Abstract

The nonlinear coupling term in the Gross-Pitiaevski equation which describes a Bose-Einstein condensate (BEC) can cause four-wave mixing (4WM) if three BEC wavepackets with momenta k 1, k 2, and k 3 interact. The interaction will produce a fourth wavepacket with momentum k 4 = k 1 + k 2 - k 3. We study this process using numerical models and suggest that experiments are feasible. Conservation of energy and momentum have different consequences for 4WM with massive particles than in the nonlinear optics case because of the different energy-momentum dispersion relations.

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References

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  1. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn and W. Ketterle, "Observation of interference between two Bose-Einstein condensates," Science 275, 637 (1997).
    [CrossRef] [PubMed]
  2. D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, "Measurements of relative phase in two-component Bose-Einstein condensates," Phys. Rev. Lett. 81, 1543 (1998).
    [CrossRef]
  3. H. Wallis, A. Rohrl, M. Naraschewski and A. Schenzle, "Phase-space dynamics of Bose condensates: Interference versus interaction," Phys. Rev. A55, 2109 (1997).
  4. A. Rohrl, M. Naraschewski, A. Schenzle and H. Wallis, "Transition from phase locking to the interference of independent Bose condensates: Theory versus experiment," Phys. Rev. Lett. 78, 4143 (1997).
    [CrossRef]
  5. J. Javanainen and M. Wilkens, "Phase and phase diffusion of a split Bose-Einstein condensate," Phys. Rev. Lett. 78, 4675 (1997).
    [CrossRef]
  6. R. Y. Chiao, E. Garmire and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  7. O. Svelto, "Self-focusing, self-trapping, and self-phase modulation of laser beams," Prog. Opt. 12, 3 (1973).
  8. R. W. Hellwarth, "Third-order optical susceptibilities of liquids and solids," Prog. Quant. Electr. 5, 1 (1977).
    [CrossRef]
  9. M.-O. Mewes, M R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee and W. Ketterle, "Bose-Einstein Condensation in a Tightly Confining dc Magnetic Trap," Phys. Rev. Lett. 77, 416 (1996).
    [CrossRef] [PubMed]
  10. E. Goldstein, K. Plattner and P. Meystre, "Atomic phase conjugation," Quantum Semiclass. Opt. 7, 743 (1995).
    [CrossRef]
  11. E. Goldstein, K. Pl"attner and P. Meystre, "Atomic phase conjugation from a Bose condensate," J. Res. Nat. Inst. Stand. Technol. 101, 583 (1996).
    [CrossRef]
  12. E. Goldstein and P. Meystre, "Phase conjugation of trapped Bose-Einstein condensates," LANL Preprint Archive Cond-mat/9806165 (1998).
  13. C. K. Law, H. Pu and N. P. Bigelow, "Quantum spins mixing of spinor Bose-Einstein condensates," LANL Preprint Archive Cond-mat/9807258 (1998).
  14. M. Kozuma, L. Deng, E. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston and W. D. Phillips, "Coherent splitting of Bose-Einstein condensed atoms with optically induced Bragg diffraction," Phys. Rev. Lett., in press (1998).
  15. E. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston and W. D. Phillips, "A well-collimated quasi-continuous atom laser," preprint (1998).
  16. P. D. Maker and R. W. Terhune, "Study of optical effects due to induced polarization third order in the electric field strength," Phys. Rev. A137, 801 (1965).
    [CrossRef]
  17. A. Yariv and D. M. Pepper, "Amplified reflection, phase-conjugate, and oscillation in degenerate four-wave mixing," Opt. Lett. 1, 16 (1977).
    [CrossRef] [PubMed]
  18. P. Nozieres and D. Pines, The Theory of Quantum Liquids, Vol. II, (Addison-Wesley, Redwood City, 1990).
  19. M. Trippenbach and Y. B. Band, "Dynamics of short pulse splitting in dispersive nonlinear media," Phys. Rev. A56, 4242 (1997)
  20. M. Trippenbach and Y. B. Band, "Effects of Self-steepening and self-frequency shifting on short- pulse splitting in dispersive nonlinear media," Phys. Rev. A57, 4791 (1998).
  21. Y. Japha, S. Choi, K. Burnett and Y. B. Band, "Coherent output, stimulated quantum evaporation and pair breaking in a trapped atomic Bose gas," Phys. Rev. Lett. (in press).
  22. J. Stenger, S. Inouye, D. M. Stamper-Kurn, H. J. Miesner, A. P. Chikkatur and W. Ketterle, "Spin domains in ground-state Bose{Einstein condensates," Nature 396, 345 (1998).
    [CrossRef]

Other (22)

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

D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, "Measurements of relative phase in two-component Bose-Einstein condensates," Phys. Rev. Lett. 81, 1543 (1998).
[CrossRef]

H. Wallis, A. Rohrl, M. Naraschewski and A. Schenzle, "Phase-space dynamics of Bose condensates: Interference versus interaction," Phys. Rev. A55, 2109 (1997).

A. Rohrl, M. Naraschewski, A. Schenzle and H. Wallis, "Transition from phase locking to the interference of independent Bose condensates: Theory versus experiment," Phys. Rev. Lett. 78, 4143 (1997).
[CrossRef]

J. Javanainen and M. Wilkens, "Phase and phase diffusion of a split Bose-Einstein condensate," Phys. Rev. Lett. 78, 4675 (1997).
[CrossRef]

R. Y. Chiao, E. Garmire and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

O. Svelto, "Self-focusing, self-trapping, and self-phase modulation of laser beams," Prog. Opt. 12, 3 (1973).

R. W. Hellwarth, "Third-order optical susceptibilities of liquids and solids," Prog. Quant. Electr. 5, 1 (1977).
[CrossRef]

M.-O. Mewes, M R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee and W. Ketterle, "Bose-Einstein Condensation in a Tightly Confining dc Magnetic Trap," Phys. Rev. Lett. 77, 416 (1996).
[CrossRef] [PubMed]

E. Goldstein, K. Plattner and P. Meystre, "Atomic phase conjugation," Quantum Semiclass. Opt. 7, 743 (1995).
[CrossRef]

E. Goldstein, K. Pl"attner and P. Meystre, "Atomic phase conjugation from a Bose condensate," J. Res. Nat. Inst. Stand. Technol. 101, 583 (1996).
[CrossRef]

E. Goldstein and P. Meystre, "Phase conjugation of trapped Bose-Einstein condensates," LANL Preprint Archive Cond-mat/9806165 (1998).

C. K. Law, H. Pu and N. P. Bigelow, "Quantum spins mixing of spinor Bose-Einstein condensates," LANL Preprint Archive Cond-mat/9807258 (1998).

M. Kozuma, L. Deng, E. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston and W. D. Phillips, "Coherent splitting of Bose-Einstein condensed atoms with optically induced Bragg diffraction," Phys. Rev. Lett., in press (1998).

E. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston and W. D. Phillips, "A well-collimated quasi-continuous atom laser," preprint (1998).

P. D. Maker and R. W. Terhune, "Study of optical effects due to induced polarization third order in the electric field strength," Phys. Rev. A137, 801 (1965).
[CrossRef]

A. Yariv and D. M. Pepper, "Amplified reflection, phase-conjugate, and oscillation in degenerate four-wave mixing," Opt. Lett. 1, 16 (1977).
[CrossRef] [PubMed]

P. Nozieres and D. Pines, The Theory of Quantum Liquids, Vol. II, (Addison-Wesley, Redwood City, 1990).

M. Trippenbach and Y. B. Band, "Dynamics of short pulse splitting in dispersive nonlinear media," Phys. Rev. A56, 4242 (1997)

M. Trippenbach and Y. B. Band, "Effects of Self-steepening and self-frequency shifting on short- pulse splitting in dispersive nonlinear media," Phys. Rev. A57, 4791 (1998).

Y. Japha, S. Choi, K. Burnett and Y. B. Band, "Coherent output, stimulated quantum evaporation and pair breaking in a trapped atomic Bose gas," Phys. Rev. Lett. (in press).

J. Stenger, S. Inouye, D. M. Stamper-Kurn, H. J. Miesner, A. P. Chikkatur and W. Ketterle, "Spin domains in ground-state Bose{Einstein condensates," Nature 396, 345 (1998).
[CrossRef]

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Figures (3)

Figure 1.
Figure 1.

Schematic view of the initial positions and momenta of the three BECs wavepackets. The inset shows the momentum of the additional wavepacket formed by the 4WM process.

Figure 2.
Figure 2.

Probability distribution |Φ(x,y,t)|2 versus x and y in length units of x 0 = 10 μm. Panels (a), (b), and (c) are for respective times t = -1, 0, and 1 t 0 before, during, and after the collision, where t 0 = 36.2 ms. The initial wavepacket started at t = -2t 0, and expanded about 8-fold by the time of panel (a). Panels (d), (e), and (f) show the corresponding momentum distribution |Φ(kx, ky, t)|2 versus kx and ky in x01 units.

Figure 3.
Figure 3.

Blowup of the probability distribution |Ψ(x, y, t)|2 of panel (c) in Fig. 2.

Equations (4)

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Ψ t = ( T x + V x t + U 0 Ψ 2 ) Ψ ,
Ψ ( x , t = 0 ) = N i = 1 3 Ψ ( x x i ) exp ( i k i · x )
Ψ t = i w 0 2 t DF ( 2 Ψ x 2 + 2 Ψ y 2 + 2 Ψ z 2 ) i 1 t NL Ψ 2 Ψ m 2 Ψ .
dz Ψ ( x , y , z , t ) 2 ,

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