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Home , Josephson effect

Kilogram, Planck Units, and Quantum

Alexander Penin

University of Alberta & TTP Karlsruhe

DESY Hamburg, May 2012

A. Penin, U of A DESY 2012 – p. 1/34 Topics Discussed

Systems of units and fundamental constants

Planck system of units vs traditional SI New definition of ampere and the value of electron charge New definition of kilogram and the value of

New SI vs Quantum Electrodynamics The role of the fine structure constant QED corrections to

A. Penin, U of A DESY 2012 – p. 2/34 System of units and fundamental constants

Historical approach define units by artefact calibrate experimental tools measure fundamental constants

Planck’s idea (Ann. Physik 1 (1900) 69 ) “ ... with the help of fundamental constants we have the possibility of establishing units of length, time, mass, and temperature, which necessarily retain their significance for all cultures, even unearthly and nonhuman ones”.

A. Penin, U of A DESY 2012 – p. 3/34 Planck’s units

Five fundamental constants G (gravity), kC (electricity), c (relativity), ~ (quantization), kB (temperature)

m1m2 q1q2 2 E = G , E = kC , E = ~ω, E = kB T, E = mc r r Combinations with the dimension of mass, length, time, temperature and charge ➪ new units

~ ~ G ∼ −35 lP ∼ −43 c ∼ −8 lP = 3 10 m, tP = 10 s, mP = 10 kg, r c c r G

2 ~c −18 mP ~c 32 qP = ∼ 10 C, TP = ∼ 10 K s kC kB

in these units ~ = c = kC = kB = G = 1 a device measuring one of the combinations is a new standard

A. Penin, U of A DESY 2012 – p. 4/34 Planck’s units

Perfect theoretical concept!

A. Penin, U of A DESY 2012 – p. 5/34 Planck’s units

Perfect theoretical concept!

➥ but as usual does not work in practice

why? the accuracy of all the fundamental constants should be equal or better than the accuracy of the “traditional” units definition based on artefact

A. Penin, U of A DESY 2012 – p. 6/34 Planck’s units

Weak link in Planck’s system

Gravitational constant from torsion balance:

4 −2 δG/G ≈ 10− (2010 CODATA, compare to 10 by Cavendish 1798)

A. Penin, U of A DESY 2012 – p. 7/34 Planck’s units

Never give up!

➥ just follow the concept with

~ = c = kC = kB = 1 or better use some predetermined numbers close to “traditional” values but with zero uncertainty

A. Penin, U of A DESY 2012 – p. 8/34 The first step: meter vs speed of light

Old definition

Current definition 1 second: the duration of 9 192 631 770 periods of the radiation corresponding to the ground state HFS transition of the caesium-133 atom 1 meter: c = 299 792 458 m/s sharp (in practice 1 m = 1 650 763.73 wavelengths of the orange-red emission line of the krypton-86) counting interference experiment

A. Penin, U of A DESY 2012 – p. 9/34 The next step - ampere vs electron charge

Traditional SI units electric charge ➪ electric current I = q/t 1A=1C/1s the constant current which will produce an attractive force of 2 · 107 newton per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum. 19 electron charge e = −1.602176487(40) · 10− C

“User friendly” Planck units 18 e = −1/6.2415093 · 10− C sharp to be formally proposed by the CIPM in 2015

A. Penin, U of A DESY 2012 – p. 10/34 The next step - ampere vs electron charge

Practical realization: how to count so many electrons? ➥ make use of macroscopic quantum effects

➊ Josephson effect through S-I-S junction

Josephson frequency- relation ν = KJ V

2e Josephson constant KJ = h

A. Penin, U of A DESY 2012 – p. 11/34 The next step - ampere vs electron charge

Practical realization: how to count so many electrons? ➥ make use of macroscopic quantum effects

➋ Quantum Hall effect 2-dimesional electron conductivity in magnetic field

Quantum Hall current I = nV/RK

h von Klitzing constant RK = e2

A. Penin, U of A DESY 2012 – p. 12/34 The next step - ampere vs electron charge

Practical realization: how to count so many electrons? ➥ make use of macroscopic quantum effects

Quantum Hall effect + Josephson effect = fundamental current-frequency converter

ν e I = KJ RK = 2ν

1 ampere ➪ 2 × 6.2415093 · 1018 Hz 11 Quantum Hall universality/precision: 10− F. Schopfer, W. Poirier (2007) 19 Josephson universality/precision: 10− A.K. Jain, J.E. Lukens, J.-S. Tsai (1987)

A. Penin, U of A DESY 2012 – p. 13/34 The last step - kilogram vs Planck constant

Current SI standard: the last artefact

Prototype mass drift: does kilogram get lighter?

A. Penin, U of A DESY 2012 – p. 14/34 The last step - kilogram vs Planck constant

Current SI: 34 2 Planck constant h = 6.62606957(29) · 10− kgm /s “User friendly” Plank units: 34 2 define kilogram so that h = 6.62606957 · 10− kgm /s sharp Practical realization Watt balance

A. Penin, U of A DESY 2012 – p. 15/34 The last step - kilogram vs Planck constant

Watt balance idea (simplified)

electric power = mechanical power: V I = m g v

ν ν Josephson quantum Hall effect: V I = = hν2 + KJ KJ RK

ν2 final equation: m = h gv

10 relative accuracy in g, ν, v better than 10− 8 relative accuracy of Watt balance: better than 10−

A. Penin, U of A DESY 2012 – p. 16/34 Watt balances

(from talk by M. Stock)

A. Penin, U of A DESY 2012 – p. 17/34 The role of QED in new SI

A. Penin, U of A DESY 2012 – p. 18/34 Question N 1

How should the values of Planck constant and electron charge be choosen?

A. Penin, U of A DESY 2012 – p. 19/34 Question N 1

How should the values of Planck constant and electron charge be choosen?

The dimensionless fine structure constant

2 e cµ0 α = h 2 ≈ 1/137 7 µ0 = 4π · 10− predetermined value in SI

The predetermined values of h and e should leave 7 µ0 = 4π · 10− unchanged

A. Penin, U of A DESY 2012 – p. 19/34 The fine structure constant

Electron proper magnetic moment µ e~ s = g 2me

Dirac theory: g = 2 α QED: g − 2 = π + ... “anomalous magnetic moment”

A. Penin, U of A DESY 2012 – p. 20/34 The fine structure constant

Determination from electron g − 2

10 experimental error: 10− G. Gabrielse et al. (2008) 10 theoretical error: 10− - requires 4-loop calculation in QED T. Kinoshita et al. (2008)

A. Penin, U of A DESY 2012 – p. 21/34 Question N 2

Which constants should be taken as fundamental? Experimental choice: Josephson and von Klitzing constants 2 e = KJ RK 4 h = 2 KJ RK Theoretical choice: Planck constant and electron charge

h RK = e2 2e KJ = h

A. Penin, U of A DESY 2012 – p. 22/34 e, ~ or RK,KJ ?

Does not matter if the above relations are exact they are exact in quantum mechanics is this true beyond quantum mechanics?

A. Penin, U of A DESY 2012 – p. 23/34 e, ~ or RK,KJ ?

Does not matter if the above relations are exact they are exact in quantum mechanics is this true beyond quantum mechanics?

Not in QED! A.Penin, Phys.Rev. B79, 113303 (2009) Phys.Rev.Lett. 104, 097003 (2010)

A. Penin, U of A DESY 2012 – p. 23/34 Gauge invariance and quantum phase

2 ~ e = c = 1, α = 4π

✔ Schrödinger equation (i∂t − H)Ψ=0

✔ 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 DESY 2012 – p. 24/34 Flux quantization

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

Superconducting ring: × S B C

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

A. Penin, U of A DESY 2012 – p. 25/34 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 DESY 2012 – p. 26/34 Quantum Hall effect

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

Wave function: x i2πm 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 DESY 2012 – p. 27/34 Gauge invariance argument

R.B. Laughlin (1981) δH B current density j ∝ E δA dE I total current I = dΦ Φ 2π Flux quantization Φ = 4π|A|/L = n2Φ0 = n e

Flux dependence ym(Φ + 2Φ0) = ym+1(Φ)

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

A. Penin, U of A DESY 2012 – p. 28/34 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 ∆θ = 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 DESY 2012 – p. 29/34 Nonlinear Electrodynamics

H. Euler, W. Heisenberg (1936) J. Schwinger (1951)

×

= + +...

× Vacuum polarization in magnetic field

A. Penin, U of A DESY 2012 – p. 30/34 QED effects

Correction to the photon dispersion:

k(ω) v = =6 c ω

Local charge renormalization:

2 α eB 1 ′ e → 1+ e ≡ e π m2 45   !

A. Penin, U of A DESY 2012 – p. 31/34 QED corrections to RK

1 2 Corrections to the filling factor RK− ∝ e → ee′

Result:

2 2 2 2 1 e 1 α ~eB e 20 B R− = 1 + ≈ 1 + 10− × K h 45 π c2m2 h 10 T "   # "   #

A. Penin, U of A DESY 2012 – p. 32/34 Testing the fundamental relations

Fundamental relations electron charge 2 e = KJ RK 4 Planck constant h = 2 KJ RK

Quantum 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 DESY 2012 – p. 33/34 Summary

In forthcoming years the International System of units will finally transform from a set of artifacts into a “user friendly” Planck system.

A. Penin, U of A DESY 2012 – p. 34/34 Summary

In forthcoming years the International System of units will finally transform from a set of artifacts into a “user friendly” Planck system.

Two messages from QED to new SI:

In the new “Planck inspired” SI the electron charge and Plank constant rather than Josephson and von Klitzing constants is the relevant choice of the fundamental constants

The predetermined values of the electron charge and Plank constant must be consistent with the measured value of the fine structure constant

A. Penin, U of A DESY 2012 – p. 34/34