Redfield theory coupling

A more general formulation of relaxation theory, suitable for systems with scalar spin-spin couplings (J-couplings) or for systems with spin quantum numbers higher than 1/2, is known as the Wangsness, Bloch and Redfield (WBR) theory or the Redfield theory 17). In analogy with the Solomon-Bloembergen formulation, the Redfield theory is also based on the second-order perturbation approach, which in certain situations (not uncommon in... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

We now come back to the simplest possible nuclear spin system, containing only one kind of nuclei 7, hyperfine-coupled to electron spin S. In the Solomon-Bloembergen-Morgan theory, both spins constitute the spin system with the unperturbed Hamiltonian containing the two Zeeman interactions. The dipole-dipole interaction and the interactions leading to the electron spin relaxation constitute the perturbation, treated by means of the Redfield theory. In this section, we deal with a situation where the electron spin is allowed to be so strongly coupled to the other degrees of freedom that the Redfield treatment of the combined IS spin system is not possible. In Section V, we will be faced with a situation where the electron spin is in... 

As in paper [5], we start from a system Hamiltonian consisting of three (one ground g) and two nonadiabatically coupled excited (j) ) and 1 states strongly coupled to a reaction mode, which in turn is weakly coupled to a dissipative environment (see Fig. 1). The bath degrees of freedom are integrated out in the framework of Redfield theory, and the signals are calculated according to the explicit formulas derived in [6,7]. 

A more general approach is required to interpret the current experiments, Jean and co-workers have developed multilevel Redfield theory into a versatile tool for describing ultrafast spectroscopic experiments [22-25], In this approach, terms neglected at the Bloch level play an important role for example, coherence transfer terms that transform a coherence between levels i and j into a coherence between levels j and k ( /t - = 2) or between levels k and l ( f - j - 2, k-j = 2) and couplings between populations and coherences. Coherence transfer processes can often compete effectively with vibrational relaxation and dephasing processes, as shown in Fig. 4 for a single harmonic well, initially prepared in a superposition of levels 6 and 7. The lower panel shows the population of levels 6 and 7 as a function of time, whereas the upper panels display off-diagonal density matrix ele-... 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

The expressions in the Redfield theory [82] is derived under the assumptions of weak coupling between the spin states and that the correlation of the perturbation, i.e. of Gg-g(r), has decayed to zero for the times of interest. This is formally expressed as... 

A possibility to overcome this limitation of the above conical-intersection models, at least in a quahtative manner, is to consider anhar-monic couplings of the active degrees of freedom of the conical intersection with a large manifold of spectroscopically inactive vibrational modes. The effect of such a couphng with an environment has been investigated for the pyrazine model in the weak-coupling limit (Redfield theory) in Ref. 19. The simplest ansatz for the system-bath interaction, which is widely employed in quantum relaxation theory assumes a coupling term which is bilinear in the system and bath operators... 

The Redfield theory is based on a weak coupling assumption and has been used extensively in nuclear magnetic resonance (NMR) theory for treating the influence of the surroundings on the spectra. Attempts to go beyond the weak coupling limit assumed in the Redfield theory have been made [78], and the requirement of complete positiveness assumed by Lindblad in the coupling between the subsystem and the reservoir has been challenged by Pechukas [79]. 

Redfield theory [18-20] is a microscopic semi-classical theory of spin relaxation in which the spin system is treated quantum mechanically whilst the coupling of the spins with the lattice is treated classically. In this classical approximation, the spin states are in equilibrium and a correction factor is needed to ensure the spin ensemble relaxes to the correct limits. This problem can be overcome by treating the lattice quantum mechanically, however, the details and nature of the computational details are beyond the scope of this thesis. In this section a brief introduction to Redfield theory for spin relaxation is provided, a more detailed analysis of the theory can be found elsewhere [21]. 

Tie, the 00sT2e dispersion having already occurred when the (Os v dispersion occurs. Actually, the validity of the SBM theory is assured only within the Redfield limit (see Section IV.A of Chapter 2) (7), i.e., in case the energy of the coupling between the spin and the lattice, E/H (in frequency units), whose modulation is responsible for the spin relaxation, is smaller than the inverse of the correlation time, Xc, for the modulation of the coupling itself, E/H x . This determines for T,e > x (1). 

The absolute value of J, which we have used all over the treatment within the Redfield limit, implies that the theory is the same for ferro- and antiferromagnetic coupling. When J kT, the sign of J is irrelevant for the nuclear shifts, the... 

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