## Four wave mixing in the scattering of Bose-Einstein condensates

Optics Express, Vol. 3, Issue 13, pp. 530-537 (1998)

http://dx.doi.org/10.1364/OE.3.000530

Acrobat PDF (872 KB)

### 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.

© Optical Society of America

## 1. Introduction

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]

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]

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]

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

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]

*et al*. [10

10. E. Goldstein, K. Plättner, and P. Meystre, “Atomic phase conjugation,” Quantum Semiclass. Opt. **7**, 743 (1995). [CrossRef]

11. E. Goldstein, K. Plättner, and P. Meystre, “Atomic phase conjugation from a Bose condensate,” J. Res. Nat. Inst. Stand. Technol. **101**, 583 (1996). [CrossRef]

**k**

_{1},

**k**

_{2}, and

**k**

_{3}collide and interact. Nonlinear 4WM can generate a new BEC wavepacket with a new momentum

**k**

_{1}+

**k**

_{2}-

**k**

_{3}. Our assumptions on geometry and number of atoms in the wavepackets are less restrictive than those of Goldstein et al. We suggest that experiments with such wavepackets should be feasible, for example, using Raman output coupling techniques which have been demonstrated experimentally by the NIST group [14, 15]

## 2. Theory of four-wave mixing

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

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]

*k*, whereas it is linear in

*k*for the case of light. Moreover, in dispersive optical media, the momentum of light waves is proportional to the product of the frequency of the light and the refractive index, and the refractive index depends upon frequency (and the propagation direction in non-isotropic media -hence conservation of energy does not in general guarantee conservation of momentum in optical 4WM). This complication involving the properties of the medium does not arise in the BEC case. For 4WM in BEC collisions, when the magnitude of the momenta of each of the wavepackets are identical, i.e., |

**k**

_{i}| = |

**k**

_{j}| for

*i*≠

*j*, conservation of momentum and energy. However, in general, when |

**k**

_{i}| ≠ |

**k**

_{j}| for

*i*≠

*j*, conservation of momentum does not imply conservation of energy for 4WM in BEC collisions. Clearly, creation of new BEC wavepackets in 4WM is limited to cases when momentum and energy conservation are simultaneously satisfied.

*F, M*

_{F}state, and any trapping potentials are turned off before propagation begins. The three condensate wavepackets are given initial momenta and positions as in Fig. 1 so that they collide at a given point. We have carried out calculations in 1D, 2D, and 3D.

*V*(

**x**,

*t*) is the potential imposed on the atoms and

**-wave scattering length,**

*s**a*

_{0}, atomic mass,

*m*, and the total number of atoms in all the wavepackets,

*N*

_{T}. The initial wavefunction is comprised of three BEC wavepackets,

*Ψ*(

**x**-

**x**

_{i}) is the solution to the GP equation with a locally harmonic potential centered around

**x**=

**x**

_{i},

*i*= 1,2,3; the normalization constant N chosen so that the norm of Ψ is unity. We assume the three inital positions

**x**

_{i}are spatially separated so the initial wavepackets are non-overlapping (one could also consider the “half-collision” case where the three

**x**

_{i}are the same and the wavepackets are generated

*in situ*from the same initial condensate). Although the initial wavepackets can have arbitrary phases multiplying the amplitudes

*Ψ*(

**x**-

**x**

_{i}) [3, 5

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

*i*(

**k**

_{i}+

**k**

_{j}-

**k**

_{l}) ·

**x**] where

*i, j*and

*l*can be 1, 2 or 3 respectively. These terms can generate a wavepacket with a central momentum that is not in the initial wavefunction Ψ(

**x**,

*t*= 0). For example, if

**k**

_{2}= -

**k**

_{1}(see Fig. (2)), then it is possible to produce a wavepacket with central momentum

**k**

_{4}=

**k**

_{1}-

**k**

_{2}+

**k**

_{3}= 2

**k**

_{1}+

**k**

_{3}.

*dE*(

*t*)/

*dt*= 0, where

*d*

**P**(

*t*)/

*dt*= 0, where

**P**(

*t*) = -

*iħ*⟨Ψ(

*t*)|∇|Φ(

*t*)⟩ is the momentum per particle.

*t*

_{DF}

*,t*

_{NL}in the following manner [19, 20]:

*t*

_{DF}= 2

*ħ*, and the nonlinear interaction time

*t*

_{NL}= (

*U*

_{0}|Ψ

_{m}|

^{2}/

*ħ*)

^{-l}, where

**x**)|

^{2}, and

*w*

_{0}stands for initial halfwidth of the colliding wavepackets. The smaller the characteristic time, the more important the corresponding term in the GP equation. We also define the collision duration time

*t*

_{col}= (2

*w*

_{0})/

*v*, where

*v*=

*k*

_{1}/

*m*is the initial group velocity of a wavepacket. The ratio

*t*

_{col}/

*t*

_{NL}gives an indication of the strength of the nonlinearity during the collision. The larger the ratio of

*t*

_{col}/

*t*

_{NL}, the stronger the effects of the nonlinearity during the overlap of the wavepackets. These characteristic times stand in the ratios

*λ*is the De Broglie wavelength associated with the wavepacket velocity

*v*. Experimental condensates with

*t*

_{col}/

*t*

_{NL}≫ 1 can be readily achieved. Thus, the nonlinear term will have time to act while the BEC wavepackets remain physically overlapped during a collision.

## 3. Numerical simulations

*x*

_{0}and time

*t*

_{0}so that (

*t*

_{0}/

*t*

_{DF})(

*w*

_{0}/

*x*

_{0})

^{2}= 1/2; once

*x*

_{0}is chosen,

*t*

_{0}is given by

*t*

_{0}=

*ħ*. Here we choose the unit of length

*x*

_{0}to be

*x*

_{0}= 10

*μ*m, so that

*t*

_{0}= 36.2 ms for

^{23}Na atoms. Consequently, the unit of energy

*E*

_{0}=

*m*(

*x*

_{0}/

*t*

_{0})

^{2}=

*ħ*/

*t*

_{0}is

*E*

_{0}= 2.91 × 10

^{-33}J =

*h*(4.39 Hz), and the unit of momentum

*p*

_{0}=

*m*(

*x*

_{0}/

*t*

_{0}) =

*ħ*/

*x*

_{0}is

*p*

_{0}= 1-05 × 10

^{-29}Kg m/s. For comparison purposes, the recoil energy and momentum for a 589 nm photon (the Na resonance transition) are 5690

*E*

_{0}and 107

*p*

_{0}respectively.

*E*=

*ħ*

^{2}

*k*

^{2}/(2

*m*), the constraints imposed by conservation of energy and momentum during the collision do not permit additional wavepackets to be created in 1D. Let us consider the following initial conditions:

*x*

_{1}= -

*x*

_{2}, and

*x*

_{3}= 0, and

*k*

_{2}= -

*k*

_{1}, and

*k*

_{3}= 0. Two wavepackets move symmetrically towards the central wavepacket whose center is at

*x*

_{3}= 0. Since the nonlinear term in the propagation equation is of third order in Ψ, and Ψ is a superposition of condensates with momenta

*k*

_{1}, -

*k*

_{1}and 0, the nonlinear term could become a source of wavepackets propagating with momentum 0, ±

*k*

_{1}, ±2

*k*

_{1}and ±3

*k*

_{1}; in addition the collision could transfer population between the condensate wavepackets. We carried out numerical experiments with different values of nonlinearity

*U*

_{0}, up to the value of the

*τ*

_{col}/

*τ*

_{NL}= 10. As expected, the only effect observed in 1D simulations was a slight delay of the maximum of the moving wavepacket peaks. No transfer of population, or additional peak creation was present, i.e., no wavepackets of momentum ±2

*k*

_{1}and ±3

*k*

_{1}were created. Even if we had taken

*k*

_{3}≠ 0, or |

*k*

_{1}| ≠ |

*k*

_{2}|, no new wavepacket would appear in a 1D calculation.

**x**

_{1}= -

**x**

_{2}= (20,0),

**k**

_{1}= -

**k**

_{2}= (10,0), and

**x**

_{3}= (20

*α*,-40

*α*),

**k**

_{3}= (-10

*α*, 20

*α*), with several different values of the parameter

*α*;

*α*= 0.5, 0.7, 1, 1.25 and 1.5. We take

*N*

_{T}= 1.4 × 10

^{6}

^{23}Na atoms equally partitioned between wavepackets with initial

*w*

_{0}= 10

*μ*m, for a typical tight trap with a mean trap frequency of 200 Hz. For this case

*t*

_{DF}= 72 ms,

*t*

_{col}= 7.2 ms, and

*t*

_{NL}= 0.031 ms. The wavepacket momenta

**k**

_{i}are significantly larger than the internal momentum. Fig. 2a shows the initial configuration of the colliding wavepackets. The initial conditions were selected such that the three wavepackets collide at

*t*= 0 at the origin of the reference frame. Fig. 2 shows results of our 2D calculations for the case of

*α*= 1.0. Fig. 2a–c shows the probability distribution |ΨΦ(

*x,y,t*)|

^{2}before, during, and after the collision, and Fig. 2 d–f shows its Fourier transform, the momentum distribution |ΨΦ(

*k*

_{x}

*, k*

_{y}

*,t*)|

^{2}. Fig. 2b illustrates the wavepacket interference during the collision.

**k**

_{4}=

**k**

_{1}-

**k**

_{2}+

**k**

_{3}. The three other wavepackets seen in Fig. 2c are the ones that pass through the collision region without changing their central momenta. Redistribution of population between different wavepackets must satisfy conservation of energy and momentum. Before or after the collision, when the wavepackets are separated in space, the population of the i-th one is given by the integral

*N*

_{i}=

*N*

_{T}∫

_{Vi}

*d*

**x**⟨Ψ

_{i}|Ψ

_{i}⟩ where the integration region

*V*

_{i}is selected to include the region around the i-th wavepacket. If we use

*N*

_{i}and

*N′*

_{i}to denote inital and final populations, conservation of energy and momentum show that

*N′*

_{4}= Δ

*N*

_{3}= Δ

*N*

_{1}= -Δ

*N*

_{2}, where Δ

*N*

_{i}=

*N′*

_{i}-

*N*

_{i}. These relationships are satisfied in our numerical simulations.

## 4. Interpretation

**x**,

*t*) consists of three wavepackets: Ψ(

**x**,

*t*) =

_{i}(

**x**,

*t*), as in Eq. (2). Hence, the nonlinear term |Ψ(

**x**,

*t*)|

^{2}Ψ(

**x**,

*t*) in Eq. (1) becomes a sum of nine contributions. Terms homogeneous in the index i describe self phase modulation (self-focusing or actually, self-defocusing for the case of positive scattering length), in analogy with nonlinear optics, and these terms can not be a source of new wavepackets. Moreover, crossed phase modulation terms of the form |Ψ

_{i}(

**x**,

*t*)|

^{2}Ψ

_{j}(

**x**,

*t*) for

*i,j*= 1,2,3 and

*j*≠

*i*also can not contribute to the formation of new peaks. Only mixed terms, containing different indices may be a source of new wavepackets, but only when the Bragg conditions (i.e., conservation of momentum and energy) are satisfied. Specifically, these conditions are: (a)

**k**

_{4}=

**k**

_{i}-

**k**

_{j}+

**k**

_{l}, and (b) conservation of energy, which for our initial configuration (see Fig. 2) is equivilant to |

**k**

_{4}| = |

**k**

_{3}| where

*i*= 3 and

*j*= 1,

*l*= 2. Referring again to the analogy with nonlinear optics, one can say that wavepackets with indices

*j*and

*l*create a grating during the collision whose grating vector is

**K**= -

**k**

_{j}+

**k**

_{l};, and the wavepacket

*i*= 3 scatters off of this grating. This is confirmed by our numerical simulations. Only when

*α*= 1 does the central

**K**vector satisfy the Bragg conditions stated above. However, since the wavepackets contain momentum components spread around the central

**K**vector, formation of the additional wavepacket may still occur for

*α*≠ 1 due to internal momentum compensating the momentum mismatch of the central

**K**vectors. As |

*α*- 1| increases, the number of atoms in the additional wavepacket decreases rapidly.

**k**

_{4}is

*not*a phase conjugate of the wavepacket with momentum

**k**

_{3}, since the wavepackets with momenta

**k**

_{1}and

**k**

_{2}do not form a static grating, but rather, form a dynamic grating which changes in time due to the evolution and interaction of the wavepackets throught the course of the collision. The new

**k**

_{4}wavepacket is cigar-shaped, whereas the initial

**k**

_{3}wavepacket is spherical. The final

**k**

_{1}and

**k**

_{3}wavepackets are mirror images of one another (about a plane containing the centers of wavepackets 2 and 4). Their final shape is distorted relative to their initial shape; a bite has been removed from wavepackets 1 and 3 by the 4WM process which (1) created the new wavepacket with momentum

**k**

_{4}and (2) added Bose atoms to the wavepacket with momentum

**k**

_{2}. Fig. 3 is an enlarged view of the final wavepackets in coordinate space with the details of their shapes shown more clearly. If we were to view the final wavepackets in the reference frame in which the new wavepacket is stationary, the trailing edges of wavepackets 1 and 3 would be reduced and the

*trailing*edge of wavepacket 2 would be enhanced. The structure of the wavepackets clearly shows that the nature of the matter waves obtained after the collision is sensitive to the details of the collision dynamics and the properties of the initial wavepackets. A static grating picture is not sufficient to explain the results.

*z*= 0. These different distributions show only a few percent difference in the ratio of number of atoms in the four final wavepackets.

*M*

_{F}is present for atoms in the condensates (e.g.,

*F*= 1,

*M*

_{F}= -1) with the z-axis perpendicular to the scattering plane. Another means of carrying out 4WM experiments is to use far off-resonance light traps to confine separate BECs [22

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]

*M*

_{F}being present in the condensates. A multi-component GP equation could be used to describe such experiments, as for example, the phase-conjugation experiment proposed by [12].

## 5. Acknowledgments

## References

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 |

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. |

3. | H. Wallis, A. Rohrl, M. Naraschewski, and A. Schenzle, “Phase-space dynamics of Bose condensates: Interference versus interaction,” Phys. Rev. |

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. |

5. | J. Javanainen and M. Wilkens, “Phase and phase diffusion of a split Bose-Einstein condensate,” Phys. Rev. Lett. |

6. | R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. |

7. | O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” Prog. Opt. |

8. | R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quant. Electr. |

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. |

10. | E. Goldstein, K. Plättner, and P. Meystre, “Atomic phase conjugation,” Quantum Semiclass. Opt. |

11. | E. Goldstein, K. Plättner, and P. Meystre, “Atomic phase conjugation from a Bose condensate,” J. Res. Nat. Inst. Stand. Technol. |

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. |

17. | A. Yariv and D. M. Pepper, “Amplified reflection , phase-conjugate, and oscillation in degenerate four-wave mixing,” Opt. Lett. |

18. | P. Noziéres and D. Pines, |

19. | M. Trippenbach and Y. B. Band, “Dynamics of short pulse splitting in dispersive nonlinear media,” Phys. Rev. |

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. |

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 |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(140.3320) Lasers and laser optics : Laser cooling

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 24, 1998

Published: December 21, 1998

**Citation**

Marek Trippenbach, Yehuda Band, and Paul Julienne, "Four wave mixing in the scattering of Bose-Einstein condensates," Opt. Express **3**, 530-537 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-13-530

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### References

- 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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