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arXiv:1212.2700 · Europhysics Letters (EPL)

Planck scale physics and Bogoliubov spaces in a Bose-Einstein condensate

Elías Castellanos

Published Aug 1, 2013• 16 citations
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Abstract

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.

DOI
10.1209/0295-5075/103/40004

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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.
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We analyze the effects caused by an anomalous single-particle dispersion relation suggested in several quantum-gravity models, upon the thermodynamics of a Bose-Einstein condensate trapped in a generic 3-dimensional power-law potential. We prove that the shift in the condensation temperature, caused by a deformed dispersion relation, described as a non-trivial function of the number of particles and the shape associated to the corresponding trap, could provide bounds for the parameters associated to such deformation. Additionally, we calculate the fluctuations in the number of particles as a criterium of thermodynamic stability for these systems. We show that the apparent instability caused by the anomalous fluctuations in the thermodynamic limit can be suppressed considering the lowest energy associated to the system in question.
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.
We analyze the corrections caused by finite size effects upon the ground state properties of a homogeneous one-dimensional (1D) Bose–Einstein condensate. We assume from the very beginning that the Bogoliubov’s formalism is valid and consequently, we show that in order to obtain a well-defined ground state properties, finite size effects of the system must be taken into account. Indeed, the formalism described in the present paper allows to recover the usual properties related to the ground state of a homogeneous 1D Bose–Einstein condensate but corrected by finite size effects of the system. Finally, this scenario allows us to analyze the sensitivity of the system when the Bogoliubov’s regime is valid and when finite size effects are present. These facts open the possibility to apply these ideas to more realistic scenarios, e.g. low-dimensional trapped Bose–Einstein condensates.
We analyze the stability of a Bose–Einstein condensate in a Lorentz violating scenario, which is characterized by a deformation in the dispersion relation. The incorporation of a Lorentz violation within the bosonic statistics has, as a consequence, the emergence of a pseudo-interaction, the one can be associated to a characteristic scattering length. In addition, we calculate the relevant parameters associated to the stability of such condensate incorporating this pseudo-interaction in the nonlinear term of the Gross–Pitaevskii equation. We show that these parameters must be corrected, as a consequence of the quantum structure of spacetime.
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.