Modified Bosonic Gas Trapped in a Generic 3-Dim Power Law Potential

Physics Letters B 731 (2014) 1–6

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Physics Letters B

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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 interactions 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). d The main goal of this work is to analyze the shift in the 3 nl γ = + (13) condensation temperature caused by a deformed dispersion re- 2 = sl lation in weakly interacting systems and also in systems con- l 1 taining a finite number of particles. We stress that these sys- is the parameter that defines the shape of the potential (4).In(12), tems could be used, in principle, to obtain criteria of viability for N0 are the particles in the ground state, Γ(y) is the Gamma func- possible signals coming from Planck scale regime, by analyzing tion, and C is a constant associated with the potential in question. some relevant thermodynamic variables, for instance, the num- In the case of Cartesian traps, and in consequence, for a three di- ber of particles, and the frequency associated with the trap, when mensional harmonic oscillator potential γ = 3andC = 8. If we |ξ1| 1. set α = 0in(12) then, we recover the result given in [22].Inthe E. Castellanos, C. Laemmerzahl / Physics Letters B 731 (2014) 1–6 3

− thermodynamic limit the conditions for condensation are given by among the constituents of the gas is about 5 × 10 2 with a 1% of μ = 0 and N0 = 0, which implies that the Bose–Einstein functions error [37]. These facts allow us to obtain a bound for the deforma- 6 become the corresponding Riemann Zeta functions ζ(x) [36].Thus, tion parameter |ξ1| 10 for the linear trap s1 = 1 (with frequen- 6 2 expression (12) at the condensation temperature is given by cies ω0/2π ∼ 10 Hz and N ∼ 10 ), to |ξ1| 10 corresponding to a →∞ 13 15 free gas in a box s1 , with densities about 10 –10 [20].In d − nl 3/2 s n nl m fact, these bounds could be improved in a system containing mas- N = C A l a l Γ + 1 ζ(γ )(κ T )γ l l 2 c sive bosons and/or lower frequencies but, where the thermody- = sl 2πh¯ l 1 namic limit is still valid, trapped in potentials where the parameter 2 m − 1 s1 is suﬃciently large. Here one important fact is that we are able − − γ 2 α ζ(γ 1/2)(κ Tc) , (14) to improve the bound associated to the deformation parameter ξ 2π 2h¯ 3 1 by the use of different classes of potentials and it is straightfor- where Tc is the condensation temperature. If we set α = 0in(14), ward to generalize this result to more general traps by using (17). we recover the usual expression for the condensation tempera- From (17) and (18), we notice that correction in the condensa- ture T0 for a gas trapped in a generic 3-dim power-law potential tion temperature depends strongly on the functional form between in the thermodynamic limit [22] the number of particles and the parameters associated to the po-

n tential in question. Finally, in the case of a harmonic oscillator l s − Tc −1/6 d l nl ¯ 2 3/2 1/γ potential we obtain ∼ αN corresponding to a shift of or- N l=1 Al al 2πh 1 T0 T0 = . (15) −6 d nl + m κ der ξ110 , which allows us to bound the parameter ξ1 up to C l=1 Γ s 1 4 l |ξ1| 10 , under typical conditions. Now, let us define 3. Weakly interacting modified bosonic gas nl s −n d A l a l l=1 l l Let us start with the modified Hartree–Fock spectrum (2).After V char = , (16) d nl + some calculations, similarly to the previous section, we obtain the C l=1 Γ s 1 l spatial density associated to the weakly interacting case as the characteristic volume associated with the system. We notice − − − →∞ n(r) = λ 3 g eβ(μeff U (r) 2U0n(r)) that if si then, V char becomes the volume associated with 3/2 a free gas in a box (in fact the inverse of the volume with this m −2 β(μeff −U (r)−2U0n(r)) definition). In this sense, V can be interpreted as the available − αλ g1 e . (21) char πh¯ volume occupied by the gas [23,25]. At this point, it is noteworthy to mention that the most general definition of thermodynamic If we set α = 0inEq.(21) we recover the usual result for the limit can be expressed as N →∞, V char → 0 keeping the prod- spatial density in the semiclassical approximation [20,21].Byusing uct NVchar constant, and is valid for all power law potentials in the properties of the Bose–Einstein functions [36], we are able to any spatial dimensionality [25]. With the criterion given above, expand expression (21) around U0 = 0, with the result the condensation temperature in the thermodynamic limit is well − − n(r) ≈ n (r) + U (2κ T ) 1λ 6 g (Z)g (Z) defined. Finally, we can express the shift in the condensation tem- 0 0 3/2 1/2 perature as a function of the number of particles N −1 −5 m + αU0(2κ T ) λ g3/2(Z)g0(Z) T − T T πh¯ c 0 ≡ c −1/2γ αΩ N , (17) + T0 T0 g1(Z)g1/2(Z) , (22) where where β(μ −U (r)) 1/2 2 3/2 −1/2γ Z = e eff , (23) 2m ζ(γ − 1/2) V char(2h¯ ) Ω = . (18) with n (r) the space density distribution for the ideal case U = 0, π γ ζ(γ ) ζ(γ ) 0 0 m For the sake of simplicity, let us analyze the case of spheri- −3 −2 n0(r) = λ g3/2(Z) − αλ g1(Z). (24) cal traps, where the corresponding potential is given by U (r) = πh¯ r s1 A1 , setting A1 = h¯ ω0/2 and a1 = h¯ /mω0. In this case, the a1 Integrating the normalization condition (8) and using expression shift in the condensation temperature is given by (22) with the corresponding potential (4), allows us to obtain an

Tc − + expression for the number of particles as a function of the chemi- s1/3(s1 2) αΩs1 N . (19) cal potential μ, the temperature T , the coupling constant U , and T 0 0 the deformation parameter α Tc −1/9 For different values of s1 we obtain, ∼ αN ,fors1 = 1, 3/2 T0 m Tc −1/6 = γ which corresponds to a linear trap. For s1 = 2, ∼ αN , NVchar gγ (zeff )(κ T ) T0 2πh¯ 2 = Tc ∼ which is an isotropic harmonic oscillator. For s1 3, T 2 0 m − −1/5 Tc −1/4 − γ 1/2 αN .Fors1 = 6, ∼ αN , and so on. We notice imme- α gγ (zeff )(κ T ) T0 2 3 →∞ 2π h¯ diately that if s1 , after some algebraic manipulation, we are 3 able to obtain the limiting case of a bosonic gas trapped in a box m γ −3/2 − U G − (z )(κ T ) 0 2 3/2,1/2,γ 3/2 eff 1/3 2πh¯ Tc 2m(V ζ(3)) −1/3 α N . (20) 5/2 T 3h¯ m m 0 + αU (κ T )γ 0 2 ¯ Additionally, the current high precision experiments in the case of 2πh¯ πh − 39 K , with a mass 15×10 26 kg, the shift in the condensation tem- 19 × + perature with respect to the ideal result, caused by the interactions G3/2,0,γ −3/2(zeff ) G1,1/2,γ −3/2(zeff ) , (25) 4 E. Castellanos, C. Laemmerzahl / Physics Letters B 731 (2014) 1–6 where finally obtain the shift in the condensation temperature in function ∞ (i+ j) of the number of particles zeff G , , −3/2(z ) = , (26) 1 η σ γ eff η σ γ −3/2 2 2 i j (i + j) Tc 1 mΛ 1 ij=1 −(aR ) 2γ N 2γ 0 2 T0 2πh¯ and we have defined an effective fugacity 1/2 (8πm) ζ(γ − 1/2) − 1 2 + 2γ = β(μ+mα /2) α (R0 N) zeff e . (27) ζ(γ )γ 1 In order to obtain the leading correction on the condensation tem- γ 4mζ(γ − 1) (R0 N) perature caused by the interactions in our deformed bosonic gas, + αa ln , (33) πh¯ ζ(γ )γ mα2 let us expand (25) to first order in T = T0, μ = 0, U0 = 0, and α = 0. Recall that T0 is the condensation temperature in the ther- where modynamic limit given by expression (15),withtheresult 2ζ(3/2)ζ(γ − 1) − G − (1) = 3/2,1/2,γ 3/2 3/2 Λ , (34) m γ ζ(γ )γ NVchar = ζ(γ )(κ T0) 2πh¯ 2 ¯ 2 3/2 = 2πh V char 3/2 R0 . (35) m − m ζ(γ ) +[T − T ] γ ζ(γ )κ(κ T )γ 1 0 2 0 2πh¯ Setting α = 0inEq.(33) we recover the usual shift on the con- 3 densation temperature caused by weakly interactions. Let us an- m +1 2 − γ / r s1 U0 G3/2,1/2,γ −3/2(1)(κ T0) alyze the case of spherical traps U (r) = A1( ) ,togetherwith 2πh¯ 2 a1 = ¯ = ¯ 3/2 A1 hω0/2 and a1 h/mω0. Notice that the possibility of de- m + − γ −1 tecting the term depending upon the deformation parameter effect μ ζ(γ 1)(κ T0) (0) ¯ 2 (δT α) requires that, if δT is the experimental error related to the 2πh c c 2 measurement of the condensation temperature when α = 0, then m − − − γ 1/2 (0) α ζ(γ 1/2)(κ T0) . (28) δT < |δT α|. In our case this entails 2 ¯ 3 c c π h 1/2 At the condensation temperature T for large N, in the mean field (8πm) ζ(γ − 1/2) − 1 c (0) 2γ = = δTc < α (R0 N) approach the chemical potential takes the value μc 2U0n0(r 0) ζ(γ )γ = = = −3 [20,21]. In the usual case α 0, n0(r 0) λc ζ(3/2) in the large 1 − γ N limit, which means that the critical density at the center of the 4mζ(γ 1) (R0 N) + αa ln . (36) trap is the same as that of the uniform model [20].However,in πh¯ ζ(γ )γ mα2 our case, we have to modified the value of μc at the condensation For spherical traps γ = 3(s1 +2)/2s1. The shift in the condensation temperature according to expression (24),duetothedivergentbe- − temperature caused by interactions is typically 5 × 10 2,witha1% havior of the Bose–Einstein functions related to n0(r = 0).When the integral associated with the Bose–Einstein functions converges, of error [37,38], then from expression (36) and the results given 39 −26 −9 2 above, in the case of K , with a mass 15 × 10 kg, a ∼ 10 m, the value mα /2 is negligible and can be replaced by zero. Nev- 19 ∼ | α| ertheless, when the integral associated to the Bose–Einstein func- and ω0 10 Hz, it allows us to obtain a criterion on δTc as a | | tions can diverge at Z → 1 the minimum of the energy associated function of the number of particles when ξ1 1. For different 33 = with the system must be taken into account [25].Inthissection values of the shape parameter γ we obtain, N > 10 for s1 1, 22 = 17 = 13 = we are interested in the corrections due to α in the large N limit, N > 10 for s1 2, N > 10 for s1 4, N > 10 for s1 9, 11 = so we will take as the minimum of energy in the system mα2/2. N > 10 for s1 18 and so on. Notice that expression (36) is = = Let us define n (r = 0) at the condensation temperature using ex- not valid for the case s1 6 (which implies γ 2) due to the 0 = pression (24) as follows divergent behavior of ζ(1). This special case, s1 6, defines a limit between a positive shift and a negative one, caused by the de- − 2 − m 2 formation parameter α. In other words, the shift caused by the = = 3 βcmα /2 − 2 βcmα /2 n0(r 0) λc g3/2 e 2αU0λc g1 e . πh¯ deformation parameter is positive when s1 < 6, for a positive α. (29) Conversely, with s1 > 6 the corresponding shift is negative. Notice

2 that, if the parameter s1 is sufficiently large then, the number of βcmα /2 We can define the Bose–Einstein functions g3/2(e ) and particles decreases, but the shift on the condensation temperature 2 βcmα /2 2 g1(e ), when (βcmα /2) → 0as[36] caused by the deformation parameter becomes negative. On the other hand, keeping the number of particles fixed, with 2 1/2 − 2 mα say N ∼ 105–106, we obtain frequencies of order ∼ 10 9 Hz for βcmα /2 + − ω0 g3/2 e ζ(3/2) Γ( 1/2) , (30) = ∼ −10 = ∼ −15 = 2κ Tc s1 1, ω0 10 Hz for s1 2, ω0 10 Hz for s1 4 and so → on. In other words, for a fixed number of particles, ω0 0ifthe β mα2/2 2κ Tc g e c ln . (31) parameter s1 grows. 1 2 mα In the case of an anisotropic three-dimensional harmonic oscil- = = ¯ = ¯ Neglecting second order terms in U0 and α allows us to write μc lator potential γ 3, with Al hωl/2, al h/mωl, from expres- using expression (29) as follows sion (33) we obtain − − − m 2κ Tc Tc a 2ζ(3/2)ζ(2) G3/2(1) μ 2U λ 3ζ(3/2) − 2αU λ 2 ln . (32) − N1/6 c 0 c 0 c 2 1/2 5/6 πh¯ mα T0 aho (2π) 3ζ(3) 3/2 −1/2 1/2 If we take the limit α → 0, then we recover the usual value for μc 2 π ζ(5/2) m − + α N 1/6 at the condensation temperature [20,21]. Inserting (32) in (28),we 3ζ(3)5/6 h¯ ω¯ E. Castellanos, C. Laemmerzahl / Physics Letters B 731 (2014) 1–6 5 ¯ 1/3 4mζ(2) 2(h¯ ω)N ω → 0 implies large values for s , when the number of particles + αa ln , (37) 0 1 3πh¯ ζ(3) ζ(3)mα2 is fixed. For an anisotropic three-dimensional harmonic oscillator poten- where we have used the usual definitions ω¯ = (ω ω ω )1/3 and 1 2 3 tial (γ = 3), we obtain that the relative correction in the conden- = h¯ 1/2 22 aho mω¯ . In this case, N > 10 which corresponds to a shift sation temperature is given by − on the condensation temperature of order ∼ 10 5,ispositive.Con- −10 5 6 versely, ω¯ ∼ 10 Hz, for a fixed N ∼ 10 − 10 corresponding to Tc ζ(2) 0 −1 3 =− N / −3 2/3 a shift on the condensation temperature of order 10 . T0 3ζ(3) h¯ ω¯ It is noteworthy to mention that ξ1 could be bounded up to 1/2 3 5 6 ζ(5/2) 8m − |ξ1| 10 by using (37) for N ∼ 10 –10 , and ω¯ ∼ 10 Hz, that + 1/6 + 2 2 α N O 0 , α . (42) is, one order of magnitude less than the bound obtained in [15], 3ζ(3)5/6 πh¯ ω¯ which is notable. If we set α = 0 in expression (37) we recover the In this case, N > 1022 which corresponds to a shift on the con- result given in [29]. − densation temperature of order ∼ 10 10. Additionally, keeping N ∼ − 103–106 implies ω¯ ∼ 8.70 × 10 10 Hz, corresponding to a shift of 4. Deformed bosonic gas and finite size corrections − order 10 8.Ifwesetα = 0 then, we recover the usual result [21, In this section let us calculate the leading correction on the 28,30,34]. condensation temperature caused by finite size effects in our modified bosonic gas. The correction to the condensation temperature 5. Conclusions originates in the zero-point motion, or equivalently, at the associated ground-state energy 0 [21,22]. Thus, at the condensation Using the formalism of the semiclassical approximation, we temperature the chemical potential is given by μ = 0. Setting have analyzed the Bose–Einstein condensation for a modified μ = 0 and a = 0in(28), we obtain that the relative shift in the bosonic gas trapped in a 3-D power law potential in three regimes, condensation temperature is given by namely, the thermodynamic limit, finite size systems, and weakly interacting systems. We have deduced the shift on the conden- − 2 3/2 −1/γ Tc ζ(γ 1) 2πh¯ sation temperature in the thermodynamic limit, in weakly in- =−0 NVchar T0 ζ(γ ) m teracting systems, and finite size systems as well, in function of 1/2 3/2 −1/2γ the number of particles and the trap parameters, which are valid 2m ζ(γ − 1/2) 2πh¯ 2 for any potential defined by the generic 3-dimensional power-law + α NVchar π ζ(γ ) m potential (4) within the semiclassical approximation. We have ob- 6 2 + 2 2 tained a bound up to |ξ1| 10 forlineartrapsto|ξ1| 10 O 0 , α . (38) corresponding to a free gas in a box, and |ξ | 104 for harmonic = 1 For spherical traps, setting 0 c1h¯ ω0 [32], the shift on the con- oscillator type potential in the ideal case, under typical conditions. densation temperature can be expressed as follows We stress here that an improvement of the precision in the condensation temperature measurement would also allow to improve Tc ζ(γ − 1) − + − ¯ 2s1/3(s1 2) c1hω0 (Ωs1 N) the bounds on ξ1. T0 ζ(γ ) For weakly interacting systems, we have obtained for the case 1/2 2m ζ(γ − 1/2) − + |ξ | 1, that if the trap parameter s is sufficiently large then, + α (Ω N) s1/3(s1 2), (39) 1 1 s1 this decreases the number of particles, but leads to corrections π ζ(γ ) − on the condensation temperature of order 10 5 for any trap pa- where rameter s1, which is approximately 3 orders of magnitude smaller 3/2 −2 2πh¯ 2 than the typical correction 10 . Conversely, keeping the num- = 5 6 Ωs1 V chars . (40) ber of particles N ∼ 10 –10 fixed leads to frequencies of order 1 m − − 10 15–10 9 corresponding to shifts on the condensation temper- − − For different values of the shape parameter γ ,weobtainfrom ature up to 10 5–10 3, for different values of the shape param- = expression (39), for instance, in the case of linear traps s1 1, eter s1. These problems could be solved, in principle, just tuning Tc −2/9 −1/9 ∼ 0 = N + αN .Fors1 = 2, which corresponds to the interaction coupling by Feshbach resonances to very small val- T0 s1 1 Tc −1/3 −1/6 ues of the scattering length a, that is, almost to the ideal case, and an isotropic harmonic oscillator, ∼ 0 = N + αN . T0 s1 2 Tc −2/5 −1/5 Tc then, reducing the contribution of interactions on the condensation For s1 = 3, ∼ 0 = N + αN .Fors1 = 4, ∼ T0 s1 3 T0 temperature below the Planck-scale induced shift for a sufficiently −4/9 −2/9 0 = N + αN , and so on. The possibility of detecting the s1 4 large parameter s1. Nevertheless, these facts could affect the ther- term depending upon the deformation parameter effect entails in modynamical equilibrium of the system, involving some technical this case difficulties. 1/2 2m ζ(γ − 1/2) On the other hand, finite size effects for sufficiently large s1 (0) −s1/3(s1+2) δTc < α (Ωs1 N) . (41) lead to a very small correction on the condensation temperature π ζ(γ ) −10 of order 10 , for any trap parameter s1 and fixed frequency The shift in the condensation temperature caused by finite size ω0, which is 8 orders of magnitude smaller than the typical cor- − − effects is typically of order 10 2 [21], then from expression (41) rection 10 2. It is a condition impossible to fulfill. Conversely, 39 ≈ ∼ 5 6 and the results given above, in the case of 19 K and ω0 10 Hz, keeping the number of particles N 10 –10 fixed leads to fre- 33 17 14 9 −15 it leads to N > 10 for s1 = 1, N > 10 for s1 = 4, N > 10 quencies up to 10 –10 corresponding to shifts on the con- 13 −8 −5 for s1 = 6, N > 10 for s1 = 9, and so on. Conversely, keeping densation temperature of order 10 –10 , for different values 3 6 the number of particles fixed with, let say N ∼ 10 –10 ,weob- of the shape parameter s1. In other words, these facts suggest −9 −10 tain, ω0 ∼ 1.62 × 10 Hz for s1 = 1, ω0 ∼ 8.70 × 10 Hz for that finite size effects are technologically impossible to be tuned −10 s1 = 2, ω0 ∼ 5.70 × 10 Hz for s1 = 4 and so on. Here the pa- below Planck-scale induced effects, at least for current experi- rameter s1 has the same behavior as in the interacting case, that is, ments. 6 E. Castellanos, C. Laemmerzahl / Physics Letters B 731 (2014) 1–6

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