By applying the standard adiabatic approximation and using the accurate analytical expression for the corresponding local chemical potential obtained in our previous work [Phys. Rev. A 75, 063610 (2007)] we derive an effective 1D equation that governs the axial dynamics of mean-field cigar-shaped condensates with repulsive interatomic interactions, accounting accurately for the contribution from the transverse degrees of freedom. This equation, which is more simple than previous proposals, is also more accurate. Moreover, it allows treating condensates containing an axisymmetric vortex with no additional cost. Our effective equation also has the correct limit in both the quasi-1D mean-field regime and the Thomas-Fermi regime and permits one to derive fully analytical expressions for ground-state properties such as the chemical potential, axial length, axial density profile, and local sound velocity. These analytical expressions remain valid and accurate in between the above two extreme regimes. Following the same procedure we also derive an effective 2D equation that governs the transverse dynamics of mean-field disk-shaped condensates. This equation, which also has the correct limit in both the quasi-2D and the Thomas-Fermi regime, is again more simple and accurate than previous proposals. We have checked the validity of our equations by numerically solving the full 3D Gross-Pitaevskii equation.

A self-adjoint operator with dimensions of time is explicitly constructed, and it is shown that its complete and orthonormal set of eigenstates can be used to define consistently a probability distribution of the time of arrival at a spatial point.Comment: 25 pages, LaTeX, no figures. A very minor correction to tex

We derive general approximate formulas that provide with remarkable accuracy the ground-state properties of any mean-field scalar Bose-Einstein condensate with short-range repulsive interatomic interactions, confined in arbitrary cylindrically symmetric harmonic traps. Our formulation is even applicable for condensates containing a multiply quantized axisymmetric vortex. We have checked the validity of our formulas by numerically solving the 3D Gross-Pitaevskii equation.

The recent experiment by Y. Shin et al. [Phys. Rev. Lett. 93, 160406 (2004)] on the decay of a doubly quantized vortex imprinted in 23 Na condensates is analyzed by numerically solving the Gross-Pitaevskii equation. Our results, which are in very good quantitative agreement with the experiment, demonstrate that the vortex decay is mainly a consequence of dynamical instability. Despite apparent contradictions, the local density approach is consistent with the experimental results. The monotonic increase observed in the vortex lifetimes is a consequence of the fact that, for large condensates, the measured lifetimes incorporate the time it takes for the initial perturbation to reach the central slice. When considered locally, the splitting occurs approximately at the same time in every condensate, regardless of its size.PACS numbers: 03.75. Kk, 03.75.Lm, 67.90.+z Since the creation of the first dilute-gas Bose-Einstein condensates there has been great interest in characterizing their superfluid properties. This has stimulated a great deal of theoretical and experimental work aimed at studying the rotational properties of dilute Bose gases [1] and, in particular, the nucleation and stability properties of vortices [2]. Numerous experiments have succeeded in generating vortices [3]. In practically all of them, the vorticity appears concentrated in a number of singly quantized vortices. This is a consequence of the fact that multiply quantized vortices are energetically unstable against their splitting in an array of vortices with unit topological charge [4]. Multiply quantized vortices are also dynamically unstable, which implies that they decay even in the zero-temperature limit [5].Recently, Leanhardt et al.[6] obtained multiply quantized vortices in a gaseous Bose-Einstein condensate by using a topological phase-imprinting technique proposed by Nakahara et al. [7]. They started from nonrotating 23 Na condensates in the |1, −1 > and |2, +2 > hyperfine states. By adiabatically inverting the magnetic bias field, the initial states were transformed into vortex states with axial angular momentum per particle 2 and −4 , respectively. This has opened the possibility for studying experimentally the stability of multiply quantized vortices, and has stimulated new theoretical works. In particular, Möttönen et al. [8] have studied numerically the stability of a doubly quantized vortex in a cylindrical condensate as a function of the (dimensionless) interaction strength per unit length along the condensate axis, an z , where a is the s-wave scattering length and n z = |Ψ(r)| 2 dx dy is the linear atom density along z. They found a series of quasiperiodic instability regions in the parameter space of an z [5]. The first two of them (the only ones relevant to this work) correspond to an z < 3 and 11.4 an z 16, respectively. By comparing with the solution of the Gross-Pitaevskii equation for a harmonically trapped cigar-shaped condensate these authors conclude that a doubly quantized vortex is dynamically unstable, and it is...

Leonhardt demonstrated (2009) that the 2D Maxwell Fish Eye lens (MFE) can perfectly focus 2D Helmholtz waves of arbitrary frequency, i.e., it can perfectly transport an outward (monopole) 2D Helmholtz wave field, generated by a point source, towards a "perfect point drain" located at the corresponding image point. Moreover, a prototype with λ/5 super-resolution property for one microwave frequency has been manufactured and tested . However, software simulations or experimental measurements for a broad band of frequencies have not yet been reported. Here we present simulations with a non-perfect drain for a device equivalent to the MFE, called the Spherical Geodesic Waveguide (SGW), that predicts up to λ /500 super-resolution close to discrete frequencies. These frequencies are directly connected with the wellknown Schumann resonance frequencies of spherical symmetric systems. Out of these frequencies, the SGW does not show super-resolution in the analysis performed.