Atomic Clocks and Fundamental Tests in Physics: Optical Frequency Standards

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2 Atomic Clocks and Fundamental Tests in Physics: Optical Frequency Standards Sébastien BIZE SYRTE, Observatoire de Paris 61 avenue de l Observatoire, Paris, France sebastien.bize@obspm.fr Cours de 3 ième cycle de la Physique en Suisse Romande March 11 th, 2010 Ecole Polytechnique Fédérale de Lausanne, Switzerland 2

3 OUTLINE Status of atomic fountains and motivations for optical clocks Optical clock transitions Optical ion clocks Optical lattice clocks Ultra stable lasers Measuring Optical Frequencies Optical Links Fundamental tests 3

4 Introduction 4

5 Principle of atomic clocks (1) Goal: deliver a signal with stable and universal frequency Bohr frequencies of unperturbed atoms are expected to be stable and universal Building blocks of an atomic clock correction macroscopic oscillator atoms output interrogation ε : fractional frequency offset Accuracy: overall uncertainty on ε y(t) : fractional frequency fluctuations Stability: statistical properties of y(t), characterized by the Allan variance σ y2 (τ) Can be done with microwave or optical frequencies, with neutral atoms, ions or molecules 5

6 Principle of atomic clocks (2) How to probe the atomic transition: 0.5 Transition probability P detuning The two main parameters: the atomic quality factor fluctuations of the measured transition probability for integration time T c : Scaling of the fractional frequency instability: Example: optimized Ramsey interrogation 6

7 Atomic fountain clocks 133 Cs levels ( 87 Rb similar) Ramsey fringes Hz detuning (Hz) Atomic quality factor: Best frequency stability (~ Quantum Projection Noise limited): Best accuracy: 4x10-16 ~ 10 fountains in operation (LNE-SYRTE, PTB, NIST, USNO, JPL, METAS, INRIM, NPL, USP, NICT, ) 7 with an accuracy a few and <10-15 for a few of them.

8 LNE-SYRTE ATOMIC CLOCK ENSEMBLE H-maser H, µw FO1 fountain Cryogenic sapphire Osc. Macroscopic oscillator Phaselock loop τ~1000 s Optical lattice clock Cs, µw Hg, opt FO2 fountain Optical lattice clock FOM transportable fountain Sr, opt Rb, Cs, µw Cs, µw 8

9 Cesium fountain comparisons FO1 vs FO2-Cs (2004) FOM vs FO2-Rb (Nov. 2007) Routine comparison down to the level. Cumulate hundreds of days of comparison supporting this uncertainty budget. Not perfect though: 1 to 1.5 σ frequency difference Some flickering at the ~4x10-16 level 9

10 Rb/Cs frequency comparison since 1998 CCTF 2004 ν Rb recommended uncertainty Weighted least square fit by a constant Chi2-goodness of fit Q = 0.75 new value (dual FO2 + FO1) previous points: FO2-Rb comparison with FOM or FO1 data and error bars are perfectly consistent Note: Errors bars are type B dominated BIPM CCTF recommended value (based on LNE-SYRTE 2002 data): ν Rb (CCTF)= (21) Hz These data and other fountain measurements have been and are still used for fundamental physics tests: Stability of fundamental constants, Local Position Invariance, Local Lorentz Invariance, Isotropy of space 10

11 Other contributions of atomic fountains Absolute frequency measurements of optical frequencies, including in remote locations with the transportable fountain. Sr, Hg lattice clocks at SYRTE. 40 Ca + in Innsbruck with transportable fountain FOM. H(1S-2S) at MPQ-Garching FOM. Support to the development of the cold atom space clock PHARAO/ACES. Ground segment of PHARAO/ACES during the mission. Remote comparisons. Contribution to testing the performance of satellite T&F transfer systems. FOM 11

12 Contributions of atomic fountains to timekeeping Calibration of TAI by atomic fountains Over the last 3 years, LNE-SYRTE fountains have made ~ 50% of all formal calibration reports to BIPM 14/28 formal reports in /45 in 2008 One report corresponds typical to a quasi continuous measurement of a H-maser frequency for 15 to 30 days u B ~ 4.5 x u A ~ 1 x u link/maser ~ 1.5 x USNO is developing several 87 Rb fountains for the GPS master clock ensemble One can safely predict a long lifetime to atomic fountains However, reaching accuracy <10-16 with atomic fountain seems unlikely 12

13 Motivations for developing optical clocks: Stability Stability of an atomic clock : Quantum limited stability : Microwave transition : υ ef ~ 10 GHz, Optical transition : υ ef =c/λ ~ Hz Example #1: cycle T C ~ 1s, linewidth ~1 Hz et σ δp ~ 10-4 : Microwave : σ y = Optical : σ y = In practice, best reported stabilities σ y = ~ Example #2: cycle T C ~ 1s, linewidth ~1 Hz et N det ~ 1 (1 atom!): Optical : σy= 3 13

14 Motivations for developing optical clocks: Accuracy Better stability implies better resolution to evaluate systematic shifts Some systematic shifts are smaller as compared to microwave clocks Effect Spectral purity of interrogation oscillator, Microwave leaks Ramsey and Rabi pulling Second order Zeeman Cold collisions Blackbody radiation Residual 1st order Doppler Quantum motion ( microwave recoil ) Uncertainty FO2 (x ) +/- 0.1 < 1 +/ / /- 0.6 < 3 < 1.4 Go down when Q at increases Do not go down when Q at increases but better atom/transition can be chosen Effects of external variables independent of the frequency: Quadratic sum < 4 Accuracy with free falling Ca atoms (NIST, PTB): ~10-14 See for instance U. Sterr et al., C.R. Physique 5, 845 (2004) or better is within reach provided a drastic solution is found 14 for the effects of external motion

15 Motional effects on free atoms Two level atom absorbing a photon Energy of the atom in g and e as a function of its momentum ħk at : E g = 2 h kat 2m 2 E e = h ω + at 2 h kat 2m 2 E(k) Atom absorbs a photon with energy ħω and momentum ħk Total momentum conservation implies that: kat kat + k Total energy conservation implies that: hω + 2 h kat 2m 2 = hω at + h 2 ( kat + k) 2m 2 The resonant frequency for absorption is: k h kat h ω = hωat + k. + m 2 h k 2m 2 kat k at + k 1st order Doppler recoil 15

16 Motional effects on confined atoms Consider an atom spatially confined to much less than the wavelength of the incoming photon In momentum space, the atomic wave-function is such that: or This implies that: the size of the wave-function in momentum space is large compared to the photon momentum The shift in momentum space implied by momentum conservation induces a minor modification of the wave-packet Small shift of the resonant frequency when the energy conservation is applied or Free space Momentum Lamb-Dicke regime Momentum 16

17 Optical Clock Transitions 17

18 Requirements for an optical clock transition Electric dipole allowed transitions typically have natural linewidth ~ or > 1 MHz. Q at ~ 10 8 or less not interesting (Q at ~ in atomic fountains) For clocks, long lived states and forbidden transitions are necessary. Several mechanisms are possible. For instance: Electric quadrupole, electric octupole transitions Intercombination transitions 2 or multi-photon transitions Blackbody radiation shift must be considered since the large ~ reduction factor between optical and microwave transitions does not apply to this shift. 18

19 Higher order transitions: quadrupole, octupole Hamiltonian for an atom in the presence of an external electromagnetic field Atom Atom-field interaction External field not to strong Traveling wave Momentum and position operator for particle i in atom Vector potential of the external field Wavelength long compared to size of atom Dipolar approximation Electric quadrupole E2 Electric octupole E3 Electric dipole transitions E1 19

20 E2 and E3 transitions used in optical clocks 199 Hg + 88 Sr + E2 E2 τ~90 ms υ~1.8 Hz υ = 1.06x10 15 Hz λ = 282 nm τ~400 ms υ~0.4 Hz υ = 4.45x10 14 Hz λ = 674 nm 171 Yb Yb + E2 τ~51 ms υ~3.1 Hz υ = 6.88x10 14 Hz λ = nm E3 τ~10 years!!! υ~0.5 nhz υ = 6.42x Hz λ = 467 nm

21 Atoms with 2 electrons on the outermost shell: Alkaline-earth and alkaline-earth-like atoms Electronic structure splits in singlet states with S=0 and triplet states with S=1 Selection rule for E1 transition: S=0 87 Sr Intercombination transitions Singlet Triplet Gross structure Fine structure interaction (L.S, S.S couplings +QED) Hyperfine interaction Nuclear spin I=9/2 (fermion only) See for instance: S. G. Porsev and A. Derevianko,Phys. Rev. A 69, (2004) 21

22 Alkaline-earth and alkaline-earth like atoms/ions considered for optical clocks Periodic table from NIST website 22

23 Intercombination transitions used in optical clocks 87 Sr 171 Yb 199 Hg τ~167s υ~1 mhz υ = 4.29x10 14 Hz λ = 698 nm 27 Al + τ~15s υ~10 mhz υ = 5.18x10 14 Hz λ = 578 nm τ~1.5s υ~100 mhz υ = 1.13x10 15 Hz λ = nm τ~377s υ~0.4 mhz υ = 1.12x Hz λ = 267 nm

24 Other schemes with alkaline-earth-like atoms Sr, Yb, Hg bosonic isotopes have no nuclear spin No hyperfine mixing 88 Sr Gross structure Fine structure interaction (L.S, S.S couplings +QED) Quenching with a static magnetic field Taichenachev et al., Phys. Rev. Lett. 96, (2006) Use a 2 photon transition 24 Zanon-Willette et al., Phys. Rev. Lett. 97, (2006)

25 Other interesting transitions Nuclear transition in Thorium 229 Th Generally, nuclear excited states are very far apart at the scale of the optical spectrum In 229 Th nucleus, an excited state lies very close to the ground state The two isomers have different nuclear magnetic moment The transition between the two states is M1 (magnetic dipole transition) Transition frequency is know from high-precision Γ-ray spectroscopy (7.6 ± 0.1) ev (163 ± 11) nm The state of ionization of 229 Th atom can be chosen to ease laser cooling and detection. Triply ionized Th is foreseen as a good choice Very interesting for fundamental physics given the different nature of this transition Transition not observed directly so far Prospects for a Nuclear Optical Frequency Standard based on Thorium-229 E. Peik et al., arxiv: v2 Nuclear laser spectroscopy of the 3.5 ev transition in Th-229 E. Peik and Chr. Tamm, Europhys. Lett.61, 181 (2003) Two photon transition in Ag J. Hall and M. Zhu, J. Opt. Soc. A. B 6, 2194 (1989) T. Badr et al., Phys. Rev. A 74, (2006) 25

26 spectral density (10-19 J.m -3.Hz -1 ) Blackbody radiation shift Spectral density of blackbody radiation at 300 K Peak: Hz (λ~17µm) 0 0 1x x x x10 14 frequency (Hz) It is mostly preferred to avoid cryogenics Already very complex experiments Long term operation required for applications A priori, no large gain compared to fountains The shift is differential in microwave clocks The shift is hardly controlled to <<10-2 Most of the effect is given by the RMS value of the Stark shift ( E 2 ) The shift varies as T 4 Atom Cs (µw) Rb (µw) Sr Ca, Yb H(1S-2S) Mg Fractional shift due to blackbody 300 K Hg Hg x10-17 Al + -8x

27 Optical Ion Clocks 27

28 Ion trapping: Field Configuration in a RF trap W. Paul, Rev. Mod. Phys. 62, 531 (1990) Use the force exerted on a charged particle by an electric field Particle at rest at the trap center: E=0 1 st derivative of potential = 0 Simplest potential: The potential must satisfy Laplace equation: Implies that confinement is not possible in all dimensions with a static potential What if the trapping potential is sinusoidally varying in time? Note: An other possibility for stable trapping is to use a large static magnetic field Penning trap H. Dehmelt, Rev. Mod. Phys. 62, 525 (1990) 28

29 Ion trapping: Practical realization of the field Equipotential surfaces are hyperboloids Ideally, one should use electrodes with hyperboloid shapes Practically, simpler geometries are often used for optical clocks and Quantum Information Processing experiments, without much drawbacks Ions are 1 or few. They are laser cooled. They only see a small region near the trap center Not very sensitive to higher order terms in the field Generally, a superposition of static and varying potential can be applied 29

30 Ion trapping: Pictures of some traps Many other groups: NPL, NRC, + QIP research groups 30

31 Ion trapping: classical motion Classical equation of motion: Mathieu equations with (and similarly: a y,a z,q y,q z ) Stable solutions exist where the trajectory is bounded: secular motion micro motion for Stability diagram Important application: Mass spectrometer 31

32 Ion trapping: quantum motion The Hamiltonian looks like: D. Leibfried et al., Rev. Mod. Phys. 75, 281 (2003) For Effective potential approximation Approximated solutions can be written as: Breathing of the wavefunction at ω rf micromotion where an effective time-independent harmonic oscillator Hamiltonian applies to where ω x is the classical secular frequencies Complete formal analogy to a static harmonic potential. The effective Hamiltonian can be written as: However, the complete states are not energy eigenstates. The micromotion periodically change the total kinetical energy of the state. 32

33 Ion trapping: spectrum of a transition D. Leibfried et al., Rev. Mod. Phys. 75, 281 (2003) Consider an ion with two internal energy levels e and g Energy levels of the trapped ion Trapped ion interacting with light propagating along x where we have introduced the Lamb-Dicke parameter: Extension on ground state wavefunction 33

34 Parameter of a 199 Hg + ion trap Nominally, no static voltage: a x =a y =a z =0 Ring inner diameter: 0.8 mm Drive frequency: 10 MHz Drive amplitude: 1 kv Radial secular frequencies: υ x, υ y ~ 1 MHz Axial secular frequency: υ z ~ 2 MHz q x2, q y 2 ~ 0.1 q z 2 ~ mm The trap is loaded by e - impact ionization or by photoionization of a vapor Wavelength of the clock transition: λ=282 nm Recoil frequency of the clock transition: υ R =13 khz The same ion can be kept for months at 4 K, for Hg+ at room T, for Yb+ Lamb-Dicke parameter: η 2 ~ 0.01, η ~ 0.1 Extension of the vibrational ground state wavefunction: z 0 ~ 5 nm 34

35 Doppler cooling and fluorescence detection Doppler cooling D. Leibfried et al., Rev. Mod. Phys. 75, 281 (2003) Transition with decay rate faster than trap frequencies Similar to free space, similar cooling limit Laser tuned ~Γ/2 to the red of the cooling resonance Hg+: cooling wavelength: 194 nm Γ=1/τ ; τ ~2 ns T min ~ 1.7 mk Average vibrational quantum number: RMS size of motion: z 2 = z 02 (2<ni> +1) z ~ 355 nm Fluorescence detection <ni> ~35 When cooling light is on, the ion is scattering photons This is providing a way to detect the ion Quantum jumps in Ba + PMT Cooling laser Counter 35

36 Temporal sequence for probing the clock transition Cooling 36

37 Temporal sequence for probing the clock transition Pumping 37

38 Temporal sequence for probing the clock transition Clock transition excitation 38

39 Temporal sequence for probing the clock transition Detection PMT Case #2 Case #1 Case #1: Fluorescence is detected Ion is in the ground state. Case #2: No fluorescence is detected Ion is in the excited state. 39

40 Temporal sequence for probing the clock transition If case #2: Depumping PMT Case #2 Case #2: Wait for fluorescence to reappear due to spontaneous decay or depumping of the excited state. 40

41 Spectroscopy of the optical clock transition in 199 Hg + Carrier and secular sidebands Not strongly in the Lamb-Dicke regime Resolved sideband regime Recoilless optical absorption and Doppler sidebands of a single trapped ion J. C. Bergquist et al., Phys. Rev. A 36, 428 (1987) n=0 n=-1 n=+1 Carrier Narrow line spectroscopy of the carrier Line width: 6.5 Hz at 1.06x10 15 Hz Q at ~ 1.6x10 14 Quantum limited detection Clock upper state lifetime: 90 ms 2 Hz Quantum limited stability Red sideband T probe = 120 ms Blue sideband 41 R. J. Rafac et al., Phys. Rev. Lett. 85, 2462 (2000)

42 Cooling to the ground state of (secular) motion Sideband cooling Narrow enough transition + narrow enough excitation laser motional sidebands are resolved n=-1 Laser excitation of the red sideband Spontaneous decay accumulation in the vibrational ground state Cooling limit: off-resonant excitation of the carrier Decay rate Raman sideband cooling is similar with Raman transition Repumper Secular frequency 2 (or more) ions in a linear trap Sideband spectrum shows eigenmodes of motion Center of mass motion, breathing mode, Sideband cooling can be applied Sympathetic cooling of 1 ion by the other through the strong coulomb interaction 42

43 Detection and spectroscopy using quantum logic Assume the following situation 2 ions in a linear trap: Spectroscopy (clock) ion (S: Al + ) + Logic ion (L: Be + ) System cooled to the vibration ground state (sympathetic Raman sideband cooling with the logic ion) Usual probe pulse on the clock transition of the S ion The scheme is a way to detect the out-coming state of the S ion using a quantum logic gate to transfer the information to the L ion Vibrational spectrum is present on both L and S transitions and can be excited either by L or S excitation P. O. Schmidt et al.,science 309, 749 (2005) L ion E State selective fluorescence detection of the L ion Fluorescence means detection in the ground state of the S ion No Fluorescence means detection in the excited state of the S ion 43

44 Frequency comparisons between ion clocks Comparison between two 171 Yb + clocks on the quadrupole transition Stability: Schneider et al., PRL 94, (2005) Fractional frequency difference: Comparison between two Al + quantum logic clocks Stability: Chou et al., PRL 104, (2010) Fractional frequency difference: Total uncertainty: 44

45 Accuracy budget in Yb+ quadrupole and Al+ clocks 171 Yb + quadrupole + Black body radiation shift correction (at 300 K): (10)x10-16 (from theory) Tamm et al., PRA 80, (2009) Al + Some shifts are specific to ion clock Micromotion Electric quadrupole shift BBR and electrode heating Synchronous perturbation of E field by the probe light Chou et al., PRL 104, (2010) 45

46 Optical Lattice Clocks 46

47 Polarizability, dipole force and dipole trapping Atom in a laser field: induced dipole Laser intensity: Strontium ground state static polarizability: 27.6x10-30 m 3 ~ 186 (a 0 ) 3 Interaction energy between induced dipole and laser field Laser intensity: 10 kw.cm -2 U/2πħ ~150 khz, U/k B ~7.2 µk Interaction can be used to trap atoms Spatial variation of laser field intensity spatial variation of U force on atom widely used in quantum gases exp., J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985) This interaction is producing a light of energy levels 150 khz ~3x10-10 fractional shift of the clock transition!!! Bad starting point for a clock? 47

48 Polarizability as a function of frequency Polarisability α( 3 P 0 ) α( 1 S 0 ) Frequency 3 P 0 1 S 0 48

49 «Non-perturbing» dipole trap H. Katori et al., Phys. Rev. Lett. 91, (2003) Light shift as a function of trap wavelength 87 Sr 679 nm 3 S nm 679 nm 813 nm 813 nm 3 P0 1 P nm 3 P nm 3 D µm light shift 1 S0 813 nm 1 S nm Clock transition (~1 mhz) 2,56 µm Dipole trapping 3 P 0 trap wavelength Magic wavelength λ magic : polarizabilities are equal for both clock states 1 S 0 Light shift of the clock frequency: Sensitivity: fractional shift vs detuning from λ magic : under typical conditions: ~10-15 /GHz = 0 at λ magic 49

50 «Non-perturbing» lattice trap H. Katori et al., Phys. Rev. Lett. 91, (2003) Lamb-Dicke regime requires strong confinement strong variation of the trapping potential over short distance use a lattice trap formed by a standing wave Trap potential Electric field number of wavelength Natural unit of energy to characterize the trap depth U 0 : recoil energy at the trap wavelength: Combines advantage of trapped ion and neutral atoms large atom number in the Lamb-Dicke regime 50

51 Parameters of a strontium lattice trap Magic wavelength: Recoil energy: Trap depth: U 0 ~1400xE R ~4 MHz ~200 µk ding analogy/difference with ions -Dicke parameter cal lattice Longitudinal trap frequency: U 0 ~1400xE R 200 µk Transverse trap frequency: Longitudinal size of vibrational ground state wavefunction: Transverse size of vibrational ground state wavefunction: 51

52 Temporal sequence for probing the clock transition 461 nm (32 MHz) Laser cooling Loading into the dipole trap 1 P 1 3 S nm (5.2 MHz) Blue MOT (2 mk) Dipole trap beam (w 0 = 90 µm, U 0 =200 µk) λ trap 1 S nm (7.6 khz) P 0 => 10 4 atoms loaded 688 nm nm : atomic drain (w 0 = 50 µm) 52

53 Temporal sequence for probing the clock transition Loading into the dipole trap 1 P 1 3 S nm (7 MHz) 679 nm (1.75 MHz) Dipole trap beam S 0 3 P nm nm : repumpers (w 0 = 200 µm) 53

54 Temporal sequence for probing the clock transition Cooling in the dipole trap 1 P 1 3 S 1 Dipole trap beam S nm Γ = 7.6 khz 3 P nm : narrow line cooling 54

55 Temporal sequence for probing the clock transition Optical pumping 3 S 1 Piège 1 P 1 3 P 1 F=9/2 B 689 nm σ + σ 1 S 0-9/2 9/2 55

56 Temporal sequence for probing the clock transition Clock transition excitation 3 S 1 Piège 1 P 1 B 3 P 0 π π 1 S 0-9/2 9/2 B=1G, Z (-9/2 9/2) 1 khz 698 nm 56

57 Temporal sequence for probing the clock transition Measure ground state 1 P 1 3 S 1 Dipole trap beam 2 1 Blue probe 1 S 0 3 P 0 CCD camera 57

58 Temporal sequence for probing the clock transition 1 P 1 3 S 1 Repump 707 nm (7 MHz) 679 nm (1.75 MHz) Dipole trap beam S 0 3 P nm nm : repumpers (w 0 = 200 µm) CCD camera 58

59 Temporal sequence for probing the clock transition Measure excited state 1 P 1 3 S 1 Dipole trap beam 2 1 Blue probe 1 S 0 3 P 0 CCD camera compute transition probability 59

60 Spectroscopy in the Lamb-Dicke regime Spectrum at high power Strongly suppressed red sideband 95% of atoms in the longitudinal ground state Sideband ratio give T z ~3 µk z =5 nm Strongly in Lamb-Dicke regime Shape of sidebands Can be modeled with non Transition probability Red sideband Blue sideband harmonic term + distribution of n x,n y And used to determine T r detuning [khz] T r ~15 µk Spectrum at optimized power Optimized for the carrier Strongly suppressed coupling to sidebands A. Brusch et al. PRL 96, (2006) R. Le Targat et al., Phys. Rev. Lett. 97, (2006) 60

61 Spectroscopy in the Lamb-Dicke regime: State of the art Narrow line spectrum of the carrier Experimental resonance in a Sr optical lattice clock (JILA, Boulder). Q at = ~ 10 4 atoms probed simultaneously in ~1s quantum projection noise limited stability would be ~ In practice: poor duty cycle (~10%) + laser noise limit to M. M. Boyd et al., Science 314, 1430 (2006) 61

62 Measurement of the magic wavelength First measurement: M. Takamoto and H. Katori, Phys. Rev. Lett. 91, (2003) Several longitudinal states populated Note: Visibly distorted line shape when λ λ magic Later: accurate measurement of the light shift residual light shift : E r 3x10-7 in fractional 10 E r : E1 term (1) L = 3 (3) 10 Close to the target accuracy 17 X. Baillard et al., EPJD 48, 11 (2008). Accurate determinations of magic wavelength: 368,554.38(45) GHz (1) nm 368,554.68(18) GHz A. Brusch et al. PRL 96, (2006) 62 A. D. Ludlow et al., Science 319, 1805 (2008)

63 Hyperpolarisability Shift of the clock transition as a function of trapping field: ( 0 ) 2 1 hν = hν α ( e, ω ) E γ ( e, ω ) E Hyperpolarizability effect of -4 (4) µhz/e R 2 (-0.4(4) mhz for 10 E R ), corresponding to a -1(1) fractional frequency shift This effect will not affect the clock accuracy down to the level A. Brusch et al. PRL 96, (2006) 63

64 Comparisons between two Sr lattice clocks Between 87 Sr and 88 Sr: isotope shift of ~ 62 MHz allows direct comparison Ultra stable light at 698 nm 429 THz Akatsuka et al., Phys. Rev. A 81, (2010) 3D lattice configuration for 88 Sr For 88 Sr, multiply populated lattice sites are eliminated using a photoassociation laser to avoid collisions Measurement of the isotope shift 64

65 Accuracy and systematic shifts Most accurate lattice clock to date ~2 to 3 times better than atomic fountains ~10 times less accurate than the best ion clock ( 27 Al + ) Tremendous improvements in only a few years after the non-perturbing lattice proposal came out Ludlow et al. Science 319, 1805 (2008) 65

66 Ultra Stable Lasers for Optical Clocks 66

67 Important applications of ultra stable lasers Optical clocks and their applications (timekeeping, General Relativity tests, stability of fundamental constants, ). Tests of Relativity (Local Lorentz Invariance, Michelson-Morley,...). Generation of ultra low phase noise microwave signals and their applications (atomic fountains, ultra high resolution VLBI, ). Transfer of stable frequencies by fiber networks. Gravitational wave detection (VIRGO, LIGO, and LISA). Deep space frequency transfer. 67

68 Ultra stable Fabry-Perot cavities Currently, the reference method for laser stabilization. A laser is stabilized with high bandwidth to a mode of an ultra stable Fabry-Perot cavity. laser Feedback Partial reflectors Cavity modes Detection The physical length of the cavity is defining the laser frequency. L=0.1 m and target fractional stability of dimensional stability of 0.1 fm (charge radius of a proton is ~0.8 fm). FSR Free spectral range: c/2l Finesse: FSR/FWHM Designs have been continuously improved to cope with: Temperature sensitivity: length changes through material expansion Acceleration sensitivity: length changes under mechanical stress Thermal noise: length fluctuations due to Brownian motion of parts Electronic noises and drifts 68

69 Acceleration sensitivity: orders of magnitude Deformation of a cylinder. Sensitivity: 2x10-9 /(m.s -2 ) L/L = ρ a L/2E~ 2x10 8 L=0.1m E~ 7x10 10 Pa (ULE) ρ ~ 2x10 3 kg/m 3 a = g = 9.81 m.s -2 L a Vibration level in a laboratory. Measured with a commercial seismometer Need to reduce sensitivity Need to reduce vibrations 69

70 Basic concepts for low vibration sensitivity What matters is the optical length L between the two mirrors Find geometry and support scheme which minimize this quantity dl L L a a L+dL Poisson effect L a L-dL dl a L 70

71 Making use of symmetries? Ideal case: Length of interest is insensitive to transverse acceleration to first order Real case: the optical axis does not coincide with the mechanical axis There is first order sensitivity Minimizing mirror tilt is important a L =K 1 a+k 2 a 2 a L =K 1 a+k 2 a 2 Due to symmetry: K 1 =0 71

72 Example of a vertical cavity design FEM simulations with two different software package Analysis of the sensitivity in all directions Aspect ratio chosen to minimize mirror tilt Also taken into account Symmetry breaking by the support points Deformation at the support points Impact of modeling of support points Impact of meshing a Sensitivity [1/m.s -2 ] FEM Measured vertical horizontal < 5x10-12 ~2x10-12 ~3.5x10-12 ~1.4x

73 Horizontal cavity designs and other designs B. Young et al., PRL 82, 3799 (1999) New ideas for horizontal cavities by T. Rosenband (2003) C. T. Taylor et al., RSI 66, 955 (1995) Ludlow et al., Opt. Lett., 32, (2007) T. Nazarova et al., Appl. Phys. B 83, 531 (2006). S.Webster et al., Phys. Rev. A 77, (2008) Millo et al., Phys. Rev. A 79, (2009) Many other groups: NRC, MPQ, Tokyo Univ., 73

74 Vibration isolation Home made passive (springs, rubber bands, ) Commercial devices available Passive Smart suspension with negative stiffness mechanism Low eigenfrequencies in a reasonable volume Active Senses vibration with piezo accelerometers Feed back to actuators David L. Platus, Proc. SPIE 3786, 98 (1999) Mechanical perturbations are also transmitted through acoustic noise acoustic enclosure 74

75 Thermal noise (1) Due to Brownian motion of mechanical parts Can be calculated using the Fluctuation-Dissipation Theorem H. B. Callen and R. F. Greene, Phys. Rev. 86, 702 (1952) C. T. Taylor et al., Rev. Sci. Instrum. 66, 955 (1995) H=X/F P. R. Saulson, Phys. Rev. D42, 2437 (1990) PSD of thermal fluctuations of position Imaginary part corresponding to dissipation Susceptibility of position to applied force 3 contributions: spacer, mirror substrate and coatings K. Numata et al., Phys. Rev. Lett. 93, (2004) Q: mechanical quality factor. Characterizes mechanical losses in material Dominant contribution at low frequency: flicker noise in position Q factor of some materials: Material Q ULE glass 6x10 4 Zerodur 3x10 3 Fused Silica 1x

76 Thermal noise (2) Example: cavity with L=100mm, R=55mm, ULE glass, w 0 = 264 µm Corresponding fractional frequency noise PSD and instability: at f=1hz Dominated by substrate: Same example with Fused Silica substrate Dominated by coating: Limit from spacer + substrate is: Solutions for further improvement Coating designed for lower noise Longer cavities H. J. Kimble et al., Phys. Rev. Lett. 101, (2008) Increased spot size w 0 Lower temperature (note however: improvement is not necessarily monotonic with temperature) Use other materials (ex: crystalline material like silicon) Forget about FP cavities and use other concepts 76

77 Temperature sensitivity: orders of magnitude Many materials (BK7 glass, aluminum alloys, ) have a coefficient of thermal expansion near ~10-5 K -1 Keeping the fractional instability <10-15 would require a T instability <0.1 nk! ULE glass is commonly used for both the spacer and mirror substrates T can be adjusted CTE ~ 10-9 K -1 T instability must be <1 µk ppm.k -1 CTE of ULE glass With Fused Silica substrates, the effective CTE is increased CTE of FS: 5.5x10-7 K -1 1D model is wrong, mirror is bending Effective CTE can be ~10-7 K -1 Can be avoided using the smart scheme of Legero et al. CTE [K -1 ] 1,5x10-7 1,0x10-7 5,0x10-8 0,0-5,0x ,0x C Mirrors totally constrained on a rigid spacer Mirrors totally constrained on ULE spacer Mirrors partially constrained on ULE spacer ULE 7.5x ,5x Temperature [ C] 77 T. Legero et al., arxiv v1

78 Allan deviation of temperature ( C) 0,1 0,01 1E-3 1E-4 1E-5 Dealing with the thermal sensitivity Orders of magnitude: effectiveness of passive shielding In vacuum and with suitable design, heat transfer is dominated by radiation heat transfer T e =300 K T 2R=100mm Defines a time constant of τ~4400 s, for ε=1 For polished, un-oxidized gold plating: ε=0.02 τ~ s ~2.5 days Example of design with active stabilization and passive shielding 1E time (s) Emitted power per unit surface (Stefan-Boltzmann law): Absorbed power per unit surface: ~0.19 W.K -1, for ε=1 Heat capacity (ULE): ~843 J.K -1, for ε=1 Cavity impulse response: ~2nd order LPF with τ>4 days T coeff. ~10-7 K -1 Variation of the cavity frequency (MHz) Time (MJD) Temperature fluctuations are reduced by ~10 s Internal chamber temperature ( C)

79 Short introduction to the Pound-Drever-Hall detection R. W. P. Drever et al., Appl. Phys. B 31, 97 (1983) Eric D. Black, Am. J. Phys. 69, 79 (2001) Phase modulated laser field: laser No intensity modulation due to cancellation of all beat notes between sidebands at Ω,2Ω,3Ω, Pound-Drever-Hall detection Input spectrum laser Cavity reflection curve LPF Reflected spectrum In the reflected signal, near resonance, the balance between the sidebands is destroyed and an intensity modulation at Ω is induced. In this method, the speed of the servo loop is NOT limited by the line width 79 of the cavity Allows tight (fast) locking to high finesse cavity.

80 Other things to care about Fluctuations of pressure Fluctuations of radiation pressure Electronic noises and environmental sensitivity of electronics Electronic offset + fluctuations of laser power Photon shot noise Stray etalon and fluctuations of optical path length Residual Amplitude Modulation in the phase modulator and its fluctuations Fluctuations of input beam pointing Interaction with other cavity modes Higher order modulation sidebands interacting with other cavity modes Most of these effects are reduced in the first place by using cavities with very finesse (close to 10 6 ) Still they need to de considered carefully 80

81 Schematic view of laser comparison system slow servo loop filter fast servo loop filter P.D.H. Detection Hg Lab Fiber Laser PZT CTRL VCO AOM E O M PD 4μW Fiber link with noise cancellation VCO Fast servo loop filter PZT CTRL AOM E O M P.D.H. Detection Fiber Laser 4μW PD PD Beat note Frequency counter OPUS Lab Frequency to voltage converter FFT Analyzer Finesse λ=1062.5nm 81

82 Comparison between two ultra stable lasers: Stability Hg system: shields + T stabilization. OPUS system: shields only Red curve: Optical to optical comparison Hg against OPUS Blue curve: Hg against ultra stable microwace reference through femtocomb Drift rate ~ -50 mhz/s at nm. Fractional drift rate ~ -1.8x10-16 s -1. J. Millo et al., Phys. Rev. A 79, (2009) S. Dawkins et al., Appl. Phys. B (2009) DOI /s Proper thermal design can accommodate the higher sensitivity of fused silica mirrors Assuming equal contributions of both laser to the flicker floor, we find the flicker floor of 1 laser at 4x10-16, only possible with FS mirrors linewidth of ~170 nm 82 High reliability: continuous operation for month

83 Comparison between two ultra stable lasers: Frequency noise and Phase noise power spectral density Frequency noise PSD Phase noise PSD Blue line: calculated thermal noise limit -19 db[rad 2 Hz for 2 lasers 83

84 Conclusions and future directions Importance: laser noise limits the stability of most optical clocks due to the Dick effect Already mentioned modifications of ultra stable Fabry-Perot cavity Lasers operating on narrow atomic lines Active Optical Clock Chen, J., Chinese Science Bulletin 54, 348 (2009) Prospects for a Millihertz-Linewidth Laser D. Meiser et al., PRL 102, (2009) Stabilization to fiber delay lines An agile laser with ultra-low frequency noise and high sweep linearity Jiang, H. et al., Opt. Express 18, 3284 (2010) Ultralow-frequency-noise stabilization of a laser by locking to an optical fiber-delay line Kéfélian, F. et al., Opt. Lett. 34, ( 2009) 84

85 Measurement of Optical Frequencies with Optical Frequency Combs 85

86 Beat note between two lasers Laser 2 Laser 1 Beam splitter Photodiode Load Electric field: Poynting vector: cst. fast cst. fast slow Average optical power over detector time constant: fast Current through the load: Beat note RF power: Beat note between the 2 lasers at frequency: 86

87 Beat note between two lasers Fast photodiodes and electronics are typically limited to <100 GHz. 100 GHz = 0.1 THz is a tiny fraction UV/VIS/IR range (~1000 THz). in practice beat note measurements are narrow band at the scale of the UV/VIS/IR spectrum. Detecting a beat note is similar to using a mixer in the RF or microwave domain. However, a mixer usually gives access to both (ω 1 - ω 2 ) and (ω 1 + ω 2 ). Mixers are broadband devices at the scale of the RF or microwave domain. Other methods to move in the optical frequency domain: Second Harmonic Generation, Sum and Difference Frequency Generation. Generally narrow band (crystals, coatings, geometry, specific to a narrow wavelength range). Metal Insulator Metal Diode used as mixer, harmonic mixer, Broadband but practically very tedious. No straightforward way to link optical frequency to microwave frequency. 87

88 Optical Frequency Measurements prior to Optical Frequency Combs Harmonic frequency chain: Ensemble of subsystems each allowing a frequency multiplication by a factor 2 or 3. Complexity. Inconvenient frequencies (THz, mid- IR). 3 or 4 chains in the world at PTB, SYRTE, NRC, JILA. Continuous operation <3h. The setup has to be redesigned if one changes the frequency to be measured. H. Schnatz et al. PRL 76, 18 (1996) 88

89 Optical Frequency Measurements with Optical Frequency Combs Harmonic chain replaced with: 1 Laser Extremely simple and cost effective in comparison. No inconvenient frequency. Measurement of several frequencies at the same time. Continuous operation for weeks. Covers the visible-near IR range. Commercially available systems. Nobel Lecture: Defining and measuring optical frequencies J. L. Hall, Rev. Mod. Phys. 78, 1279 (2006) Nobel Lecture: Passion For Precision T. W. Hänsch, Rev. Mod. Phys. 78, 1297 (2006) 89

90 Output of a mode-locked Laser Ideal train of light pulses in time domain E(t) T T R =1/f R f R : repetition rate of the light pulses Pulse train inside the laser cavity with: Carrier envelop phase slip φ v g v φ 2 φ t Frequency domain Components are phase coherent Fourier limit: T. ν 1/4π I(f) f o f 0 = f r φ 2π f r ν Comb shifted by f 0 0 f n = nf r + f o 90

91 Requirements for mode-locked operation Gain medium with broad gain curve to support a large number of modes Controlled dispersion to allow short pulses to remain short through round trip propagation in the cavity Chirped mirrors or prism pairs to compensate normal dispersion Mechanism for (passive) mode-locking: Coupling between longitudinal modes so that they oscillate with a well defined phase with respect to each others (Intra cavity non linear effect: Saturable absorber, Kerr lens) 91

92 Example of a femtosecond Ti:Sa laser Gain medium is a Titanium doped sapphire crystal. Pumped with ~6 W of laser light at 532 nm. Cavity made of 6 chirped mirrors. Intra cavity wedge for coarse tuning f 0. Fast and slow piezo actuators. Fast pump power control through an AOM. 30 fs pulses at 850 nm. Repetition rate: ~760 MHz. ~500 mw output power. PZT wedge Acts on f R PZT AOM Acts on f 0 Acts on f R (slightly) 92

93 Femtosecond lasers used as comb generators Ti:Sa Th. Udem, et al., PRL 82, 3568 (1999), Opt. Lett. 24, 881 (1999) S. Diddams, et al. PRL, 84, 5102 (2000) Fiber lasers Erbium doped fiber Ytterbium doped fiber Chromium Forsterite Diode pumped Yb:KYW Microresonators Newbury, N.R. and Washburn, B.R., IEEE J. Quantum Electron. 41, 1388 (2005) P. Kubina et al., Opt. Express 13, 904 (2005) P. Pal et al., Opt. Express 15, (2007) K. A. Tillman et al., Appl. Opt. 48, 6980 (2009) S. A. Meyer et al., Eur. Phys. J. D 48, 19 (2008) P. Del Haye et al., Nature 450, 1214 (2007) Note: here the comb generation mechanism is different (4-wave mixing mediated by Kerr non-linearity) Repetition rate is very high Note: Ti:Sa and fiber femtosecond lasers well suited for optical frequency metrology are commercially available, as well as full optical frequency measurement systems 93

94 Measuring the comb parameters f R and f 0 Measurement of f R : detect the pulse train with a fast photodiode Note: gives directly access to harmonics of f R Measurement of f 0 with an octave spanning comb: The self-referencing method Note: many modes contribute to the signal H.R. Telle, et al., Appl. Phys. B 69, 327 (1999) 94

95 Generation of an octave spanning optical frequency comb Photonic crystal fiber Core: 2µm Short input pulses with high peak power Small core: high peak intensity Pulses remain short through propagation Large non-linear effects Dispersion management allows propagation of short pulses Time domain: Self phase modulation, cross-phase modulation, Frequency domain: Four wave mixing Coherent spectral broadening Note: pulses are broadened Spectral coherence but no Fourier limited pulses 95

96 Octave spanning optical frequency combs Ti:Sa, Er-doped fiber broadened with photonic crystal fiber S. Diddams et al., Phys. Rev. Lett. 84, 5102 (2000) Ultra broadband Ti:Sa T. M. Fortier et al., Opt. Lett. 31, 1011 (2006) Toroidal micro resonators P. Del Haye et al., arxiv v1 (2009) 96

97 Application to the measurement of optical frequencies T. W. Hänsch, Rev. Mod. Phys. 78, 1297 (2006) J. L. Hall, Rev. Mod. Phys. 78, 1279 (2006) The Optical Frequency Comb can be used in many different ways with pros and cons depending on the situation. For instance: f R can be locked to the RF reference f 0 can be stabilized to a reference, f b and f 0 can be counted and f 0 subtracted from f b digitally In optical frequency ratio measurements, f R can be eliminated from the measurement electronically in a smart combination of f b1 and f b2, H.R. Telle et al., Appl. Phys. B 74, 1 (2002) 97

98 Absolute frequency measurement of a Sr lattice clock Sr lattice clock vs atomic fountain F J E? = = J J E? A 5 H = J I. A J I B H A G K A? O? > ) ) 5-4 B H A G K A? O ) - + ( $ ' & X. Baillard et al., Eur. Phys. J. D 48, (2008) >15 days of ~continuous operation of the full metrological chain from optical clock to PFS Overall stability Best stability for comparing a microwave clock to an optical clock. Overall fractional uncertainty of the measurement 2.6x Overview of all Sr measurements validation of the non-perturbing lattice scheme S. Blatt et al., Phys. Rev. Lett. 100, (2008) 98

99 199 Hg + Some other recent absolute frequency measurements H (1S-2S) 171 Yb+ quadrupole Stalnaker et al., Applied physics. B, 89, 167 (2009) W. H. Oskay et al., Phys. Rev. Lett. 97, (2006) T. M. Fortier et al., Phys. Rev. Lett. 98, (2009) M. Fischer et al., Phys. Rev. Lett. 92, (2004) Niering et al., Phys. Rev. Lett. 84, 5496 (2000) Tamm et al., PRA 80, (2009) 171 Yb+ octupole K. Hosaka et al., Phys. Rev. A 79, (2009) 88 Sr+ 40 Ca+ 174 Yb 171 Yb 40 Ca H. Margolis et al., Phys. Rev. A 67, (2003) M. Chwalla et al., Phys. Rev. Lett. 102, (2009) N. Poli et al., Phys. Rev. A 77, (2008) N. D. Lemke et al., PRL 103, (2009) G. Wilpers et al., Metrologia 44, 146 (2007) Mg 199 Hg, 201 Hg Note: 27 Al + Friebe et al., Phys. Rev. A 78, (2008) M. Petersen et al., Phys. Rev. Lett. 101, (2008) 99

100 Frequency ratio of Al + and Hg + single ion clocks T. Rosenband et al., Science 319, 1808 (2008) Fractional uncertainty: 100

101 Applications: Ultra low noise microwave generation 101

102 Other Applications of Optical Frequency Combs Test of comb accuracy Ultraprecise Measurement of Optical Frequency Ratios Jörn Stenger et al., Phys. Rev. Lett. 88, (2002) Optical clockwork with an offset-free difference-frequency comb: accuracy of sum- and difference-frequency generation M. Zimmermann et al., Opt. Lett. 29, 301 (2004) Ultra low noise microwave generation The Stability of an Optical Clock Laser Transferred to the Interrogation Oscillator for a Cs Fountain B. Lipphardt et al., IEEE Trans. Instrum. Meas. 57, 1119R (2008) Ultralow noise microwave generation with fiber-based optical frequency comb and application to atomic fountain clock J. Millo et al., Appl. Phys. Lett. 94, (2009) Astrocombs: Use of OFC as reference for high resolution spectrometers used in astronomical telescopes Nature 452, 610 (2008) Eur. Phys. J. D 48, (2008) Science 321, 1335 (2008.) Direct Frequency Comb Spectroscopy Molecular fingerprinting Direct frequency comb spectroscopy, Matthew C. Stowe et al., Adv. Atom. Mol. Opt. Phys. 55, 1-60 (2008) Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb, S.A. Diddams et al., Nature 445, 627 (2007) Extension to the DUV/XUV range A deep-uv optical frequency comb at 205 nm Peters, E. et al., Optics Express 17, 9183 (2009) 102

103 Optical Links 103

104 Toward an optical link across Europe (1) Principle : heterodyne Michelson interferometer to detect optical path length fluctuation in the fiber Demonstration of ultra stable optical carrier transfer in telecom fiber over ~ km demonstrated in several groups (PTB, JILA/NIST, SYRTE/LPL, NMIJ/Tokyo Univ., ) Over a single such segment stability is when satellite systems are at Allan Deviation σ y (τ) τ[s] = 13 km SYRTE H. Jiang et al., JOSA B 25, 2029 (2008) telecom optical fibre of the urban france telecom paris network (44 km) 104

105 Demonstration of compatibility with internet traffic Fractional frequency stability Toward an optical link across Europe (2) Floor 86 km compensated measurement time[s] 108 km free running 86 km free running 108 km compensated SYRTE, LPL, RENATER Test with purposely developed WMDM devices + RENATER 11km test fiber F. Kéfélian et al., Opt. Lett. 34, 1573 (2009) Develop repeater stations which meet reliability, robustness, cost requirements Develop appropriate collaborations, to access fibers, spaces for repeater stations,... over the relevant ~1000 km paths 105

106 Application of Atomic Clocks to Testing the Stability of Fundamental Constants 106

107 Atomic transitions and fundamental constants (1) Atomic transitions and fundamental constants Hyperfine transition Electronic transition See also, molecular vibrational and rotation => (m e /m p ) 1/2, m e /m p Actual measurements: ratio of frequencies Electronic transitions test α alone (electroweak interaction) Hyperfine and molecular transitions bring sensitivity to the strong interaction 107

108 Atomic transitions and fundamental constants (2) m p, g (i) are not fundamental parameters of the Standard Model m p, g (i), can be related to fundamental parameters of the Standard Model (m q /Λ QCD, m s /Λ QCD, m q =(m u +m d )/2) It is often assumed that : δ ( m ( m s s / Λ / Λ QCD QCD ) ) δ ( m = ( m q q / Λ / Λ QCD QCD ) ) V. V. Flambaum et al., PR D69, (2004) Recent accurate calculations have been done for some relevant transitions V. V. Flambaum and A. F. Tedesco, PR C73, (2006) Any atomic transition (i) has a sensitivity to one particular combination of only 3 parameters (α, m e /Λ QCD, m q /Λ QCD ) 108

109 Sensitivity coefficients of some transitions Rb hfs κ α 2.34 κ q κ e 1 K α, K e : accuracy at the percent level or better Cs hfs K q : accuracy? H opt Yb + opt Hg + opt Dy comb PR C73, (2006) Dysprosium : RF transition between 2 accidentally degenerated electronic states Dzuba et al., PRL 82, 888 (1999) In some diatomic molecules: cancellation between hyperfine and rotational energies also leads to large (2-3 orders of magnitude enhancement) 109 Flambaum, PRA 73, (2006)

110 Overview of recent measurements LNE-SYRTE (EFTF, IFCS, CPEM 2008) MPQ + LNE-SYRTE (PRL 2004) Tokyo, JILA, LNE-SYRTE, (PRL 2008) NIST, (PRL 2007) Berkeley, (PRL 2007) PTB, (PRL 2004), (arxiv 2006) NIST, (Science 2008) Least squares fit INDEPENDENT OF COSMOLOGICAL MODELS 110

111 Improvement of clock accuracy with time Several optical clock transitions are recognized by BIPM as secondary representations of the SI second In the future, optical clocks will offer a large number of possibilities 111 for a redefining the SI second and testing fundamental physical laws.