Manipulating the rotational properties of a two-component Bose gas
Manipulating the rotational properties of a two-component Bose gas
J. Christensson1, S. Bargi1, K. Kärkkäinen1, Y. Yu1, G. M. Kavoulakis1, M. Manninen2, and S. M. Reimann1
Mathematical Physics, Lund Institute of Technology, P.O. Box 118, SE-22100 Lund, Sweden
Nanoscience Center, Department of Physics, FIN-40014 University of Jyväskylä, Finland
(Dated: October 25, 2018)
A rotating, two-component Bose-Einstein condensate is shown to exhibit vortices of multiple
quantization, which are possible due to the interatomic interactions between the two species. Also,
persistent currents are absent in this system. Finally, the order parameter has a very simple structure
for a range of angular momenta.
PACS numbers: 05.30.Jp, 03.75.Lm, 67.40.-w
When a superfluid is set into rotation, it demonstrates
many fascinating phenomena, such as quantized vortex
states and persistent flow [1]. The studies of rotational
properties of superfluids originated some decades ago,
mostly in connection with liquid Helium, nuclei, and neu-
tron stars. More recently, similar properties have also
been studied extensively in cold gases of trapped atoms.
Quantum gases of atoms provide an ideal system for
studying multi-component superfluids. At first sight, the
rotational properties of a multi-component gas may look
like a trivial generalization of the case of a single compo-
nent. However, as long as the different components inter-
act and exchange angular momentum, the extra degrees
of freedom associated with the motion of each species is
not at all a trivial effect. On the contrary, this coupled
system may demonstrate some very different phenomena,
see, e.g., Refs. [2, 3, 4]. Several experimental and theo-
retical studies have been performed on this problem, see,
e.g., Refs. [5, 6, 7, 8, 9, 10, 11], as well as the review
article of Ref. [12].
In this Letter, the rotational properties of a super-
fluid that consists of two distinguishable components are
examined. Three new and surprising conclusions result
from our study:
Firstly, under appropriate conditions, one may achieve
vortex states of multiple quantization. It is important
to note that these states result from the interaction be-
tween the different species, and not from the functional
form of the external confinement. It is well known from
older studies of single-component gases, that any external
potential that increases more rapidly than quadratically
gives rise to vortex states of multiple quantization, for
sufficiently weak interactions [13]; on the contrary, in a
harmonic potential, the vortex states are always singly-
quantized. In the present study, vortex states of multiple
quantization result purely because of the interaction be-
tween the different components, even in a harmonic ex-
ternal potential. Therefore, our study may serve as an
alternative way to achieve such states [14].
Secondly, our simulations indicate that multi-
component gases do not support persistent currents, in
agreement with older studies of homogeneous superflu-
ids [3]. Essentially, the energy barrier that separates the
(metastable) state with circulation/flow from the non-
rotating state, is absent in this case, as the numerical
results, as well as the intuitive arguments presented be-
low, suggest.
Finally, we investigate the structure of the lowest state
of the gas, in the range of the total angular momentum L
between zero and Nmin = min(NA, NB), where NA and
NB are the populations of the two species labelled as A
and B. In this range of L, only the single-particle states
with m = 0 and m = 1 are macroscopically occupied, as
derived in Ref. [15] within the approximation of the low-
est Landau level of weak interactions. Remarkably, our
numerical simulations within the mean-field approxima-
tion, which go well beyond the limit of weak interactions,
show that this result is more general.
For simplicity we assume equal masses for the atoms
of the two components, MA = MB = M . Also, we
model the elastic collisions between the atoms by a con-
tact potential, with equal scattering lengths for colli-
sions between the same species and different species,
aAA = aBB = aAB = a (except in Fig. 4). Our results
are not sensitive to the above equality and hold even if
aAA ≈ aBB ≈ aAB, as in Rubidium, for example. For the
atom populations we assume NA 6= NB, but NA/NB <∼ 1
(without loss of generality). The trapping potential is
assumed to be harmonic, Vext(r) = M(ω
2ρ2 + ω2zz
2)/2.
Our Hamiltonian is thus
NA+NB
+ Vext(ri) +
NA+NB
i6=j=1
δ(ri − rj),(1)
where U0 = 4πh̄
2a/M . We consider rotation around the
z axis, and also assume that h̄ωz ≫ h̄ω, and h̄ωz ≫ n0U0,
where n0 is the typical atom density. With these assump-
tions, our problem becomes effectively two-dimensional,
as the atoms reside in the lowest harmonic oscillator state
along the axis of rotation. Thus, there are only two quan-
tum numbers that characterize the motion of the atoms,
the number of radial nodes n, and the quantum number
m associated with the angular momentum. The corre-
sponding eigenstates of the harmonic oscillator in two
dimensions are labelled as Φn,m.
Within the mean-field approximation, the energy of
the gas in the rest frame is
i=A,B
h̄2∇2
+ Vext(r)
http://arxiv.org/abs/0704.0447v1
(|ΨA|
4 + |ΨB|
4 + 2|ΨA|
2|ΨB|
2) d3r, (2)
where ΨA and ΨB are the order parameters of the two
components. By considering variations in Ψ∗A and Ψ
we get the two coupled Gross-Pitaevskii-like equations,
h̄2∇2
+ Vext + U0|ΨB|
ΨA + U0|ΨA|
2ΨA = µAΨA,
h̄2∇2
+ Vext + U0|ΨA|
ΨB + U0|ΨB|
2ΨB = µBΨB,
where µA and µB is the chemical potential of each com-
ponent. We use the method of relaxation [16] to minimize
the energy of Eq. (2) in the rotating frame, E′ = E−LΩ,
where Ω is its angular velocity.
For the diagonalization of the many-body Hamilto-
nian, we further assume weak interactions, n0U0 ≪ h̄ω,
and work within the subspace of the states of the lowest
Landau level, with n = 0. This condition is not neces-
sary, however it allows us to consider a relatively larger
number of atoms and higher values of the angular mo-
mentum. We consider all the Fock states which are eigen-
states of the number operators N̂A, N̂B of each species,
and of the operator of the total angular momentum L̂,
and diagonalize the resulting matrix.
Combination of the mean-field approximation and of
numerical diagonalization of the many-body Hamiltonian
allows us to examine both limits of weak as well as strong
interactions. For obvious reasons we use the diagonal-
ization in the limit of weak interactions, and the mean-
field approximation (mostly) in the limit of strong in-
teractions. The interaction energy is measured in units
of v0 = U0
|Φ0,0(x, y)|
4|φ0(z)|
4 d3r = (2/π)1/2h̄ωa/az,
where φ0(z) is the lowest state of the oscillator potential
along the z axis, and az = (h̄/Mωz)
1/2 is the oscilla-
tor length along this axis. For convenience we introduce
the dimensionless constant γ = Nv0/h̄ω =
2/πNa/az,
with N = NA + NB being the total number of atoms,
which measures the strength of the interaction.
We first study the limit of weak coupling, γ ≪ 1, and
use numerical diagonalization. Considering NA = 4 and
NB = 16 atoms, we use the conditional probability dis-
tributions to plot the density of the two components,
for L = 4, 16, 28, and 32, as shown in Fig. 1. When
L = 4 = NA, and L = 16 = NB, the component whose
population is equal to L forms a vortex state at the center
of the trap, while the other component does not rotate,
residing in the core of the vortex. This is a so-called
“coreless” vortex state [5, 6, 17]. As L increases beyond
L = NB = 16, a second vortex enters component B, and
for L = 2NB = 32, this merges with the other vortex to
form a doubly-quantized vortex state. For this value of
L = 32, the smaller component A does not carry any an-
gular momentum (apart from corrections of order 1/N).
The fact that this is indeed a doubly-quantized vortex
state, is confirmed by the occupancy of the single-particle
FIG. 1: The conditional probability distribution of the two
components, with NA = 4 (higher row), and NB = 16 (lower
row). Each graph extends between −2.4a0 and 2.4a0 in
both directions. The reference point is located at (x, y) =
(1.25a0, 0) in the lower graphs (B component). The angular
momentum L increases from left to right, L = 4(= NA), 16(=
NB), 28, and 32(= 2NB).
states. By increasing NA, NB, and L = 2NB proportion-
ally, we observe that the occupancy of the single-particle
state withm = 2 of componentB approaches unity, while
the occupancy of all the other states are at most of order
1/NB. The same happens for the single-particle state
with m = 0 of the non-rotating component A.
A similar situation emerges for the case of stronger cou-
pling, γ = 50, where we have minimized the mean-field
energy of Eq. (2) in the rotating frame (in the absence of
rotation the two clouds do not phase separate). For ex-
ample, we get convergent solutions, shown in Fig. 2, for
NB/NA = 2.777, and (i): LA = NA, LB = 0, for Ω/ω =
0.35 (top left), (ii) LA = 0, LB = NB, for Ω/ω = 0.45
(top middle), (iii) LA = 0.755NA, LB = 1.171NB, for
Ω/ω = 0.555 (top right), (iv) LA = 0, LB = 2NB, for
Ω/ω = 0.60 (bottom left), (v) LA = 0.876NA, LB =
2.057NB, for Ω/ω = 0.69 (bottom middle), and (vii)
LA = 0, LB = 3NB, for Ω/ω = 0.73 (bottom right).
Here, LA and LB is the angular momentum of each com-
ponent, with L = LA + LB. Again, when L = 2NB, and
L = 3NB the phase plots show clearly a doubly-quantized
and a triply-quantized vortex state in component B, and
a non-rotating cloud in component A.
The picture that appears from these calculations is
intriguing: as Ω increases, a multiply-quantized vortex
state of multiplicity κ splits into κ singly-quantized ones,
and on the same time, one more singly-quantized vortex
state enters the cloud from infinity. Eventually all these
vortices merge into a multiply-quantized one of multiplic-
ity equal to κ + 1. Figure 2 shows the above results for
various values of Ω.
The mechanical stability of states which involve the
gradual entry of the vortices from the periphery of the
cloud is novel. This behavior is absent in one-component
systems, in both harmonic, as well as anharmonic trap-
ping potentials. In one component gases, only vortex
phases of given rotational symmetry are mechanically
stable [18, 19]. In the present problem, the mechanical
stability of states with no rotational symmetry (shown
in Fig. 2) is a consequence of the non-negative curva-
ture of the dispersion relation (i.e., of the total energy)
FIG. 2: The density (higher graphs of each panel) and the
phase (lower graphs of each panel) of the order parameters
ΨA (left graphs of each panel) and ΨB (right graphs of each
panel), with NB/NA = 2.777 and a coupling γ = 50. Each
graph extends between −4.41a0 and 4.41a0 in both directions.
The values of the angular momentum per atom and of Ω in
each panel are given in the text.
E(L). This observation also connects with the (absence
of) metastable, persistent currents (i.e., the second main
result of our study), which we present below.
In Ref. [15] we have given a simple argument for the
presence of vortex states of multiple quantization within
the mean-field approximation. At least when the ra-
tio between NA and NB is of order unity (but NA 6=
NB), there are self-consistent solutions of Eqs. (3) of vor-
tex states of multiple quantization. Within these solu-
tions, the smaller component (say component A), does
not rotate, providing an “effective” external potential
Veff,B(r) = Vext(r) + U0nA(r) for the other one (compo-
nent B), which is anharmonic close to the center of the
trap. This effectively anharmonic potential is responsible
for the multiple quantization of the vortex states. There-
fore, we conclude that for a relatively small population
imbalance, the “coreless vortices” are vortices of multiple
quantization.
The second aspect of our study is the absence of
metastable currents (in the laboratory frame, for Ω = 0).
A convenient and physically-transparent way to think
about persistent currents is that they correspond to
metastable minima in the dispersion relation E = E(L)
[20]. A non-negative curvature of E(L) for all values of
L implies the absence of metastability. For all the cou-
plings we have examined, both within the numerical diag-
onalization, as well as the mean-field approximation, we
have found a non-negative second derivative of the dis-
persion relation. Figure 3 shows LA/NA, LB/NB, and
L/N versus Ω, for γ = 50. These curves are calculated
by minimizing the energy E(L) in the rotating frame for
a fixed Ω, and plotting the angular momentum per par-
ticle of the corresponding state for the given rotational
frequency.
Again, our argument for the effective anharmonic po-
tential is consistent with this positive curvature. Let us
 0.3  0.4  0.5  0.6  0.7
 0.3  0.4  0.5  0.6  0.7
FIG. 3: Higher graph: The angular momentum LA/NA
(crosses) and LB/NB (dots), as function of Ω. Lower graph:
L/N as function of Ω. All curves result from the minimiza-
tion of the energy in the rotating frame, within the mean-field
approximation, for γ = 50.
consider for simplicity aAA = aBB = 0, and aAB 6= 0.
Then, the problem of solving Eqs. (3) becomes essen-
tially a (coupled) eigenvalue problem. If E0,m are the
(lowest) eigenvalues of the effective (anharmonic) po-
tential felt by the rotating component for a given an-
gular momentum mh̄, then ∂2E0,m/∂m
2 is always posi-
tive. For example, if one considers a weakly-anharmonic
effective potential, Veff(ρ) = Mω
2ρ2[1 + λ(ρ/a0)
2k]/2,
where k = 1, 2, . . . is a positive integer, a0 = (h̄/Mω)
is the oscillator length, and 0 < λ ≪ 1 is a small di-
mensionless constant, according to perturbation theory,
E0,m = h̄ω|m|+ λ(|m|+ 1) . . . (|m|+ k)/2, which clearly
has a positive curvature.
One may gain some physical insight into the absence
of persistent currents by understanding the difference be-
tween a gas with one and two components. In the case
of a single component, for sufficiently strong (and repul-
sive) interactions, an energy barrier that separates the
state with circulation from the vortex-free state may de-
velop. In the simplest model where the atoms rotate in
a toroidal trap, in order for them to get rid of the circu-
lation, they have to form a node in their density, which
costs interaction energy, and this creates the energy bar-
rier [20, 21]. On the other hand, in the presence of a
second component, this node may be filled with atoms of
the other species, and therefore the system may get rid
of the circulation with no energy expense. This physical
picture is also supported by the density plots in Figs. 2
and 4. For example, in the case of coreless vortices, the
core of the vortex is filled by the other (non-rotating)
component [12]. More generally, the density minima of
FIG. 4: The density (higher graphs) and the phase (lower
graphs) of the order parameters ΨA (left graphs of each panel)
and ΨB (right graphs of each panel), with NB/NA = 2.777.
Here Ω/ω = 0.6 and γ = 50. In the left panel LA = 0,
and LB/NB = 2. In the right panel, the scattering length
aBB is twice as large as in the left panel, aBB = 2a. In this
case, LA/NA = 0.05, and LB/NB = 1.936. All graphs extend
between −4.41a0 and 4.41a0.
the one component coincide, roughly speaking, with the
density maxima of the other component, resulting in a
total density ntot = |ΨA|
2 + |ΨB|
2 which does not have
any local minima or nodes.
Our third result is based on the mean-field approxima-
tion. For 0 ≤ L ≤ Nmin, where Nmin = min(NA, NB),
the only components of the order parameters ΨA and ΨB
are the single-particle states with m = 0 and m = 1, i.e.,
cn,0Φn,0 + cn,1Φn,1,
dn,0Φn,0 + dn,1Φn,1, (4)
where cn,0, cn,1, dn,0 and dn,1 are functions of L and of
the coupling. The numerical simulations that we perform
within a range of couplings γ ≤ 50 that extend well be-
yond the lowest-Landau level approximation, reveal this
very simple structure for the lowest state of both com-
ponents. Also, the corresponding dispersion relation is
numerically very close to a parabola, as in the case of
weak interactions [15]. Again, one may attribute these
facts to the effective potential that arises from the inter-
action between the two species [15].
In the studies that have examined a single-component
gas in an external anharmonic potential, it has been
shown that as the strength of the interaction increases,
there is a phase transition from the phase of multiple
quantization to the phase of single quantization [19]. In
the present case the situation is more complex, since the
effective anharmonic potential is generated by the in-
teraction between the two species as a result of a self-
consistent solution. Still, a similar phase transition takes
place here, as, for example, one keeps the scattering
lengths aAA and aAB fixed, and increases aBB that cor-
responds to the rotating component. Figure 4 shows the
density and the phase of both species, for aAA = aBB =
aAB = a (left panel), and aBB = 2aAA = 2aAB = 2a
(right panel). Component B undergoes a phase transi-
tion from a doubly-quantized vortex state to two singly-
quantized vortices.
To conclude, mixtures of bosons demonstrate numer-
ous novel superfluid properties and provide a model sys-
tem for studying them. Here we have given a flavor of
the richness of this problem. Many of the results pre-
sented in our study are worth investigating further, as,
for example, one changes the ratio of the populations, the
coupling constant between the same and different species,
or the masses.
We acknowledge financial support from the Euro-
pean Community project ULTRA-1D (NMP4-CT-2003-
505457), the Swedish Research Council, the Swedish
Foundation for Strategic Research, and the NordForsk
Nordic Network on “Low-dimensional physics”.
[1] A. J. Leggett, Rev. Mod. Phys. 71, S318 (1999).
[2] N. D. Mermin and T.-L. Ho, Phys. Rev. Lett. 36, 594
(1976).
[3] P. Bhattacharyya, T.-L. Ho, and N. D. Mermin, Phys.
Rev. Lett. 39, 1290 (1977).
[4] T.-L. Ho, Phys. Rev. Lett. 49, 1837 (1982).
[5] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S.
Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.
83, 2498 (1999).
[6] A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard,
and W. Ketterle, Phys. Rev. Lett. 90, 140403 (2003).
[7] V. Schweikhard, I. Coddington, P. Engels, S. Tung, and
E. A. Cornell, Phys. Rev. Lett. 93, 210403 (2004).
[8] J. E. Williams and M. J. Holland, Nature (London) 401,
568 (1999).
[9] E. J. Mueller and T.-L. Ho, Phys. Rev. Lett. 88, 180403
(2002).
[10] K. Kasamatsu, M. Tsubota, and M. Ueda Phys. Rev.
Lett. 91, 150406 (2003).
[11] K. Kasamatsu, M. Tsubota, and M. Ueda Phys. Rev. A
71, 043611 (2005).
[12] K. Kasamatsu, M. Tsubota, and M. Ueda, Int. J. of Mod.
Phys. B, 19, 1835 (2005).
[13] See, e.g., A. L. Fetter, Phys. Rev. A 64, 063608 (2001); E.
Lundh, Phys. Rev. A 65, 043604 (2002), and references
therein.
[14] V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys.
Rev. Lett. 92, 050403 (2004).
[15] S. Bargi, J. Christensson, G. M. Kavoulakis, and S. M.
Reimann, Phys. Rev. Lett. 98, 130403 (2007).
[16] See, e.g., S. A. Chin and E. Krotscheck, Phys. Rev. E 72
036705 (2005).
[17] R. Blaauwgeers, V. B. Eltsov, M. Krusius, J. J. Ruohio,
R. Schanen, and G. E.Volovik, Nature (London) 404, 471
(2000).
[18] D. A. Butts, and D. S. Rokhsar, Nature (London) 397,
327 (1999).
[19] See, e.g., G. M. Kavoulakis and G. Baym, New Journal
of Phys. 5, 51.1 (2003), and references therein.
[20] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).
[21] D. Rokhsar, e-print cond-mat/9709212.
http://arxiv.org/abs/cond-mat/9709212
