Dynamics of an entangled pair of spin-1/2 particles under correlated random magnetic fields.

The dynamics of a maximally entangled pair of spin-1/2 particles is obtained in the presence of random magnetic fields which are correlated. The two spin-1/2 particles are assumed to be maximally entangled initially and are then disturbed by the magnetic fields modeled as Gaussian vector random processes whose corresponding spatial components are correlated. The dynamics is derived in terms of the joint density matrix of the entangled pair using the ideas of stochastic calculus, from which the steady-state density matrix and the associated timescale for it to be reached are obtained. The asymptotic density matrix represents a state of (partial) disentanglement.

Dynamics of entangled pair of spin-1/2 particles under independent random magnetic fields.

The dynamics of a pair of spatially separated spin-1/2 particles which are initially entangled and subjected to independent random magnetic fields is obtained by considering a pure joint state of entangled spin-1/2 particles and using stochastic calculus. The disentanglement time defined as the time taken for the particles to become disentangled, is obtained from the dynamics. In view of entanglement-based quantum communication, this analysis could help to assess the feasibility to send and receive quantum information within the disentanglement time.

Dynamical theory of spin noise and relaxation: Prospects for real-time NMR measurements.

Recent developments in theoretical aspects of spin noise and relaxation and their interrelationship reveal a modified spin density, distinct from the density matrix, as the necessary object to describe fluctuations in spin systems. These fluctuations are to be viewed as an intrinsic quantum mechanical property of such systems immersed in random magnetic environments and are observed as "spin noise" in the absence of any radio frequency excitation. With the prospect of ultrafast digitization, the role of spin noise in real-time parameter extraction for (NMR) spin systems, and the advantage over standard techniques, is of essential importance, especially for systems containing a small number of spins. In this article we outline prospects for harnessing the recent dynamical theory in terms of spin-noise measurement, with attention to real-time properties.

Singularities in the lineshape of a second-order perturbed quadrupolar nucleus. The magic-angle spinning case.

For a nucleus with a half-integral spin and a strong quadrupole coupling, the central transition (from magnetic quantum number -1/2 to +1/2) in the spectrum shows a characteristic lineshape. By strong coupling, we mean an interaction strong enough so that second-order perturbation theory is needed, yet still sufficient. The spectrum of a static sample is well-known and the magic-angle-spinning (MAS spectrum) is different, but still can be calculated. The important features of both these spectra are singularities and steps in the lineshape, since these are the main tools in fitting the calculated spectrum to experimental data. A useful tool in this investigation is a plot of the frequency as a function of orientation over the surface of the unit sphere. These plots have maxima, minima and saddle points, and these correspond to the features of the spectrum. We used these plots to define both the positions and derive new formulae for the heights of the features and we now extend this to the magic-angle spinning case. For the first time, we identify the orientations corresponding to the features of the MAS spectra and derive formulae for the heights. We then compare the static and MAS cases and show the relationships between the features in the two spectra.

Singularities in the lineshape of a second-order perturbed quadrupolar nucleus and their use in data fitting.

Even for large quadrupolar interactions, the powder spectrum of the central transition for a half-integral spin is relatively narrow, because it is unperturbed to first order. However, the second-order perturbation is still orientation dependent, so it generates a characteristic lineshape. This lineshape has both finite step discontinuities and singularities where the spectrum is infinite, in theory. The relative positions of these features are well-known and they play an important role in fitting experimental data. However, there has been relatively little discussion of how high the steps are, so we present explicit formulae for these heights. This gives a full characterization of the features in this lineshape which can lead to an analysis of the spectrum without the usual laborious powder average. The transition frequency, as a function of the orientation angles, shows critical points: maxima, minima and saddle points. The maxima and minima correspond to the step discontinuities and the saddle points generate the singularities. Near a maximum, the contours are ellipses, whose dimensions are determined by the second derivatives of the frequency with respect to the polar and azimuthal angles. The density of points is smooth as the contour levels move up and down, but then drops to zero when a maximum is passed, giving a step. The height of the step is determined by the Hessian matrix-the matrix of all partial second derivatives. The points near the poles and the saddle points require a more detailed analysis, but this can still be done analytically. The resulting formulae are then compared to numerical simulations of the lineshape. We expand this calculation to include a relatively simple case where there is chemical shielding anisotropy and use this to fit experimental (139)La spectra of La2O3.

Dynamical theory of spin relaxation.

The dynamics of a spin system is usually calculated using the density matrix. However, the usual formulation in terms of the density matrix predicts that the signal will decay to zero, and does not address the issue of individual spin dynamics. Using stochastic calculus, we develop a dynamical theory of spin relaxation, the origins of which lie in the component spin fluctuations. This entails consideration of random pure states for individual protons, and how these pure states are correctly combined when the density matrix is formulated. Both the lattice and the spins are treated quantum mechanically. Such treatment incorporates both the processes of spin-spin and (finite temperature) spin-lattice relaxation. Our results reveal the intimate connections between spin noise and conventional spin relaxation.

Design, synthesis, and evaluation of proline and pyrrolidine based melanocortin receptor agonists. A conformationally restricted dipeptide mimic approach.

The design, synthesis, and structure-activity relationships (SAR) of a series of novel proline and pyrrolidine based melanocortin receptor (MCR) agonists are described. To validate a conformationally constrained Arg-Nal dipeptide analogue strategy, we first synthesized and evaluated a test set of cis-(2R,4R)-proline analogues (21a-g). All of these compounds showed significant binding and agonist potency at the hMC1R, hMC3R, and hMC4R. Potent cis-(2S,4R)-pyrrolidine based MCR agonists (35a-g) were subsequently developed by means of this design approach. A SAR study directed toward probing the effect of the two chiral centers in the pyrrolidine ring on biological activity revealed the importance of the (S) absolute configuration at the 2-position for binding affinity, agonist potency, and receptor selectivity. Among the four sets of the pyrrolidine diastereomers investigated, analogues with the (2S,4R) configuration were the most potent agonists across the three receptors, followed by those possessing the (2S,4S) configuration.

Design, synthesis, and evaluation of proline based melanocortin receptor ligands.

A series of proline based melanocortin ligands has been developed on the basis of initial piperazine leads by using a more conformationally rigid scaffold. A number of these novel ligands showed significant binding affinity for MC3 and MC4 receptors.