for its time behavior at small or large times. The present formalism may find useful applications in problems like, e.g., chemically induced polarization phenomena, triplet-triplet annihilation (delayed fluorescence) and laser theory.

Evolution matrix in a coherence vector formulation for quantum Markovian master equations of N-level systems

Quantum Markovian master equation in N dimensions are systematically transformed into vector form by using the Lie algebra of the special unitary group SU(N). Density operators are represented through coherence vectors and the infinitesimal Kossakowski generator of completely positive quantum dynamical semigroups appears as a real evolution matrix that is not completely diagonalisable, in general. A complete classification of the spectrum of the latter and its Jordan canonical form, together with all types of associated stationary states, is given in detail. The results of this analysis are particularly suited for computations in practical applications.

Influence of a magnetic field on delayed fluorescence of aromatic hydrocarbons in solution: III. Application of perturbation theory

Abstract A recently developed superoperator perturbation theory is used to derive an analytical expression from a generalized Liouville equation for the influence of a magnetic field on the delayed fluorescence intensity of molecules in liquid solution. This expression is valid at high temperature (room temperature) and low viscosity and allows for a quantitative explanation of observed magnetic field effects using parameter values which are physically acceptable and compatible with those formerly used by us in the explanation of the low temperature effects. It is particularly shown which relations between the parameters lead to positive or negative low field effects.

Dynamical invariants for time-evolution of open quantum systems in finite dimensions

Various equivalent representations of dynamical invariants (or constants of motion) are derived for N-level systems in mixed states with particular emphasis on so-called coherence vector invariants. They appear as certain homogeneous forms in the real solutions of the von Neumann equation with coefficients given by multilinear forms in the completely symmetric structure constants of the Lie-algebra of SU(N). The treatment is motivated by the close analogy between Lax pairs For classical dynamical systems and pairs given by density and Hamilton operators. In both cases the underlying mathematical structure is essentially determined by the properties of symmetric functions. However, in the quantum case more theorems of general validity can be derived due to the peculiar properties of density operators. In particular, the maximum number of functionally independent invariants in relation to the spectral properties is obtained, as well as bounds on their order of magnitude, the latter from an extremal property analysis. From group-theoretical methods a complete classification of all possibilities of completely incoherent state dynamics is deduced. As a by-product a simple algorithm for the explicit determination of all (N-1)-dimensional irreducible representation matrices of the symmetric group SN and an associated construction of hyperpolyhedra is worked out. Finally, the importance of invariants is stressed for the control of numerical accuracy in large-scale computations in very high dimensions which, after taking partial traces, can be used For the description of irreversible processes. In summary, the results will be of practical relevance for applications to problems of short-time dynamics in molecular laser spectroscopy, quantum optics and magnetic resonance.

Regularization of Quantum Relative Entropy in Finite Dimensions and Application to Entropy Production

The fundamental concept of relative entropy is extended to a functional that is regular-valued also on arbitrary pairs of nonfaithful states of open quantum systems. This regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution. On this basis a generalized formula for entropy production is proposed, the applicability of which is tested in models of irreversible processes. The dynamics of the latter is determined by either Markovian or non-Markovian master equations and involves all types of states.

Virtues and limitations of markovian master equations with a time-dependent generator

A Markovian master equation with time-dependent generator is constructed that respects basic constraints of quantum mechanics, in particular the von Neumann conditions. For the case of a two-level system, Bloch equations with time-dependent parameters are obtained. Necessary conditions on the latter are formulated. By employing a time-local counterpart of the Nakajima–Zwanzig equation, we establish a relation with unitary dynamics. We also discuss the relation with the weak-coupling limit. On the basis of a uniqueness theorem, a standard form for the generator of time-local master equations is proposed. The Jaynes–Cummings model with atomic damping is solved. The solution explicitly demonstrates that reduced dynamics can be described by time-local master equations only on a finite time interval. This limitation is caused by divergencies in the generator. A limit of maximum entropy is presented that corroborates the foregoing statements. A second limiting case demonstrates that divergencies may even occur for small perturbations of the weak-coupling regime.

Quantum Theory of Dissipative Processes: The Markov Approximation Revisited

Adopting the standard mathematical framework for describing reduced dynamics, we derive two formal identities for the density operator of an open quantum system. Each of these is equivalent to the old Nakajima-Zwanzig equation. The first identity is local in time. It contains the inverse of the dynamical map which govern the evolution of the density operator. We indicate a time interval on which this inverse exists. The second identity constitutes a suitable starting point for going beyond the Markov approximation in a controlled way. On the basis of the Bloch equations we argue once more that in studying quantum dissipation one has to pay attention to the von Neumann conditions. In the Nakajima-Zwanzig equation we make the first Born approximation. The ensuing master equation possesses the correct weak-coupling limit. While proving this rather obvious but at the same time important statement, we elucidate the mathematical methods which underlie the weak-coupling limit. Moving to a two-dimensional Hilbert space, we find that both for short and for long times our approximate master equation respects the von Neumann conditions. Assuming exponential decay for correlation functions, we propose a physical limit in which the solutions for the density operator become Markovian in character. We confirm the well-known statement that, as seen from a macroscopic standpoint, the system starts from an effective initial condition. The approach to equilibrium is exponential. The accessory relaxation constants can differ from the usual Bloch parameters γ⊥ and γ∥ by more than 50%.