Equation of motion for the density matrix

The first term on the righthand side of Eq. (5.1) describes the effect of light absorption on mm at reduced absorption rate 

The fourth term characterizes the relaxation of the density matrix /mm  

The diagonal elements of density matrices /mm and as already mentioned in Section 3.1, characterize the population of the magnetic sublevels M and p for the states J and J , whilst the non-diagonal elements /mm and (pm, describe the phase coherence of the corresponding wave functions J M) and J M ), or of J p) and J p ). Sometimes even special normalized magnitudes 

The intensity of the fluorescence appearing in the radiational transition of the molecule from level J upon level J (see Fig. 3.14) may be calculated by means of the formula [133] 

In the analysis of optical pumping of atoms, when the values of the quantum numbers J , J and J are, as a rule, of the order of unity, and hence the number of magnetic sublevels is small, the system of equations 

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The basic equation [8] is tlie equation of motion for the density matrix, p, given in equation (B2.4.25), in which H is the Hamiltonian. 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

Having established the Hamiltonian, the next step is to derive an equation of motion for the density matrix. One can start with the Liouville-von Neumann equation for the complete system-plus-bath density matrix a(t)... 

K. Burnett and J. Cooper. Collisional redistribution of radiation II. The effects of degeneracy on the equations of motion for the density matrix. Phys. Rev. A, 22 2027-2043... 

It is also of relevance in the present context to note that the perturbation theory of Gorling and Levy [61] has been applied by Holas and March [62] to provide a calculational scheme for exact exchange and correlation potentials based on the equation of motion for the density matrix. 

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... 

An alternative way of viewing the process of the reduction of the dressed trapping state to the state a) in a very strong field is to analyze the equations of motion for the density matrix elements in terms of the symmetric and antisymmetric superpositions of the atomic excited states... 

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]... 

Using the ground-state density matrix as an input, the CEO procedure - computes vertical transition energies and the relevant transition density matrices (denoted electronic normal modes ( v)mn = g c j Cn v ), which connect the optical response with the underlying electronic motions. Each electronic transition between the ground state and an electronically excited state v) is described by a mode which is represented hy K x K matrix. These modes are computed directly as eigenmodes of the linearized time-dependent Hartree—Fock equations of motion for the density matrix (eq A4) of the molecule driven by the optical field. 

A time reversible BO molecular dynamics scheme based on the propagation of one-particle density matrices has been proposed by Niklasson [8]. The equation of motion for the density matrix is... 

The time dependence of the magnetization is then calculated by solving the equation of motion for the density matrix a to the second order in The results are... 

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