Equilibrium Density Matrix

With the approximations used in Eq. 11.33, we can represent the relative populations of the four energy levels in terms of e, = (E3 — Ex)/kT and es = (E2 — E /kT. We already know that the density matrix at equilibrium p(0) is diagonal, with elements that describe the excess populations  

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A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

A quantitative evaluation of the relaxivities as a function of the magnetic field Bo requires extensive numerical calculations because of the presence of two different axes (the anisotropy and the external field axis), resulting in non-zero off-diagonal elements in the Hamiltonian matrix (15). Furthermore, the anisotropy energy has to be included in the thermal equilibrium density matrix. Figures 7 and 8 show the attenuation of the low field dispersion of the calculated NMRD profile when either the crystal size or the anisotropy field increases. 

We now derive the expression for the fluorescence signal in terms of the doorway and window wavepackets instead of the four-point correlation function. We start with Eq. (3.1) and write the four-point correlation function F(4) explicitly as the trace with respect to the equilibrium density matrix. We then use the cyclic invariance of the trace and obtain... 

We now write the equilibrium density matrix in the coordinate represen-... 

The most simple procedure can be carried out in the case of path integral for the equilibrium density matrix, i.e., with it = j6. Specifically, the Metropolis algorithm [Metropolis et al., 1953] applies to n-dimension-al integrals of the general form... 

A spin system placed in a constant homogeneous magnetic field B0 finally attains thermal equilibrium with the lattice. This relaxation is induced by fluctuations of local magnetic fields which result from molecular motions. At equilibrium, the spin system is described by the equilibrium density matrix p0 ... 

The average <(poz)3>=, or order parameter, is calculated over the equilibrium density matrix, p=e H/T/Tr[e H/T], where Tr denotes the trace (sum over diagonal elements). The Ising Hamiltonian can be expressed as ... 

Let s now look at the density matrix as it evolves for the pulse sequences that we outlined in Section 11.5. We have illustrated INEPT and related pulse sequences by the two-spin system in which I = H and S = 13C.The equilibrium density matrix can be considered as the sum of two matrices ... 

We will now prepare the initial density matrix superket to be the equilibrium state or equivalently the equilibrium density matrix, by a minimum free energy principle, with the free energy given by ... 

The next step in the calculation is to choose pw to) to be the equilibrium density matrix, pwe- One of the differences between quantum and quantum-classical response theories appears at this stage. In quantum mechanics, the quantum canonical equilibrium density is pj = exp(—/ .ff) which, when... 

In the high-field approximation the space-dependent thermodynamic-equilibrium density matrix po(r) is proportional to l fr) (cf. eqn (2.2.58)). A 90° pulse converts this density matrix into p (0+, r) oc U(r) = [l+(r) -I- l (r)] /2, where U is the x-component of the spin operator, 1+ = 1 + il>-, and L = 1 — il. In the subsequent free-precession period the density matrix evolves for a time t under the spin Hamiltonian Hx(r) (cf. eqn (3.1.1)) and under the influence of the applied gradient, which introduces a phase evolution given by kr = —yGrti, where k is the experimental variable,... 

The thermal equilibrium density matrix is calculated from the definition of the density matrix and the Boltzmann equilibrium spin distribution.39 Ignoring all constants, equilibrium density matrix, is simply equal to Iz, as expected. 

Before the n/2x pulse the density matrix is given by o(0), the thermal equilibrium density matrix. After the pulse,... 

The commutator and anticommutator operations in Hilbert space can thus be implemented with a single multiplication by a and + superoperator, respectively. We further introduce the Liouville space-time ordering operator T. This is a key ingredient for extending NEGFT to superoperators when applied to a product of superoperators it reorders them so that time increases from right to left. We define (A(t)) = Tr A(f)Peq where peq = p(t = 0) represents the equilibrium density matrix of the electron-phonon system. It is straightforward to see that for any two operators A and B we have... 

The standard methods of obtaining equations of motion for reduced systems are based on the projection-operator techniques developed by Zwanzig and Mori in the late 1950s [23,24]. In this approach one defines an operator, that acts on the full density matrix p to project out a direct product of cr and the thermalized equilibrium density matrix of the... 

It should be noted that the centroid density is distinctly different from the coordinate (or particle) density p( ) = ((jlexp(-)3//) (5f). The particle density function is the diagonal element of the equilibrium density matrix in the coordinate representation, while the centroid density does not have a similar physical interpretation. However, the integration over either density yields the quantum partition function. 

Markov operator for the various rotational processes and Qeq is the equilibrium density matrix. 

In the basis of the Hamiltonian eigenstates, the thermal equilibrium density matrix constructed from Equation (2.5.7) is diagonal ... 

Conventional NMR deals with a large ensemble of spins. It means that the state of the system is in a statistical mixture, which is obviously inadequate for QIP. However, the NMR ability for manipulating spins states worked out by Cory et al. [24] and Chuang et al. [23] resulted in elegant methods for creating the so called effectively pure or pseudo-pure states. Behind the idea of the pseudo-pure states is the fact that NMR experiments are only sensitive to the traceless deviation density matrix. Thus, we might search for transformations that, applied to the thermal equilibrium density matrix, produce a deviation density matrix with the same form as a pure state density matrix. Once such state is created, all remaining unitary transformations will act only on such a deviation density matrix, which will transform as a true pure state. 

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