Reduced dimensionality and spatial entanglement in highly anisotropic Bose-Einstein condensates

We investigate the reduced dimensionality of highly anisotropic Bose-Einstein condensates (BECs) in connection to the entanglement between its spatial degrees of freedom. We argue that the reduced-dimensionality of the BEC is physically meaningful in a regime where spatial correlations are negligible. We handle the problem analytically within the mean-field approximation for general quasi-one-dimensional and -two-dimensional geometries, and obtain the optimal reduced-dimension, pure-state description of the condensate mean field. We give explicit solutions to the case of harmonic potentials, which we compare against exact numerical integration of the three-dimensional Gross-Pitaevskii equation.Comment: 15 pages, 3 figures. Minor changes in text to be in agreement with published versio

Nonlinear interferometry with Bose-Einstein condensates

This dissertation investigates the possibility of using a two-mode Bose-Einstein condensate (BEC) of N atoms to implement a nonlinear Ramsey interferometer whose detection uncertainty scales better than the optimal 1/N Heisenberg scaling of linear interferometry. Our theoretical analysis discusses several phenomena that get in the way of achieving the desired super-Heisenberg\u27 scaling, such as the spatial degrees of freedom and the temporal evolution of the gas. Using heuristic estimates, we discuss the conditions for observing nonlinear- enhanced scalings in this system. The expansion of the condensate gets in the way of the scaling, and we deal with that by going to highly anisotropic traps. In view of realistic experimental parameters, we further investigate such issues by means of numerical simulations for a quasi-1D BEC. In this situation, there are still position-dependent phase shifts that need to be modeled precisely. This brings into question the accuracy of the quasi-1D approximation, both spatially and temporally. We study effects associated with the emergence of 3D behavior in highly anisotropic BECs. We study the ground-state properties of the gas analytically, by performing a perturbative Schmidt decomposition of the condensate wave function between the tightly confined and the loosely confined directions. Our approach provides a straightforward way, first, to derive corrections to the transverse and longitudinal wave functions of the reduced-dimension approximation and, second, to calculate the amount of entanglement that arises between the transverse and longitudinal spatial directions. Numerical integration of the 3D Gross-Pitaevskii equation for experimentally accessible parameters reveals good agreement with our analytical model even for relatively high nonlinearities. Lastly, we study the dynamics of two-mode BECs in highly anisotropic traps, also by means of perturbative techniques. We derive equations that effectively simulate the BEC interferometry protocol, which show how the corrections to the reduced- dimension approximation propagate in time and affect the dynamics of the condensate. We compare these theoretical results with the exact numerical results for the evolution of the two-mode BEC. This analysis leads to an improved model which provides a considerably refined account of the interference signal

‘Prestation Economy’: a model for Late Bronze Age and Early Iron Age burial deposition in Central-Western Europe

Desde os anos de 1970, enterramentos ricos como aqueles de Vix e Hochdorf têm sido analisados como ícones de um fenômeno pan-regional – a emergência de Estados tradicionais, a qual é largamente sustentada pelo impacto de sua interação com as sociedades mediterrâneas. Contudo, nas últimas duas décadas, tem havido uma crítica significativa a um tal modelo interpretativo. No seio desse debate, o presente artigo afirma que o registro arqueológico fornece evidência de que depósitos rituais (também e juntamente com outras formas de dom e contra-dom) constituíam parte significativa do que podemos chamar “economia de prestações”. A fim de tal demonstrar, este artigo analisa os enterramentos e respectivos depósitos funerários das regiões de quatro “centros principescos” (Fürstensitze) – a saber: Bourges, Châtillon-sur-Glâne, Mont Lassois e Hohenasperg.Since the 1970’s, rich burials like those of Vix and Hochdorf have been analysed as the icons of a pan-regional phenomenon – the emergence of traditional states, which is largely underpinned by the impact of their interaction with Mediterranean societies. However, during the last two decades, there has been significant criticism of such an interpretative model. In the core of such a debate, this paper argues that the archaeological record provides evidence that ritual deposits (likewise and alongside other forms of gift-giving) constituted a significant part of what I name as ‘prestation economy’. In order to demonstrate that, this paper analyses the burials and their grave-goods from the regions of four Fürstensitze, namely: Bourges, Châtillon-sur-Glâne, Mont Lassois, and Hohenasperg

Mean-field dynamics of two-mode Bose-Einstein condensates in highly anisotropic potentials: Interference, dimensionality, and entanglement

We study the mean-field dynamics and the reduced-dimension character of two-mode Bose-Einstein condensates (BECs) in highly anisotropic traps. By means of perturbative techniques, we show that the tightly confined (transverse) degrees of freedom can be decoupled from the dynamical equations at the expense of introducing additional effective three-body, attractive, intra- and inter-mode interactions into the dynamics of the loosely confined (longitudinal) degrees of freedom. These effective interactions are mediated by changes in the transverse wave function. The perturbation theory is valid as long as the nonlinear scattering energy is small compared to the transverse energy scales. This approach leads to reduced-dimension mean-field equations that optimally describe the evolution of a two-mode condensate in general quasi-1D and quasi-2D geometries. We use this model to investigate the relative phase and density dynamics of a two-mode, cigar-shaped $^{87}$Rb BEC. We study the relative-phase dynamics in the context of a nonlinear Ramsey interferometry scheme, which has recently been proposed as a novel platform for high-precision interferometry. Numerical integration of the coupled, time-dependent, three-dimensional, two-mode Gross-Pitaevskii equations for various atom numbers shows that this model gives a considerably more refined analytical account of the mean-field evolution than an idealized quasi-1D description.Comment: 35 pages, 10 figures. Current version is as publishe

Nonlinear interferometry with Bose-Einstein condensates

We analyze a proposed experiment [Boixo et al., Phys. Rev. Lett. 101, 040403 (2008)] for achieving sensitivity scaling better than 1/N in a nonlinear Ramsey interferometer that uses a two-mode Bose-Einstein condensate (BEC) of N atoms. We present numerical simulations that confirm the analytical predictions for the effect of the spreading of the BEC ground-state wave function on the ideal 1/N^(3/2) scaling. Numerical integration of the coupled, time-dependent, two-mode Gross-Pitaevskii equations allows us to study the several simplifying assumptions made in the initial analytic study of the proposal and to explore when they can be justified. In particular, we find that the two modes share the same spatial wave function for a length of time that is sufficient to run the metrology scheme

Entanglement-based perturbation theory for highly anisotropic Bose-Einstein condensates

We investigate the emergence of three-dimensional behavior in a reduced-dimension Bose-Einstein condensate trapped by a highly anisotropic potential. We handle the problem analytically by performing a perturbative Schmidt decomposition of the condensate wave function between the tightly confined (transverse) direction(s) and the loosely confined (longitudinal) direction(s). The perturbation theory is valid when the nonlinear scattering energy is small compared to the transverse energy scales. Our approach provides a straightforward way, first, to derive corrections to the transverse and longitudinal wave functions of the reduced-dimension approximation and, second, to calculate the amount of entanglement that arises between the transverse and longitudinal spatial directions. Numerical integration of the three-dimensional Gross-Pitaevskii equation for different cigar-shaped potentials and experimentally accessible parameters reveals good agreement with our analytical model even for relatively high nonlinearities. In particular, we show that even for such stronger nonlinearities the entanglement remains remarkably small, which allows the condensate to be well described by a product wave function that corresponds to a single Schmidt term.Comment: 12 pages, 7 figure