Angular momentum of a magnetically trapped atomic condensate.

Abstract

For an atomic condensate in an axially symmetric magnetic trap, the sum of the axial components of the orbital angular momentum and the hyperfine spin is conserved. Inside an Ioffe-Pritchard trap (IPT) whose magnetic field (B field) is not axially symmetric, the difference of the two becomes surprisingly conserved. In this Letter we investigate the relationship between the values of the sum or difference angular momentums for an atomic condensate inside a magnetic trap and the associated gauge potential induced by the adiabatic approximation. Our result provides significant new insight into the vorticity of magnetically trapped atomic quantum gases.

Full text

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arXiv:cond-mat/0611376v2 [cond-mat.other] 19 Jan 2007
The angular momentum of a magnetically trapped atomic condensate
P. Zhang,1H. H. Jen,1C. P. Sun,2and L. You1,3
1School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
2Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100080, China
3Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China
(Dated: February 4, 2008)
For an atomic condensate in an axially symmetric magnetic trap, the sum of the axial components
of the orbital angular momentum and the hyperfine spin is conserved. Inside an Ioffe-Pritchard trap
(IPT) whose magnetic field (B-field) is not axially symmetric, the difference of the two becomes
surprisingly conserved. In this paper we investigate the relationship between the values of the sum
or difference angular momentums for an atomic condensate inside a magnetic trap and the associated
gauge potential induced by the adiabatic approximation. Our result provides significant new insight
into the vorticity of magnetically trapped atomic quantum gases.
PACS numbers: 03.75.Mn, 03.75.Lm, 03.65.Vf, 67.57.Fg
A magnetic trap constitutes one of the key enabling
technologies for the recent successes in atomic quan-
tum gases [1]. The most commonly employed mag-
netic traps includes the quadrupole trap (QT) as in the
magneto-optical-trap (MOT) configuration [2] and the
Ioffe-Pritchard trap (IPT) [3]. The direction of the B-
field forming the magnetic trap is generally a function of
the spatial position. For a trapped atom, its hyperfine
spin adiabatically follows the changing B-field direction,
and the atom remains aligned (or anti-aligned) with re-
spect to the local B-field. As a result of the adiabatic
approximation, the center of mass motion of a magneti-
cally trapped atom experiences an induced gauge poten-
tial from the changing B-field [4, 5].
In a variety of magnetic traps, e.g., in a QT, the B-field
is invariant with respect to rotations along a fixed z-axis.
As the cse of single particle dynamics [6], such a SO(2)
symmetry leads to the conservation of the z-component
Jz (= Lz+ Fz) of the total angular momentum or the
sum of the z-components of the atomic spatial angular
momentum?L and the hyperfine spin?F. A different sym-
metry exists for an IPT giving rise to a corresponding
conserved quantity Dz (= Lz− Fz), the difference of
Lz and Fz. To our knowledge, this surprising property
has never before been identified explicitly. We thus feel
obliged to present this letter because it significantly af-
fects the vortical properties of a global condensate ground
state in a magnetic trap.
Our work is focused on a detailed investigation of the
relationship between the gauge potential and the associ-
ated values of Jz(Dz) in a magnetic trap. For a spin-F
condensate, due to the appearance of the adiabatic gauge
potential, the possible values of Jzor Dzare restricted to
a definite region [−F,F]. The gauge potential of our for-
mulation is directly related to the effective trap rotation
studied earlier in Ref. [5]. While Ho and Shenoy mainly
studied the orbital angular momentum component in an
IPT [5], instead we concentrate on the conserved quan-
tity, the sum (Jz) or difference (Dz) for a QT or a IPT.
This paper is organized as follows. We first consider
a spin-1 condensate in a QT. Making use of an effective
energy functional appropriate for the adiabatic approxi-
mation [5], we prove Jz ∈ [−1,1] with the actual value
determined by the angle between the z-axis and the di-
rection of the B-field. We then generalize to the spin-F
case. Finally, our result is extended to an IPT.
The Hamiltonian of a spin-1 atomic condensate with
N atoms in a magnetic trap is H = HS+ HI with the
single atom part
HS=
?
ˆψ†(? r)
?
−∇2
2M+ µBgFB(? r)?F · ˆ n(? r)
?
ˆψ(? r)d? r,
and the atom-atom interaction Hamiltonian
HI=
?
m,n,p,q
?
ˆψ(z)†
m (? r)ˆψ(z)†
n
(? r′)Vmn
pq(? r,? r′)ˆψ(z)
p(? r)ˆψ(z)
q (? r′)d? rd? r′.
ˆψ(? r) = [ˆψ(z)
tion field operator for the z-quantized Fz-component of
m,n,p,q = 0,±1. M is the atomic mass, and µBis the
Bohr magneton. B(? r) and ˆ n(? r) denote the strength and
direction of the local B-field. ? = 1 is assumed. The
Lande g factor is gF=1= −1/2.
Within the mean field approximation, the field opera-
torˆψ(? r) is replaced by its average ?ˆψ(? r)?. To introduce
the adiabatic approximation, we define a group of nor-
malized scalar wave functions ϕu(? r):
√
NξB(b,? r)ϕb(? r),
−1(? r),ˆψ(z)
0(? r),ˆψ(z)
1(? r)]Tdenotes the annihila-
?ˆψ(? r)? =
?
b=0,±1
(1)
where ξB(b,? r) is the eigenstate of the B-quantized spin
component?F ·ˆ n(? r) with eigenvalue b, satisfying the rela-
tions?F ·ˆ n(? r)ξB(b,? r) = bξB(b,? r) and ξB†(b,? r)ξB(b′,? r) =
δb,b′. In the z-quantized representation, it takes the form
ξB(b,? r) = [ξB
it is important to distinguish ξB(b,? r) from the eigen-
states ξz(0,±1) of Fzwith eigenvalues 0,±1. In explicit
−1(b,? r),ξB
0(b,? r),ξB
1(b,? r)]T. In this study,

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form, we have ξz(−1) = [1,0,0]T, ξz(0) = [0,1,0]T, and
ξz(1) = [0,0,1]T.
A magnetic dipole precesses around the direction of a
B-field. Majorona transitions between different ξB(b,? r)
states can be neglected when B(? r) is large enough. Thus
the atomic hyperfine spin adiabatically freezes in the low-
field seeking state ξB(−1,? r) during the trapped center
of mass motion, and ϕ−1(? r) = ϕ(? r) and ϕ0,+1(? r) = 0.
Similarly the z-quantized mean field becomes
√NξB(−1,? r)ϕ(? r),
with ϕ(? r) a B-quantized scalar function.
Substituting Eq. (2) into the expression of H, we can
obtain the expression of the condensate energy Ead as
a functional of the scalar wave function ϕ(? r): Ead[ϕ] =
N?ϕ∗(? r)Eadϕ(? r)d? r. Here Eaddefined as [4, 5]:
Ead=[−i∇ +?A(? r)]2
2M
ˆψ(? r) ≈ ?ˆψ(? r)? =
(2)
+ µBB(? r) + W + Vo+g2
2|ϕ|2. (3)
The gauge potential induced by the adiabatic approx-
imation is?A(? r) = −iξB†(−1,? r)∇ξB(−1,? r) and W =
(|∇ξB†(−1,? r)∇ξB(−1,? r)|−?A·?A)/(2M). The trap poten-
tial µBB(? r) is augmented by Vo(ρ,z) from other sources,
e.g., an optical potential with rotational symmetry along
the z-axis. Under the approximation of contact pseudo-
potentials, trapped atoms collide in the same spin aligned
state. Thus the g2term is proportional to the scattering
length a2in the total spin channel Ftot= 2.
We show below that the gauge potential?A(? r) con-
strains the values of Jz for a condensate ground state.
In cylindrical coordinate (ρ,φ,z), the B-field of a QT is
expressed as?B(? r) = B′(ρˆ eρ− 2zˆ ez). An optical poten-
tial Vo(ρ,z) is introduced to push atoms away from the
region of small B(? r), eliminating the deadly Majorona
transitions [7, 8]. Because this B-field is cylindrically
symmetric,?F · ˆ n(? r) commutes with the z-component of
the total atomic angular momentum Jz= Lz+Fz. This
allows us to choose ξB(−1,? r) as the common eigenstate
of?F · ˆ n(? r) and Jzwith any possible eigenvalues. In this
study we choose ξB(−1,? r) for convenience to satisfy
JzξB(−1,? r) = 0.
This constraint on Jzlimits the respective values for Lz
or Fz, in a sense equivalent to a gauge choice. The ac-
tual value of Jz for the ground state of a condensate is
determined by appropriate system parameters. Explic-
itly, a simple rotation gives ξB(−1,? r) = exp[iφ]exp[−i?F ·
ˆ eφβ(ρ,z)]ξz(−1). The angle between the z-axis and ˆ n(? r)
is β(ρ,z) = arccos[−2z/?ρ2+ 4z2].
The B-quantized ground state scalar mean field wave
function is denoted as ϕg(? r), determined from a mini-
mization of Ead[ϕ]. The cylindrical symmetry of both
?B(? r) and Voassures Ead[ϕ(ρ,φ,z)] = Ead[ϕ(ρ,φ + θ,z)]
for any θ. Due to this SO(2) symmetry, if there exists
(4)
only one normalized ground state (up to global phase fac-
tors) as in the situation considered here for a condensate,
it has to be a common eigenstate for all rotation opera-
tors exp[θ∂/(∂φ)], i.e., an eigenstate of i∂/(∂φ). Thus,
we take ϕg(? r) = ˜ ϕg(ρ,z)exp(isφ). On substituting into
Eq. (2), we obtain the z-quantized mean field ?ˆψ(? r)?gfor
the ground state
√NξB(−1,? r)˜ ϕg(ρ,z)exp(isφ),
which is an eigenstate of Jzwith an eigenvalue s, i.e.,
?ˆψ(? r)?g=
(5)
Jz?ˆψ(? r)?g= s?ˆψ(? r)?g,(6)
because of Eq. (4).
Equation
?
?ˆψ(z)
with bm= ξ†
components of state Eq. (7) naturally carry topological
windings as a direct result of the conservation of Jz.
To determine the value of s or Jz for the ground
state, we first compute the gauge potential?A(? r) for a
QT. With the expression of ξB(−1,? r), we find?A(? r) =
cosβ(ρ,z)ˆ eφ/ρ. Using the expressions for?A(? r), Ead, and
Ead, it is easy to show that for a scalar wave function
˜ ϕg(ρ,z)exp(imφ) with any integer m, the energy func-
tional Eadsatisfies
(5)andtheexpansion?ˆψ(? r)?g
=
m=0,±1?ˆψ(z)
m (? r)?gξz(m) gives
√NξB
m?g=
m(−1,? r)ϕg(? r) = bm(ρ,z)˜ ϕg(ρ,z)ei(s−m)φ,(7)
z(m)e−iFyβ(ρ,z)ξz(−1). The individual spin
Ead[˜ ϕg(ρ,z)eimφ] = Ead[˜ ϕg] + ∆Em[˜ ϕg], (8)
with Em[˜ ϕg] =?d? r|˜ ϕ∗
In addition to the centrifugal term proportional to m2, a
term linear in m appears due to the?A · ∇ term in Ead.
In the following we show that the above linear term
is important for the value s, which we determine with
a variational approach. Because Ead takes its minimal
value in the state ˜ ϕgexp[isφ], we have Ead[˜ ϕgeisφ] ≤
Ead[˜ ϕgei(s±1)φ]. Together with Eq. (8), we find the nec-
essary condition satisfied by s: ∆Es[˜ ϕg] ≤ ∆Es±1[˜ ϕg] or
|s + C| ≤ 1/2, where the coefficient C is defined as
?d? r|˜ ϕ∗
?d? r|˜ ϕ∗
When the correlation between cosβ(ρ,z) and ρ−2
is neglected, the factor is approximated by C
?d? r|˜ ϕ∗
cosβ(ρ,z) for state ˜ ϕg. Since |C| < 1, we have |s+C| ≥
|s| − 1, which is the same as |s| ≤ 1 or s ∈ [−1,1].
Without the gauge potential?A(? r), we would always have
∆Em[˜ ϕg] = m2?d? r|˜ ϕ∗
value of s definitely would be zero. Therefore, the ap-
pearance of a non-zero valued s ∈ [−1,1] arises due to
the induced gauge potential.
g(ρ,z)|2[(m2+ 4mcosβ)/(2Mρ2)].
C =
g(ρ,z)|2[cosβ(ρ,z)/ρ2]
g(ρ,z)|2[1/ρ2]
.
≈
g(ρ,z)|2cosβ(ρ,z), i.e., the expectation value of
g(ρ,z)|2(2Mρ2)−1≥ 0. Thus the

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We now generalize our result to atoms with an arbi-
trary F and inside any axially symmetric B-fields. Anal-
ogously we can prove that the value s of Jzin the ground
state satisfies the necessary condition
|s − ηFFC| ≤ 1/2,(9)
and s ∈ [−F,F] with ηF = sign(gF). The result of s ∈
[−F,F] and the conservation of Jzis independent of the
form of the atomic interaction potential. Although its
strength g2does affect the wave function shape, thus can
influence the value of s through the factor C.
The condition Eq. (9) also allows for a rough esti-
mate of Lz. A straightforward calculation gives ?Lz? =
s − ηFF?d? r|˜ ϕ∗
field ?ˆψ(? r)?g.
and cosβ(ρ,z) as before, the value of ?Lz? becomes
approximately s − ηFFC, which lies always in the re-
gion [−1/2,1/2] according to Eq. (9). Therefore, the
value ?Lz?, or the weighted average of the winding num-
bers, is generally a small number, despite the wind-
ing number s − m itself, for the component ?ˆψ(z)
may take any integer in the region [−2F,2F]. We find
?Fz? = ηFF?d? r|˜ ϕ∗
sion of ?Lz?, a qualitative reflection that atomic hyper-
fine spin is aligned (gF > 0) or anti-aligned (gF < 0)
with respect to the local B-field.
Our result above allows for the direct creation of vortex
states in a quadrupole trapped atomic condensate. For
example, assume a spin-1 condensate in a QT plus an
“optical plug” [8] satisfies Vo(ρ,z) = Vo(ρ,−z), then we
find C = 0 and s = 0 due to the spatial reflection symme-
try about the x−y plane. The ground state components
?ˆψ(z)
with winding numbers ∓1 according to Eq. (7). In addi-
tion, the low field seeking atoms are trapped near the x-y
plane at z = 0 because |B(? r)| is an increasing function
of z. The populations for the three z-quantized states,
determined by ξB
order of magnitudes. Therefore, when a ground state
condensate in the “plugged” QT is created, its ±1 com-
ponents ?ˆψ(z)
can be directly resolved with a Stern-Gerlach B-field as
used in Ref. [9].
The qualitative example above is confirmed by the nu-
merical solution for a condensate of 5 × 106 23Na atoms
in a quadrupole plus a plug trap.
Gauss/cm and Vo= Uoexp[−ρ2/σ2] with Uo= (2π)8 ×
104Hz and σ = 7.4µm. The ground state distribution
pi =
?d? r|?ˆψ(z)
p0= 45.6%. The phase and density distributions for the
three components ?ˆψ(z)
We also can expand the ground state ?ˆψ(? r)?gin terms
of the eigenstates ξx(m) of Fx with eigenvalues m:
?ˆψ(? r)?g =?
g(ρ,z)|2cosβ(ρ,z) for the spinor mean
Neglecting the correlation between ρ−2
m (? r)?g,
g(ρ,z)|2cosβ(ρ,z) from the expres-
±1(? r)?g then automatically carry persistent currents
0(−1,? r) and ξB
±1(−1,? r), are of the same
±1(? r)?gare single quantized vortex states and
We take B′= 22
i (? r)?|2is found to be p±1 = 27.2% and
0,±1(? r)?gare shown in Fig. 1 (a, b).
m=0,±1
√N?ˆψ(x)
m (? r)?gξx(m). We then im-

0

π

0
−5
(c)
0 5

10
15
−10 0 10
−15
0
15
−5 0 5 −5 0 5
−10 0 10 −10 0 10
(a)
(b)
Fz=1
Fz=−1
0
π
π
2π
−π
0
π
Fx=1
Fx=−1
z
ρ
x
y
φ
θ(φ)
FIG. 1: (Color online). (a) The phases of the z-quantized
components ?ˆψ(z)
panel), as functions of the azimuth angle φ; (b) the density
distributions |?ˆψ(z)
(right panel) of the z-quantized components as functions of
ρ and z; (c) the integrated density distributionsR|?ˆψ(x)
(left),
R|?ˆψ(x)
of the x-quantized components as functions of x and y. The
units for x, y, ρ, and z in (b) and (c) are all arbitrary.
1? (left), ?ˆψ(z)
0? (middle), and ?ˆψ(z)
−1? (right
1?|2(left), |?ˆψ(z)
0?|2(middle), and |?ˆψ(z)
−1?|2
1 ?|2dz
0 ?|2dz (middle), and
R|?ˆψ(x)
−1?|2dz (right panel)
mediately note that ?ˆψ(x)
tex states with definite winding numbers 0 or ±1, e.g.,
?ˆψ(x)
density distribution of ?ˆψ(x)
clearly illustrates the interference pattern along the ˆ eφ
direction.As is demonstrated in Fig.
dle panel for |?ˆψ(x)
structure along the azimuthal direction, arising from the
interference of the terms proportional to ei±φ. Thus, if a
Stern-Gerlach B-field is used to separate the x-quantized
components ?ˆψ(x)
different winding numbers would be obtained.
We now extend our result for an axially symmetric
magnetic trap to the widely used IPT whose B-field pos-
sesses a different symmetry. In the region near the z-
axis,?B(? r) = B′[cos(2φ)ˆ eρ− sin(2φ)ˆ eφ+ hˆ ez], the an-
gle β(ρ,z) between the local B-field and the z-axis sat-
isfies cosβ(ρ,z) = h/?ρ2+ h2. In this case Jz is no
longer conserved due to the lack of the SO(2) symme-
try.However, we find that Dz is now conserved be-
cause it commutes with?F ·?B(? r). Therefore, we can se-
lect the low field seeking hyperfine spin state ξB(ηFF,? r)
as the eigenstate of Dz with an eigenvalue −ηFF, the
same spin state as used in [5], again defined through
m (? r)?gis a superposition of vor-
0(? r)?g= (√N/√2)[b−1(ρ,z)eiφ− b1(ρ,z)e−iφ]. The
0,±1(? r)?gas shown in Fig. 1(c)
1(c), the mid-
0(? r)?|2clearly displays the double peak
m (? r)?g, a superposition of vortices with

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a rotation ξB(ηFF,? r) = exp[−i?F · ˆ n⊥β]ξz(ηFF).
h > 0, we find the induced gauge potential becomes
?A(? r) = −ηFF(1 − cosβ(ρ,z))ˆ eϕ/ρ.
Adopting the same notation as before we denote
?ˆψ(? r)?g =
Ead[ϕ(ρ,φ,z)] = Ead[ϕ(ρ,φ+θ,z)] remains satisfied, and
the ground state takes the form ϕg= ˜ ϕg(ρ,z)exp(iuφ).
Therefore ?ˆψ(? r)?g is the eigenstate of Dzwith an eigen-
value d = u − ηFF, and its components ?ˆψ(z)
√Nb′
current with a winding number m + u − ηFF.
b′
the ground state vortex phase diagram for an F = 1 con-
densate found numerically in the z = 0 plane of an IPT.
The conservation of Dzas found by us, however, calls for
a simpler labelling of each vortex phase because only one
of three integers (m1, m0, m−1) is independent, as with
Eq. (15) of Ref. [10].
Following the same reasoning as before, we find
For
√Nϕg(? r)ξB(ηFF,? r).Interestingly we find
m (? r)?g =
m(ρ,z)ei(m+u−ηFF)φanalogously carry a persistent
Here
m= ξ†
z(m)e−iFyβξz(ηFF). This result is consistent with
??d + ηFF(1 − C)??≤ 1/2,(10)
for d ?= ηFF, and d ∈ [−F,F] or the value of Dzin the
ground state lies in the region [−F,F].
In an IPT, atoms are trapped near the z-axis where
the B-field is essentially along the z-axis direction and
ξB(ηFF,? r) is approximately the eigenstate ξz(ηFF). Lz
then is essentially always zero corresponding to a ground
state without a vortex. The angular momentum differ-
ence Dzthen becomes d = −ηFF.
Several previous proposals [11] and experiments [12]
on creating vortex states unknowingly have used the fact
that ?ˆψ(? r)?gin an IPT is an eigenstate of Dz. For exam-
ple, in the experiment of Ref. [12], spin-1 atoms initially
were prepared in the internal state ξz(−1) (Fz = −1)
with no vorticity in its spatial mode. This corresponds
to Dz= 1. To create a vortex state, the bias field along
the z-axis was adiabatically inverted from the +z to the
−z direction. In this process the atomic internal state
was changed from ξz(−1) to ξz(1). If the whole opera-
tion is adiabatic, the commutator [?F·?B(? r,t),Dz] = 0 will
be maintained. Consequently Dzis conserved. In the end
when the internal state was changed to ξz(1) (Fz = 1),
its orbital angular momentum became Lz= Dz+Fz= 2
or a double vortex spatial mode as was observed [12].
In Ref. [9] when the bias field is adiabatically switched
off, the direction of the B-field adiabatically changes to
lie in the x-y plane. Dz is again conserved during this
process (= 1).When complete, the atomic hyperfine
spin state is changed from ξz(−1) to a superposition of
all three components ξz(0,±1). As a result, different Fz
components are then associated with vortex states with
corresponding Lz = Dz+ Fz. When the three internal
states are separated, two of them are observed to contain
vortices with non-zero winding numbers.
Creating a ground state condensate with Dz?= −ηFF
is quite challenging inside an IPT. This was considered
quantitatively by Ho and Shenoy [5], who obtained an
approximate gauge potential resembling an effective trap
rotation when expanded to the first order of ρ. How-
ever, their expansion easily fails away from the z-axis.
A numerical discussion on this challenge is provided in
[10]. The more general constraint of Dz?= −ηFF, i. e.,
the necessary condition Eq. (10) of the angular momen-
tum difference Dz, was not obtained in [5, 10]. From
a direct calculation, it can be proved that if C is ap-
proximated by
?d? r|˜ ϕ∗
we have ?Lz? = d + ηFF?d? r|˜ ϕ∗
which can never be greater than 1/2 according to Eq.
(10).
g(ρ,z)|2cosβ(ρ,z) and d ?= ηFF,
g(ρ,z)|2[1 − cosβ(ρ,z)],
In summary, we have investigated the angular momen-
tum of a magnetically trapped condensate.
axially symmetric trap such as a QT, the total angular
momentum Jzalong the symmetric z-axis is found to be
conserved, while the angular momentum difference Dzis
conserved in an IPT. Both conservation laws reflect the
underlying symmetries of the traps’ magnetic fields, and
the values of Jzor Dzin the ground states are determined
by the gauge potential?A(? r). In the global ground state,
the corresponding eigenvalues of Jz and Dz are limited
to ∈ [−F,F] with the precise values directly related to
the angle between the local B-field and the z-axis.
Inside an
Our results provide significant insights into the study
of magnetically trapped condensates. The conservation
laws we discuss reveal an important observational con-
sequence: the components ?ˆψ(z)
ground state ?ˆψ(? r)?gautomatically carry persistent cur-
rents with different winding numbers. Furthermore, ac-
cording to the conditions Eqs. (9) and (10), the values
of Jz or Dz, or the topological winding numbers of the
spatial wave functions, can be controlled through the an-
gle β(ρ,z) between the local B-field and the z-axis. We
have shown explicitly for a condensate in a QT with an
optical plug, where the atomic populations for the 2F +1
components have approximately the same order of mag-
nitude. Therefore, vortex states can be present already
in a ground state condensate without requiring adiabatic
operations as in [9, 12].
m (? r)?g of a condensate
We thank Dr. D. L. Zhou, Mr. B. Sun, Profs. C.
Raman and W. Ketterle for helpful discussions.
work is supported by NSF, NASA, and NSFC.
This
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