Penning trapを用いた
電子g因子測定とDark Photon/Axion探索

Xing FanNorthwestern University
京都大学 高エネルギー研究室セミナー

2022.Dec.28

Masason Foundation

今日の二つの話
g因子測定
Measurement of the Electron Magnetic Moment
X. Fan,1, 2, ⇤ T. G. Myers,2 B. A. D. Sukra,2 and G. Gabrielse2, †1

2

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
Center for Fundamental Physics, Northwestern University, Evanston, Illinois 60208, USA(Dated: September 28, 2022)

arxiv: 2209.13084

The electron magnetic moment in Bohr magnetons, µ/µB = 1.001 159 652 180 59 (13) [0.13 ppt],
accepted
byprecisely
Phys. Rev.
Lett.
is consistent with a 2008 measurement and is 2.2 times more precise.
The mostmeasured
property of an elementary particle agrees with the most precise prediction of the Standard Model(SM) to 1 part in 1012 , the most precise confrontation of all theory and experiment. The SM test
will improve further when discrepant measurements of the fine structure constant ↵ are resolved,
FERMILAB-PUB-22-482-SQMS-T
Dark
Photon/Axion
since theprediction
is a function of ↵. The magnetic moment measurement
and SM theory togetherandaxion
1predict ↵ = 137.035 999 166 (15) [0.11 ppb]
One-Electron Quantum Cyclotron as a Milli-eV Dark-Photon Detector

Xing Fan,1, 2, ⇤ Gerald Gabrielse,2, † Peter W. Graham,3, 4, ‡ Roni Harnik,5, 6 Thomas G. Myers,2The quest to find physics
beyondRamani,
the Standard
magnetic
momentoperator
3, §2
3Harikrishnan
BenedictModel
A. D. Sukra,The
SamuelS. Y. Wong,and Yawen
Xiao3 for a spin-1/2 electron,

Particle Physics (BSM) is1 Department
well motivatedtheof Physics,because
Harvard University,
Cambridge, Massachusetts 02138, USAgS2Centerfor
FundamentalPhysics,
Northwestern
University, Evanston, Illinoisµ60208,
USA
M is incomplete.3 No CP violation mechanism is large=261801µB (2022),(1Phys.
Rev.
Lett.
129,
Stanford Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA94305,
USA
2~/2
4 produced in theough to keep matter and antimatter
Kavli Institute for Particle Astrophysics & Cosmology,
Departmentof Physics,
Stanford
Stanford, CA 94305, USAg Bang [1] from annihilating
as theuniverse
cooled[2], University,
is proportional
to its spin S normalized to its spin eigen
5Superconducting
Quantum
Materials
andSystems Center(SQMS), Fermilab, Batavia, IL 60510, USAFor
Englishversion,
therearevideos
inrk matter [3, 4] has 6not
beenPhysics
identified,
en-Accelerator
value~/2. For
electroncharge
Theoretical
Division,andFermidark
National
Laboratory,
Batavia,
Illinois 60510,
USA e and mass m, dimen(Dated: August 16, 2022)
http://cfp.physics.northwestern.edu/gabrielse-group/gabrielse-home.html
gy [5, 6] and inflation [7, 8] have no SM explanation.
sional analysis gives a Bohr magneton, µB =2 e~/(2m)電子の磁気モーメント(g因子)• 円運動する電子の磁気モーメント
eAngular momentum L−𝑒𝝁=𝑳2𝑚ℏ!"ℏ• 電子のスピン角運動量は𝑺 = なので， 𝝁 = #$ × # ?
"ℏ
S= #e-−𝑒 ℏ
𝝁=× ×𝑔2𝑚 23

電子のg因子と標準模型の歴史
古典論:

g=1

Dirac’s theory:

g=2

QED+繰り込み群＋QCD:e-

g=2.002 319 304 36 …

e-B=?
𝒈𝟐e-=+𝟏++…

𝟏𝜶𝜶 𝟐+×+ −𝟎. 𝟑𝟐𝟖 …×+⋯𝟐𝝅𝝅fine structure constant: Rb および Csを用いて独立に測られる

5 loopsまで+ 仮想μ, τ+ハドロン4標準模型の予測と乖離Aoyama, Kinoshita, Nio,
Atoms 2019, 7(1), 28
and references therein

5次のQED計算

hadron loop

α 測定の誤差𝑔(Rb) = 1.001 159 652 180 254 (12) (11) (93)2Nature 588, 61 (2020)
𝑔(Cs) = 1.001 159 652 181 598 (12) (11) (234)
2Science 360 191 (2018)

this work
g/2 2022
g/2 20082008
g/2測定

予測(Rb)
SM(Rb) 2020
SM(Cs) 2018
予測(Cs)
2018−1−0.5

2.1σ
3.8σ
00.5
(g/2 - 1.001 159 652 180 59)×1012

11.55なぜ電子のg因子なのか？
• 電子のg因子なんかより高精度に測れているものはいくらでもある• 水素原子1s-2s: 15桁• 光格子時計の周波数: 18桁• 重力波によるひずみh: 24桁→しかし理論的にその値を予測できない(原子核モデル，束縛系・多体系，そもそもパラメータが不明)• 高精度の予測・測定のためにはシンプルさが大事→Penning trap中の電子のg因子
6g因子測定の原理
理想的には電子1つを磁場中に置き𝒈 ℏ𝒆𝑩∆𝑬𝒔 = 𝟐𝝁 & 𝑩 = ×𝟐𝒎

Beor

spin歳差運動
エネルギー 𝒉𝝂𝒔

cyclotron運動
エネルギー 𝒉𝝂𝒄

実際には，電子をPenning Trapで捕獲する
magnetic field

electric field

平面電極

e-B

リング型電極横方向の閉じ込め

+e- +
平面電極

縦方向の閉じ込め7

Penning trap中での運動

e-B

electric field+Bmagnetic field

T<100mKelectrodee- +spacerCyclotron 運動 νcAxial 運動 νz
Magnetron 運動 νm

例 B=5.3T, V=33V
νc~150GHz >> νz~100MHz >> νm~50kHz
νs= νc× g/2 ~150.17 GHz

8単一電子の検出B

axial振動が電極に誘起する 電流を検出(~100MHz, 10-100 fA)

axial motion νzelectron gun(表面電界放出)-1000V印加

excitated amplitude(a.u.)
detectedpower (a.u.)

0.50.4

アンプ出力をモニターしながら
electron gun弱く駆動

0.3
0.2
0.1
0−0.1
0

単一電子の非破壊検出!
100

200

300

400time (s)

500

600

700

9

cyclotron運動は量子的な描像
トラップ内での電子の準位(Landau準位)|↓>

B=5.3T, T=100mKe-νc =150 GHzhνc/kB =7.2K

nc=2

|↑>
cyclotron

anomalyνa = νs - νc=170MHz

νcnc=1γc ~5snc=0nc=1
νs =νc×g/2~150.17GHz
nc=010なぜνsではなくνaを測るのか?
𝑔2=𝜈!
𝜈"~1.001
😣 これを 10-13 精度で測るより

=

𝜈#1+
𝜈"

~0.001

🤗 これを10-10 精度で測る方が遥かに楽11

遷移をどう観測するか？
zz

axial frequency νzをモニター

nc=0
nc=1z軸に沿うポテンシャル

V(z) = eVz2 + μtotB2z2e- +2

νc drive applied

nc=0
axial frequency

nc=2

1

nc=1

0

nc=0zpowerνz shiftΔω(/1.3Hz)/δnc=1V(z)磁場からの電場からのNickel
ポテンシャル ポテンシャル
B(z) = B0 + B2z2μtot:∝ (nc+ms)

020

4060time (s)80100122008年から装置を一新
真空チェンバー

希釈冷凍機

NbTi磁石入りLHeデュワー

T<100mK

上から希釈冷凍機挿入

2m2m

B

5 cm

Ag50 cm

B

Penning trap

gold plated
silvertitanium
チェンバー

底にNbTi ソレノイド
13

2008年からの改善点
1. 再現性・安定性の向上
Ø長期の安定したデータ取得のため，モジュラーで，冷却で壊れないハードウェアを多く製作した

2. 磁場の一様性・ドリフトの改善

統計量5倍(本日は省略)Ø新しく3Heを用いた磁力計を製作
X.Fan, et al., Rev. Sci. Instrum. 90, 083107 (2019)

Ø一様性も5倍改善

3. Penning trapによるνcのずれ(cavity QED)の理解
14g因子の測定サイクル4s

3s

3s

冷却

cyclotron
ドライブ

遷移したか測定

1s

4s

30s

60s

冷却

anomalyドライブ

遷移したか測定νc• cyclotron と anomaly を交互に測定→長期的なBドリフトをキャンセル→遷移確率 vs ドライブ周波数を記録• 今回anomaly drive周波数をblindしたνatrue = νaset in computer + X

νa =νs - νctime

|↓>
nc=1

|↑>
𝜈a

νc

nc=1
nc=0

nc=0

RF generator

15共鳴曲線(遷移確率vsドライブ周波数)times) after drives are turned o↵. A maximum quantum
jump rate of 40% suggests a slight power broadening, but⌫¯a iscyclotron
still determined faranomaly
more precisely than f¯c .
transition probability

transition probability(a)0.4(b)

0.4

0.3
0.2
0.1
0 −4−2024drive detuning (ppb)

点線(理解されてる形)がなまった形をしている

FWHM 0.3ppb

0.3
0.2

noise power (a.u.)

2008年よりx4倍細く

2008年よりx1.1倍細く𝑔 𝜈𝜈(c) = s = 1 + a2 𝜈c𝜈c0.80.6
Ø cyclotron の共鳴曲線は理解できてる形(点線)より太かった0.2→ 主要な系統誤差の一つ

0.4

0.1
0 −4−202

4drive detuning (ppb)

L. S. Brown, PRL 52, 2013 (1984)
X.0Fan and G. Gabrielse, PRA 103, 022824 (2021)−40 −2002040frequency - ν z (Hz)FIG. 3. Quantum jump cyclotron (a) and anomaly (b) line
shapes that are measured (points), predicted (dashed) and16

cavity QED効果
zxy平面で見ると円筒型
マイクロ波 cavity

Penning トラップの壁

e-

鏡像電荷𝑔
𝜈s
=
2 𝜈c + Δ𝜈ccav
Ø Penning trapの壁の効果でνcだけが変わる(νsは変わらない!!!)Ø ずれの大きさはνc とcavityの共振周波数で決まる→cavityの共振周波数を測り,そこから離れた周波数νcでgを測定
17

(g/2 - 1.001 159 652 180 59)×1012

新しく補正の方法を開発し，正しくcavity効果を評価(したつもり)
2cyclotron frequency (GHz)10012014080160

1.5
1SM(Cs)0.5short bar: stat only
long bar: total

0−0.5SM(Rb)

−1

−1.5
−22.5

近い測定点同士はエラーに相関アリχ2/ndf=13.05/10, p=0.22
33.5

4

4.5

55.5

6B (T)

Ø 11 fields from 3.1 T to 5.4T
相関を考慮して全データをアベレージ

18

結果
(a)

−0.5

0ppt

0.5

1

g/2 2022
g/2 2008
Rb
Cs−1−0.5(b)−100.5(g/2 - 1.001 159 652 180 59) ×1012−0.5 ppb

11.5

00.5

g/2 2022
g/2 2008
Rb
Cs

−0.2

−0.15

−0.1
−0.05
06-1(α - 137. 035 999 166) ×100.05g/2 = 1.001 159 652 180 59 (13)stat.
0.029×10-12
sys. (line shape)
0.094×10-12
sys. (cavity効果)
0.090×10-12

0.1

19

トラップされた電子を用いたDark
Photon/Axion探索
PHYSICAL REVIEW LETTERS 129, 261801 (2022)
PHYSICAL REVIEW LETTERS 129, 261801 (2022)

One-Electron Quantum Cyclotron as a Milli-eV Dark-Photon Detector
Xing Fan ,1,2,* Gerald Gabrielse,2,† Peter W. Graham ,3,4,‡ Roni Harnik,5,6 Thomas G. Myers,2 Harikrishnan Ramani ,3,§
One-Electron Quantum Cyclotron
as a Milli-eV Dark-Photon Detector
Benedict A. D. Sukra ,2 Samuel S. Y. Wong ,3 and Yawen Xiao 3
1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

Xing
Fan ,1,2,* Gerald Gabrielse,2,† Peter W. Graham ,3,4,‡ Roni Harnik,5,6 Thomas G. Myers,2 Harikrishnan Ramani ,3,§2
Center for Fundamental Physics, Department of Physics
and Astronomy, Northwestern
University, Evanston,
Illinois 60208, USA
3Benedict A.
D. Sukra,2 SamuelS. Y. Wong,3 University,and YawenStanford,
Xiao 3 California 94305, USA
Stanford Institute for TheoreticalPhysics,
Departmentof Physics,
Stanford

Collaboration work with
Stanford University
Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics, Stanford University, Stanford, California 94305, USA
Quantum
MaterialsandSystems5 Center(SQMS),Illinois2022)
60510, USA(Received23 August2022;
accepted
December2022;Fermilab,
publishedBatavia,
23 December• SuperconductingPeter
GrahamTheoretical Physics Division, Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
We propose using trapped electrons as high-Q resonators for detecting meV dark photon dark matter.
(Received 23 August2022; accepted 5 December 2022; published 23 December 2022)• Harikrishnan
Ramani
When the rest
energy of the dark
photonmatches the energy splitting of the two lowest cyclotron levels, thefirstexcitedstateof theelectron
cyclotron
will be resonators
resonantly forexcited.
A proof-of-principle
We
proposeusing
trappedelectrons
as high-Q
detectingmeV dark photonmeasurement,dark matter.
• SamuelS.
Y.
Wong
carriedthe
outrestwith
one electron,
demonstrates
that thebackgrounda 7.4
day search.
It sets
Whenenergyof the dark
photon matches
themethodenergyissplittingof thefree
twooverlowest
cyclotronlevels,thea limit
on dark
matterat 148 will
GHzbe(0.6meV) thatis around
75 times better measurement,
than previous
firstexcitedstatephotonof the dark
electron
cyclotron
resonantlyexcited.
A proof-of-principle
constraints.
Dark
photondarkdemonstrates
matter in the that
0.1–1
massisrange(20–200GHz)
be detected• YawenXiao
carriedout withoneelectron,themeVmethodbackgroundfree
overcould
a 7.4 likely
day search.
It setsa similar
sensitivity
an matterapparatus
designedfor dark
photonaat limit
on dark
photon indark
at 148GHz (0.6meV)that isdetection.
around 75 times better than previous
constraints. Dark
photon dark matterin the 0.1–1 meV mass range (20–200 GHz) could likely be detected
Fermilab
NationalLaboratory
DOI: 10.1103/PhysRevLett.129.261801
at a similar sensitivity in an apparatus designed for dark photon detection.
• Roni Harnik

14
Departmentof Physics,
HarvardDepartment
University, ofCambridge,
Massachusetts
02138,USA California 94305, USAKavli
InstituteforParticle
Astrophysics
and Cosmology,
Physics, Stanford
University,Stanford,
2Center for 5Fundamental
Physics,
Department
ofPhysics
andAstronomy,
Northwestern
University,
Evanston,
IllinoisSuperconducting Quantum Materials and Systems Center (SQMS), Fermilab, Batavia, Illinois 60510,
USA60208, USA
36
Stanford Institutefor Theoretical
Physics,
Department
Physics, Stanford
University,
Stanford,
94305, USA
Theoretical
Physics Division,
Fermi
NationalofAccelerator
Laboratory,
Batavia,
Illinois California
60510, USA
456

DOI: 10.1103/PhysRevLett.129.261801

The particle nature of dark matter (DM) and its inter-of an apparatus designed for making the most accurate20

Dark Matter/Dark Photon/Axion
ダークマターハローの性質• SM 粒子との結合は弱い• 速度 𝑣/𝑐 ≈ 10!"• local 密度 ρDM ~0.4 GeV/cm3Axion

Dark Photon

strong CP問題の解に付随する粒子

U(1)ゲージ場

Ø mass ma

Ø mass mA’Ø ℒ = 𝜒 𝐁 5 𝐁< − 𝐄 5 𝐄<
dark photon A’χØ ℒ = 𝑔=>> 𝑎𝐄 5 𝐁
SM photon AAxion
agaγγ

SM photonAA21

探索の仕方• Dark Photon
DPに付随するE fieldを検出
dark photon
A’ordinary photonAχdetectorman diagram showing the kinetic mixing of DP field and the ordinaryeld.

• Axionordinary photonAB field

gaγγthe model. For the model that the DP arises fromwidelyAxion
depending ony, 1012a<< 103

is predicted [134]. There are stringent limits on

偏光E//Bdetector, for example from cosmological observation [130] and solar lifetime [151],

ide range of mA0andA, especially around mA0 ⇠meV and< 10 9 , is22B

トラップされた電子はmeV光子検出器

dark photon in galaxyelectron(s)left at nc=0
cyclotron motion
e-nc=0nc=1• そもそもmeVは他の領域に比べデバイスが少ない• チューナブルな検出エネルギー 0.1-1 meV• nc=0 → nc=1 遷移はバックグラウンドフリー• 強磁場下で何の問題もなく動く (=Axionに応用できる)23

meVは狙い目
1−810frequency (GHz)
3102
101051041010−9cyclotron 周波数は自然に
χ10−1020 GHz – 200 GHz
0.1 meV – 1 meV
0.7 T – 7 T

10−12DP limits

10−1310−14−610−51010−410

1−8−310−210mA' (eV)frequency (GHz)
3102
101010−1

15104

1010−9aγ γ(Gev-1)

10−10g𝑒𝐵
𝜔" =𝑚

10−1110−1110−12QC

10−13nxioaD

10−14axion limits

10−15−1610−610−51010−4−310ma (eV)10−210−124 1

B

バックグラウンド フリー
量子サイクロトロン準位
T=100mKnc=2
平均量子数
𝑛8 𝑐 =

nc=1= 1.8×10!?@exp ℏ𝜔𝑐 − 1𝑘𝐵 𝑇
blackbody photonからの励起レート

νcnc=01

blackbodyphotonΓ)*+,-)./0 = 10-31 s-1-> 1013 ×(age of universe)25

Demonstration Search
電子を基底状態nc=0に放置してモニターするだけ
expected DP excitation

1Δω z / δ

nc = νz shift /(1.3Hz)

monitor νz

0−10

サイクロトロン周波数測定
51015time (h)2026得られたDP Limit(a)−20010−7••

合計: 638870s =7.4 days
DP による信号ナシωA' /2π - 148 047 782 (kHz)−100010020010−8Cosmology
XENON1Tχ

10−910−10ü x75 improved limit at 0.6 meV
ü g因子用の装置での制限
10

this work−6×10−11−1−0.500.5mA' - 0.612 276 43 (meV)1

ただしあまりに細すぎる
DP/Axionに特化した装置を作ると?
→ next page27

いろんな改良アイデア
アイデアüopen endcap trapを使い
ü外部にfocusing antennaをおく
ü10個くらい電子をトラップする−81010−9χ

M
10−10 D1

frequency (GHz)31021010olomcos

1050cmexternal foucusing antenna
エンハンスメント: K ~105solar
K=105, N=10
T=1 year−121010−135104gy

10−11n =0
nc=0 nc=0 c
nc=1 nc=0n=0c
nc=0n =0n =0 nc=0 cdemonstrated
expected

Haloscopesc−141010−6

10−510−4

open endcap trap

10−3
mA' (eV)

10−2

10−1

1

28

アクシオン用の実験も考え中z

Bℒ = 𝑔!"" 𝑎𝐄 & 𝐁
AxionからのE field
cyclotron motion
e-nc=0nc=1

Axionからの電場(磁場と並行)をcyc.運動平面(磁場と垂直)に変換する必要あり29

Shamelessly Borrowing an Idea
BREAD collaborationの偏光変換アンテナ
PRL 128, 131801 (2022)

だがそれでも…

open endcap trap
10−81

no B sweep
Δω/ω=10-6frequency (GHz)
3102
10105104

10CAST(solar)

10−9gaγ γ(Gev-1)

10−10
10−1110−12QC

Haloscopes

ledoDMs10−13

B=10T, K=105,
N=10, T=1 year

10−14
10−15
10−16
10−6

10−510−4

10−3ma (eV)10−210−1130Summary
1. 新しく建設した装置で電子のg因子を測定した
Ø 非常に広く磁場を変えて系統誤差をより詳しくスタディ.

2. meV Dark Photonサーチをdemonstration
Ø 0.6 meVでのバックグラウンドフリーサーチ
Ø 感度向上案とアクシオンへの応用も設計中!

Northwestern
G. Gabrielse
X. FanT. Myers
B.A.D. Sukra

Masason Foundation
31

Summary
1. 新しく建設した装置で電子のg因子を測定したØ 非常に広く磁場を変えて系統誤差をより詳しくスタディ.

PhD学生
Ø 0.6 meVでのバックグラウンドフリーサーチ
ポスドク
Ø 感度向上案とアクシオンへの応用も設計中!
急募!!!

2. meV Dark Photonサーチをdemonstration

Northwestern
G. Gabrielse
X. FanT. Myers
B.A.D. Sukra

Masason Foundation32