Josephson & Quantum Hall Effect vs Quantum Electrodynamics

Alexander Penin

University of Alberta & INR Moscow

CERN, October 2009

A. Penin, U of A & INR CERN 2009 – p. 1/21 Josephson & Quantum Hall Effect vs Quantum Electrodynamics

or how to measure the vacuum polarization with a voltmeter

Alexander Penin

University of Alberta & INR Moscow

CERN, October 2009

A. Penin, U of A & INR CERN 2009 – p. 1/21 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 effect Theory Experiment

A. Penin, U of A & INR CERN 2009 – p. 2/21 Exact result: an example

Determination of the fine structure constant α

electron g − 2 10 experimental error: 10− 10 theoretical error: 10− - requires 4-loop calculation in QED

quantum Hall effect 8 12 experimental error: 10− (could be 10− ) theoretical error: 0 - for free!

A. Penin, U of A & INR CERN 2009 – p. 3/21 Gauge invariance and quantum phase

2 ~ e = c = 1, α = 4π

✔ Schrödinger equation (i∂t − H)Ψ=0

✔ Wave function Ψ = |Ψ|eiθ

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

r1 ➥ If ∇ × A = 0 then θ(r1) − θ(r2) = q A(r′) · dr′ Zr2 r ➥ Also φ(r) = θ(r) − q A(r′) · dr′ is gauge invariant Z

A. Penin, U of A & INR CERN 2009 – p. 4/21 Flux quantization

F. London (1948) Superconductivity ➪ coherent state of Cooper pairs ➪ q = 2e Meissner effect ➪ B = 0 inside superconductor

Superconducting ring: × S B C

➪ Single-valued wave function ∆φ = 2e C A · dx = 2πn H

A. Penin, U of A & INR CERN 2009 – p. 5/21 Flux quantization

F. London (1948) Superconductivity ➪ coherent state of Cooper pairs ➪ q = 2e Meissner effect ➪ B = 0 inside superconductor

Superconducting ring: × S B C

➪ Single-valued wave function ∆φ = 2e C A · dx = 2πn 2 ➪ πn C A(r) · r = S B(r) · d s ≡ Φ Φ = e ≡H nΦ0

H R h Flux quantum: Φ0 = 2e

A. Penin, U of A & INR CERN 2009 – p. 6/21 Hall effect

E. Hall (1879)

B I Lorentz force vs electrostatic force z x eE = −ev × B y E

ρe current density j = ρev = E B ρe total current per length I = V B 1 ρe Hall conductivity R− = B

A. Penin, U of A & INR CERN 2009 – p. 7/21 Quantum Hall effect

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

Wave function: 2 x i πn L L Ψ(x,y) = e ψ(y − ym)

ψ(y − ym) ➪ harmonic oscillator 2πm centered at ym = eBL 2π 1 eBL √eB Density of quantum states with eB n Landau levels filled: ρ = n 2π

1 Quantum Hall conductivity: R− = 2nα = n/RK h von Klitzing constant: RK = e2

A. Penin, U of A & INR CERN 2009 – p. 8/21 Gauge invariance argument

R.B. Laughlin (1981) δH B current density j ∝ E δA dE I total current I = dΦ Φ Flux quantization Φ = 4π|A|/L = n2Φ0

Flux dependence ym(Φ + 2Φ0) = ym+1(Φ)

for dΦ = 2Φ0 ➪ dE = neV neV nV ➥ I = = 2Φ0 RK

A. Penin, U of A & INR CERN 2009 – p. 9/21 Vacuum polarization

× J. Schwinger (1951)

= + +... q → ×

2 Charge renormalization δe ∝ Πµν(q)/q |q2 0 → Vacuum polarization in magnetic field

2 α eB 1 2 2 2 δΠµν (q)= − 2 gµν q − qµqν − 7 gµν q − qµqν + 4 gµν q − qµqν π m2 45 k ⊥    


Lorentz invariance broken ➪ different couplings

A. Penin, U of A & INR CERN 2009 – p. 10/21 QED corrections to electromagnetic couplings

Coulomb potential

3 2 2 δΠ00(q) −iq·r d q α α eB 2 7 2 VC (r)= e 1 − e = 1+ − sin θ q2 (2π)3 |r| π m2 45 90 Z   "    #

Interaction with the homogeneous electric field

2 α eB 11 1+ e r·E ≡ e∗r·E π m2 180   !

Hall current 2 α eB 1 e′′ 1+ I ≡ I π m2 15 e   !

A. Penin, U of A & INR CERN 2009 – p. 11/21 QED corrections to RK

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

1 2 Origin of the corrections RK− ∝ e → e∗e′′

Result:

2 2 2 2 1 e 23 α ~eB e 20 B R− = 1 + ≈ 1 + 10− × K h 180 π c2m2 h 10 T "   # "   #

A. Penin, U of A & INR CERN 2009 – p. 12/21 Josephson effect

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

Tunneling Josephson current I |Ψ|eiθ1 ✔ no voltage drop if the current

is time independent E B

✔ 2π periodic in θ1 − θ2 |Ψ|eiθ2 ✔ simplest solution I = Ic sin(θ1 − θ2) ✔ with voltage V across the junction the current oscillates at frequency ν

A. Penin, U of A & INR CERN 2009 – p. 13/21 Josephson effect

Time evolution of the phase:

i t Ψ(t) ∝ e E ➪ θ1 − θ2 =(E1 − E2)t = 2eV t ➪ ω = 2eV

Josephson frequency-voltage relation ν = KJ V

2e Josephson constant KJ = h

A. Penin, U of A & INR CERN 2009 – p. 14/21 Gauge invariance argument

F. Bloch (1968)

r1 Observables depend on ∆φ = θ1 − θ2 − 2e A(r′) · dr′ Zr2

Gauge transformation µ µ µ A ′(r) = A (r) − ∂ ξ(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

A. Penin, U of A & INR CERN 2009 – p. 15/21 QED corrections to KJ

Correction to the coupling with scalar potential

4 δΠ0ν (q) ν iqr d q δv.p.H = −e A˜ (q)e = 2δeA0(r) q2 (2π)4 Z

here δe = e∗ − e gauge invariant!

A. Penin, U of A & INR CERN 2009 – p. 16/21 QED corrections to KJ

A.P., (2009) Correction to the coupling with scalar potential

4 δΠ0ν (q) ν iqr d q δv.p.H = −e A˜ (q)e = 2δeA0(r) q2 (2π)4 Z

here δe = e∗ − e gauge invariant! δe can be removed by δξ(r) = e ξ(r) ➥ ω = 2e∗V

A. Penin, U of A & INR CERN 2009 – p. 17/21 QED corrections to KJ

A.P., (2009) Correction to the coupling with scalar potential

4 δΠ0ν (q) ν iqr d q δv.p.H = −e A˜ (q)e = 2δeA0(r) q2 (2π)4 Z

here δe = e∗ − e gauge invariant! δe can be removed by δξ(r) = e ξ(r) ➥ ω = 2e∗V Result:

2 2 2e 11 α ~eB 2e 20 B K = 1 + ≈ 1 + 10− × J h 180 π c2m2 h 10 T "   # "   #

A. Penin, U of A & INR CERN 2009 – p. 17/21 Experiment: quantum Hall effect

F. Schopfer, W. Poirier (2009)

gallium arsenide heterostructures ➥ two-dimensional electron gas

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

A. Penin, U of A & INR CERN 2009 – p. 18/21 Experiment: Josephson effect

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

J1 MW J2

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

19 ✔ accuracy 10− – sensitive to gravitational red shift

A. Penin, U of A & INR CERN 2009 – p. 19/21 Metrology

Fundamental relations fine structure constant α = cµ0 2RK 4 Planck constant h = 2 KJ RK

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 (?)

A. Penin, U of A & INR CERN 2009 – p. 20/21 Summary

A. Penin, U of A & INR CERN 2009 – p. 21/21 Summary

Vacuum polarization results in a weak dependence of the Josephson and von Klitzing constant on the magnetic field strength

A. Penin, U of A & INR CERN 2009 – p. 21/21 Summary

Vacuum polarization results in a weak dependence of the Josephson and von Klitzing constant on the magnetic field strength

This remarkable manifestation of a fine nonlinear quantum field effect in collective phenomena in condensed matter can be observed in a dedicated experiment

A. Penin, U of A & INR CERN 2009 – p. 21/21 Summary

Vacuum polarization results in a weak dependence of the Josephson and von Klitzing constant on the magnetic field strength

This remarkable manifestation of a fine nonlinear quantum field effect in collective phenomena in condensed matter can be observed in a dedicated experiment

One literally can measure the vacuum polarization with a voltmeter!

A. Penin, U of A & INR CERN 2009 – p. 21/21