Lindblad equation

Fast dissipation is treated numerically within the Markoff approximation, which leads to differential equations in time, and dissipative rates most commonly written in the Redfield [9,10] or Lindblad [11,12] forms. Several numerical procedures have been introduced for dissipative dynamics within the Markoff approximation. The differential equations have been solved using a pseudospectral method [13], expansions of the Liouville propagator in terms of polynomials, [14-16] and continued fractions. [17]... 

As an illustration of the numerical treatment, the instantaneous dissipation due to fast electronic motions was constructed from the Lindblad expression in the treatment of CO/Cu(001), from decay and transition rates. The delayed dissipation, present for slow atomic vibrations of the medium, was given in terms of a memory kernel in the integrodifferential equation, calculated to second order in the coupling of the adsorbate and its environment. 

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... 

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. 

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. 

In the theory of semi-groups, it is proved that the most general acceptable evolution equation has to be of the form (9), [Lindblad 1976 (b)], which is called the Lindblad form. 

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. 

In quantum information applications one often treats the system of two qubits being manipulated as part of some information processing. This is modelled by a couple of interacting two-level systems. Following the approach in the present paper, we consider the one system to be strongly damped. In that case it serves as a faked continuum for the other one, and we desire to derive an equation of motion for the originally undamped system. The damping is described by a Markovian term of the Lindblad type. 

However, the solution of the system does also apply in the strong driving limit /teir < 2 /2 a. As the original equation of motion is of the Lindblad form, no unphysical features should emerge. In the case of no damping k —> 0, we obtain the simple result... 

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. 

In the subspace spanned by the preferred coordinates (also denoted "pointer basis"), we assume the relevant density matrix to obey a Lindblad-type equation of the form [Lindblad 1976 (b) Barnett 2001]... 

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

The final form of the Redfield equation [Eq. (20)] is superficially similar to the equation of motion that arises in the axiomatic semigroup theory of Lindblad, Gorini et al. [48,54-57]. They showed that the most general equation of motion that preserves the positivity of the density matrix must have the general form... 

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. 

Atrial muscle The first detailed model of the excitation-contraction coupling mechanism in heart cells was constructed by Hilgemann and Noble [1987] and a single cell model was completed by Earm and Noble [1990]. A similarly detailed model of rabbit atrial cells was constructed by Lindblad et al. [1996] and modified to fit data from human atrial cells by Nygren et al. [1998]. Another model of human atrial cells based on the ventricular ceU model of Luo and Rudy [1994] but with improved calcium handling equations was formulated by Courtemanche et al. [1998]. The main differences between the Nygren and... 

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... 

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... 

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