Rotational matrix, time-dependent

In the language of quanPim meehanies, the time-dependent B -field provides a perturbation with a nonvanishing matrix element joining the stationary states a) and P). If the rotating field is written in temis of an amplitude a perturbing temi in tlie Hamiltonian is obtained... 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

After transformation into the interaction picture and application of the rotating-wave approximation [46, SO, 54] the population dynamics can be calculated numerically by solving the time-dependent three-level Schrodinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by... 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

So far, the equations in this chapter are based on the laboratory frame of reference. In Section 2.8 we saw that the description of magnetic resonance can often be simplified by using a frame rotating with angular frequency coabout the z axis, where pulse frequency (and reference frequency) used to observe the spin system. Now we want to express the density matrix in the rotating frame in order to facilitate our handling of time-dependent Hamiltonians that arise when radio frequency fields are applied. 

P indicates that the components are those for the principal axis system (PAS) of the tensor. The terms o(QpL(f)) are Wigner rotation matrix elements. They are functions of the set of Euler angles, Qpf (f), which relates the PAS of the chemical shift to the laboratory frame. Due to MAS, these angles are time dependent. A full treatment of the orientation dependence of the chemical shift requires the transformation between several different reference frames. 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

It is convenient to transform into a rotating frame via the time-dependent unitary matrix... 

The time-dependent matrixelements (( ) 13/at (jtj) can be rewritten in the form of rotational- and radial-matrix elements which couple the amplitudes of the levels in the correlation diagram with the same projection of the angular momentum of the levels in case of the radial matrix elements and with -1-1 or -1 in case of the rotational matrix elements. For details on this procedure see Ref. 2. 

The relationship between T and and the J s can be derived through time dependent perturbation theory as was done originally by Bloembergen, Purcell, and Pound or by the use of the density matrix which turns out to be equivalent. See the general references in Appendix A, especially Chapter 5 of Slichter. For our case of two spins on a rotating molecule, the relaxation rates are given by... 

The BPP expression was derived under the assumptions that the dipolar interactions formed a perturbation on the Zeeman levels and that the time dependent part of the dipolar interaction could be treated by time dependent perturbation theory or equivalently the density matrix approach to determine the relaxation expression. This does not rule out a BPP type expression for relaxation in the rotating frame. Specifically, a BPP type approach can be used to derive the following expression for T due to rotational motion under basically... 

The HNCA pulse sequence has groups of three pulse sequences optimized for the J-coupling constants. The rotating frame description can be applied to parts of the pulse sequence (Fig. 7.22), but is insufficient to describe the entire sequence. With tools such as Mathematica or Matlab, it is straightforward to simulate the pulse sequence (Fig. 7.23). These tools allow for the evaluation of the exponent of a Hermitian matrix, a common step in time dependent quantum mechanics. 

Deviations from the above simple model will be used to show the further application of laser PES to the study of state perturbations and non-radiative processes. If a selected resonant intermediate level is significiantly mixed with another level, the Franck-Condon factors for both "isolated levels will contribute to the vibrational structure in the PES [3]. Variations in the PES as the laser is scanned over rotation band structure can be used to determine the rotational dependence of the interstate mixing matrix elements. State-to-state relaxation can also be seen by the time dependence of the PES. As the time between the state preparation and its subsequent ionization is increased, the non-radiative flow of energy amongst the coupled levels will be illustrated by the changes in the PES [4]. 

A solid matrix supplies an external field in which the potential energy U(r, 6, ) of the diatomic molecule depends on both the position r of its center of mass and its spatial orientation giwn by two angles 0 and < >. The translational and rotational degrees-of-fieedom may be coupled by means of the external field and it has to be determined to which extent tlte organic polymer matrices are capable of influencing the rotation of diatomic molecides. R ults of a Molecular Dynamics study of O2 dynamics in poly(isobutylene) (PIB) at 300 K indicate [59] that the rotation of the O2 molecules is weU separated from tlteir translational motion and the rotational correlation time x, i%0.1 ps derived from the Molecular Dynamics trajo tories [59] agrees well with the value of 0.15 ps deduced above one can conclude diat the PIB matrix d )es not affect the... 

Time-Dependent Treatment of SVRT Model for Polyatom-Polyatom Reaction where is the normalized rotation matrix defined as... 

The line shape of the ESR spectrum can be calculated using the spin Hamiltonian H (f) for the nitroxide radical. The time dependence of the spin Hamiltonian describes the orientation-dependent part which contains terms characterizing the anisotropy of the A- and g-tensors, and terms characterizing the rotational reorientation of the free radical as a classic stochastic process using the Wigner rotation matrix elements. A comprehensive theory of ESR spectra of nitroxides has been described step-by-step in contributions to two monographs and papers cited therein. Computer programs suitable for calculation of theoretical line shapes of ESR spectra measured in continuous wave (CW) and pulse experiments have become indispensable tools for practical applications. The ESR spectra of nitroxides are also described in chapters 1 and 3 in this volume. 

The density matrix representation of spin and orbital angular momentum is capable of expressing a static state of matter and its time-dependent response to an external perturbation. Our application necessitates that we follow the response of the orbital and spin momenta subject to full or partial excitations, and the density matrix provides a direct solution to the stochastic Liouville equation. But the density matrix representation in a rotating operator is algebraically ambiguious, and we must also clarify the algebraic description of selective excitation of multiquantum systems. 

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