Measurement scheme and analysis for weak ground state hyperfine transition moments through two-pathway coherent control

1.6 MB·

by  J. Choi

APS/123-QED
Measurement scheme and analysis for weak ground state hyperfine transition
moments through two-pathway coherent control
J. Choi1 and D. S. Elliott1,2,3
1School of Electrical and Computer Engineering,
2Department of Physics and Astronomy, and 3Purdue Quantum Center
Purdue University, West Lafayette, IN 47907
(Dated: January 13, 2016)
We report our detailed analysis of a table-top system for the measurement of the weak-force-
6 inducedelectricdipolemomentofagroundstatehyperfinetransitioncarriedoutinanatomicbeam
1 geometry. Wedescribeanexperimentalconfigurationofconductorsforapplicationoforthogonalr.f.
0 andstaticelectricfields,withcavityenhancementofther.f.fieldamplitude,thatallowsconfinement
2 ofther.f.fieldtoaregioninwhichthestaticfieldsareuniformandwell-characterized. Wecarryout
detailednumericalsimulationsofthefieldmodes,andanalyzetheexpectedmagnitudeofstatistical
n
and systematic limits to the measurement of this transition amplitude in atomic cesium. The
a
J combination of an atomic beam with this configuration leads to strong suppression of magnetic
dipole contributions to the atomic signal. The application of this technique to the measurement of
1
extremely weak transition amplitudes in other atomic systems, especially alkali metals, seems very
1
feasible.

]
h PACSnumbers:
p
-
m I. INTRODUCTION as evidenced by the many efforts underway world-
wideinavarietyofsystems. Laboratoryeffortshave
o
t sought, or are currently underway, to determine the
a
Laboratory measurements of very weak atomic anapole moment of other nuclei, including Tl [24],
.
s transitions that violate the usual parity selection Yb [10, 25–27], Fr [28–30], Ba+ [31–33], Ra+ [34–
c
i rules are a means of determining the weak force 37], and Yb+ [38], and several molecular systems
ys at low collision energies [1–5]. The component of as well [8, 12]. Differences between EPNC on various
this electric dipole transition moment E that is hyperfinelinesforthesesystemscouldrevealthenu-
h PNC
p induced by the weak-force coupling between nucle- clearanapolemomentofthesesystems. Comparison
[ ons has become of great interest in recent years [6– between different isotopes of the same species could

12]. These Nuclear Spin Dependent (NSD) contri- removethedependenceofthedeterminationonpre-
1
butions to E are expected to result from the cise atomic theory, subject to the ability to correct
v PNC
0 nuclear anapole moment of the nucleus, with addi- forvariationsinthenuclearstructureamongtheiso-
9 tional smaller contributions from the weak neutral topes [39–43].
6 axial-vectornucleonvectorelectron(A ,V )current, Measurements performed on a hyperfine tran-
n e
2 and the combined effect of the hyperfine interaction sition between components of an atomic ground
0 and the (V ,A ) current [9, 13–16]. To date, the state present an attractive alternative to the above
. n e
1 only non-zero determination of NSD contributions schemes for determining the NSD contributions to
0 to E in any element was based upon the differ- E . This moment contains only the NSD contri-
PNC PNC
6 encebetweenmeasurementsofE /βinatomicce- bution, simplifying the measurement, and in many
PNC
1
sium[17],whereβ isthevectorpolarizabilityforthe cases, the value of E on ground state transitions
: PNC
v transition, on two different hyperfine components of ispredictedtobelargerthantheweakamplitudebe-
i the6s→7stransition; theF =3→F(cid:48) =4andthe tween different electronic states [11]. Of particular
X
F = 4 → F(cid:48) = 3 lines. E /β on these lines dif- interest is a large program on francium [7], one goal
PNC
r
a fered by ∼5% of their average value. This NSD fac- of which is to measure EPNC on transitions between
torwasmuchlargerthanwasexpected,andtheoret- hyperfine levels of the ground state of this unsta-
ical efforts [3, 6, 9, 13, 18] to understand this result ble heavy element at TRIUMF. To carry out these
have not been successful. Meson exchange coupling measurements, development of techniques for cool-
constants of the so-called DDH model [19] derived ing and trapping these species in a magneto-optical
fromthisresultdonotagreewellwithresultsderived trap (MOT) and carrying out the measurements in
from measurements of the asymmetry in the high- this restricted space is necessary.
energy scattering of light nuclei [3, 20–23]. While The measurement in atomic cesium that we have
the applicability of the DDH model to such a large under development in our laboratory, which we de-
atom is questionable, there is none-the-less strong scribe in this work, has several features in common
interest in understanding the NSD of large nuclei, with those of the francium effort. As a ground state
2
transition, atomic coherences are long-lived, and we PPTL structure, and describe the field modes sup-
exploit the interference between the direct transi- ported by it. Finally, we analyze the magnitudes
tion driven by a radio frequency (r.f.) field and the of the dominant residual contributions to the mea-
Raman process driven by a two-frequency cw laser surement of E , and consider the effects of the
PNC
field, in a derivation of the two-pathway coherent distribution of atomic velocities in the beam.
control techniques that we have developed for sim-
ilar measurements [44, 45]. Atomic cesium offers
several benefits over the francium system that are II. THE COHERENT CONTROL SCHEME
derived from an atomic beam geometry: that is, a
greater atomic density, the capability of sequential
We employ the two-pathway coherent control
preparation, interaction, and detection, and a less
schemeforsensitivemeasurementofweakmoments.
restrictive experimental environment. Furthermore,
This technique is based on the interference between
thebeamgeometryallowustospatiallyseparatethe
various optical interactions driven by two or more
interaction regions for the different coherent fields,
coherently-related fields. We developed and em-
and to highly suppress the magnetic dipole contri-
ployed this technique on measurements of the mag-
butions to the atomic signal, a primary challenge in
neticdipoletransitionmomentM onthe6s2S →
ground-state measurements of weak signals. In this 1/2
7s2S transitioninatomiccesium[44,45]. TheFr
work, we discuss how the two-pathway interference 1/2
collaborationbasestheirmeasurementsonthistech-
method can be used to determine the ratio of the
niquealso[7]. Inthissection,wedescribetheprinci-
PNCamplitudetotheStarkvectorpolarizabilityβ.
plesbehindthistechnique,withparticularattention
While our primary interest is in atomic cesium, the
paid to a transition between hyperfine components
techniqueisgenerallyapplicableinanyofthestable
of a ground state system, in which both states are
alkali metal species.
longlived. Weshowhowthismeasurementcanyield
We describe in detail the measurement require- a determination of E /β, independent of the pro-
PNC
ments, and the capability of our technique. The op- fileoramplitudeofther.f.fieldthatdrivesthetran-
timal arrangement uses r.f. and static electric fields sition.
that are oriented in perpendicular directions, and We consider a sinusoidal wave of amplitude εrf
the r.f. field should be confined to a space within and frequency ωrf, incident upon a two-level atom
whichthestaticfieldisuniform. Theserequirements
with hyperfine components ψ and ψ of the ground
i f
can be satisfied by a parallel plate transmission line
state, of energy E and E , respectively. We choose
i f
(PPTL)configurationtowhichcylindricalreflectors
the field to be continuous wave, but spatially vary-
(to form an r.f. resonant cavity) and isolated con-
ing, such that as the atoms move across the interac-
ducting pads (for application of the orthogonal d.c.
tion region, they effectively see a time-varying field.
field) have been added. We report the results of our
When the atoms are initially prepared in a single
detailed numerical analysis of the electric and mag-
hyperfine component ψ , and when the field com-
i
netic fields supported by this structure, and we use
ponents are chosen so as to couple the initial state
the magnitudes of the field components to estimate
to a single final state ψ , the atomic system is very
f
the residual systematic effects that one should ex-
closely described as a two-level system, and we can
pect in a determination of E in atomic cesium.
PNC write the state of the atoms using the time-varying
This paper is organized as follows. In Sec. II, amplitudes c (t) and c (t) as
i f
we discuss the transition probability of a two-level
atom interacting with a resonant r.f. field and a ψ(t)=ci(t)ψie−iωit+cf(t)ψfe−iωft.
two-frequency optical field through a Raman inter-
action. We show that, when a variable d.c. elec- The time evolution of the system is described in
tric field is applied, this coherent control process al- termsoftheHamiltonianH +Vint,whereH isthe
0 0
lows one to determine E /β. We then discuss in atomic Hamiltonian and Vint describes the interac-
PNC
Sec. III the various transition amplitudes, including tion between the atom and the field. In this work,
the magnetic dipole, Stark-induced electric dipole, we consider the weak-force induced electric dipole
and weak-induced electric dipole, for the transition interaction Vint , the Stark-induced electric dipole
PNC
between hyperfine levels of the ground state of an interactionVint,andthemagneticdipoleinteraction
St
alkali metal atom. We present an estimate of the Vint of the atom with the r.f. field, plus a Raman
M
signalsizeinSection IV,withanestimateofthesta- interaction Vint of the atom with a two-frequency
Ram
tisticaluncertainty,andreviewthebenefitsofcarry- laser field, all of which we describe in more detail
ing out the measurement in a standing wave cavity later, and write Vint as the sum of the individuals
for suppression of magnetic dipole contributions in
SectionV.Inthefollowingsection,weintroducethe Vint =Vint +Vint+Vint+Vint .
PNC St M Ram
3
fields propagate in the y-direction, the d.c. electric
E0 and magnetic B0 fields are oriented in the z-
direction, and the electric field εrf of the r.f. field is
directed in the x-direction. (Parallel propagation of
the r.f. and Raman fields is necessary to maintain a
uniform phase difference between interactions.) Not
shown in this figure are the two components of the
laser electric field that drive the Raman transition,
each linearly polarized, one in the x-direction, the
other in the z-direction. In this geometry, the pri-
maryr.f.andRamanfieldseachindependentlydrive
a ∆F = ±1, ∆m = ±1 transition, the magnetic
dipole contribution on this transition is suppressed,
andtheStark-inducedandthePNCinteractionsare
FIG. 1: (Color online) An abbreviated energy level dia- inquadrature-phasewithoneanother. Theprimary
gram showing the relevant ground state levels. We pre- contributionshere,undertheprecise(idealized)con-
pare the cesium atoms one hyperfine component of the ditions specified in Fig. 2, are
ground state, (F,m), where m = ±F. Through the in-
teractions with the r.f. field and the optical field, some
oftheatomsaretransferredtothelevel(F(cid:48),m(cid:48)). Inthis Vint =βE0εrf ei(ωrft−ky−φrf) CF(cid:48)m±1 (3)
figure, we show (3,3) as the initial state, and (4,4) as St z x Fm
the final state.
and
We illustrate these interactions schematically in
Fig. 1. Vint =∓i Im{E }εrfei(ωrft−ky−φrf)CF(cid:48)m±1.
When the atoms exit the interaction region, the PNC PNC x Fm
(4)
probability that they are in state ψ is
f InEq.(3),β isthevectorpolarizabilityandCF(cid:48)m±1
Fm
(cid:32)(cid:12) (cid:12)(cid:33) isafactorrelatedtotheClebsch-Gordoncoefficients,
(cid:12)(cid:88) (cid:12)
|c (∞)|2 =f(δ)sin2 (cid:12) Θ (cid:12) , (1) defined in detail in Ref. [47]. Note that we have
f (cid:12) i(cid:12)
(cid:12) i (cid:12) explicitly included the phase of the r.f. field in these
expressions.
wheretheΘ aretheintegratedinteractionstrengths
i
of any of the individual interactions In addition to these primary amplitudes, extra
contributionsduetomagneticdipoletransitionsand
(cid:90) ∞
Θ = Ω (t)dt.
i i
−∞
The Rabi frequencies of the various interactions are
Ω = Vint/(cid:126), and f(δ) represents the reduction in
i i
amplitude when the r.f. frequency is detuned from E0
the resonant frequency by δ = ωrf −|E −E |/(cid:126).
f i k
f(δ) depends on the temporal shape of the ‘pulse’ y B0
as the atoms travel through the interaction region
Cs
inanon-trivialway,andwewilllimitourdiscussion x z εrf atomic
to resonant excitation, δ =0, for which f(δ)=1. beam
In an atomic beam, collisions are infrequent, and
theatomstravelthroughtheinteractionregionwith
FIG.2: (Coloronline)Thefieldorientationsforthemea-
a constant velocity v. In this case, the interaction
surementofE /βonthe∆F =±1,∆m=±1ground
PNC
strength can be rewritten state hyperfine transition. The static electric and mag-
netic fields are oriented in the z-direction, while the po-
1(cid:90) ∞
Θ = Ω (z)dz. (2) larization of the r.f. field is in the x-direction. The po-
i v i larizations of the laser field components that drive the
−∞
Raman interaction, not shown, are aligned with the x-
We use notation similar to that of Gilbert and and z-axes. The r.f. and Raman fields propagate paral-
Wieman [46] for each of the various interactions, leltooneanother,shownasthedirectionk,asrequired
and show the optimal field geometry for this mea- to maintain a uniform phase difference between interac-
surement in Fig. 2. That is, the r.f. and Raman tions.
4
fieldmisalignmentscanarise. Thelargestoftheseis limitoftheRamaninteractionstrengthΘ being
Ram
much greater than any of the interactions driven by
(cid:40)
the r.f. field Θ , Θ , and Θ . Under these con-
Vint = η M (cid:2)∓hrf +ihrf(cid:3)CF(cid:48)m±1 (5) St M PNC
M 0 x y Fm ditions, andwiththedetuningδ =0, Eq.(1)canbe
expanded to the form
(cid:32)±B0+iB0CF(cid:48)mCF(cid:48)m
+hrzf xB0 y Fmg F(cid:48)m±1 |cf(∞)|2 = sin2(|ΘRam|)+sin(2|ΘRam|) (7)
z F(cid:48)
×sin[|Θ +Θ +Θ |cos(∆φ+δφ(E ))].
∓B0+iB0CFm±1CF(cid:48)m±1(cid:33)(cid:41) St M PNC z
+ xB0 y Fm g Fm±1 ei(ωrft−ky−φrf) ∆φ = φrf − φRam is the controllable phase differ-
z F
ence between the r.f. field and the phase difference
φRam, and δφ(E )=tan−1(E /βE0) is the phase
for ∆m = ±1 transitions, where the hrf are the z PNC z
i shift introduced by the quadrature combination of
components of the magnetic field of the r.f. wave,
E and βE0. (In writing this phase shift, we pre-
M is the magnetic dipole transition moment, η = PNC z
(cid:112) 0 sumethatthemagneticdipolecontributionsaresup-
µ /(cid:15) =120πΩ is the impedance of vacuum, and
0 0 pressed, as we show later.) We see from this expres-
g and g are the gyromagnetic ratio of the ini-
F F(cid:48) sion a feature that is similar to that of the coherent
tial and final states. For cesium, g is −1/4 for
F controlschemeonashort-livedstate[44,45];thatis,
the F = 3 level and +1/4 for the F = 4 level of
that the signal consists of a d.c. term resulting from
the ground state. The first terms in Eq. (5) are the
the Raman interaction alone, plus a sinusoidally-
magneticdipoleamplitudedrivenbythehrf andhrf
x y varying contribution that varies with the phase dif-
fieldcomponents, whilethelasttermsinhrf andB0
z x ference ∆φ between the Raman field and the one-
or B0 arise from Zeeman mixing of the hyperfine
y photon r.f. field. Furthermore, the amplitude of the
componentsbythestaticmagneticfield. Toinvesti-
modulating term is the magnitude of the sum of in-
gatepossibleinterferencesfrom∆m=0transitions,
teractionangles|Θ +Θ +Θ |≈|Θ +Θ |,
St M PNC St PNC
we also present the magnetic dipole transition am-
where we have omitted the small magnetic dipole
plitude for these transitions
integrated angle in the final step. A laboratory
(cid:40) measurement of this population modulation ampli-
VMint = η0M hrzfCFFm(cid:48)m+(cid:88)(cid:2)∓hrxf +ihryf(cid:3) tude as a function of the d.c. electric field Ez0 yields
E /β. We see this as follows.
± PNC
×(cid:34)(cid:32)∓Bx0B+z0iBy0(cid:33)CFFm(cid:48)m±g1FC(cid:48)FF(cid:48)(cid:48)mm±1 (6) |ΘSt+ΘPNC|= v1(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)−∞∞[ΩSt(z)+ΩPNC(z)] dz(cid:12)(cid:12)(cid:12)(cid:12),
+ (cid:32)±Bx0+iBy0(cid:33)CFFmm∓1CFFm(cid:48)m∓1(cid:35)(cid:41)ei(ωrft−ky−φrf). which, using Eqs. (3) and (4) becomes
B0 g
In additiozn to these tranFsitions driven by the r.f. |ΘSt+ΘPNC| = (cid:126)1v (cid:12)(cid:12)βEz0∓i Im{EPNC}(cid:12)(cid:12)
(cid:90) ∞
field, we consider the Raman transition of the form ×CF(cid:48)m±1 εrf(z)dz, (8)
Fm x
−∞
Vint =β˜εR1(εR2)∗ei(ωrft−φRam) CF(cid:48)m±1
Ram z x Fm valid when E0 is uniform in the interaction region.
z
where εR1 and εR2 are the electric field amplitudes Since the Stark and PNC moments add in quadra-
z x
of the two laser components, and ωrf = ωR1−ωR2, ture, the amplitude of the sinusoidal modulation of
where ωR1 and ωR2 are the optical frequencies. The the signal scales as
phase φRam is the phase difference between the
(cid:113)
pmhaansepsoolafrtihzaebtiwlitoycβo˜mdpeponenendtssoφnRt1h−e dφeRt2u.niTnhge∆Rao-f (cid:12)(cid:12)βEz0∓i Im{EPNC}(cid:12)(cid:12)= (βEz0)2+|EPNC|2. (9)
these field components from the D transition fre- Atsmalld.c.field,themodulationamplitudeispro-
2
quency, and the Raman transition can be enhanced portional to Im{E } alone, while at large field,
PNC
by making ∆ small. the modulation amplitude is nearly proportional to
We will analyze these r.f. transition amplitudes βE0. Bymeasuringthisamplitudeofthepopulation
z
later using electric and magnetic field amplitudes modulation as a function of the d.c. field, therefore,
that we expect to encounter for our parallel plate one can determine the ratio E /β.
PNC
structure to place limits on unwanted magnetic To optimize the amplitude of the signal modula-
dipolecontributionstothePNCsignal. Beforewedo tion in Eq. (7), one should adjust the amplitude of
this, we return to Eq. (1), which we examine in the the Raman interaction to |Θ | = π/4. At this
Ram
5
shows the state amplitudes when the interactions
are π out of phase with one another. The peak
Rabi frequency, center position, and beam radius
are Ω = 23.9 ms−1, z = -4 cm, and w =
Ram,0 c Ram
0.5cmfortheRamanbeam, andΩ =0.61ms−1,
w,0
z = 0, and w = 2.5 cm for the r.f.-driven inter-
c rf
action. We use 270 m/s, the peak velocity of the
atoms in our atomic beam for v. The duration of
the interaction is w /v (cid:39) 19 µs for the Raman
Ram
beam,andw /v (cid:39)93µsforther.f.field. Whenthe
rf
amplitudes are in phase with one another, |c (z)|
f
growsmonotonically,whilewhentheinteractionsare
out of phase, the amplitude decreases after its ini-
tial preparation by the Raman beam. The value of
|c (∞)| after the atoms have exited the interaction
f
(cid:112)
region is 1/2+sin(|Θ |) for in-phase interactions
w
(cid:112)
FIG.3: (Coloronline)Thevariationofstateamplitudes and 1/2−sin(|Θw|) for out-of-phase interactions.
|c (z)| (red solid) and |c (z)| (blue dashed) versus z as When the PNC and Stark-induced terms are driven
f i
the atoms pass through the interaction region from left by the r.f. field, then |Θ | is |Θ +Θ |, where
w St PNC
to right. The atoms are prepared by the Raman beams the PNC interaction angle is
in a superposition state before entering the broad r.f.
field. BothfieldsareGaussianinshape,withpeakRabi Θ = (cid:16)∓iIm{E }CF(cid:48)m±1/(cid:126)v(cid:17)(cid:90) ∞ εrf(z)dz
frequency and beam radii of Ω = 23.9 ms−1 and PNC PNC Fm x
Ram,0 −∞
20..55ccmmffoorrtthheeRr.fa.m-darinvebneainmt,eraancdtioΩnw.,0In=(a0).6,1thmesR−a1manand =(cid:16)∓iIm{EPNC}CFFm(cid:48)m±1/(cid:126)v(cid:17)√π wrf εrxf,0. (10)
andr.f.interactionsareinphasewithoneanother,while
in (b), the interactions are out of phase. In either case, Similarly, the integrated area of the Stark-induced
thedurationoftheinteractionisw /v(cid:39)19µsforthe interaction angle for this Gaussian-shaped profile is
Ram √
Raman beam, and wrf/v(cid:39) 93 µs for the r.f. field. ΘSt =βEz0CFFm(cid:48)m±1 π wrf εrxf,0/(cid:126)v. The term 1/2 in
theexpressionsfor|c (∞)|comesfromsin2(|Θ |)
√ f Ram
with |Θ | = πw |Ω |/v. The weak sig-
value, the factor sin(2|Θ |) is equal to 1, and Ram Ram√Ram,0
Ram nal strength is |Θ | = πw |Ω |/v in this ex-
the atomic population due to the Raman interac- w rf w,0
ample is 0.10. Any interaction of the atoms with
tion alone is equal to 1/2, i.e. equal probability in
the r.f. field therefore is evident as a modulation of
the initial and final states. Any additional interac-
this signal as we vary the phase difference between
tions of the atom with the r.f. field add (slightly) to
the fields. We illustrate this in Fig. 4, which shows
thepopulationintheψ statewhenthisinteraction
f the sinusoidal modulation of the final state popula-
isinphasewiththeRamaninteraction,andsubtract
tion as a function of ∆φ. Here the parameters are
when out-of-phase.
as they were in Fig. 3, with the exception of Ω
We can gain some insight into the interference w,0
which we have decreased to 0.061 ms−1 for this fig-
by following the evolution of the amplitudes |c (t)|
f ure. The amplitude of the modulation of |c (∞)|2
(red solid) and |c (t)| (blue dashed) as the atoms √ f
i is |Θ | = πw Ω /v = 0.010, in agreement with
move across the interaction region, which we show w rf w,0
thenumericaldatainthefigure. Inoursimulations,
inFig.3. Forthisillustration, theatomsmovefrom
theamplitudeofthemodulationscaleslinearlywith
left to right, and encounter the Raman field first,
the weak amplitude.
centered at z = -4 cm, which prepares them in a
Important conditions and features of this mea-
coherent superposition state. The atoms then enter
surement technique include:
the broad r.f. field. We use Gaussian profiles for the
r.f. and Raman fields. For the former, the peak am- 1. Mutualcoherenceofthedifferenttime-varying
plitudeisεrf andbeamradiusw intheinteraction fields is required. This can be implemented
x,0 rf
region, in the laboratory by using non-linear mixing,
injection locking of diode lasers, or frequency
εrf(z)=εrf e−(z/wrf)2. modulation techniques.
x x,0
We show this for two values of the phase ∆φ in 2. Thecoherentbeamsthatdrivetheinteractions
Fig. 3. Fig. 3(a) shows the magnitudes of the state must propagate in the same direction in order
amplitudeswhentheRamanandr.f.-driveninterac- to maintain a uniform phase difference for all
tions are in phase with one another, while Fig. 3(b) atoms in the interaction region.
6
III. EXPECTED MAGNITUDES OF M, β,
AND E
PNC
In order to design a measurement system and un-
derstand the effect of stray fields and the magni-
tude of unwanted contributions to the signal, we
mustfirstknowtheexpectedmagnitudesofthePNC
moment, E , the vector polarizability β, and the
PNC
magnetic dipole moment M for the transition.
The PNC amplitude for this transition is calcu-
lated [11] to be
FIG. 4: (Color online) The sinusoidal variation of the
signal as a function of the phase difference between the E =1.82×10−11iea , (11)
PNC 0
r.f. and Raman interactions. The peak Rabi frequency
of the r.f.-driven interaction is Ω = 0.061 ms−1 for whereeanda aretheelectronchargeandtheBohr
w,0 0
this plot. Other parameters are as given in the caption radius,respectively. ThisislargerthanE forthe
PNC
to Fig. 3. moment on the 6s → 7s transition in cesium by a
factor of 2.2.
The vector polarizability has not previously been
3. TheRamanandther.f.fielddistributionneed
calculated, but we can estimate its approximate
not overlap one another. Since the ground
magnitude using the sum-over-states expansion of
state is long lived, the final level retains its
Refs. [1] and [47],
coherence, and the net excitation of the final
statedependsontheaccumulatedeffectacross (cid:34) (cid:18) (cid:19)
the interaction region. β = e (cid:88)r2 1 − 1
6(cid:126) n,1/2 ∆ ∆
4. We control the phase difference between the n 4;n,1/2 3;n,1/2
(cid:18) (cid:19)(cid:21)
transition amplitudes with r.f. devices, com- 1 1 1
+ r2 − ,
pletely external to the interaction region. 2 n,3/2 ∆ ∆
4;n,3/2 3;n,3/2
5. We select the particular interactions that con-
where r represents the reduced dipole matrix ele-
tribute to the measurement by choosing the n,j
ments(cid:104)np ||r||6s (cid:105)forj =1/2or3/2,and(cid:126)∆
orientation of the various fields in the interac- j 1/2 F;n,j
are the energy differences E −E for the two
tion region. 6s,F npj
hyperfine states F = 3 or 4 of the ground 6s2S
1/2
6. The measurement uses only modest d.c. elec- andtheexcitednp2P states. Then=6termdomi-
j
tric fields, (cid:46) 100 V/cm. This allows flexibility natesthissum,andthegroundstatehyperfinesplit-
in the experimental configuration. ting ∆ is small compared to the energy of the 6p
hfs
states, so the polarizability is approximately
7. Since the interactions Ω and Ω are π/2
PNC St
out of phase with one another, these ampli- (cid:34)(cid:12) (cid:12)2
tudes add in quadrature. This indicates that β (cid:39) e∆hfs (cid:12)(cid:104)6p1/2||r||6s1/2(cid:105)(cid:12)
the amplitude of the modulating signal is at 6 (E6s−E6p1/2)2
a minimum when the static electric field is (cid:12) (cid:12)2(cid:35)
turned off, and increases when a static field +1(cid:12)(cid:104)6p3/2||r||6s1/2(cid:105)(cid:12) .
of either polarity is applied. 2 (E6s−E6p3/2)2
8. Usingdifferentfieldorientations,thiscoherent
We use (cid:104)6p ||r||6s (cid:105) = 4.5062 a and
control technique may be used to determine 1/2 1/2 0
(cid:104)6p ||r||6s (cid:105) = 6.3400 a [48–53] to estimate
M/β. This may be a useful means of deter- 3/2 1/2 0
the vector polarizability for this transition as β (cid:39)
mining the vector polarizability β, but we de-
0.00346a3. Based on these expected magnitudes of
fer any further discussion of this to a future 0
β and E , the ratio E /β is about 27 V/cm;
report. PNC PNC
uponapplicationofastaticelectricfieldofthismag-
In the following sections, we will discuss the ex- nitude, the magnitudes of the Stark-induced ampli-
pected magnitudes of the different interactions, and tudeandthePNCamplitudeareequivalent. Sinceβ
present an experimental assembly of conductors for is so small for this transition, we conclude that sys-
such a measurement in an atomic beam configura- tematic errors due to uncontrolled electric fields in
tion. Finally, we will analyze the effect of expected the interaction region, due to surface contamination
magnetic dipole contributions to the measurement. and patch effects and estimated to be (cid:46)0.1 V/cm,
7
are inconsequential in these ground state measure- phase with one another (|c |2 = 1 +|Θ |), and
f 2 PNC
ments. This is in strong contrast to measurements N the total count of excitations when the r.f. and
−
of E on the 6s → 7s transition [17], for which Raman interactions are π out of phase with one an-
PNC
uncontrolled electric fields were of major concern. other (|c |2 = 1 −|Θ |). Then
f 2 PNC
In addition to these two relatively weak ampli-
1 N −N
tudes driven by the r.f. field, the magnetic dipole Θ = + −.
moment is active on this transition. The amplitude PNC 2 N++N−
for this transition is Vint = (cid:104)6s2S F(cid:48)m(cid:48)|−µ ·
M 1/2 m To use this result to determine E , however, one
brf|6s2S Fm(cid:105), where µ =µ (g L+g S+g I) PNC
1/2 m B L S I must also have an accurate determination of the r.f.
isthemagneticmomentoftheatom,µ =e(cid:126)/2mis
B beam profile and field amplitude. Alternatively, one
theBohrmagneton,andbrf isthemagneticfluxden-
canapplyad.c.electricfieldtotheatoms,andmea-
sityofther.f.wave. L,SandIaretheusualorbital,
sure the amplitude of the modulation as a function
spin,andnuclearangularmomenta,andgL,gS,and of the field amplitude E0, as suggested in Eqs. (8)
g are the respective gyromagnetic ratios. For the z
I and (9).
transition of this work, the orbital angular momen-
When the precision of N and N is limited by
+ −
tum is zero, and g is much less than g (which is
I S counting statistics, then√the uncertainty in either of
≈ 2) due to the heavy mass of the nucleus. For the
these counts is σ = N, where N represents ei-
N
ground state transition, the spatial parts of ψ and
i ther N or N (which are essentially the same).
ψ are the same, and using εrf/brf = c, the mag- + − √
f The uncertainty in Θ is σ = 1/ 8N, and
netic dipole amplitude is M = µ g /2c (cid:39) µ /c. PNC PNC
B S B to achieve a 3% measurement of Θ , one must
But µ /c = ea α/2, where α (cid:39) 1/137 is the fine PNC
B 0 count N = 1/8σ2 = 3 × 1012 atoms for each
structure constant, so M (cid:39) ea α/2, and the ratio PNC
0 individual measurement. In a counting interval T,
M/E (cid:39) 2×108. The magnetic dipole contribu-
PNC the number of counts is N = 1ρ AvT, where 1 is
tions to the signal must be suppressed for a success- 2 Cs 2
the average excitation probability, ρ is the num-
ful measurement of E , representing the primary Cs
PNC berdensityoftheatomicbeam(109 cm−3),Aisthe
challenge of these measurements. The orientations
cross sectional area of the atomic beam (1 mm2),
ofthefieldcomponentsthatwehaveshowninFig.2
andv isthepeakvelocityoftheatomsinthebeam.
areanimportantfirststepinmeetingthischallenge.
The counting time T to achieve the required statis-
tics is 20 seconds per data point. During the course
of a measurement, one must repeat the process at
IV. MAGNITUDE OF SIGNAL
many different phases, not just two, and one must
varythed.c.electricfieldstrengthE0andrepeatthe
z
In this section, we will use the results of the anal- measurement. Regardless, the estimate of the inte-
ysis of Sec. II, in particular Eqs. (7) and (10), and gration time shows that the measurement is feasible
thecalculatedvalueofEPNC giveninEq.(11),toes- in the beam geometry.
timate the magnitude of the PNC signal, and from We conclude this section with an estimate of
thistheintegrationtimerequiredtoachieveauseful the maximum value of the d.c. field amplitude E0
statistical uncertainty of the measurement. To cal- needed. As discussed in the previous section, we
culatethesignalsize,wewilluse|CF(cid:48)m±1|=(cid:112)7/8, expect that the ratio E /β is approximately 27
Fm PNC
εrf = 250 V/cm, and w = 2.50 cm. The value of V/cm. In carrying out the measurements, we must
x,0 rf
CF(cid:48)m±1 is valid for cesium ground state transitions vary the Stark-induced angle ΘSt over the range
Fm from zero to ∼ ±3|Θ |. This requires a variable
(F,m) = (3,±3) → (4,±4) or (4,±4) → (3,±3), PNC
field strength of maximum value 3E /β ≈ ±80
and we will show in Sec. VI that the values of PNC
V/cm.
the peak field amplitude and radius are reasonable.
Then using the cesium atomic beam peak velocity v
= 270 m/s, we estimate that the interaction angle
V. STANDING WAVE CAVITY
for the PNC interaction is
Θ =±i5.6×10−6. In the previous section, we estimated the mag-
PNC
nitude of the hyperfine ground state PNC coherent
To measure this amplitude, one can drive the inter- controlsignal,basedonexpectedatomicparameters
fering Raman and PNC interactions, and count the andreasonablefieldstrengthsthatcanbegenerated
transition rate as a function of the phase difference in the laboratory. Among the latter was an r.f. field
between the transitions. A minimal measurement amplitude εrf of 250 V/cm. This field amplitude
x,0
may consist of N , the total count of atomic exci- canbeachievedeitherinsidearesonantpowerbuild-
+
tations when the r.f. and Raman interactions are in upcavity,orbyusingaverylarger.f.amplifier. Use
8
+V -V y
x z
d
Cs
w
-V
+V
FIG. 6: (Color online) The electrode configuration that
supports the standing wave r.f. field ε and the static
x
electric field E0.
z
FIG. 5: (Color online) The standing wave pattern of
the r.f. electric field εrf and magnetic field hrf, with the
x z
atomic beam located at the node of the magnetic field. basedonaparallelplatetransmissionlinestructure,
whichallowsspatialconfinementofther.f.fieldand
generation of a transverse d.c. electric field.
of a resonant cavity also helps to suppress the mag-
neticdipolecontributionstothemeasuredsignal,as
we now discuss. This approach is also discussed in VI. PARALLEL PLATE TRANSMISSION
Ref. [7]. LINE STRUCTURE
As we discussed earlier, the large magnetic dipole
amplitude is suppressed to first order by the choice Themeasurementthatwehavedescribedpresents
of orientations of the primary fields. (The hrf com- several experimental challenges. First, we must ap-
z
ponent drives a ∆m=0 transition, whereas the in- ply r.f. and static electric fields that are oriented
terference that we have discussed takes place on a in directions that are perpendicular to one another.
∆m = ±1 transition.) Still, due to the large mag- Second, we require that the r.f. field is in a standing
nitude of the ratio M/E and reasonable limits wave configuration for suppression of the magnetic
PNC
in the field uniformity and experimental alignment, dipole contributions. And third, we must minimize
additionalmeasuresarerequiredtosuppressthisin- the unwanted field components of the r.f. field, as
teraction further. This additional suppression can thesealsoleadtosystematicmagneticdipolecontri-
be achieved by working in a standing wave config- butionstothesignal. Inthissection,wedescribean
uration, in which the nodes of the magnetic field electrode configuration that allows us to meet these
coincide with the anti-nodes of the electric field, as requirements.
we illustrate in Fig. 5. At this point, the interac- In Fig. 6, we show a section of a parallel plate
tionsVint andVint aremaximized,andVint ismin- transmissionline, withwavespropagatinginthe±y
PNC St M
imized. To take best advantage of this, one should directions, that is modified in two regards. First,
(1) use a cavity geometry in which the amplitudes we have isolated several conducting pads on the top
of the traveling waves propagating in the +y and andbottomconductorsforapplicationofad.c.bias,
−y directions, εrf and εrf, respectively, are equal, and secondly, we have inserted cylindrical reflectors
+ −
and (2) keep the radius b of the atomic beam small. toeithersideoftheinteractionregiontoformanr.f.
Thefirstrequireseitherthatthecavityissymmetric cavity, open on the z faces, allowing power build-up
(thereflectivitiesofthetwoendreflectorsareequal, ofthecavitymodeattheresonantfrequency. When
and the cavity is excited by equal amplitude inputs wehavebiasedthed.c.padsprogressively,atavolt-
on each side), or that one of the reflectors has unit age +V on one side to −V on the other, we can
reflectivity. The choice of beam radius b is a com- generate an electric field E0 in the central region
promise between large atom number, improving the between the plates that is primarily directed in the
counting statistics, or small magnetic dipole ampli- ±z-direction. We capacitively couple each of the
tudeforatomsattheedgeofthebeam,whichscales bias pads to the transmission line structure so that
as sin(kb) = sin(2πb/λ), where λ = 3.2 cm is the they carry the a.c. components without any signifi-
wavelength of the 9.2 GHz wave. For b = 0.5 mm, cantperturbation. Foratransmissionlinecharacter-
this reduction factor is ∼0.1. Furthermore, the sign istic impedance Z = 50 Ω, this requires a coupling
0
of the magnetic dipole amplitude is opposite on the capacitance of C (cid:38) 30 pF.
twosidesofthenode,furtherreducingthiscontribu- We can model the cavity modes that are sup-
tion. WewillreturntothisreductioninSectionVII. ported by the parallel-plate structure in the region
In the next section, we will discuss the design and between the cylindrical reflectors approximately us-
analysis of a symmetric r.f. power build-up cavity ing the elliptical Hermite-Gaussian modes as de-
9
scribed in Yariv [54]. These modes are nearly Gaus- (cid:96) = 11.9 cm, the cavity has a resonance at the ce-
c
sian in shape in the z-direction, but uniform in the sium hyperfine transition frequency (9.2 GHz), its
x-dimension, in the limit of an infinite beam size in free spectral range (FSR) is c/2(cid:96) = 1.26 GHz, the
c
this dimension. Within the cavity, the spatial mode beamradiusatthewaistis2.50cm,thebeamradius
is described by the superposition of waves traveling atthereflectorsis3.53cm,andthetransversemode
in the +y and −y directions, spacingis0.2487timestheFSR,orabout313MHz.
Weestimatethefieldamplitudeattheinteraction
εrxf(y,z)=εr+f(y,z)+εr−f(y,z), (12) regionasfollows. Wechoosethespacingbetweenthe
parallel plates of the transmission line to be 1 cm,
and
and the conductor width 7.5 cm. These dimensions
hrf(y,z)= 1 (cid:0)εrf(y,z)−εrf(y,z)(cid:1), (13) yield a characteristic impedance of the transmission
z η + − line of 50 Ω, and allow for a reasonable clearance
0
of the atomic beam in the space between the con-
where ductors. With a copper thickness on the reflectors
of 170 nm, we calculate a reflection coefficient of
(cid:114) w (cid:26)
εrf(y,z) = εrf 0 exp ∓i[ky−η(y)] 0.9992. Note that this thickness is smaller than the
± 0,± w(y) skindepthδ=680nmofcopperatthisfrequency,so
(cid:20) 1 ik (cid:21)(cid:27) thetransmissionlossesaresmall, butnotnegligible.
−z2 + ,
w2(y) 2R(y) With this reflectivity, thecavity losses due toreflec-
tion are of the same magnitude as the losses L due
Intheseexpressions,w isthe1/e2 (intensity)beam to other mechanisms, primarily conduction losses in
0
radius at the focus, the beam profile radius a dis- the upper and lower conducting plates, and diffrac-
tance y from the focus is tion losses due to the finite size of the conductor.
(These results come from our numerical analysis of
(cid:112)
w(y)=w 1+(y/y )2, thecavitymodes,whichwediscussnext.) Foranr.f.
0 0
input power of 250 mW incident on the cavity from
y0 is the confocal parameter either side, the incident voltage amplitude is 5.0 V,
and the electric field of the traveling wave incident
y0 =πw02/λ, on the cavity is ε+ = 5.0 V/cm. The amplitude of
in
the traveling wave inside the cavity is
R(y) is the radius of curvature of the wavefronts
R(y)=y(cid:2)1+(y /y)2(cid:3), ε+ =2 t ε+in =125 V/cm,
0
1−r2(1−L)
and η(y)
whereweuset=0.04forthetransmissioncoefficient
1 ofthereflectorand(1−L)=r2. Thefactor2results
η(y)= tan−1(y/y )
2 0 from symmetric inputs from the two sides. At the
anti-node of the field, where the amplitudes of the
istheslowphaseshift(theGuoyphase)throughthe two traveling waves inside the cavity add in phase,
focal region. For a symmetric cavity constructed of thefieldamplitudeistwicethisvalue, or250V/cm.
cylindrical reflectors of radius of curvature R sep- This is the value of the r.f. field amplitude that we
arated by a distance (cid:96)c, the confocal parameter is usedinSec.IVtoestimatethesignalsize. Inmaking
(cid:112)
y0 = ((cid:96)c/2) 2R/(cid:96)c−1, the beam radius at the this estimate, we have not included the absorption
center is w = (λ(cid:96) /2π)1/2(2R/(cid:96) −1)1/4, and the ofthecopperreflector,whichreducestheamplitude,
0 c c
beam radius at the reflectors is w(y = ±(cid:96) /2) = or the increase of the wave amplitude as the wave
c
(λR/π)1/2(2R/(cid:96) −1)−1/4. The cavity mode has an propagates to the waist of the Gaussian profile.
c
electric field anti-node (and magnetic field node) at In order to determine more-detailed field parame-
thecenterwhenthecavitylength(cid:96) isapproximately ters,wehavecarriedoutaseriesofnumericalsimula-
c
(n+1/2)λ, where n is an integer. The r.f. beam ra- tionsofthecavitymodeusingComsolMultiPhysics.
dius w(y = ±(cid:96) /2) at the reflectors is minimized These simulations allow us to determine the effects
c
when the reflector spacing is confocal, i.e. (cid:96) = R. ofresistivelossesoftheparallelplates,thethickness
c
By adjusting the reflector slightly away from the of the reflective copper layers, and the finite width
confocalspacing, onecanretainthesmallmodesize of the cavity on the cavity Q; the effect of the gaps
w(±(cid:96) /2) at the reflectors, but shift the frequencies intheconductorbetweenthed.c.biaspads;andthe
c
of the transverse modes away from the frequency of uniformity of the static electric field in the interac-
the lowest order mode, improving the selectivity of tionregion. Weshowthethreeprimarycomponents,
cavity modes. We calculate that for R=12 cm and Re[εrf(y,z)], Im[hrf(y,z)], and Im[hrf(y,z)], of the
x z y
10
simulated r.f. field mode in Fig. 7. We note very electric field E0, and shown that with an array of
close agreement of the components εrf(y,z) with 10 bias pads and ∆V = 100 V between pads, we
x
the analytic result in Eq. (12) and hrf(y,z) with can generate a relatively uniform field of magnitude
z
Eq. (13). The component Im[hrf(y,z)] would be E0(z)∼140V/cm. Weshowthisfield,normalizedto
y z
negligible for a weakly focused beam, but since in its maximum value, as the red dashed line in Fig. 8.
our geometry, w ∼ λ, this component survives. We also show E0(z) in the plot (black dotted line),
rf x
For this figure, the separation between the upper which is small in magnitude, and has an average
and lower conducting planes of the PPTL and the value of zero. The non-uniform part of E0(z), seen
z
width of the conductors are as before, 1.0 cm and in Fig. 8 as a nearly sinusoidal modulation of am-
7.5 cm, respectively, as are the radius of curvature plitude ∼7% of the constant part, has little impact
of the cylindrical reflectors R = 12.0 cm, and the onthemeasurement. Wecanseethisbyintegrating
reflector separation (cid:96) = 11.9 cm. With the thick- theproductE0(z)εrf(z)acrosstheinteractionregion
c z x
ness of the copper reflector layers equal to 200 nm, in z. For the case of ten bias pads, as shown, the
we determine a cavity Q of 9000, while for a 1.5 µm correction to the signal due to the sinusoidal modu-
∼ 2δ layer, the Q increases to 13,000. In the lat- lation is less the 0.7% of the signal. We can also see
ter case, the Q is limited primarily by the resistive in this figure that the width of the Gaussian shaped
losses in the conductors and diffraction losses of the r.f. field profile is somewhat less than the width of
finitewidthofthereflectors. ForacavityQof9000, the d.c. field, allowing us to avoid fringe effects of
the linewidth of the transmission peak of the cavity the d.c. field near the edges of the conductors.
is ∆ν = ν /Q ∼ 1 MHz. We show the computed We have used these simulations of the field am-
0
Gaussian r.f. field amplitude, εrf(0,z) across the in- plitudes, andtheirvariationthroughtheinteraction
x
teractionregionasthesolidbluecurveinFig.8. The region, to estimate systematic contributions to the
diameter of the cavity mode agrees well with 2w = PNC signal. We discuss these contributions in the
0
5.0 cm that we determined analytically earlier. next section.
We used the Eigenfrequency module and fre-
quency domain analysis to carry out these calcula-
tions,anddeterminedthequalityfactorofthecavity VII. ESTIMATION OF MAGNETIC DIPOLE
as the ratio of the energy stored inside the cavity to CONTRIBUTIONS TO THE PNC SIGNAL
the diffraction and dissipation losses. We obtained
the field patterns by launching a 9.2 GHz plane-
In this section, we will make use of the field sim-
wave-likeelectricfieldontheparallelplatetransmis-
ulations of the previous section in order to estimate
sion line towards the cavity, exciting a TE cavity
q,n the expected systematic contributions to the PNC
mode, where indices q and n label the transverse
signal. Theprimarycontributionsthatmustbecon-
andlongitudinalmodes. Themodespacingbetween
sidered are the magnetic dipole terms, due to the
the TE and TE mode agrees well with the
q,n q,n+1 relatively large magnetic dipole moment M on this
1.26 GHz FSR that we determined earlier. We used
transition. As we have shown, the primary mag-
a trial-and-error approach to reduce the diffraction
netic field components of the r.f. field are h(cid:48)(cid:48)(y,z)
losses by varying the cavity parameters, such as the z
and h(cid:48)(cid:48)(y,z), where we use primed (double-primed)
width and height of the cavity, while maintaining y
variables for the real (imaginary) part of the field
the resonant mode frequency close to 9.2 GHz.
quantities, and omit the superscript ‘rf’. By set-
In order to calculate the r.f. field distributions in ting up the geometry of the experiment to make the
a more refined manner in the interaction region, we atomic beam cross the r.f. field at the center of the
added about ten thousand times more mesh points cavity, where the component h(cid:48)(cid:48)(y) is minimal, the
z
in the vicinity of the interaction region. Higher magneticdipolecontributionstothesignalfromany
mesh point density helped to reduce errors that are individual atom can be reduced. Furthermore, the
present in the interpolation schemes, without com- contributions from atoms on one side of the node
promising the eigenfrequency calculations. We used are of opposite sign to those on the other side of
ten bias pads, with the spacing between the pads the node, and the net magnetic dipole contribution
about one tenth the width of the pads. As long as can be suppressed even further. In this section, we
the transmission lines are thin (less than 0.1 mm), usethenumericalsimulationsofthefieldssupported
the gaps have little impact on the r.f. fields. We by the resonant cavity to explore the magnitude of
found that neither horizontal nor vertical misalign- magnetic dipole contributions to the PNC signal.
mentofthecylindricalreflectorsaffectsthefieldpat- The net contribution of the h(cid:48)(cid:48)(y,z) term can be
z
terns or the Q factor, for misalignment less than 1 minimized by adjusting the relative position ∆y of
degree. the center of the atomic beam relative to the node
Wehavealsomodeledallcomponentsofthestatic of the magnetic field. (No control of the x-position