Parity violation in atomic ytterbium: experimental sensitivity and systematics

1.4 MB·English

by  K. Tsigutkin

Parity violation in atomic ytterbium: experimental sensitivity and systematics
K. Tsigutkin,1,∗ D. Dounas-Frazer,1 A. Family,1 J. E. Stalnaker,1,† V. V. Yashchuk,2 and D. Budker1,3
1Department of Physics, University of California at Berkeley,
Berkeley, CA 94720-7300
2Advanced Light Source Division, Lawrence Berkeley National Laboratory, Berkeley CA 94720
3Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720
(Dated: January 4, 2010)
Wepresentadetaileddescriptionoftheobservationofparityviolationinthe6s21S → 5d6s3D
0 1
408-nmforbiddentransitionofytterbium,abriefreportofwhichappearedearlier. Linearlypolarized
0
408-nm light interacts with Yb atoms in crossed E- and B-fields. The probability of the 408-
1 nm transition contains a parity violating term, proportional to (E ·B)[(E×E)·B], arising from
0
interference between the parity violating amplitude and the Stark amplitude due to the E-field (E
2
istheelectricfieldofthelight). Thetransitionprobabilityisdetectedbymeasuringthepopulation

n of the 3P state, to which 65% of the atoms excited to the 3D state spontaneously decay. The
0 1
a population of the 3P state is determined by resonantly exciting the atoms with 649-nm light to
0
J the 6s7s3S state and collecting the fluorescence resulting from its decay. Systematic corrections
1
4 due to E-field and B-field imperfections are determined in auxiliary experiments. The statistical
uncertainty is dominated by parasitic frequency excursions of the 408-nm excitation light due to

] imperfect stabilization of the optical reference with respect to the atomic resonance. The present
h
uncertainties are 9% statistical and 8% systematic. Methods of improving the accuracy for the
p
future experiments are discussed.
-
m
PACSnumbers: 11.30.Er,32.90.+a
o
t
a
. I. INTRODUCTION M1-(Stark-induced)E1 interference technique [5]. The
s
c ytterbium atomic system, where transition amplitudes
i and interferences are well understood, has proven use-
s In an earlier paper [1], we reported on observa-
y tion of the atomic parity violation (PV) effect in the ful for gaining insight into the Jones-dichroism effects
h 6s2 1S 5d6s 3D 408-nm forbidden transition of that had been studied in condensed-matter systems at
p 174Yb.0 W→e measured t1he PV induced transition matrix extreme conditions and whose origin had been a matter
[
element to be 8.7 1.4 10−10 ea , which confirms the of debate (see Ref. [6] and references therein).
1 theoretically antic±ipated×PV enhan0cement in Yb [2] and An experimental and theoretical study of the dynamic
v constitutes the largest atomic parity violation effect ob- (ac) Stark effect on the 6s2 1S 5d6s 3D forbid-
7 0 → 1
served so far. However, the measurement accuracy is den transition was also undertaken [7]. A model was de-
8
not yet sufficient for the observation of the isotopic and veloped to calculate spectral line shapes resulting from
5
0 hyperfine differences in the PV amplitude, the study of resonant excitation of atoms in an intense standing light
. which is the main goal of the present experiments. Here waveinthepresenceofoff-resonantac-Starkshifts. Abi-
1
we describe the impact of the apparatus imperfections product of this work was an independent determination
0
0 and systematic effects on the accuracy of the measure- (from the saturation behavior of the 408-nm transition)
1 ments and discuss ways of improving it. oftheStarktransitionpolarizability, whichwasfoundto
: During the initial stage of the experiment, an effort be in agreement with the earlier measurement [4].
v
i was invested into measuring various spectroscopic prop- The present Yb APV experiment involves a measure-
X erties of the Yb system of direct relevance to the PV ment using an atomic beam. An alternative approach
r measurement, including determination of radiative life- would involve working with a heat-pipe-like vapor cell.
a
times,measurementoftheStark-inducedamplitudes,hy- Variousaspectsofsuchanexperimentwereinvestigated,
perfine structure, isotope shifts, and dc-Stark shifts of including measurements of collisional perturbations of
the 6s21S0 5d6s3D1 transition [3]. In addition, the relevant Yb states [8], nonlinear optical processes in a
→
6s21S0 3D2 transition at 404 nm has been observed, dense Yb vapor with pulsed UV-laser excitation [9], and
→
andtheelectricquadrupoletransitionamplitudeandten- an altogether different scheme for measuring APV via
sortransitionpolarizabilityhavebeenmeasured[4]. The optical rotation on a transition between excited states
forbiddenmagnetic-dipole(M1)amplitudeofthe408nm [10].
transition was measured to be 1.33 10−4 µ using the
× B The present paper addresses the issues of sensitivity
and systematics in the Yb APV experiment. In Sec-
tions II and III the experimental technique and its ap-
plication in the present experiment are discussed. In
∗Electronicaddress: [email protected]
†Presentaddress: DepartmentofPhysicsandAstronomy,Oberlin Section IV a method of analyzing the impact of various
College,Oberlin,OH44074 apparatus imperfections is described based on theoret-
2
ical modeling of signals recorded by the detection sys- Stark interference. In addition, this effect is minimized
tem in the presence of imperfections. In Section V a de- bytheuseofapower-build-upcavitytogenerateastand-
taileddescriptionoftheexperimentalapparatusisgiven, inglightwave. Sinceastandingwavehasnonetdirection
alongwithadiscussionoftheoriginsoftheimperfections, of propagation any transition rate which is linear in the
which is followed by an account of the measurements of M1 amplitude, will cancel out (see below).
theimperfectionsinSectionVI. InSectionsVIIandVIII Theadvantagesofthepresentexperimentalconfigura-
we discuss measurements and analysis of the PV ampli- tioncanbedemonstratedbyconsideringYbatomsinthe
tude and systematic effects, and ways of improving the presence of static electric, E, and magnetic, B, fields in-
accuracy of the PV measurements to better than 1% in teractingwithastandingmonochromaticwaveproduced
order to measure the difference in the APV effects be- by two counter-propagating coherent waves in an optical
tween different isotopes and hyperfine components. cavity. Theelectricfieldinthestandingwave,E,isasum
of the fields of the two waves. For resonant atoms, the
transition rate from the ground state 1S to the excited
0
II. EXPERIMENTAL TECHNIQUE FOR THE state 3D1 is (see e.g. [13], Eq. (3.127))
APV MEASUREMENT
4
R = A 2, (1)
M ~2Γ| M|
As discussed in Ref. [1], the idea of the experiment is
to excite the forbidden 408-nm transition (Fig. 1) with where Γ is the natural linewidth of the transition, A
M
resonant laser light in the presence of a quasi-static elec- is the transition amplitude, and M = 0, 1 is the mag-
±
tric field. The PV amplitude of this transition arises netic quantum number of the excited state. Here and
due to PV mixing of the 5d6s3D and 6s6p1P states. in the rest of this section it is assumed that the individ-
1 1
The purpose of the electric field is to provide a refer- ual magnetic sublevels of the 3D state are resolved. For
1
ence transition amplitude, which is due to Stark mixing convenience, we set ~ = 1 henceforth and measure the
of the same states interfering with the PV amplitude. transition rate in units of Γ.
In such an interference method [11, 12], one is measur- ThetransitionamplitudeA isthesumoftheelectric-
M
ing the part of the transition probability that is linear in (E1) and magnetic- (M1) dipole transition amplitudes:
both the reference Stark-induced amplitude and the PV
amplitude. In addition to enhancing the PV dependent A =A(E1)+A(M1). (2)
M M M
signal, the Stark-PV interference technique provides for
all-importantreversalsthatseparatethePVeffectsfrom The E1 amplitude has two contributions corresponding
the systematics. to the Stark- and PV- mixing of the 3D1 and 1P1 states.
That is,
A(E1) =A(Stark)+A(APV)
M M M
=iβ( 1)M(E E) +iζ( 1)M , (3)
− × −M − E−M
where β is the vector transition polarizability, ζ is re-
lated to the reduced matrix element of the Hamilto-
nian describing the weak interaction, and are the
0,±1
spherical components of the vector E. AltEhough Stark-
inducedtransitionsaregenerallycharacterizedbyscalar,
vector, and tensor polarizabilities [4, 11], for the case of
a J = 0 1 transition, only the vector polarizability
→
contributes. Equation (3) is derived in Appendix A.
Similarly, the M1 transition amplitude has two com-
ponents: one for each of the two counter-propagating
laser beams. Let E = Eˆ and E = Eˆ denote the
+ + − −
E E
electric fields of the beams traveling in the k and k di-
rections, respectively. Then E = E +E and t−he M1
+ −
amplitude is given by
FIG.1: (coloronline)Low-lyingenergyeigenstatesofYband
transitions relevant to the APV experiment. A(M1) = ( 1)M(k E ) + ( 1)M( k E )
M M − × + −M M − − × − −M
= ( 1)M(δk E) , (4)
Even though the APV effect in Yb is relatively large, M − × −M
and the M1 transition is strongly suppressed, the M1 where is the reduced matrix element of the M1 tran-
M
transition amplitude is still three orders of magnitude sition and k is a unit vector in the direction of the
largerthanthePVamplitude. Asaresult, thegeometry wavevector. Here we have introduced δk = δkk with
oftheexperimentwasdesignedtosuppressspuriousM1- δk =( )/ . Foraperfectstandingwave, =
+ − + −
E −E E E E
3
and hence δk = 0 and the M1 transition is completely With this field orientation (see Fig. 2), Eqs. (1) through
suppressed. In practice, = δ due to the small (4) yield
− +
E E − E
butnonzerotransmissionofthebackmirrorinthecavity.
Since δ , δk δ / 1. Thus, although the R =4 2 β2E2sin2θ+2ζβEsinθcosθ , (7)
0
| E| ≪ E | | ≈ | E E| ≪ E
M1 transition amplitude is not strictly zero, it is highly R±1 =2 2(cid:0)β2E2cos2θ 2ζβEsinθcosθ(cid:1), (8)
suppressed. E −
Without loss of generality, the quantities β, ζ, and M where terms of o(cid:0)rder ζ2 and higher are neglect(cid:1)ed, and
are assumed to be real. In general, the rate R given
M δk =0 is assumed.
byEq.(1)includestermsproportionaltoβ (Stark-M1
M InordertoisolatetheStark-PVinterferencetermfrom
interference) and βζ (Stark-PV interference).
thedominantStark-inducedtransitionrate,wemodulate
Acarefulchoiceoffieldgeometryallows foradditional the electric field: E =E +E˜ cos(ωt), where E˜ is the
dc 0 0
suppression of undesirable Stark-M1 interference. From
modulation amplitude, ω is the modulation frequency,
Eq. (3), it is evident that the Stark-PV interference is
and E provides a DC bias. Then Eqs. (7) and (8)
dc
proportional to the rotational invariant
become
( E ·B)[(E×E)·B], (5) RM =R[M0]+R[M1]cos(ωt)+R[M2]cos(2ωt), (9)
PV Stark
which is P-odd and|{Tz}-even|. In{zthe} present experimental where R[Mn] is the amplitude of the nth harmonic of the
transition rate R . The dominant Stark-induced con-
apparatustheelectric field, E, isappliedorthogonally to M
tribution, which oscillates at twice the modulation fre-
themagneticfield,B,andcollinearlywiththeaxisofthe
quency, is
linearly-polarizedstandinglightwave,asshowninFig.2.
[2] =2β2E˜2 2sin2θ, (10)
R0 0E
[2] =β2E˜2 2cos2θ. (11)
R±1 0E
On the other hand, the amplitude [1] contains the
RM
Stark-PV interference term:
[1] =8 2 β2E˜ E sin2θ+ζβE˜ sinθcosθ , (12)
R0 E 0 dc 0
(cid:16) (cid:17)
[1] =4 2 β2E˜ E cos2θ ζβE˜ sinθcosθ . (13)
R±1 E 0 dc − 0
(cid:16) (cid:17)
The term [0] is a constant “background”:
RM
[0] =4 2 β2(E˜2+E2 )sin2θ+4ζβE sinθcosθ ,
R0 E 0 dc dc
FIG. 2: (color online) Orientation of fields for PV-Stark in- (cid:16) (cid:17)
[0] =2 2 β2(E˜2+E2 )cos2θ 4ζβE sinθcosθ .
terference experiment and schematic of the present APV ap- R±1 E 0 dc − dc
paratus. Notshownisthevacuumchambercontainingallthe (cid:16) (cid:17)
depictedelements,exceptthephotomultiplier(PMT)andthe For an arbitrary polarization angle θ, all three Zee-
photodiode (PD). PBC–power buildup cavity. Light is ap- man components of the transition, as shown in Fig. 3a,
plied collinearly with x.
arepresentwhilescanningoverthespectralprofileofthe
transition. Thefirst-harmonicsignalduetoStark-PVin-
This geometry is such that the M1 and Stark-induced terference has a characteristic signature: the sign of the
amplitudes are out of phase. Thus, they do not interfere oscillating terms for the two extreme components of the
and therefore do not produce spurious PV-mimicking ef- transition is opposite to that of the central component.
fects (see Section IV). The second-harmonic signal provides a reference for the
lineshape since it is free from interference effects linear
in E (Fig. 3b). With a non-zero DC component present
III. PV SIGNATURE: IDEAL CASE in the applied electric field, a signature identical to that
in the second harmonic will also appear in the first har-
In the ideal case where we neglect the apparatus im- monic, Fig. 3c. The latter can be used to increase the
perfections, the static magnetic and electric fields are first-harmonic signal above the noise, which makes the
B=Bzˆ and E=Exˆ, respectively, and the light stand- profile analysis more reliable.
ing wave has an electric field To obtain the PV term from the measured first-
and second-harmonic transition rates, we first normalize
E = (sinθyˆ+cosθzˆ). (6) the first-harmonic signals [1] by their second-harmonic
E RM
4
of the out-of-phase AC components was analyzed within
a complete model of the systematics and found to be
negligible. TheACcomponentsareduetomisalignments
of the applied E-field with respect to the light wave axis
aswellastothequantizationaxiszˆ. TheDCcomponents
ariseduetoamisalignment oftheDC-biasfieldandalso
due to stray electric fields in the interaction region.
The magnetic-field imperfections are defined within
FIG.3: DiscriminationofthePVeffectbyE-fieldmodulation
the same frame of reference by taking analogously
understaticmagneticfield. TheZeemanpatternisshownfor
the polarization angle θ=π/4.
B=B˜ +B′,
where
counterparts [2] and combine the results in the follow-
RM
ing way: B˜ = ˜b xˆ+˜b yˆ+Bzˆ
x y
[1] [1] [1] 16ζ B′ = b′xxˆ+b′yyˆ+b′zzˆ,
= R−1 + R+1 2R0 = . (14)
K R[−2]1 R[+21] − R[02] ∓βE˜0 where B˜ and B′ are reversing and stray non-reversing
magnetic fields, respectively.
Here we take θ = π/4, which are optimal polarization Equations (1) through (4) apply when the quantiza-
±
angles for the PV measurements (see next section). This tion axis is along the magnetic field, thus a rotation D is
method has the advantage that is independent of Edc, applied to each of the vectors E, B, E, and k such that
K
sothattheE-fieldbiasmaybechosenbasedontechnical DB zˆ. That is, we take
requirements of the experimental apparatus. ∝
B DB, E DE, E DE, and k Dk, (16)
→ → → →
IV. PV SIGNATURE: IMPACT OF where
APPARATUS IMPERFECTIONS
D=D( α ,yˆ)D(α ,xˆ). (17)
y x
−
While the current Yb-APV apparatus has been de- Here the matrix D(α,nˆ) represents a rotation about an
signed to minimize systematic effects, the PV mimicking
axis nˆ through angle α. The angles α and α are given
x y
systematics may be a result of a combination of multiple
by
apparatus imperfections. In order to understand the im-
portance of these effects, the electric and magnetic field α =(B b′)(b′ +˜b )/B2. (18)
misalignmentsandstrayfieldswereincludedinatheoret- x,y − z y,x y,x
icalmodelofthetransitionratesaswellastheexcitation Thus, the electric field E and the polarization vector
light’s deviations from linear polarization. In addition, E acquire additional components after the rotation (be-
we relax the assumption that δk = 0 and include the sides, for example, e and e˜ ).
y y
effects of the residual M1 transition. Due to the imperfections, the normalized-rate modu-
Thequantizationaxisisdefinedalongzˆ,andfollowing lation amplitudes now include additional terms besides
the ideal case model, the axis of the standing light wave the Stark- and the PV effects:
is collinear with xˆ. We added a small ellipticity to the
[1]
light polarization by taking RM r =r(Stark)+r(APV)+r(M1)+r(φ), (19)
[2] ≡ M M M M M
E = (yˆsinθ+zˆeiφcosθ), (15) RM
E
where r(Stark) is the Stark contribution due to the DC-
where φ is a small phase. For φ 1, the ellipticity M
of the light is 2φsin(2θ). The ele|ctr|ic≪field imperfections bias and the field imperfections, r(APV) is the PV-Stark-
M
are included by taking interferenceterm,r(M1) istheM1-Stark-interferencecon-
M
E=E˜ +E′, tribution, and rM(φ) is a contribution due to the distorted
linear polarization of the light (which is a Stark contri-
bution,butweexplicitlyseparatethecontributionlinear
where
in φ). Expressions for the lowest-order terms are sum-
E˜ = (E˜ xˆ+e˜ yˆ+e˜ zˆ)cos(ωt) marized in the Table I.
0 y z
E′ = E xˆ+e yˆ+e zˆ, The normalized-amplitude combination (14) has been
dc y z
chosen to determine the PV asymmetry. Since the M1
aretheACandDCcomponentsoftheelectricfield. Itis and ellipticity terms have opposite signs for M = 1,
±
assumedthatthey-andz-componentsoftheACfieldare their contributions to cancel, while the contributions
K
in phase with the leading oscillating E-field. The impact from r(APV) add.
M
5
PV asymmetry, 16ζ/βE˜, is 4 10−4, for θ = π/4 and
TABLEI:Lowest-ordertermscontributingtothenormalized ∼ ·
E˜ = 2 kV/cm. For a typical value of misalignments
transition-rate modulation amplitudes rM. an0d“parasitic”fields, e /E˜ and˜b /B (ontheorderof
y,z 0 x
r(APV) r(M1) r(φ) 10−3 in the present apparatus), the contribution of the
M M M
“parasitic”termsmaybeuptoafewpercentofthetotal
4ζcotθ
M =0 + 0 0 valueof . Waysofmeasuringthecontributionofthese
βE˜ K1
0 imperfections are discussed in the following sections.
M =−1 −4ζtanθ +4δkM(e˜y−e˜ztanθ) +4ezφtanθ
βE˜ βE˜2 E˜
0 0 0
M =+1 −4ζtanθ −4δkM(e˜y−e˜ztanθ) −4ezφtanθ V. EXPERIMENTAL APPARATUS
βE˜ βE˜2 E˜
0 0 0
Theforbidden408-nmtransitionisexcitedbyresonant
laser light coupled into the power-buildup cavity in the
TheStark-contribution,r(Stark),hasseveraltermsthat presence of the magnetic and electric fields. The transi-
M tion rates are detected by measuring the population of
are produced due to different imperfections and impacts
the 3P state, where 65% of the atoms excited to the
all three Zeeman components, M = 0, 1. In order to 0
± 3D state decay spontaneously (Fig. 1). This is done
determine which terms could potentially mimic the PV 1
by resonantly exciting the atoms with 649-nm light to
asymmetry in , we discriminate the PV contribution
K the 6s7s 3S state downstream from the main interac-
to with respect to the B-field reversal and flip of the 1
K tion region, and by collecting the fluorescence resulting
polarization angle, θ. Switching to a different Zeeman
from the decay of this state to the 3P and 3P states
componentofthetransitionisalsoareversal,whichisin- 1 2
and subsequently, from the decay of the 3P state to the
corporatedintheexpressionfortheasymmetry, . Anal- 1
K groundstate1S (556nmtransition). Aslongasthe408-
ysisofthenoiseaffectingtheaccuracyofPV-asymmetry 0
nm transition is not saturated, the fluorescence intensity
measurements demonstrate that the highest signal-to-
measured in the probe region is proportional to the rate
noise ratio is achieved when θ = π/4, and therefore,
± of that transition.
the polarization flip is a change of the polarization an-
AschematicoftheYb-APVapparatusisshowninFig.
gle by π/2. Thus, the normalized-amplitude combina-
2. A beam of Yb atoms is produced (inside of a vacuum
tion (14) must be determined for four different combina-
chamberwitharesidualpressureof 3 10−6Torr)with
tions of the B-field directions and light-polarization an-
≈ ×
an effusive source: a stainless-steel oven loaded with Yb
gles: (+B,+π/4), ( B,+π/4), (+B, π/4), and
K K − K − metal, operating at 500 600◦C. The oven is outfitted
( B, π/4),sothattermshavingdifferentsymmetries
−
K − − with a multi-slit nozzle, and there is an external vane
with respect to the reversals can be isolated:
collimatorreducingthespreadoftheatomicbeaminthe
1 1 +1 +1 (+B,+θ) horizontal direction. The resulting Doppler width of the
1
K − − K 408-nm transition is 12MHz [7].
K2 = 1−1 +1 +1 −1 K(−B,+θ) . ≈
4 +1 1 +1 1 · (+B, θ)
3
K   − −  K − 
  +1 +1 +1 +1  ( B, θ) 
K4    K − − 
     (20)
The result of this procedure is summarized in Table II.
TABLEII:Listofthelowest-ordertermscontributingtothe
asymmetry K for |θ| = π/4 sorted with respect to their re-
sponsetothereversals. K correspondingtoaratherlonglist
4
oftermsthatareinvariantwithrespecttoallreversals,isnot
shown in the table.
K1 K2 K3 FIG.4: (coloronline)Schematicofthepowerbuildupcavity.
8(e˜yezE˜+2e˜zey) + 1B6˜bEx˜ey + β1E6˜ζ 1B6bE′x˜ey 1B6bE′x˜ez Downstream from the collimator, the atoms enter the
0 0 0 0 0 maininteractionregionwheretheStark-andPV-induced
transitionstakeplace. Upto80mWoflightatthetran-
ThePVasymmetrycontributingto isB-fieldeven, sition wavelength of 408.345nm in vacuum is produced
1
K
θ-flip odd. It competes with the second-order terms that by frequency doubling the output of a Ti:Sapphire laser
cw
areacombinationoftheE-fieldandB-fieldalignmentim- (Coherent 899) using the Wavetrain ring-resonator
perfections and stray fields. Using the theoretical value doubler. After shaping and linearly polarizing the laser
of ζ 10−9 ea [14, 15] combined with the measured beam, 10mWofthe408-nmradiationiscoupledintoa
0
β =≃2.24+0.07 10−8 ea /(V/cm) [4, 7], the expected powerb≈uildupcavity(PBC)insidethevacuumchamber.
| | −0.12 × 0
6
FIG. 5: (color online) Schematic of the optical setup. Light at 408-nm is produced by frequency doubling the output of a
cw
Ti:Sapphire laser (Coherent 899) using the Wavetrain ring resonator doubler. The laser is locked to the PBC using the
FM-sideband technique. The PBC is locked to a confocal Fabry-P´erot ´etalon. This scannable ´etalon provides the master
frequency. The 649-nm excitation light is derived from a single-frequency diode laser (New Focus Vortex). The diode laser is
locked to a frequency-stabilized He-Ne laser using another scanning Fabry-P´erot´etalon.
The cavity was designed to operate as an asymmet- routinely monitored during the PV measurements. De-
ric cavity with flat input mirror and curved back mirror tails of the characterization of the PBC are addressed in
with a 25-cm radius of curvature and 22-cm separation Appendix B.
between the mirrors. The atomic beam intersects the We found that the use of the 408-nm-PBC in vacuum
cavity mode in the middle of the cavity, where the 1/e2 isaccompaniedbysubstantialdegradationofthemirrors.
radiusofthemodeinintensityis172µm. Theasymmet- Typically after 6 hours of operation, the finesse drops by
ricconfigurationhastheadvantageoflargermoderadius a factor of two. This is a limiting factor for the duration
attheinteractionpositioncomparedtoasymmetriccav- of the measurement run. The degradation of the finesse
ity. Alargermodeallowsustoreducetheac-Starkshifts, is due to the increased absorption and scattering losses.
consequentlyreducingthewidthofthe408nmtransition. This effect is reversible: the mirror parameters can be
Alternatively,thecavitycanbemodifiedtooperateinthe restored by operating the PBC for several minutes in
symmetric confocal condition where multiple transverse air, which makes performing a number of runs possible
modes can be excited, thereby increasing the effective without replacing the mirrors. However, it takes several
“mode” size. However, we were unable to obtain high hours with the present apparatus to reach the desired
power and stable lock for the confocal configuration. vacuum after exposing the PBC to air. Presently, this
ThecavitymirrorswerepurchasedfromResearchElec- effect is under investigation aiming for longer-duration
troOptics,Inc. Fortheflatinputmirrorthetransmission experiments and shorter breaks in between.
is 350 ppm with the absorption and scattering losses of A schematic of the PBC setup is presented in Fig. 4.
150 ppm total at 408 nm. The curved back mirror is de- The mirrors are mounted on precision optical mounts
signed to have a lower transmission of 50 ppm in order (Lees mounts) with micrometer adjustments for the hor-
to additionally suppress the net light wave vector and, izontal and vertical angles and the pivot point of the
therefore, the M1 transition amplitude. The absorption mirror face. The mirror mounts are attached to an Invar
and scattering losses in the curved mirror are 120 ppm. rodsupportedbyadjustabletablerestingonleadblocks.
The finesse and the circulating power of the PBC are The input mirror is mounted on a piezo-ceramic trans-
up to = 9000 and P = 8 W. These parameters were ducer allowing cavity scanning.
F
7
plated wires. This design allows us to reduce the stray
chargesaccumulatedontheelectrodesbyminimizingthe
surfaceareafacingtheatomicbeam, therebyminimizing
stray electric fields. AC voltage of up to 10 kV at a
frequency of 76.2 Hz is supplied to the electrodes via a
high-voltage amplifier. An additional DC bias voltage of
up to 100 V can be added.
The result of the electric field non-uniformity calcula-
tions is shown in Fig. 6. These calculations demonstrate
thaterrorsinthecenteringofthelightbeamwithrespect
to the E-field plates may induce substantial parasitic
components as large as, for example, e 5 10−3E˜ ,
z 0
| |∼ ×
producing parasitic effects comparable to the PV asym-
metry. In order to measure and/or compensate the
impact of the parasitic fields, additional electrodes de-
signed to simulate stray E-field components were added
to the interaction region. By applying high-voltage to
these electrodes (“correction electrodes” in Fig. 6), the
parasitic-field components may be exaggerated and ac-
curately measured as described in the following sections.
Light at 556 nm emitted from the interaction region
is collected with a light guide and detected with a pho-
tomultiplier tube. This signal is used for initial selec-
tion of the atomic resonance and for monitoring pur-
poses. Fluorescent light from the probe region is col-
lected onto a light guide using two optically polished
curved aluminum reflectors and registered with a cooled
photodetector (PD). The PD consists of a large-area
FIG. 6: (color online) Schematic of the E-field electrodes as- (1 1cm2)Hamamatsuphotodiodeconnectedtoa1-GΩ
×
sembly, and a result of the E-field modeling showing an X-Z transimpedancepre-amplifier,bothcontainedinacooled
slice of the amplitude of the E-field z-component, ez, in a housing (temperatures down to 15◦C). The pre-amp’s
midplane (Y=0) of the assembly normalized by the total E- −
bandwidth is 1 kHz and the output noise is 1 mV
field amplitude, E. The voltage is applied to electrode 1, and ∼
(rms). The 649-nm excitation light is derived from a
electrode 2 and the correction electrodes are grounded.
single-frequencydiodelaser(NewFocusVortex)produc-
ing 1.2 mW of cw output, high enough to saturate
the ≈6s6p3P 6s7s3S transition. Due to the sat-
0 1
ThelaserislockedtothePBCusingtheFM-sideband uration of this→transition, 3 fluorescence photons per
technique [16]. In order to remove frequency excursions atom exited to the 3P sta∼te are emitted at the probe
0
of the PBC in the acoustic frequency range, the cavity region. The natural width of the 649-nm transition is
is locked to a more stable confocal Fabry-P´erot ´etalon, 70 MHz, thus, its profile covers all transverse velocity
onceagainusingtheFM-sidebandtechnique. Thisstable groups(v )intheatomicbeam( 8MHzDopplerwidth
x
scannablecavityprovidesthemasterfrequency, withthe at 649 nm). A drift of the laser≈frequency ( 100 MHz
power-build-upcavityservingasthesecondarymasterfor per minute) is eliminated by locking the diod∼e laser to a
the laser. A schematic of the optical system is presented frequency-stabilizedHe-NelaserusingascanningFabry-
in Fig. 5. P´erot ´etalon with the scanning rate of 25 Hz. The spec-
The magnetic field is generated by a pair of rectangu- traldistancebetweenthe´etalontransmissionpeaksfrom
larcoilsdesignedtoproduceamagneticfieldupto100G the two lasers is measured during each scan and main-
with a 1% non-uniformity over the volume with the di- tained constant within an accuracy of 3 MHz, good
±
mensionsof1 1 1cm3 intheinteractionregion. Addi- enough to eliminate any degradation of the probe-region
× ×
tional coils placed outside of the vacuum chamber com- signal.
pensate for the external magnetic fields down to 10 mG The signals from the PMT and PD are fed into lock-
at the interaction region. The residual B-field of this in amplifiers for frequency discrimination and averaging.
magnitude does not have an impact on the PV measure- A typical time of a single spectral-profile acquisition is
ments since its contribution is discriminated using the 20 s. The signals at the first and the second harmonic of
field reversals. the electric-field modulation frequency are registered si-
Theelectricfieldisgeneratedwithtwowire-frameelec- multaneously, which reduces the influence of slow drifts,
trodesseparatedby2.1cm(seeFig.6). Thecopperelec- such as instabilities of the atomic-beam flux. The mod-
trode frames support arrays of 0.2-mm diameter gold- ulation frequency is limited by several factors. Thermal
8
distribution of atomic velocities in the beam causes a zero within the statistical uncertainty, which is the same
spread (of 2 ms) in the time of flight between the in- as that of the PV-asymmetry (see below).
≈
teraction region and the probe region. This, along with As can be seen from Table II, terms in associated
1
K
the finite bandwidth of the PD, leads to a reduction of with the fields imperfection are of crucial importance:
the signal-modulation contrast (see below). The choice
of the modulation frequency of 76.2 Hz is a tradeoff be- 16 e˜z ˜bx e˜y
e + +e .
tween this contrast degradation and the frequent E-field E˜0 " y 2E˜0 B! z2E˜0#
reversal.
Inordertomeasurethecontributionoftheseterms,arti-
ficially exaggerated E-field imperfections both static and
oscillating, eex, eex, e˜ex and e˜ex, are imposed by use of
VI. RESULTS AND ANALYSIS z y y z
the “correction electrodes” (see Fig. 6), and two sets of
the experiments were performed. In the first one, a DC-
In Fig. 7 a profile of the B-field-split 408-nm spectral voltage was applied to the correction electrodes, and the
lineofthe174Ybisshown. The649-nm-light-inducedflu- measurements were done reversing eex and eex. These
y z
orescence was recorded during a single profile scan. Sta- experiments yield values of e˜ and e˜ +2E˜ ˜b /B. In the
y z 0 x
tistical error bars determined directly from the spread of
secondset,anAC-voltagemodulatedsynchronouslywith
dataaresmallerthanthepointsinthefigure. Thepecu-
the main E-field is applied to the correction electrodes.
liar asymmetric line shape of the Zeeman components is
In order to reverse the sign of the parasitic terms a π-
a result of the dynamic Stark effect [7]. During a typical
phase-shiftofe˜ex ande˜ex withrespecttothemodulation
y z
signal is employed by switching the wiring of the correc-
tionelectrodes. Thus,valuesoftheDC-imperfections,e
y
and e , are determined. The magnitudes of the applied
z
electricfieldsandtheirdistributionsarecalculatedusing
a 3D-numerical-model of the interaction region. The re-
sults of the experiments are presented in Table III. The
TABLE III: Results of measurements of the electric field
imperfections using artificially exaggerated AC- and DC-
components, e˜ex and eex. These fields were generated by
y,z y,z
use of the correction electrodes, Fig. 6. E˜ =2000(2) V/cm.
0
DC-Set AC-Set
FIG. 7: (color online) A profile of the B-field-split 408-nm
Exaggerated imperfections (V/cm)
spectral line of 174Yb recorded at 1st- and 2nd-harmonic of
themodulation. AlsoshownisasimulatedPVcontributionin eex =−140(2) e˜ex =−120(2)
y y
the first-harmonic signal. E˜=5 kV/cm; DC offset=40 V/cm; eex =20(2) e˜ex =30(2)
z z
θ=π/4; the effective integration time is 200 ms per point.
Measurements (mV/cm)
eex e˜ex
experimental run 100 profiles are recorded for each com- e˜y2Ez˜ =16(10) ey2Ez˜ =4(5)
0 0
b(4in00atpiornofiolfetshceanmsaingnteottiacl)fi.elIdnaonrddetrhteopcoolmarpizuatteiotnheannogrle- (2E˜0˜bBx +e˜z)2eEey˜x =442(10) ez2e˜Eey˜x =40(5)
0 0
malized amplitude, r , of a selected Zeeman component,
q Parasitic fields (V/cm)
the actual first-harmonic signal near the Zeeman peak
is divided by the respective second-harmonic signal and e˜y =3.2(2) ey =0.5(0.6)
then averaged over a number of the data points1. Then, (2E˜0˜bBx +e˜z)=−12.6(0.3) ez =−1.3(0.2)
the combination of Eq. (14) is computed for each pro-
K
filescanfollowedbyaveragingtheresultoverallthescans
at a given B-θ configuration. This procedure is repeated net contribution of these imperfections to 1 in the ab-
K
for all four reversals, and all B-θ symmetrical contribu- sence of the exaggerated fields is found to be 2:
tions, , are determined. In the present experiment,
thevalKue1−s4of 2,3,4-termsarefoundtobeconsistentwith e e˜z + ˜bx +e e˜y =
K y 2E˜0 B! z2E˜0
= 2.6(1.6)stat.(1.5)syst. mV/cm. (21)
−
1 In the normalized rate calculations only data points having in-
tensity higher that 1/3 of the respective Zeeman peak are used
to avoid excessive noise from spectral regions with low signal
intensity. 2 ComparewiththePVasymmetryparameterζ/β≈40mV/cm.
9
The systematic uncertainty comes from a sensitivity of
the numerical model of the E-field, which is used for cal-
culating the exaggerated fields in the interaction region,
to an imperfect approximation of the electrode-system
geometry. These experiments suggest that this field’s
imperfection cannot mimic the PV-effect entirely, never-
theless,itappearstobeamajorsourceofsystematicun-
certainty impacting the accuracy of the PV-asymmetry
measurements. Themostprominentcontributionisgiven
by a combined effect of the parasitic components of the
electric field and the non-zero projection of the lead-
ing magnetic field on the direction of the electric field:
e (e˜ /2E˜ +˜b /B). The PV-asymmetry parameter, ζ/β
y z 0 x
isobtainedfromthemeasuredvalueof bycompensat- FIG. 8: (color online) The PV interference parameter ζ/β.
1
K Mean value: 39(4) (3) mV/cm, |ζ| = 8.7 ± 1.4 ×
ing for the influence of these magnetic- and electric-field stat. syst.
10−10 ea .
imperfections, Eq. (21). 0
There is another effect that cannot, by itself, mimic
the PV-asymmetry, but needs to be taken into account
model of the transition rates employing the field geome-
forpropercalibration. ThiseffectisrelatedtotheE-field
tryshowninFig.2. Thedirectionand, thus, thesignsof
modulationimplementedinthepresentexperiment. The
theelectricandmagneticfieldsaswellasthepolarization
atoms are excited to the metastable state, 6s6p3P , by
0 angle θ were calibrated prior to the PV measurements.
the light beam in the interaction region and then travel
Special care was taken of detecting parasitic phase shifts
20cmuntiltheyaredetecteddownstreamintheprobe
∼ in the lock-in amplifier. A signal from an arbitrary func-
region. Due to the spread in the time-of-flight between
tion direct digital synthesis (DDS) generator simulating
theinteractionandproberegions,thephasemixingleads
the output of the probe region photodetector was fed
to a reduction of the signal modulation contrast at the
into the amplifier. The signal is comprised of a sum of
probe region, and it depends on the signal-modulation
twosinusoidalwaveforms,onefrequencydoubled,attenu-
frequency. Since the signal comprises two time-scales of
ated,andphaseshiftedwithrespecttotheother. Results
interest, first- and second-harmonic of the E-field mod-
of the signal parameters measurement from the lock-in,
ulation, the contrast reduction is different for the two.
suchasthefirst-to-secondharmonicamplituderatio,rel-
Therefore,theratioofthesignalmodulationamplitudes,
ativephaseshiftanditssign,arecomparedtothoseused
r , on which we base the PV-asymmetry observation,
M in the DDS generator to simulate the signal. The differ-
appear altered in the probe region compared to what it
ence in the measured and generated amplitude ratio is
would be at the interaction region. The amplitude com-
found to be below 0.01%, and the relative phase shift is
bination, , and, therefore, the PV-parameter, ζ/β, are
K detectedwithin 1.5◦. Norelativesignflipsbetweenthe
similarly affected. In our data analysis, a correction co- ±
first- and second-harmonic amplitudes were detected.
efficient, C , is introduced, which has been calculated
0
theoretically:
ζ ζ VII. ERROR BUDGET
= C .
0
β β
(cid:20) (cid:21)probe reg. (cid:20) (cid:21)real
Thepresentmeasurementaccuracyisnotyetsufficient
Under present experimental conditions, this coefficient, to observe the isotopic and hyperfine differences in the
C ,isfoundtobe1.028(3),andthemeasuredPVparam- PV amplitude, which requires an accuracy better than
0
eter is corrected accordingly. Principles of its derivation 1% for PV amplitude in a single transition [17–19]. In
≈
are given in Appendix C. the present apparatus the signal levels achieved values
In Fig. 8, the PV interference parameter ζ/β is shown high enough to reach the signal-to-noise ratio (SNR) of
as determined in 19 separate runs ( 60 hours of integra- 2/ τ(s) for the PV asymmetry if the noise were domi-
∼
tionintotal). Itsmeanvalueis39(4) (3) mV/cm, nated by the fluorescence-photon shot-noise (τ is the in-
stat. syst.
p
which is in agreement with the theoretical predictions tegration time). This is good enough to reach the sub-
[14, 15]. The value of the PV parameter was extracted percent accuracy in a few hours of integration. However,
using the expression given in the first column of Ta- a number of additional factors limit the accuracy.
ble II, taking into account the calibration correction Oneofthemostimportantnoisesourcesisthefluctua-
C0. Thus, ζ = 8.7 1.4 10−10 ea0, which is the tionsofthemodulating-andDC-fieldparametersduring
| | ± ×
largest APV amplitude observed so far (here we used the experiment. The first- and the second-harmonic sig-
|β|T=he2s.2ig4n−+00o..f0172th×e1P0V−8inetaer0f/e(rVen/ccemp)a[r4a,m7e])t.erζ/β isfound namalsplditeupdeen,dE˜di,ffearnednttlyheonDCth-ebimaso,dtuhluast,inag neloeicsteriicnfitehlde
0
by comparing the measurements with the theoretical electronics controlling the fields contaminates the first-
10
to-second harmonic ratio directly. A substantial effort resonance is expected to be a factor of about √2 higher
was made to cope with this problem: a home-built high- than at the wing due to larger signal. It was found,
voltage amplifier used in the first 13 runs was replaced however, that the SNR went up by a factor of 4 by tun-
by a commercial Trek 609B amplifier and a circuit con- ing from the wing to the peak of the resonance. This
trolling the DC-bias was upgraded. This allowed us to demonstratesthatthemainsourceofnoiseisnotphoton
controltheDC-biasandE˜ withmV-scaleaccuracythat statistics but fluctuations in the spectral reference. In-
0
would make the SNR to approach the shot-noise limit if deed,thefrequencyexcursionsatthewingofthespectral
this were the only source of the noise. As seen in Fig. 8, line produces substantially more intensity noise due to a
the last six measurements exhibit higher accuracy than steeper spectral slope than that at the peak, where the
the rest. These are the runs after the HV-system up- slope is nominally zero. It must be emphasized, that in
grade. However, the present SNR of 0.03/ τ(s) is the case of slow frequency excursions (compared to the
≈
worse by almost two orders of magnitude than the shot- E-fieldmodulationperiod),thenoiseinthefirst-andthe
p
noise limit. second-harmonicchannelswouldbecanceledintheratio.
Thereareotherfluctuationsinthesystemparameters, However, fast excursions can generate noise in the signal
such as light intensity fluctuations in the PBC, fluctua- ratio.
tions of the spectral position of the PBC resonance with The factors affecting the measurement accuracy men-
respect to the frequency reference, and noise in the de- tioned above have an impact on the statistical error of
tection system. All of them contribute to the noise in theresult. Thepresentsystematicerrors(summarizedin
the first- and the second- harmonic signals but we found Table IV) has nearly the same significance as the statis-
that such noise largely canceled in the ratio rM. tical one and also comprises a number of factors.
However, there is a noise source, which is not canceled One of the most significant factors is the uncertainty
in the ratio. The following experiments demonstrated in the field-imperfection contributions, Eq. (21). This
that this noise source is related to frequency excursions uncertainty is mostly due to statistical factors such as
of the Fabry-P´erot ´etalon serving as the frequency ref- laser drifts, nevertheless, it provides an offset to the PV-
erence for the optical system. In these experiments the parameter. Since the measurement of this contribution
excitation light was frequency tuned to a wing of the is actually the same measurement as the PV-effect, any
atomicresonance,andthefirst-andsecond-harmonicsig- improvementsofthestabilityreducestheoverallsystem-
nals were recorded without scanning over the resonance. atic uncertainty. We would like to emphasize also that
Then, thesamewasdonewhenthespectralpositionwas Eq.(21)representsameanvalueoftheimperfectioncon-
setatthepeakoftheresonance,andachangeoftheSNR tributionovernumerousexperimentsaveragingoverpos-
for the harmonics ratio was determined. These experi- sible fluctuations of the field-imperfection contribution.
ments were performed using the upgraded HV-system. These fluctuations may be partially responsible for the
ResultsofthemeasurementsarepresentedinFig.9. For variance in the PV-parameter, and, thus, the statisti-
cal uncertainty of its value. This fact demonstrates that
the elimination of the field-imperfections is an essential
requirementforimprovingtheoverallaccuracyoftheex-
periments.
Another significant source of the systematic uncer-
tainty is the uncertainty in the value of the electric field
in the interaction region. While the voltage applied to
the E-field plates and the correction electrodes is con-
trolled precisely, the actual E-field value used in the PV
parameterdeterminationdependsontheaccuracyofthe
numericalmodelingoftheelectric-fielddistributioninthe
particular geometry. There are two factors in the model
contributing to the uncertainty: finite accuracy of mea-
surements of the interaction region geometrical parame-
ters,andtheimperfectapproximationofthegeometryin
the numerical simulation.
However,whilethissystematicuncertaintyplaysasig-
FIG. 9: (color online) Impact of the frequency excursions of
nificant role for measurements of the PV parameter of a
the Fabry-P´erot ´etalon on the noise level in the harmonics
single isotope, for the isotope ratios this uncertainty will
ratio. A change in the noise level when the optical system
cancel (or will be substantially reduced), if the measure-
was tuned from the wing of the atomic resonance to its peak
mentsobservingdifferentisotopesareperformedwithout
is shown. In the inserts above the excitation light spectral
position is shown schematically with respect to the atomic changing the E-field geometry. The same is true for the
resonance. Arrows denote the fluctuations. calibration parameter, C0, which also cancels in the iso-
tope ratios.
a shot-noise-limied signal, the SNR at the peak of the There are other, rather minor, factors contributing to