frequency and quantum metrology research group at uwa · rotating bulk acoustic wave oscillators....

Quantum Technologies for Axion Dark Matter Detection IncubatorProf. Michael Tobar

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Frequency and Quantum Metrology Research Group at UWA

Precision measurement => Phase, Frequency, Energy, Time

Technology: High-Q -> Narrow Line Width Systems: Low Noise Techniques Classical and Quantum (SQL)

New Tests of Fundamental Physics

Applications: Sensors, Clocks, Radar etc.

A Short History of My Attempt to Measure Nothing with Precision Measurement

CSO

CARRIER SUPPRESSION INTERFEROMETER

Sapphire Loaded Cavity (SLC) Resonators

Temperature Q-factor (10 GHz)

300 K 2 10 577 K 3 10 74. 2 K 4 10 9

Key features

Very high Q-factors at WG- modes

Frequency-temperature turning points

Electric field density plot ( H16,1,1 – mode)

FrequencyFrequency vs vs Temperature Temperature

-4 10-8

-3 10-8

-2 10-8

-1 10-8

0

1 10-8

2 10-8

2 4 6 8 10 12 14 16

Frac

tiona

l Fre

quen

cy o

ffse

t

Temperature (K)

SOPHIE

FAITH

Microwave OscillatorsRoom temperature oscillator with interferometric

signal processing

Cryogenic Sapphire Oscillators

developed @ UWA

developed @ UWA

approaching 10-16

Microwave OscillatorsLowest phase and amplitude noise Exceptional spectral purity Low spurious content Low vibrational sensitivity Unequalled short-term stability.

SAPPHIRE LOADED CAVITYOSCILLATORS

Compact low noise oscillator: Defence Radar Applications

National Metrology Laboratories

Oscillator/Clock ZooPhotons Phonons Magnons Atoms

LC-circuits

Metallic Cavities

Dielectric Cavities

SAW

BAW

Structures

Bulk

Spin-Torque

Electron transitions

Nuclear transitions

Collaboration with SYRTE at Paris Observatory

Test Relativity

VaryingFundamentalConstants

Long term operation of CSO -> 5.5 yrs76% Duty cycle since August 2003

Before 2003 with less duty cycle

Planck energy suppressed by the energy scale of electroweak unification (100 GeV) dimensionless ratio of

~ 8x1018

Current goal

Rotating Quartz Oscillators

Rotating Bulk Acoustic Wave Oscillators

Trying to Enter the Game to Search for the Axion

Frequency Metrology in Paraphoton Detection

New alternative to Light Shining through a Wall

Paraphoton coupling to the 2nd cavity modulate resonance frequency

coupled mode system

Trying to Enter the Game to Search for the Axion

2011-2017 renewed 2018-2024My lab $350,000 / year 14 years!

7 T Magnet (10 cm bore)

Funding History For Axion Dark Matter at UWA

PARTICLE PHYSICS DID NOT GET THIER CENTRE RENEWED: New Plan for Centre of Excellence on Dark matter Particle Physics

Funded Biggest BlueFors DilFridge 14 T Magnet: Arrive June 2019

FUNDED: $530,000 Gray Rybka ADMX, Frank Wilczek

Applying $43 M AUD 7 Years

$2.8 M for Axion Wisp$1.2 M Quantum Tech WIMPs

The ORGAN Experiment:

McGillivary Organ at UWA

What is ORGAN?● High frequency/high mass axion haloscope ● Oscillating Resonant Group AxioN Experiment ● Designed to probe promising high mass

window

Multiple cylindrical resonators to scan over multiple frequencies

What is ORGAN?

● ORGAN compared to ADMX:

● 15 – 50 GHz rather than ~1 GHz

● 14 T smaller bore magnet rather than

● ~8 T custom magnet

● Been in construction/design phase

● Hosted at UWA

● Part of EQUS CoE program!

Who is ORGAN?

● Key UWA Personnel: − Michael Tobar − Ben McAllister − Maxim Goryachev − Jeremy Bourhill − Eugene Ivanov − Graeme Flower

− Catriona Thomson

−

EQuS Collaborators

Buying Expensive item Can be a Problem 625,000 Euro

-$63,000

-$15,000

When is ORGAN?● Now! ● Has been in development ● Path-finding run complete ● Time-line of long term operation:

− Narrow search (26 – 27 GHz) − Wider search in 5 GHz chunks

First Experiment

First run complete

TM020 mode

sampling frequency of the digitizer is 1GHz, the 26.54GHz

Where is ORGAN?

10-5 10-4

10- 15

10- 14

10- 13

10- 12

10- 11

10- 10

Axion Mass(eV)

Axio

n-ph

oton

coup

ling,

g aγγ

(GeV

-1)

UF

RBF

DFSZKSVZ

CAST

ADMX

HAYSTAC

ALPs DM CANDIDATES

QCD Axions DM Candidates

ORGAN 14 T

ORGAN 28 T

ORGAN 14 T + Low Q Noise

ORGAN 28 T + Low Q Noise

High Mass Haloscopes: Problems

● Signal power in axion haloscope

Pa ∝ g2ayyB2CVQ

℘a

ma

● Shows half of the problem

● High frequency: − Low volume for cavities − Lossier materials → Lower quality factor − Inverse dependence on axion mass

● SQL increases with frequency for amplifiers

● This is a big issue in the community

Form Factor

● Signal power in axion haloscope

Pa ∝ g2ayyB2CVQ

℘a

ma

● Geometric integral

● Means we can only use specific modes

ORGAN Phase I and II: Resonator Design● Think about ways to boost C, for example

● Dielectric materials suppress electric field ● Reduce the electric field where there are out of

phase field lobes ● We can Apply this to TM modes → Dielectric Rings ● Tuning mechanisms naturally included

ORGAN Phase I and II: Resonator Design● TM030 mode Ez field looks like this:

ORGAN Phase I and II: Resonator Design● We can calculate where these things need to go ● E-field looks like:

● So, required thickness and location:

ORGAN Phase I and II: Resonator Design● If we do it right, we get this

● Finite element simulations → Form factor ~0.45, improved from 0.053

● Can use higher order modes and maintain C while boosting V

ORGAN Phase I and II: Resonator Design● Even better, we can tune this structure

● TM030 and TM031 modes

● Axial “super-modes”

ORGAN Phase I and II: Resonator Design● Most sensitive “symmetric super-mode” retains

sensitivity as the gap increases ● Frequency tuning grater than 20% of central

frequency

ORGAN Phase I and II: Resonator Design● We can compare this with an “ADMX-style” tuning

rod structure at the same frequency

HYBRID QUANTUM SYSTEMS RESEARH WITH SPINS AND PHOTONS

Types of Cavities:WG Modes

TE + TM Cylindrical modes

Reentrant Lattice

Reentrant

Goryachev and Tobar, 2014, Patent, PNo. AU2014,903,143

arXiv:1408.2905 [quant-ph]

Magnons

Cavity-Magnon polaritons

Photons Magnons Interaction

First results

White Dwarf Cooling

Our work

DFSZ model

36MHz in 6 MHz blocks from 8hrs of averages

3MHz static range 7x6hrs of averages

Centred at 14GHz

Centred at 8.2GHz

Maxim Goryachev Ben McAllister Mike Tobar

Axion Detection with Precision Frequency Metrology

System for Axion Detectionphotonic cavity with two mutually

orthogonal modesoptical or microwave

Axion Electrodynamics

normal ED

arXiv:1806.07141

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Hamiltonian Density

Axion Mediated Mode-Mode Interactionbased on axion Electrodynamics we derive axion induced coupling between two cavity modes

(B)

(A)

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<latexit sha1_base64="wILVa+5ksjgZjoJOP7NGQfPv1Bg=">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</latexit>

Axion UpConversion<latexit sha1_base64="lsPFkjbCnPQorEv6YdR1fEH6eOM=">AAACB3icbZDLSgMxFIYz9VbrbdSlIMEiuLHMFEE3QtGNywr2Au0wZNJMG5rLkGSEMnTnxldx40IRt76CO9/GtB1BW38IfPznHE7OHyWMauN5X05haXllda24XtrY3NrecXf3mlqmCpMGlkyqdoQ0YVSQhqGGkXaiCOIRI61oeD2pt+6J0lSKOzNKSMBRX9CYYmSsFbqHXclJH4UIXsIcq/D0B/3QLXsVbyq4CH4OZZCrHrqf3Z7EKSfCYIa07vheYoIMKUMxI+NSN9UkQXiI+qRjUSBOdJBN7xjDY+v0YCyVfcLAqft7IkNc6xGPbCdHZqDnaxPzv1onNfFFkFGRpIYIPFsUpwwaCSehwB5VBBs2soCwovavEA+QQtjY6Eo2BH/+5EVoViu+V/Fvz8q1qzyOIjgAR+AE+OAc1MANqIMGwOABPIEX8Oo8Os/Om/M+ay04+cw++CPn4xuPx5fR</latexit>

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Axion DownConversion<latexit sha1_base64="gUJINFByBuCTSv1yvMx67iAccL4=">AAACB3icbZDLSgMxFIYz9VbrbdSlIMEiCEKZKYJuhKIblxXsBdphyKSZNjSXIckIZejOja/ixoUibn0Fd76NaTuCtv4Q+PjPOZycP0oY1cbzvpzC0vLK6lpxvbSxubW94+7uNbVMFSYNLJlU7QhpwqggDUMNI+1EEcQjRlrR8HpSb90TpakUd2aUkICjvqAxxchYK3QPu5KTPgoRvIQ5VuHpD/qhW/Yq3lRwEfwcyiBXPXQ/uz2JU06EwQxp3fG9xAQZUoZiRsalbqpJgvAQ9UnHokCc6CCb3jGGx9bpwVgq+4SBU/f3RIa41iMe2U6OzEDP1ybmf7VOauKLIKMiSQ0ReLYoThk0Ek5CgT2qCDZsZAFhRe1fIR4ghbCx0ZVsCP78yYvQrFZ8r+LfnpVrV3kcRXAAjsAJ8ME5qIEbUAcNgMEDeAIv4NV5dJ6dN+d91lpw8pl98EfOxzeMrZfP</latexit>

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Rotating Wave Approximation

beam splitter

parametric amplificationarXiv:1806.07141

Dimensionless Orthogonality Form Factors Effective Coupling

<latexit sha1_base64="YVF1plRbqkxwS3N5AV2Q2pohzKQ=">AAACAHicbZDLSsNAFIZP6q3WW9WFCzeDRXBVkiLoRii6cVnBXqAJYTKdtENnkjAzEUvoxldx40IRtz6GO9/GSZuFth4Y5uP/z2Hm/EHCmdK2/W2VVlbX1jfKm5Wt7Z3dver+QUfFqSS0TWIey16AFeUsom3NNKe9RFIsAk67wfgm97sPVCoWR/d6klBP4GHEQkawNpJfPXIfme8mAl2hnByD+d3wqzW7bs8KLYNTQA2KavnVL3cQk1TQSBOOleo7dqK9DEvNCKfTipsqmmAyxkPaNxhhQZWXzRaYolOjDFAYS3MijWbq74kMC6UmIjCdAuuRWvRy8T+vn+rw0stYlKSaRmT+UJhypGOUp4EGTFKi+cQAJpKZvyIywhITbTKrmBCcxZWXodOoO3bduTuvNa+LOMpwDCdwBg5cQBNuoQVtIDCFZ3iFN+vJerHerY95a8kqZg7hT1mfP4tClbc=</latexit>

allows optical search at microwaves and mm-wave

allows microwave search at mm-wave

Axion Mediated Mode-Mode Interaction

Experimental Approaches

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arXiv:1806.07141

beam splitter parametric amplification




Power Detection

arXiv:1806.07141

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P. Sikivie: arXiv:1009.0762





Power Detection

arXiv:1806.07141

Cross Correlation

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Power Detection

arXiv:1806.07141

Cross Correlation Eigenfrequency Shift

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Phase Shift

Phase Shift


Axion

Design of Cavity

FREQUENCY DOMAIN AND TUNABILITY

• Tunable cavity height (lid attached to micrometer)

• TM020 mode frequency fixed by cavity radius

• TE011 mode frequency tuned by cavity height

Design of Cavity

TM020 TE011

Frequency Space

Frequency Space

DC – 1.77 GHz

16.3 – 19.5 GHz

Frequency Space

DC – 7.32 µeV

67.5 – 80.47 µeV

Coupling to the Fields


TM Probe TE Probe


TE ProbeWeakly Coupled

TE ProbeStrongly Coupled

TM ProbeStrongly Coupled

TM ProbeWeakly Coupled

Quality Factor• Q ~15,000 for two probe copper cavity• Most significant loss is from cavity wall surface resistance• Q ~ 10,000 for two probe cavity after silver coating• Silver coating provides a theoretically superior quality factor (smaller

surface resistance)• However, silver coating deteriorated the quality factor due to

introducing impurities/asymmetry (poor silver coating application)

• Q ~ 10,000 for four probe copper cavity

Theoretical Sensitivity LimitsSensitivity of the degenerate/broadband mode

experiment

Maxim Goryachev, Ben McAllister, Michael Tobar, 2018

Theoretical Sensitivity LimitsSensitivity of the loop oscillator

experiment

Maxim Goryachev, Ben McAllister, Michael Tobar, 2018

Free Running Loop Oscillator

β : Probe couplings

Low Noise Amplifier

PhaseShifter

Free Running Loop Oscillator

β : Probe couplings

Low Noise Amplifier

LOOP

1LOOP 2

TM MODE

TE MODE

LNA

LNA

PHASE SHIFTER

PHASE SHIFTER

TE IN

TE OUT

TM OUT

TM IN

TM = 9GHz

Searching for Axion in Fourier Spectrum of Phase Noise

f = 400 MHz

TE = 8GHzTE = 9GHz

fa = 9±8+0.4 GHZ= 17.4 GHz or 1.4GHz

Phase Noise DetectionFREQUENCY DISCRIMINATOR

LOOP 2TM

MODE

FREQUENCY DISCRIMINATO

R

LNA

PHASE SHIFTER

TM OUT

TM IN

EXTERNAL RESONATOR

DIRECTIONAL COUPLER

PHASE SHIFTER

MIXER FFTJFET AMP

Voltage Noise Detection FREQUENCY DISCRIMINATOR

FREQUENCY DISCRIMINATO

R

EXTERNAL RESONATOR

DIRECTIONAL COUPLER

PHASE SHIFTER

MIXER FFTJFET AMP

ExternalResonatorFrom

Loop To FFT

• External resonator phase fluctuations into frequency fluctuations

• Frequency discriminator turns frequency fluctuations into voltage fluctuations when phase across the mixer is set to quadrature

• Contains noise of loop, external resonator and tunable cavity

• FFT can produce a voltage noise spectral density

Initial Phase Noise Data TE

= 9.5

GHz


= 8.5

GHz


= 8 G

Hz

Modified Axion Electrodynamics∇ ⋅ D = ρf + gaγγ

ϵ0

μ0

B ⋅ ∇ a

∇ × H = Jf +∂ D∂t

− gaγγϵ0

μ0 ( B∂a∂t

+ ∇ a × E )∇ ⋅ B = 0

∇ × E = −∂ B∂t

D = ϵ0E + P

H =B

μ0− M

B ⋅ ∇ a = ∇ ⋅ (a B ) + a( ∇ ⋅ B )

∇ a × E = ( ∇ × (a E )) − a( ∇ × E )

Vector Identities

Modified Gauss’ Law and Ampere’s Law

∇ ⋅ D = ρf + gaγγϵ0

μ0

∇ ⋅ (a B )

∇ × H = Jf +∂ D∂t

− gaγγϵ0

μ0 ( ∂(a B )∂t

+ ∇ × (a E ))

∇ ⋅ B = 0

∇ × E = −∂ B∂t

Reformulate Modified Electrodynamics∇ ⋅ Da = ρf

∇ × Ha = Jf +∂ Da

∂t∇ ⋅ B = 0∇ × E = −

∂ B∂t

Modification in the Constitutive RelationsDa = ϵ0

E + P + PaP a = − gaγγaϵ0(c B )

Ha =1μ0

B − M − MaMa = gaγγa

1μ0

Ec

Similar to Standard Model Extension Modifications for Lorentz Invariance

Violations

Vair &,tMI: 5H 4 MAY 1987 Nt. MHFR 1H

Two Applications of Axion Electrodynamics

Frank Wilczekinstitute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California 93106

(Received 27 January 1987)

The equations of axion electrodynamics are studied. Variations in the axion field can give rise topeculiar distributions of charge and current. These eAects provide a simple understanding of the frac-tional electric charge on dyons and of some recently discovered oddities in the electrodynamics of anti-phase boundaries in PbTe. Some speculations regarding the possible occurrence of related phenomena inother solids are presented.

PACS numbers: 14.80.Gt, 05.30.Fk, 14.80.Hv, 71.50.+t

Whether or not axions' have any physical reality, theirstudy can be a useful intellectual exercise. For by havinga field which modulates the eA'ects of anomalies and in-stantons and calculating the consequences of its variationin space and time, we can get some intuitive feeling forthese important, but often subtle and obscure, things.Also, it is (I shall argue) not beyond the realm of possi-bility that fields whose properties partially mimic thoseof axion fields can be realized in condensed-matter sys-tems. In this spirit, I will consider in this paper two situ-ations where the equations of axion electrodynamicsseem to illuminate otherwise surprising phenomena, andthen speculate briefly on potential generalizations.To begin, let us recall the equations of axion electro-

dynamics. They are generated by adding to the ordinaryMaxwell Lagrangean an additional term

hL=xaE 8,where K. is a coupling constant. The resulting equationsare

al VaxE+aB. The form of these terms reflects thediscrete symmetries of a: a is P and T odd. Also, theseterms depend only on space-time gradients of the axionfield. This is because with a =const, hX in Eq. (1) be-comes a perfect derivative, and does not aAect the equa-tions of motion.Dyon charge. —Consider a magnetic monopole sur-

rounded by a spherical ball in which a=O, modulatingwithin a thin shell into a =0 at large distances (Fig. l).Now because of the axion term in (2) one finds that thedomain wall carries electric charge density —KVa. B, orcharge/unit length —tcVaC& when integrated over direc-tion, where @ is the magnetic flux. The total charge seenby observers far from the monopole is

q = —rc&k (6)

The Witten eflect, that in a 0 vacuum magnetic mono-poles become dyons with fractional charge to their mag-netic charge and to 0, is essentially contained in (6). Byour introducing axions, and allowing 0 to become a

V E=p —xVa B,V x E = —BB/Bt,V. B=0V x 8 =BE/Bt +j+ rc(a B+Va x E),

(3)(4)

(s)where P,j are the ordinary (nonaxion) charge andcurrent. We see that there is an extra charge densityproportional to —Va B, and current density proportion-

FIG. l. Monopole surrounded by a shell of axion domainwall

1987 The American Physical Society 1799

Vair &,tMI: 5H 4 MAY 1987 Nt. MHFR 1H

Two Applications of Axion Electrodynamics

Frank Wilczekinstitute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, California 93106

(Received 27 January 1987)

The equations of axion electrodynamics are studied. Variations in the axion field can give rise topeculiar distributions of charge and current. These eAects provide a simple understanding of the frac-tional electric charge on dyons and of some recently discovered oddities in the electrodynamics of anti-phase boundaries in PbTe. Some speculations regarding the possible occurrence of related phenomena inother solids are presented.

PACS numbers: 14.80.Gt, 05.30.Fk, 14.80.Hv, 71.50.+t

Whether or not axions' have any physical reality, theirstudy can be a useful intellectual exercise. For by havinga field which modulates the eA'ects of anomalies and in-stantons and calculating the consequences of its variationin space and time, we can get some intuitive feeling forthese important, but often subtle and obscure, things.Also, it is (I shall argue) not beyond the realm of possi-bility that fields whose properties partially mimic thoseof axion fields can be realized in condensed-matter sys-tems. In this spirit, I will consider in this paper two situ-ations where the equations of axion electrodynamicsseem to illuminate otherwise surprising phenomena, andthen speculate briefly on potential generalizations.To begin, let us recall the equations of axion electro-

dynamics. They are generated by adding to the ordinaryMaxwell Lagrangean an additional term

hL=xaE 8,where K. is a coupling constant. The resulting equationsare

al VaxE+aB. The form of these terms reflects thediscrete symmetries of a: a is P and T odd. Also, theseterms depend only on space-time gradients of the axionfield. This is because with a =const, hX in Eq. (1) be-comes a perfect derivative, and does not aAect the equa-tions of motion.Dyon charge. —Consider a magnetic monopole sur-

rounded by a spherical ball in which a=O, modulatingwithin a thin shell into a =0 at large distances (Fig. l).Now because of the axion term in (2) one finds that thedomain wall carries electric charge density —KVa. B, orcharge/unit length —tcVaC& when integrated over direc-tion, where @ is the magnetic flux. The total charge seenby observers far from the monopole is

q = —rc&k (6)

The Witten eflect, that in a 0 vacuum magnetic mono-poles become dyons with fractional charge to their mag-netic charge and to 0, is essentially contained in (6). Byour introducing axions, and allowing 0 to become a

V E=p —xVa B,V x E = —BB/Bt,V. B=0V x 8 =BE/Bt +j+ rc(a B+Va x E),

(3)(4)

(s)where P,j are the ordinary (nonaxion) charge andcurrent. We see that there is an extra charge densityproportional to —Va B, and current density proportion-

FIG. l. Monopole surrounded by a shell of axion domainwall

1987 The American Physical Society 1799

VOLUME 58, NUMBER 18 PHYSICAL REVIEW LETTERS 4 MAY 1987

"Im rn

la+ /6 Arg rn= Tr

———+———y g———E.————6Arg m= —Tr Re m

FIG. 3. Expectation of the current in a background field isderived from the vacuum polarization.

FIG. 2. Phase changes through ~ n depending on the signof the imaginary part.

will depend on small perturbations —including bothterms, such as Zeeman splitting, not retained in the mod-el Lagrangean (9), and effects of impurities and doping—when the Fermi level is near midgap.For truly complex-valued masses the mechanism

whereby charges and currents are generated need not beconnected with the existence of 0 modes, or midgapstates. For instance, if

I m (x) Iis constant, and its

phase varies slowly in the sense that I Bm/9x I/Im I(I m I, then the local magnitude of the gap will remain—2 Im

I everywhere. Nevertheless, the charges andcurrents discussed above will be produced; they are asso-ciated with the behavior of the Fermi sea as a whole.Their magnitude in this case is robust.The current associated with an electric field in the

plane of the wall is so remarkable that it deserves furtherdiscussion. First, let us note its properties:(i) Since j~E, the current is voltage controlled(ii) Since jJ E, the current is nondissipative Of.

course I have ignored impurities, etc. , so that this state-ment is only approximate. At the level of analysis in thispaper many-body eAects have been ignored, and thecurrent is not a supercurrent.(iii) The direction of the current is determined by the

unit vector B/I BI. This form follows from time-reversalsymmetry. It indicates that the direction of current canreverse, ideally, in response to changes in magnetic fieldsof tiny magnitude. This peculiar dependence is associat-ed, in the axion picture [Eq. (12)], with the fact that achange in the sign of the mass is ambiguous; it can mean3, =+z or —z across the wall.And now let me discuss in more detail how these prop-

erties arise. I will first discuss the case where m is anodd, real function of z and m(~) is positive. 1 use theconventions of Bjorken and Drell. The zero-mode solu-tions of the Dirac equation are then of the form

p2y=exp J~ m(z)dz [Z~f(x,y)+Ezg(x, y)l, (13a)

Other modes have energy —I m(~) I and can be

neglected. The eff'ective two-dimensional dynamics ofthe low-energy modes are described by the efrectiveHamil ton ian

AH P(73, P ~xk,

as we see by sandwiching the eAective coupling

(i4)

030 Q 33,

in (is).So in this case the interaction of the low-energy modes

with a planar electric field reduces to a problem in theelectrodynamics of (2+ 1)-dimensional massive fer-mions. Now there is a peculiarity in the vacuum polar-ization of (2+1)-dimensional electrodynamics that isrelevant here. ' That is, the current induced by anexternal field, calculated from the Feynman diagram inFig. 3, is of the form

H =io3cr 8

acting on the two-component spinor ri =(gI). A magneticfield normal to the wall leads to a series of Landau levelsat energies n(eB/2~) 'i with degeneracy eB/2x per unitarea. If the Fermi level is slightly above 0, then one hascharge accumulated on the wall, as discussed above, be-cause the n =0 mode is composed half of positive-energyand half of negative-energy states relative to the free(i.e., no domain wall) Hamiltonian. The independenceof the magnitude of j on the magnitude of B is now easi-ly understood heuristically as follows. The basic phe-nomena is a drift of the zero-mode plasma in crossedelectric and magnetic fields. In this situation, the driftvelocity of a charged particle is proportional to E and in-versely proportional to 8. But since the number of statesin the zero-mode plasma, as discussed above, is itselfproportional to 8 this dependence cancels out.An infinitesimal complex (3+ 1)-dimensional mass,

i.e., an effective axion field, generates a (2+1)-dimen-sional mass p, leading to similar results. Indeed, theefIect of the axion field in the z direction is simply to in-duce an effective mass term

where j, (p/ I u I ) ~...F., (is)1

0

0

0

(i3b)This gives us the current discussed above, with the prop-erties (i)- (iii).Note that the (2+1)-dimensional mass p is P and T

odd; it is this which explains how it can be proportional

1801

arXiv:hep-th/9609099 v1 11 Sep 1996

arX

iv:h

ep-th

/960

9099

v1

11

Sep

1996

IASSNS 96/95

hep-th/9609099

September 1996

Asymptotic Freedom⋆

Frank Wilczek†

School of Natural Sciences

Institute for Advanced Study

Olden Lane

Princeton, N.J. 08540

⋆ Lecture on receipt of the Dirac medal for 1994, October 1994.† Research supported in part by DOE grant DE-FG02-90ER40542. [email protected]

intuition that the effective charge strength would decrease with distance, as for

an ordinary dielectric. And indeed this sort of behavior was shown to occur in a

variety of examples by Landau and his school. As I already mentioned, Landau

attached tremendous significance to these results, arguing that any finite value

for the charge attached to a pointlike particle would be completely neutralized

at finite distances. Since it is difficult to implement relativistic invariance except

for pointlike objects⋆, Landau concluded that the only truly consistent relativistic

quantum field theories are trivial, non-interacting ones. Later systematic work us-

ing the mature techniques of quantum field theory in a very wide class of examples

confirmed that screening is the generic behavior [9, 10] if one excludes nonabelian

gauge interactions.

1.2. Antiscreening as Paramagnetism: The Importance of Spin

It is ironic that the physical behavior to which we will ultimately trace asymp-

totic freedom was well known to Landau, and central to one of his major interests,

the quantum theory of magnetism. In a relativistic theory we must have the re-

lationship ϵµ = 1 between the dielectric constant and the magnetic susceptibility.

(Indeed one can think of ϵ as the coefficient of the electric term E · D ∝ ϵFoiF oi

and µ−1 as the coefficient of the magnetic term B · H ∝ µ−1FijF ij in the action,

and these will form an invariant combination only if ϵ = µ−1.) Thus normal elec-

tric screening behavior ϵ ≥ 1 is associated with diamagnetism, µ ≤ 1. However

one knows that diamagnetism is not the universal response of matter to an ap-

plied magnetic field. Indeed it is a familiar and important result in the theory of

metals [11], that for an ideal Fermi gas of non-interacting electrons the Landau

diamagnetism associated with moments generated by orbital motion is dominated

by the Pauli paramagnetism arising from alignment of elementary spin moments.

Of course this analogy is far from conclusive, since in metals one is dealing with a

non-relativistic system of real particles rather than a relativistic system of virtual

⋆ Modern string theory does this, but also illustrates how difficult it is to do!

8

Vacuum acts as Dielectric and Paramagnet!

•hep-ph/0306230

Signals for Lorentz violation in electrodynamics

V. Alan Kostelecky and Matthew MewesPhysics Department, Indiana University, Bloomington, Indiana 47405

!Received 20 May 2002; published 23 September 2002"

An investigation is performed of the Lorentz-violating electrodynamics extracted from the renormalizablesector of the general Lorentz- and CPT-violating standard-model extension. Among the unconventional prop-erties of radiation arising from Lorentz violation is birefringence of the vacuum. Limits on the dispersion oflight produced by galactic and extragalactic objects provide bounds of 3!10"16 on certain coefficients forLorentz violation in the photon sector. The comparative spectral polarimetry of light from cosmologicallydistant sources yields stringent constraints of 2!10"32. All remaining coefficients in the photon sector aremeasurable in high-sensitivity tests involving cavity-stabilized oscillators. Experimental configurations inEarth- and space-based laboratories are considered that involve optical or microwave cavities and that could beimplemented using existing technology.

DOI: 10.1103/PhysRevD.66.056005 PACS number!s": 11.30.Cp, 03.30.#p, 12.60."i, 13.40."f

I. INTRODUCTION

Lorentz symmetry underlies the theory of relativity andall accepted theoretical descriptions of nature at the funda-mental level. A crucial role in establishing both the rotationand boost components of Lorentz symmetry has been playedby experimental studies of the properties of light. In the clas-sic tests, rotation invariance is investigated in Michelson-Morley experiments searching for anisotropy in the speed oflight, while boost invariance is studied via Kennedy-Thorndike experiments seeking a variation of the speed oflight with the laboratory velocity #1–3$.In this work, a theoretical study is performed of various

experiments testing Lorentz symmetry with light and otherelectromagnetic radiation. The analysis is within the contextof the Lorentz- and CPT-violating standard-model extension#4$, developed to allow for small general violations in Lor-entz and CPT invariance #5$. The Lagrangian of this theoryincludes all observer Lorentz scalars formed by combiningstandard-model fields with coupling coefficients having Lor-entz indices. At the level of quantum field theory, the viola-tions can be regarded as remnants of Planck-scale physicsappearing at attainable energy scales. The coefficients forLorentz violation may be related to expectation values ofLorentz tensors or vectors in an underlying theory #6$. Todate, experimental tests of the standard-model extensionhave been performed with hadrons #7–10$, protons and neu-trons #11$, electrons #12,13$, photons #14,15$, and muons#16$.In the present context of studies of electrodynamics, the

standard-model extension is of interest because it provides ageneral field-theoretic framework for investigating the Lor-entz properties of light. The theory contains as a subset ageneral Lorentz-violating quantum electrodynamics !QED",which includes a general Lorentz-violating extension of theMaxwell equations. We study experiments that can measurethe coefficients for Lorentz violation in this generalized elec-trodynamics. Our attention is restricted here to exceptionallysensitive experiments that could be in a position to detect theminuscule effects motivating the standard-model extension.A basic feature of Lorentz-violating electrodynamics is

the birefringence of light propagating in vacuo. This resultsin several potentially observable effects, including pulse dis-persion and polarization changes. One goal of this work is toconsider the implications of these effects for the propagationof radiation on astrophysical scales. We use available obser-vations to constrain certain coefficients for Lorentz violation.Another goal of this work is to analyze modern versions

of some classic tests of special relativity based on resonant-cavity oscillators #17–19$, which have extreme sensitivity tothe properties of electromagnetic fields. These experimentsdepend on the Earth’s sidereal and orbital motion. However,the advent of the International Space Station !ISS" makes itfeasible to perform laboratory experiments in space, wherethe orbital motion can yield different sensitivity to Lorentz-violating effects #20$. We consider here both space- andEarth-based laboratory experiments with resonant cavities.The structural outline of this paper is as follows. Section

II presents some basic results and definitions for the generalLorentz-violating electrodynamics and outlines the connec-tion to some test models. We then consider birefringenceexperiments, beginning in Sec. III A with some general is-sues. Constraints stemming from the resulting effects onpulse dispersion from astrophysical sources are addressed inSec. III B, while those from polarization changes over cos-mological scales are treated in Sec. III C. A general analysisfor laboratory-based experiments on the Earth and in space ispresented in Sec. IV A. Sections IV B and IV C apply thisanalysis to experiments with optical and microwave resonantcavities. We summarize in Sec. V. Throughout this work, weadopt the conventions of Ref. #4$.

II. LORENTZ-VIOLATING ELECTRODYNAMICS

This section provides some background and contextualinformation about the general Lorentz-violating electrody-namics. The basic formalism is presented, and some defini-tions used in later sections are introduced. We also discussthe connection between this theory and some test models forLorentz violation.

PHYSICAL REVIEW D 66, 056005 !2002"

0556-2821/2002/66!5"/056005!24"/$20.00 ©2002 The American Physical Society66 056005-1

where E! and B! are the electric and magnetic fields obtainedfrom solving the modified Maxwell equations !2". The 3!3 matrices #DE , #HB , #DB , and #HE are defined by

!#DE" jk"#2!kF"0 j0k,

!#HB" jk"12 $ jpq$krs!kF"pqrs,

!#DB" jk"#!#HE"k j"!kF"0 jpq$kpq.!5"

The double-trace condition on (kF)#%&' translates to thetracelessness of (#DE$#HB), while (kF)#[%&']"0 impliesthe tracelessness of #DB"#(#HE)T. This leaves #DE and#HB with eleven independent elements and the matrix #DB"#(#HE)T with eight, which together represent the 19 in-dependent components of kF . Note also that #DE and #HBare parity even, while #DB"#(#HE)T is parity odd.With these definitions, the modified Maxwell equations

!2", !3" take the familiar form

(! !H! #)0D! "0, (! •D! "0,

(! !E! $)0B! "0, (! •B! "0. !6"

As a consequence, many results from conventional electro-dynamics in anisotropic media also hold for this Lorentz-violating theory. For example, the energy-momentum tensortakes the standard form in terms of E! , B! , D! and H! . Thisimplies the usual Poynting theorem, which can be applied inconjunction with the symmetries of the matrices in Eq. !4" toshow that the vacuum is lossless.For the applications to be addressed in later sections, it is

convenient to introduce the following decomposition of(kF)#%&' coefficients:

! #e$" jk"12 !#DE$#HB" jk,

! #e#" jk"12 !#DE##HB" jk#

13 * jk!#DE" ll,

! #o$" jk"12 !#DB$#HE" jk,

! #o#" jk"12 !#DB##HE" jk,

# tr"13 !#DE" ll. !7"

The first four of these equations define traceless 3!3 matri-ces, while the last defines a single coefficient. All parity-evencoefficients are contained in #e$ , #e# and # tr , while allparity-odd coefficients are in #o$ and #o# . The matrix #o$

is antisymmetric while the other three are symmetric.

The form of this decomposition helps in determining theportion of the parameter space to which experiments are sen-sitive and how different experiments might overlap. For ex-ample, typical laboratory experiments with electromagneticcavities search for rotation-violating parity-even observables.The sensitivity of such experiments is therefore expected tobe dominantly to the ten rotation-violating parity-even coef-ficients #e$ and #e# . For those observables depending atleading order on the velocity, the eight coefficients #o$ and#o# can be expected to play a role. Finally, at second orderin the velocity one can expect the sole rotation-invariantquantity # tr to affect measurements. These considerations areconfirmed by the results of the detailed analysis in the sec-tions below.As another example of the use of the decomposition !7",

recall that birefringence is known to depend on ten linearlyindependent combinations of the components of kF , whichcan be chosen as +15,

ka"+!kF"0213, !kF"0123, !kF"0202#!kF"1313,

!kF"0303#!kF"1212, !kF"0102$!kF"1323,

!kF"0103#!kF"1223, !kF"0203$!kF"1213,

!kF"0112$!kF"0323, !kF"0113#!kF"0223,

!kF"0212#!kF"0313]. !8"

Relating these to the # matrices, we find

! #e$" jk"#! #!k3$k4" k5 k6

k5 k3 k7

k6 k7 k4" ,

! #o#" jk"! 2k2 #k9 k8

#k9 #2k1 k10

k8 k10 2!k1#k2"" . !9"

In this way, we can see directly that birefringence is con-trolled by the matrices #e$ and #o# .In terms of the # matrices defined in Eq. !5", and assum-

ing as before that (kAF)-"0, the Lagrangian !1" becomes

L"12 !E! 2#B! 2"$

12E

! •!#DE"•E! #12B

! •!#HB"•B!

$E! •!#DB"•B! . !10"

Similarly, using instead the # matrices defined in Eq. !7", wefind

L"12 +!1$# tr"E! 2#!1## tr"B! 2,$

12E

! •! #e$$#e#"•E!

#12B

! •! #e$##e#"•B! $E! •! #o$$#o#"•B! . !11"

SIGNALS FOR LORENTZ VIOLATION IN ELECTRODYNAMICS PHYSICAL REVIEW D 66, 056005 !2002"

056005-3

A. Basic theory

The standard model of particle physics is believed to bethe low-energy limit of a fundamental theory that includes allthe forces in nature. The natural scale of this fundamentaltheory is likely to be determined by the Planck mass. Thepossibility that Lorentz- and CPT-violating signals from thistheory may be observable at energies attainable today led tothe development of the standard-model extension !4", whichis a general theory based on the standard model but allowingfor violations of Lorentz and CPT symmetry !5". The addi-tional terms must be small because the usual standard modelagrees well with experiment. They may originate from spon-taneous symmetry breaking in the fundamental theory !6".The standard-model extension can be defined as the usual

standard-model Lagrangian plus all possible additionalLorentz- and CPT-violating terms involving standard-modelfields that maintain invariance under Lorentz transformationsof the observer’s inertial frame. This invariance ensures thatthe physics is independent of the choice of coordinates. TheLorentz violation is associated with rotations and boosts ofparticles or localized field configurations in a fixed observerinertial frame.Many of the detailed investigations of the standard-model

extension have been performed under the simplifying as-sumption that the additional Lorentz- and CPT-violatingterms preserve the SU(3)!SU(2)!U(1) local gauge sym-metry of the usual standard model. Another widely adoptedsimplifying assumption is that the coefficients for Lorentzviolation are independent of position. This implies the viola-tion is restricted to the Lorentz symmetry instead of the fullPoincare symmetry and has several useful consequences forexperiment, including the conservation of energy and mo-mentum. It is also often convenient to restrict attention to therenormalizable sector of the theory, since this is expected todominate the physics at low energies. However, nonrenor-malizable terms are known to play an important role athigher energies !21".Extracting terms involving the photon fields from the

standard-model extension yields a Lorentz- andCPT-violating extension of QED !4". The fermion sector ofthis theory has been widely studied. Here, we focus attentionon the pure-photon sector and limit attention to the renormal-izable terms, which involve operators of mass dimensionfour or less. The relevant Lagrangian is !4"

L"#14 F#$F#$$

12 %kAF&'(')#$A)F#$

#14 %kF&')#$F')F#$, %1&

where F#$*+#A$#+$A# . This theory maintains the usualU%1& gauge invariance under the transformations qA#→qA#$+#, . The Lagrangian contains the standard Max-well term and two additional Lorentz-violating terms. Thefirst of these extra terms is CPT odd, and its coefficient(kAF)' has dimensions of mass. The other is CPT even. Itscoefficient (kF)')#$ is dimensionless and has the symmetries

of the Riemann tensor and a vanishing double trace, whichimplies a total of 19 independent components.The CPT-odd term has received much attention in the

literature !22". This term provides negative contributions tothe canonical energy and therefore is a potential source ofinstability. One solution is to set the coefficient to zero,(kAF)'"0. This is theoretically consistent with radiativecorrections in the standard-model extension and is well sup-ported experimentally: stringent constraints on kAF havebeen set by studying the polarization of radiation from dis-tant radio galaxies !14".In contrast, much less is known about the CPT-even co-

efficient kF . Theoretical studies show that it provides posi-tive contributions to the canonical energy and that it is radia-tively induced from the fermion sector in the standard-modelextension !4,23". Constraints on some components have re-cently been obtained from optical spectropolarimetry of cos-mologically distant sources !15". In the present work, wefocus on the experimental implications of this CPT-eventerm. The coefficient (kAF)' is set to zero for the analysis.The equations of motion from Lagrangian %1& are

+-F#-$%kF&#-./+-F./"0. %2&

These are modified source-free inhomogeneous Maxwellequations. The homogeneous Maxwell equations,

+#F#$*12 (#$')+#F')"0, %3&

remain unchanged.Although it lies beyond our present scope, the techniques

presented here and the results obtained can be generalized tothe nonrenormalizable sector. The nonrenormalizable termscan be classified according to their mass dimension. The di-mensions of the corresponding coefficients are inverse pow-ers of mass, and it is plausible that these coefficients aresuppressed by corresponding powers of the Planck scale.Terms of this type appear in various special Lorentz-violating theories, including noncommutative field theoriesincorporating QED !24". Indeed, any coordinate-independenttheory with a photon sector containing nonrenormalizableLorentz-violating terms must be a subset of the standard-model extension. It would be interesting to provide a detailedstudy of the nonrenormalizable terms in the Lorentz-violating electrodynamics and their experimental signals.

B. Analogy and definitions

A useful analogy exists between the Lorentz-violatingelectrodynamics in vacuo and the conventional situation inhomogeneous anisotropic media !4". The idea is to definefields D! and H! by the six-dimensional matrix equation

! D!H!" "! 1$'DE 'DB

'HE 1$'HB" ! E!B!" , %4&

V. ALAN KOSTELECKY AND MATTHEW MEWES PHYSICAL REVIEW D 66, 056005 %2002&

056005-2New methods of testing Lorentz violation in electrodynamics

Michael Edmund Tobar,1,* Peter Wolf,2,3 Alison Fowler,1 and John Gideon Hartnett11University of Western Australia, School of Physics, M013, 35 Stirling Highway, Crawley 6009 WA, Australia

2Bureau International des Poids et Mesures, Pavillon de Breteuil, 92312 Sevres Cedex, France3BNM-SYRTE, Observatoire de Paris, 61 Avenue de l’Observatoire, 75014 Paris, France

(Received 1 September 2004; published 7 January 2005)

We investigate experiments that are sensitive to the scalar and parity-odd coefficients for Lorentzviolation in the photon sector of the standard model extension (SME). We show that of the classic tests ofspecial relativity, Ives-Stilwell (IS) experiments are sensitive to the scalar coefficient, but at only parts in105 for the state-of-the-art experiment. We then propose asymmetric Mach-Zehnder interferometers withdifferent electromagnetic properties in the two arms, including recycling techniques based on travellingwave resonators to improve the sensitivity. With present technology we estimate that the scalar and parity-odd coefficients may be measured with a sensitivity better than parts in 1011 and 1015 respectively.

DOI: 10.1103/PhysRevD.71.025004 PACS numbers: 03.30.+p, 06.30.Ft, 11.30.Cp

I. INTRODUCTION

The postulate of Lorentz invariance (LI) is at the heart ofspecial and general relativity and therefore one of thecornerstones of modern physics. The central importanceof this postulate has motivated tremendous work to experi-mentally test LI with ever increasing precision [1].Additionally, many unification theories (e.g., string theoryor loop gravity) are expected to violate LI at some level,[2–4] which further motivates experimental searches forsuch violations.

Numerous test theories that allow the modelling andinterpretation of experiments that test LI have been devel-oped. Kinematical frameworks [5,6] postulate a simpleparameterization of the Lorentz transformations with ex-periments setting limits on the deviation of those parame-ters from their values in special relativity. A morefundamental approach is offered by theories that parame-terize the coupling between gravitational and nongravita-tional fields (e.g., TH!" formalisms [1,7]). Formalismsbased on string theory [2,3] have the advantage of beingwell motivated by theories of physics that are at presentgood candidates for a unification of gravity and the otherfundamental forces of nature. Fairly recently a generalLorentz violating extension of the standard model of par-ticle physics (standard model extension, SME) has beendeveloped [8–10] whose Lagrangian includes all parame-terized Lorentz violating terms that can be formed fromknown fields. Many of the theories mentioned above areincluded as special cases of the SME [11,12]. In this paperwe restrict our attention to the photon sector of the SME.Within this framework we analyze past experiments thatcan be shown to set limits on SME parameters that have notbeen determined previously, and propose new experimentsthat could significantly improve those limits.

As shown in [11] the photon sector of the SME can beexpressed in the form of modified source free Maxwell

equations, which take their familiar form

r:D ! 0; (1a)

r:B ! 0; (1b)

r"E# @tB ! 0; (1c)

r"H$ @tD ! 0; (1d)

but with modified definitions of D and H

DH

! "!

!0%e!r # #DE&#####!0"0

q#DB#####

!0"0

q#HE "$1

0 % e"r$1 # #HB&

0B@

1CA E

B

! ":

(2)

Here #DE, #DB, #HE and #HB are all 3" 3 matrices,which parameterize possible Lorentz violating terms asdescribed in [11]. If we suppose the medium of interesthas general magnetic or dielectric properties, then e!r ande"r are also 3" 3 matrices. In vacuum e!r and e"r areidentity matrices. For experimental tests it is convenientto further define linear combinations of the # coefficients

%e#e#&jk! 12%#DE # #HB&jk;

%e#e$&jk! 12%#DE $ #HB&jk$ 1

3$jk%#DE&ll;

%e#o#&jk! 12%#DB # #HE&jk;

%e#o$&jk! 12%#DB $ #HE&jk; %e#tr& ! 1

3%#DE&ll:

(3)

The first four of these equations define traceless 3" 3matrices, while the last defines a single coefficient. All e#matrices are symmetric except e#o# which is antisymmetric(odd parity). There are 19 independent coefficients of the #tensors, which are generally used to quote and compareexperimental results [11–15].

The # tensors in (2) and (3) are frame dependent andconsequently vary as a function of the coordinate systemchosen to analyze a given experiment. In principle they*Electronic address: [email protected]

PHYSICAL REVIEW D 71, 025004 (2005)

1550-7998=2005=71(2)=025004(15)$23.00 025004-1 © 2005 The American Physical Society

New methods of testing Lorentz violation in electrodynamics

Michael Edmund Tobar,1,* Peter Wolf,2,3 Alison Fowler,1 and John Gideon Hartnett11University of Western Australia, School of Physics, M013, 35 Stirling Highway, Crawley 6009 WA, Australia

2Bureau International des Poids et Mesures, Pavillon de Breteuil, 92312 Sevres Cedex, France3BNM-SYRTE, Observatoire de Paris, 61 Avenue de l’Observatoire, 75014 Paris, France

(Received 1 September 2004; published 7 January 2005)

We investigate experiments that are sensitive to the scalar and parity-odd coefficients for Lorentzviolation in the photon sector of the standard model extension (SME). We show that of the classic tests ofspecial relativity, Ives-Stilwell (IS) experiments are sensitive to the scalar coefficient, but at only parts in105 for the state-of-the-art experiment. We then propose asymmetric Mach-Zehnder interferometers withdifferent electromagnetic properties in the two arms, including recycling techniques based on travellingwave resonators to improve the sensitivity. With present technology we estimate that the scalar and parity-odd coefficients may be measured with a sensitivity better than parts in 1011 and 1015 respectively.

DOI: 10.1103/PhysRevD.71.025004 PACS numbers: 03.30.+p, 06.30.Ft, 11.30.Cp

I. INTRODUCTION

The postulate of Lorentz invariance (LI) is at the heart ofspecial and general relativity and therefore one of thecornerstones of modern physics. The central importanceof this postulate has motivated tremendous work to experi-mentally test LI with ever increasing precision [1].Additionally, many unification theories (e.g., string theoryor loop gravity) are expected to violate LI at some level,[2–4] which further motivates experimental searches forsuch violations.

Numerous test theories that allow the modelling andinterpretation of experiments that test LI have been devel-oped. Kinematical frameworks [5,6] postulate a simpleparameterization of the Lorentz transformations with ex-periments setting limits on the deviation of those parame-ters from their values in special relativity. A morefundamental approach is offered by theories that parame-terize the coupling between gravitational and nongravita-tional fields (e.g., TH!" formalisms [1,7]). Formalismsbased on string theory [2,3] have the advantage of beingwell motivated by theories of physics that are at presentgood candidates for a unification of gravity and the otherfundamental forces of nature. Fairly recently a generalLorentz violating extension of the standard model of par-ticle physics (standard model extension, SME) has beendeveloped [8–10] whose Lagrangian includes all parame-terized Lorentz violating terms that can be formed fromknown fields. Many of the theories mentioned above areincluded as special cases of the SME [11,12]. In this paperwe restrict our attention to the photon sector of the SME.Within this framework we analyze past experiments thatcan be shown to set limits on SME parameters that have notbeen determined previously, and propose new experimentsthat could significantly improve those limits.

As shown in [11] the photon sector of the SME can beexpressed in the form of modified source free Maxwell

equations, which take their familiar form

r:D ! 0; (1a)

r:B ! 0; (1b)

r"E# @tB ! 0; (1c)

r"H$ @tD ! 0; (1d)

but with modified definitions of D and H

DH

! "!

!0%e!r # #DE&#####!0"0

q#DB#####

!0"0

q#HE "$1

0 % e"r$1 # #HB&

0B@

1CA E

B

! ":

(2)

Here #DE, #DB, #HE and #HB are all 3" 3 matrices,which parameterize possible Lorentz violating terms asdescribed in [11]. If we suppose the medium of interesthas general magnetic or dielectric properties, then e!r ande"r are also 3" 3 matrices. In vacuum e!r and e"r areidentity matrices. For experimental tests it is convenientto further define linear combinations of the # coefficients

%e#e#&jk! 12%#DE # #HB&jk;

%e#e$&jk! 12%#DE $ #HB&jk$ 1

3$jk%#DE&ll;

%e#o#&jk! 12%#DB # #HE&jk;

%e#o$&jk! 12%#DB $ #HE&jk; %e#tr& ! 1

3%#DE&ll:

(3)

The first four of these equations define traceless 3" 3matrices, while the last defines a single coefficient. All e#matrices are symmetric except e#o# which is antisymmetric(odd parity). There are 19 independent coefficients of the #tensors, which are generally used to quote and compareexperimental results [11–15].

The # tensors in (2) and (3) are frame dependent andconsequently vary as a function of the coordinate systemchosen to analyze a given experiment. In principle they*Electronic address: [email protected]

PHYSICAL REVIEW D 71, 025004 (2005)

1550-7998=2005=71(2)=025004(15)$23.00 025004-1 © 2005 The American Physical Society

2

assume a(t) = a0 cos(!at). Note that all terms containingga�� are sometimes presented with the opposite sign, buthas no impact on this work as both representation arecorrect.

By substituting the following vector identities, ~B·~ra =~r · a ~B + a(~r · ~B) and ~ra ⇥ ~E = (~r⇥ a ~E) � a(~r⇥ ~E)along with (5) and (6), into equations (3) and (4), themodified Gauss’ and Ampere’s Law become

✏r~r · ~E =

⇢f

✏0+ ga��c

~r · (a ~B), (7)

~r⇥ ~B

µr

=✏r

c2

@ ~E

@t+ µ0

~Jf � ga��

c

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)

which is a more convenient and consistent way of ex-pressing modified axion electrodynamics. In general, it isbetter to represent the photon-axion interaction term asthe product of the axion scalar amplitude, a(t,~r), multi-plied by either the applied ~E-field or the applied ~B-field.This is similar to the form of the equations in [14], butwithout the magnetic monopole duality. Moreover, thisrepresentation directly satisfies Faraday’s Law (Eqn..(5))and ~r · ~B = 0 (Eqn.(6)). The former representation,Eqns. (3)-(6) may lead to confusion, with Faraday’s Lawseemingly sometimes only approximately satisfied whenthe applied field ~E has been set to zero. This is becausethe last term in Eqn.(4), actually has a term that de-pends on the time derivative of the ~B field. With furthermanipulation one can show that the modified Maxwell’sequations maintain a similar form to the non-modifiedequations, given by

~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t, (12)

with the constitutive relations redefined as

~Da = ~D + ~Pa = ✏0✏r~E � ga��a

r✏0

µ0

~B, (13)

~Ha = ~H � ~Ma =1

µ0µr

~B + ga��a

r✏0

µ0

~E. (14)

Here ~D is the usual electric flux density (or ~D-field), ~H

the usual magnetic field intensity (or ~H-field), with ~Da

and ~Ha the modified definitions of these fields that sat-isfy the equations (9) to (12) due to the additional axionpolarization, ~Pa and axion magnetization ~Ma (which wewill shortly define).

This is a similar approach to that which is adoptedwhen deriving modified Maxwell’s equations for the pho-ton sector Standard Model Extension (SME)[15], which

includes in the Lagrangian all possible Lorentz invari-ance violations. By comparison to SME modified elec-trodynamics, it is apparent that ga��a is similar to anoscillating odd-parity Lorentz invariance violation, DB

or HE . This type of Lorentz invariance violation is dis-cussed in detail in [16] and is also presented in SI unitsin this work.With the modification defined thusly, it is straightfor-

ward to show that the continuity equation is satisfied.From equations (7) and (8) we may define

⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)

In the situation where there are no free charges or freeconducting electrons, we interpret ⇢a as a vacuum boundcharge and ~Ja as a polarization current. Taking the timederivative of Eqn.(15) and the divergence of Eqn. (16)we obtain

r · ~Ja = �@⇢a

@t, (17)

demonstrating that the continuity equation is satisfied.This means we may interpret the e↵ective current den-sity, ~Ja, due to the displacement of e↵ective boundcharge, ⇢a, as an oscillating vacuum polarization Pa givenby;

~Ja =@ ~Pa

@t; ~Pa = �ga��

r✏0

µ0a ~B. (18)

In general, the total displacement current is defined by

~JDa =@ ~Da

@t= ✏0✏r

@ ~E

@t� ga��

r✏0

µ0

@(a ~B)

@t. (19)

Note, there is also an axion bound current associatedwith the induced magnetization given by;

~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E) (20)

Since the axion modifications are in the source termsit is instructive to think of the oscillating bound chargesand currents as providing oscillations in the magnetiza-tion and polarization of the vacuum, in a similar way thatfree electron charge and spin cause vacuum polarization(due to electric screening) and vacuum magnetization(due to magnetic anti-screening), which also causes ”run-ning” of the fine structure constant at high energies andsmall distance scales [17, 18]. Thus, the oscillating mag-netization and polarization could be interpreted as anoscillation of the fine structure constant, ↵ (i.e. Eqn.(1)shows axion-photon coupling is proportional to ↵), or

2

assume a(t) = a0 cos(!at). Note that all terms containingga�� are sometimes presented with the opposite sign, buthas no impact on this work as both representation arecorrect.By substituting the following vector identities, ~B·~ra =

~r · a ~B + a(~r · ~B) and ~ra ⇥ ~E = (~r⇥ a ~E) � a(~r⇥ ~E)along with (5) and (6), into equations (3) and (4), themodified Gauss’ and Ampere’s Law become

✏r~r · ~E =

⇢f

✏0+ ga��c

~r · (a ~B), (7)

~r⇥ ~B

µr

=✏r

c2

@ ~E

@t+ µ0

~Jf � ga��

c

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)


~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t, (12)


~Da = ~D + ~Pa = ✏0✏r~E � ga��a

r✏0

µ0

~B, (13)

~Ha = ~H � ~Ma =1

µ0µr

~B + ga��a

r✏0

µ0

~E. (14)



and ~Ha the modified definitions of these fields that sat-isfy the equations (9) to (12) due to the additional axionpolarization, ~Pa and axion magnetization ~Ma (which wewill shortly define).This is a similar approach to that which is adopted

when deriving modified Maxwell’s equations for the pho-ton sector Standard Model Extension (SME)[15], which


or HE . This type of Lorentz invariance violation is dis-cussed in detail in [16] and is also presented in SI unitsin this work.With the modification defined thusly, it is straightfor-


⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)


r · ~Ja = �@⇢a

@t, (17)


~Ja =@ ~Pa

@t; ~Pa = �ga��

r✏0

µ0a ~B. (18)


~JDa =@ ~Da

@t= ✏0✏r

@ ~E

@t� ga��

r✏0

µ0

@(a ~B)

@t. (19)


~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E) (20)


2

assume a(t) = a0 cos(!at). Note that all terms containingga�� are sometimes presented with the opposite sign, buthas no impact on this work as both representation arecorrect.


✏r~r · ~E =

⇢f

✏0+ ga��c

~r · (a ~B), (7)

~r⇥ ~B

µr

=✏r

c2

@ ~E

@t+ µ0

~Jf � ga��

c

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)


~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t, (12)


~Da = ~D + ~Pa = ✏0✏r~E � ga��a

r✏0

µ0

~B, (13)

~Ha = ~H � ~Ma =1

µ0µr

~B + ga��a

r✏0

µ0

~E. (14)



and ~Ha the modified definitions of these fields that sat-isfy the equations (9) to (12) due to the additional axionpolarization, ~Pa and axion magnetization ~Ma (which wewill shortly define).

This is a similar approach to that which is adoptedwhen deriving modified Maxwell’s equations for the pho-ton sector Standard Model Extension (SME)[15], which


or HE . This type of Lorentz invariance violation is dis-cussed in detail in [16] and is also presented in SI unitsin this work.

With the modification defined thusly, it is straightfor-ward to show that the continuity equation is satisfied.From equations (7) and (8) we may define

⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)


r · ~Ja = �@⇢a

@t, (17)


~Ja =@ ~Pa

@t; ~Pa = �ga��

r✏0

µ0a ~B. (18)


~JDa =@ ~Da

@t= ✏0✏r

@ ~E

@t� ga��

r✏0

µ0

@(a ~B)

@t. (19)


~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E) (20)


~ Axion Interaction similar to odd parity Lorentz Invariance Violation

www.physics.indiana.edu/~kostelec/

Sidereal Modulations of Constant Background Fields

Oscillating Background Fields Create EM Radiation

Axion is similar to an oscillating odd parity background SME Lorentz invariance violation field.

Cannot shield against these type of violations -> Source Terms.

Axion Induced Vacuum Bound Charges and Currents

2

the e↵ective wave vector related to the axion Comptonwavelength. However, for most experiments it is su�-cient to ignore the axion spatial dependence and assumea(t) = a0 cos(!at). Note that all terms containing ga��

are sometimes presented with the opposite sign, but doesnot have an impact on this work as both representationare correct.


~r · ~D = ⇢f + ga��

r✏0

µ0

~r · (a ~B), (7)

~r⇥ ~H = ~Jf +@ ~D

@t� ga��

r✏0

µ0

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)

which is a more consistent way of expressing modifiedaxion electrodynamics. In general, it is better to rep-resent the photon-axion interaction term as the productof the axion scalar amplitude, a(t,~r), multiplied by ei-ther the applied ~E-field or the applied ~B-field. This issimilar to the form of the equations in [14], but withoutthe magnetic monopole duality. Moreover, this repre-sentation directly satisfies Faraday’s Law (Eqn. (6)) andGauss’ Law for Magnetism (Eqn.(5)). The former rep-resentation, Eqns. (3)-(6) may lead to confusion, withFaraday’s Law seemingly sometimes only approximatelysatisfied when the applied field ~E has been set to zero.This is because the last term in Eqn. (4), actually has aterm that depends on the time derivative of the ~B field.

With further manipulation one can show that the mod-ified Maxwell’s equations maintain a similar form to thenon-modified equations, given by

~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t(12)


~Da = ✏0~E + ~P + ~Pa where ~Pa = �ga��

r✏0

µ0(a ~B), (13)

~Ha =~B

µ0� ~M � ~Ma where ~Ma = ga��

r✏0

µ0(a ~E). (14)

Here ~Da and ~Ha the modified definitions of these fieldsthat satisfy the equations (9) to (12) due to the addi-tional axion induced vacuum polarization, ~Pa and axioninduced vacuum magnetization ~Ma.

This description is similar to that which is adoptedwhen deriving modified Maxwell’s equations for the pho-ton sector Standard Model Extension (SME)[15], whichincludes in the Lagrangian all possible Lorentz invarianceviolations. By comparison to SME modified electrody-namics, it is apparent that ga��a is similar to an oscillat-ing odd-parity Lorentz invariance violation, DB or HE .This type of Lorentz invariance violation is discussed indetail in [16] and was also presented in SI units.


⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)


r · ~Ja = �@⇢a

@t, (17)

demonstrating that the continuity equation is satisfied.This means we may interpret the e↵ective current den-sity, ~Ja, due to the displacement of e↵ective boundcharge, ⇢a, as an oscillating vacuum polarization Pa given

by ~Ja = @ ~Pa@t

.Note, there is also an axion bound current associated

with the induced magnetization given by

~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E). (18)

Since the axion modifications are in the source terms,it is instructive to think of the oscillating bound chargesand currents (which are due to virtual particles) as pro-viding oscillations in the magnetization and polarizationof the vacuum. Vacuum polarization and magnetiza-tion e↵ects also causes the running of the fine structureconstant, ↵, due to equal components of electric screen-ing (polarization of vacuum) and magnetic anti-screening(magnetization of vacuum). These e↵ects cause the per-ceived quantum of electric charge to increase at smalldistances, while the preceived quantum of magnetic fluxdecreases[17, 18]. Thus the fine structure constant in-creases at small distances and high energy scales, withthe vacuum e↵ectively acting as a dielectric and param-agnetic medium. The oscillating magnetization and po-larization could be interpreted as a tiny oscillation of thefine structure constant due to oscillations in the screen-ing and anti-screening process, or equivalently, an oscil-lation in the refractive index of the vacuum (similar to a

2


are sometimes presented with the opposite sign, but doesnot have an impact on this work as both representationare correct.By substituting the following vector identities, ~B·~ra =


~r · ~D = ⇢f + ga��

r✏0

µ0

~r · (a ~B), (7)

~r⇥ ~H = ~Jf +@ ~D

@t� ga��

r✏0

µ0

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)



~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t(12)



r✏0

µ0(a ~B), (13)

~Ha =~B


r✏0

µ0(a ~E). (14)




⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)


r · ~Ja = �@⇢a

@t, (17)


by ~Ja = @ ~Pa@t



~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E). (18)


2


are sometimes presented with the opposite sign, but doesnot have an impact on this work as both representationare correct.By substituting the following vector identities, ~B·~ra =


~r · ~D = ⇢f + ga��

r✏0

µ0

~r · (a ~B), (7)

~r⇥ ~H = ~Jf +@ ~D

@t� ga��

r✏0

µ0

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)

which is a more consistent way of expressing modifiedaxion electrodynamics. In general, it is better to rep-resent the photon-axion interaction term as the productof the axion scalar amplitude, a(t,~r), multiplied by ei-ther the applied ~E-field or the applied ~B-field. This issimilar to the form of the equations in [14], but withoutthe magnetic monopole duality. Moreover, this repre-sentation directly satisfies Faraday’s Law (Eqn. (6)) andGauss’ Law for Magnetism (Eqn.(5)). The former rep-resentation, Eqns. (3)-(6) may lead to confusion, withFaraday’s Law seemingly sometimes only approximatelysatisfied when the applied field ~E has been set to zero.This is because the last term in Eqn. (4), actually has aterm that depends on the time derivative of the ~B field.With further manipulation one can show that the mod-

ified Maxwell’s equations maintain a similar form to thenon-modified equations, given by

~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t(12)



r✏0

µ0(a ~B), (13)

~Ha =~B


r✏0

µ0(a ~E). (14)


This description is similar to that which is adoptedwhen deriving modified Maxwell’s equations for the pho-ton sector Standard Model Extension (SME)[15], whichincludes in the Lagrangian all possible Lorentz invarianceviolations. By comparison to SME modified electrody-namics, it is apparent that ga��a is similar to an oscillat-ing odd-parity Lorentz invariance violation, DB or HE .This type of Lorentz invariance violation is discussed indetail in [16] and was also presented in SI units.With the modification defined thusly, it is straightfor-


⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)


r · ~Ja = �@⇢a

@t, (17)


by ~Ja = @ ~Pa@t



~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E). (18)


2




~r · ~D = ⇢f + ga��

r✏0

µ0

~r · (a ~B), (7)

~r⇥ ~H = ~Jf +@ ~D

@t� ga��

r✏0

µ0

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)



~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t(12)



r✏0

µ0(a ~B), (13)

~Ha =~B


r✏0

µ0(a ~E). (14)




⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)


r · ~Ja = �@⇢a

@t, (17)


by ~Ja = @ ~Pa@t



~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E). (18)


Vacuum Bound Charge

Vacuum Polarization Current

Satisfies the Continuity Equation

P a = − gaγγaϵ0(c B )

Pa = − ϵ0(gaγγac B ) E a screen = gaγγac B V/m

E screenE 0

E = E 0 + E screenV0 V

+ + + + + + + + + + + + + + + + +

− − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − −

+ + + + + + + + + + + + + + + + +

E screen = −χe

χe + 1E 0 = − χe

E

D = ϵ0E + P P = ϵ0 χe

E

Consider a Capacitor with Linear Dielectric

Axion Electrodynamics

= − ϵ0E screen

P

Note if then E a screen = 0 P a = 0

2




~r · ~D = ⇢f + ga��

r✏0

µ0

~r · (a ~B), (7)

~r⇥ ~H = ~Jf +@ ~D

@t� ga��

r✏0

µ0

@(a ~B)

@t+ ~r⇥ (a ~E)

!,

(8)



~r · ~Da = ⇢f , (9)

~r⇥ ~Ha = ~Jf +@ ~Da

@t, (10)

~r · ~B = 0, (11)

~r⇥ ~E = �@ ~B

@t(12)



r✏0

µ0(a ~B), (13)

~Ha =~B


r✏0

µ0(a ~E). (14)




⇢a = ga��

r✏0

µ0

~r · (a ~B), (15)

~Ja = �ga��

r✏0

µ0

@(a ~B)

@t. (16)


r · ~Ja = �@⇢a

@t, (17)


by ~Ja = @ ~Pa@t



~Jba = ~r⇥ ~Ma = �ga��

r✏0

µ0

~r⇥ (a ~E). (18)


= 0

fa =F a

qa= − E a screen = − gaγγa0c B0 Cos[ωat] V/m

E = 0 B = B0 a(t) = a0Cos[ωat]

Effective axion forces

Pa = − ϵ0gaγγa0(cB0)e−jωat θd P a

dt= jωaϵ0gaγγa0(cB0)e−jωat θ

E = 0 B = B0θ

r=r1

l = 2πr1

Cross section area A

fa =F a

qa= − E a screen =

1ϵ0

Pa

ℰ = ∮P

fa ⋅ d l =1ϵ0 ∮P

Pa ⋅ d l

∇ × Pa ≠ 0

I = Ad P a

dt= jωaϵ0gaγγa0cB0Ae−jωat θ

V = ℰ = − gaγγa0cB0le−jωat

VI

= −1ϵ0

lA

1jωa

COULD NOT DO THIS IN NORMAL ELECTRODYNAMICS

FACULTY OF SCIENCE

Perth, Australia

Broadband Electric-field Axion Sensing Technique(BEAST)

BT McAllister, M Goryachev, J Bourhill, EN Ivanov, ME Tobar

BEAST: First LimitsHigher resolution search was conducted around 5 kHz, with the minimal spectral resolution of 4.5 mHz (increasing at higher frequencies)

All sharp peaks greater than 4.4 standard deviations from the mean originating from the SQUID were able to be excluded, due to a similar signal appearing in the flux line

Using this data, we may place the 95 % confidence exclusion limits on axion-photon coupling

frequency and quantum metrology research group at uwa · rotating bulk acoustic wave oscillators....

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