Abstract

We report on directly measuring the atom number in a Bose-Einstein condensate by the method of optical pumping. Only the branching ratio of the spontaneous decay in the system and the absorption energy of a probe laser beam are required to determine the atom number. The measured absorption energy is not affected by the measurement condition such as the intensity, detuning, and polarization of the probe beam, the magnetic field, etc. We have shown that atom numbers as low as a few thousands can be measured. The atom number is an important parameter in the studies of Bose condensates and its accuracy is greatly improved by this sensitive and robust method.

© 2007 Optical Society of America

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References

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  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, 198-201 (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, 3969-3973 (1995).
    [CrossRef] [PubMed]
  3. 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, 637-641 (1997).
    [CrossRef] [PubMed]
  4. M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature (London) 415, 39-44 (2002).
    [CrossRef]
  5. S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, and R. Grimm, "Bose-Einstein Condensation of Molecules," Science 302, 2101-2103 (2003).
    [CrossRef] [PubMed]
  6. M. Greiner, C. A. Regal, and D. S. Jin, "Emergence of a molecular Bose-Einstein condensate from a Fermi gas," Nature (London) 426, 537-540 (2003).
    [CrossRef]
  7. M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, "Observation of Bose-Einstein Condensation of Molecules," Phys. Rev. Lett. 91, 250401 (2003).
    [CrossRef]
  8. L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
    [CrossRef]
  9. C. Liu, Z. Dutton, C. H. Behroozi, L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
    [CrossRef]
  10. N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature (London) 445, 623-626 (2007).
    [CrossRef]
  11. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, "Vortices in a Bose- Einstein Condensate," Phys. Rev. Lett. 83, 2498-2501 (1999).
    [CrossRef]
  12. A. J. Leggett, "Bose-Einstein condensation in the alkali gases: Some fundamental concepts," Rev. Mod. Phys. 73, 307-356 (2001).
    [CrossRef]
  13. C. F. Li and G. C. Guo, "Quantum nondemolition measurement of the atom number of a Bose-Einstein condensate," Phys. Lett. A 248, 117-123 (1998).
    [CrossRef]
  14. In a two-level system, the absorption cross section of a laser field is equal to the imaginary part of [3λ 2/(2π)]ρegΓ′/Ω, where ρeg is the amplitude of the density matrix element between the excited state |e> and the ground state |g>, Γ′ is the spontaneous decay rate from |e> to |g>, and Γ is the Rabi frequency of the laser field. For the solution of ρeg, see M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997), Sec. 5.3.2 and Eq. (17.1.20).
  15. Y. C. Chen, Y. A. Liao, L. Hsu, and I. A. Yu, "Simple technique for directly and accurately measuring the number of atoms in a magneto-optical trap," Phys. Rev. A  64, 031401(R) (2001).
    [CrossRef]
  16. J. Dalibard and C. Cohen-Tannoudji, "Laser cooling below the Doppler limit by polarization gradients: simple theoretical models," J. Opt. Soc. Am. B 6, 2023-2045 (1989).
    [CrossRef]
  17. W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, "Stable, Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms," Phys. Rev. Lett. 74, 3352-3355 (1995).
    [CrossRef] [PubMed]
  18. H. F. Hess, "Evaporative cooling of magnetically trapped and compressed spin-polarized hydrogen," Phys. Rev. B 34, 3476-3479 (1986).
    [CrossRef]
  19. K. B. Davis, M. O. Mewes, M. A. Joffe, M. R. Andrews, and W. Ketterle, "Evaporative Cooling of Sodium Atoms," Phys. Rev. Lett. 74, 5202-5205 (1995).
    [CrossRef] [PubMed]

2007

N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature (London) 445, 623-626 (2007).
[CrossRef]

2003

S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, and R. Grimm, "Bose-Einstein Condensation of Molecules," Science 302, 2101-2103 (2003).
[CrossRef] [PubMed]

M. Greiner, C. A. Regal, and D. S. Jin, "Emergence of a molecular Bose-Einstein condensate from a Fermi gas," Nature (London) 426, 537-540 (2003).
[CrossRef]

M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, "Observation of Bose-Einstein Condensation of Molecules," Phys. Rev. Lett. 91, 250401 (2003).
[CrossRef]

2002

M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature (London) 415, 39-44 (2002).
[CrossRef]

2001

A. J. Leggett, "Bose-Einstein condensation in the alkali gases: Some fundamental concepts," Rev. Mod. Phys. 73, 307-356 (2001).
[CrossRef]

C. Liu, Z. Dutton, C. H. Behroozi, L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

1999

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, "Vortices in a Bose- Einstein Condensate," Phys. Rev. Lett. 83, 2498-2501 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

1998

C. F. Li and G. C. Guo, "Quantum nondemolition measurement of the atom number of a Bose-Einstein condensate," Phys. Lett. A 248, 117-123 (1998).
[CrossRef]

1997

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, 637-641 (1997).
[CrossRef] [PubMed]

1995

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, 198-201 (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, 3969-3973 (1995).
[CrossRef] [PubMed]

K. B. Davis, M. O. Mewes, M. A. Joffe, M. R. Andrews, and W. Ketterle, "Evaporative Cooling of Sodium Atoms," Phys. Rev. Lett. 74, 5202-5205 (1995).
[CrossRef] [PubMed]

W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, "Stable, Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms," Phys. Rev. Lett. 74, 3352-3355 (1995).
[CrossRef] [PubMed]

1989

1986

H. F. Hess, "Evaporative cooling of magnetically trapped and compressed spin-polarized hydrogen," Phys. Rev. B 34, 3476-3479 (1986).
[CrossRef]

J. Opt. Soc. Am. B

Nature (London)

M. Greiner, C. A. Regal, and D. S. Jin, "Emergence of a molecular Bose-Einstein condensate from a Fermi gas," Nature (London) 426, 537-540 (2003).
[CrossRef]

M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature (London) 415, 39-44 (2002).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, "Light speed reduction to 17 meters per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

C. Liu, Z. Dutton, C. H. Behroozi, L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature (London) 445, 623-626 (2007).
[CrossRef]

Phys. Lett. A

C. F. Li and G. C. Guo, "Quantum nondemolition measurement of the atom number of a Bose-Einstein condensate," Phys. Lett. A 248, 117-123 (1998).
[CrossRef]

Phys. Rev. B

H. F. Hess, "Evaporative cooling of magnetically trapped and compressed spin-polarized hydrogen," Phys. Rev. B 34, 3476-3479 (1986).
[CrossRef]

Phys. Rev. Lett.

K. B. Davis, M. O. Mewes, M. A. Joffe, M. R. Andrews, and W. Ketterle, "Evaporative Cooling of Sodium Atoms," Phys. Rev. Lett. 74, 5202-5205 (1995).
[CrossRef] [PubMed]

W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, "Stable, Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms," Phys. Rev. Lett. 74, 3352-3355 (1995).
[CrossRef] [PubMed]

M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, "Observation of Bose-Einstein Condensation of Molecules," Phys. Rev. Lett. 91, 250401 (2003).
[CrossRef]

M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, E. A. Cornell, "Vortices in a Bose- Einstein Condensate," Phys. Rev. Lett. 83, 2498-2501 (1999).
[CrossRef]

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, 3969-3973 (1995).
[CrossRef] [PubMed]

Rev. Mod. Phys.

A. J. Leggett, "Bose-Einstein condensation in the alkali gases: Some fundamental concepts," Rev. Mod. Phys. 73, 307-356 (2001).
[CrossRef]

Science

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, 637-641 (1997).
[CrossRef] [PubMed]

S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, and R. Grimm, "Bose-Einstein Condensation of Molecules," Science 302, 2101-2103 (2003).
[CrossRef] [PubMed]

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, 198-201 (1995).
[CrossRef] [PubMed]

Other

In a two-level system, the absorption cross section of a laser field is equal to the imaginary part of [3λ 2/(2π)]ρegΓ′/Ω, where ρeg is the amplitude of the density matrix element between the excited state |e> and the ground state |g>, Γ′ is the spontaneous decay rate from |e> to |g>, and Γ is the Rabi frequency of the laser field. For the solution of ρeg, see M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997), Sec. 5.3.2 and Eq. (17.1.20).

Y. C. Chen, Y. A. Liao, L. Hsu, and I. A. Yu, "Simple technique for directly and accurately measuring the number of atoms in a magneto-optical trap," Phys. Rev. A  64, 031401(R) (2001).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

The three images in the right column show the spatial distribution of the optical density for atom clouds of different temperatures. From top to bottom the images show thermal atoms above the transition temperature Tc , thermal atoms mixed with a condensate near Tc (≈250 nK) and a pure condensate below Tc , respectively, at a time-of-flight of 21 ms. The size of each image is 850×850 µm2. In the left column, each plot shows the optical density versus the horizontal axis intersecting the center of the image of the same row. The solid lines are the best Gaussian fits of the thermal (blue) and condensed (red) atoms.

Fig. 2.
Fig. 2.

(a) Relevant energy levels in the OP measurement. The spontaneous decay rates from the excited state to the two ground states are both equal to 1/2. (b) The timing diagram of the measurement sequence.

Fig. 3.
Fig. 3.

Experimental setup of the detection system. The image which is taken by the CMOS camera shows a condensate (the dark shadow in the bright spot) in the center of the pinhole (the bright spot in the image).

Fig. 4.
Fig. 4.

(a) Power of the probe beam transmitted through the pinhole with (red line) and without (black line) the atoms in the optical path versus time. In the inset, blue line is the difference of the two signals and black dashed line is the best fit of the exponential decay function of y(x)=y0 exp(-x/τ). The data have been averaged over nine measurements for the same condensate. The product of y 0 and τ of the best fit indicates that the number of atoms is 4.9×104. We note that the spike on the rising edge of the pulse is the transient behavior due to the APD bandwidth. Its size is about 9% of the pulse amplitude. As the initial data points in the inset do not deviate from the exponential decay curve very much, the effect of the spike is negligible. (b) Transmitted power of the probe beam versus time for an atom number of 6×103. All the legends are the same as those in (a).

Equations (4)

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α ( x , y ) = n ( x , y , z ) σ d z .
α ( x , y ) d x d y = N σ
σ = 3 λ 2 2 π C ij 2 1 + I ( x , y , z ) I 0 , i j + 4 Δ 2 Γ 2 ,
N p = n = 1 n ( 1 p ) n 1 p = 1 p .

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