Physikalisch-Technische Bundesanstalt

Time and Frequency Department

Braunschweig, Germany

Optical Clocks and Tests of Fundamental Principles

Les Houches, Ultracold Atoms and Precision Measurements 2014

Ekkehard Peik

Physikalisch-Technische Bundesanstalt:

the National Metrology Institute of Germany

Metrology: The science of measurement

with applications for science, technology, economy, society

Meteorology

Metrology

Clock Hall in Kopfermann-Building of PTB, Braunschweig

• Legal time for Germany• 4 primary caesium clocks• Time distribution via long-wave radio DCF77 / internet / telephone• Time transfer via satellites for international atomic time scales TAI and UTC

• Caesium fountain clocks• Optical clocks with trapped ions• Optical frequency measurements

Outline of the Lectures

1.Introduction to atomic clocks2.Optical clocks with laser-cooled trapped atoms and ions3.Clocks and relativity

Modern Textbook:C. Audoin, B. Guinot: The Measurement of Time, Cambridge Univ. Press

General publications on Time and Frequency fromNIST: http://tf.nist.gov/timefreq/general/generalpubs.htmPTB:http://www.ptb.de/time/

General reading (from NIST): J. Jespersen, J. Fitz-Randolph:From Sundials to Atomic clocks

Recommended literature:Review article: Optical Atomic ClocksA. D. Ludlow, M. M. Boyd, Jun Ye, E. Peik, P. O. SchmidtarXiv 1407.3493

Atomic Clocks

PrinciplesClock characteristics

Microwave and optical clocks

James Clerk Maxwell, 1870: If, then, we wish to obtain standards of length, time, and mass which shall beabsolutely permanent, we must seek them not in the dimensions, or the motion,or the mass of our planet, but in the wave-length, the period of vibration, and theabsolute mass of these imperishable and unalterable and perfectly similarmolecules.

A Brief History of Atomic Time

Postulate: Atomic energies are natural constants and do not depend onspace or time (apart from relativistic effects).(Einstein‘s Equivalence Principle)

1955: Cs beam clock, Essen and Parry, NPL1955-58: Measurement of Cs frequency in terms of the ephemeris second

ν=9 192 631 770 (20) Hz (NPL and USNO)1967: Definition:

“The second is the duration of 9 192 631 770 periods of the radiationcorresponding to the transition between the two hyperfine levels ofthe ground state of the caesium 133 atom.“

from: C. Audoin, B. Guinot: The Measurement of Time

Improvement in the Accuracy of Clocks

ν ≅ 1/s

ν ≅ 104/s

ν ≅ 1010/s

ν ≅ 1015/sOptical clocks

Uncertainties in the Realization of the SI Base Units

Second 3*10-16

Meter 10-11 (definition linked to the second)Kilogramm 0 (for prototype, 10-9 for comparisons)Ampere 4*10-8

Kelvin 3*10-7

Candela 10-4

Mol 8*10-8

Oszillator

ν0

Atome, Moleküle oder Ionen

Detektor

Regelungs-elektronik

νν0

νν0

S

Absorptions- signal

FehlersignaldSdν

oscillator detector

servo -electronics

absorptionsignal

error signal

Absorber (ions, atoms, molecules)

Principle of Atomic Clocks

ννout

Optical clockwork:femtosecond laser

Caesium Clocks

Caesium Beam Clock with Magnetic State Selection

„flop-in“ detection of atomsthat have made thehyperfine transition

Magnetic dipole forstate selection(Stern-Gerlach configuration)

Ramsey interactionregion withhomogeneousmagnetic field.

Detection viasurface ionsationof caesium

Oven at T≈100oC

Magnetic State Selection

In the Paschen-Back region:

(F=4,mF=0) atoms aremagnetic low-field seekers

(F=3,mF=0) atoms aremagnetic high-field seekers

A lens based on magnetic field gradients(e.g. a hexapole) will focus low-field seekersand defocus high-field seekers.

Rabi- and Ramsey-Excitation

2-level system with pulsed excitation.χ: res. Rabi frequency, δ: detuning

Excitation probability after the pulse

(quadatic Fourier transform of the pulse)

1 Pulse (Rabi)

P2 =

P2 =

2 Pulses (Ramsey)

Advantages of Ramsey excitation

about 0,5x narrower resonancefor the same interaction time

Resonance not broadened by perturbations between the interaction zones(like B field inhomogeneity)

Influence of the velocity distribution: leads to a distribution in TCentral peak: δ=0, therefore δT=0, independent of vFurther peaks: phase δt=nπ, will be washed out

The Caesium Fountain

Early fountain experiments:1953 Zacharias, MIT (hot Cs, failed)1989 Chu, Stanford (cold Na)1991 ENS/LPTF (cold Cs)

Use laser cooled atomsinstead of a thermal atomic beam

PTB‘s fountain clock CSF1 (1999)

Ramsey fringes in a caesium fountain

Instability: a few 10-13 in 1 sAccuracy: a few 10-16

(requires ≈3 days of averaging; central fringe split by factor 106)Dominant systematic shifts from: Cs-Cs „cold collisions“, cavity phase shift

The most important specifications of a clock: Stability and Accuracy

NI tutorial

ideal:Primary clock,agrees with an unperturbedreference value

Useful referenceif calibrated

e.g. hydrogen maser

bad!problematic:long measuring time,Long evaluation

Stability analysis using the Allan Deviation

- Perform sequence of frequency measurements over time interval τ- Calculate normalized frequencies yk- Calculate variance of differences yk+1-yk , Allan variance σy

2

(two-sample variance, named after David Allan, NIST, avoids divergences for drifting sources)- produce double-log plot of σy(τ)-Slope ρ indicates spectral shape of

dominant noise sources

Stability analysis using the Allan Deviation

-Slope ρ indicates spectral shape of dominant noise sources

(Atom) Shot noise,E.g. thermal, 1/f noise

Good atomic clock:averages down σy(τ) like 1/τ1/2

until it hits the „flicker floor“

Typical Allan deviation curves

Commercial Rb

Commercial Cs (5071A)

CS1 vs. CS2 (8 years)Passive H maserActive H maser

Cs fountains (CSF1, FO-2)

Sr lattice clock

Stability of atomic frequency standards

microwave optical frequencyν0 increases by 5 orders of magnitude

An optical single-ion frequency standard with ∆ν=1 Hz provides higherstability than a caesium fountain clock with 106 atoms.

Measuring population of a two-level systemin a single atom yields a random sequence ofvalues 0 and 1 (state projection)

Variance for a measurement with N atoms:

Optical Frequency Standard / Optical Clock

AtomicReference

„forbidden“ transition of atomsin a laser-cooled, trapped samplein the Lamb-Dicke regime

fs-CombGenerator

„optical clockwork“, provides radiofrequencyoutput and means for comparison with otheroptical frequencies

Laserlocked to atomic resonance,short-term stabilized to passiveFabry-Perot cavity

Forbidden transitions as reference transitions for the clock

„Forbidden“: based on the selection rules for electric dipole radioation.

Photon carries angular momentum: L=1, 2, 3, … (not 0 !)Atomic electron makes transition: J → J‘

| J – J‘ | ≤ L ≤ J + J‘

L=1 dipole radiation ∆J=0, ±1, not 0→0L=2 quadrupole radiation ∆J=0, ±1, ±2 not 0→0L=3 octupole radiation ∆J=0, ±1, ±2, ±3 not 0→0Etc.

An indication on the radiative lifetime of excited states:Power emitted by an antenna of size r in multipole order L:

AEL ≈ ω (r/λ)2L r/λ ≈ 10-2 ... 10-3 for visible lightDipole decay: nano– … microsecondsQuadrupole: milliseconds … secondsOctupole: hours … months

The Lamb-Dicke Regime

R. H. Dicke, Phys. Rev. 89, 472 (1953)The linear Doppler shift may be suppressed if the motion of the emitting orabsorbing atom is restrained to a region of size <λ

Reaching the Lamb-Dicke regime is a prerequisite for a precise atomic clock.It is relatively easy for microwaves, but harder for an optical clock.

Emission spectrum of an atomin a box

Doppler and recoil free line

Doppler shift of the free atom

Heterodyne detection ofFluorescence in a 1D lattice

Lamb-Dicke Regime,Transitions between vibrational levels

σ+σ−: Lamb-Dicke confinement,but no lattice structure

Sideband - Strengths

Classical harmonic oscillator: Frequency modulated spectrum (via the Doppler effect),Modulation index: kx

Bessel functions

Lamb-Dicke condition, carrier dominates

Quantum harm. Osc.:Lamb-Dicke condition:

Emission spectrum in the LD limit:

carrier dominates,high frequency sidebandvanishes for n → 0

Lamb-Dicke confinement: recoil-free absorption and emission

Harmonic oscillator ground state extension

Lamb-Dicke condition for the ground state

Recoil energy < 1 oscillator quantum

Resonant scattering is elastic and recoil free.But: random momentum transfer is possible in non-resonant scattering events.Close to <n>=0: absorption and emission spectrum are different!

Transitions:(A) Doppler cooling(B) Sideband cooling(C)Quench

After Doppler coolingafter sideband cooling

<n>=0.051(12)T=47(3) µK

Two systems for optical clocks with atoms in traps

Absorption images of trapped Sr atoms and of an expanding cloud of free atoms

8 msT = 6 ms2 12 ms10 ms 16 ms14 ms

g

• Optical lattice: Dipole trap at the “magic” wavelength• ~106 atoms interrogatedsimultaneously

5 Yb+ Ions

• Storage with minimal perturbationfrom the trap potential• unlimited observation time• one ion: no collisional shift

Single ion in an ion trapOptical lattice with neutral atoms

Problem for neutral atoms: Trap shifts energy levels.

Possible solution: Optical lattice of dipole traps with „magic“detuning, so that both levels of the reference transition shift by the sameamount. (Hidetoshi Katori, 2001)

x

E

see: T. Ido, H. Katori, Phys. Rev. Lett. 91, 53001 (2003)

Optical clock with trapped neutral atoms

The „magic“ wavelength

A. Brusch et al. PRL 96, 103003 (2006)

Light shift as a function of the lattice wavlength (Sr clock, SYRTE group Paris)

J=0→0 forbidden in all multipole orders, because of conservation of angular momentum.

States with J=0: two-electron systems: Mg, Sr …Al+, In+ etc. 1S0 →

3P0: favorable reference transition because both states possess high symmetry and are not easily polarized by external E fields(original proposal by H. Dehmelt)

J=0→0: (weak) electric dipole transition is possible under loss of rotational symmetry, e.g. from a magnetic field:Internal (nuclear spin, hyperfine interaction) or external field mixes states with different J.

Two types of lattice clocks:

Fermionic isotopes with: half-integer nuclear spin (e.g. 87Sr)J=0→0: transition induced by hyperfine mixingCollisions (s-wave) suppressed even in 1D lattice with many atoms per siteProblem: no mF=0 component (small linear Zeeman shift)

Bosonic isotopes without nuclear spin (e.g. 88Sr)J=0→0: transition induced by external magnetic fieldCollisions suppressed in 3D lattice with one atom per siteProblem: quadratic Zeeman shift from strong external field

Optical clockswith trapped ions

Paul trapsPrinciple of operation of a single-ion clock

Systematic frequency shifts, uncertainty budgetExample: Yb+

Optical Clock with a Single Laser-Cooled Ion in a Paul Trap

~ QuadrupoleElectrodes

The spectroscopist‘s ideal: an isolated atom at restin free space

• Lamb-Dicke confinement withsmall trap shifts

• unlimited interaction time• no collisions

Very low uncertainty is possible (to 10-18)proposed by Hans Dehmelt 1975

Experiments with Hg+ , Al+ (NIST), Yb+ (PTB, NPL), In+ (U Wash.,NICT),Sr+ (NRC, NPL), Ba+, Ca+ (Innsbruck, Marseille), ....

Hans Dehmelt Wolfgang Paul Norman Ramsey

Physics Nobel Laureates from 1989

Single electrons and positrons in Penning trapLaser coolingSingle-ion optical clock

Quadrupole mass spectrometerPaul trap

Ramsey spectroscopyMicrowave atomic clocksHydrogen maser

The Paul Trap

storage in anoscillatingquadrupole field

preferred operation region as a trap

highly mass-selective (e/m) operation

Linear Paul trap

Open ends: quadrupole mass spectrometer(main commercial application of Paul‘s idea)

Closed ends: linear trap(trap string of ions on field-free line without micromotion)

animation: Wolfgang Lange, MPQ

Particle trajectories in a Paul trap

single particle:secular oscillation withsuperposed micromotion

Coulomb crystal of many particles:particle in center at rest,outer particles with micromotion.

R. Wuerker, H. Shelton, R. LangmuirJ. Appl. Phys. 30, 342 (1959)

Time average over many 1/Ω results in a pseudopotential:(ponderomotive potential)

Particles minimize the kinetic energyin the driven micromotion and aredriven to regions of low field strength E(r)

Quadrupole Paul trap: E(r) ∞ rtime averaged pseudopotential is harmonic

Many other potential shapes are possible.

trap electrodes

ytterbium ovenelectron source

Paul trap for Yb+

d=1.4 mmU=600 V at 16 MHz

J=0 -- J=0 transition, hyperfine-quenched27Al+ ,115In+ small field-induced shiftsAll neutral atom lattice clocks: Sr, Yb, Hg

S -- D electric quadrupole transition40Ca+, 88Sr+, 171Yb+, 199Hg+ convenient laser systems

S -- F electric octupole transition171Yb+ narrow linewidth, dα/dt test case

nuclear magnetic dipole transition229Th3+ small field-induced shifts

Ions, atoms and types of transition under study

State Detection in a Single Ion via Electron Shelving

Coolingtransition(dipoleallowed) "forbidden"

transition

Time (s)

Pho

ton

Cou

nt R

ate

Single ion data (In+):Observation of a „Quantum Jump“

171Yb+ Single Ion Optical Frequency Standard

Measurement cycle

171Yb+ Optical Frequency Standard

High-resolution spectroscopy of the Yb+ quadrupole transition

motional sidebands≈ Doppler cooling limit,

<nvib> ≈ 15

π−pulse, τ = 1 ms~ Fourier-limited

Zeeman structureB ≈ 1 µT∆m=0 shift: ≈ 0.05 Hz

∼π−pulse, τ = 30 ms~ Resolution limit

Qua

ntum

jum

p pr

obab

ility

Detuning at 436 nm

-νr νr

∆m=012 −1 -2

30 HzAll measurements:single scan, 20 cycles/pointscan time 3...6 minutes

High resolution excitation spectrum of the Yb+ octupole transition

Results of absolute frequency measurements of Yb+ transitionswith caesium fountain clocks at PTB, 2000-2012

Relative uncertainty of recent measurements: ≈ 6×10-16 (Cs-limited)ν(E2)=688 358 979 309 307.82(36) Hzν(E3)=642 121 496 772 645.34(25) Hz

Highly accurate and stable optical clocks

Recent rapid progress on a variety of systems!

Next: optical frequency ratio measurements for consistency checksand tests of fundamental physics