Measuring Vacuum Polarization with Josephson Junctionspenin/talks/eth10.pdfVacuum polarization...

Measuring Vacuum Polarizationwith Josephson Junctions

Alexander Penin

University of Alberta & INR Moscow

ETH Zürich, May 2010

A. Penin, U of A & INR ETH 2010 – p.1/??

Topics Discussed

Exact results in quantum mechanics

Flux quantization

Quantum Hall effect

Josephson effect

Vacuum polarization in strong magnetic field

QED corrections to quantum Hall and Josephson effectTheory

Experiment


Exact result: an example

Determination of the fine structure constant α ∝ e2/h

electron g − 2

experimental error: 10−10

theoretical error: 10−10 - requires 4-loop calculation in QED

quantum Hall effectexperimental error: 10−8 (could be 10−12)

theoretical error: 0 - for free!


Gauge invariance and quantum phase

~ = c = 1, α = e2

4π

✔ Schrödinger equation (i∂t −H) Ψ = 0

✔ Wave function Ψ = |Ψ|eiθ

✔ Hamiltonian H = qA0 + F (B,E,D)D = ∂ − iqA

➥ If ∇ × A = 0 then θ(r1) − θ(r2) = q∫ r1

r2A(r′) · dr′

➥ Also φ(r) = θ(r) − q∫ r

A(r′) · dr′ is gauge invariant


Flux quantization

F. London (1948)

Superconductivity ➪ coherent state of Cooper pairs ➪ q = 2e

Meissner effect ➪ B = 0 inside superconductor

×B

C

Superconducting ring: S

Single-valued wave function ➪ ∆θ = 2e∮

CA · dx = 2πn


Flux quantization

F. London (1948)

Superconductivity ➪ coherent state of Cooper pairs ➪ q = 2e

Meissner effect ➪ B = 0 inside superconductor

×B

C

Superconducting ring: S

Single-valued wave function ➪ ∆θ = 2e∮

CA · dx = 2πn

∮

C A(r) · r =∫

S B(r) · d2s ≡ Φ ➪ Φ = πne ≡ nΦ0

Flux quantum: Φ0 = h2eA. Penin, U of A & INR ETH 2010 – p.6/??

Hall effect

E. Hall (1879)

B

E

I

z xy

Lorentz force vs electrostatic force

eE = −ev × B

current density j = ρev =ρe

BE

total current per length I =ρe

BV

Hall conductivity R−1 =ρe

B


Quantum Hall effect

K. von Klitzing, G. Dorda, M. Pepper (1980)

L

2πeBL

1√eB

Wave function:Ψ(x, y) = ei2πm

xL ψ(y − ym)

ψ(y − ym) ➪ harmonic oscillatorcentered at ym = 2πmeBL

Density of quantum states with

n Landau levels filled: ρ = neB2π

Quantum Hall conductivity: R−1 = 2nα = n/RK

von Klitzing constant: RK = he2


Gauge invariance argument

R.B. Laughlin (1981)

B

E

I

Φ

current density j ∝δH

δA

total current I =dE

dΦ

Flux quantization Φ = 4π|A|/L = n2Φ0

Flux dependence ym(Φ + 2Φ0) = ym+1(Φ)

for dΦ = 2Φ0 ➪ dE = neV

➥ I =neV

2Φ0=nV

RK


Vacuum polarization

J. Schwinger (1951)×

×

= + + . . .q →

Charge renormalization δe ∝ Πµν(q)/q2|q2→0

Vacuum polarization in magnetic field

δΠµν(q) = −α

π

(

eB

m2

)

2 1

45

[

2(

gµνq2 − qµqν

)

−7(

gµνq2 − qµqν

)

‖+4

(

gµνq2 − qµqν

)

⊥

]

Lorentz invariance broken!


QED corrections to electromagnetic coupling

Coulomb potential

VC(r) = e2

∫(

1 −δΠ00(q)

q2

)

e−iq·r

q2

d3q

(2π)3=

α

|r|

[

1 +α

π

(

eB

m2

)

2(

2

45−

7

90sin2 θ

)]

correction to the photon dispersion

λ 6= cν

local charge renormalization(

1 +α

π

(

eB

m2

)

2 1

45

)

e ≡ e∗



Coulomb potential

VC(r) = e2

∫(

1 −δΠ00(q)

q2

)

e−iq·r

q2

d3q

(2π)3=

α

|r|

[

1 +α

π

(

eB

m2

)

2(

2

45−

7

90sin2 θ

)]


1

q20− (1 + Ciα)q2i

➪ νλ 6= c

local charge renormalization

(

1 +α

π

(

eB

m2

)

2 1

45

)

e ≡ e∗



Coulomb potential

VC(r) = e2

∫(

1 −δΠ00(q)

q2

)

e−iq·r

q2

d3q

(2π)3=

α

|r|

[

1 +α

π

(

eB

m2

)

2(

2

45−

7

90sin2 θ

)]


1

q20− (1 + Ciα)q2i

➪ νλ 6= c

local charge renormalization

e ➪

(

1 +α

π

(

eB

m2

)21

45

)

e ≡ e∗


QED corrections to RK

A.P., Phys.Rev. B 79, 113303 (2009)

Origin of the corrections R−1K ∝ e2 ➪ (e∗)2

Result:

R−1K =e2

h

[

1 +2

45

α

π

(

~eB

c2m2

)2]

≈e2

h

[

1 + 10−20 ×

(

B

10 T

)2]


Josephson effect

B.D. Josephson (1962); S. Shapiro (1963)

E B

|Ψ|eiθ1

|Ψ|eiθ2

Tunneling Josephson current I

✔ no voltage drop if the currentis time independent

✔ 2π periodic in θ1 − θ2

✔ simplest solution I = Ic sin(θ1 − θ2)

✔ with voltage V across the junctionthe current oscillates at frequency ν


Josephson effect

Time evolution of the phase:

Ψ(t) ∝ eiEt ➪ θ1 − θ2 = (E1 − E2)t = 2eV t ➪ ω = 2eV

Josephson frequency-voltage relation ν = KJV

Josephson constant KJ = 2eh


Gauge invariance argument

F. Bloch (1968)

Observables depend on ∆φ = θ1 − θ2 − 2e∫ r1

r2A(r′) · dr′

Gauge transformation

Aµ′(r) = Aµ(r) − ∂µξ(r), ξ(r) = tA0(r)A′0(r) = 0, A

′(r) = A(r) + t∇A0(r)

Gauge invariant phase difference∆φ = ∆φ0 − 2et (A0(r1) −A0(r2))

➥ ω = 2eV


QED corrections to KJ

A.P., Phys.Rev.Lett. 104, 097003 (2010)

Correction to the coupling with scalar potential

δv.p.H = −e

∫

δΠ0ν(q)

q2Ãν(q)eiqr

d4q

(2π)4= 2δeA0(r)

here δe = e∗ − e gauge invariant!



A.P., Phys.Rev.Lett. 104, 097003 (2010)


δv.p.H = −e

∫

δΠ0ν(q)

q2Ãν(q)eiqr

d4q

(2π)4= 2δeA0(r)


can be removed by δξ(r) = δeeξ(r)

➥ ω = 2e∗V



A.P., Phys.Rev.Lett. 104, 097003 (2010)


δv.p.H = −e

∫

δΠ0ν(q)

q2Ãν(q)eiqr

d4q

(2π)4= 2δeA0(r)


can be removed by δξ(r) = δeeξ(r)

➥ ω = 2e∗V

Result:

KJ =2e

h

[

1 +1

45

α

π

(

~eB

c2m2

)2]

≈2e

h

[

1 + 10−20 ×

(

B

10 T

)2]


Experiment: quantum Hall effect

F. Schopfer, W. Poirier (2007)

gallium arsenide heterostructures

➥ two-dimensional electron gas

✘ thermal Johnson-Nyquist noise ➪ accuracy limit 10−12


Experiment: Josephson effect

A.K. Jain, J.E. Lukens, J.-S. Tsai (1987)

SQUID

MWJ1 J2

linearly growing current I = t(V1 − V2)/L

✔ accuracy 10−19 – sensitive to gravitational red shift


Metrology

Fundamental relations

electron charge e = 2KJRKPlanck constant h = 4

K2JRK

Quantum metrology triangle K.K. Likharev, A.B. Zorin (1985)

quantum Hall effect V/I = RK

Josephson effect V = ν/KJ

Single electron tunneling I = eν (?)


Summary


Summary

Vacuum polarization results in a weak dependence ofthe Josephson and von Klitzing constant on themagnetic field strength


Summary


This remarkable manifestation of a fine nonlinearquantum field effect in collective phenomena incondensed matter can be observed in a dedicatedexperiment


Summary


This remarkable manifestation of a fine nonlinearquantum field effect in collective phenomena incondensed matter can be observed in a dedicatedexperiment

One literally can measure the vacuumpolarization with a voltmeter!


ormalsize bl Topics Discussed{ormalsize bl Exact result: an example}{ormalsize bl Gauge invariance and quantum phase}{ormalsize bl Flux quantization}{ormalsize bl Flux quantization}{ormalsize bl Hall effect}{ormalsize bl Quantum Hall effect}{ormalsize bl Gauge invariance argument }{ormalsize bl Vacuum polarization}{ormalsize bl QED corrections to electromagnetic coupling}{ormalsize bl QED corrections to electromagnetic coupling}{ormalsize bl QED corrections to electromagnetic coupling}{ormalsize bl QED corrections to $R_K$}{ormalsize bl Josephson effect}{ormalsize bl Josephson effect}{ormalsize bl Gauge invariance argument}{ormalsize bl QED corrections to $K_J$}{ormalsize bl QED corrections to $K_J$}{ormalsize bl Experiment: quantum Hall effect}{ormalsize bl Experiment: Josephson effect}{ormalsize bl Metrology}{ormalsize bl Summary}

Measuring Vacuum Polarization with Josephson Junctionspenin/talks/eth10.pdfVacuum polarization...

Documents

Transcript of Measuring Vacuum Polarization with Josephson Junctionspenin/talks/eth10.pdfVacuum polarization...