We demonstrate coherence amplification in off-diagonal density matrix elements for a decoherence model for spin ½ particles with applied radio-frequency pulse without relaxation. The method of coherence amplification developed may be of general utility in applications in NMR/MRI.
Detecting quantum coherence Observing quantum coherence is a tricky business. Even before we fire up the femtosecond laser systems and start our search, we need to ask ourselves some basic questions: What is coherence? How does coherence manifest in measurements? What methods can we use to detect and measure coherence? Only then, can we begin to interpret data to search for coherence and to try to make sense of the observed signals. In this section, I will discuss detecting quantum coherence both in theory and in practice. The goal of this section is to present a unified view of experimental approaches in broad strokes and to provide useful references to guide further exploration. What is ‘quantum coherence’? Firstly, the precise meaning of ‘quantum coherence’ must be pinned down. I define quantum coherences to be off-diagonal elements of the density matrix representing the ensemble. (I am explicitly setting aside questions of ‘quantum’ versus ‘classical’ coherences; let us simply assume that the system in question is best described quantum mechanically and presume that the coherences in such a system are also quantum in nature. The simple fact is that we don't yet have the tools to answer this debate definitively.) This definition, however, is still insufficient. The density matrix and the magnitudes of its off-diagonal elements in particular depend on the basis set used to write down the matrix. For the purposes of this chapter, I will consider the density matrix only in the Hamiltonian eigenbasis.
A method for performing quantum state tomography for quadrupole nuclei is presented in this paper. First, it is shown that upon appropriate phase cycling, the nuclear-magnetic-resonance (NMR) intensities of quadrupole nuclei depend only on diagonal elements of the density-matrix. Thus, a method for obtaining the density-matrix elements, which consists of dragging off-diagonal elements into the main diagonal using fine phase-controlled selective radio-frequency pulses, was derived. The use of the method is exemplified through 23Na NMR (nuclear spin I=3∕2) in a lyotropic liquid crystal at room temperature, in three applications: (a) the tomography of pseudopure states, (b) the tomography of the quadrupole free evolution of the density matrix, and (c) the unitary state evolution of each qubit in the system over the Bloch sphere upon the application of a Hadamard gate. Further applications in the context of pure NMR and in the context of quantum information processing, as well as generalizations for higher spins, are discussed.
Decoherence and relaxation processes in quantum systems are studied on the basis of an exactly solvable model. That is, a generalized Coleman-Hepp model is analyzed in a framework of a spin coherent state representation which enables us to determine time evolution of the whole system as successive applications of rotation operators. A density matrix of the whole system and reduced density matrices of constituent subsystems are obtained to yield quasi-probability densities and averages of observables of an incident particle and a detector (reservoir) system. Numerical evaluation of these quantities is performed to find decoherence like phenomena for larger values of N (a number of detector spins) and S l (magnitude of a detector spin). An appearance of a mixed state (i.e., vanishing of off-diagonal elements of a reduced matrix) is explicitly shown when N and/or S l become infinity. Namely, this occurs in the thermodynamic limit ( N →∞) and/or in the classical apparatus limit ( S l →∞).