Abstract

For a dilute, interacting Bose gas of magnetically-trapped atoms at temperatures below the critical temperature T 0 for Bose-Einstein condensation, we determine the second-order coherence function g (2)(r 1, r 2) within the framework of a finite-temperature quantum field theory. We show that, because of the different spatial distributions of condensate and thermal atoms in the trap, g (2)(r 1, r 2) does not depend on |r 1 - r 2| alone. This means that the experimental determinations of g (2) reported to date give only its spatial average. Such an average may underestimate the degree of coherence attainable in an atom laser by judicious engineering of the output coupler.

© Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  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, 198 (1995).
    [CrossRef] [PubMed]
  2. 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, 3969 (1995).
    [CrossRef] [PubMed]
  3. C. C. Bradley, C. A. Sackett, and R. G. Hulet, "Bose-Einstein condensation in lithium: observation of limited condensate number," Phys. Rev. Lett. 78, 985 (1997).
    [CrossRef]
  4. M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, "An output coupler for Bose condensed atoms," Phys. Rev. Lett. 78, 582 (1997).
    [CrossRef]
  5. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Observation of interference between two Bose condensates," Science 275, 637 (1997).
    [CrossRef] [PubMed]
  6. H. M. Wiseman, "Defining the (atom) laser," Phys. Rev. A (1997 in press).
    [CrossRef]
  7. E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, "Coherence, correlation and collisions: what one learns from Bose-Einstein condensates from their decay," Phys. Rev. Lett. 79, 337 (1997).
    [CrossRef]
  8. W. Ketterle and H.-J. Miesner, "Coherence properties of Bose condensates and atom lasers," Phys. Rev. A 57, 3291 (1997).
    [CrossRef]
  9. R. J. Dodd, K. Burnett, M. Edwards, and C. W. Clark, "Two-gas description of dilute Bose-Einstein condensates at finite temperature," Phys. Rev. A (submitted).
  10. O. Penrose and L. Onsager, "Bose-Einstein condensation and liquid helium," Phys. Rev. 104, 576 (1956).
    [CrossRef]
  11. D. F. Walls, "Evidence for the quantum nature of light," Nature 280, 451 (1979).
    [CrossRef]
  12. R. Glauber, "Optical coherence and photon statistics," in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965).
  13. A. L. Fetter, "Nonuniform states of an imperfect Bose gas," Ann. Phys. (NY) 70, 67 (1972).
    [CrossRef]
  14. E. M. Lifshitz and L. P. Pitaevski, Statistical Physics Part 2 (Butterworth Heinemann, Oxford, 1995).
  15. V. N. Popov, Functional Integrals and Collective Modes (Cambridge University Press, New York, 1987), Chapter 6.
  16. A. Griffin, "Conserving and gapless approximations for an inhomogeneous Bose gas at finite temperatures," Phys. Rev. B 53, 9341 (1996).
    [CrossRef]
  17. M. Houbiers and H. T. C. Stoof, "Stability of Bose condensed atomic 7 Li," Phys. Rev. A 54, 5055 (1996).
    [CrossRef] [PubMed]
  18. Yu. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, "Effect of Bose condensation on inelastic processes in gases," JETP Lett. 42, 210 (1985).
  19. H. T. C. Stoof, A. M. L. Janssen, J. M. V. A. Koelman, and B. J. Verhaar, "Decay of spin-polarized atomic hydrogen in the presence of a Bose condensate," Phys. Rev. A 39, 3157 (1989).
    [CrossRef] [PubMed]
  20. R. Feynman, Statistical Mechanics (W. A. Benjamin, Reading, MA, 1972).
  21. S. R. de Groot, G. J. Hooyman, and C. A. ten Seldam, "On the Bose-Einstein condensation," Proc.R. Soc. London,Ser. A203, 266 (1950).
    [CrossRef]
  22. V. Bagnato, D. E. Pritchard, and D. Kleppner, "Bose-Einstein condensation in an external potential," Phys. Rev. A 35, 4354 (1987).
    [CrossRef] [PubMed]
  23. D. A. W. Hutchinson, E. Zaremba, and A. Griffin, "Finite temperature excitations of a trapped Bose gas," Phys. Rev. Lett. 78, 1842 (1997).
    [CrossRef]
  24. M. J. Holland and J. Cooper, "Expansion of a Bose-Einstein condensate in a harmonic potential," Phys. Rev. A 53, R1954 (1996)
    [CrossRef] [PubMed]
  25. M. Edwards, R. J. Dodd, C. W. Clark, P. A. Ruprecht, and K. Burnett, "Properties of a Bose-Einstein condensate in an anisotropic harmonic potential," Phys. Rev. A 53, R1950 (1996).
    [CrossRef] [PubMed]
  26. M. Edwards P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, "Collective excitations of Bose-Einstein condensates," Phys. Rev. Lett. 77, 1671 (1996)
    [CrossRef] [PubMed]
  27. S. Stringari, "Collective excitations of a trapped Bose-condensed gas" Phys. Rev. Lett. 77, 2360 (1996)
    [CrossRef] [PubMed]
  28. Y. Castin and R. Dum, "Bose-Einstein condensates in time-dependent traps," Phys. Rev. Lett. 77, 5315 (1996).
    [CrossRef] [PubMed]
  29. M. J. Holland, D. S. Jin, M. L. Chiofalo, and J. Cooper, "Emergence of interaction effects in Bose-Einstein condensation," Phys. Rev. Lett. 78, 3801 (1997).
    [CrossRef]
  30. S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Condensate fraction and critical temperature of a trapped interacting Bose gas," Phys. Rev. A 54, R4633 (1996).
    [CrossRef] [PubMed]
  31. A. Minguzzi, S. Conti, and M. P. Tosi, "The internal energy and condensate fraction of a trapped interacting Bose gas," J. Phys.: Condens. Matter 9, L33 (1997).
    [CrossRef]
  32. R. J. Dodd, M. Edwards, C. W. Clark, and K. Burnett, "Collective excitations of Bose-Einstein condensed gases at finite temperatures," Phys. Rev. A (in press).

Other (32)

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, 198 (1995).
[CrossRef] [PubMed]

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, 3969 (1995).
[CrossRef] [PubMed]

C. C. Bradley, C. A. Sackett, and R. G. Hulet, "Bose-Einstein condensation in lithium: observation of limited condensate number," Phys. Rev. Lett. 78, 985 (1997).
[CrossRef]

M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, "An output coupler for Bose condensed atoms," Phys. Rev. Lett. 78, 582 (1997).
[CrossRef]

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, "Observation of interference between two Bose condensates," Science 275, 637 (1997).
[CrossRef] [PubMed]

H. M. Wiseman, "Defining the (atom) laser," Phys. Rev. A (1997 in press).
[CrossRef]

E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E. Wieman, "Coherence, correlation and collisions: what one learns from Bose-Einstein condensates from their decay," Phys. Rev. Lett. 79, 337 (1997).
[CrossRef]

W. Ketterle and H.-J. Miesner, "Coherence properties of Bose condensates and atom lasers," Phys. Rev. A 57, 3291 (1997).
[CrossRef]

R. J. Dodd, K. Burnett, M. Edwards, and C. W. Clark, "Two-gas description of dilute Bose-Einstein condensates at finite temperature," Phys. Rev. A (submitted).

O. Penrose and L. Onsager, "Bose-Einstein condensation and liquid helium," Phys. Rev. 104, 576 (1956).
[CrossRef]

D. F. Walls, "Evidence for the quantum nature of light," Nature 280, 451 (1979).
[CrossRef]

R. Glauber, "Optical coherence and photon statistics," in Quantum Optics and Electronics, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965).

A. L. Fetter, "Nonuniform states of an imperfect Bose gas," Ann. Phys. (NY) 70, 67 (1972).
[CrossRef]

E. M. Lifshitz and L. P. Pitaevski, Statistical Physics Part 2 (Butterworth Heinemann, Oxford, 1995).

V. N. Popov, Functional Integrals and Collective Modes (Cambridge University Press, New York, 1987), Chapter 6.

A. Griffin, "Conserving and gapless approximations for an inhomogeneous Bose gas at finite temperatures," Phys. Rev. B 53, 9341 (1996).
[CrossRef]

M. Houbiers and H. T. C. Stoof, "Stability of Bose condensed atomic 7 Li," Phys. Rev. A 54, 5055 (1996).
[CrossRef] [PubMed]

Yu. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, "Effect of Bose condensation on inelastic processes in gases," JETP Lett. 42, 210 (1985).

H. T. C. Stoof, A. M. L. Janssen, J. M. V. A. Koelman, and B. J. Verhaar, "Decay of spin-polarized atomic hydrogen in the presence of a Bose condensate," Phys. Rev. A 39, 3157 (1989).
[CrossRef] [PubMed]

R. Feynman, Statistical Mechanics (W. A. Benjamin, Reading, MA, 1972).

S. R. de Groot, G. J. Hooyman, and C. A. ten Seldam, "On the Bose-Einstein condensation," Proc.R. Soc. London,Ser. A203, 266 (1950).
[CrossRef]

V. Bagnato, D. E. Pritchard, and D. Kleppner, "Bose-Einstein condensation in an external potential," Phys. Rev. A 35, 4354 (1987).
[CrossRef] [PubMed]

D. A. W. Hutchinson, E. Zaremba, and A. Griffin, "Finite temperature excitations of a trapped Bose gas," Phys. Rev. Lett. 78, 1842 (1997).
[CrossRef]

M. J. Holland and J. Cooper, "Expansion of a Bose-Einstein condensate in a harmonic potential," Phys. Rev. A 53, R1954 (1996)
[CrossRef] [PubMed]

M. Edwards, R. J. Dodd, C. W. Clark, P. A. Ruprecht, and K. Burnett, "Properties of a Bose-Einstein condensate in an anisotropic harmonic potential," Phys. Rev. A 53, R1950 (1996).
[CrossRef] [PubMed]

M. Edwards P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, "Collective excitations of Bose-Einstein condensates," Phys. Rev. Lett. 77, 1671 (1996)
[CrossRef] [PubMed]

S. Stringari, "Collective excitations of a trapped Bose-condensed gas" Phys. Rev. Lett. 77, 2360 (1996)
[CrossRef] [PubMed]

Y. Castin and R. Dum, "Bose-Einstein condensates in time-dependent traps," Phys. Rev. Lett. 77, 5315 (1996).
[CrossRef] [PubMed]

M. J. Holland, D. S. Jin, M. L. Chiofalo, and J. Cooper, "Emergence of interaction effects in Bose-Einstein condensation," Phys. Rev. Lett. 78, 3801 (1997).
[CrossRef]

S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Condensate fraction and critical temperature of a trapped interacting Bose gas," Phys. Rev. A 54, R4633 (1996).
[CrossRef] [PubMed]

A. Minguzzi, S. Conti, and M. P. Tosi, "The internal energy and condensate fraction of a trapped interacting Bose gas," J. Phys.: Condens. Matter 9, L33 (1997).
[CrossRef]

R. J. Dodd, M. Edwards, C. W. Clark, and K. Burnett, "Collective excitations of Bose-Einstein condensed gases at finite temperatures," Phys. Rev. A (in press).

Supplementary Material (1)

» Media 1: MOV (152 KB)     

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)

Figure 1.
Figure 1.

A false-color plot of g (2)(r)vs. reduced temperature, T/T 0, and radial trap coordinate r, for N = 40000 87Rb atoms in a spherical trap with v = 200 Hz. The coherence length R(T) is given by Eq. 11. Blue corresponds to g (2) = 1 viz. coherence characteristic of a laser source; red to g (2) = 2 viz. coherence characteristic of a thermal source. It is apparent that R(T) defines the typical length scale over which laser-like coherence is maintained.

Figure 2.
Figure 2.

A plot of g (2)(r) versus r/d for 2000 87Rb atoms in the JILA TOP trap at 40 nK, with ωρ /(2π) = 74 Hz. The figure displays the variation in the plane z = 0, with r being the cylindrical radius.

Figure 3.
Figure 3.

Animation of surface plots of the scaled total densities, n(r)d 3 , versus ρ/d and z/d, for 2000 87Rb atoms in the JILA TOP trap with vρ = 74 Hz, at temperatures as labelled in each frame. The critical temperature T 0 for this system is ≈ 59 nK. The height of the surface is proportional to the density, the peak value displayed in these frames being ≈ 5 × 1013cm-3; the (dimensionless) z and ρ coordinates attain maximum values of 6 and 12, respectively; and the color shading represents the zero-separation, second-order coherence function of the system, g (2)(r,r). Blue indicates high coherence (g (2) ~ 1), while red indicates low coherence (g (2) ~ 2). [Media 1]

Equations (17)

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

g ( 2 ) ( r 1 , r 2 ) = ψ ̂ ( r 1 ) ψ ̂ ( r 2 ) ψ ̂ ( r 2 ) ψ ̂ ( r 1 ) ψ ̂ ( r 1 ) ψ ̂ ( r 1 ) ψ ̂ ( r 2 ) ψ ̂ ( r 2 ) ,
[ ψ ̂ ( r 1 ) , ψ ̂ ( r 2 ) ] = δ ( r 1 r 2 ) , [ ψ ̂ ( r 1 ) , ψ ̂ ( r 2 ) ] = [ ψ ̂ ( r 1 ) , ψ ̂ ( r 2 ) ] = 0 .
K H μ N = d r ψ ̂ ( r ) ( H 0 μ ) ψ ̂ ( r )
+ U 0 2 d r ψ ̂ ( r ) ψ ̂ ( r ) ψ ̂ ( r ) ψ ̂ ( r ) ,
ψ ̂ ( r ) = ψ ( r ) + ψ ͂ ( r ) = ψ ( r ) + j [ u j ( r ) α j + v j * ( r ) α j ] .
[ α j , α k ] = δ j k , [ α j , α k ] = [ α j , α k ] = 0 .
K ̂ = j E j α j α j .
O ̂ = Tr [ O ̂ e β K ̂ ] Tr [ e β K ̂ ] .
g ( 2 ) ( r , r ) = 1 + 1 n ( r ) { 2 ψ ( r ) 2 n ͂ ( r ) + n ͂ 2 ( r ) } .
g ( 2 ) ( r , r ) = 2 f 2 ( r ) ,
f ( r ) = [ 1 N 0 k = 1 e β k ( ε 0 μ ) [ 1 e 2 βkħω ] 3 / 2 e ( r d ) 2 { 1 tanh ( βkhω 2 ) } ] 1 ,
R ( T ) = d e βħω 2 ( ln [ N 0 ] 2 ) 1 / 2 ,
{ H 0 + U 0 [ N 0 ϕ ( r ) 2 + 2 n ͂ ( r ) ] } ϕ ( r ) = μ ϕ ( r ) ,
L u j ( r ) + N 0 U 0 ϕ ( r ) 2 v j ( r ) = E j u j ( r )
L v j ( r ) + N 0 U 0 ϕ ( r ) 2 u j ( r ) = E j v j ( r ) ,
n ͂ ( r ) = j { [ u j ( r ) 2 + v j ( r ) 2 ] N j + v j ( r ) 2 } ,
N = d r n ( r ) = N 0 + d r n ͂ ( r ) .

Metrics