Extremal values of the (fractional) Weinstein functional on the hyperbolic space

Abstract We study Weinstein functionals, first defined in [33], mainly on the hyperbolic space ℍ n {\mathbb{H}^{n}} . We are primarily interested in the existence of Weinstein functional maximizers or, in other words, existence of extremal functions for the best constant of the Gagliardo–Nirenberg inequality. The main result is that the supremum of the Weinstein functional on ℍ n {\mathbb{H}^{n}} is the same as that on ℝ n {\mathbb{R}^{n}} and the related fact that the said supremum is not attained on ℍ n {\mathbb{H}^{n}} , when functions are chosen from the Sobolev space H 1 ⁢ ( ℍ n ) {H^{1}(\mathbb{H}^{n})} . This proves a conjecture made in [8] (see also [3]). We also prove an analogous version of the conjecture for the Weinstein functional defined with the fractional Laplacian.</jats:p

Quantum metrology to probe atomic parity nonconservation

An entangled state prepared in a decoherence free sub-space together with a Ramsey type measurement can probe parity violation in heavy alkali ions like Ba+ or Ra+. Here we propose an experiment with Ba+ as an example to measure the small parity violating effect in this system. It has been shown that a measurement on a maximally correlated system will reduce the uncertainty as compared to that on a single ion measurement. In addition it also provides a feasible solution to measure the nuclear spin dependent part of the total parity violating light shift in an ionic system which has so far not been addressed.Comment: 4 pages, 3 figure

Dynamics of ion cloud in a linear Paul trap

A linear ion trap setup has been developed for studying the dynamics of trapped ion cloud and thereby realizing possible systematics of a high precision measurement on a single ion within it. The dynamics of molecular nitrogen ion cloud has been investigated to extract the characteristics of the trap setup. The stability of trap operation has been studied with observation of narrow nonlinear resonances pointing out the region of instabilities within the broad stability region. The secular frequency has been measured and the motional spectra of trapped ion oscillation have been obtained by using electric dipole excitation. It is applied to study the space charge effect and the axial coupling in the radial plane.Comment: Colloquium paper, 9 pages, 6 figure

Non-equilibrium phonon dynamics in trapped ion systems

We propose a concrete experiment to probe the non-equilibrium local dynamics of the one-dimensional Bose-Hubbard model using a trapped ion system consisting of a linear chain of few Ba^+ ions prepared in a state of transverse motional mode which corresponds to a fixed number of phonons per ion. These phonons are well-known to be described by an effective Bose-Hubbard model. We propose a protocol which leads to a sudden local sign reversal of the on-site interaction strength of this Hubbard model at one of the sites and demonstrate that the subsequent non-equilibrium dynamics of the model can be experimentally probed by measuring the time-dependent phonon number in a specific motional state of the Ba+ ions. We back our experimental proposal with exact numerical calculation of the dynamics of a Bose-Hubbard model subsequent to a local quench.Comment: The submission contains 5 pages and 4 figure