Pis'ma v ZhETF, vol. 101, iss. 8, pp. 631-636
© 2015 April 25
Weakly interacting Bose-Einstein condensates in temperature-dependent generic traps
Dedicated to the loving memory father El'ias Castellanos de Luna
E. Castellanos+1\ F. Bríscese*x M. Grether0M. de Llanov ^
+ Mesoarnerican Centre for Theoretical Physics (ICTP regional headquarters in Central America, the Caribbean and Mexico), Universidad Autónoma de Chiapas, Ciudad Universitaria, Real del Bosque (Terán), 29040 Tuxtla Gutiérrez, Chiapas, México
* Departamento de Física, CCEN, Universidade Federal da Paraiba, Cidade Universitária, 58051-970 Joáo Pessoa, PB, Brazil
x Istituto Nazionale di Alta Matematica Francesco Severi, Gruppo Nazionale di Física Matematica, Cittá Universitaria,
00185 Rome, EU
°Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México, DF, México
vInstituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, A. P. 70-360, 04510 México, DF, México
Submitted 25 February 2015 Resubmitted 16 March 2015
We study the shift ATc in the condensation temperature oí an atomic Bose-Einstein condensate trapped in a temperature-dependent three-dimensional generic potential. With no assumptions other than the mean-field approach and the semiclassical approximation, it is shown that the inclusion ol a T-dependent trap improves upon the pure semiclassical result giving better agreement between the predicted ATc and its experimental value. However, despite this improvement, the effect ol a T-dependent trap is not sufficient to lully reduce the discrepancy between theoretical prediction and data.
DOI: 10.7868/S0370274X15080123
I. Introduction. Since its theoretical prediction by Bose and Einstein [1, 2] in the 1920s until its laboratory observation with magneto-optical traps [3-6] from 1995 onwards Bose-Einstein condensation (BEC) of dilute atomic gases has stimulated enormous efforts of related work. Among the issues addressed one finds, e.g., rigorous mathematical questions related to BEC [7], diverse theoretical and heuristic aspects [8, 9], and is now even viewed as a viable tool for precision tests in gravitational physics [10-20].
The study of its associated thermodynamic properties is naturally also a pertinent aspect of BECs [21-25]. Indeed, the condensation temperature Tc, i.e., the critical temperature below which a macroscopic quantum state of matter appears, has been the subject of considerable discussion, see Ref. [8, 26] and Refs. therein. In particular, the influence of interparticle interactions on Tc turns out to be a deep nontrivial matter, see e.g. Refs. [27-29].
Interboson interactions produce a shift ATC/TC° = = (Tc— Tc°)/Tc° in the condensation temperature Tc with
-^e-mail: ecastellanos@mctp.mx; fabio.briscese@sbai.uniromal.it; mdgg@hp.fciencias.unam.mx; dellano@unam.mx
respect to that of the ideal noninteracting case Tc° in the thermodynamic limit. For instance, the contributions to ATC/TC° due to interactions in a uniform dilute gas originate in the fact that the associated many-body system is affected by long-range critical fluctuations rather than from purely mean-field (MF) considerations [26, 30, 31]. However, it is generally accepted that ATC/TC° for this system behaves like ciS+(c'2 In S+c'^S2, with the dimen-sionless variable S = where p is the corresponding boson number density, a the S-wave two-body scattering length [30] related to the pair interaction, and the Cj's are dimensionless constants. A good fit [27] gives ci ~ 1.32, c'2 ~ 19.75, and 4 ~ 75.7.
It is noteworthy that these ideas can be extended to more general traps [32-34] in which the relative shift ATC/TC° on the condensation temperature explicitly exhibits a sensitive trap-dependence. This extension to generic traps allows summarizing the corrections on ATc/T® as function of a simple index parameter describing the trap shape.
On the other hand, when interactions are considered for the more common harmonic traps one finds a shift in Tc up to second order in the S-wave scattering length a within the MF approach given by [28, 29]
A Tc
~ bi(a/XTo) + b2(a/XTo
where
kBT° = hu[N/a 3)]V3
(1)
(2)
(with £(3) ~ 1.202) is the condensation temperature associated with the ideal system (a = 0) in the thermodynamic limit [22], and 61 ~ -3.426 [35] while b2 ~ 11.7 [29], together with ATo = (2-kH2/mkT®)1!2 the thermal wavelength. Furthermore, these results seem to contrast with the results reported, e.g., in Refs. [36, 37] since, as mentioned in Ref. [28], the well-known logarithmic corrections to (1) are not discernible within the error bars.
Note that from (1) ATc is negative for repulsive interactions, i.e., a > 0 since 61 is negative. The result (1) is in excellent agreement with laboratory measurements of ATc/T° [29, 38-40] to first order in (a/XTo) but differs somewhat with data to second order (a/XTo)2. In Ref. [28], high precision measurements of the condensation temperature of the bosonic atom 39 K vapor in the range of parameters N ~ (2—8) • 105, ui ~ (75—85) Hz, 10~3 < a/XTo < 6 • 10-2 and Tc ~ (180-330) nK have detected second-order effects in ATc/T°. The measured ATC/TC° is well fitted by a quadratic polynomial (1) with best-fit parameters 5®xp ~ —3.5 ± 0.3 and b^ — 46 ± 5 so that the value b2 — H-7 [29] is strongly excluded by data. This discrepancy between (1) and data may be due to beyond-MF effects (see Ref. [29]). Beyond-MF effects are expected to be important near criticality, where the physics is often nonperturbative. It would therefore seems reasonable that a beyond-MF treatment might give a correct estimation of 62- However, this is not certain since beyond-MF effects have been calculated in the case of uniform condensates [37, 41] but are still poorly understood for trapped BECs [36,42,43-45]. It thus seems that it is currently not possible to ascertained whether the discrepancy between 62 and b2*p can be explained in the MF context or arises from beyond-MF effects.
Nevertheless, the effect of interactions on the condensation temperature Tc of a Bose-Einstein condensate trapped in a harmonic potential was recently discussed [35]. In the latter paper it was shown that, within the MF Hartree-Fock (HF) and semiclassical approximations, interactions among the particles produce a shift ATc/T° ~ 61 (a/ATo) + b2(a/XTo)2 + V [a/XTo] with XTo = (2iih2/mkT®)1/2 the thermal wavelength, and ip [a/XTo] a non-analytic function such that ip [0] = = V>' [0] = iS' [0] = 0 but \ip'" [0] | = 00. Therefore, with only the usual assumptions of the HF and semiclassical approximations, interaction effects are perturbative to second order in a/XTo and the expected nonperturba-
tivity of physical quantities at the critical temperature emerges only at third order. Indeed, in Ref. [35] an analytical estimation for 62 — 18.8 was obtained which improves the previous numerical fit-parameter value of 62 — 11-7 obtained in Ref. [29]. Even so, the value for 62 obtained in Ref. [35] still differs substantially from the empirical value ~ 46 ± 5 [28].
We mention that the temperature shift ATC/TC° induced by interparticle interactions obtained in Ref. [35] seems to contradict, for instance, the result reported in Ref. [36] where the interaction induced temperature shift is estimated as
A Tc
= b1(a/XTo)+ b'2 + 6^'ln(a/ATo) (a/ATo)2 (3)
with 61 ~ -3.426, b'2 ~ -45.86, and b'{ ~ -155.0 [37] (see also Ref. [27] for a discussion). This result has been obtained using lattice simulations and a technique based on a scalar field analogy, but is questionable (see discussion in Ref. [35]) besides being in striking contradiction to the data. It is thus clear that these results differ substantially from the estimations obtained in Ref. [35] and the results obtained here (see below), but also conflict with the results obtained in Ref. [29] as well as experiment [28].
Also, it was recently proposed [46] that accounting for a nonlinear quadratic Zeeman effect gives a value of b2 which depends on the properties of the atomic species of the condensate, which for a 39 K condensate gives a value b2 ~ 42.3 in much better agreement with measurements obtained in Ref. [28]. But this result is based on a physical mechanism completely different from the one considered here. Furthermore, to confirm whether that the quadratic Zeeman effect actually plays such an important role in the physics of atomic condensates, one should repeat the measurements performed in Ref. [28] for different atomic species and compare the results with the predictions obtained in Ref. [46]. However, to our knowledge, Ref. [28] is the only reported measurement of the nonlinear coefficient b2.
We therefore propose that before addressing beyond-MF effects these facts suggest that MF effects might still be well-understood and deserve further analysis.
In fact, in a recent paper [47] the use of an effective temperature-dependent trapping potential was suggested in order to calculate the condensation temperature of noninteracting systems; see also Ref. [48] for a wide-ranging justification of T-dependent Hamiltoni-ans. Hence, it might be useful to explore this idea in the context of the effects on the condensation temperature caused by interparticle interactions.
These considerations drove us into the novel terrain of T-dependent Hamiltonians, and more specifically to T-dependent trapping potentials. We note that this it is not the first time that such a terrain has been reached, e.g., we find the successful use of T-dependent dynamics in: a) superconductivity in the work of Bogoliubov, Zubarev, and Tserkovnikov, as mentioned by Blatt [49]; b) an explanation [50] of the empirical law in superconductors HC(T) = Hc(0)[1 - (T/Tc)2] where HC(T) is the critical magnetic field at T; c) finite-T behavior [24,25, 51-54] of a class of relativistic field theories (RFTs) to address the question of restoration of a symmetry which at T = 0 is broken either dynamically or spontan
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