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We introduce the notion of a partial corepresentation of a given Hopf algebra $H$ over a coalgebra $C$ and the closely related concept of a partial $H$-comodule. We prove that there exists a universal coalgebra $H^{par}$, associated to the original Hopf algebra $H$, such that the category of regular partial $H$-comodules is isomorphic to the category of $H^{par}$-comodules. We introduce the notion of a Hopf coalgebroid and show that the universal coalgebra $H^{par}$ has the structure of a Hopf coalgebroid over a suitable coalgebra., Comment: 41 papes, v2:several small corrections, final version as accepted for publication in Journal of Algebra

We introduce the notion of a dilation for a partial representation (i.e. a partial module) of a Hopf algebra, which in case the partial representation origins from a partial action (i.e.a partial module algebra) coincides with the enveloping action (or globalization). This construction leads to categorical equivalences between the category of partial $H$-modules, a category of (global) $H$-modules endowed with a projection satisfying a suitable commutation relation and the category of modules over a (global) smash product constructed upon $H$, from which we deduce the structure of a Hopfish algebra on this smash product. These equivalences are used to study the interactions between partial and global representation theory., Comment: 25 pages. Corrected several typos, final version to appear in Journal of the London Mathematical Society

We apply the global thermodynamic variables approach to experimentally determine the isothermal compressibility parameter $\kappa_T$ of a trapped Bose gas across the phase transition. We demonstrate the behavior of $\kappa_T$ around the critical pressure, revealing the second order nature of the phase transition. Compressibility is the most important susceptibility to characterize the system. The use of global variables shows advantages with respect to the usual local density approximation method and can be applied to a broad range of situations., Comment: 6 pages, 3 figures

Phase transitions are well understood and generally followed by the behavior of the associated thermodynamic quantities, such as in the case of the $\lambda$ point superfluid transition of liquid helium, which is observed in its heat capacity. In the case of a trapped Bose-Einstein condensate (BEC), the heat capacity cannot be directly measured. In this work, we present a technique able to determine the global heat capacity from the density distribution of a weakly interacting gas trapped in an inhomogeneous potential. This approach represents an alternative to models based on local density approximation. By defining a pair of global conjugate variables, we determine the total internal energy and its temperature derivative, the heat capacity. We then apply the technique to a trapped $^{87}$Rb BEC a $\lambda$-type transition dependent on the atom number is observed, and the deviations from the non-interacting, ideal gas case are discussed. Finally we discuss the chances of using this method to study the heat capacity at $T \rightarrow 0$., Comment: 9 pages, 4 figures

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which has the universal property that each partial representation of $H$ can be factorized by an algebra morphism from $H_{par}$. We define also the category of partial modules over a Hopf algebra $H$, which is the category of modules over its associated Hopf algebroid $H_{par}$. The Hopf algebroid structure of $H_{par}$ enables us to enhance the category of partial $H$ modules with a monoidal structure and such that the algebra objects in this category are the usual partial actions. Some examples of categories of partial $H$ modules are explored. In particular we can describe fully the category of partially $\mathbb{Z}_2$-graded modules., Comment: 39 pages, version 2: some additions in the section on partial gradings and minor corrections elsewhere

In this work, the notion of a twisted partial Hopf action is introduced as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras. The conditions on partial cocycles are established in order to construct partial crossed products, which are also related to partially cleft extensions of algebras. Examples are elaborated using algebraic groups.

In this note, we present an analogue for codes in vector spaces with a Rosenbloom-Tsfasman metric of the well-known standard form of generator matrices for codes in spaces with the Hamming metric.

We have created a long-lived (~ 40 s) persistent current in a toroidal Bose-Einstein condensate held in an all-optical trap. A repulsive optical barrier creates a tunable weak link in the condensate circuit, which can affect the current around the loop. Superflow stops abruptly at a barrier strength such that the local flow velocity exceeds a critical velocity. The measured critical velocity is consistent with dissipation due to the creation of vortex-antivortex pairs. This system is the first realization of an elementary closed-loop atom circuit., Comment: 5 pages, 3 figures. Accepted for publication in PRL

We study the phase-fluctuating condensate regime of ultra-cold atoms trapped in a ring-shaped trap geometry, which has been realized in recent experiments. We first consider a simplified box geometry, in which we identify the conditions to create a state that is dominated by thermal phase-fluctuations, and then explore the experimental ring geometry. In both cases we demonstrate that the requirement for strong phase fluctuations can be expressed in terms of the total number of atoms and the geometric length scales of the trap only. For the ring-shaped trap we discuss the zero temperature limit in which a condensate is realized where the phase is fluctuating due to interactions and quantum fluctuations. We also address possible ways of detecting the phase fluctuating regime in ring condensates., Comment: 10 pages, 5 figures, minor edits

Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to prove the existence of enveloping actions, i.e., every partial Hopf action on a algebra A is induced by a Hopf action on a algebra B that contains A as a right ideal. This globalization theorem allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we prove a dual version of the globalization theorem: that every partial coaction of a Hopf algebra admits an enveloping coaction. We also show how this works on a series of examples which go beyond partial group actions. Finally, we explore some consequences of globalization theorems in order to present versions of the duality theorems of Cohen-Montgomery and Blattner-Montgomery for partial Hopf actions., Comment: A second version of the duality theorems of Blattner-Montgomery and Cohen-Montgomery is added. Paper to be published in Contemporary Mathematics

Let V be an n-dimensional vector space over a finite field F_q. We consider on V the $\pi$-metric recently introduced by K. Feng, L. Xu and F. J. Hickernell. In this short note we give a complete description of the group of symmetries of V under the $\pi$-metric.

Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we explore some properties of the fixed point subalgebra with relations to a partial action of a Hopf algebra. We also construct, for partial actions of finite dimensional Hopf algebras a Morita context relating the fixed point subalgebra and the partial smash product. This is a generalization of a well known result in the theory of Hopf algebras for the case of partial actions. Finally, we study Hopf-Galois extensions and reobtain some classical results in the partial case., Comment: revised version with a new section on Partial Hopf-Galois theory added. To be published in "Algebra and Discrete Mathematics"

Motivated by partial group actions on unital algebras, in this article we extend many results obtained by Exel and Dokuchaev to the context of partial actions of Hopf algebras, according to Caenepeel and Jansen. First, we generalize the theorem about the existence of an enveloping action, also known as the globalization theorem. Second, we construct a Morita context between the partial smash product and the smash product related to the enveloping action. Third, we dualize the globalization theorem to partial coactions and finally, we define partial representations of Hopf algebras and show some results relating partial actions and partial representations., Comment: 27 pages; revised version with minor corrections, mainly typos

We have realized a conical matter wave lens. The repulsive potential of a focused laser beam was used to launch a Bose-Einstein condensate into a radially expanding wavepacket whose perfect ring shape was ensured by energy conservation. In spite of significant interactions between atoms, the spatial and velocity widths of the ring along its radial dimension remained extremely narrow, as also confirmed by numerical simulations. Our results open the possibility for cylindrical atom optics without the perturbing effect of mean-field interactions., Comment: 11 pages, 5 figures, Multimedia files (movies) available in our website: http://www.physics.gatech.edu/chandra/index.htm

Gaseous Bose-Einstein condensates (BECs) have become an important test bed for studying the dynamics of quantized vortices. In this work we use two-photon Doppler sensitive Bragg scattering to study the rotation of sodium BECs. We analyze the microscopic flow field and present laboratory measurements of the coarse-grained velocity profile. Unlike time-of-flight imaging, Bragg scattering is sensitive to the direction of rotation and therefore to the phase of the condensate. In addition, we have non-destructively probed the vortex flow field using a sequence of two Bragg pulses., Comment: 13 pages, 5 figures. Invited paper submitted to a special issue on "Nonlinear Waves" of the (Elsevier) journal 'Math. Comput. Simul.', for participants in the 4th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena (2005). Visit our website at http://www.physics.gatech.edu/chandra for additional information

We have created a Bose-Einstein condensate whose spin orientation is metastable. Condensates were transferred into a quadrupole magnetic trap, where Majorana transitions limited the lifetime to a few hundred milliseconds, about 30 times the trapping period. Atoms held in the trap frequently displayed a ring-shaped time-of-flight distribution. We speculate that such a ring could be either a quantized vortex or a feature of the Majorana loss dynamics in the quantum regime.

In this paper we extend previous hydrodynamic equations, governing the motion of Bose-Einstein-condensed fluids, to include temperature effects. This allows us to analyze some differences between a normal fluid and a Bose-Einstein-condensed one. We show that, in close analogy with superfluid He-4, a Bose-Einstein-condensed fluid exhibits the mechanocaloric and thermomechanical effects. In our approach we can explain both effects without using the hypothesis that the Bose-Einstein-condensed fluid has zero entropy. Such ideas could be investigated in existing experiments., Comment: 19 pages, RevTex 4

Using a system composed of a Quadrupole and Ioffe Configuration (QUIC) trap loaded from a slowed atomic beam, we have performed experiments to observe the Bose-Einstein Condensation of Na atoms. In order to obtain the atomic distribution in the trap, we use an in situ out of resonance absorption image through a probe beam, to determine temperature and density. The phase space density (D) is calculated using the density profile and the temperature. We have followed D as a function of the final evaporation frequency. The results show that at 1.65 MHz we crossed the value for D expected to correspond to the critical point to start de Bose-Condensation of the sample. Due to the low number of atoms remaining in the trap at the critical point, the interaction produces minor effects and therefore an ideal gas model explains well the observations. We analyze the obtained low number in terms of efficiency of evaporation. The utility of an in situ detection is illustrated by measuring the harmonic gas pressure of the trapped gas in the route to condensation., Comment: 13 pages, 8 figures

We have measured the velocity field of a vortex lattice within a sodium Bose-Einstein condensate using Bragg scattering. The phase gradient of the macroscopic wavefunction was mapped into the spatial structure of the diffracted atom cloud, allowing for single shot measurement of the rotation parameters. A combination of spectral and spatial information yields a complete description of the superfluid flow, coarse-grained over the lattice structure, including direct and independent measurements of the rate and sense of rotation. Signatures of the microscopic quantum rotation have also been observed., Comment: 5 pages, 5 Figures, A movie built from the CM data is available in our Webpage: http://www.physics.gatech.edu/chandra/index.htm; added Fig.5 presents new data, showing signatures of the microscopic vortex structure in the diffracted cloud

This communication describes the observation of a new type of dark spontaneous-force optical trap (dark SPOT) obtained without the use of a mask blocking the central part of the repumper laser beam. We observe that loading a magneto-optical trap (MOT) from a continuous and intense flux of slowed atoms and by appropriately tuning the frequency of the repumper laser is possible to achieve basically the same effect of the dark SPOT, using a simpler apparatus. This work characterizes the new system through measurements of absorption and fluorescence imaging of the atomic cloud and presents a very simple model to explain the main features of our observations. We believe that this new approach may simplify the current experiments to produce quantum degenerated gases., Comment: 13 pages, 8 figures, Submitted to Optics Communications (30/10/2003), accepted for publication (Feb/2004)

In a recent paper, we have used the dark-spot Zeeman tuned slowing technique [Phys. Rev. A 62, 013404-1, (2000)] to measure the capture velocity as a function of laser intensity for a sodium magneto optical trap. Due to technical limitation we explored only the low light intensity regime, from 0 to 27 mW/cm^2. Now we complement that work measuring the capture velocity in a broader range of light intensities (from 0 to 400 mW/cm^2). New features, observed in this range, are important to understant the escape velocity behavior, which has been intensively used in the interpretation of cold collisions. In particular, we show in this brief report that the capture velocity has a maximum as function of the trap laser intensity, which would imply a minimum in the trap loss rates., Comment: 2 pages, 2 figures

Neutral atoms may be trapped via the interaction of their magnetic dipole moment with magnetic field gradients. One of the possible schemes is the cloverleaf trap. It is often desirable to have at hand a fast and precise technique for measuring the magnetic field distribution. We introduce a novel diagnostic tool for instantaneous imaging the equipotential lines of a magnetic field within a region of space (the vacuum recipient) that is not accessible to massive probes. Our technique is based on spatially resolved observation of the fluorescence emitted by a hot beam of sodium atoms crossing a thin slice of resonant laser light within the magnetic field region to be investigated. The inhomogeneous magnetic field spatially modulates the resonance condition between the Zeeman-shifted hyperfine sublevels and the laser light and therefore the amount of scattered photons. We demonstrate this technique by mapping the field of our cloverleaf trap in three dimensions under various conditions., Comment: 8 pages, 8 figures