Preliminary studies for anapole moment measurements in rubidium...

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.


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)


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


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),


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.


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


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


-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


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


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


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


−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


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.


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.


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

N. Davidson, S. Schnelle, J. A. Behr, G. D. Sprouse, and B. A. Brown for helpful

discussions.[1] J A Behr and G Gwinner. Satndard model tests with trapped radioactive atoms. J. Pys G; Nucl.

and Prt. Phys., 36:033101, 2009.

[2] P. A. Vetter, D. M. Meekhof, P. K. Majumder, S. K. Lamoreaux, and E. N. Fortson. Precise test

of electroweak theory from a new measurement of parity nonconservation in atomic thallium.

Phys. Rev. Lett., 74:2658, 1995.

[3] C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, J. L. Roberts, C. E. Tanner, and C. E.

Wieman. Measurement of parity nonconservation and an anapole moment in cesium. Science,

275:1759, 1997.

[4] J. Guena, M. Lintz, and M. A. Bouchiat. Atomic parity violation: Principles, recent results,

present motivations. Mod. Phys. Lett. A, 20:375, 2005.

[5] K. Tsigutkin, D. Dounas-Frazer, A. Family, J. E. Stalnaker, V. V. Yashchuk, and D. Budker.

Observation of a large atomic parity violation effect in ytterbium. Phys. Rev. Lett., 103:071601,

2009.

[6] V. A. Dzuba, V. V. Flambaum, A. Ya. Kraftmakher, and O. P. Sushkov. Summation of the high

orders of perturbation theory in the correlation correction to the hyperfine structure and to the

amplitudes of e1-transitions in the caesium atom. Phys. Lett. A, 142:373, 1989.

[7] S. A. Blundell, J. Sapirstein, and W. R. Johnson. High-accuracy calculation of parity

nonconservation in cesium and implications for particle physics. Phys. Rev. D, 45:1602, 1992.

[8] M. S. Safronova and W. R. Johnson. High-precision calculation of the parity-nonconserving

amplitude in francium. Phys. Rev. A, 62:022112, 2000.

[9] S G Porsev, K Beloy, and A Derevianko. Precision determination of electroweak coupling from

atomic parity violation and implications for particle physics. Phys. Rev. Lett., 102:181601, 2009.

[10] V. A. Dzuba, V. V. Flambaum, and M. S. Safronova. Breit interaction and parity nonconservation

in many-electron atoms. Phys. Rev. A, 73:022112, 2006.

[11] J S M Ginges and V V Flambaum. Violations of fundamental symmetries in atoms and tests of

unification theories of elementary particles. Phys. Rep., 397:63, 2004.


[12] E. Gomez, L. A. Orozco, and G. D. Sprouse. Spectroscopy with trapped francium: advances and

perspectives for weak interaction studies. Rep. Prog. Phys., 66:79, 2006.

[13] Marie-Anne Bouchiat and Claude Bouchiat. Parity violation in atoms. Rep. Prog. Phys., 60:1351,

1997.

[14] Y B Zel’dovich. Parity nonconservation in the 1st order in the weak-interaction constant in electron

scattering and other effects. Sov. Phys.-JETP, 9:682, 1959.

[15] V. V. Flambaum, I. B. Khriplovich, and O. P. Sushkov. Nuclear anapole moments. Phys. Lett.

B, 146:367, 1984.

[16] W. C. Haxton and C. E. Wieman. Atomic parity nonconservation and nuclear anapole moments.

Annu. Rev. Nucl. Part. Sci., 51:261, 2001.

[17] W. R. Johnson, M S Safronova, and U I Safronova. Combined effect of coherent z exchange and the

hyperfine interaction in the atomic parity-nonconserving interaction. Phys. Rev. A, 67:062106,

2003.

[18] V. F. Dmitriev and I. ßB. Khriplovich. P and todd nuclear moments. Phys. Rep., 391:243–260,

2004.

[19] I. B. Khriplovich. Parity Nonconservation in Atomic Phenomena. Gordon and Breach, New York,

1991.

[20] V V Flambaum and D W Murray. Anapole moment and nucleon weak interactions. Phys. Rev.

C, 56:1641, 1997.

[21] C S Wood, S C Bennett, J L Roberts, D Cho, and C E Wieman. Precision measurement of parity

nonconservation in cesium. Can. J. Phys., 77:7, 1999.

[22] V F Dmitriev and V B Telitsin. Many-body corrections to the nuclear anapole moment. Nucl.

Phys. A, 613:237 – 252, 1997.

[23] V F Dmitriev and V B Telitsin. Many-body corrections to the nuclear anapole moment ii. Nuclear

Physics A, 674:168, 2000.

[24] J. S. Grossman, L. A. Orozco, M. R. Pearson, J. E. Simsarian, G. D. Sprouse, and W. Z. Zhao.

Hyperfine anomaly measurements in francium isotopes and the radial distribution of neutrons.

Phys. Rev. Lett., 83:935, 1999.

[25] B A Brown (private communication), 2009.

[26] E. J. Angstmann, T. H. Dinh, and V. V. Flambaum. Parity nonconservation in atomic zeeman

transitions. Phys. Rev. A, 72:052108, 2005.

[27] A Bohr and V. F. Weisskopf. The influence of nuclear structure on the hyperfine structure of

heavy elements. Phys. Rev., 77:94, 1950.

[28] H. H. Stroke, R. J. Blin-Stoyle, and V. Jaccarino. Configuration mixing and the effects of

distributed nuclear magnetization on hyperfine structure in odd-a nuclei. Phys. Rev., 123:1326,

1961.

[29] M. F. Crawford and A. L. Schawlow. Electron-nuclear potential fields from hyperfine structure.

Phys. Rev., 76:1310, 1949.

[30] A.-M. Martensson-Pendrill. Magnetic moment distributions in tl nuclei. Phys. Rev. Lett., 74:2184,

1995.

[31] T. Kuhl, A. Dax, M. Gerlach, D. Marx, H. Winter, M. Tomaselli, T. Engel, M. Wurtz, V. M.

Shabaev, P. Seelig, R. Grieser, G. Huber, P. Merz, B. Fricke, and C. Holbrow. New access to

the magnetic moment distribution in the nucleus by laser spectroscopy of highly charged ions.

Nucl. Phys. A., A626:235, 1997.

[32] S. Buttgenbach. Hyp. Int., 20:1, 1984.

[33] J R Crespo Lopez-Urrutia, P Beiersdorfer, K Widman, B B Birkett, and A-M Martensson-Pendrill.

Nuclear magnetization distribution radii determined by hyperfine transitions in the 1s level of

h-like ions 185re74+ and 187re74+. Phys. Rev. A, 57:879, 1998.

[34] A. Perez-Galvan, , Y. Zhao, L. A. Orozco, E. Gomez, F. J. Baumer, A. D. Lange, and G. D.

Sprouse. Comparison of hyperfine anomalies in the 5S1/2 and 6S1/2 levels of 85Rb and 87Rb.

PLB, 655:114, 2007.


[35] A. Perez Galvan, Y. Shao, and L. A. Orozco. Measurement of the hyperfine splitting of the 6S1/2

level in rubidium. Phys. Rev. A, 78:012502, 2008.

[36] H. T. Duong, C Ekstrom, M Gustafsson, T T Inamura, P Juncar, P Lievens, I Lindgren, S Matsuki,

T Murayama, R Neugart, T Nilsson, T Nomura, M Pellarin, S Penselin, J Persson, J Pinard,

I Ragnarsson, O Redi, H H Stroke, J L Vialle, and the ISOLDE Collaboration. Atomic beam

magnetic resonance apparatus for systematic measurement of hyperfine structure anomalies

(bohr-weisskopf effect). Nuc. Instr. and Meth. A, 325:465, 1993.

[37] H. Kopfermann. Nuclear Moments. Academic Press, New York, 1958.

[38] C Thibault, F Touchard, S Buttgenbach, R Klapisch, M de Saint Simon, H T Duong, P Jacquinot,

P Juncar, S Liberman, P Pillet, J Pinard, J L Vialle, A Pesnelle, and G Huber. Hyperfine

structure and isotope shift of the d2 line of 76−98rb and some of their isomers. Phys. Rev. C,

23:2720, 1981.

[39] I. Angeli. A consistent set of nuclear rms charge radii: properties of the radius surface r(n,z). At.

Data Nucl. Data Tables, 87:185, 2004.

[40] E. Arimondo, M. Inguscio, and P. Violino. Experimental determinations of the hyperfine structure

in the alkali atoms. Rev. Mod. Phys., 49:31, 1977.

[41] N J Stone. Table of nuclear magnetic dipole and electric quadrupole moments. At. Data Nucl.

Data Tables, 90:75, 2005.

[42] C. Ekstrom, L. Robertsson, and A. Rosen. Nuclear and electronic g-factors of 211fr, nuclear ground

spin of 207fr and the nuclear single-particle structure in the range 207−228fr. Physica Scripta,

34:624, 1986.

[43] E. Gomez, S Aubin, L. A. Orozco, G. D. Sprouse, E. Iskrenova-Tchoukova, and M. Safronova. The

nuclear magnetic moment of 210fr, a combined theoretical and experimental approach. Phys.

Rev. Lett., page in press, 2008.

[44] J R Persson. Extraction of hyperfine anomalies without precise values of the nuclear magnetic

dipole moment. Eur. Phys. J. A, 2:3, 1998.

[45] A. Coc, C. Thibault, F. Touchard, H. T. Duong, P. Juncar, S. Liberman, J. Pinard, J. Lerme, J. L.

Vialle, S. Buttgenbach, A. C. Mueller, A. Pesnelle, and the ISOLDE Collaboration. Hyperfine

structures and isotope shifts of 207−−213,220−−228fr; possible evidence of octupolar deformation.

Phys. Lett. B, 163B:66, 1985.

[46] A. Coc, C. Thibault, F. Touchard, H. T. Duong, P. Juncar, S. Liberman, J. Pinard,

M. Carre, J. Lerme, J. L. Vialle, S. Buttgenbach, A. C. Mueller, A. Pesnelle, and the

ISOLDE Collaboration. Isotope shifts, spins and hyperfine structures of 118,146cs and of some

francium isotopes. Nuc. Phys. A, 468:1, 1987.

[47] B. A. Brown, A. Derevianko, and V. V. Flambaum. Calculations of the neutron skin and its effect

in atomic parity violation. Phys. Rev. C, 79:035501, 2009.

[48] B A Brown. Private Communication, 2009.

[49] E. N. Fortson, Y. Pang, and L. Wilets. Nuclear-structure effects in atomic parity nonconservation.

Phys. Rev. Lett., 65:2857, 1990.

[50] J Pollock, E N Fortson, and L Wilets. Atomic parity nonconservation: Electroweak parameters

and nuclear structure. Phys. Rev. C, 46:2587, 1992.

[51] B. Desplanques, J. F. Donoghue, and B. R. Holstein. Unified treatment of the parity violating

nuclear-force. Ann. Phys. (N.Y.), 124:449, 1980.

[52] E. Gomez, S Aubin, G. D. Sprouse, L. A. Orozco, and D. P. DeMille. Measurement method for the

nuclear anapole moment of laser-trapped alkali-metal atoms. Phys. Rev. A, 75:033418, 2007.

[53] S. Aubin, E. Gomez, L. A. Orozco, and G. D. Sprouse. High efficiency magneto-optical trap for

unstable isotopes. Rev. Sci. Instrum., 74:4342, 2003.

[54] C Chin, V Leiber, V Vuletic, A J Kerman, and S Chu. Measurement of an electron’s electric

dipole moment using cs atoms trapped in optical lattices. Phys. Rev. A, 63:033401, 2001.

[55] M. V. Romalis and E. N. Fortson. Zeeman frequency shifts in an optical dipole trap used to search

for an electric-dipole moment. Phys. Rev. A, 59:4547, 1999.


[56] J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nagerl, D. M. Stamper-Kurn, and

H. J. Kimble. State-insensitive cooling and trapping of single atoms in an optical cavity. Phys.

Rev. Lett., 90(13):133602, Apr 2003.

[57] S Kulin, S Aubin, S Christe, B Peker, S L Rolston, and L A Orozco. A single hollow-beam optical

trap for cold atoms. J. Opt. B: Quantum Semiclass. Opt., 3:353, 2001.

[58] N. Friedman, A. Kaplan, and N. Davidson. Dark optical traps for cold atoms,. Advances in

Atomic, Molecular, and Optical Physics, 48:99, 2002.

[59] N. Friedman, L. Khaykovich, R. Ozeri, and N. Davidson. Compression of cold atoms to very high

densities in a rotating-beam blue-detuned optical trap. Phys. Rev. A, 61:031403, 2000.

[60] S. J. M. Kuppens, K L Corwin, K W Miller, T E Chupp, and C E Wieman. Loading an optical

dipole trap. Phys. Rev. A, 62:013406, 2000.

[61] R. Ozeri, L. Khaykovich, and N. Davidson. Long spin relaxation times in a single-beam blue-

detuned optical trap. Phys. Rev. A, 59:R1750, 1999.

[62] N Friedman, A Kaplan, D Carasso, and N Davidson. Observation of chaotic and regular dynamics

in atom-optics billiards. Phys. Rev. Lett., 86:1518–1521, 2001.

[63] R. A. Williams, J. D. Pillet, S. Al-Assam, B. Fletcher, M. Shotter, and C. J. Foot. Dynamic optical

lattices: two-dimensional rotating and accordion lattices for ultracold atoms. Opt. Express,

16(21):16977–16983, 2008.

[64] S. K. Schnelle, E. D. van Ooijen, M. J. Davis, N. R. Heckenberg, and H. Rubinsztein-Dunlop.

Versatile two-dimensional potenstials for ultra-cold atoms. Optics Express, 16:1405, 2008.

Preliminary studies for anapole moment measurements in rubidium...

Documents

Transcript of Preliminary studies for anapole moment measurements in rubidium...