Redfield equation

Consider a general system described by the Hamiltonian of Eq. (5), where = Huif) describes the interaction between the spin system (7) and its environment (the lattice, L). The interaction is characterized by a strength parameter co/i- When deriving the WBR (or the Redfield relaxation theory), the time-dependence of the density operator is expressed as a kind of power expansion in Huif) or (17-20). The first (linear) term in the expansion vanishes if the ensemble average of HiL(t) is zero. If oo/z,Tc <5c 1, where the correlation time, t, describes the decay rate of the time correlation functions of Huif), the expansion is convergent and it is sufficient to retain the first non-zero term corresponding to oo/l. This leads to the Redfield equation of motion as stated in Eq. (18) or (19). In the other limit, 1> the expan-... 

The stochastic Liouville equation, in the form relevant for the ESR line shape calculation, can be written in a form reminiscent of the Redfield equation in the superoperator formulation, Eq. (19) (70-73) ... 

The best one can hope is that there is an approximate equation of type (3.6), called Redfield equation 510 - or in the present context the quantum master equation . The approximation requires an expansion parameter the obvious choice is the parameter a. To prepare for this expansion we transform pT to its interaction representation [Pg.437]

B. A. Hess The application of the Redfield equation requires that the time scales involved are well separated. Is this the case in the example shown by Prof. Fleming ... 

In the coherent (Hamiltonian) approach to the four-state spin system, the state populations are just the diagonal elements of the corresponding density matrix p(r, f), which obeys the Bloch or Redfield equation [211] ... 

Time evolution equations for reduced density operators Combining all terms we get the so called Redfield equation... 

The Redfield equation describes the time evolution of the reduced density matrix of a system coupled to an equilibrium bath. The effect of the bath enters via the average coupling V = and the relaxation operator, the last sum on the right of Eq. (10.155). The physical implications of this term will be discussed below. 

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

If the bath is kept at thennal equilibrium, the system should approach the same thermal equilibrium at long time. In practical situations we often address this distribution in the representation defined by the system eigenstates, in which case the statement holds rigorously in the limit of zero coupling. Detailed-balance relationships such as Eq. (10.161) indeed imply that a Boltzmann thermal distribution is a stationary (da/dt = 0) solution of the Redfield equation. 

The fact that the lineshape (18.49) is Lorentzian is a direct consequence of the fact that our starting point, the Redfield equations (10.174) correspond to the limit were the thermal bath is fast relative to the system dynamics. A similar result was obtained in this limit from the stochastic approach that uses Eq. (10.171) as a starting point for the classical treatment of Section 7.5.4. In the latter case we were also able to consider the opposite limit of slow bath that was shown to yield, in the model considered, a Gaussian lineshape. 

The Redfield equations that lead to Eqs (18.43) or (18.46) were obtained under the assumption that thennal environment is fast relative to the system and therefore correspond to this homogeneous limit. Consequently the absorption spectrum (18.49) obtained from these equations corresponds to a homogeneous lineshape. In contrast, the classical stochastic theory of lineshape, Section 7.5.4, can account for both limits and the transition between them. We will see in the next section that an equivalent theory can be also constructed as an extension of the Bloch equations (18.43). 

Returning now to a discussion of the Redfield equations (40) we first note that, with our choice in (39), the only nondiagonal elements that are coupled to each other are p,2 and P34. Except for the driving-field terms these equations therefore have a form very similar to the modified Bloch equations postulated by McConnell. The quantum-mechanical derivation of (40), however, leads to new insight and restrictions in the use of these equations in the optical domain (Section IV). 

THE REDFIELD EQUATION IN CONDENSED-PHASE QUANTUM DYNAMICS... 

This result is the Redfield-Liouville-von Neumann equation of motion or, simply, the Redfield equation [29,30,49-53]. Here the influence of the bath is contained entirely in the Redfield relaxation tensor, 3i, which is added to the Liouville operator for the isolated subsystem to give the dissipative Redfield-Liouville superoperator (tensor) that propagates (T. Expanded in the eigenstates of the subsystem Hamiltonian, H, Eq. (9) yields a set of coupled linear differential equations for the matrix... 

To evaluate the correlation functions in Eqs. (12) and (13), it is usual to complete the separation of the system and bath by decomposing the system-bath coupling into a sum of products of pure system and bath operators. This allows the correlation functions of the system-bath coupling to be replaced, without loss of generality, by correlation functions of bath operators alone, evolving under the uncoupled bath Hamiltonian. Moreover, as we have previously pointed out [39,40], this decomposition of the system-bath coupling make it possible to write the Redfield equation in a highly compact form, without explicit reference to the Redfield tensor at all. 

In most applications of the theory to date, the solution of the Redfield equation has required first the explicit calculation of the Redfield tensor elements [Eq. (11)] given these, Eq. (10) could be solved as an ordinary set of linear differential equations with constant coefficients, either by explicit time stepping [41, 42] or by diagonalization of the Redfield tensor [37,38]. Since there are such tensor elements for an A -state subsystem, the number of these quantities can become quite large. Because of this, until recently most applications of Redfield theory have been limited to small systems of two to four states, or else assumptions, such as the secular approximation, have been used to neglect large classes of tensor elements. 

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... 

Here the W are operators of the subsystem and the superscript dagger denote the Hermitian conjugate. The Redfield equation can be written in this form only when an additional symmetrization of the bath correlation functions is performed [48]. Note that this alternative equation also expresses the dissipative evolution of the density matrix in terms of N x N... 

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