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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
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 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
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
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
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)
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.
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+ 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
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