Thermal equilibrium density matrix

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. 

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 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... 

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. 

The pseudo-pure state preparation by temporal averaging in quadrupolar nuclei can be done in a similar way. To illustrate the procedure lets take a two-qubit system implemented by spin 3/2 nuclei. The corresponding thermal equilibrium density matrix is given by ... 

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 ... 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

Now that the entanglement of the XY Hamiltonian with impurities has been calculated at Y = 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble p = jZ, where = l/k T, and Z = Tr is the partition function. The thermal density matrix is diag-... 

The total system is subject to the Hamiltonian (3.1). Its thermal equilibrium has the density matrix... 

C. Spin density matrix of a spin system at thermal equilibrium with a lattice. 231... 

If a spin system in thermal equilibrium with the lattice (i.e. that described by the vector p,) is subjected, immediately prior to the moment t = 0, to a strong radio-frequency pulse, then, after the pulse ceases, the vector corresponding to the density matrix of the system becomes (23,99)... 

In a pulse-type NMR experiment the density matrix p(t) of a nonexchanging system approaches the thermal equilibrium p which is described satisfactorily by equation (16). The same matrices p0 approximate, with good accuracy, the time-independent solution of the system of differential equations (72). This can be inferred from equation (99) which is derived from equations (95a) and (95b) ... 

In the above, the principles of how to study the dynamics of an isolated system by using the density matrix method have been shown. However, most experiments are performed for the system (or subsystem) embedded in a heat bath in this case the isolated system consists of the system plus heat bath. In the following the MEs shall be derived for the system embedded in heat bath. In this case, instead of pm, pnsnbfls b will be employed. Here s and b describe the system and heat bath, respectively. For the case in which the bath is much larger than the system, it may be assumed that the bath maintains thermal equilibrium and... 

Besides, observe that the probabilities pk are at thermal equilibrium with the matrix elements of the Boltzmann density operator that is diagonal with respect to k), ... 

The strong role of collisions in decohering a system is readily seen by considering fie density matrix pj of a system that has reached thermal equilibrium at tempera-rs T through collisional relaxation, that is, pf = Qexp(—Hs./kgT). Here Hs is the system Hamiltonian, Q is a normalization factor, and kB is the Boltzmann constant, fmsiderable insight is obtained if we cast the density matrix in the energy repre-... 

If the oscillator is weakly coupled to the bath, in canonical thermal equilibrium the probability of finding the oscillator in the nth state is of course P q = e / En/Zq, where ft = I/kT and the oscillator s canonical partition function is Zq = e In addition, the oscillator s off-diagonal (in this energy representation) density matrix elements are zero. The average oscillator energy (in thermal equilibrium) is Eeq = n13nPnq-... 

The computational procedure follows closely the steps of an actual m.p. experiment see Fig. 1. The spin system, which is initially in thermal equilibrium, is hit by a preparation pulse Pp. Thereafter, one component of the transverse nuclear magnetization created by Pp, say My, is measured and the measurement is repeated at intervals of the cycle time The resulting time series My(qtJ,q = 0,...,(2 " - 1), if Fourier transformed. For simulations we accordingly first specify the initial condition of the spin system, that is, the initial value of the spin density matrix g(t) in the rotating frame. Our standard choice Pp, = P implies p(0) fy == the sum running over k = We then follow the evolution... 

We will show that this evolution equation guarantees that the equivalence relation (Equation (120)) is fulfilled, the density matrix proceeds to an equilibrium state and that the entropy of the ensemble of the damped oscillator proceeds the maximum value over time, which corresponds to thermal equilibrium. Indeed, the proof of the relations in Equation (120) proceeds as follows... 

If a system has reached thermal equilibrium, the amplitudes obtained from equation (9) multiplied with their complex conjugates, IC P will represent the statistical weight of the corresponding basis state in the equihbrium density matrix. This is a natural result for the Q C scheme. A statistical approach, however, ignores the role played by the excited states (among them, the present diabatic TSs) throng their coupling to closed-shell chemical species represented as electronic class-1 GED states. 

When discussing the general aspects of FTNMR, we have to remember that all principal statements about Fourier methods have been introduced for a strictly linear system (mechanical oscillator) in Chapter 1. In Chapter 2, on the other hand, we have seen that the nuclear spin system is not strictly linear (with Kramer-Kronig-relations between absorption mode and dispersion mode signal >). Moreover, the spin system has to be treated quantummechanically, e.g. by a density matrix formalism. Thus, the question arises what are the conditions under which the Fourier transform of the FID is actually equivalent to the result of a low-field slow-passage experiment Generally, these conditions are obeyed for systems which are at thermal equilibrium just before the initial pulse but are mostly violated for systems in a non-equilibrium state (Oberhauser effect, chemically induced dynamic nuclear polarization, double resonance experiments etc.). 

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