American Institute of Mathematical Sciences

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June  2014, 19(4): 1119-1128. doi: 10.3934/dcdsb.2014.19.1119

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, Taiwan
2. 

Department of Applied Mathematics and Center of Mathematical Modeling, and Scientific Computing, National Chiao Tung University, Hsinchu, 30010, Taiwan

Received  October 2012 Revised  January 2014 Published  April 2014

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.
Keywords: mass redistribution., ground state, single-mode approximation, Spin-1 BEC, nonlinear system.
Mathematics Subject Classification: Primary: 35A15, 35Q55; Secondary: 35Q4.
Citation: Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces, Journal d'Analyse Mathématique, 80 (2000), 37-86. doi: 10.1007/BF02791533.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

T.-L. Ho, Spinor bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), 742-745. doi: 10.1103/PhysRevLett.81.742.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

J. Bourgain, H. Brezis and P. Mironescu, Lifting in Sobolev spaces, Journal d'Analyse Mathématique, 80 (2000), 37-86. doi: 10.1007/BF02791533.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[16]

T.-L. Ho, Spinor bose condensates in optical traps, Phys. Rev. Lett., 81 (1998), 742-745. doi: 10.1103/PhysRevLett.81.742.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP, 13 (1961), 451-454. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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