Liouville equation coherences

A second problem with the GME derived from the contraction over a Liouville equation, either classical or quantum, has to do with the correct evaluation of the memory kernel. Within the density perspective this memory kernel can be expressed in terms of correlation functions. If the linear response assumption is made, the two-time correlation function affords an exhaustive representation of the statistical process under study. In Section III.B we shall see with a simple quantum mechanical example, based on the Anderson localization, that the second-order approximation might lead to results conflicting with quantum mechanical coherence. 

The current transfer described here is a coherent phenomenon, sensitive to dephasing. To investigate environmental dephasing effects, we incorporate additional relaxation of coherences in the site representation of the Liouville equation for the system s density matrix (DM) p ... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

Reineker, P. (1982). Exciton Dynamics in Molecular Crystals and Aggregates. Stochastic Liouville Equation Approach Coupled Coherent and Incoherent Motion, Optical Lineshapes, Magnetic Resonance Phenomena. Springer Tracts in Modern Physics, Vol. 94, Springer, Berlin, Heidelberg. 

C. Coherently Excited Systems Stochastic Liouville Equation... 

The time evolution of the atomic system can be calculated by the Liouville equation. It is found that population oscillations, represented by the diagonal elements of the density matrix, show up between states with identical values of mj = m + mg. Their oscillation frequencies are exactly given by the fine structure splitting. In our case, due to the excitation only coherence between the substates (mqlms) = (ll- ) and (0 ) ig ex-... 

P18 and P36 (with a light), while the free evolution of the previously induced coherences between these states is perturbed. For a quantitative description again the Liouville equation has to be solved for the excitation and free evolution phase, however, with initial conditions at time t2, which result from the evolution after the first excitation. 

The subscript W refers to this partial Wigner transform, N is the eoordinate space dimension of the bath and X = R, P). In this partial Wigner representation, the Hamiltonian of the system takes the form Hw R,P) = P /2M + y-/2m+ V q,R). If the subsystem DOF are represented using the states of an adiabatic basis, a P), which are the solutions of hw R) I R)=Ea R) I where hw K)=p /2m+ V q,R) is the Hamiltonian for the subsystem with fixed eoordinates R of the bath, the density matrix elements are p i -, 0 = ( I Pw( 01 )- From the solution of the quantum Liouville equation given some initial state of the entire quantum system, the reduced density matrix elements of the quantum subsystem of interest can be obtained by integrating over the bath variables, p f t) = dX p X,t), in order to find the populations and off-diagonal elements (coherences) of the density matrix. 

In the last section, we used the stochastic Liouville equation to find the steady-state rate of transitions between two weakly-coupled quantum states, on the assumption that coherence decayed rapidly relative to the rate of the transitions. The resulting expression (Eq. 10.36) reproduces Fermi s golden rule. We can use the same... 

Analytical expressions for coherence- and polarization-transfer functions can be derived using various approaches. These approaches are based on the Liouville-von Neumann equation [Eq. (47)]. If relaxation is neglected, the evolution of the density operator under an effective Hamiltonian is governed by the differential equation... 

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