preliminary studies for anapole moment measurements in rubidium...
TRANSCRIPT
Preliminary studies for anapole moment
measurements in rubidium and francium with
atomic spectroscopy.
D. Sheng and L. A. Orozco
Joint Quantum Institute, Dept. of Physics, University of Maryland and National
Institute of Standards and Technology, College Park, MD 20742-4111, USA
E-mail: [email protected]
E. Gomez
Instituto de Fısica, Universidad Autonoma de San Luis Potosı, San Luis
Potosı,78290, Mexico
Abstract. The calculated anapole moments in a chain of Fr isotopes are highly
correlated with indirect measurements of the hyperfine anomaly on the same isotopes.
The same correlation reproduces on a chain of Rb isotopes where hyperfine anomalies
directly measured. The anapole moment predictions in Rb include a change in sign
for the two bosonic stable isotopes 85Rb and 87Rb. The correlation shows that the
nuclear magnetization plays an important role on the single-particle model of the
anapole model. Preparations for the anapole measurement in Fr show the possibility
of performing a similar measurement in a chain of Rb. The sensitivity analysis to
magnetic fields shows new regions of operation at much lower fields than found before.
TAnalysis of other systematic effects show the effect will be eighty times smaller as
measured by the amplitude of an equivalent electric field, but the stability of the
isotopes and the current performance of the dipole trap in the apparatus signal towards
performing the measurement.
Preliminary study for anapole moment measurements in rubidium and francium 2
1. Introduction
The constraints obtained from Atomic Parity Non-Conservation (PNC) on the structure
of weak interaction and its manifestation both at high energy and in hadronic
environments are unique [1]. The information it provides is complementary to that
obtained with high energy experiments. The last twenty years have been seen steady
progress in the experimental advances [2, 3, 4, 5] together with theoretical calculations
[6, 7, 8] having a precision better 1% [9, 10, 11].
As we prepare for a new generation of PNC experiments with radioactive isotopes,
[12], we continue to study the measurement and find advances both in the understanding
of the important parameters and on the specific approaches that we are taking to
achieve the ultimate goal. This paper presents progress on both fronts. We explore the
possibility of measurements in Rb and Fr and study our results on precision spectroscopy
that extract the hyperfine anomaly together with the calculated anapole moments using
current single particle nuclear models. We find the possible constraints in the nuclear
weak interaction parameters that such measurements can bring and present the current
performance of our atomic trap.
2. The anapole moment in atoms
Parity nonconservation in atoms appears through two types of weak interaction: Nuclear
spin independent and nuclear spin dependent [13]. Nuclear spin dependent PNC
occurs in three ways [14, 15, 11]: An electron interacts weakly with a single valence
nucleon (nucleon axial-vector current AnVe), the nuclear chiral current created by
weak interactions between nucleons (anapole moment), and the combined action of the
hyperfine interaction and the spin-independent Z0 exchange interaction from nucleon
vector currents (VnAe). [16, 17, 18].
Assuming an infinitely heavy nucleon without radiative corrections, the
Hamiltonian is [19]:
H =G√2(κ1iγ5 − κnsd,iσn · α)δ(r), (1)
where G = 10−5 m−2p is the Fermi constant, mp is the proton mass, γ5 and α are Dirac
matrices, σn
are Pauli matrices, and κ1i and κnsd,i (nuclear spin dependent) with i = p, n
for a proton or a neutron are constants of the interaction. At tree level κnsd,i = κ2i, and
in the standard model these constants are given by
κ1p =1
2(1 − 4 sin2 θW ), κ1n = −1
2,
κ2p = −κ2n = κ2 = −1
2(1 − 4 sin2 θW )η, (2)
with sin2 θW ∼ 0.23 the Weinberg angle and η = 1.25. κ1i (κ2i) represents the coupling
between nucleon and electron currents when the electron (nucleon) is the axial vector.
Preliminary study for anapole moment measurements in rubidium and francium 3
The contribution from Eq. 1 for all the nucleons in an atom add. Work in
the approximation of shell model with a single valence nucleon of unpaired spin, the
first term of Eq. 1 gives a contribution that is independent of the nuclear spin and
proportional to the weak charge QW . For the standard model values, the weak charge
is almost equal to the number of neutrons in the isotope -N. The second term is
nuclear spin dependent and due to the pairing of nucleons its contribution has smaller
dependence on Z [20]:
HnsdPNC =
G√2
KI · α
I(I + 1)κnsdδ(r), (3)
where K = (I + 1/2)(−1)I+1/2−l, l is the nucleon orbital angular momentum, I is the
nuclear spin. The terms proportional to the anomalous magnetic moment of the nucleons
and the electrons have been neglected.
The interaction constant is then:
κnsd = κa −K − 1/2
Kκ2 +
I + 1
KκQW
, (4)
where κ2 ∼ −0.05 from Eq. 2 is the tree level approximation, and we have two
corrections, the effective constant of the anapole moment κa and κQWthat is generated
by the nuclear spin independent part of the electron nucleon interaction together with
the hyperfine interaction.
Flambaum and Murray show that[20]:
κa =9
10gαµ
mpr0A2/3,
κQW= −1
3QW
αµN
mpr0AA2/3, (5)
where α is the fine structure constant, µ and µN are the magnetic moment of the external
nucleon and of the nucleus respectively in nuclear magnetons, r0 = 1.2 fm is the nucleon
radius, A = Z + N, and g gives the strength of the weak nucleon-nucleon potential
with gp ∼ 4 for protons and 0.2 < gn < 1 for neutrons [19]. For a heavy nucleus like
francium, the anapole moment contribution is the dominant one (κa,p/κQW= 14 and
κa,p/κ2,n = 9). Rubidium is heavy enough that the anapole moment contribution still
dominates (κa,p/κQW= 20 and κa,p/κ2,n = 5).
Vetter et al. [2] set an upper limit on the anapole moment of Thallium and Wood
et al.[3, 21] measured with an uncertainty of 15% the anapole moment of 133Cs by
extracting the dependence of atomic PNC on the hyperfine levels involved. Following
Khriplovich [19] the anapole moment is:
a = −π∫
d3rr2J(r), (6)
with J the electromagnetic current density.
Flambaum, Khriplovich and Sushkov [15] by including weak interactions between
nucleons in their calculation of the nuclear current density, estimate the anapole moment
from Eq. 6 of a single valence nucleon to be
a =1
e
G√2
Kj
j(j + 1)κa = Canj, (7)
Preliminary study for anapole moment measurements in rubidium and francium 4
where j is the nucleon angular momentum. The calculation is based on the shell model
for the nucleus, under the assumption of homogeneous nuclear density and a core with
zero angular momentum leaving the valence nucleon carrying all the angular momentum.
Dimitrev and Telitsin [22, 23] have looked into many body effects in anapole moments
and find strong compensations among many-body contributios, making the resulting
values of the anapole moment close to the initial single-particle ones.
2.1. Calculation of anapole moments in a chain of francium and rubidium
We have used Eqs. 5 and 7 to estimate the anapole moments of sofive francium isotopes
on the neutron deficient side with approximately one minute lifetimes. These are the
isotopes that we have studied extensively in Ref. [24]. In even-neutron isotopes, the
unpaired valence proton generates the anapole moment, whereas in the odd-neutron
isotopes both the unpaired valence proton and neutron participate. Fr has an unpaired
h9/2 proton for all the isotopes and an f5/2 neutron for the odd-neutron isotopes. In
the latter case, one must add vectorially the contributions from both the proton and
the neutron to obtain the anapole moment as stated in Eq. 8. Figure 1 shows also
the effective constant of the anapole moment for different isotopes of rubidium (open
squares) for isotopes on both sides of the stability region. The even-odd neutron number
alternation is no longer evident due to changes in the orbitals for the valence nucleons.
In particular the value of κa has a different sign for the two stable isotopes of rubidium
(85Rb and 87Rb). The nucleon orbitals used for rubidium are πf5/2 for isotopes 84 and
85, πp3/2 for 86-88, νg9/2 for 84 and 86 and νf5/2 for 88 [25]. The lighter of the isotopes,85Rb, has the spin angular momentum of the valence proton anti-aligned with its orbital
angular momentum while 87Rb adds them both together. The two neutron holes deform
the nucleus very slightly and change the order of the orbitals that are almost degenerate.
The sign change comes from the addition of two extra neutrons that change the valence
proton orbital from πf5/2 to πp3/2. Our result for 87Rb is different from Ref. [26] as we
take into account the nuclear structure; however, for 85Rb we reproduce their results
since the nuclear structure of this isotope follows well the filling of the shells.
a =Can
p jp · I + Cann jn · I
I2I, (8)
with Cani ji the anapole moment for a single valence nucleon i in a shell model description
as given by Eq. 7, with jp=9/2 and jn=5/2.) using gn=1.
3. Hyperfine anomaly
Precise measurements of hyperfine structure can probe the nuclear magnetization
distribution. The hyperfine splitting is due mainly to the coupling between the
magnetization of the nucleus with the magnetic field created by the electrons. The
point-like coupling has corrections usually called hyperfine anomalies, that arise from the
finite magnetic and charge distributions of the nucleus. Bohr and Weisskopf [27, 28] first
Preliminary study for anapole moment measurements in rubidium and francium 5
208 209 210 211 212
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
84 85 86 87 88
A (Fr)
κa
A (Rb)
Figure 1. Anapole moment effective constant for different isotopes of francium
(triangles) and rubidium (open squares). The lines are only to guide the eye.
discussed the finite magnetization effect in the anomaly. The modified charge potential
that the valence electron sees as it gets closer to the nucleus, the Breit-Crawford-
Schawlow [29] effect, is the other source for a hyperfine anomaly. Recent experimental
and theoretical advances allow now more broadly based and detailed investigations of
the Bohr Weisskopf effect [30, 31, 32, 33, 24, 34, 35]. Comparison of adjacent isotopes
allows extraction of the contribution to the nuclear magnetization distribution of the
last neutron [36, 24].
The hyperfine shift for a level with electronic angular momentum J = 1/2 is given
by EHF = A(F (F +1)−I(I+1)−J(J+1))/2 where F is the total angular momentum,
A is the magnetic dipole constant, and I is the nuclear spin. Derivations of A assume
that the nucleus is a point particle with magnetic moment µN and charge Z. The value
of the magnetic dipole constant has to be modified to include the effect of the charge and
magnetic distribution on the electronic wave function. We can write A for an extended
(ext) nucleus as a function of the point value (point) as [37]
Aext =16π
3
µ0
4πhgIµNµB|ψ(0)|2
× fR(1 + ǫBCS)(1 + ǫBW )
Aext = Apoint(1 + ǫBCS)(1 + ǫBW ), (9)
where ψ(0) is the electronic wave function evaluated at the center of the nucleus, µB
is the Bohr magneton, µN is the nuclear magneton, gI is the nuclear g-factor, and fR
represents the relativistic enhancement. The last two terms of this expression modify
the hyperfine interaction to account for an extended nucleus. The first of them (ǫBCS),
Preliminary study for anapole moment measurements in rubidium and francium 6
the Breit-Crawford-Schawlow (BCS) correction, accounts for the fact that the valence
electron sees a modified potential as it gets closer to the nucleus. The second one (ǫBW ),
the Bohr Weiskopf (BW) effect, focuses on the magnetic distribution of the nucleus.
The fractional difference in the mean charge radius between the two isotopes of
interest (85Rb and 87Rb) is less than 10−3 and justifies neglecting the BCS correction
[38, 39]. 87Rb has a closed-neutron shell that makes the nuclear charge distribution
insensitive to the addition or subtraction of neutrons [38].
The magnetic moments and the A coefficients of the Rb nuclei are well known
[40, 38, 36, 41]. We can use a ratio to extract the Bohr-Weisskopf effect difference with
reference to the stable isotope 85Rb for the i isotope:
g85Ai
giA85≃ 1 +i δ85. (10)
We recently showed that it is possible to extract an anomaly not only on the ground
state but also on an electronic excited state using this procedure [34, 35].
When taking Fr, there are only two measurements of the magnetic moment of Fr
[42, 43]. The first is for 211Fr and the second for 210Fr. They give consistent numbers,
but their uncertainty does not allow their use in the direct measurement of hyperfine
anomalies. We have used for that a different approach [44]: Precision measurements
of the hyperfine structure in atomic states with different radial distributions can give
information on the hyperfine anomaly and be sensitive to the nuclear magnetization
distribution. The hyperfine anomaly differences of the ground state in francium were
measured by our group [24] by comparing the ratio of the hyperfine splittings of the
7P1/2 (which has a very small anomaly) and the 7S1/2 (which has the large anomaly)
done by Coc et al. [45, 46].
Part of the preparation of the PNC experiment is the understanding of the nuclear
structure of the different isotopes. Recent theoretical work by Brown et al. [47] has
demonstrated that the neutron skin contributions to the spin independent part of atomic
PNC, the part that directly relates to extensions of the Standard Model.
Figure 2 shows our results on the hyperfine anomaly difference ∆ with Fr [24]
normalized to 210Fr. The results for Rb are referenced to 85Rb based on our work
[34, 35] together with results from the work of Thibault et al. [38] and Stone [41].
The nuclear distribution of both Fr and Rb for the isotopes in Fig. 2 is well
understood [48]. The calculations that B. A. Brown provided us for the Fr work [24]
predict the anomaly with comparable accuracy to our measurements. The numbers for
Rb should be similar.
3.1. Correlation between anapole moment and hyperfine anomaly
We calculate the linear correlation r between the normalized hyperfine anomaly ǫ and the
anapole moment κa for the two chains of isotopes. We find that r(Fr) = 0.992 ± 0.038
and r(Rb) = 0.986 ± 0.132. These numbers are strong evidence that for the nuclear
anapole moment, reflecting details on the structure of the nucleus.
Preliminary study for anapole moment measurements in rubidium and francium 7
0.992
0.996
1.000
1.004
84 85 86 87 88
A (Rb)
0.988
208 209 210 211 212
A (Fr)
iδ8
5,2
09
Figure 2. Hyperfine anomalies for different isotopes of francium (triangles) and
rubidium (open squares). See text for references to the numbers.
The existence of this strong correlation is most likely a reflection on the angular
momentum algebra used for the calculation of the anapole moment that reflects itself on
the way the nuclear magnetization influences the hyperfine anomaly. Both parameters
are proportional to the nuclear magnetic moment of the isotope, but there is no simple
linear correlation between the magnetic moment and either the anomaly of the anapole
moment as calculated. The correlation is an open question. It brings again the nuclear
structure to the front of the discussion in a similar way as Fortson and coworkers have
done in Ref. [49, 50]; but this time the effects are on the spin dependent part of atomic
PNC and its influence on the measurements in a chain of isotopes. It would be great to
have a study, such as the one recently published by Brown, Flambaum, and Derevianko
[47] about the nuclear structure effects in the extraction of standard model limits from
the spin independent part of atomic PNC.
4. Constraints to weak meson-nucleon interaction constants from anapole
measurements
The anapole moment constant (κa) depends on the strength of the weak nucleon-nucleus
potential, characterized by gp for a proton and gn for a neutron. Equation 18 of
Ref. [20] gives a relation between the weak nucleon-nucleus constants (gp and gn) that
appear in the anapole Eq. 5 and the meson-nucleon parity nonconserving interaction
Preliminary study for anapole moment measurements in rubidium and francium 8
constants formulated by Desplanques, Donoghue, and Holstein (DDH) [51]. Evaluating
the relations with some of the DDH “best” values for the weak meson-nucleon constants
we arrive at the following expressions
Y = 3.61(−X + 1.77gp + 0.26), (11)
Y = 2.5(X + 2.65gn − 0.29), (12)
with X = (fπ − 0.12h1ρ − 0.18h1
ω) × 107 and Y = −(h0ρ + 0.7h0
ω) × 107 combinations of
weak meson-nucleon constants. Figure 3 shows the expected constraints on the weak
meson-nucleon constants from an anapole moment measurement using Eqs. 11 and 12.
The figure is analogous to Fig. 8 in Ref. [16] that shows the constraints obtained from
different experiments. This figure complements the one that appears in the review of
Behr and Gwinner [1] as it adds the rubidium numbers to the constraints obtained from
the anapole moment measurement in Cs considering only the experimental uncertainty
[3, 21] and the calculations for Fr.
The main contribution to the anapole moment in bosonic alkali atoms (even number
of neutrons) comes from the valence proton, and the experiment provides a measurement
of gp. Using equation 11 we obtain constraints in the direction of the Cs result of Fig.
3. The fractional experimental uncertainty in gp is the same as the measurement one.
The actual final uncertainty for gp is considerably higher and depends on the accuracy
of the theoretical calculations [20]. A measurement in any other bosonic alkali atom
generates constraints in the same direction as the Cs result, with a confidence band
size proportional to the measurement uncertainty. As an example we show in Fig. 3
the expected constraint from a 3% measurement in a bosonic atom (rubidium, francium
or other). Here is where having a chain of isotopes becomes attractive. The fermionic
alkali atoms have contributions from the valence neutron. The constraints obtained
(Eq. 12) are not colinear to those of Cs and generate a crossing region as shown in
Fig. 3. The size of the error band depends on the relative contributions of the valence
proton and neutron to the anapole moment which depends on the atom. Figure 3 shows
the expected constraints for a 3% measurement in francium and rubidium. All the
calculations assume gp = 6 and gn = −0.9 [20].
5. Experimental requirements
This section presents the experimental requirements to measure the anapole moment in
chains of Rb and Fr isotopes. Most of the details are in Ref [52], but here we focus on
the differences for Rb and new ways that we have to perform the measurement. The
measurement relies on driving an anapole allowed electric dipole (E1) transition between
hyperfine levels of the ground state. The transition probability is very small and is
enhanced by interfering it with an allowed Raman (or magnetic dipole (M1)) transition.
The excitation is coherent and allows for long interaction times. The signal to noise ratio
is linear with time (up to the coherence time). We report on the progress towards an
anapole measurement, the possibility of making the measurement in rubidium and the
Preliminary study for anapole moment measurements in rubidium and francium 9
-5
0
5
10
15
20
25
30
-2 3 8 13
-(hρ0+
0.7
hω
0) x1
07
(fπ-0.12hρ -0.18h’ω ) x107X=
Y=
‘
Figure 3. Constraints to weak meson-nucleon interaction constants from an anapole
moment measurement. The isotopes with even number of neutrons give constraints
aligned with the Cs result, while those with odd number of neutrons give constraints
in an different direction. Cs result (dotted line), francium 3% measurement (solid line)
and rubidium 3% measurement (dashed line).
expected constraints from the measurement.
5.1. Sources of atoms
We take the numbers for the production of unstable isotopes from what is available at
TRIUMF in the Isotope Separator and Accelerator (ISAC), where we are an approved
experiment. A 500 MeV proton beam collides with an uranium carbide target to produce
francium as fission fragments. A voltage up to 60 kV extracts the atoms as ions from the
target. The beam goes through a mass separator and into the trapping area. The yield
is up to 2 × 1011 s−1 for Rb, and 2 × 106 s−1 for Fr, but it is expected to reach 108-109
for Fr once the accelerator runs at full capacity. A high efficiency magneto-otpical trap
working in batch from a neutralizer, as described in Ref. [53], can capture and cool the
atoms.
5.2. Measurement
The anapole moment constant scales with the atomic mass number as A2/3 as seen in
Eq. 5. The anapole induced electric dipole (E1) transition scales faster than that due
to additional enhancement factors and it is about 83 times bigger in francium than in
rubidium. The magnetic dipole (M1) transition between hyperfine levels on the other
Preliminary study for anapole moment measurements in rubidium and francium 10
Table 1. Operating point for the magnetic field (B0), resonant frequency (νm) and
Zeeman sublevels (m1, m2) for the transition.
Atom Isotope Spin m1 m2 B0 (G) νm (Mhz)
Rb 84 2 -0.5 0.5 0.2 3084
85 5/2 0 -1 186.1 2992
86 2 0.5 -0.5 0.3 3947
87 3/2 0 -1 654.2 6602
88 2 0.5 -0.5 0.03 1191
Fr 208 7 0.5 -0.5 3.3 49880
210 6 0.5 -0.5 3.4 46768
212 5 0.5 -0.5 4.5 49853
hand has about the same value for both species. It is an allowed transition and does
not depend on the nuclear size through the weak force. The M1 transition gives the
main systematic error contribution, and the important quantity is the ratio of the two
transition amplitudes |AE1/AM1| ∼ 1 × 10−9 for francium and |AE1/AM1| ∼ 1 × 10−11
for rubidium. In order to do a measurement in rubidium it becomes more important to
understand and suppress the M1 contribution.
The experiment starts by pumping all the atoms to a particular level |F1, m1〉.The atoms interact with two fields for a fixed time. One field corresponds to a Raman
transition and the other to an anapole induced electric dipole (E1) transition. Both
fields are resonant with the hyperfine transition |F1, m1〉 → |F2, m2〉 in the presence of a
static magnetic field. This triad of vectors defines the coordinate axis. The interference
of both transitions gives a signal linear in the anapole moment [52]. The field driving
the E1 transition is produced inside a microwave Fabry-Perot cavity. The signal we
measure is the population in |F2, m2〉 at the end of the interaction.
The specific magnetic field and transition levels reduce the sensitivity to
fluctuations. There is an operating point where the transition frequency varies
quadratically with the magnetic field for a |∆m| = 1 transition. The operating field
grows with the hyperfine separation and it is bigger in francium than in rubidium. Table
1 shows the magnetic field for different isotopes of francium and rubidium. For fermions
one can use a m1 = 0.5 → m2 = −0.5 transition that has a very small magnetic field.
The electronic contribution to the linear Zeeman effect cancels for the two levels at
low magnetic fields, but the nuclear magnetic contribution remains since the two states
belong to different hyperfine levels. The magnetic field for fermionic francium is smaller
than previosly reported (∼ 2000 G) [52]. this will require circular polarization on
the Fabry Perot? That is much harder than with linear
The dimensions of the microwave cavity scale with the wavelength of the transition
(λm ∼ 6 mm for Fr and λm ∼ 6 cm for Rb). The mirror separation of the Fabry-Perot
cavity should be at least 20 cm for Rb. The anapole signal remains unchanged between
Fr and Rb by putting more power in the microwave cavity to compensate for the loss
Preliminary study for anapole moment measurements in rubidium and francium 11
in the nuclear size.
We hold the atoms in place for the measurement using a far off resonance trap
(FORT). The same FORT can be used for both rubidium and francium. The dipole
trap causes an ac Stark shift that is different for the two hyperfine levels. The differential
shift causes a change of the resonant frequency and eventually leads to decoherence. The
Stark shift in a FORT is in the same direction for both hyperfine levels but of different
size due to the different detuning. The differential shift is proportional to the hyperfine
splitting, and it is reduced by an order of magnitude in rubidum.
The measurement in both species depends on the effectiveness of the suppression
mechanisms [52]. The first suppression mechanism works by having the atoms in the
magnetic field node (electric field antinode). The reduction depends on the magnitude
of the field at the edges of the atomic cloud. Since the wavelength increases by an
order of magnitude in rubidium, the suppression improves by the same amount. The
second suppression mechanism works by forcing the M1 transition to have the wrong
polarization for the levels considered. It remains unchanged in rubidium. The atoms
oscillate around the magnetic field node for the third suppression mechanism. The
suppression is proportional to the M1 field, and since it got reduced because of the
better positioning in the node, we gain an order of magnitude (assuming no increase
in the driving field power). In summary, the suppression mechanisms work better in
rubidium than in francium by two orders of magnitude because of better positioning to
the magnetic field node. The improvement compensates the two orders of magnitude
loss in the ratio of the transitions (|AE1/AM1|).We compare the requirements in rubidium to those stablished on Table III of Ref.
[52] for francium. We assume an increase in the microwave power to keep the same
E1 transition amplitude. The magnetic field stability is still about 10−5 but since now
the magnetic field is smaller this means that the field has to be controled to about 10
µG. The requirements on all the systematic effects that depend on the M1 transition
produced by the microwave cavity increase by two orders of magnitude. This is because
by increasing the microwave field we increase the E1 and M1 transition by the same
amount. The systematic effects introduced by the dipole trap or Raman beams remain
the same.
6. Optical Dipole trap
We report next on the experimental progress towards the implementation of the optical
dipole trap. The dipole trap is a convenient way to capture the atoms almost without
perturbing them. Still there is some photon scattering and differential ac Stark shifts
introduced by the dipole trap that contribute as sources of signal dephasing. The dipole
trap design aims to decrease these effects e.g. Refs. [54, 55].
We choose a far-off resonance trap (FORT) to reduce the photon scattering rate.
The ac Stark shift depends on the position in the trap and the atomic state. The shift
changes with time as the atoms move in the trap. The problem can be solved in a red
Preliminary study for anapole moment measurements in rubidium and francium 12
detuned trap by adding an extra laser at a “magic wavelength” to compensate for the
shift [21, 56]. We choose a blue detuned trap instead to reduce the shift where the atoms
are confined on the dark region of the trap.
There are many ways to generate blue detuned dipole trap [57, 58]. We choose
the rotating dipole trap because we can control the shape and size dynamically. A laser
rotating faster than the motion of the atoms creates a time averaged potential equivalent
to a hollow beam potential [59]. The laser beam propagating in the z direction goes
through two acousto-optical modulators (AOMs) placed back-to-back in the x and y
directions respectively. We use the output beam that corresponds to the first diffraction
order in both directions, the (1,1) mode. We scan the modulation frequency of both
AOMs with two phase locked function generators (Stanford Research SR345) to generate
different hollow beam shapes.
The general expression of the time averaged potential U0 in the radial direction for
linearly polarized light and a detuning larger than the hyperfine structure splitting, but
smaller than the fine structure splitting is [60]:
U0 =hγI024IS
[
1
δ1/2
+2
δ3/2
]
(13)
where γ is the natural linewidth, IS is the saturation intensity defined as IS =
2π2hcγ/(3λ3), and I0 is the peak intensity in terms of the laser power P , such that
I0 = 2P/(πw20) with w0 the waist of the laser beam. The detunings δ1/2 and δ3/2 in
units of γ
Tightly focusing the laser at the position of the atoms confines them along the
beam axis. Figure 4 shows the shape of the potential both along the radial and axial
directions for a circular shaped trap.
We study the lifetime and spin relaxation rate of such a circular trap with a beam of
400 mW and blue detuned 2.5 nm from the 87Rb D2 line. The beam rotating frequency
is 50 kHz, which is much faster than the oscillation frequency of the trap (1 kHz). The
trap has a transverse diameter of 150 µm and an axial diameter of ?? mm. We measure
the atom number in the dipole trap after some hold time by shinning a 100 µs long
resonant pulse and imaging the the fluorescence into a photomultiplier tube (Fig. 5a).
We image the fluorescence from a region of radius of 2 mm. We see a fast decay during
the first 100 ms folowed by an slow decay. We attribute the fast decay to the diffusion
of the atoms out of the imaging region which is smaller than the axial confinement of
the dipole trap and bigger than the MOT size. The fast decay is followed by a slower
decay (2.5s lifetime). The slow decay is shorter than the MOT lifetime (30 s) so we
attribute it to the continous difussion of the atoms out of the imaging region (?? is
this the idea? have we made a model to explain it? or is there another explanation,
like the loss of F=2 atoms). We follow the method of Ref. [61] to measure the spin
relaxation rate. We load the atoms from a magneto-optical trap (MOT) to the dipole
trap, turn off the magnetic field and MOT beams and pump the atoms to the F = 1
ground state. We get the relaxation rate due to the interaction of the atoms with the
dipole trap laser by comparing the populations of the atoms in both hyperfine levels as
Preliminary study for anapole moment measurements in rubidium and francium 13
−10
0
10
20
−100−50
050
100150
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
axial dire
ction (m
m)
radial direction ( µm)
no
rma
lize
d p
ote
nti
al
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4. (Color online) Time averaged potential along the axial and radial directions
for a cigar shaped trap. The aspect ratio is not to scale in the figure.
a function of time. Figure 5b shows the fraction of atoms in the F = 2 ground state as
a function of time. A fit to the data gives a spin relaxation time of 840 ms, similar to
previous measurements [61]. (what about the scape from the imaging region, is this not
a problem?).
We reduce the diffusion of the atoms in the axial direction by adding a one-
dimensional (1D) blue denuned standing wave where its detuning may not be the same
as the rotating trap. The combination of tight radial confinement from the rotating trap
and confinement from the standing wave gives a higher density dipole trap (Fig. 6). It
also opens the possibility to study the motion of atoms in 2D billiards with arbitrary
transverse shape [62]. We study the transition from two independent trapping regions
to a single one as we gradually overlap them. The motion can be described by classical
mechanics and will be published elsewhere.
The symmetry of the trap is an important point for the anapole moment
measurement. As we scan the beam, there will be diffraction power changes on the
AOMs. This has been pointed out in the study with Bose-Einstein condensates where
the uniformity is required to avoid parametric excitation [63]. We feedforward on the RF
power to reduce the diffraction variations [64]. We managed to decrease the diffraction
fluctuations by more than an order of magnitude using this method (Fig. 7).
7. Conclusions
There is a strong correlation between the current predictions for the anapole moment
based on a single particle model and the measured hyperfine anomalies in chains of
Preliminary study for anapole moment measurements in rubidium and francium 14
0 2 4 6 8
0.01
0.1
1
Raw data
Exponential fit curve
Velocity escaping modelN
orm
aliz
ed
ato
m n
um
be
r
Time (s)
0 1000 2000 3000 4000 5000 6000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
F=
3 s
tate
fra
ctio
n
Spin relaxation time t=840ms
a
b
Time (ms)
Figure 5. a) Lifetime measurement of the atoms in the dipole trap. b) Fraction of
the atoms in the F = 2 state to measure the spin relaxation time of the dipole trap.
Preliminary study for anapole moment measurements in rubidium and francium 15
g
ba
Figure 6. Fluorescence image of the atoms 35 ms after turning off the magnetic field
and MOT beams. a) rotating dipole trap. b) 1D blue detuned standing wave. c)
rotating dipole trap and 1D blue detuned standing wave.
.
0 5 10 15 20 25 30 35 40 45 50
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Am
plit
ud
e (
a.
u.)
Time (µs)
With feedforward
without feedforward
Figure 7. Intensity profile of the laser within a few oscilation periods with (solid line)
and without (dashed line) feedforward on the RF power.
Preliminary study for anapole moment measurements in rubidium and francium 16
isotopes in Fr and Rb. The measurement of the anapole moment in a chain of isotopes
can constraint the values of weak meson-nucleon interaction constants. The constraints
obtained from measurements in even and odd neutron isotopes in alkali atoms are not
colinear and produce a crossing region in the weak DDH meson-nucleon constants space.
The measurement of the anapole moment is possible in any of the heavy alkali atoms
(rubidium, cesium or francium) but it becomes increasingly difficult with decreasing
atomic number. The case of rubidium is particularly interesting since it has two stable
isotopes available. The signal for the two isotopes has the opposite sign which can be
useful for the study of systematic effects. We found a particular transition for neutron
odd isotopes where the experiment can be done at a much smaller static magnetic field
that does not require the instalation of very big coils. We implemented and characterized
a rotating blue detuned dipole trap necessary to hold the atoms for the duration of the
measurement. The trap shows a spin relaxation time of 840 ms. The dipole trap will
be used in future measurements in both francium and rubidium.
Acknowledgments
Work supported by NSF. E. G. acknowledges support from CONACYT. We thank
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