Quantum-mechanical signal processing

Dz. Belkic, Quantum-Mechanical Signal Processing and Spectral Analysis, Institute of Physics Publishing, Series in Atomic and Molecular Physics, Bristol, 2005. 

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... 

Similar transient signals were obtained from time-dependent quantum mechanical calculations performed by Meier and Engel, which well reproduce the observed behavior [49]. They show that for different laser field strengths the electronic states involved in the multiphoton ionization (MPI) are differently populated in Rabi-type processes. In Fig. 13 the population in the neutral electronic states is calculated during interaction of the molecule with 60-fs pulses at 618 nm. For lower intensities the A state is preferentially populated by the pump pulse, and the A state wavepacket dominates the transient Na2+ signal. However, for the higher intensities used in the... 

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. 

At this point we need to consider that there is another process operating in this system. When the populations of the spin states have been disturbed from their equilibrium values, as in this case by irradiation of the proton signal, relaxation processes will tend to restore the populations to their equilibrium values. Unlike excitation of a spin from a lower to a higher spin state, relaxation process are not subject to the same quantum mechanical selection rules. Relaxation involving changes of both spins simultaneously (called double-quantum transitions) are allowed in fact they are relatively important in magnitude. The relaxation pathway labeled W2 in Fig. 4.6 tends to restore equilibrium populations by relaxing spins from state N4 to Ni. We shall represent the number of spins that are relaxed by this pathway by the symbol d. The populations of the spin states thus become as follows ... 

Spectrum Intensity of a signal due to a process such as optical absorption or emission displayed as a function of some varying characteristic such as wavelength, energy, or mass. Also used in quantum mechanics and applied mathematics to specify the pattern of eigenvalues of a linear operator and in electrodynamics to specify the range of frequencies of electromagnetic radiation. 

It is worth mentioning in advance a major difficulty that occurs while establishing the connection between the measured signal and the quantities of interest while the scattering process can only be treated quantum mechanically, the desired - and tractable - quantities (like the static and dynamic structure factor) are classical statistical averages. 

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