Density matrix Bloch equation

Bloch s Equation for the Density Matrix.—Bloch s equation will be derived here both for the equilibrium ensemble density matrix of Eq. (8-204), and for the equilibrium grand ensemble density matrix of Eq. (8-219). 

To evaluate the density matrix at high temperature, we return to the Bloch equation, which for a free particle (V(x) = 0) reads... 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

Before developing the theory when interactions are present, it is convenient to derive Bloch s equation for the density matrix. 

Density matrix and ensembles, 465 Bloch s equation, 475 Derogatory form, 73... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

Using the long time-weak coupling approximation and the hypothesis of random phases for the thermostat, Bloch and Wangsness find, after taking the trace over the heat bath, the following equations for the reduced density matrix a ... 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

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] ... 

The Bloch equation gives the time derivative of the density matrix p in terms of its commutator with the Hamiltonian for the system, and the decay rate matrix T. Each of the matrices, p, H, and T are n x n matrices if we consider a molecule with n vibration-rotation states. We so ve this equation by rewriting the n x n square matrix p as an n -element column vector. Rgwrit ng p in this way transforms the H and V matrices into an n x n complex general matrix R. We obtain... 

The classical treatment using the Bloch equations is very useful but has limitations when quantum effects must be considered. Hence, when spin—spin coupling is present (which depends on the quantized spin states of the nuclear moments), the simple treatment is inadequate. In some instances, as we shall see in later chapters, consideration of spin states can be easily grafted onto the classical treatment, whereas in others a full density matrix treatment (Chapter 11) is needed. 

Most earlier theoretical work on multiple resonance used either the Bloch equations or the spin-Hamiltonian. That is, relaxation is considered separately from other effects. The tendency in recent years has been to adopt the density-matrix approach but on some occasions simpler methods have still been appropriate. In the interpretation of multiple resonance experiments it is often necessary to consider the labelling of the energy level diagram in detail. 

Meakin and Jesson (48) used the Bloch equations in part of their work on the computer simulation of multiple-pulse experiments. They find that this approach is efficient for the effect upon the magnetization vector of any sequence of pulses and delays in weakly coupled spin systems. However, relaxation processes and tightly coupled spin systems cannot be dealt with satisfactorily in this way and require the use of the density matrix. 

Let us turn immediately to illustrating the use of the canonical density matrix for treating free electrons in a uniform magnetic field. As follows from the definitions in Eqs. (1) and (2), the canonical density matrix satisfies the so-called Bloch equation, as is readily verified, namely... 

To gain some insight into the shape of the solution for the canonical density matrix, let us consider first a one-dimensional problem of electrons moving in a potential V(x). Writing the Bloch equation (Eq. (6)) for the Hamiltonians and and subtracting them to remove the derivative, one obtains the so-called equation of motion of the density matrix as... 

Below, therefore, the solution of the Bloch equation in Eq. (6) for the canonical density matrix C(r, Tq, P, F, co) for independent electrons in a constant electric field of strength F, with harmonic restoring force corresponding to an oscillator angular frequency to, will be presented. In Sect. 7.1 below, the electric field is taken as the z axis. Then this solution can readily be generalized to include harmonic restoring forces also in the x and y directions. 

Upon using Eq. (171) in (168), we obtain our generalized Bloch equations for the components of the TLS density matrix (compare with [Cohen-Tannoudji 1992])... 

The Bloch equations describe the motion of the transverse magnetization in the static magnetic field in terms of a precession around the axis of the field. Similarly (2.2.65) describes a rotation of the density matrix around the z-axis by an angle (Wo(t to)- The effects of rf pulses are consequently described by rotations of the density matrix around axes in the transverse plane. For instance, a rotation around the y-axis by an angle a is expressed by... 

The density matrix satisfies the Bloch differential equation... 

The detection of NMR signals is based on the perturbation of spin systems that obey the laws of quantum mechanics. The effect of a single hard pulse or a selective pulse on an individual spin or the basic understanding of relaxation can be illustrated using a classical approach based on the Bloch equations. However as soon as scalar coupling and coherence transfer processes become part of the pulse sequence this simple approach is invalid and fails. Consequently most pulse experiments and techniques cannot be described satisfactorily using a classical or even semi-classical description and it is necessary to use the density matrix approach to describe the quantum physics of nuclear spins. The density matrix is the basis of the more practicable product operator formalism. 

In NMR-SIM the simulation of an NMR experiment is based on the density matrix approach with relaxation phenomena implemented using a simple model based on the Bloch equations. Spectrometer related difficulties such as magnetic field inhomogenity, acoustic ringing, radiation damping or statistical noise cannot be calculated using the present approach. Similarly neither can some spin system effects such as cross-relaxation and spin diffusion can be simulated. 

The linear response of a system is determined by the lowest order effect of a perturbation on a dynamical system. Formally, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical commutators into classical Poisson brackets, or vice versa. Suppose that the system is described by Hamiltonian H + where denotes an external perturbation that may depend on time and generally does not commute with H. The density matrix equation for this situation is given by the Bloch equation [32]... 

We complete our discussion of density matrices by mentioning that the density matrix in any representation satisfies the well-known Bloch equation [11,33,66] ... 

The significance of this is that the solutions to the Bloch equation are known for certain simple systems. For example, for a free particle (F = 0) in a one-dimensional space we find from the Bloch equation that the density matrix is [66]... 

Electron exchange reactions may also be followed using line-width measurements. The relaxation of a diamagnetic nucleus will be modified if the species picks up or loses an electron, the magnitude of the effect depending upon ai and the time the unpaired electron spends in the ion or radical, Such exchange reactions have been analysed using both modified Bloch equations and a density matrix formalism. ... 

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