A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates

Liren Lin; I-Liang Chern

doi:10.3934/dcdsb.2014.19.1119

Article Contents

2014, Volume 19, Issue 4: 1119-1128. Doi: 10.3934/dcdsb.2014.19.1119

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A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates

Liren Lin^1, and
I-Liang Chern^2,

1.
Institute of Mathematics, Academia Sinica, Taipei, 10617
2.
Department of Applied Mathematics and Center of Mathematical Modeling, and Scientific Computing, National Chiao Tung University, Hsinchu, 30010

Received: September 30, 2012

Revised: December 31, 2013

Published: May 2014

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Abstract

We justify some characterizations of the ground states of spin-1 Bose-Einstein condensates exhibited from numerical simulations. For ferromagnetic systems, we show the validity of the single-mode approximation (SMA). For an antiferromagnetic system with nonzero magnetization, we prove the vanishing of the $m_F=0$ component. In the end of the paper some remaining degenerate situations are also discussed. The proofs of the main results are all based on a simple observation, that a redistribution of masses among different components will reduce the kinetic energy.

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Mathematics Subject Classification: Primary: 35A15, 35Q55; Secondary: 35Q40.

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References

[1]	M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science, 269 (1995), 198-201.doi: 10.1126/science.269.5221.198.
[2]	W. Bao and Y. Cai, Ground states of two-component Bose-Einstein condensates with an internal atomic Josephson junction, East Asian J. Appl. Math, 1 (2011), 49-81.doi: 10.4208/eajam.190310.170510a.
[3]	W. Bao and F. Y. Lim, Computing ground states of spin-1 Bose-Einstein condensates by the normalized gradient flow, SIAM J. Sci. Comput., 30 (2008), 1925-1948.doi: 10.1137/070698488.
[4]	M. D. Barrett, J. A. Sauer and M. S. Chapman, All-optical formation of an atomic Bose-Einstein condensate, Phys. Rev. Lett., 87 (2001), 010404.
[5]	F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., 80 (1988), 60-75.doi: 10.1016/0022-1236(88)90065-1.
[6]	J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces, Journal d'Analyse Mathématique, 80 (2000), 37-86.doi: 10.1007/BF02791533.
[7]	C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett., 75 (1995), 1687-1690.doi: 10.1103/PhysRevLett.75.1687.
[8]	D. Cao, I.-L. Chern and J. Wei, On ground state of spinor Bose-Einstein condensates, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 427-445.doi: 10.1007/s00030-011-0102-9.
[9]	J.-H. Chen, I.-L. Chern and W. Wang, Exploring ground states and excited states of spin-1 Bose-Einstein condensates by continuation methods, J. Comput. Phys., 230 (2011), 2222-2236.doi: 10.1016/j.jcp.2010.11.048.
[10]	F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512.doi: 10.1103/RevModPhys.71.463.
[11]	K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), 3969-3973.doi: 10.1103/PhysRevLett.75.3969.
[12]	L.-M. Duan, J. I. Cirac and P. Zoller, Quantum entanglement in spinor Bose-Einstein condensates, Phys. Rev. A, 65 (2002), 033619.doi: 10.1103/PhysRevA.65.033619.
[13]	E. V. Goldstein and P. Meystre, Quantum theory of atomic four-wave mixing in Bose-Einstein condensates, Phys. Rev. A, 59 (1999), 3896-3901.doi: 10.1103/PhysRevA.59.3896.
[14]	A. Görlitz, T. L. Gustavson, A. E. Leanhardt, R. Löw, A. P. Chikkatur, S. Gupta, S. Inouye, D. E. Pritchard and W. Ketterle, Sodium Bose-Einstein condensates in the $f=2$ state in a large-volume optical trap, Phys. Rev. Lett., 90 (2003), 090401.
[15]	E. P. Gross, Structure of a quantized vortex in boson systems, Il Nuovo Cimento Series 10, 20 (1961), 454-477.doi: 10.1007/BF02731494.
[16]	T.-L. Ho, Spinor bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), 742-745.doi: 10.1103/PhysRevLett.81.742.
[17]	T.-L. Ho and S. K. Yip, Fragmented and single condensate ground states of spin-1 bose gas, Phys. Rev. Lett., 84 (2000), 4031-4034.doi: 10.1103/PhysRevLett.84.4031.
[18]	C. K. Law, H. Pu and N. P. Bigelow, Quantum spins mixing in spinor Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5257-5261.doi: 10.1103/PhysRevLett.81.5257.
[19]	E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001.
[20]	E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A, 61 (2000), 043602.doi: 10.1103/PhysRevA.61.043602.
[21]	L.-R. Lin, Mass Redistribution and Its Applications to the Ground States of Spin-1 Bose-Einstein Condensates, Ph.D thesis, National Taiwan University, 2013.
[22]	H.-J. Miesner, D. M. Stamper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur and W. Ketterle, Observation of metastable states in spinor Bose-Einstein condensates, Phys. Rev. Lett., 82 (1999), 2228-2231.doi: 10.1103/PhysRevLett.82.2228.
[23]	T. Ohmi and K. Machida, Bose-Einstein condensation with internal degrees of freedom in alkali atom gases, Journal of the Physical Society of Japan, 67 (1998), 1822-1825.doi: 10.1143/JPSJ.67.1822.
[24]	L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454.
[25]	H. Pu, C. K. Law and N. P. Bigelow, Complex quantum gases: spinor Bose-Einstein condensates of trapped atomic vapors, Physica B: Condensed Matter, 280 (2000), 27-31.doi: 10.1016/S0921-4526(99)01429-5.
[26]	H. Pu, C. K. Law, S. Raghavan, J. H. Eberly and N. P. Bigelow, Spin-mixing dynamics of a spinor Bose-Einstein condensate, Phys. Rev. A, 60 (1999), 1463-1470.doi: 10.1103/PhysRevA.60.1463.
[27]	D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger and W. Ketterle, Optical confinement of a Bose-Einstein condensate, Phys. Rev. Lett., 80 (1998), 2027-2030.doi: 10.1103/PhysRevLett.80.2027.
[28]	J. Stenger, S. Inouye, D. M. Stamper-Kurn, H. Miesner, A. P. Chikkatur and W. Ketterle, Spin domains in ground-state Bose-Einstein condensates, Nature, 396 (1998), 345-348.
[29]	S. Yi, O. E. Müstecapliǧlu, C. P. Sun and L. You, Single-mode approximation in a spinor-1 atomic condensate, Phys. Rev. A, 66 (2002), 011601.doi: 10.1103/PhysRevA.66.011601.