OSA's Digital Library

Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 5, Iss. 9 — Oct. 25, 1999
  • pp: 188–195
« Show journal navigation

Instability of a mixed atom-molecule condensate under photoassociation

J. Javanainen and M. Koštrun  »View Author Affiliations


Optics Express, Vol. 5, Issue 9, pp. 188-195 (1999)
http://dx.doi.org/10.1364/OE.5.000188


View Full Text Article

Acrobat PDF (1561 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

In a process of photoassociation (PA), a pair of atoms and a photon combine to make a molecule. Recently several authors have recognized that the conditions conducive to Bose-Einstein condensation of atoms also favor fast and efficient PA [1

1. 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]

4

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

]. In particular, we have studied quantum mechanically a model that includes two modes of the joint atom-molecule system, atoms with zero momentum and molecules with momenta equal to the momentum of the photoassociating photons. The results suggest that, by simply sweeping the frequency of the light, it might be possible to convert a condensate of atoms adiabatically, yet rapidly, into a condensate of molecules [4

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

].

Suppose now that, by halting the adiabatic sweep of the laser frequency or otherwise, one sets up a situation in which the atomic and molecular condensates are in equilibrium in the presence of the photoassociating light. Within our two-mode model [4

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

] the joint atom-molecule condensate is trivially stable, since it is the quantum mechanical ground state. However, when complete sets of spatial wave functions are taken into account for both the atoms and the molecules, the ground state might instead be, say, a quantum soliton [5

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]

]. The joint atom-molecule condensate may or may not be stable against small perturbations, including quantum fluctuations.

In this paper we study the stability of the atom-molecule condensate in the presence of photoassociating light. For simplicity, we consider the spatially homogeneous case with no trapping potentials for either atoms or molecules. We derive the counter-part of the Gross-Pitaevskii equation (GPE) for the present system, and then linearize the GPE around the stationary solution. It turns out that only a state in which only molecules are present is unconditionally stable. Otherwise, in analogy with the modulational instability in second-harmonic generation [6

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

,7

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]

], the atom-molecule condensate tends to break up into high-density and low-density domains.

We assume that the condensates of free atoms and molecules are being photoas-sociated/photodissociated by a plane wave of light characterized by the wave vector q. The detuning of light from the photodissociation (PD) threshold of the molecule is denoted by 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=(2m) 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]

]. Under the ordinary densities and temperatures of alkali vapor condensates, laser-driven evolution should then be much faster than the evolution resulting from the collisional interactions. Consequently, the effects of atom-atom interactions should be negligible, at least over short enough time scales.

We begin with the field theory put forth in Ref. [4

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

]. The evolution of the boson fields for atoms and molecules, φ(r) and ψ(r), is governed by the Hamiltonian density H(r),

𝓗=ψ(δ24m)ψ+φ(22m)φ12[E+·ψd3r'φ(r+12r')d(r')φ(r12r')+H.C.].
(1)

We have omitted the position argument of a field when it is the default r. The mass of an atom is denoted by m, and E+(r)=12E0eiq·r stands for the positive frequency part of the electric field of the driving laser light.

The dipole coupling matrix element d(r) is defined by

dkk'=1Vd3rei(kk')2·rd(r),
(2)

where 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

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

], hence the explicit quantization volume V. Nonetheless, in the limit V→∞, the matrix element d(r) is independent of the quantization volume. We will discuss the matrix element in detail elsewhere [8

8. M. Koštrunet al., unpublished.

], 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 Δr 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 Δr, or roughly equivalently, that the energy scale for the atoms is smaller than ħ 2=[mr)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

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

] gives the contact interaction as

12E+·ψd3r'φ(r+12r')d(r')φ(r12r')
12E+·[d3r'd(r')]ψφφ=Ω2ρeiq·rψφφ.
(3)

The quantity 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

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

],

Ω=limυ02π2Γ(υ)ρμ2υ.
(4)

Here Γ(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.

By virtue of the Wigner threshold law for 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 ρλ3~1, the characteristic frequency of PA, Ω, is comparable to the photon recoil frequency, R, for laser intensities of the order of tens of W/cm2. This is, not entirely coincidentally, a typical light intensity in experimental PA spectroscopy.

As it takes two atoms to make one molecule, the conserved quantity analogous to the number of atoms in an ordinary condensate is ∫d 3 r(φ φ+2ψ ψ). We thus introduce the chemical potential µ by adopting the eective Hamiltonian density K=H-µ(φ φ+2ψ 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.

We simplify the GPE with a few of notational tricks. First, although we do not indicate this momentum translation explicitly, we eliminate the position dependence of the driving plane wave of light by always considering the field 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

1Vd3r(φ2+2ψ2)=1.
(5)

Third, while our notation again does not account for this explicitly, we use as the unit of frequency, mΩ as the unit of length, and m as the unit of mass. The GPE finally becomes

iφ˙=122φμφφ*ψ,
(6)
iψ˙=142ψ+12iq·ψ(2μδ)ψ12φ2.
(7)

The vector q corresponds to photon recoil in the process of PA, and δ=δ+14q2 is the effective detuning corrected for photon recoil energy.

For any wave vector 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 φe ik·r, ψe 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

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

]. When the detuning is far enough below the PD threshold, everything is molecules; for δ̄≤-√2 we have

ψ0=12,φ0=0,μ=12δ̅.
(8)

When the detuning is increased, atoms begin to emerge, and in the limit δ̄→∞ the stationary state is atoms only. Specifically, for δ̄>√2, the stationary fields are

ψ0=μ=6+δ2δ6,φ0=12ψ02.
(9)

We study the stability of the stationary solution 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]

], analyses of excitation spectra of double condensates [12

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

14

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]

], and indeed the second quantized Bogoliubov theory for condensate excitations [15

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

]. Thus, we consider a small excitation with the wave vector 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

φ(r,t)=φ0+uφei(p·rωt)+υφ*ei(p·rω*t),
(10)

and similarly for the molecular field ψ. This form with constant uφ, etc., gives a solution to the linearized (in uφ, etc.) GPE, provided the frequency ω and the coefficients uφ, etc., satisfy the eigenvalue equations

[12p2+μ+ω]uφ+ψ0υφ+φ0uψ=0,
(11)
[12p2+μ+ω]υφ+ψ0uφ+φ0υψ=0,
(12)
[14(p2+2p·q)+2μδ+ω]uψ+φ0uφ=0,
(13)
[14(p2+2p·q)+2μδ+ω]υψ+φ0υφ=0.
(14)

If an eigenvalue 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.

It may be seen from Eqs. (11)(14) that the evolution frequencies 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 ξ=(q·p)2mΩ(p·p). For instance, if the evolution frequency Ω is of the order of the recoil frequency R, then the value of 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.

We first examine potential instabilities for perpendicular excitation modes, with 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]

] and [7

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]

].

Figure 1. The largest imaginary part among the four evolution frequencies of small perturbations of the joint atom-molecule condensate is plotted as a function of the laser detuning δ̄ and the wave number of the plane-wave like excitation p. The excitation and the light are taken to propagate in orthogonal directions.

The largest imaginary part (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 mΩ and mΩ. For instance, if the PA frequency is of the order of the recoil frequency, Ω~∊R, the length scale is of the order of the wavelength of the driving light.

We have seen that the all-molecule state with 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.

Linearized analysis is a legitimate tool for the studies of stability of a mixed atom-molecule condensate, but it tells nothing about the outcome after the instability has run its course. An analogous issue has been discussed in connection with trapped two-component atomic condensates [16

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

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]

], where the symmetry of the stationary state tends to be lower than the manifest symmetry of the trap. Conversely, the dynamics of a multicomponent condensate starting far away from equilibrium has recently attracted some attention [19

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

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]

].We study the atom-molecule condensate by integrating the GPE explicitly in time. At this time, we only have one-dimensional results. We use a split-operator method, whereby we alternatingly integrate the GPE by ignoring the position derivatives altogether, then integrate the evolution due to the position derivatives only. The former step is carried out by means of a second-order Runge-Kutta method, the latter is implemented using the Fast Fourier Transformation (FFT) [21

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]

].

A typical result is shown in Fig. 2. We plot the absolute square of the atomic field as a function of the one-dimensional position x and time 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.

While Fig. 2 only shows atoms, molecules and atoms behave in the same way, and the molecular field peaks together with the atomic field. The nature of the instability is such that atoms and molecules concentrate into clumps. For the parameter values of Fig. 2, Eqs. (6) and (7) are known to have solitary-wave solutions [10

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]

]. We have not studied the precise relations between the atom-molecule clumps and solitary waves, but is seems that, between the collisions, the peaks in Fig. 2 are remarkably stable.

A number of topics has to be addressed before a comprehensive picture of the instabilities of the joint atom-molecule condensate is in place. First, up to this point, one-color PA has been studied experimentally only in transitions between two different electronic manifolds. At least one of the states of the system, atoms or molecules, then has to be unstable against spontaneous emission. In real experiments one has to deal with spontaneous-emission losses. This problem is well known in quantum optics, and so is the solution: add another laser-driven transition from the spontaneously decaying state to a stable state. Exactly analogously, two-color Raman PA, a free-bound transition followed by a bound-bound transition of the molecule, may take place between non-decaying atomic and molecular states. If the intermediate state of the Raman scheme is far-off resonance, one is back to an effective two-state scheme and (ideally) has any desired degree of control over the spontaneous emission [3

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

]. In the effective two-level system the analog of Rabi frequency simply has to be replaced by Ωξ/Δ, where ξ is the Rabi frequency in the bound-bound step and is the detuning from intermediate resonance.

Figure 2. Evolution of the absolute square of the atomic field is plotted as a function of position x and time t. At t=0 the fields start with the equilibrium values, though a small amount of noise is articially added to bring forth the instability. The parameters are δ̄=0, ξ=0.

As may be seen from Fig. 2, the densities of the atom-molecule peaks tend to increase when the peaks merge. As far as we can tell from this kind of simulations, within our model the mergers could continue ad infinitum. With time the peak density of atoms and molecules increases, and collisions may become increasingly relevant in, say, limiting the peak density [9

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.

]. Similarly, the approximation that atom-molecule conversion takes places via a contact interaction may break down [5

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]

]. Nonetheless, we emphasize once more that in our “typical case” Ω~R collisions should not have much effect on the onset of the instability.

So far our numerical simulations of the instability are one-dimensional. On the other hand, we know that in three dimensions the system is most unstable in the directions perpendicular to the propagation direction of light. We speculate that in three dimensions the gas tends to break up into filaments along the direction of light propagation. Conclusive numerical verification of this scenario has to be carried out in full 3+1 dimensions, though, which we expect to be a demanding task.

Finally, extrapolating from the present-day experimental techniques, at least the atoms and possibly also the molecules will be trapped rather than free. So far we have done no detailed studies of the effects of trapping on instabilities. Nevertheless, we surmise that, if the trapped condensates are substantially larger than the length scale of the prospective instability, and if the time scale of the instability is much shorter than the characteristic time for the motion of the atoms and molecules in the trap, the instability at least sets in like in a free gas. Conversely, an instability that is too large to fit in the trapped gas is obviously suppressed.

In sum, we predict that a joint-atom molecule condensate under photoassociating light is prone to an instability analogous to the modulational instability in second-harmonic generation. From our one-dimensional model simulations it appears that in a free gas the atoms and molecules together aggregate in clumps, which locally increases the density. While the question of potential instabilities in trapped atomic and molecular gases remains largely open, similar instabilities as in free space should occur if the length and time scales of the trapped gas can accommodate them.

This work is supported in part by NSF, Grant Nos. CHE-9612207, PHY-9801888, and by NASA, Grant No. NAG8-1428.

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 58, R797–800 (1998). [CrossRef]

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]

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]

8.

M. Koštrunet al., unpublished.

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 54, 896–911 (1996). [CrossRef]

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]

13.

B. D. Esry and C. H. Greene, “Low-lying excitations of double Bose-Einstein condensates, ” Phys. Rev. A 57, 1265–71 (1998). [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]

16.

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

17.

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]

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]

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


Sort:  Journal  |  Reset  

References

  1. 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]
  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]
  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]
  8. M. Kostrun et al., unpublished.
  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 54, 896-911 (1996). [CrossRef]
  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]
  13. B. D. Esry and C. H. Greene, "Low-lying excitations of double Bose-Einstein condensates," Phys. Rev. A 57, 1265-71 (1998). [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]
  16. P. �hberg and S. Stenholm, "Hartree-Fock treatment of the two-component Bose-Einstein condensate," Phys. Rev. A 57, 1272-9 (1998). [CrossRef]
  17. 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]
  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]

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.

Figures

Figure 1. Figure 2.
 

OSA is a member of CrossRef.

CrossCheck Deposited