Lindblad, master equation

It is interesting to notice that Eq. (325) can also be derived from the Lindblad master equation using the same subordination approach as that adopted to derive Eq. (318). Here, however, the memory kernel of this master equation does not have the meaning of a correlation function. 

The three-pulse EOM-PMA can be formulated not only in terms of density matrices and master equations but also in terms of wavefunctions and Schrodinger equations [29]. The EOM-PMA can therefore be straightforwardly incorporated into computer programs which provide the time evolution of the density matrix or the wavefunction of material systems. Besides the multilevel Redlield theory, the EOM-PMA can be combined with the Lindblad master equation [49], the surrogate Hamiltonian approach [49], the stochastic Liouville equation [18], the quantum Fokker-Planck equation [18], and the density matrix [50] or the wavefunction [14] multiconfigurational time-dependent Hartree (MCTDH) methods. When using the... 

Abstract Interaction between a quantum system and its surroundings - be it another similar quantum system, a thermal reservoir, or a measurement device - breaks down the standard unitary evolution of the system alone and introduces open quantum system behaviour. Coupling to a fast-relaxing thermal reservoir is known to lead to an exponential decay of the quantum state, a process described by a Lindblad-type master equation. In modern quantum physics, however, near isolation of individual quantum objects, such as qubits, atoms, or ions, sometimes allow them only to interact with a slowly-relaxing near-environment, and the consequent decay of the atomic quantum state may become nonexponential and possibly even nonmonotonic. Here we consider different descriptions of non-Markovian evolutions and also hazards associated with them, as well as some physical situations in which the environment of a quantum system induces non-Markovian phenomena. 

Letting At go to zero and assuming all limits exist, we re-derive a Master Equation of the Lindblad form. The problem is that the assumption about the limiting process requires some additional assumptions. These are usually taken to imply that the system 2, in some way, acts as a huge reservoir. 

Even if we found that the Master Equation is not of the Lindblad form, we may not actually encounter the trouble. In this section we will show that the trouble cannot be safely ignored. 

The problem is, however, that we need to fix the exact conditions of validity of this approximation, this was attempted already in Ref. [Fano 1954], In particular, it has turned out that introduction of the memory effect is a very sensitive issue [Barnett 2001], Highly reasonable but unprecise approximations may lead to non-physical time evolution. An additional problem is that the procedure does not necessarily lead to Master Equations of the Lindblad type, see above. If this is not its form, we may find well known complications, which have to be avoided if we want to escape unphysical results. 

We have found that even in these simple cases things can go wrong. A physically reasonable Master Equation may not be of the Lindblad form. The corresponding Lindblad form may, on the other hand, violate simple rules like the fluctuation-dissipation theorem. 

Another problem is that memory kernels seem to be delicate entities. Erroneous kernels can destroy the physical sense of the time evolution of an initially acceptable density matrix. We do not have a general criterion to help us judge from the Master Equation with memory if the evolution is acceptable. In the Markovian limit, we know that the Lindblad form is certain to preserve the physical interpretation. It is a challenge for the theory of irreversibility in quantum systems to find such a criterion when memory effects are important. 

We here assume a simple Lindblad-type ansatz for the master equation in the relevant subspace. We set... 

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