Evolution of the Density Matrix

The equation of motion for a is obtained from the Schrodinger equation for IV ), 

This can be verified by substitution of Eq. (2.11) into Eq. (2.10). Alternatively, an orthonormal basis of eigenkets of H 

This indicates that the off-diagonal elements of the density matrix do not decay but oscillate as a function of time. By combining Eqs. (2.13) and (2.15), the solution given by Eq. (2.11) is recovered for the special case of a time-independent Hamiltonian. 

The density matrix in thermal equilibrium is hence given by 

If in the domain of high temperature ( 1 K ), i.e., p = huJo/ksT 1, the equilibrium density matrix for our system is approximated by 

We now have a formula for constructing the density matrix for any system in terms of a set of basis functions, and from Eq. 11.6 we can determine the expectation value of any dynamical variable. However, the real value of the density matrix approach lies in its ability to describe coherent time-dependent processes, something that we could not do with steady-state quantum mechanics. We thus need an expression for the time evolution of the density matrix in terms of the Hamiltonian applicable to the spin system. 

Provided IK. is not an explicit function of time, P can be expanded as in Eq. 11.1, except that the ck now become functions of time. Insertion of the expansion into 

Multiplying both sides of Eq. 11.11 by f m integrating, and taking into account the orthonormality of the basis functions (f k, we obtain 

To develop an expression for the time dependence of p, we begin by examining the time derivative of c c  

Multiplying Eq. 11.13 by the populations pt and summing, we obtain an expression for the time derivative of the density matrix  

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Note that, since the von Neumann equation for the evolution of the density matrix, 8 j8t = — ih H, / ], differs from the equation for a only by a sign, similar equations can be written out for p in the basis of the Pauli matrices, p = a Px + (tyPy -t- a p -t- il- In the incoherent regime this leads to the master equation [Zwanzig 1964 Blum 1981]. For this reason the following analysis can be easily reformulated in terms of the density matrix. 

The spin Hamiltonian also forms the theoretical basis for describing the temporal response of the spin system to a pulse sequence and/or mechanical manipulations of the sample via calculations of the evolution of the density matrix. Computer... 

Poliak and Eckhardt have shown that the QTST expression for the rate (Eq. 52) may be analyzed within a semiclassical context. The result is though not very good at very low temperatures, it does not reduce to the low temperature ImF result. The most recent and best resultthus far is the recent theory of Ankerhold and Grabert," who study in detail the semiclassical limit of the time evolution of the density matrix and extract from it the semiclassical rate. Application to the symmetric one dimensional Eckart barrier gives very good results. It remains to be seen how their theory works for asymmetric and dissipative systems. 

It is worth mentioning that constant term disappears in Eq. (6) because of the suitable choice of the basis in the form (4). In the following, the evolution of the density matrix of the relevant system will be examined by means of the standard projection technique. The matrix elements pa/it) of the reduced density matrix operator are defined as follows... 

Because of the quantum origin of the Stark effect (see Section 5.2), the evolution of the density matrix elements /mm must be considered for the ground state molecules. In the field region we must choose the... 

So to obtain expectation values relevant to any particular experiment one needs an estimate of the density matrix at the time of measurement. For an NMR experiment, this typically requires the ability to estimate the time evolution of the density matrix for the pulse sequence used for the experiment. The time dependent differential equation that describes the time evolution of the density matrix, known as the Liouville-von Neumann equation is given by... 

Simulation packages such as GAMMA take advantage of the fact that evolution of the density matrix under the Liouville-von Neumann equation is well approximated by a small number of easily applied transformations of the density matrix, namely free evolution can be represented by a simple unitary transformation and application of ideal RF pulses can be represented by a simple rotation. Real RF pulses can be effectively modelled as a succession of ideal RF pulses. The beauty of this method is that fairly complex, realistic effects, such as evolution of coupled spin systems through complex pulses, can be modelled by a straightforward combination of these simple building blocks. 

In Section 11.3 we find that the density matrix for a spin system at equilibrium can be separated into the unit matrix 1 and other terms pertaining to populations of spin states. The unit matrix is unaffected by rf pulses or any other evolution of the density matrix hence it is conventional to delete it, and we do so. Some authors introduce a new symbol for the truncated matrix, but most do not. We continue to use the symbol p for this truncated density matrix. 

We now look at the evolution of the density matrix for the one spin system as the magnetization precesses in the rotating frame. Once more, we apply Eq. 11.16 in the same manner we did in Section 11.2 to take into account the effect of the rotating frame. In this case we obtain... 

Hazards of parameter mismatch. In the present model all results are solvable. We know the exact memory kernels in (152) and we can solve for the time evolution from (156). However, in realistic applications, one must ordinarily resort to approximations. In this situation, we get equations of the type (150) but with incorrect values of the parameters in the memory function. In order to display the possible disasters that may ensue, we modify the functions /i and /2 arbitrarily and look at the time evolution of the density matrix. 

In the following, we show how an MQMAS echo is formed. The quadmpolar Hamiltonian under MAS will be derived as a perturbation to the strong static magnetic field Bq. The evolution of the density matrix under this Hamiltonian will be then shown to form an echo under a suitable selection of symmetric coherences. The manipulation of spin coherences by RF pulses will be assumed to be ideal here. The effects of nonidealities were discussed by the group of Vega. From the expression for the elements of the quadmpolar Hamiltonian, line narrowing by DOR experiment and echo formation by two alternative experiments, DAS and STMAS, will also be demonstrated and briefly discussed. Finally, some additional line narrowing schemes will be mentioned. 

In order to calculate the evolution of the density matrix and consequently the positions of the spectral lines, the transition frequencies have to be calculated. The hrst-order transition frequencies are given by [following Eq. (14)]... 

Since we are interested in the time evolution of the density matrix elements, we will need explicit expressions for the components X of the vector X(f) in terms of their initial values. This can be done by a direct integration of (83). Thus, if to denotes an arbitrary initial time, the integration of (83) leads to the following formal solution for X(f)... 

Substituting (21) into (17) shows the effect of a hard pulse of duration t on the evolution of the density matrix, i.e. 

The coherent evolution of the density matrix is described by the corresponding Liouville operators ... 

Mixing interaction Q and level splitting E form the z and x components of a vector TC, the y component being zero. The time evolution of the density matrix is then a precession of q around the axis f. ... 

Figure 3. Exact vector model for the evolution of the density matrix of a radical pair. The meaning of the components of q is given by Eq. 33 or Eq. 34, depending on the intersystem crossing mechanism. For further explanation, see the text. [Adapted from ref. [lOi] with permission. Copyright 1995 John Wiley Sons, Inc.]...

After the encounter, those radicals that have not reacted enter on a diffusive excursion, during which intersystem crossing can take place. The evolution of the density matrix up to the moment of the next encounter is described by the action of a mixing matrix M, for example, the one given in Eq. 37, on o/.after> which also takes into account that a fraction of the radical pairs separates permanently. In the new basis of Eq. 47, Eq. 37 becomes... 

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... 

The evolution of the density matrix in time requires the solution of the equation of motion, Eq. (10) or (20) in a basis set of N states, this represents a system of coupled linear differential equations for the individual density matrix elements. It is most natural to consider the equation in Liouville space, where ordinary operators (N x N matrices) are treated as vectors (of length N ) and superoperators such as and which act on operators to create new operators, become simple matrices (of size X N ). In the Liouville space notation, Eq. (9) would... 

The subsequent time evolution of the density matrix elements pAt) is governed by the stochastic Liouville equations (Section I.B). In the electric dipole approximation, the intensity of fluorescence subsequently emitted by such a dimer with polarization e, will be... 

In order to describe the time evolution of the density matrix Q(t) during some arbitrary pulse sequence, we divide the sequence into regions, where a pulse is present and regions where there is no pulse. The action of the different non-selective puls (including a single 90° pulse for the FID which after FT yields the CW frequency spectrum) is considered by unitary transformations employing Wigner rotation matrices [10, 49]. After the pulse the density matrix is assumed to obey the stochastic Liouville equation [85, 86]... 

The family of the density-matrix spectral moments is defined as S = L"t] which are the expansion coefficients in the short-time evolution of the density-matrix response function. These moments are used to construct the main DSMA equations - °°... 

The evolution of the density matrix, p t), that describes such processes is given by the quantum Liouville equation ... 

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