## A basis-set based fortran program to solve the Gross-Pitaevskii equation for dilute Bose gases in harmonic and anharmonic traps.(English)Zbl 1196.81039

Summary: Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators $$x, y$$, and $$z$$. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas-Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way.

### MSC:

 81-04 Software, source code, etc. for problems pertaining to quantum theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

EISPACK; bose.x
Full Text:

### References:

 [1] Anderson, M.H.; Ensher, J.R.; Matthews, M.R.; Wieman, C.E.; Cornell, E.A., Science, 269, 198, (1995) [2] Bradley, C.C.; Sackett, C.A.; Tollett, J.J.; Hulet, R.G., Phys. rev. lett., 75, 1687, (1995) [3] Davis, K.B.; Mewes, M.-O.; Andrews, M.R.; van Druten, N.J.; Durfee, D.S.; Kurn, D.M.; Ketterle, W., Phys. rev. lett., 75, 3969, (1995) [4] Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S., Rev. modern phys., 71, 463, (1999), For a review, see, e.g. [5] Gross, E.P., Nuovo cimento, 20, 454, (1961) [6] Pitaevskii, L.P., Zh. eksp. teor. fiz., 40, 646, (1961) [7] Edwards, M.; Burnett, K., Phys. rev. A, 51, 1382, (1995) [8] Edwards, M.; Dodd, R.J.; Clark, C.W.; Ruprecht, P.A.; Burnett, K., Phys. rev. A, 53, R1950, (1996) [9] Dalfovo, F.; Stringari, S., Phys. rev. A, 53, 2477, (1996) [10] Esry, B.D., Phys. rev. A, 55, 1147, (1997) [11] Schneider, B.I.; Feder, D.L., Phys. rev. A, 59, 2232, (1999) [12] Adhikari, S.K., Phys. lett. A, 265, 91, (2000) [13] Adhikari, S.K., Phys. rev. E, 62, 2937, (2000) [14] Chiofalo, M.L.; Succi, S.; Tosi, M.P., Phys. rev. E, 62, 7438, (2000) [15] Bao, W.; Tang, W., J. comput. phys., 187, 230, (2003) [16] Dion, C.M.; Cancès, E., Phys. rev. E, 67, 046706, (2003) [17] Ziegler, K.; Shukla, A., Phys. rev. A, 56, 1438, (1997) [18] Szabo, A.; Ostlund, N.S., Modern quantum chemistry: introduction to advanced electronic structure theory, (1996), Dover Publications New York, See, e.g. [19] Busbridge, I.W., J. London math. soc., 23, 135, (1948) [20] Shannon, C.E., Bell system tech. J., 27, 379, (1948), 623 [21] Gadre, S.R.; Gadre, S.R.; Bendale, R.D., Phys. rev. A, Phys. rev. A, 36, 1932, (1987), See, e.g. [22] Ziesche, P., Int. J. quantum chem., 56, 363, (1995) [23] Moustakidis, Ch.C.; Massen, S.E., Phys. rev. B, 71, 045102, (2005) [24] Smith, B.; Boyle, J.; Dongarra, J.; Garbow, B.; Ikebe, Y.; Klema, V.; Moler, C., Matrix eigenvalues routines, EISPACK guide, Lecture notes in computer science, vol. 6, (1976), Springer-Verlag · Zbl 0325.65016 [25] These are the parameters used to describe the experiment of Anderson et al. on ^{87}Rb [1]. They correspond to a frequency ratio ωzωx=\sqrt{8}, and the scattering length a=4.33×10−3ax. These parameters were also used by Dalfovo et al. in their calculations [9] [26] Fetter, A.L.; Lundh, E., Phys. rev. A, Phys. rev. A, 65, 043604, (2002), See, e.g. [27] Pullen, R.A.; Edmonds, A.R., J. phys. A, 14, L319, (1981) [28] Pullen, R.A.; Edmonds, A.R., J. phys. A, 14, L477, (1981) [29] Ho, T.-L., Phys. rev. lett., 81, 742, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.