Lindblad operator

The Lindblad operators become matrices Lpfc and the Lindblad rate is then... 

The striking result of the derivation may be summarized as follows the irreversible time-evolution (owing to the Lindblad operator X) may cause a reduction of the transition rate of the system (from its initial to its final state). In "experimental" terms, this is tantamount to an effective reduction of the system s cross-section density and thus a shortfall of scattering intensity. 

Each dissipation channel k is described by a so-called Lindblad operator, C/c, which is used to represent the energy exchange via inelastic collision with the electrons of the metal. In this particular case, the operators take the form of a transition between two vibrational states of... 

This form of the Lindblad operators suggests that, provided their computation remains numerically tractable, the eigenstates of the vibrational Hamiltonian would provide the most efficient basis to represent the evolution of the reduced density matrix, eqn (16). Indeed, in such a basis, the equations of motion for the matrix elements take the particularly simple and elegant form... 

Models for the dissipative dynamics can frequently be based on the assumption of fast decay of memory effects, due to the presence of many degrees of freedom in the s-region. This is the usual Markoff assumption of instantaneous dissipation. Two such models give the Lindblad form of dissipative rates, and rates from dissipative potentials. The Lindblad-type expression was originally derived using semigroup properties of time-evolution operators in dissipative systems. [45, 46] It has been rederived in a variety of ways and implemented in applications. [47, 48] It is given in our notation by... 

The operators can be constructed as combinations of position and momentum operators in the p-region, [45] or from empirical transition rates ka <-a between orthonormal eigenstates and 4>, of Hp. [47] The index L then refers to a given transition a —> a, and the corresponding operator is C(pL) = This simplifies the form of the Lindblad dissipa-... 

Wpla)]. The Lindblad form of the dissipation rate can be constructed from state-to-state transition rates r(ft <— a) in the p-region, induced by its interaction with the s-region, and the operator for the transition k = (/ <— a) is... 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

We consider a most simplified damping by introducing for a two-level system the Lindblad type relaxation operator... 

Thus we obtain a Lindblad time evolution for the operators too by writing... 

Alternatively, the equation of motion can he obtained from the PWT Hamiltonian Fp, using its eigenstates 4 P,R)) for quantum states T at each phase space point to construct the operators Cp P, R) of the Lindblad expression, with semiempirical rates kj i P,R). The result is the equation... 

The DM equation for the Lindblad formulation follows along lines similar to the ones mentioned for propagation of the quantum-classical DOp without dissipation. It starts with an expansion of operators in the p-region in terms of a basis set of functions and a discretization of quasiclassical variables in the p-phase space, and leads to a matrix equation which can be written as... 

We have also outlined a treatment of systems showing dissipative dynamies. These ean be treated with a natural extension of the formulation shown here. The density operator equation differs in a signifieant way when dissipation is present, because it eontains terms whieh are not derived from a Hamiltonian, and must be written as rates involving superoperators. We have shown here one such situation, with the Lindblad form of the dissipation rate, and other forms can be treated equally well with the present eombination of a PWT and a quasielassieal approximation in phase spaee. 

We finally mention that the system/bath picture for studying the pyrazine spectrum has been adopted by two other publications. There, however, the bath was treated implicitly by dissipative operators of the Lindblad type. 

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