Abstract
We have considered a Bose gas in an anisotropic potential. Applying the the Gross-Pitaevskii Equation (GPE) for a confined dilute atomic gas, we have used the methods of optimized perturbation theory and self-similar root approximants, to obtain an analytical formula for the critical number of particles as a function of the anisotropy parameter for the potential. The spectrum of the GPE is also discussed.
Bose gas; Anisotropic potential; Critical number of particles; Optimized perturbation theory; critical coupling parameter
REGULAR ARTICLES
An analytical solution for the critical number of particles for stable bose-einstein condensation under the influence of an anisotropic potential
A. G. de SousaI, II; V. S. BagnatoI; A. B. F. da SilvaIII
IInstituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, São Carlos, SP, 13560-970, Brazil
IIFaculdade Independente do Nordeste, Av. Luís Eduardo Magalhães 1305, Candeias, Vitória da Conquista, Ba, 45055-030, Brazil
IIIInstituto de Química de São Carlos, Universidade de São Paulo, Caixa Postal 780, São Carlos, SP, 13560-970, Brazil
ABSTRACT
We have considered a Bose gas in an anisotropic potential. Applying the the Gross-Pitaevskii Equation (GPE) for a confined dilute atomic gas, we have used the methods of optimized perturbation theory and self-similar root approximants, to obtain an analytical formula for the critical number of particles as a function of the anisotropy parameter for the potential. The spectrum of the GPE is also discussed.
Keywords: Bose gas; Anisotropic potential; Critical number of particles; Optimized perturbation theory and critical coupling parameter
I. INTRODUCTION
The experimental realization of BEC with alkali atoms [1-5] opened a new and exciting field involving quantum atomic fluids. This new field has been launched with many interesting theoretical predictions and ideas. When the system is spatially confined with atomic interaction effectively attractive, BEC is stable only below a certain critical value. In this case, the zero-point motion of the atoms serves as a kinetic obstacle against collapse, allowing a metastable BEC to be formed [6-10]. Above the critical number of particles the colapse is unavoidable.
When the ground-state energy becomes complex, the system becomes unstable. That is a general way to identify the critical value for the coupling parameter in the system as well as identifying the critical number of particles.
It was shown by Gammal et al [11-12] within the Gross-Pitaevskii formalism, that the critical number of particles for BEC in cylindrical traps can be obtained numerically for the case attractive interactions. It was also calculated in reference [13] the spectrum of the Gross-Pitaevskii Equation (GPE) for a system composed of attractive bosons confined in a harmonic trap through the Controlled Perturbation Theory. In reference [13], the critical number of particles that ensures real values for the energy spectrum was obtained as a function of the potential anisotropic parameter.
The aim of this work is to consider the analytical spectrum of the Gross-Pitaevskii equation through approximants crossover roots [16] with negative effective interaction strength in a cylindrical symmetry. An analytical formula was derived to analyze the stable critical number of particles, Nc as a function of the anisotropy of the confining potential. Here we used controlled perturbation theory [14] to derive numerically the spectrum of the Gross-Pitaevskii equation, and approximants crossover roots [15] to obtain the analytical expression for the spectrum of the Gross-Pitaevskii equation.
In fact, we used controlled perturbation theory instead of standard (ordinary) perturbation theory since the first one uses control functions that optimize the convergence [13].
II. MODEL
Bose system is confined by a trapping potential given by
The interaction potential is considered as a Fermi contact potential as
with as being the s-wave scattering length. With the confining potential and the interaction potential, we get the nonlinear Hamiltonian
Due to the presence of this confining potential, the spectrum of the stationary Gross-Pitaevskii Equation (GPE) is discrete, being defined by the eigenvalue problem
When the confining presents cylindrically symmetric, so that
hence anisotropy parameter is defined as
The radial oscillator length
serves to define the dimensionless cylindrical variables, and
Then one may define the dimensionless coherent wave function
depending on the cylindrical variables r Î (0, ¥), j Î (0, 2p) and z Î (¥, -¥) and the Hamiltonian, H, is defined as
The atom-atom coupling parameter can be expressed as
Thus we can write the Hamiltonian in the form
where
and the coupling parameter, g , is the perturbed Hamiltonian term.
The eigenproblem (4) takes the form HYnml = Enml Ynml , in which n = 0,1,2,... is radial quantum number, m = 0,±1,±2,... is the azimuthal quantum number, and l = 0,1,2,... is the axial quantum number.
The equilibrium Bose-Einstein condensate corresponds to the ground-state solution of the stationary GPE, when n = m = l = 0.
The nonlinear eigenproblem cannot be solved exactly. We can employ the optimized perturbation theory to find the accurate expressions for the spectrum of GPE and for arbitrary values of the coupling parameter. This theory has been applied in several physical problems [13-14]. This approach has been used for calculating the critical temperature of Bose-Einsten condensation in a Dilute Bose Gas [14]. The introduction of optimized perturbation theory provides us the control functions that are optimal choice and provoke the optimal convergence of a calculational procedure [14], this theory was introduced by Yukalov [16].
In applying the optimized perturbation theory to the GPE, the spectrum, in first approximation, can be written as:
e(g) º E(g, u(g), v(g))
with the expression
where p and q are defined as
The related eigenfunction are [17]
where (.) are the Laguerre polynomials and Hk(.) are Hermite polynomials.
The fixed-point conditions are, therefore, obtained from conditions
which yield the control function equations
and the effective interaction strength is represented by
Equations (5)-(7) define the spectrum (4) for all quantum numbers n, m and l and for arbitrary values of the coupling parameter g. The term Inml can be related with the coefficients an through a recursion formula showed in reference [13]. We derive the expressions for the spectrum of GPE and find weak-coupling
limit for the spectrum as
with the coefficients
For the strong-coupling limit
where the coefficients are given by the equations
and the exponents are
interpolating the asymptotic limits (15) and (16) by means of the self-similar root approximants [17-18], that results in the following approximants crossover roots function
Here, depending on the approximation order b = 1,2,... we have the values of A and n in the first order
in the second order
and in the third order
This model was described in several references [16-18].
III. RESULTS
Analysis for the critical number of particles in different traps are performed using an analytical expression from the self-similar root approximants to get an expression for Nc. We need to find the negative value of sc in which the spectrum (10) become complex. We limit the first order of root approximant, (g), which is complex for s < sc , with
Using the eq. (14), we get the equation for the critical coupling parameter
The equation g = , gives us the expression for the critical number of particles
Using the analytical expression (18) with the values of 7Li Condensate [18-19] and considering the ground state, with n = m = l = 0, p = l = 1 and I000 = 0.063494,we can obtain Nc in Figure 1.
From these results, we observe that there are two regimes: cigar-shape (l << 1) and the disc-shape (l >> 1). Both situations increase Nc, while in a spherical-shape Nc reaches its the minimal value. For example, in a cigar-shape trap l = 0.1 and Nc = 7157, in a disc-shape trap l = 10 and Nc = 5586 and in a spherical-shape trap l = 1 with Nc = 1350.These values are in a good agreement with those in references [11,13].
IV. CONCLUSION
The method of the self-similar root approximants resulted in analytical expressions for the spectrum of energy levels, for the critical coupling parameter and for the critical number of particles. We have analytically studied the behaviour of the critical number of particles as a function of the anisotropy parameter, which introduced a better stability in the system.
Acknowledgments
We are grateful to Yukalov for important contributions and discussion to this work and to CNPQ and CAPES for the financial support.
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[8] M. Ueda and K. Huang, Phys. Rev. A 60, 3317 (1999).
[9] A. Eleftheriou and K. Huang, Phys. Rev A 61, 043601 (2000).
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[11] A. Gammal, T. Frederico, and L. Tomio, Phys. Rev. A 64, 055602 (2001).
[12] A. Gammal, L. Tomio, and T. Frederico, Phys. Rev. A 66, 043619 (2002).
[13] A. G. De Sousa, A. B. F. da Silva, G. C. Marques, and V. S. Bagnato, Phys. Rev. A 70, 063608 (2004).
[14] V. I. Yukalov, Ann. Physik 38, 419 (1981).
[15] F. F. de Souza Cruz, M. B. Pinto, R. O. Ramos, and P. Sena, Phys. Rev. A 65, 053613 (2002).
[16] V. I. Yukalov, Theor. Math. Phys 28, 652 (1976).
[17] Ph. W. Courteille, V. S. Bagnato, and V. I. Yukalov, Laser Physics, vol. II, nº 6, 659 (2001).
[18] V. I. Yukalov and E. P. Yukalova, Chaos Solitons Fractals 14, 839 (2002).
[19] S. Gluzman and V. I. Yukalov, Phys. Rev. E 58, 4197 (1998).
Received on 24 July, 2007
- [1] M. H. Anderson, J. R. Ensher, M. R. Mathews, C. E. Wieman, and E.A. Cornell, Science 269, 198 (1995).
- [2] K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995).
- [3] D. G. Fried et al., Phys. Rev. Lett. 81, 3811 (1998).
- [4] A. Robert et al., Science 292, 461 (2001).
- [5] C. C. Bradley, C. A. Sackett, J. J. Tollet, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C. C. Bradley, C. A.Sackett, and R. G. Hulet, ibid. 78, 895 (1997).
- [6] Y. E. Kim and A.L. Zubarev, Phys. Lett. A 246, 389 (1998).
- [7] M. Wadati and T. Tsurumi, Phys. Lett. A 247, 287 (1998).
- [8] M. Ueda and K. Huang, Phys. Rev. A 60, 3317 (1999).
- [9] A. Eleftheriou and K. Huang, Phys. Rev A 61, 043601 (2000).
- [10] M. Ueda and A.J. Leggett, Phys. Rev. Lett. 80, 1576 (1998).
- [11] A. Gammal, T. Frederico, and L. Tomio, Phys. Rev. A 64, 055602 (2001).
- [12] A. Gammal, L. Tomio, and T. Frederico, Phys. Rev. A 66, 043619 (2002).
- [13] A. G. De Sousa, A. B. F. da Silva, G. C. Marques, and V. S. Bagnato, Phys. Rev. A 70, 063608 (2004).
- [14] V. I. Yukalov, Ann. Physik 38, 419 (1981).
- [15] F. F. de Souza Cruz, M. B. Pinto, R. O. Ramos, and P. Sena, Phys. Rev. A 65, 053613 (2002).
- [16] V. I. Yukalov, Theor. Math. Phys 28, 652 (1976).
- [17] Ph. W. Courteille, V. S. Bagnato, and V. I. Yukalov, Laser Physics, vol. II, nş 6, 659 (2001).
- [18] V. I. Yukalov and E. P. Yukalova, Chaos Solitons Fractals 14, 839 (2002).
- [19] S. Gluzman and V. I. Yukalov, Phys. Rev. E 58, 4197 (1998).
Publication Dates
-
Publication in this collection
27 Mar 2008 -
Date of issue
Mar 2008
History
-
Received
24 July 2007