## Abstract

We describe our recent progress on the investigation of two-species Bose-Einstein condensation. From a theoretical analysis we show that there is a new rich phenomenology associated with two-species Bose-Einstein condensates which does not exist in a single-species condensate. We then describe results of a numerical model of the evaporative cooling process of a trapped two-species gas.

© Optical Society of America

## 1. Introduction

The realization of Bose-Einstein condensation (BEC) in dilute alkali vapors^{1–3} has opened the field of weakly-interacting degenerate Bose gases. During the
past two and a half years, substantial experimental and theoretical progress has
been made on the study of the properties of this new state of matter. Indeed, the
physics of trapped diluted condensates has emerged as one of the most exciting
fields of physics in this decade. Recently, the remarkable experimental realization
of a condensate mixture composed of two spin states of ^{87}Rb^{4} has prompted significant interest in the physics of a new class of quantum
fluids: the two-species Bose-Einstein condensate (TBEC)^{5–12}. Multi-component and, particularly, multi-species condensates offer new
degrees of freedom, which give rise to a rich set of new phenomena that do not exist
in a one-species condensate. Furthermore, the TBEC offers new and interesting
experimental challenges.

The investigation of a TBEC requires progress on several different fronts. On the
theoretical side, our effort is rooted in the study of the properties of a set of
coupled non-linear Schrödinger equations which can be used to describe
the TBEC within the mean field limit. Of equal importance is our development of an
experimental strategy for preparing a high-density laser cooled atomic mixture, for
loading magnetic trap where evaporativecooling can be applied to form a condensate.
We first discuss our contributions to the theoretical description of the TBEC with
an emphasis on those phenomenological features which distinguish the TBEC from the
single component BEC *and* which are experimentally accessible.
Finally, we describe results derived from a numerical model of the evaporative
cooling process developed to help determine the optimal strategy for cooling into
the doubly condensed phase.

In this paper we focus our discussion on the particular system composed of a mixture
of sodium (Na) and rubidium (Rb) atoms. We have selected this system because (a)
both species have been successfully Bose condensed and (b) because other candidate
systems, such as a sodium-cesium mixture, have already been found by us^{13} to be less desirable due to large inter-species trap loss rates. We stress,
however, that many of our results are generalizable to other atomic mixtures,
spin-state mixtures and Fermi-Bose mixtures.

## 2. Properties of the TBEC: some theoretical predictions

In the following, we give a brief description of the novel properties of a TBEC confined in an isotropic spherical potential at zero temperature. Throughout our discussion, we assume that the effect of gravity can be compensated for by choosing appropriate atomic species and spin states or by using the proper magnetic/optical trapping fields. Neglecting gravitational effects not only simplifies our calculations, it allows us to focus on the more essential intrinsic couplings within the TBEC.

At zero temperature, the self-consistent nonlinear Schrödinger equations,
known as Gross-Pitaevskii equations (GPEs), for a TBEC may be written as^{5,6,10,11}:

where
*ψ*_{i}
(*r*,*t*)
denotes the macroscopic condensate wave function for species *i*,
with *r* being the radial coordinate. *N*_{i}
,
*m*_{i}
and *ω*_{i}
are particle number, mass and trap frequency, respectively. The interaction between
particles are described by a self-interaction term *U*_{i}
=
4*πħ*
^{2}
*a*_{i}
/*m*_{i}
and a term that corresponds to the interaction between different species
*U*
_{12} =
2*πħ*
^{2}
*a*
_{12}/*m*
(with m being the reduced mass of the two species), where
*a*_{i}
is the scattering length of species
*i* and *a*
_{12} between species 1 and 2. The
time-independent form of the nonlinear Schrödinger equations are obtained
by replacing the left hand sides of Eqs. (1),(2) with
*μ*_{i}*ψ*_{i}
(*r*)
(*i*=1,2), with *μ*_{i}
being
the chemical potential.

The ground state wavefunctions *ψ*
_{1} and
*ψ*
_{2} can be obtained^{10} by solving the coupled GPEs (Eqs. (1)and (2)) iteratively. For negative *a*
_{12}
(i.e., attractive inter-species interaction), both wave functions are compressed as
compared to the case of two independent condensates (i.e.,
*a*
_{12} = 0). If the magnitude of
*a*
_{12} becomes larger than a certain value, the condensates
mixture will eventually collapse. By contrast, for small positive
*a*
_{12}, two coupled, interpenetrating condensates are
formed and each of the individual condensate wavefunctions become somewhat flattened
due to the mutual repulsion between the species. However, for large positive
*a*
_{12}, the ground state is no longer necessarily a
mixture of two overlapping condensates. Instead, the system “phase
separates” into two distinct condensates^{5,10}: one forming a core at the center of the trap and
the other forming a surrounding shell. Fig. 1 illustrates the ground state density distribution of a
Na-Rb condensates mixture at different values of *a*
_{12}. At
large *a*
_{12}, we see a phase separated TBEC with a Rb core
and a Na shell. Another interesting phenomenon arising from a large repulsive
inter-species interaction is the possibility of the formation of a metastable state
of the TBEC^{11}. Our simulations (see below) show that, in the Na-Rb system, the Na
condensate will form prior to the Rb condensate such that the Rb atoms condense in
the presence of the repulsive Na condensate core into a metastable shell around the
Na. However, from previous calculations, we know that the more stable state in this
case should be comprised of a Rb core and a Na shell as shown in Fig. 1. We investigate the mechanical stability of this
Na-core/Rb-shell system by externally perturbing the trapping potential and find
that it is indeed not unconditionally stable: under a sufficiently strong external
perturbation it will make a *macroscopic quantum jump* to the more
stable Rb-core/Na-shell system. These macroscopic metastable states arise from the
inter-species interactions and hence are unique for the multi-component condensates.

One of the fundamental properties of the confined condensate lies in the nature of
the collective excitations. Excitation frequencies of a single-species Rb or Na
condensate have been experimentally measured^{14,15} and theoretically calculated^{16–19}, and good agreement has been found between the two. We have generalized the
standard Bogoliubov-Hartree theory^{20} for one-species BEC to the case of the TBEC^{11} and find that inter-species coupling dramatically modifies the excitation
spectrum. We identified two types of isotropic breathing modes: in-phase and
out-of-phase modes. We have also found that for large repulsive coupling, some
non-isotropic modes possess imaginary frequencies indicating that the TBEC is unstable^{7}. Recently, Öhberg^{21} showed that, under these conditions, a symmetry-breaking state is more stable
and hence might represent the true ground state. However, that work is done for a 2d
condensate with a very small number of particles, and the relative energy difference
between the symmetric and un-symmetric states is only a few percent. More detailed
studies with realistic parameters in a 3d trap is needed for the full understanding
the dynamics of the transition between a symmetric and non-symmetric state. The
recent observation of Feshbach resonance^{22} provides us with the exciting possibility of tuning the value of
*a*
_{12} and studying such transitions experimentally.
The instability induced by large repulsive coupling is reminiscent of the
cross-phase modulation (XPM) instability in nonlinear optics^{23}. In fact, the GPEs for BEC are very similar to the nonlinear
Schrödinger equations describing wave propagation inside optical fibers.
When one light field is present, a modulation instability occurs in the case of
anomalous group velociy dispersion, analogous to the negative scattering length
instability for a one-species BEC. When two light fields co-propagate, instability
can be induced by XPM in both the anomalous- and normal-dispersion regimes. Such
modulation instability can lead to the break up of intense cw radiation into
ultrashort pulses and formation of solitons. Extensive work in the field of
nonlinear optics may then help us understand the physics of BEC, such as the
detailed dynamics of how quantum fluctuations will affect a TBEC with imaginary
modes.

For a more rigorous treatment of the TBEC fluctuation and stability character, we
must go beyond the mean-field theory. Starting from a second quantized grand
canonical Hamiltonian, we have identified an eigenvalue associated with the TBEC
which plays the same role as the sign of the scattering length in a one-species BEC
in that it is the determiner of condensate stability^{9}. We predict that there is a finite range of inter-species interaction
strengths in which a Na-Rb double condensate can be stable in a harmonic trap,
beyond this range, however, we find that the TBEC is unstable against particle
number fluctuations. A phenomenon closely related to fluctuations is the diffusion
of quantum phases of the condensate wavefunctions^{25}. We have found that at the exact boundary of the stable/unstable region, the
relative phase of the two condensate components of the TBEC can become locked,
evolving without relative diffusion, while the individual phases continue to diffuse
over time^{12}.

As an important property of superfluidity, quantum vortices in alkali BEC have
attracted several theoretical investigations. In his recent work, Rokhsar argued
that a vortex state is unstable in a one-species condensate due to the presence of a
bound core state^{24}. In the case of a Rb-Na TBEC, one may produce a system comprised of a
vortex-free Rb condensate at the center of the trap surrounded by a Na condensate in
a vortex state such that a repulsive inter-species interaction may prevent the
existence of the bound state and hence, stabilize the system. A detailed study of
vortices in the TBEC is currently under way. In the above, we have briefly discussed
some novel properties of the TBEC. There are still many open questions. More
theoretical investigations are in progress to deepen our understanding of this
unique macroscopic quantum system. Of equal importance is the realization of an
experiment that will test these theoretical predictions.

## 3. Models of forced evaporative cooling in the Na-Rb system

Schematically, Bose-Einstein condensation in trapped alkalis is realized via the
following steps: first, a large laser cooled sample of atoms is prepared in a
magneto-optical trap (MOT); next this sample is loaded into a magnetic trap; and
then finally forced evaporative cooling is used to increase the phase space density
up to the critical values where the condensate forms. To date, we have made
significant progress on the first two steps in this process - namely, we have
produced a dense heteronuclear alkali vapor using a MOT and, most recently, we have
successfully magnetically trapped this ultra-cold mixture. A detailed account of
these achievements is however beyond the scope of the present work. In this review,
we will concentrate on our numerical model of the evaporative cooling process in the
Na-Rb system and show that efficient cooling can happen in a time scale that is
satisfactory for the production of a TBEC. To investigate the the two-species
evaporative cooling process we have developed a numerical model based on
Bird’s method for fluid dynamics^{28} which is very similar to that employed by Wu, Arimondo and Foot to study
evaporative cooling and Bose condensation in the single species case^{29}. The method can be described as follows: two atomic samples, each at a given
temperature, are generated and accordingly accommodated by a harmonic trap (with
possibly different characteristic frequencies). The samples are then separated into
cells according to their position. Collisions are carried out at each cell for a
time *δt*, much smaller than the average collision time.
These are hard-sphere collisions such that pairs of atoms with high relative
velocity have higher probability of collision. After that, the atoms are allowed to
evolve in the trap with their post-collision velocities during the same time
*δt*. Evaporative cooling is modeled by requiring that
atoms with a distance from the center of the trap larger than
*R*_{in}
exp(-*t*/*τ*) are ejected. We used
*R*_{in}
=0.4 cm and *τ*= 12
s.

In order to perform the simulations, we used an estimate for the cross-section
between Na and Rb obtained from our experimental studies of the inter-species^{22} collisions. From our result, the sodium loss rate due to the collision with
rubidium is *β*
^{*}
*n*_{Rb}
~ 0.1/s (with a density of *n*_{Rb}
= 8
× 10^{8} cm^{-3}). And using a relative velocity of
~ 15 cm/s, we obtain the estimated Na-Rb cross section
*σ*_{Na-Rb}
= 1.8 ×
10^{-12}cm^{2}. However, it should be noted that, even though
this estimate is the best that can be inferred from the available data from
two-species experiments, the true value of the inter-species cross-section is
dependent on the yet undetermined inter-species scattering length. If the actual
cross-section is much smaller than our estimate, the evaporative cooling process may
need to be carried out at a slower rate in order to allow for efficient
thermalization of the atomic samples during the evaporation process. As a first step
in this investigation, we have modeled the thermalization of the two atomic clouds
assuming that they are loaded into themagnetic trap with each species at its
respective Doppler limited temperature (240 *μ*K for Na
and 120 *μ*K for Rb). As shown in Fig. 3, we find that thermal relaxation occurs in a few
hundred ms. Naturally, larger the inter-species cross section
*σ*_{Na-Rb}
is, the faster the system
reaches its equilibrium.

The results of the evaporative cooling for the Na-Rb system are shown in Fig. 4. In Fig. 4(a), we see that after 50s (approximately 500 average
collision times), more than 1% of the initial number of atoms remain in the trap. In Fig. 4(b), we note that due to the inter-species interaction,
the system remains in equilibrium while evaporation takes place (except during the
first second, where sympathetic cooling takes place). Fig. 4(c) displays the time dependence of the size of the
samples together with the RF cut-off modeled in the manner described above. In Fig. 4(d), we show the time evolution of phase-space density
(*n*${\mathrm{\lambda}}_{\mathit{\text{dB}}}^{3}$) for the Na-Rb system undergoing evaporative cooling.

The model discribed here assumes purely classical hard-sphere collisions and does not
take into account the (1+N) Bose-Einstein enhancement factor^{29}. Hence the penetration into the critical BEC boundary (defined by
*n*${\mathrm{\lambda}}_{\mathit{\text{dB}}}^{3}$=2.61) should be considered as a conservative estimate. In sum, our
simulations clearly indicate that evaporative cooling has the power to increase
phase-space density enough to expect that TBEC formation will occur for such
conditions. In the animation (Fig. 5), the evaporative cooling path can be followed in
phase space (temperature vs. density) so that the effectiveness of the process can
be fully appreciated.

In order to more realistically study the formation of the two-species condensate, a
version of the code that includes Bose statistics (which affect the collisional rate^{29}) is currently being utilized. Moreover, we are also taking full advantage of
our multi-species capability to model the physics of sympathetic cooling, formation
of Fermi-Bose mixtures and mixtures of Bose gases with various combinations of
scattering lenghts.

## 4. Conclusions and prospects

In summary, we have discussed some of the properties of the TBEC. We have shown that the nonlinear coupling between the two components gives rise to a rich set of new phenomena. We also modeled a two-species evaporative cooling process, indicating that it can lead to the formation of a TBEC. The territory of two inter-penetrating quantum fluids is still wide open and we believe that many exciting and unexpected new phenomena are waiting to be discovered.

## Acknowledgements

The experiments mentioned in this paper have benefited from the crucial work of M. Banks, J. Janis, P. Rudy and S. B. Weiss. Aspects of the theoretical work were carried out in collaboration with J. H. Eberly. This work was supported by the National Science Foundation, the David and Lucile Packard Foundation and the Army Research Office. RE is grateful for the financial support from CNPq during part of this work.

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