## Instability of a mixed atom-molecule condensate under photoassociation

Optics Express, Vol. 5, Issue 9, pp. 188-195 (1999)

http://dx.doi.org/10.1364/OE.5.000188

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### Abstract

Using the counterpart of the Gross-Pitaevskii equation, we study a system of atomic and molecular condensates in equilibrium in the presence of photoassociating light. All equilibria except a special case with only molecules are prone to an analog of the modulational instability in second-harmonic generation. The nature of the instability is such that the atoms and molecules aggregate in dense clumps.

© Optical Society of America

4. J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A **59**, R3186–9 (1999). [CrossRef]

5. P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. **81**, 3055–8 (1998). [CrossRef]

6. S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. **20**, 438–40 (1995). [CrossRef]

7. H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. **123**, 394–402 (1996). [CrossRef]

*δ*, with

*δ*>0 when the PD channel is open. We ignore all atom-atom, molecule-molecule and atom-molecule collisions. For PA, we anticipate a characteristic frequency of the order of the photon recoil frequency

*∊*=

_{R}*ħ*

**q**

^{2}=(2

*m*) of laser cooling [4

4. J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A **59**, R3186–9 (1999). [CrossRef]

4. J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A **59**, R3186–9 (1999). [CrossRef]

*φ*(

**r**) and

*ψ*(

**r**), is governed by the Hamiltonian density

*H*(

**r**),

**r**. The mass of an atom is denoted by

*m*, and

**d**(

**r**) is defined by

**d**

_{kk′}is the dipole matrix element for PA of two atoms with the wave vectors

**k**and

**k**′. The expression (2) is written down within the framework of our quasicontinuum method [2

2. J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A **58**, R789–92 (1998). [CrossRef]

3. M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A **60**, 3174–87 (1999). [CrossRef]

*. Nonetheless, in the limit*

**V***→∞, the matrix element*

**V****d**(

**r**) is independent of the quantization volume. We will discuss the matrix element in detail elsewhere [8], and here only cite two results relevant for our subsequent analysis. First,

**d**(

**r**) is peaked in the neighborhood of

**r**=0, and has a characteristic range Δ

*r*that is of the order of the larger of the two: absolute value of the scattering length, or the maximum interatomic separation in the bound state of the molecule. In what follows we assume that the relevant length scale of the atomic field is larger than Δ

*r*, or roughly equivalently, that the energy scale for the atoms is smaller than

*ħ*

^{2}=[

*m*(Δ

*r*)

^{2}]. Then the nonlocal interaction between atoms and molecules in Eq. (1) may be replaced by a contact interaction. Second, our quasicontinuum approach [2

2. J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A **58**, R789–92 (1998). [CrossRef]

3. M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A **60**, 3174–87 (1999). [CrossRef]

*ρ*represents the average density of atoms if all molecules were to dissociate. We introduce it in order to bring out explicitly the analog of Rabi frequency for PA [4

**59**, R3186–9 (1999). [CrossRef]

*υ*) is the PD rate of molecules that the laser with field strength

**E**

_{0}would cause when tuned in such a way that the relative speed of the molecular fragments is

*υ*, and

*µ*=

*m*/2 is the reduced mass of the two atoms.

*s*-wave PD, the

*v*→0 limit in Eq. (4) behaves well. The numerical value of the frequency Ω depends on light intensity, atom density, and on the specific transition in the specific molecule. Quick estimates show that in an atomic condensate where the density

*ρ*and wavelength of driving light λ satisfy

*ρ*λ

^{3}~1, the characteristic frequency of PA, Ω, is comparable to the photon recoil frequency,

*∊*, for laser intensities of the order of tens of W/cm

_{R}^{2}. This is, not entirely coincidentally, a typical light intensity in experimental PA spectroscopy.

*d*

^{3}

*r*(

*φ*

^{†}

*φ*+2

*ψ*

^{†}

*ψ*). We thus introduce the chemical potential

*µ*by adopting the eective Hamiltonian density

*K*=

*H*-

*µ*(

*φ*

^{†}

*φ*+2

*ψ*

^{†}

*ψ*). in lieu of the Hamiltonian density

*H*. We write down the Heisenberg equations of motion for the fields

*φ*and

*ψ*that ensue from

*K*, and finally posit that we treat the fields in the equations of motion as classical fields. This procedure yields the present GPE.

*ψe*

^{-iq·r}in lieu of the field

*ψ*. Second, we rescale atomic and molecular fields in such a way that if all molecules were to dissociate, the mean atomic density would be unity. In other words, the rescaled classical fields satisfy

*m*as the unit of mass. The GPE finally becomes

**q**corresponds to photon recoil in the process of PA, and

**k**, it is possible to find a chemical potential

*µ*in such a way that the GPE (6) and (7) along with the constraint (5) have a stationary solution of the form

*φ*∝

*e*

^{ik·r},

*ψ*∝

*e*

^{2ik·r}. Since the solutions for different

**k**are Galilean transformations of one another, it suffices to consider stationary fields

*φ*

_{0}and

*ψ*

_{0}that are constants in space as well. Physically, these correspond to stationary atoms, and molecules with the momentum

*ħ*

**q**each. Depending on the detuning of the laser

*δ̄*, there may be several stationary states. Here we consider the solution that mimics the ground state of the quantum mechanical two-mode version of the same problem, as discussed in Ref. [4

**59**, R3186–9 (1999). [CrossRef]

*δ̄*≤-√2 we have

*δ̄*→∞ the stationary state is atoms only. Specifically, for

*δ̄*>√2, the stationary fields are

*φ*

_{0},

*ψ*

_{0}by linearizing the GPE. Variations of this general theme include early studies of excitation spectra of trapped alkali condensates [11

11. P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A **54**, 4178–87 (1996). [CrossRef] [PubMed]

12. E. V. Goldstein and P. Meystre, “Quasiparticle instabilities in multicomponent atomic condensates, ” Phys. Rev. A **55**, 2935–40 (1997). [CrossRef]

14. H. Pu and N. P. Bigelow, “Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate, ” Phys. Rev. Lett. **80**, 1134–7 (1998). [CrossRef]

15. A. L. Fetter, “Nonuniform states of an imperfect Bose gas, ” Ann. Phys. (N.Y.) **70**, 67–101 (1972). [CrossRef]

**p**and frequency

*ω*, the latter possibly a complex number. The nonlinear GPE couples the fields and their complex conjugates, which is accounted for by the Ansatz

*ψ*. This form with constant

*u*, etc., gives a solution to the linearized (in

_{φ}*u*, etc.) GPE, provided the frequency ω and the coefficients

_{φ}*u*, etc., satisfy the eigenvalue equations

_{φ}*ω*with a positive imaginary part is encountered for a given stationary solution

*φ*

_{0},

*ψ*

_{0}and

*any*excitation wave vector

**p**, the stationary solution is unstable.

*ω*depend on photon recoil only through the projection of the excitation wave vector

**p**onto the wave vector of light

**q**. We call the corresponding dimensionless parameter

*ξ*, and express it in terms of the dimensional quantities as

*∊*, then the value of

_{R}*ξ*ranges from zero to a number of the order of unity as we consider excitation modes whose propagation directions vary from perpendicular to parallel with respect to the propagation direction of light.

*ξ*=0. Although in this case an analytical treatment is feasible, we solve the eigenvalue problem (11)–(14) directly numerically. In Fig. 1 we plot the largest imaginary part among the four eigenvalues

*ω*of Eq. (10) as a function of the light detuning

*δ̄*(hence, the stationary solution

*φ*

_{0},

*ψ*

_{0}) and the wave number of the excitation

*p*. It may be seen that for every detuning

*δ̄*>-√2 there exist excitation wave numbers

*p*for which the imaginary part of a corresponding evolution frequency

*ω*is positive. Thus, for any

*δ̄*>-√2, the stationary solution is unstable. This is an exact counterpart of the modulational instability in second-harmonic generation, as analyzed in Refs. [6

6. S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. **20**, 438–40 (1995). [CrossRef]

7. H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. **123**, 394–402 (1996). [CrossRef]

*ℑ*(

*ω*)=0.24256 is found for

*δ̄*=-0.154496 and

*p*=±0.771324. In dimensional units this means that at atom-molecule resonance,

*δ̄*≃0, the system is unstable on a time scale ~4/Ω. The momentum and length scales of the corresponding instability are

_{R}, the length scale is of the order of the wavelength of the driving light.

*δ̄*>-√2 is stable, and any atom-molecule mixtures encountered for any

*δ̄*>-√2 is unstable. However, with increasing detuning, the time scale of the instability eventually grows longer.

16. P. Öhberg and S. Stenholm, “Hartree-Fock treatment of the two-component Bose-Einstein condensate, ” Phys. Rev. A **57**, 1272–9 (1998). [CrossRef]

18. B. D. Esry and C. H. Greene, “Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates, ” Phys. Rev. A **59**, 1457–60 (1999). [CrossRef]

19. A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. **82**, 251–4 (1999). [CrossRef]

20. H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose-Einstein condensate, ” Phys. Rev. A **60**, 1463–70 (1999). [CrossRef]

21. D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media, ” Opt. Commun. **148**, 79–84 (1998). [CrossRef]

*t*. We choose the detuning

*δ̄*=0, and set

*ξ*=0. At time

*t*=0 we seed the GPE with the appropriate stationary solution

*φ*

_{0},

*ψ*

_{0}, plus a small amount of Gaussian noise to precipitate the instability. Because of the FFT method, the solution is periodic in the spatial coordinate

*x*. Moreover, the range of

*x*, 24.6282, has been intentionally chosen in such a way that it fits precisely three wavelengths of the most unstable plane wave mode. It is then no surprise that in this example the instability initially produces six density peaks for the atoms. The peaks move and oscillate in height. Two peaks quickly collide and merge, and two more peaks do the same at a later time. At the end of the integration time, at

*t*=255, four peaks remain.

10. H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E **54**, 896–911 (1996). [CrossRef]

3. M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A **60**, 3174–87 (1999). [CrossRef]

*ξ*/Δ, where

*ξ*is the Rabi frequency in the bound-bound step and is the detuning from intermediate resonance.

9. P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015.

5. P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. **81**, 3055–8 (1998). [CrossRef]

*∊*collisions should not have much effect on the onset of the instability.

_{R}## References and links

1. | P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A |

2. | J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A |

3. | M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A |

4. | J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A |

5. | P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. |

6. | S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. |

7. | H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. |

8. | M. Koštrun |

9. | P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015. |

10. | H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E |

11. | P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A |

12. | E. V. Goldstein and P. Meystre, “Quasiparticle instabilities in multicomponent atomic condensates, ” Phys. Rev. A |

13. | B. D. Esry and C. H. Greene, “Low-lying excitations of double Bose-Einstein condensates, ” Phys. Rev. A |

14. | H. Pu and N. P. Bigelow, “Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate, ” Phys. Rev. Lett. |

15. | A. L. Fetter, “Nonuniform states of an imperfect Bose gas, ” Ann. Phys. (N.Y.) |

16. | P. Öhberg and S. Stenholm, “Hartree-Fock treatment of the two-component Bose-Einstein condensate, ” Phys. Rev. A |

17. | D. Gordon and G. M. Savage, “Excitation spectrum and instability of a two-species Bose-Einstein condensate, ” Phys. Rev. A |

18. | B. D. Esry and C. H. Greene, “Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates, ” Phys. Rev. A |

19. | A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. |

20. | H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose-Einstein condensate, ” Phys. Rev. A |

21. | D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media, ” Opt. Commun. |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(190.2620) Nonlinear optics : Harmonic generation and mixing

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 16, 1999

Published: October 25, 1999

**Citation**

Juha Javanainen and M. Kostrun, "Instability of a mixed atom-molecule condensate under
photoassociation," Opt. Express **5**, 188-195 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-9-188

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### References

- P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, "Stimulated Raman molecule production in Bose-Einstein condensates," Phys. Rev. A 58, R797-800 (1998). [CrossRef]
- J. Javanainen and M. Mackie, "Probability of photoassociation from a quasicontinuum approach," Phys. Rev. A 58, R789-92 (1998). [CrossRef]
- M. Mackie and J. Javanainen, "Quasicontinuum modeling of photoassociation," Phys. Rev. A 60, 3174-87 (1999). [CrossRef]
- J. Javanainen and M. Mackie, "Coherent photoassociation of a Bose-Einstein condensate," Phys. Rev. A 59, R3186-9 (1999). [CrossRef]
- P. D. Drummond, K. V. Kheruntsyan, and H. He, "Coherent molecular solitons in Bose-Einstein condensates," Phys. Rev. Lett. 81, 3055-8 (1998). [CrossRef]
- S. Trillo and P. Ferro, "Modulational instability in second-harmonic generation," Optics Lett. 20, 438-40 (1995). [CrossRef]
- H. He, P. D. Drummond, and B. A. Malomed, "Modulational stability in dispersive optical systems with cascaded nonlinearity," Opt. Commun. 123, 394-402 (1996). [CrossRef]
- M. Kostrun et al., unpublished.
- P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, "Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems," preprint http://xxx.lanl.gov/abs/cond-mat/9804015.
- H. He, M. J. Werner, and P. D. Drummond, "Simultaneous solitary-wave solutions in a nonlinear parametric waveguide," Phys. Rev. E 54, 896-911 (1996). [CrossRef]
- P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, "Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap," Phys. Rev. A 54, 4178-87 (1996). [CrossRef] [PubMed]
- E. V. Goldstein and P. Meystre, "Quasiparticle instabilities in multicomponent atomic condensates," Phys. Rev. A 55, 2935-40 (1997). [CrossRef]
- B. D. Esry and C. H. Greene, "Low-lying excitations of double Bose-Einstein condensates," Phys. Rev. A 57, 1265-71 (1998). [CrossRef]
- H. Pu and N. P. Bigelow, "Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate," Phys. Rev. Lett. 80, 1134-7 (1998). [CrossRef]
- A. L. Fetter, "Nonuniform states of an imperfect Bose gas," Ann. Phys. (N.Y.) 70, 67-101 (1972). [CrossRef]
- P. �hberg and S. Stenholm, "Hartree-Fock treatment of the two-component Bose-Einstein condensate," Phys. Rev. A 57, 1272-9 (1998). [CrossRef]
- D. Gordon and G. M. Savage, "Excitation spectrum and instability of a two-species Bose-Einstein condensate," Phys. Rev. A 58, 1440-4 (1998). [CrossRef]
- B. D. Esry and C. H. Greene, "Spontaneous spatial symmetry breaking in two-component Bose- Einstein condensates," Phys. Rev. A 59, 1457-60 (1999). [CrossRef]
- A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, "Dynamics of two interacting Bose-Einstein condensates," Phys. Rev. Lett. 82, 251-4 (1999). [CrossRef]
- H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, "Spin-mixing dynamics of a spinor Bose-Einstein condensate," Phys. Rev. A 60, 1463-70 (1999). [CrossRef]
- D. V. Skryabin and W. J. Firth, "Generation and stability of optical bullets in quadratic nonlinear media," Opt. Commun. 148, 79-84 (1998). [CrossRef]

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