Gross-Pitaevskii equation: Variational approach

phys. stat. sol. (c) 2, No. 10, 3665 – 3668 (2005) / DOI 10.1002/pssc.200461762 Gross-Pitaevskii equation: Variational approach Julio C. Drake Perez*1, C. Trallero-Giner1, V Lopez Richard2, C. Trallero-Herrero3, and Joseph L. Birman4 1 2 3 4 Faculty of Physics, University of Havana, San Lázaro y L, Vedado, CP 10400, Havana, Cuba FFCLRP, Departamento de Física, USP, 14040-901, Ribeirão Preto-SP, Brazil Physics and Astronomy Department, SUNY at Stony Brook, NY 11794-3800, USA Department of Physics, The City College of CUNY, New York, NY 10031, USA Received 11 October 2004, revised 18 February 2005, accepted 18 February 2005 Published online 29 July 2005 PACS 03.75.Kk, 03.75.Lm, 31.15.Pf In this paper we present an analytical method for solving the Gross-Pitaevskii equation for the BoseEinstein condensation in the dilute atomic alkali gases. Using a variational ansatz, we are able to obtain an analytical solution for the order parameter and for the chemical potential as a function of a unique universal parameter: λ/ (ћ ω ℓ) (ℓ is the oscillator length, ω the trap frequency and λ an effective coupling constant). We also perform a comparative analysis between our method of solution and two other methods with defined range of validity: the perturbation theory and the Thomas-Fermi Approximation (TFA). The presented method is valid for negative and for positive self interactions as well. In the positive coupling constant region it is consistent with the TFA, extending its validity beyond the limits of validity of this well known approximation, towards small and negative values of the self-interaction coupling constant. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Almost 80 years ago Bose and Einstein [1] predicted that a system of non-interacting bosons would undergo a phase transition to a state having a macroscopic population of the ground state at finite temperature. More than 50 years after those predictions, laser and evaporative cooling techniques, combined with the development of novel traps, have led to the first unambiguous observations of Bose-Einstein condensation (BEC) of a weakly interacting atomic Bose gas in the laboratory [2–4]. From the theoretical point of view the dynamics of a BEC has been studied using mainly the main field theory leading to the GrossPitaevskii equation (GPE), a non-linear Schrödinger equation (NLSE) for the order parameter. Most of the theoretical work aimed at solving the GPE equation has focused on the Thomas-Fermi Approximation (TFA), which corresponds to the case where the non-linear interaction term is much larger than the kinetic energy one. In the present paper we present a method of solution of the time independent GPE, based on a variational procedure which allows us to obtain general analytical expressions for the order parameter and for the chemical potential, valid for repulsive and also for the nowadays more interesting case of attractive interactions. The resulting expressions are compared with those corresponding to the TFA which constitute the preferred analytical solution in most of the reports in the literature. * Corresponding author: e-mail: jcdrake@fisica.uh.cu, Phone: +53 787 889 56, Fax: +53 787 834 71 © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 3666 J. C. Drake Perez et al.: Gross-Pitaevskii equation 2 Variational solution of the GPE The stationary NLSE, written in terms of the order parameter Ψ(x) is the T = 0 K time independent GPE [5], which in the one-dimensional case reads: − ℏ 2 d 2Ψ ( x) 2 + V ( x)Ψ ( x) ± λ Ψ ( x) Ψ ( x) = µΨ ( x), 2m dx 2 (1) where λ is an effective one-dimensional self-interaction parameter describing the interaction between the particles, V(x) is the trapping potential which can be safely approximated by an harmonic oscillator potential with angular frequency ω, and µ is the chemical potential. If a zero trap potential and attractive self-interaction between the particles is considered, the GPE admits a soliton normalized bound solution given by: ψ ( x, A) = A sech ( Ax ) , 2 (2) mλ A= 2 . 4ℏ To get an explicit analytical solution of equation (1) we propose a variational “soliton” type wave function (2) with A considered as a variational parameter [6]. The variational mathematical procedure provides for the chemical potential the expression: 2 2 µ (b, λ ) 1   λ b  3π 2  ℏ ω ℓ   = −  −     3   ℏω ℓ  4  λb   ℏω   (3) and for the order parameter: ℓ ψ ( x) =  λb x λb Sech  , 2 ℏω ℓ  ℏω ℓ ℓ  (4) where the coefficient b must fulfil the transcendental equation:  π ℏω ℓ  b3 (b ± 1) =    2 λ  4 (5) with positive (negative) sign applying for repulsive (attractive) interaction. 3 Results Seeking for comparison points for our variational solution, we perform a perturbation theory (PT) calculation based on the fact that in the non-interacting case, when λ ≡ 0, equation (1) becomes the Schrödinger equation for the harmonic oscillator, with a well-known solution. For small values of λ, we found the chemical potential up to second order PT. This solution is expected to be correct for values of λ /(ћ ω ℓ) smaller than or of the order of 1. Figure 1 shows the chemical potential (in units of ћω), as a function of the universal parameter λ /(ћ ω ℓ) for repulsive interaction. As can be seen, the agreement between our variational solution and the PT is remarkably good for values of λ /(ћ ω ℓ) smaller than 1.5. We find that for λ / (ћ ω ℓ) = 0 the exact solution is µ/(ћω) = 0.50, whereas our variational solution yields µ/(ћω) ≈ 0.52, giving a difference of about 4%. This is a remarkable result, especially if we take into account that the starting point of our variational procedure is the solution of the problem for the zero trap case. With respect to the TFA, it is well known that this method is appropriated only in the strong repul© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim phys. stat. sol. (c) 2, No. 10 (2005) / www.pss-c.com 3667 sive limit. Nevertheless for large values of λ/(ћ ω ℓ), we observe a shift between the values given by the TFA and the variational soliton solution (4). As λ /(ћ ω ℓ) → ∞, the asymptotic behaviour of the TFA reads: 2 2 1  3 3  λ 3 µTF =     2  2   ℏω ℓ  while equations (3) and (5) yield: (6) 2 µVar 1  π λ 3 =   2  2 ℏω ℓ  (7) = 3 ( π3 ) 2 , explaining the shift observed in the Fig. 1. Using equations (6) and (7) we obtain that µµVar TF 4 3.5 Var TFA PT 3 2.5 Μ 
Ω 2 1.5 1 0.5 2 4 6 8 10 12 Λ 
Ω
Fig. 1 The chemical potential in units of ћω as a function of the parameter λ/(ћ ω ℓ) for repulsive interaction. It can be seen that the PT is only valid for small values of the parameter. In the strong repulsive limit the variational solution has the same behaviour as the Thomas-Fermi approximation. Figure 2 presents the dependence µ as a function of λ / (ћ ω ℓ) for the attractive interaction in the condensate. It can be seen that the PT agree very well for –1.5 < λ / (ћ ω ℓ) < 0 (see inset), while for λ / (ћ ω ℓ) << –1.5 or the strong attractive limit, equations (3)–(5) give a straightforward evaluation of the condensate behavior. It is worth noting that beyond this limit, in the zone where the TFA is no longer valid, our variational solution retains its excellent agreement with the PT near the non-interacting limit case (inset). Furthermore, for attractive interaction, and especially in the strong attractive limit case, equations (3)–(5) yield simple analytical representations for the estimation of the BEC properties. 3 Conclusions We have presented a variational procedure which allows to solve the GPE and to describe the BE condensate parameters. We derived simple analytical expressions for the chemical potential and condensate wave function valid for attractive and repulsive interactions. Our results are given in universal units, which allow validating quantitatively the TFA for the positive coupling constant case and PT in the weak interaction limit (–1.5< λ / (ћ ω ℓ) < 1.5). The present theoretical model can be straightforward generalized for the realistic three-dimensional BEC and also to attempt the dynamics of new BEC systems consistent of mixtures of two kinds of atoms or hyperfine structure species [7]. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 3668 J. C. Drake Perez et al.: Gross-Pitaevskii equation 2 0 Var 2 TFA 4 Μ 
Ω 1 0.8 6 PT 0.6 0.4 8 0.2 10 0 12 0.2 1.5 7 6 5 4 Λ  
Ω
1 0.5 3 0 2 0.5 1 1 1.5 0 Fig. 2 The chemical potential in units of ћω as a function of the parameter λ / (ћ ω ℓ) for attractive interaction. The inset shows the zone corresponding to small values of the parameter λ / (ћ ω ℓ), for both repulsive and attractive interactions. Acknowledgements E. Altshuler for his carefully reading of the present manuscripts is gratefully acknowledged. References [1] S. N. Bose, Z. Phys. 26, 178 (1924). A. Einstein, Sitzungsber. K. Preuss. Akad. Wiss. 1924, 261 (1924). [2] E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). [3] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle. Phys. Rev. Lett. 75, 3969 (1995). [4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995). [5] E.P. Gross, Nuovo Cimento 20, 454 (1961). L.P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961). [6] J. C. Drake Perez, C. Trallero-Giner, V. López, C. Trallero-Herrero, and J. L. Birman (unpublished). [7] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 78, 586 (1997). © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim