Vortex lattice stability and phase coherence in three-dimensional rapidly rotating Bose condensates

We establish the general equations of motion for the modes of a vortex lattice in a rapidly rotating Bose-Einstein condensate in three dimensions, taking into account the elastic energy of the lattice and the vortex line bending energy. As in two dimensions, the vortex lattice supports Tkachenko and gapped sound modes. In contrast, in three dimensions the Tkachenko mode frequency at long wavelengths becomes linear in the wavevector for any propagation direction out of the transverse plane. We compute the correlation functions of the vortex displacements and the superfluid order parameter for a homogeneous Bose gas of bounded extent in the axial direction. At zero temperature the vortex displacement correlations are convergent at large separation, but at finite temperatures, they grow with separation. The growth of the vortex displacements should lead to observable melting of vortex lattices at higher temperatures and somewhat lower particle number and faster rotation than in current experiments. At zero temperature a system of large extent in the axial direction maintains long range order-parameter correlations for large separation, but at finite temperatures the correlations decay with separation.Comment: 10 pages, 2 figures, Changes include the addition of the particle density - vortex density coupling and the correct value of the shear modulu

Dislocation-Mediated Melting in Superfluid Vortex Lattices

We describe thermal melting of the two-dimensional vortex lattice in a rotating superfluid by generalizing the Halperin and Nelson theory of dislocation-mediated melting. and derive a melting temperature proportional to the renormalized shear modulus of the vortex lattice. The rigid-body rotation of the superfluid attenuates the effects of lattice compression on the energy of dislocations and hence the melting temperature, while not affecting the shearing. Finally, we discuss dislocations and thermal melting in inhomogeneous rapidly rotating Bose-Einstein condensates; we delineate a phase diagram in the temperature -- rotation rate plane, and infer that the thermal melting temperature should lie below the Bose-Einstein transition temperature.Comment: 9 pages, 2 figure

Моделювання рецептур сироватково-полуничних напоїв профілактичного призначення

Expediency of the development of formulae and innovative technologies for production of prophylactic application drinks possessing antioxidant, probiotic and hepatoprotective properties with the use of the secondary dairy product – whey, as well as the domestic vegetable raw materials having a high content of bioactive substances has been substantiated. Formulation composition of the prophylactic drinks based on cheese whey, extract of Tagetes patula flowers and the berry filler “Strawberry” with the use of the response surface method has been developed. Bioactivity of the drinks and the complex quality indicator which accounts for the total influence of the bioactivity, organoleptic assessment and weight coefficients of the specified unit indicators were taken as the optimization criteria; as the independent factors that were varied in the course of the experiment, the mass fractions of the marigold flowers extract and the strawberries filler were selected. It is recommended that the mass fractions of the berry filler “Strawberry” and the extract of Tagetes patula flowers in the prophylactic drinks are set as 7 and 20 % of the finished product, accordingly. The practical mass fraction of the citric acid of 0.2 % was determined as it ensures high organoleptic characteristics of the finished drinks. Recommendations are given concerning development of innovative technologies of unfermented and fermented strawberry whey drinks of prophylactic application enriched with the extract of Tagetes patula flowers.Обґрунтовано доцільність розроблення рецептур та інноваційних технологій напоїв профілактичного призначення з антиоксидантними, пробіотичними та гепатопротекторними властивостями з використанням вторинної молочної сировини – сироватки, а також вітчизняної сировини рослинного походження з високим вмістом біологічно активних речовин. Розроблено рецептурний склад напоїв профілактичного призначення на основі сирної сироватки, екстракту квітів Tagetes patula і ягідного наповнювача «Полуниця» з використанням методології поверхні відгуку. У якості критеріїв оптимізації обрано біологічну активність напоїв і комплексний показник якості, який враховує сукупний вплив біологічної активності, органолептичної оцінки і коефіцієнтів вагомості зазначених одиничних показників; незалежними факторами, які варіювалися, в експерименті – масові частки екстракту квітів чорнобривців і полуничного наповнювача. Рекомендовано масову частку ягідного наповнювача «Полуниця» та екстракту квітів Tagetes patula в напоях профілактичного призначення встановити 7 і 20 % від маси готового продукту відповідно. Визначено раціональну масову частку лимонної кислоти – 0,2 %, яка забезпечує високі органолептичні характеристики готових напоїв. Наведено рекомендації з розробки інноваційних технологій неферментованих і ферментованих сироватково-полуничних напоїв профілактичного призначення, збагачених екстрактом квітів Tagetes patula

Pinning and collective modes of a vortex lattice in a Bose-Einstein condensate

We consider the ground state of vortices in a rotating Bose-Einstein condensate that is loaded in a corotating two-dimensional optical lattice. Due to the competition between vortex interactions and their potential energy, the vortices arrange themselves in various patterns, depending on the strength of the optical potential and the vortex density. We outline a method to determine the phase diagram for arbitrary vortex filling factor. Using this method, we discuss several filling factors explicitly. For increasing strength of the optical lattice, the system exhibits a transition from the unpinned hexagonal lattice to a lattice structure where all the vortices are pinned by the optical lattice. The geometry of this fully pinned vortex lattice depends on the filling factor and is either square or triangular. For some filling factors there is an intermediate half-pinned phase where only half of the vortices is pinned. We also consider the case of a two-component Bose-Einstein condensate, where the possible coexistence of the above-mentioned phases further enriches the phase diagram. In addition, we calculate the dispersion of the low-lying collective modes of the vortex lattice and find that, depending on the structure of the ground state, they can be gapped or gapless. Moreover, in the half-pinned and fully pinned phases, the collective mode dispersion is anisotropic. Possible experiments to probe the collective mode spectrum, and in particular the gap, are suggested.Comment: 29 pages, 4 figures, changes in section