Density matrix elements master equation

At the electric-dipole approximation (EDA) and in the fast-modulation limit of the chromophore-bath interaction, the molecular density matrix element, p , (t), in the dark molecular basis, (Im), satisfies the master equation [6],... 

The traditional method to analyze conditions for modification of spontaneous emission is to derive equations of motion for the probability amplitudes or density matrix elements and solve them by direct integration, or by a transformation to easily solvable algebraic equations. Here, we discuss an alternative approach proposed by Akram et al. [24] that allows us to identify conditions for a modification of spontaneous emission directly in the master equation of two arbitrary systems. In this approach, we introduce linear superpositions of the dipole operators... 

The spectrum of the fluorescence field emitted on the 1) —> 2) and 13) —> 12) transitions is given by the Fourier transform of the average two-time correlation function of the dipole moments of the transitions that, according to the quantum regression theorem [33], satisfy the same equations of motion as the density matrix elements pj2( ) and P32W- Using the master equation (64) with the Hamiltonian (82), we obtain the following set of coupled equations of motion for the density matrix elements... 

When the atomic transitions 1) —> 2) and 3) —> 2) are directly driven by a laser field, the master equation (64) leads to the following set of equations of motions for the density matrix elements... 

We find the density matrix elements from the master equation of the system. In the frame rotating with the laser frequency a>L and within a secular approximation, in which we ignore all terms oscillating with (colrf - a>L) and ((]>2d — the master equation for the density operator of the system is given by... 

The master equation (165) leads to a closed system of 25 equations of motion for the density matrix elements. Since the laser field does not couple to the level d), the system of equations splits into two subsystems a set of 17 equations of motion directly coupled to the driving field and the other of 8 equations of motion not coupled to the driving field. It is not difficult to show that the steady-state solutions for the 8 density matrix elements are zero. Using the trace property, one of the remaining equations can be eliminated, and the system of equations reduces to the 16 coupled linear inhomogeneous equations. 

After transforming to the collective state basis, the master equation (31) leads to a closed system of 15 equations of motion for the density matrix elements [46]. However, for a specifically chosen geometry for the driving field, namely, that the field is propagated perpendicularly to the atomic axis (k rn = 0), the system of equations decouples into 9 equations for symmetric and 6 equations for antisymmetric combinations of the density matrix elements [45-50]. In this case, we can solve the system analytically, and find that the steady-state values of the populations are [45,46]... 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

We have already mentioned in Section II.G that the Greens function approach predicts nonexponential decay when the coupling matrix element is allowed to vary with energy. Lin used a very different approach to investigate the limitations of the master equation description. He showed, using a density matrix formalism, that memory effects appear if perturbation theory is extended to fourth order in the coupling between two vibrational levels. 

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