Physics Letters B 731 (2014) 1–6 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Modified bosonic gas trapped in a generic 3-dim power law potential ∗ E. Castellanos , C. Laemmerzahl ZARM, Universitaet Bremen, Am Fallturm, 28359 Bremen, Germany article info abstract Article history: We analyze the consequences caused by an anomalous single-particle dispersion relation suggested in Received 2 November 2012 several quantum-gravity models, upon the thermodynamics of a Bose–Einstein condensate trapped in a Received in revised form 2 February 2014 generic 3-dimensional power-law potential. We prove that the condensation temperature is shifted as Accepted 2 February 2014 a consequence of such deformation and show that this fact could be used to provide bounds on the Available online 6 February 2014 deformation parameters. Additionally, we show that the shift in the condensation temperature, described Editor: S. Dodelson as a non-trivial function of the number of particles and the trap parameters, could be used as a criterion to analyze the effects caused by a deformed dispersion relation in weakly interacting systems and also in finite size systems. © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 1. Introduction parameters, ξ1,ξ2,ξ3, and particularly that produces a linear- momentum term in the non-relativistic limit [7,9]. Unfortunately, The possibility of a deformation in the dispersion relation of as it is usual in quantum gravity phenomenology, the possible microscopic particles appears in connection with the quest for a bounds associated with the deformation parameters open a wide quantum theory of gravity [1–9]. This entails, in some schemes, range of possible magnitudes, which is translated to a significant that a possible spacetime quantization has as a consequence a challenge. modification of the classical spacetime dispersion relation between On the other hand, the use of Bose–Einstein condensates, as a energy E and (modulus of) momentum p of a microscopic parti- possible tool in the search of quantum-gravity manifestations (for cle with mass m [2,3,5]. A deformed dispersion relation emerges as instance, in the context of Lorentz violation or to provide phe- an adequate tool in the search for phenomenological consequences nomenological constrains on Planck-scale physics) has produced an caused by this type of quantum gravity models. Nevertheless, the enormous amount of interesting publications [10–19].Itturnsout principal difficulty in the search of quantum gravity manifestations to be rather exciting to look for the effects in the thermodynamic in our low energy world is the smallness in the predicted effects properties associated with Bose–Einstein condensates caused by [3,4]. If this kind of deformations is characterized, for instance, by the quantum structure of space–time. some Planck scale, then the quantum gravity effects become very In a previous report [15],wewereabletoprovethatthecon- small [2,5]. In the non-relativistic limit, the deformed dispersion densation temperature of the ideal bosonic gas is corrected as a relation can be expressed as follows [5,6] consequence of the deformation in the dispersion relation. More- p2 1 p3 over, this correction described as a non-trivial function of the num- + + + 2 + E m ξ1mp ξ2 p ξ3 , (1) ber of particles and the shape associated with the corresponding 2m 2M p m trap could provide representative bounds for the deformation pa- in units where the speed of light c = 1, with M 1.2 × 1028 eV p rameter ξ1.Wehaveprovedthatthedeformationparameterξ1 can the Planck mass. The three parameters ξ , ξ , and ξ are model 6 2 1 2 3 be bounded, under typical conditions, from |ξ1| 10 to |ξ1| 10 , dependent [2,5], and should take positive or negative values close by using different classes of trapping potentials in the thermody- to 1. There is some evidence within the formalism of Loop quan- namic limit. In the case of a harmonic oscillator-type potential, tum gravity [5–8] that indicates non-zero values for the three 4 we have obtained a bound up to |ξ1| 10 .InRefs.[5,6] it was suggested the use of ultra-precise cold-atom-recoil experiments to constrain the form of the energy–momentum dispersion relation in * Corresponding author. E-mail addresses: [email protected] (E. Castellanos), the non-relativistic limit. There, the bound associated with ξ1 is at [email protected] (C. Laemmerzahl). least four orders of magnitude smaller than the bound associated http://dx.doi.org/10.1016/j.physletb.2014.02.002 0370-2693/© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 2 E. Castellanos, C. Laemmerzahl / Physics Letters B 731 (2014) 1–6 with a Bose–Einstein condensate trapped in a harmonic oscillator 2. Condensation temperature in the thermodynamic limit; in the ideal case obtained in [15]. U0 = 0 In a more realistic system, finite size effects and interac- tions among the constituents of the gas must be taken into Due to an extensive use of some results, let us briefly summa- account. To this aim, let us propose a particularly simple modified rize the results obtained in [15].From(2),thecaseU0 = 0leadsto Hartree–Fock type spectrum, in the semi-classical approximation, 2 which basically consists in the assumption that the constituents p p = + αp + U(r). (6) of the gas behave like non-interacting bosons moving in a self- 2m consistent mean field, valid when the semiclassical energy spec- In the semiclassical approximation, the single-particle phase-space trum p is bigger than the associated chemical potential μ,for distribution may be written as [20,21] dilute gases [20,21] = 1 2 n(r, p) − , (7) p β(p μ) − p = + αp + U(r) + 2U0n(r), (2) e 1 2m where β = 1/κ T , κ is the Boltzmann constant, T is the tempera- where p is the momentum, m is the mass of the particle, and the ture, and is the chemical potential. The number of particles in mc μ term αp,withα = ξ1 in ordinary units, is the leading order 2M p the 3-dimensional space obeys the normalization condition [20,21], modification in expression (1),withc the speed of light. The term 1 3 3 2U0n(r) is a mean field generated by the interactions with other N = d rd pn (r, p), (8) 3 constituents of the bosonic gas, with n(r) the spatial density of (2πh¯ ) the cloud [20]. The coupling constant U0 is related to the s-wave where scattering length a through the following expression 3 2 n(r) = d pn (r, p) (9) 4πh¯ U0 = a. (3) m is the spatial density. Using expression (6), and integrating ex- The potential term pression (7) over momentum, with the help of (9),wegetthe spatial distribution associated with our modified semi-classical d si ri spectrum (6) U(r) = Ai (4) a = i i 1 − − − m − 3 β(μeff U (r)) 2 β(μeff U (r)) n(r) = λ g3/2 e − αλ g1 e is the generic 3-dimensional power-law potential, where Ai and ai πh¯ are energy and length scales associated with the trap [22].Ad- 2 2 −1 m β(μ −U (r)) ditionally, ri are the d radial coordinates in the ni -dimensional + α λ g e eff (10) 2 1/2 subspace of the 3-dimensional space. The sub-dimensions ni sat- 2πh¯ isfy the following expression in three spatial dimensions 2 = 2πh¯ 1/2 = where λ mκ T is the de Broglie thermal wavelength, μeff d μ + mα2/2 is an effective chemical potential, and g (z) is the so- = ν ni 3. (5) called Bose–Einstein function defined by [36] = i 1 ∞ = = = = ν−1 If d 3, n1 n2 n3 1, then the potential becomes the Carte- = 1 x dx = = = gν(z) − . (11) sian trap. If d 2, n1 2 and n2 1, then we obtain the cylindri- Γ(ν) z 1ex − 1 cal trap. If d = 1, n1 = 3, then we have the spherical trap. If si → 0 ∞ , we have a free gas in a box. The external potential included If we set α = 0inEq.(10) we recover the usual result for the spa- in (2) is quite general. Different combinations of these parameters tial density in the semiclassical approximation [20,21].Byusing give different classes of potentials, according to (4).Itisnotewor- the properties of the Bose–Einstein functions [36], assuming that thy to mention that the use of these generic potentials opens the mα2/2 κ T and integrating the normalization condition (8),we possibility to adiabatically cool the system in a reversible way, by obtain an expression for the number of particles N to first order changing the shape of the trap [20]. The analysis of a Bose–Einstein in α condensate in the ideal case, weakly interacting, and with a finite number of particles, trapped in different potentials shows that the d − nl 3/2 s n nl m N − N = C A l a l Γ + 1 g (z)(κ T )γ main properties associated with the condensate, and in particu- 0 l l 2 γ = sl 2πh¯ lar the condensation temperature, strongly depend on the trapping l 1 potential under consideration [22–35]. Additionally, the character- 2 m γ −1/2 istics of the potential (in particular, the parameter that defines the − α gγ −1/2(z)(κ T ) , (12) 2 ¯ 3 shape of the potential) have a strong impact on the dependence of 2π h the condensation temperature with the number of particles (or the where associated density).