## Characterizing the coherence of Bose-Einstein condensates and atom lasers

Optics Express, Vol. 1, Issue 10, pp. 284-292 (1997)

http://dx.doi.org/10.1364/OE.1.000284

Acrobat PDF (537 KB)

### 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

## 1. Introduction

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]

^{87}Rb gas in the JILA TOP trap. [1

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. Coherence of matter waves

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]

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

*vs*. those of matter waves from a thermal source.

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

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

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

*ψ̂*(

**r**

_{i}) is the Bose field operator which annihilates an atom at position

**r**

_{i}. The Bose field operators obey the usual commutation relations

17. M. Houbiers and H. T. C. Stoof, “Stability of Bose condensed atomic ^{7}Li,” Phys. Rev. A **54**, 5055 (1996). [CrossRef] [PubMed]

*N*, the total number of trapped atoms;

*T*, the absolute temperature; and either

*N*

_{0}, the number of condensate atoms, or

*μ*, the chemical potential. The system Hamiltonian has the form

*V*

_{trap}(

**r**) =

*M*(

*ρ*

^{2}+

*z*

^{2}) /2, with

*ω*

_{ρ}and

*ω*

_{z}= √8

*ω*

_{ρ}being the radial and axial trap frequencies; and

*U*

_{0}= 4

*πħ*

^{2}

*a*/

*M*is a measure of the interaction strength between atoms, with

*a*being the scattering length for zero-energy binary atomic collisions.

*c*-number condensate wave function,

*ψ*(

**r**), and a fluctuation term,

*ψ͂*(

**r**) which can be decomposed into a sum of quasi-particle modes

*α*

_{j}are operators that, respectively, create and destroy the

*j*

^{th}mode. These operators satisfy the following commutation relations

*u*

_{j}(

**r**) and

*v*

_{j}(

**r**) satisfy Eqs. (13), the Popov Hamil-tonian has the diagonal form

*Ô*is given by [13

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

*ψ͂*〉 = 0 and products of three

*ψ͂*operators will vanish as well as they consist of sums containing products either one or three quasi-particle creation and annihilation operators.

**r**

_{1}=

**r**

_{2}≡

**r**) and take the ensemble average. Under the Popov approximation anomalous averages such as 〈

*ψ͂*ψ͂〉 are assumed negligible, after some algebra we find that

*n͂*(

**r**) = 〈

*ψ͂*

^{†}(

**r**)

*ψ͂*(

**r**)〉 is the thermal-atom density and

*n*(

**r**) = |

*ψ*(

**r**)|

^{2}+

*n͂*(

**r**) is the total density. The coherence function can then be written in the simple form

*f*(

**r**) = |

*ψ*(

**r**)|

^{2}/

*n*(

**r**), the spatially resolved condensate fraction, is the ratio of the condensate density to the total density at position

**r**. This equation is relevant to recent experiments such as the determination of decay rates [7

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]

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]

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

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]

*f*, the condensate fraction, is independent of

**r**; in that case,

*g*

^{(2)}is simply a function of temperature

*T*. However, as we have suggested above and shall show below, in current experiments

*f*(

**r**) depends strongly upon

**r**. Note that

*g*

^{(2)}(

**r**,

**r**) = 2 for a thermal gas without condensation and

*g*

^{(2)}(

**r**,

**r**) = 1 for a pure BEC.

## 3. Second-order coherence of the ideal trapped Bose gas

*M*confined in a spherical harmonic potential with angular frequency

*ω*; the next section treats the effects of interactions and trap anisotropy. The spatially-resolved condensate fraction

*f*(

**r**) ≡

*f*(

**r**) for the noninteracting trapped gas at temperature

*T*can be computed directly from known results for the harmonic oscillator system: [20]

*N*

_{0}is the number of condensate atoms,

*β*= 1/

*k*

_{B}

*T*with

*k*

_{B}being the Boltzmann constant,

*μ*is the chemical potential, and

*βħω*≫ 1, we find that

*f*(

**r**) reduces to a form in which we can identify a characteristic length,

*R*(

*T*), at which

*f*(

*R*) = 1/2:

*g*

^{(2)}(

*r*) =

*g*

^{(2)}(

**r**,

**r**)

*vs*. reduced temperature,

*T*/

*T*

_{0}, for

*N*= 40, 000

^{87}Rb atoms in a spherical trap with

*v*=

*ω*/(2

*π*) = 200 Hz. Here

*T*

_{0}is the critical temperature for an ideal trapped gas in the semiclassical limit;[21

21. S. R. de Groot, G. J. Hooyman, and C. A. ten Seldam, “On the Bose-Einstein condensation,” Proc. R. Soc. London, Ser. A **203**, 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]

*N*systems are somewhat lower than

*T*

_{0}. The sharp blue → red transition shows that

*R*(

*T*) defines an appropriate length scale over a large range of temperatures, even though it was derived only in the low temperature limit,

*T*≪

*ħω*/

*k*

_{B}, with

*N*

_{0}≫ 1.

## 4. Interacting trapped gas

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

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]

^{87}Rb atoms in the JILA TOP trap. Following the approach of Sec. 2, we decompose the Bose field operator into a

*c*-number condensate wave function plus an operator describing the non-condensate part:

*ψ̂*(

**r**) =

*ϕ*(

**r**) +

*ψ͂*(

**r**) and insert into Eq. (3). Linearization of this equation in the operators

*ψ͂*(

**r**), leads [15, 16

**53**, 9341 (1996). [CrossRef]

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]

*ϕ*(

**r**)

*u*

_{j}(

**r**) and

*v*

_{j}(

**r**),

*L*≡

*H*

_{0}+ 2

*U*

_{0}

*n*(

**r**) -

*μ*and the

*E*

_{j}, the quasi-particle energies, are determined as eigenvalues. The quasi-particle amplitudes generate

*ψ͂*(

**r**) via

*ψ*(

**r**) = Σ

_{j}(

*u*

_{j}(

*r*)

*α*

_{j}+

**r**)

*α*

_{j},

*n͂*(

**r**) can be written in terms of the quasi-particle amplitudes as

*N*

_{j}= (

*e*

^{βEj}- 1)

^{-1}. The total number of trapped atoms,

*N*, is given by

*et al*. [23

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]

*v*

_{j}(

**r**)). Numerical solution of these equations proceeds by choice of state variables {

*T*,

*μ*,

*N*}: for fixed

*T*and

*μ*,

*N*is determined by solving the HFB-Popov equations, iteratively, to self-consistency.

*g*

^{(2)}(

**r**,

**r**) that emerges from such calculations for a sample of 2000

^{87}Rb atoms in the JILA TOP trap with

*ω*

_{ρ}/(2

*π*) = 74 Hz at

*T*= 40 nK, a temperature at which the condensate fraction is approximately 50%. The solid line shows the value of

*g*

^{(2)}for the interacting sample, calculated by the HFB-Popov theory, and the dashed line shows the same calculation for a noninteracting sample. As can be seen, the repulsive atomic interactions extend the coherence length of the sample. This is due to two mechanisms: repulsive interactions between atoms of the condensate cause it to swell; and condensate atoms repel thermal atoms, leading to a more sharply defined variation of

*f*(

**r**). These mechanisms are clearly manifested in the animated sequence of figures that follows below. Near the trap center, however, the ideal gas has a coherence closer to unity, since the peak density of a ideal gas condensate is substantially higher than that of an interacting one.

*total*trapped-atom density in a plane that contains the trap axis, while the false-color shading exhibits the value of

*g*

^{(2)}(

**r**,

**r**). The blue-colored areas indicate laser-like coherence (

*g*

^{(2)}= 1), while red-colored areas depict thermal-like coherence (

*g*

^{(2)}= 2). Each frame of the animated sequence exhibits the density and coherence of the trapped atoms at a lower temperature than the previous one. The effect of the thermal-atom density in this sequence of frames is quite subtle and can be discerned by watching the

*ρ*and

*z*axes as the animation progresses through the frames. One will see the

*ρ*axis (white dotted line) being uncovered first (since the radial direction is more weakly confined than the axial direction in the JILA TOP trap) as

*T*decreases followed by the uncovering of the

*z*axis.

*T*decreases. Furthermore, it is clear from this sequence that the condensate density (roughly the blue regions) far exceeds the thermal-atom density. The repulsion mechanisms discussed above are clearly in play.

## 5. Implications for experiment

*T*

_{0}, the second-order coherence function

*g*

^{(2)}(

**r**

_{1},

**r**

_{2}) does not depend on |

**r**

_{1}-

**r**

_{2}| alone, as would be the case for a homogenous system. Instead, it exhibits significant dependence upon the trap spatial coordinates, and it reflects the relative purity of the condensate. For cases of repulsive interaction, regions of high relative purity are more extensive than those encountered in the ideal Bose gas.

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

*U*of a Bose gas satisfies

*U*∝ ∫

*d*

**r**

*g*

^{(2)}(

**r**,

**r**)

*n*

^{2}(

**r**). Experimental measurement of

*U*can be made by observing the expansion of the gas upon dropping the trap. [29

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]

*g*

^{(2)}(

**r**,

**r**) is independent of

**r**,

*i*.

*e*.

*g*

^{(2)}(

**r**,

**r**) =

*g*

^{(2)}(0), then

*g*

^{(2)}(0) can be obtained directly from experimental measurements of

*U*and

*n*(

**r**). A similar idea is employed by Burt

*et al*., [7

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]

*g*

^{(3)}(0) from a three-body decay rate. However, according to the picture we have developed in this paper, the release-energy measurements actually determine a spatially-averaged value of

*g*

^{(2)}(

**r**,

**r**). Thus, the analysis of Ref. [8

**57**, 3291 (1997). [CrossRef]

*g*

^{(2)}(0) = [∫

*d*

**r**

*g*

^{(2)}(

**r**,

**r**)

*n*

^{2}(

**r**) / [∫

*d*

**r**

*n*

^{2}(

**r**)], rather than a value of

*g*

^{(2)}that characterizes the sample as a whole. For very cold samples that are nearly pure condensates, the density-weighted variance in the distribution of

*g*

^{(2)}(

**r**,

**r**) will be small, but quantitative investigations of the coherence properties of partially-condensed gases will have to account for effects of spatial dependence.

*g*

^{(2)}, so there is a higher rate of collisional loss per unit density in the thermal cloud than in the condensate.

## Acknowledgements

## Footnotes

† | Also at the Physics Laboratory, National Institute of Standards and
Technology, Technology Administration, U. S. Department of Commerce,
Gaithersburg, MD 20899. |

## References and links

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 |

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. |

3. | C. C. Bradley, C. A. Sackett, and R. G. Hulet, “Bose-Einstein condensation in lithium: observation of limited condensate number,” Phys. Rev. Lett. |

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. |

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 |

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. |

8. | W. Ketterle and H.-J. Miesner, “Coherence properties of Bose condensates and atom lasers,” Phys. Rev. A |

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. |

11. | D. F. Walls, “Evidence for the quantum nature of light,” Nature |

12. | R. Glauber, “Optical coherence and photon statistics,” in |

13. | A. L. Fetter, “Nonuniform states of an imperfect Bose gas,” Ann. Phys (NY) |

14. | E. M. Lifshitz and L. P. Pitaevski, |

15. | V. N. Popov, |

16. | A. Griffin, “Conserving and gapless approximations for an inhomogeneous Bose gas at finite temperatures,” Phys. Rev. B |

17. | M. Houbiers and H. T. C. Stoof, “Stability of Bose condensed atomic |

18. | Yu. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, “Effect of Bose condensation on inelastic processes in gases,” JETP Lett. |

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 |

20. | R. Feynman, |

21. | S. R. de Groot, G. J. Hooyman, and C. A. ten Seldam, “On the Bose-Einstein condensation,” Proc. R. Soc. London, Ser. A |

22. | V. Bagnato, D. E. Pritchard, and D. Kleppner, “Bose-Einstein condensation in an external potential,” Phys. Rev. A |

23. | D. A. W. Hutchinson, E. Zaremba, and A. Griffin, “Finite temperature excitations of a trapped Bose gas,” Phys. Rev. Lett. |

24. | M. J. Holland and J. Cooper, “Expansion of a Bose-Einstein condensate in a harmonic potential,” Phys. Rev. A |

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 |

26. | M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, “Collective excitations of Bose-Einstein condensates,” Phys. Rev. Lett. |

27. | S. Stringari, “Collective excitations of a trapped Bose-condensed gas” Phys. Rev. Lett. |

28. | Y. Castin and R. Dum, “Bose-Einstein condensates in time-dependent traps,” Phys. Rev. Lett. |

29. | M. J. Holland, D. S. Jin, M. L. Chiofalo, and J. Cooper, “Emergence of interaction effects in Bose-Einstein condensation,” Phys. Rev. Lett. |

30. | S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Condensate fraction and critical temperature of a trapped interacting Bose gas,” Phys. Rev. A |

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 |

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). |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(030.1640) Coherence and statistical optics : Coherence

(140.3320) Lasers and laser optics : Laser cooling

**ToC Category:**

Focus Issue: Fluctuations and oscillations of Bose-Einstein

**History**

Original Manuscript: September 10, 1997

Published: November 10, 1997

**Citation**

Robert Dodd, Charles Clark, Mark Edwards, and Keith Burnett, "Characterizing the coherence of Bose-Einstein
condensates and atom lasers," Opt. Express **1**, 284-292 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-10-284

Sort: Journal | Reset

### References

- 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).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.