Geometric phase of an atom inside an adiabatic radio frequency potential
Geometric phase of an atom inside an adiabatic radio frequency potential
P. Zhang1 and L. You1, 2
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
Center for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China
(Dated: November 4, 2018)
We investigate the geometric phase of an atom inside an adiabatic radio frequency (rf) potential
created from a static magnetic field (B-field) and a time dependent rf field. The spatial motion
of the atomic center of mass is shown to give rise to a geometric phase, or Berry’s phase, to the
adiabatically evolving atomic hyperfine spin along the local B-field. This phase is found to depend
on both the static B-field along the semi-classical trajectory of the atomic center of mass and an
“effective magnetic field” of the total B-field, including the oscillating rf field. Specific calculations
are provided for several recent atom interferometry experiments and proposals utilizing adiabatic rf
potentials.
PACS numbers: 03.65.Vf, 39.20.+q, 03.75.-b, 39.25.+k
I. INTRODUCTION
Magnetic trapping is an important enabling technol-
ogy for the active research field of neutral atomic quan-
tum gases. A variety of trap potentials can be developed
using magnetic (B-) fields with different spatial distri-
butions and time variations. For instance, the widely
used quadrupole trap and the Ioffe-Pritchard trap [1] are
usually created with static B-fields, while the time aver-
aged orbiting potential (TOP) [2] and time orbiting ring
trap (TORT) [3, 4] are created using oscillating B-fields
with frequencies larger than the effective trap frequen-
cies. Atom chips [5] have brought further developments
to magnetic trap technology, as they can provide larger
B-fields and gradients at reduced power-consumptions
or electric currents using micro-fabricated coils. Today,
magnetic trapping is a versatile tool used in many labo-
ratories around the world for controlling atomic spatial
motion in regions of different scales and geometric shapes,
e.g., 3D or 2D traps, double well traps, and storage ring
traps [1, 2, 3, 4, 5, 6, 7].
Recently, magnetic traps based on adiabatic microwave
[8, 9] and adiabatic radio frequency (rf) potentials
(ARFP) [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26] have attracted considerable attention.
An ARFP is typically created with the combination of
a static B-field and an rf field. The idea for an ARFP
has been around for some time [10, 11, 12], and experi-
mental demonstrations recently have been carried out for
confining both thermal [13] and Bose condensed atoms
[14, 15, 16, 17]. Further development of improved ARFP
with atom chip technology likely will assist in practi-
cal applications of atom interferometry. For instance,
a double well potential was constructed recently using
low order multi-poles capable of atomic beam splitting
while maintaining tight spatial confinement [21]. Sev-
eral interesting recent proposals outline the construc-
tion of small storage rings with radii of the order 1µm
[18, 19, 20, 21, 22], which could become useful if imple-
mented for atom Sagnac interferometry [27] setups on
atom chips.
When a neutral atom is confined in a magnetic poten-
tial, its hyperfine spin is assumed to follow adiabatically
the spatial variation of the B-field direction during its
spatial translational motion. As a result of this adia-
batic approximation, the center of mass motion for the
atom experiences an induced gauge field [28], giving rise
to a geometric phase (or Berry’s phase) to the atomic
internal spin state [29, 30]. The effect of this geometric
phase is widely known, and is first addressed carefully
in a meaningful way for atomic quantum gases by an
explicit calculation of the resulting geometric phase in a
static or a time averaged magnetic trap in Ref. [31]. Sev-
eral important consequences are predicted to occur for a
magnetically trapped atomic condensate in a quadrupole
trap, a Ioffe-Pritchard trap [31, 32], or a TORT based
storage ring [33]. To our knowledge, this geometric phase
effect has not been investigated in any detail for an atom
inside an ARFP.
In a recent paper, we show that this geometric phase
causes an effective Aharonov-Bohm-type [34] phase shift
in a magnetic storage ring based atom interferometer [33].
In addition, our studies imply that the spatial fluctuation
of the geometric phase can lead to a reduction of the
visibility of the interference contrast. In view of this, we
decided to carry out this study as reported here for the
atomic geometric phase in an ARFP in order to shed light
on the proposed high precision atom Sagnac interference
experiment [27].
Analytical derivations for this study at some places
become rather tedious and complicated. We therefore
first will summarize our major results here for readers
who may not be interested in the intricate details. We
find that the geometric phase in an ARFP generally takes
a more complicated form in comparison to the case of a
static trap or a time averaged trap. In an ARFP, this
phase factor is found to be determined by the trajectory
of the time independent component of the trap field as
well as an “effective B-field” that depends on the total
B-field. In contrast to the earlier result found for a static
trap or a time averaged trap [31, 33], the final result turns
out to be not expressible as a functional of the trajectory
http://arxiv.org/abs/0704.0476v1
for the direction of the total B-field in the parameter
space.
This paper is organized as follows. In sec. II, we gener-
alize the semi-classical approach as outlined in Ref. [21]
for the operating principle of an ARFP to a form more
convenient for discussing the geometric phase. Section
III parallels that of sec. II by reformulating a full quan-
tum theory for discussing the geometric phase inside an
ARFP [22]. The explicit expression for the geometric
phase inside an ARFP is given in a readily adaptable
form for specifical calculations. In sec. IV, we discuss the
effect of the geometric phase in several types of ARFP
recently proposed for atomic splitters and storage rings
[18, 19, 20, 21, 22] and beam splitters [14, 21]. Finally,
concluding remarks are given in sec. V.
II. A SEMI-CLASSICAL APPROACH
In this section, we provide a semi-classical formula-
tion for calculating the atomic geometric phase inside an
ARFP. The semi-classical working principle for an ARFP
is described in Ref. [21], although only for the special
case when the atomic center of mass is assumed at a fixed
location. In order to calculate the geometric phase, our
formulation allows for the explicit consideration of atomic
center of mass motion classically. In our approach, the
geometric phase is obtained naturally, and the validity
conditions for both the adiabatic and the rotating wave
approximations are clearly shown for an ARFP.
Inside an ARFP [21], the total B-field ~B(~r, t) is the
sum of a static field component ~Bs(~r) and an oscillatory
rf field ~Bo(~r, t), which conveniently is expressed as
~Bo(~r, t) = ~B
rf (~r, t) cos(ωt) +
rf (~r, t) cos(ωt+ η). (1)
where ~r is the spatial position vector of the atom, ω is
frequency of the rf field, and η is a relative phase factor.
In this section, we will assume that the atomic spatial
motion is pre-determined, i.e., ~r(t) is given (as a slowly
varying function of time t). For weak B-fields, the system
Hamiltonian is simply the linear Zemman interaction
H(t) = gFµB ~F · ~B[~r(t), t], (2)
where gF is the corresponding Lande g-factor and µB
denotes the Bohr magneton. ~ = 1 is assumed.
For a static or a time averaged magnetic trap, the
Hamiltonian (2) varies slowly over time scales of the Lar-
mor precession of the atomic spin in the total B-field.
During the effectively slow trapped motion, the atomic
hyperfine spin is assumed to be fixed at the instanta-
neous eigenstate of the Hamiltonian (2). The geometric
phase then can be calculated straightforwardly from the
variation of the B-field direction in the parameter space
[31, 33].
In an ARFP, the situation is more complicated. Al-
though the variation of ~Bs[~r(t)] remains much slower
than the Larmor precession, the rf frequency ω usually
is assumed to be nearly resonant with the precession fre-
quency. Thus, the Hamiltonian (2) contains both fast
and slow time varying components, making the direct
calculation of the geometric phase a more involved task.
In the following, we will proceed step by step, clarifying
the various approximations adopted along the way.
To understand the working principle for an ARFP, we
first decompose the Hamiltonian H(t) (2) into the fol-
lowing form
H(t) = Hs[~r(t)] +H+[~r(t)]e
−iωt +H−[~r(t)]e
iωt, (3)
where Hs and H± are all slow varying functions of time
and are given by
Hs[~r(t)] = gFµB ~F · ~Bs[~r(t)],
H+[~r(t)] =
gFµB ~F ·
rf [~r(t)] + e
−iη ~B
rf [~r(t)]
H−[~r(t)] = H
+[~r(t)]. (4)
Hs is diagonal in the spin angular momentum ba-
sis defined along the local direction of the static B-
field ~Bs[~r(t)]. The eigenstate takes the familiar form
|mF [~r(t)]〉s, quantized along the direction of ~Bs[~r(t)],
with the eigenvalue mF |Bs[~r(t)]| for ~Bs[~r(t)] · ~F and
mF ∈ [−F, F ], in analogy with the usual case of the z-
quantized representation result of Fz |mF 〉z = mF |mF 〉z.
Next we introduce a unitary transformation
U(t) =
mF=−F
|mF 〉z s〈mF [~r(t)]|eimF κωt, (5)
with κ = sign(gF ) for the rotating wave approxima-
tion. The quantum state in the interaction picture
|Ψ(t)〉I = U(t)|Ψ(t)〉 defined by U(t) is governed by the
Schroedinger equation i∂t|Ψ(t)〉I = HI(t)|Ψ(t)〉I , with
the Hamiltonian in the interaction picture given by
HI(t) = UHU
† + i(∂tU)U
mκ∆[~r(t)]|m〉z z〈m| − i
m,n=−F
|m〉z s〈m[~r(t)]|
|n[~r(t)]〉s z〈n|ei(m−n)κωt
m=−F+1
h(+)m [~r(t)]|m〉z z〈m− 1|+ h(−)m [~r(t)]|m〉z z〈m− 1|e2iκωt + h.c.
hm[~r(t)]|m〉z z〈m|eiκωt + h.c.
, (6)
where the time dependent parameters are defined as
∆[~r(t)] = µB|gF ~B[~r(t)]| − ω,
h(±)m [~r(t)] = s〈m[~r(t)]|H±[~r(t)]|(m− 1)[~r(t)]〉s, and(7)
hm[~r(t)] = s〈m[~r(t)]|H±[~r(t)]|m[~r(t)]〉s.
The above result is obtained easily if we note that
the matrix element s〈m[~r(t)|H±(t)|m′[~r(t)〉s is non-zero
only when m − m′ = 0,±1. So far, we have always as-
sumed that |m[~r(t)]〉s is a single valued function of the
atomic position ~r. A careful examination shows that the
eigenstate |m[~r(t)]〉s cannot be determined uniquely be-
cause of the presence of the U(1) gauge freedom for se-
lecting a local phase factor exp{iφ[~r(t)]}, which conse-
quently affects the resulting expressions for h
m (t) and
s〈m[~r(t)]|d/dt|m′[~r(t)]〉s.
The rotating wave approximation neglects of the
oscillating terms proportional to eimωt (m 6= 0) in the
Hamiltonian HI (6). The error for this approximation
is estimated easily from a time dependent perturbation
calculation. The sufficient condition for its validity
requires that all factors such as
dt′hm(t
′) exp[iκωt′],
′)ξm,m−1(t
′) exp[iκ(2ω + ∆)t′], and
dt′〈m[~r(t′)]|d/dt′|n[~r(t′)]〉ξmn(t′) exp[i(m − n)κ(ω +
∆)t′] are negligible, where
ξmn(t) = exp
dt′ s〈m[~r(t′)]|
|m[~r(t′)]〉s
dt′ s〈n[~r(t′)]|
|n[~r(t′)]〉s
. (8)
Thus, the gauge independent factors h
m ξm,m−1,
〈ms|d/dt|ns〉ξmn , hm, and ∆ should all vary slowly with
time and with the modulus of their amplitudes much less
than ω.
The effective Hamiltonian in the interaction picture
under the rotating wave approximation then becomes
eff (t) = µBgF
~F · ~Beff [~r(t)]
|m〉z s〈m[~r(t)]|
|m[~r(t)]〉s z〈m|, (9)
where the first term resembles a coupling between the
atomic spin and an “effective B-field” ~Beff(~r), whose
components in real space are given by
Beffx (~r) = Re
2 s〈m(~r)|H±(~r)|(m− 1)(~r)〉s
(F +m)(F −m− 1)
Beffy (~r) = −Im
2 s〈m(~r)|H±(~r)|(m− 1)(~r)〉s
(F +m)(F −m− 1)
, and
Beffz (~r) =
~Bs(~r)
µB|gF |
. (10)
Clearly, the x- and y-components of the effective field
~Beff(~r) depend on the explicit form of the eigenstate
|m(~r)〉s. In fact, it easily can be seen that different
choices of the local phase factor for the |m(~r)〉s actually
lead to different values of ~Beff(~r) related to each other
through ~r-dependent rotations in the x-y plane.
In practice, the eigenstate |n(~r)〉s and the effective field
~B eff can sometimes be constructed more simply, as in
Ref. [21]. For any spatial position ~r, we first choose a
rotation R[m̂(~r), χ(~r)] along the axis m̂(~r) with an angle
χ(~r) that satisfies R[~n(~r), χ(~r)] ~Bs(~r) = | ~Bs(~r)|êz. It is
then easy to show that the eigenstate |n[~r(t)]〉s can be
chosen as
|n(~r)〉s = exp
i ~F · m̂(~r)χ(~r)
|n〉z. (11)
Unfortunately, the choice for R is not unique in a given
static field ~Bs(~r), an analogous result to the U(1) gauge
freedom for the the egienstate |n(~r)〉s. Corresponding
to the choice (11) given above for |n(~r)〉s, the unitary
transformation U defined in (5) would become
U(t) = exp(−iFzωt) · exp
−i ~F · m̂(~r)χ(~r)
, (12)
and the transverse components of the “effective B-field”
given by B effx,y(~r) = Bx,y(~r)/2 [21] with
~B(~r) = R[m̂(~r), χ(~r)] ~B
rf (~r)
+R[êz,−κη]R[m̂(~r), χ(~r)] ~B(b)rf (~r). (13)
In earlier discussions of an ARFP [21, 22], the atomic
internal state is assumed uniformly to remain adiabati-
cally in a certain eigenstate of the first term of H
eff (t).
To fully appreciate this adiabatic approximation and to
calculate the geometric phase, we expand |Ψ(t)〉I into
the instantaneous eigenstate basis |n[~r(t)]〉eff quantized
along the direction of the effective B-field ~Beff accord-
ing to |Ψ(t)〉I =
n Cn(t)|n[~r(t)]〉eff . The first term of
eff (t) is simply the effective Zemman interaction be-
tween the atomic hyperfine spin and the effective B-field.
The corresponding Schroedinger equation for the Hamil-
tonian H
eff (t) of (9) then becomes
Cn(t) = [ǫ
I (t) + νnn(t)]Cn(t) +
m 6=n
νnm(t)Cm(t),
I (t) = nµBgF | ~B
eff(t)|,
νpq(t) = −i
eff〈p[~r(t)]|l〉z
s〈l[~r(t)]|
|l[~r(t)]〉s z〈l|q[~r(t)]〉eff
−i eff〈p[~r(t)]|
|q[~r(t)]〉eff . (14)
Under the adiabatic approximation, the atomic inter-
nal state remains in a given eigenstate |n[~r(t)]〉eff with
transitions to states |m[~r(t)]〉eff (m 6= n) being negligi-
bly small. Thus, the transition probability, as estimated
from the first order perturbation theory,
dt′νnm(t
dt′′[ǫ
(t′′)+νnn(t
′′)−ǫm
(t′′)−νmm(t
should be much less than one. As before, we find the
sufficient condition for the validity of the adiabatic ap-
proximation is given by
|νmn(t)|
|ǫmI (t′)− ǫ
≪ 1, (15)
provided that νmn(t) exp[i
[νnn(t
′′) − νmm(t′′)]dt′],
which is independent of the local phase factor for
|n[~r(t)]〉eff and |n[~r(t)]〉s, remains a slowly varying func-
tion of time.
A straight forward calculation from the effective
Hamiltonian (9) then gives the general expression for the
geometric phase in an ARFP
γn(t) =
νnn(t
′)dt′
| eff〈n[~r(t′)|l〉z|
s〈l[~r(t′)]|
|l[~r(t′)]〉sdt′
+γ(I)n (t), (16)
and γ
n (t) = −i
dt′ eff〈n[~r(t′)|d/dt′|n[~r(t′)〉eff . During
the adiabatic motion in a given internal state, the time
evolution of the coefficient Cn(t) takes the form
Cn(t) = Cn(0)e
(t′)dt′e−iγn(t). (17)
Equation (16) is the central result of this work. The
geometric phase of an atom inside an ARFP is shown to
contain two parts. The second part, γ
n (t), is clearly
due to the interaction term µBgF ~F · ~Beff in H(I)eff (9),
with its value determined by the trajectory of the direc-
tion for the “effective B-field” ~Beff . The first part arises
from the second term of H
eff (9). It is determined by
the trajectories of both the static field ~Bs and the ef-
fective B-field ~Beff . The expression for γn in an ARFP
is complicated because the internal quantum state in an
ARFP is assumed to be adiabatically kept in an eigen-
state of µBgF ~F · ~Beff , rather than an eigenstate of the
total interaction Hamiltonian H
eff (t).
In section IV, we will perform explicit calculations for
several examples of ARFP proposed for various applica-
tions: e.g., as atomic storage rings or atomic beam split-
ters. Most often we find that only the first part of Eq.
(16) contributes a non-zero value to the geometric phase.
Before proceeding to the next section for a quantal
treatment of the geometric phase, we find the time evo-
lution of the atomic spin state in the Schroedinger picture
|Ψ(t)〉 =
Cl(0) z〈m|l[~r(t)]〉eff ×
(t′)dt′e−iγl(t)e−imωt|m[~r(t)]〉s, (18)
obtained directly from |Ψ(t)〉 = U †(t)|Ψ(t)〉I after the
applications of the rotating wave and adiabatic approxi-
mations. When the atom is prepared initially in a specific
adiabatic state |n[~r(t)]〉eff of the interaction picture, we
arrive at the simple case of Cl(0) = δln.
III. A QUANTUM MECHANICAL
TREATMENT
In the previous section, we provided the result for the
geometric phase γn(t) in an ARFP based on a semi-
classical approach, where the atomic center of mass mo-
tion is described classically. A clear physical picture ex-
ists in this case for the appearance of the geometric phase
in a certain parameter space. The validity conditions for
the rotating wave and the adiabatic approximations as
obtained above are all formulated in terms of gauge inde-
pendent forms. However, if the influence of the geomet-
ric phase on the atomic spatial motion is to be included,
e.g., as in the Aharonov-Bohm-type, phase shift, inter-
ference arrangement in an atomic Sagnac interferometer
discussed earlier [33], we would need an improved de-
scription where both the atomic spin and its center of
mass motion are treated quantum mechanically.
In a full quantum treatment of the atomic motion, the
quantum state of an atom can be expressed as |Φ(t)〉 =
φl(~r, t)|l〉z, where φl(~r, t) is the atomic spatial wave
function for the internal state |l〉z of Fz. The state |Φ(t)〉
then satisfies the Schroedinger equation governed by the
Hamiltonian
+ gFµB ~F · ~B(~r, t), (19)
with ~P being the kinetic momentum and M the atomic
mass.
The rotating wave and adiabatic approximations can
be introduced now by defining the interaction picture
with the unitary transformation
U(t) =
|m〉z eff〈m(~r)|
|n〉z s〈n(~r)|einκωt
. (20)
The state in the interaction picture |Φ(t)〉I = U(t)|Φ(t)〉
now is governed by the Schroedinger equation with the
Hamiltonian Heff = UHU†. Under the rotating wave
and adiabatic approximations, we neglect transitions be-
tween states |m〉z and |n〉z (m 6= n) as well as the rapidly
oscillating terms. We then obtain
Heff ≈
|n〉z z〈n|Heff |n〉z z〈n|
ad |n〉z z〈n|, (21)
where the adiabatic Hamiltonian H
ad for the n-th adia-
batic branch is defined as
~P − ~An
I (~r), (22)
with the effective gauge potential
~An(~r) = −i
| eff〈n(~r)|l〉z|2 s〈l(~r)|∇|l(~r)〉s
−i eff〈n(~r)|∇|n(~r)〉eff . (23)
In this form, it is well known that the geometric phase
γn can be expressed as the integral of the gauge poten-
tial ~An along the spatial trajectory for the atomic center
of mass in an ARFP, i.e., one would expect generally
that γn =
~An · d~r. Similar to the result of the semi-
classical approach, the gauge potential ~An(~r) can be ex-
pressed as the sum of two parts. The first part in Eq.
(23) is the weighted sum of the atomic gauge potential
−i s〈l(~r)|∇|l(~r)〉s from the static field ~Bs, while the sec-
ond term is the atomic gauge potential from the “effective
B-field” ~Beff .
A full quantum treatment for atomic motion in an
ARFP has been attempted earlier [22]. In fact, many of
our formulations are identical to the results of Ref. [22].
For instance, it is easy to show that the unitary trans-
formations US, UR, and UF in [22] are related directly
to ours as U
S = U . The only difference concerns
the gauge potential ~An that was neglected in Ref. [22].
Thus, they did not give the expression for the gauge po-
tential, and the result for the geometric phase was not
obtained either [22]. Our study shows that the neglect of
the adiabatic gauge potential potentially can give rise to
a final result, dependent on the choice of the local phase
factors for the internal eigenstate.
IV. GEOMETRIC PHASES IN ARFP BASED
APPLICATIONS
In the above two sections, we obtain the expression for
the atomic geometric phase in an ARFP. This section is
devoted to the calculations of the geometric phases for
several proposed applications of ARFP, such as storage
rings or beam splitters for neutral atoms [18, 19, 20, 21,
Before presenting our results for the more specific
cases, we provide some general discussions of the geo-
metric phases in several ARFP based storage rings. As
was pointed out earlier, the geometric phase γn is given
by the line integral of the gauge potential ~An along the
trajectory for the atomic center of mass motion. For a
closed path in the storage ring at a fixed ρ = ρc and
z = zc, this can be further reduced to
γn = q
A(φ)n (ρ, φ, z)ρdφ, (24)
where the integer q is the winding number of the path
and A
n is the component of ~An along the azimuthal
direction êφ of the familiar cylindrical coordinate system
(ρ, φ, z). Without loss of generality, we take q = 1 in this
paper. For the storage rings proposed in Refs. [18, 19,
20, 21, 22], the gauge potentials A
n (ρ, φ, z) are actually
independent of the angle φ. Therefore, the geometric
phase is simply given by
γ(c)n = 2πρcA
n (ρc, zc), (25)
given out in explicit forms for different storage ring
schemes [18, 19, 20, 21, 22].
In reality, because of thermal motion or when the
atomic transverse motional state is considered, the center
of mass for an atom can deviate from (ρc, zc) even for a
closed trajectory. This uncertainty in the exact shape of
the closed trajectory gives rise to a fluctuating geometric
phase and is usually difficult to study. Assuming a simple
closed path at fixed ρ and z, we have found previously
that the subsequently fluctuations could decrease the vis-
ibility of the interference pattern [33]. Quantum mechan-
ically, such destructive interference can be explained as
resulting from entanglement between the freedoms for φ
and (ρ, z) because of the dependence of the gauge po-
tential Aφn on ρ and z. Therefore, it is important to
investigate this dependence near the trap center.
For simplicity, our discussions below will focus on the
closed loops where ρ and z are φ-independent constants.
In this case, the geometric phase can be expressed as
γn(ρ, z) = 2πρA
n (ρ, z). We will show numerically the
distributions for γn(ρ, z) obtained this way near the cen-
tral region of (ρc, zc). If needed, a more rigorous ap-
proach can be developed to investigate the fluctuations
of the resulting geometric phase from the gauge potential
n (ρ, z).
A. The storage ring proposals of Refs. [18, 19, 20]
This subsection is devoted to a detailed calculation of
the geometric phases for the ARFP storage ring propos-
als of Refs. [18, 19, 20]. We will derive the analyti-
cal expressions for the azimuthal component A
n of the
gauge potential that arises in both cases from cylindri-
cally symmetric static B-field and rf fields. Because of
the cylindrical symmetry, the angle βs(ρ, z) between the
local static B-field and the z-axis is required to be ana-
lytical in the region near the storage ring. Therefore, the
eigenstate |n(~r)〉s of ~F · ~Bs can be chosen as
|n(~r)〉s = exp{−i[~F · êφβs(ρ, z) + nφ]}|n〉z . (26)
Consequently, ~B eff(~r) is also cylindrically symmetric,
which leads to the eigenstate |n(~r)〉eff of ~F · ~B eff as
|n(~r)〉eff = exp{−i[~F · n̂ eff⊥ (~r)βeff(ρ, z) + nφ]}|n〉z, (27)
with the unit vector n̂ eff⊥ (~r) in the x-y plane orthogonal to
~B eff(~r) and β eff(ρ, z) denoting the angle between ~B
eff(~r)
and the z-axis. We note that the unit vector field n̂ eff⊥ (~r)
also possesses cylindrical symmetry, i.e., remains invari-
ant under rotation around the z-axis. The expressions of
(26) and (27) allow us to obtain the simple expression of
the gauge potential
A(φ)n (ρ, z) = −
cosβeff(ρ, z) cosβs(ρ, z), (28)
after straightforward calculations.
In the scheme of Ref. [18], the static B-field is a “ring-
shaped quadrupole field” that vanishes along a circle of
a radius ρ0 in the x-y plane. Near ρ = ρ0, the B-field is
given approximately by
~Bs(~r) = B
′(ρ− ρ0)êρ −B′zêz, (29)
like a quadrupole field, while the rf-field takes a compli-
cated form
~Bo(~r, t) =
cos(ωt) +
cos(ωt+ ϕ)
sin(ωt) +
sin(ωt+ ϕ)
êz, (30)
with constants a and b independent of ~r.
From the expression of (26) for the eigenstate |n(~r)〉s,
the “effective B-field” ~B eff becomes
~B eff(~r) = B′[
(ρ− ρ0)2 + z2 − r0]êz
cos(θ + ϕ) +
cos θ
sin(θ + ϕ)− a√
sin θ
êφ, (31)
FIG. 1: (Color online) A cross-sectional view for the storage
ring of Ref. [18]. The static field is zero in the ring at the
fixed radius ρ0. The addition of rf-fields creates an ARFP
centered at a ring through (ρc, zc). The distance from the
trap center to the ring with radius ρ0 in the plane z = 0 is r0.
where r0 and θ are given by
|µBgFB′|
cos θ(ρ, z) =
ρ− ρ0
(ρ− ρ0)2 + z2
sin θ(ρ, z) =
(ρ− ρ0)2 + z2
. (32)
In an ARFP, as discussed here, the trap center
at (ρc, zc) is determined by minimizing both the z-
component and the transverse component of ~B eff . With-
out loss of generality, we will assume a, b > 0. Then,
(ρc, zc) is found to satisfy
θ(ρc, zc) = −ϕ/2,
(ρc − ρ0)2 + z2c = r0, (33)
i.e., the trap center lies on the surface of the “resonance
toroid” at ρ = ρ0 with a radius r0 as shown in Fig. 1.
The relative angle of the trap center with respect to the
center of the toroid cross-section is given by −ϕ/2. On
this “resonance toroid,” the rf-field is resonant with the
static field, i.e., B effz vanishes. As a result, the “effective
B-field” lies again in the x-y plane on the “resonance
toroid,” which gives cosβeff(ρc, zc) = 0 and leads to the
result A
n (ρc, zc) = γn = 0 as shown in the trap center
for the storage ring considered before in Ref. [18].
From the expression (29) of the static field and the def-
inition of the angle θ(ρ, z), we find a simple relationship
βs(ρ, z) = π/2 + θ(ρ, z), with which the gauge potential
n (ρ, φ) in (28) can be further simplified as
A(φ)n (ρ, z) =
cosβeff(ρ, z) sin θ(ρ, z)
cosβeff(ρ, z) sin θ(ρc, zc), (34)
near the trap center. Thus, the spatial fluctuation of
the gauge potential A
n (ρ, z) in the region around the
trap center is closely related to the angle θ(ρc, zc) of the
trap center, or the parameter ϕ of the oscillating field
~Bo. When ϕ = 0, the atom is trapped in the region
with θ ≈ 0 or π, where the fluctuation of A(φ)n (ρ, z) is
suppressed significantly due to the small value of sin θ.
On the other hand, if the angle ϕ is set to π with the
trap center located in the region with θ ≈ ±π/2, the
fluctuation of the gauge potential becomes amplified.
In Fig. 2, we illustrate numerical results for the dis-
tribution of the geometric phase γ1(ρ, z) = 2πρA
1 (ρ, z)
in the region near the trap center at ϕ = 0, π/2, π. We
see clearly decreased fluctuations of γ1 when the absolute
value of sin θ(ρc, zc) = − sin(ϕ/2) is decreased.
Next we turn to the storage ring of Ref. [19] con-
structed from a quadrupole static B-field ~Bs(~r) =
B′(x, y,−2z) and an ~r-independent rf field ~Bo =
Brf cos(ωt)êz along the z direction. The resulting ARFP
provides a 2D ring shaped trap in the x-y plane. In ad-
dition, a 1D optical potential along the z direction is em-
ployed to confine atoms in the transverse plane at z = 0
[19]. The “effective B-field” takes the form
~B eff(~r) = B′(ρ− ρ0)êz −
Brf êρ, (35)
in the plane at z = 0, with ρ0 = ω/|µBgFB′|. Because
the strength of ~B eff is near minimum at the ring ρ = ρ0,
the trap center for this storage ring is located at ρc =
ρ0 and zc = 0. At the trap center, the “effective B-
field” is along the direction of êρ. Thus, according to Eq.
(28), the geometric phase γ
n at the trap center again
vanishes.
In Fig. 3, we show the distribution of the geometric
phase γ1 in the region near the trap center for Brf =
0.05|B′|ρ0 and Brf = 0.15|B′|ρ0. We see that the fluctu-
ation is relatively small when the strength of the rf-field
is large. This can be explained by Eq. (28), which shows
that A
n is proportional to cosβeff and can be approx-
imated as 2B effz /Brf near the trap center. When Brf is
large, the gauge potential becomes a relatively slow vary-
ing function of ρ and z. In this case, the presence of a
1D optical potential allows for the possibility of tuning
the trap center position to a nonzero value of z, with the
storage ring remaining in the x-y plane. Then cosβs is
assumed to a nonzero value, leading to increased fluctu-
ations for the geometric phase.
Finally, we discuss the geometric phase in the “time
averaged” ARFP storage ring proposed in Ref. [20]. Un-
like previously considered ARFP based storage rings, the
time dependence now exists in both the “static B-field”
and the frequency of the rf field given by
~Bs(~r, t) = B
′ρêρ − 2B′zêz +Bm sin(ωmt)êz,
−0.1 
0    
0.1  −0.1 
0    
0.1  
−0.1 
0    
0.1  −0.1 
0    
0.1  
 (ρ−ρc)/ρ0 
−0.1 
0    
0.1  −0.1 
0    
0.1  
FIG. 2: (Color online) The distribution of the geometric phase
γ1 near the trap center (ρc, zc) of the storage ring proposed
in Ref. [18] at (a) ϕ = 0, (b) ϕ = π/2, and (c) ϕ = π, clearly
displaying the sin(ϕ/2) dependence.
~Bo(t) = Brf sin[ω(t)t]êz,
ω(t) = ω0
1 + (Bm/B′ρ0)2 sin
2(ωmt) . (36)
The frequency ωm is assumed to be much smaller than ω0
but much larger than the trap frequency. The radius ρ0
is now defined as ρ0 = ω0/|µBgFB′|, and the “effective
−0.05
0.05 −0.05
0.05 
−0.05
0    
0.05 −0.05
0.05 
FIG. 3: (Color online) The geometric phase γ1 for the storage
ring of Ref. [19] with (a) Brf = 0.15B
′ρ0 and (b) Brf =
0.05B′ρ0.
B-field” takes the form
~B eff(~r, t) = ∆(~r, t)êz −
|2 ~Bs(~r, t)|
Brf êφ . (37)
The operating principle for the time averaged storage
ring of Ref. [20] is similar to the well-known TOP [2]
and TORT traps [3, 4]. The effective trap potential ex-
perienced by the atom is proportional to the time aver-
aged value of the “effective B-field”
∫ 2π/ωm
| ~B eff(~r, t)|dt.
When Brf and Bm are much smaller than B
′ρ0, the
center of the storage ring is located approximately at
ρc = ρ0, zc = 0. Using the earlier result [33], we find that
in the time averaged storage ring, the effective gauge po-
tential Ã
n (ρ, z) is reduced simply to the time averaged
instantaneous gauge potential
Ã(φ)n (ρ, z) =
∫ 2π/ωm
A(φ)n (ρ, z, t)dt, (38)
with A
n (ρ, z, t) given in (28). The geometric phase then
is given approximately by γn(ρ, z) = 2πÃ
n (ρ, z). In this
case, we find that the geometric phase always vanishes at
−0.05 
−0.05
−0.2 
0 (ρ−ρ0)/ρ0
−0.05
−0.05
 z/ρ0 
FIG. 4: (Color online) The geometric phase γ1 for the storage
ring of Ref. [20] at (a) Brf = 0.3B
′ρ0 and (b) Brf = 0.1B
Bm = 0.05B
the trap center (ρc, zc). Figure 4 illustrates the distribu-
tion of the geometric phase in the region near the trap
center for two different values of the rf-field amplitude
Brf . Similar to the storage ring of Ref. [19], the fluctua-
tion of the geometric phase is suppressed in this case for
large Brf .
B. The storage ring proposals of Refs. [21, 22]
Next we consider the ARFP based storage ring pro-
posed in Refs. [21, 22]. In this case, the static B-field
is that of a Ioffe-Pritchard trap on an atom chip. In the
Cartesian coordinate (x, y, z), it takes the form
~Bs = B
′xêx −B′yêy +B′Lêz, (39)
where B′ is the B-field gradient and the bias field along
the z-direction is denoted as B′L. The amplitudes ~B
and ~B
rf (z) of the rf field are
rf = [Brf(z)/
2]êx and
rf = [Brf(z)/
2]êy with
Brf(z) = B
rf +B
′′z2. (40)
In the schemes of Ref. [21, 22] considered earlier, the
phase η of the rf field is assumed to be κπ/2. The x-
and y-components of the “effective B-field” ~B eff(~r) then
become
B effx (~r) =
Brf(z)
(1 + cosβs(ρ, z)),
B effy (~r) = 0, (41)
according to Eq. (10). Then the strength of the “effective
B-field” ~B eff has its minimum along a circle with a non-
zero radius ρc, provided a positive detuning ∆ exists at
the origin (0, 0, 0) [21, 22]. The “effective B-field” ~B eff is
easily shown to lie in the x-z plane along the trap bottom
mapped out by the atomic center of mass motion. This
gives rise to a vanishing γ
F . With a proper choice for
the local phase of |n(~r)〉eff , the gauge potential A(φ)n takes
the form
A(φ)n (ρ, z) =
cosβeff(ρ, z) (1− cosβs(ρ, z)) . (42)
Figure 5 displays the geometric phase along a closed
path for a spin-1 atom as a function of ρc for the ARFP
storage ring proposed in Refs. [21, 22]. The parameter λ
is defined as
∆[~r = 0]
|gFµBB(0)rf |
. (43)
To assure the validity of the rotating wave approx-
imation, we find that the maximal values of ∆[~r =
0]/(|gF |µB) and B(0)rf /
2 must be restricted to the re-
gion of λ ∈ [0, 0.15].
As shown in Fig. 1, the geometric phase remains much
smaller than 2π in this situation. This fact can be appre-
ciated easily if we look at the distribution of the “effec-
tive B-field” ~B eff . According to Eq. (41), the component
B effx has a nonzero minimal value Brf/(2
2), while |Beffz |
can become arbitrarily small, although not necessarily
zero in general. Therefore, at the trap center where | ~B eff |
is a minimum, the value of cosβeff = B
z /| ~B eff | can be-
come very small, leading to small geometric phases. Yet,
despite the relatively small geometric phase found here,
our result remains important because it could represent
a systematic error if not properly included in a Sagnac
interference experiment.
In Fig. 6, we show the spatial distribution of the ge-
ometric phase γ1 around the trap center with λ = 1/3
and λ = 3. The fluctuation is found to be relatively
small when λ is small or when the rf-field amplitude Brf
is large.
Although not discussed in Refs. [21, 22], a ring shaped
trap also can be realized if we take η = −κπ/2. The
“effective B-field” ~B eff still lies in the x-y plane
~B effx (~r) = −
Brf(z)
cos(2φ)(1 − cosβs(ρ, z)),
~B effy (~r) =
Brf(z)
sin(2φ)(1 − cosβs(ρ, z)), (44)
0  0.1 0.2 0.3 0.4 0.5 0.6
−0.1 
−0.08
−0.06
−0.04
−0.02
0    
λ=1/3 
FIG. 5: (Color online) The geometric phase γ1 is plotted
against the radius ρc for the ARFP storage ring of Ref.
[21, 22] with η = κπ/2 at λ = 3, λ = 1, and λ = 1/3.
To assure the validity of the rotating wave approximation, in
the solid lines, the maximal value of ∆[~r = 0]/|gFµBB′ρ0|
or Brf/(
2B′ρ0) are restricted to be smaller than 0.15. The
extending dashed line is beyond the rotating wave approxi-
mation for λ = 1/3 and Brf/(
2B′ρ0) ∈ [0.15, 0.3].
clearly giving rise to a non-zero solid angle with respect to
a closed path along the storage ring. Therefore, the term
(eff)
F is non-zero in this case. We choose the eigenstates
|n(~r)〉s and |n(~r)〉eff as
|n(~r)〉s = exp[−i ~F · n̂s(~r)βs(ρ, z)]|n〉z,
|n(~r)〉eff = exp[−i ~F · n̂ eff⊥ (~r)βeff(ρ, z)]|n〉z . (45)
with the unit vector n̂s⊥(~r) in the x-y plane orthogonal
to ~Bs(~r). In this case, the gauge potential A
n can be
expressed as
A(φ)n (ρ, z) =
cosβeff(ρ, z)[1 + cosβs(ρ, z)]. (46)
In Figure 7, we show the fluctuation of the geometric
phase γ1 for a closed path with a new parameter
λ′ = 6
∆[~r = 0]
|gFµBB(0)rf |
, (47)
equal to 3 and 1/3. The fluctuation for γ1 is found to
be much larger than the case of η = κπ/2, which can be
explained by the transverse components B effx,y of the “ef-
fective B-field.” Because cosβs is always close to unity.
In the case of η = −κπ/2, B effx,y can take only small pos-
itive values. Therefore, at the minimum of the ARFP
trap ρ = ρ0 of | ~B eff |, both B effz and B effx,y have to be
close to zero. In this case the value for cosβeff becomes
a rapidly changing function of ρ in the region near ρc.
Our above calculations have obtained analytical ex-
pressions of the geometric phases in an ARFP based stor-
age ring for η = ±κπ/2. We have further investigated the
−0.05
0.05 
−0.05
0.05 
−0.06
−0.02
−0.05
0.05 
−0.05
0.05 
FIG. 6: (Color online) The spatial distribution of the geo-
metric phase γ1 for the storage ring of Refs. [21, 22] at (a)
λ = 1/3 and (b) λ = 3. η = κπ/2. B
rf = 0.08B
′L and
B′′ = 10−12B′/L are assumed.
fluctuations of the geometric phase for the two cases of
η = ±κπ/2. It seems one benefits from implementing
a Sagnac interferometer in the discussed ARFP storage
ring with η = κπ/2 and operating at a relatively large λ.
Before proceeding onto the concluding section, we will
discuss the geometric phase in an ARFP based beam
splitter created via a double potential [14, 21]. In such an
implementation, the static field ~Bs is created from a Ioffe-
Pritchard trap, while the oscillating rf field components
are ~B
rf = Brf [z]êx and
rf = 0. By spatially tuning
the amplitude of Brf from zero to a significant value,
in the x-y plane, an ARFP can be tuned from a single
well centered near the origin to a double well with two
minimal points at the point with nonzero radius ρ0 and
φ = 0, π. Therefore, a Y-shaped atom beam splitter
can be accomplished when the Brf [z] initially is increased
along the z-axis to a large value, and then decreased to
zero. In such an arrangement, the atom beam moving
along the z direction can be separated into two beams
that move along the z-axis at φ = 0, π for a while, and
then can be recombined again into a single beam.
−0.05
0.05 
−0.05
0.05 
−0.05
0.05 
−0.05
0    
0.05 
FIG. 7: (Color online) The spatial distribution of the geo-
metric phase γ1 for the storage ring of Refs. [21, 22] at (a)
λ′ = 3 and (b)λ′ = 1/3. η = −κπ/2. B(0)rf = 0.08B
′L and
B′′ = 10−12B′/L are assumed.
In the atom interferometer considered above, both the
static field ~Bs and the “effective B-field” ~B
eff are limited
to the x-z plane. Therefore, for motion along the closed
path of the trap bottom, the solid angle enclosed by the
trajectory of ~B eff is zero. Thus, the geometric phase in
(16) can be expressed as
γn(t) = −i
| eff〈n(~r)|l〉z|2 s〈l(~r)|∇|l(~r)〉s · ~vdt′. (48)
We can show that the product
eff〈n(~r)|l〉z|2 s〈l(~r)|∇|l(~r)〉s is a function of ρc and
is independent of z. Thus, the geometric phase can be
expressed as an integral of this function with respect to
ρc, from zero to a large value and then back to zero.
Therefore, the value of the geometric phase would be
zero in the end.
V. CONCLUSION
In this study, we develop theoretical formalisms for
the calculation of the atomic geometric phase inside an
ARFP. We show that, due to the complexity of the
ARFP, the geometric phase depends on the spatial vari-
ation of both the static field and an “effective B-field”
~B eff . We provide general expressions for the geometric
phase and the corresponding adiabatic gauge potential in
Eqs. (16) and (23), respectively.
To shed light on actual applications of the atomic ge-
ometric phase, we investigate the distribution of atomic
geometric phases for several proposed or ongoing exper-
iments with ARFP based storage rings and atom beam
splitters. We prove rigorously that the geometric phase in
the center of the storage rings proposed in Refs. [18, 19]
is always zero. In addition, we find that in the storage
ring of Ref. [18], the spatial fluctuation of the geometric
phase sensitively depends on the position of the trap cen-
ter on the “resonance toroid.” In the proposals of Refs.
[19, 20, 21, 22], the fluctuation for the geometric phase
becomes significantly suppressed when the amplitude Brf
of the rf-field is large. In the proposals of [21, 22], the
fluctuations of the geometric phase also is suppressed if
the angle η is set to be κ2π. In the beam splitter real-
ized with the double well potential ARFP [14, 21], the
geometric phase is shown to be zero.
Our work helps to clarify the working principle of trap-
ping neutral atoms in an ARFP and the validity condi-
tions for the various approximations involved. We hope
our results will shine new light on the proposed inertial
sensing experiments based on trapped atoms in ARFP.
Acknowledgments
We thank Dr. T. Uzer and Dr. B. Sun for helpful dis-
cussions. This work is supported by NASA, NSF, CNSF,
and the 863 and 973 programs of the MOST of China.
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