Time-dependent Liouvillian

Here /L(t) = r(t) dldT is the fully time dependent Liouvillian that evolves f T, t) in accordance with Eqs. [91]. Furthermore, it is the same Liouvillian used for... 

Sg denotes the adjoint electron spin operator. One should notice that the expression [exp(—iLtx)Sp Do i p(f2ML)] results in the S-operators and the ml being (implicitly) time-dependent. In order to continue any further, we need to specify the lattice and its Liouvillian. 

The previous discussion shows that the relaxation processes emerge from the quantum dynamics under appropriate circumstances leading to the formation of time-dependent quasiclassical parts in the observable quantities. Let us add that quasiclassical and semiclassical methods have been recently applied to the optical response of quantum systems in several works [65, 66] where the relation to the Liouville formulation of quantum mechanics has been discussed, without however pointing out the existence of Liouvillian resonances as we discussed here above. The connection between the property of chaos and n-time correlation functions or the nth-order response of a system in multiple-pulse experiments has also been discussed [67, 68]. 

Here /L( ) is the Liouvillian corresponding to the fully time-dependent dynamics of Eqs. [91], and the subscript R denotes a right-time-ordered product, with later times appearing to the right. ... 

According to (1.22) and (1.26), the eigenvalues v of the Liouvillian L are distributed symmetrically around the point v = 0, and this implies that, even if the Hamiltonian H in physics is bounded from below, H > a 1, the Liouvillian L is as a rule unbounded. Except for this difference, practically all the Hilbert-space methods developed to solve the Hamiltonian eigenvalue problem in exact or approximate form may be applied also to the Liouvillian eigenvalue problem. In the time-dependent case, the L2 methods developed to solve the Schrodinger equation are now also applicable to solve the Liouville equation (1.7). 

The seeond so-called trajectory based approach, that has been employed for EPR simulations, is based on the Liouville von Neumann equation (LvN) in the semi-classical approximation, often called the Langevin form of SLE. This method was first introdueed for EPR simulations by Robinson and co-workers in 1992. In this approach the SLE is transformed into a system of coupled stoehastie differential equations with explieit time dependence in the spin-lattiee coupling of the Liouvillian ... 

One can present the time dependent part of the Liouvillian in the following form ... 

A = and M.22 z) is the matrix element (2 M(z) 2). All the other matrix elements ot the memory operator are zero [63]. The appearance ot the tactor i corresponds to the Hermitian definition ot the Liouvillian which is not universal. Equation (98) is an exact z-dependent expression. This z-dependence can be eliminated by approximating M22(z) by a positive constant r (Markovian approximation). This approximation is justified since the microscopic correlation time is much shorter than the regression time ot the fluctuation. Then Eq. (98) transtorms into the z-independent effective Liouvillian... 

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