## 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 [1–4]. 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].

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] 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]. 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 [6,7], 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}=(2

*m*) of laser cooling [4]. 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]. The evolution of the boson fields for atoms and molecules,
*φ*(**r**) and
*ψ*(**r**), is governed by the Hamiltonian
density *H*(**r**),

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 ${\mathbf{E}}^{+}\left(\mathbf{r}\right)=\frac{1}{2}{\mathbf{E}}_{0}{e}^{i\mathbf{q}\xb7\mathbf{r}}$ 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

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 [2,3], hence the explicit quantization volume * V*. Nonetheless, in the limit

*→∞, 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 [2,3] gives the contact interaction as

$$\simeq \frac{1}{2\hslash}{\mathbf{E}}^{+}\xb7[\int {d}^{3}r\text{'}\mathbf{d}\left(\mathbf{r}\text{'}\right)]{\psi}^{\u2020}\phi \phi =\frac{\Omega}{2\sqrt{\rho}}{e}^{i\mathbf{q}\xb7\mathbf{r}}{\psi}^{\u2020}\phi \phi .$$

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],

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 *ρ*λ^{3}~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/cm

^{2}. 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*+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.

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

Third, while our notation again does not account for this explicitly, we use as the
unit of frequency, $\sqrt{\hslash \u2044m\Omega}$ as the unit of length, and *m* as the unit of mass.
The GPE finally becomes

The vector **q** corresponds to photon recoil in the process of PA, and $\stackrel{\u2013}{\delta}=\delta +\frac{1}{4}{q}^{2}$ is the effective detuning corrected for photon recoil energy.

Related field theories have been proposed to model PA [4,5], and to describe the Feshbach resonance [9]. Equations (6) and (7) also correspond to a case of genuinely three dimensional second-harmonic generation with a group velocity difference between the fundamental and second harmonic, and besides, the parameters of dispersion happen to be such that the group velocity difference cannot be removed by a coordinate transformation [10]. Although we have not seen an explicit three-dimensional treatment of Eqs. (6) and (7), there is an enormous literature on closely related problems in second-harmonic generation. Particularly relevant here is the stability analysis in Refs. [6] and [7].

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

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

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], analyses of excitation spectra of double condensates [12–14], and indeed the second quantized Bogoliubov theory for
condensate excitations [15]. 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

and similarly for the molecular field *ψ*. This form with
constant *u _{φ}*, etc., gives a solution to the
linearized (in

*u*, etc.) GPE, provided the frequency ω and the coefficients

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

_{φ}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 $\xi =\sqrt{{\hslash (q\xb7p)}^{2}\u2044m\Omega (p\xb7p)}$. 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] and [7].

The largest imaginary part
*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 $\sim \sqrt{\hslash m\Omega}$ and $\sim \sqrt{\hslash \u2044m\Omega}$. 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.

Let us next consider excitation modes that do not propagate in a direction
perpendicular to the propagation direction of light,
*ξ*≠0. We have found that, for any fixed
detuning *δ̄*>-√2 but
varying *ξ*, the largest rate of instability is always
encountered at *ξ*=0. Photon recoil thus tends to
stabilize the system. Nonetheless, photon recoil has no effect on the small
excitations that propagate perpendicular to the propagation direction of light.
Instabilities in the perpendicular direction are unavoidable once
*δ̄*>-√2.

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 [16–18], 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 [19,20].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].

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

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

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