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arXiv:1507.00331 · Modern Physics Letters A

Modified uncertainty principle from the free expansion of a Bose–Einstein condensate

Elías Castellanos, Celia Escamilla‐Rivera

Published Dec 15, 2016• 8 citations
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Abstract

In this paper, we present a theoretical and numerical analysis of the free expansion of a Bose–Einstein condensate, where we assume that the single particle energy spectrum is deformed due to a possible quantum structure of spacetime. Also, we consider the presence of interparticle interactions in order to study more realistic and specific scenarios. The modified free velocity expansion of the condensate leads in a natural way to a modification of the uncertainty principle, which allows us to investigate some possible features of the Planck scale regime in low-energy earth-based experiments.

DOI
10.1142/s0217732317500079

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In this paper, we present a theoretical and numerical analysis of the free expansion of a Bose–Einstein condensate, where we assume that the single particle energy spectrum is deformed due to a possible quantum structure of spacetime. Also, we consider the presence of interparticle interactions in order to study more realistic and specific scenarios. The modified free velocity expansion of the condensate leads in a natural way to a modification of the uncertainty principle, which allows us to investigate some possible features of the Planck scale regime in low-energy earth-based experiments.
The free expansion of a Bose–Einstein condensate is analyzed, assuming that the single particle energy spectrum is modified due to some quantum structure of space time. The condensate’s free velocity expansion leads in a natural way to a deformed uncertainty principle. We show that large expansion times are required in order to amplify some possible traces of the Planck scale regime. The analysis presented in this work opens up the opportunity to use Bose–Einstein condensates as tools in quantum gravity phenomenology with a solid experimental basis.
We analyze the possible effects arising from the Planck scale regime upon the interference pattern of two noninteracting Bose-Einstein condensates. We start with the analysis of the free expansion of a condensate, and have taken into account the effects produced by a deformed dispersion relation, suggested in several quantum-gravity models. The analysis of the condensate free expansion, in particular, the modified free velocity expansion, suggests in a natural way, a modified uncertainty principle that could lead to new phenomenological implications related to the quantum structure of space-time. Finally, we analyze the corresponding separation between the interference fringes after the two condensates overlap, in order to explore the sensitivity of the system to possible signals caused by the Planck scale regime.
Abstract In this study, an analytical investigation was conducted to assess the effects of the extended uncertainty principle (EUP) on a Bose–Einstein condensate (BEC) described by the deformed one-dimensional Gross–Pitaevskii equation (GPE). Analytical solutions were derived for null potential while we used variational and numerical methods for a harmonic oscillator potential. The effects of EUP on stability, probability density, position, and momentum uncertainties of BEC are analyzed. The EUP is found to be applicable for the free dark soliton solution and in the presence of a harmonic potential within specific ranges of the deformation parameter α, while it is not valid for the free bright soliton solution.
The effects of a Planck-scale deformation of the Minkowski energy-momentum dispersion relation on the phenomenology of non-trapped Bose-Einstein condensates (BECs) are examined. Such a deformation is shown to cause a shift in the condensation temperature TcT_{c}Tc​ of the BEC and, for a specific functional form of deformation, this shift can be as large as the current measured precision on TcT_{c}Tc​. For a 3785Rb_{37}^{85}Rb3785​Rb cold-atom BEC with a particle density n≃1012cm−3n\simeq 10^{12}cm^{-3}n≃1012cm−3 we find a fractional shift of order 10−410^{-4}10−4, but this can be much larger for even more dilute BECs. We discuss the possibility of planning specific experiments with BECs that might provide phenomenological constraints on Planck-scale physics. These corrections to TcT_{c}Tc​ are found to be extremely small for ultrarelativistic BECs implying that, in some cases, Planck-scale effects may be more important in low- rather than high-energy processes.
Although atomic Bose gases are investigated in the dilute gas regime, the physical properties of the Bose–Einstein condensates are affected by interparticle interactions and the fractal nature of the space where the Bose systems are evolving. Theoretical predictions of the traditional Bose–Einstein thermostatistics do not account for the deviations from the experimental results, which are related to internal energy, specific heat, transition temperature, etc. On the other hand, in this study, fractional calculus is introduced where effects of the fractality of space are taken into account. Meanwhile, the order of the fractional derivative α is handled as a measure of the fractality of space. In this content, some improvements which take into account the effects of the fractal nature of the system are made in the standard physical results of the Bose–Einstein condensation phenomena. As an example, for the dilute atomic gas 7Li, the measured transition temperature of Bose–Einstein condensation could be obtained for the value of α ≈ 0.976, and one could predict that the interparticle interactions have a weak attractive nature consistent with experiment (Bradley et al 1995 Phys. Rev. Lett. 75 1687). Thus, a fractional mathematical theory is established in coherence with experimental results of Bose–Einstein condensation.
We analyze the consequences caused by a deformed dispersion relation, suggested in several quantum gravity models, upon a bosonic gas. Concerning the ground state of the Bogoliubov space of this system, we deduce the corrections in the pressure, the speed of sound, and the corresponding healing length. Indeed, we prove that the corrections in the relevant thermodynamic properties associated with the ground state, defines a non trivial function of the density of particles and the deformation parameters, allowing us to constrain, in principle, the form of the modified energy-momentum dispersion relation.