Abstract

By developing an approach by which we are able to quickly obtain spectra and eigenstates, we reveal for what is believed to be the first time the two novel phenomena of magic numbers and erratic level crossings in double-well Bose–Einstein condensates of N atoms. For N27 and values of UJ that are not too small (U is the two-body interaction strength, and J is the hopping parameter), systems with even atoms are shown to be much more stable than those with odd atoms, and hence even integers play a role in such systems as if they were the magic numbers of nuclei. For N30, erratic level crossings occur as NUJ.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
    [CrossRef]
  2. M. J. Steel and M. J. Collett, Phys. Rev. A 57, 2920 (1998).
    [CrossRef]
  3. J. R. Anglin and A. Vardi, Phys. Rev. A 64, 013605 (2001).
    [CrossRef]
  4. J. Javanainen and M. Yu. Ivanov, Phys. Rev. A 60, 2351 (1999).
    [CrossRef]
  5. S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999).
    [CrossRef]
  6. A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997).
    [CrossRef]
  7. M. Jääskeläinen and P. Meystre, Phys. Rev. A 71, 043603 (2005).
    [CrossRef]
  8. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
    [CrossRef] [PubMed]
  9. G. Kalosakas and A. R. Bishop, Phys. Rev. A 65, 043616 (2002).
    [CrossRef]
  10. J. A. Dunningham and K. Burnett, Phys. Rev. A 70, 033601 (2003).
    [CrossRef]
  11. A. I. Streltsov, L. S. Cederbaum, and N. Moiseyev, Phys. Rev. A 70, 053607 (2004).
    [CrossRef]
  12. A. P. Tonel, J. Links, and A. Foerster, J. Phys. A 38, 6879 (2005).
    [CrossRef]
  13. R. V. F. Janssens, Nature 435, 897 (2005).
    [CrossRef] [PubMed]
  14. T. Guhr, A. Müller-Groeling, and H. A. Weidenmüer, Phys. Rep. 299, 189 (1998).
    [CrossRef]
  15. Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
    [CrossRef] [PubMed]
  16. Y. Wu and X. Yang, Opt. Lett. 28, 1793 (2003).
    [CrossRef] [PubMed]
  17. Y. Wu and X. Yang, Opt. Lett. 29, 839 (2004).
    [CrossRef] [PubMed]
  18. Y. Wu and X. Yang, J. Opt. Soc. Am. B 21, 73 (2004).
    [CrossRef]
  19. The Mathematica code is ⪡ Linear Algebra 'Tridiagonal'; n=to be designated; p[j−]≔j(j−1)(n+2−j)(n+1−j); q[j−]≔2j+un(n−1)+2j(n−j)u−lambda; k=Floor[n/2]+1; m=Floor[(n−1)/2]+1; a=Table[Switch[j−i,−1,up[2j],0,q[2j−2],1,u, −,0],i,k,j,k]; b=Table[Switch[j−i,−1,up[2j+1],0,q[2j−1],1,u, −,0],i,m,j,m]; Solve[Det[a]==0,lambda]; Solve[Det[b]==0,lambda] with the total atoms N denoted as n.
  20. The code {n=10,f[j−]≔Plot[Root[exp1&,j],{u,0,100}]; g[j−]≔Plot[Root[exp2&,j],{u,0,100}]; Show[f[1],...,f[6],g[1],...,g[5]]} plots in a single figure all 11 eigenvalues lambda versus u∊[0,100] for N=10. Root[exp1&,j] and Root[exp2&,j] are the solutions to Eqs. , respectively.

2005

M. Jääskeläinen and P. Meystre, Phys. Rev. A 71, 043603 (2005).
[CrossRef]

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

A. P. Tonel, J. Links, and A. Foerster, J. Phys. A 38, 6879 (2005).
[CrossRef]

R. V. F. Janssens, Nature 435, 897 (2005).
[CrossRef] [PubMed]

2004

2003

Y. Wu and X. Yang, Opt. Lett. 28, 1793 (2003).
[CrossRef] [PubMed]

J. A. Dunningham and K. Burnett, Phys. Rev. A 70, 033601 (2003).
[CrossRef]

2002

G. Kalosakas and A. R. Bishop, Phys. Rev. A 65, 043616 (2002).
[CrossRef]

2001

J. R. Anglin and A. Vardi, Phys. Rev. A 64, 013605 (2001).
[CrossRef]

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

1999

J. Javanainen and M. Yu. Ivanov, Phys. Rev. A 60, 2351 (1999).
[CrossRef]

S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999).
[CrossRef]

1998

M. J. Steel and M. J. Collett, Phys. Rev. A 57, 2920 (1998).
[CrossRef]

T. Guhr, A. Müller-Groeling, and H. A. Weidenmüer, Phys. Rep. 299, 189 (1998).
[CrossRef]

1997

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[CrossRef]

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997).
[CrossRef]

Albiez, M.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Anglin, J. R.

J. R. Anglin and A. Vardi, Phys. Rev. A 64, 013605 (2001).
[CrossRef]

Bishop, A. R.

G. Kalosakas and A. R. Bishop, Phys. Rev. A 65, 043616 (2002).
[CrossRef]

Burnett, K.

J. A. Dunningham and K. Burnett, Phys. Rev. A 70, 033601 (2003).
[CrossRef]

Cederbaum, L. S.

A. I. Streltsov, L. S. Cederbaum, and N. Moiseyev, Phys. Rev. A 70, 053607 (2004).
[CrossRef]

Collett, M. J.

M. J. Steel and M. J. Collett, Phys. Rev. A 57, 2920 (1998).
[CrossRef]

Corney, J.

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[CrossRef]

Cristiani, M.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Dunningham, J. A.

J. A. Dunningham and K. Burnett, Phys. Rev. A 70, 033601 (2003).
[CrossRef]

Fantoni, S.

S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999).
[CrossRef]

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997).
[CrossRef]

Foerster, A.

A. P. Tonel, J. Links, and A. Foerster, J. Phys. A 38, 6879 (2005).
[CrossRef]

Fölling, J.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Gati, R.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Giovanazzi, S.

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997).
[CrossRef]

Guhr, T.

T. Guhr, A. Müller-Groeling, and H. A. Weidenmüer, Phys. Rep. 299, 189 (1998).
[CrossRef]

Hunsmann, S.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Ivanov, M. Yu.

J. Javanainen and M. Yu. Ivanov, Phys. Rev. A 60, 2351 (1999).
[CrossRef]

Jääskeläinen, M.

M. Jääskeläinen and P. Meystre, Phys. Rev. A 71, 043603 (2005).
[CrossRef]

Janssens, R. V. F.

R. V. F. Janssens, Nature 435, 897 (2005).
[CrossRef] [PubMed]

Javanainen, J.

J. Javanainen and M. Yu. Ivanov, Phys. Rev. A 60, 2351 (1999).
[CrossRef]

Kalosakas, G.

G. Kalosakas and A. R. Bishop, Phys. Rev. A 65, 043616 (2002).
[CrossRef]

Links, J.

A. P. Tonel, J. Links, and A. Foerster, J. Phys. A 38, 6879 (2005).
[CrossRef]

Meystre, P.

M. Jääskeläinen and P. Meystre, Phys. Rev. A 71, 043603 (2005).
[CrossRef]

Milburn, G. J.

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[CrossRef]

Moiseyev, N.

A. I. Streltsov, L. S. Cederbaum, and N. Moiseyev, Phys. Rev. A 70, 053607 (2004).
[CrossRef]

Müller-Groeling, A.

T. Guhr, A. Müller-Groeling, and H. A. Weidenmüer, Phys. Rep. 299, 189 (1998).
[CrossRef]

Oberthaler, M. K.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

Raghavan, S.

S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999).
[CrossRef]

Shenoy, S. R.

S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999).
[CrossRef]

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997).
[CrossRef]

Smerzi, A.

S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999).
[CrossRef]

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997).
[CrossRef]

Steel, M. J.

M. J. Steel and M. J. Collett, Phys. Rev. A 57, 2920 (1998).
[CrossRef]

Streltsov, A. I.

A. I. Streltsov, L. S. Cederbaum, and N. Moiseyev, Phys. Rev. A 70, 053607 (2004).
[CrossRef]

Tonel, A. P.

A. P. Tonel, J. Links, and A. Foerster, J. Phys. A 38, 6879 (2005).
[CrossRef]

Vardi, A.

J. R. Anglin and A. Vardi, Phys. Rev. A 64, 013605 (2001).
[CrossRef]

Walls, D. F.

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[CrossRef]

Weidenmüer, H. A.

T. Guhr, A. Müller-Groeling, and H. A. Weidenmüer, Phys. Rep. 299, 189 (1998).
[CrossRef]

Wright, E. M.

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[CrossRef]

Wu, Y.

Xiao, Y.

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

Yang, X.

J. Opt. Soc. Am. B

J. Phys. A

A. P. Tonel, J. Links, and A. Foerster, J. Phys. A 38, 6879 (2005).
[CrossRef]

Nature

R. V. F. Janssens, Nature 435, 897 (2005).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rep.

T. Guhr, A. Müller-Groeling, and H. A. Weidenmüer, Phys. Rep. 299, 189 (1998).
[CrossRef]

Phys. Rev. A

G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[CrossRef]

M. J. Steel and M. J. Collett, Phys. Rev. A 57, 2920 (1998).
[CrossRef]

J. R. Anglin and A. Vardi, Phys. Rev. A 64, 013605 (2001).
[CrossRef]

J. Javanainen and M. Yu. Ivanov, Phys. Rev. A 60, 2351 (1999).
[CrossRef]

S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 620 (1999).
[CrossRef]

M. Jääskeläinen and P. Meystre, Phys. Rev. A 71, 043603 (2005).
[CrossRef]

G. Kalosakas and A. R. Bishop, Phys. Rev. A 65, 043616 (2002).
[CrossRef]

J. A. Dunningham and K. Burnett, Phys. Rev. A 70, 033601 (2003).
[CrossRef]

A. I. Streltsov, L. S. Cederbaum, and N. Moiseyev, Phys. Rev. A 70, 053607 (2004).
[CrossRef]

Phys. Rev. Lett.

M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005).
[CrossRef] [PubMed]

A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett. 79, 4950 (1997).
[CrossRef]

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

Other

The Mathematica code is ⪡ Linear Algebra 'Tridiagonal'; n=to be designated; p[j−]≔j(j−1)(n+2−j)(n+1−j); q[j−]≔2j+un(n−1)+2j(n−j)u−lambda; k=Floor[n/2]+1; m=Floor[(n−1)/2]+1; a=Table[Switch[j−i,−1,up[2j],0,q[2j−2],1,u, −,0],i,k,j,k]; b=Table[Switch[j−i,−1,up[2j+1],0,q[2j−1],1,u, −,0],i,m,j,m]; Solve[Det[a]==0,lambda]; Solve[Det[b]==0,lambda] with the total atoms N denoted as n.

The code {n=10,f[j−]≔Plot[Root[exp1&,j],{u,0,100}]; g[j−]≔Plot[Root[exp2&,j],{u,0,100}]; Show[f[1],...,f[6],g[1],...,g[5]]} plots in a single figure all 11 eigenvalues lambda versus u∊[0,100] for N=10. Root[exp1&,j] and Root[exp2&,j] are the solutions to Eqs. , respectively.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

( N + 1 ) eigenenergies λ versus u for N = 21 . The bottom right inset shows the eigenenergy difference ( λ 1 λ 0 ) of the first excited state and ground state.

Fig. 2
Fig. 2

Same as Fig. 1 except N = 22 .

Fig. 3
Fig. 3

Same as Fig. 1 except N = 30 . Notice that the erratic level crossings occur even for an even N!

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

̂ = J ( c ̂ 1 c ̂ 2 + c ̂ 2 c ̂ 1 ) + 1 2 U ( c ̂ 1 c ̂ 1 c ̂ 1 c ̂ 1 + c ̂ 2 c ̂ 2 c ̂ 2 c ̂ 2 ) ,
H ̂ = 2 a ̂ 2 a ̂ 2 + u [ ( a ̂ 1 2 + a ̂ 2 2 ) ( a ̂ 1 2 + a ̂ 2 2 ) + 4 a ̂ 1 a ̂ 1 a ̂ 2 a ̂ 2 ] ,
a ̂ j = c ̂ 1 + ( 1 ) j + 1 c ̂ 2 2 c ̂ j = a ̂ 1 + ( 1 ) j + 1 a ̂ 2 2 ,
2 y F y + u ( x 2 + y 2 ) 2 F + 4 u x y 2 F x y = λ F ,
F ( x , y ) = n = 0 N α n ( N n ) ! n ! x N n y n n ! ( N n ) ! ,
u α n + 2 + q n α n + u p n α n 2 = 0 , n = 0 ; 1 , , N ,
q n = q n ( λ ) = 2 n + [ N ( N 1 ) + 2 n ( N n ) ] u λ ,
p n = n ( n 1 ) ( N n + 2 ) ( N n + 1 ) ,
det A ( n + 1 ) ( λ 2 n ) = 0 , n = 0 , 1 , , n ,
det B ( n + 1 ) ( λ 2 n + 1 ) = 0 , n = 0 , 1 , , n ,
Ψ 2 n = β 0 ( n ) k = 0 n α ̃ k ( n ) N 2 k , 2 k ,
Ψ 2 n + 1 = β 1 ( n ) j = 0 n β ̃ j ( n ) N 2 j 1 , 2 j + 1 ,

Metrics