1. Tests of Lorentz Invariance with atomic clocks and optical cavities Fundamental Physics Laws: Gravity, Lorentz Symmetry and Quantum Gravity - 2 & 3 June 2010 - Paris, France P. Wolf 1, F. Chapelet 1, S. Bize 1, A. Clairon 1 1 LNE-SYRTE, Observatoire de Paris

2. Contents Introduction The Lorentz violating Standard Model Extension (SME) - Photon sector and present limits - Matter sector Frequency shift of the Cs hyperfine transition Experimental strategy Data and analysis Systematic effects Results Outlook and Conclusion.

3. Introduction Lorentz Invariance (LI): invariance of physics in inertial frames under change of velocity or orientation. Founding postulate of relativity  cornerstone of all of modern physics. Unification theories (string theory, loop quantum gravity, …) hint towards LI violation.  strong motivations for experimental LI tests. Michelson-Morley, Kennedy-Thorndike, Ives-Stilwell, Hughes-Drever,…. Comprehensive framework for all tests of LI developed (Kostelecky et al.), the minimal Standard Model Extension (SME). Photon and matter sectors. Photon sector is tested by astrophysical observations (birefringence), and laboratory experiments (cavities, clocks) Search for a dependence of atomic transition frequencies on the orientation of the involved spins. We test a previously “unexplored” region of the SME parameter space: the proton quadrupole SME energy shift (related to the 7/2 nuclear spin of Cs). First measurements of 4 parameters, improvement by 11 and 12 orders of magnitude on 4 others  good probability of finding a LI violating signal. Wolf et al. (2004), Stanwix et al. (2006), Müller et al. (2007). Wolf et al., Phys. Rev. Lett. 96, 060801, (2006).

4. The Standard Model Extension (SME): photon sector Generalization of the SM Lagrangian including all Lorentz violating terms that can be formed from known fields (photons, p +, e -, n, etc..). The photon sector of the SME is equivalent to usual Maxwell equations with: Experiments generally set limits on linear combinations of the  tensors: 10 components: limited by astrophysical tests to  10 -32 5 components: limited by resonator tests at ≈ 10 -17 3 components: limited by resonator tests at ≈ 10 -13 1 component: limited by IS exp. at ≈ 10 -7

5. The Standard Model Extension (SME): photon sector Spectropolarimetry of distant sources (0.1 – 2 Gpc) in IR – UV band. Search for polarization change proportional to L/. [Kostelecky and Mewes, PRL 2001] 10 components limited to  10 -32 Michelson-Morley experiments with rotating optical cavities. [Herrmann et al. PRD 2009, Eisele et al. PRL 2009] 8 components limited to  10 -17 and  10 -13 Ives-Stilwell experiments with Li ions at 0.06c in particle accelerators [Reinhardt et al. Nature Physics 2007] 1 component limited to  10 -7

6. The Standard Model Extension (SME): matter sector The matter sector of the SME can be expressed as a perturbation of the standard model hamiltonian, parametrised by 44 parameters (40 at first order in v/c) for each known particle (p +, e -, n, in atomic physics). Leads to shifts of atomic energy levels as function of the atomic state. In the atom frame: -  w,  w,  w,  w, w are specific to the atom and the particular state. - the tilde coefficients are combinations of SME parameters, to be determined by experiment. They are in general time dependent due to the movement of the atom with respect to a cosmological frame. in GeV Wolf et al., Phys. Rev. Lett. 96, 060801, (2006).

7. Cs hyperfine Zeeman transitions in the SME Using the results of Bluhm et al. [PRD 68, 125008], the perturbation of a |F=3, m F >  |F=4, m F > transition in Cs is: SME part Classical part: Z (1) B ≈ m F 1400 Hz mFmF BpBp DpDp GpGp CpCp BeBe DeDe GeGe +3-1/6+1/22 K p -2/33 K p +1/14 K p -3/2+1/2 K e -1/2 K e -3+1/6-1/22 K p +2/33 K p +1/14 K p +3/2-1/2 K e +1/2 K e K e ≈ 10 -5 ; K p ≈ 10 -2 (Schmidt nuclear model). Direct measurement limited by first order Zeeman shift (fluctuations of B).  measure “simultaneously” 3, -3, and 0 : cancellation of first order Zeeman second order Zeeman ≈ -2 mHz

8. Atomic fountains: Principle of operation N at ~2  10 9 ;  r ~  mm; T ~1  K V launch ~ 4m.s -1 ; H ~1m; T ~500ms T c ~0.8-2s; B ~ 200 nT Stability (m F =0): 1.5 x 10 -4 Hz/√  1.5 x 10 -6 Hz observed Accuracy (m F =0): 6.0 x 10 -6 Hz transition probability B Quantization field

9. Experimental strategy Alternate m F = 3 and m F = -3 measurement every second (interleaved servo-loops). Measure m F = 0 clock transition every 400 s (reference). Limited by stability of magnetic field at  < 4 s. Reduce launching height to optimize stability of observable. A, C i, S i, are functions of the 8 proton components: 3 proton components ( ) are suppressed by v  /c ≈ 10 -4 Search for offset, sidereal and semi-sidereal signatures in the observable Transforming to sun-frame SME parameters:

10. 21 days of data in April 2005, 14 days in September 2005. Least squares fit: in mHz

11. Residual First order Zeeman Shift Magnetic field gradients and non-identical trajectories of m F =+3 and m F =-3 atoms can lead to incomplete cancellation of Z (1). Confirmed by TOF difference ≈ 158  s (  623  m). Variation of B with launching height ≈ 0.02 pT/mm (at apogee).  MC simulation gives offset of only ≈ 6  Hz. Contrast as function of m F : 0.94, 0.93, 0.87, 0.75 MC simulation with only vertical B gradient cannot reproduce the contrast  horizontal B gradient of ≈ 6 pT/mm (≈ 2 pT/mm from tilt measurements). Complete MC simulation, assuming horizontal asymmetry between trajectories is same as vertical (worst case) gives offset ≈ 25 mHz. Fitting sidereal and semi-sidereal variations to the TOF difference and using the above gradients we obtain no significant effect within the statistical uncertainties (≈ 0.03 mHz at both frequencies). We take this as our upper limit of the time varying part of the residual first order Zeeman.

12. Results Sensitivity to c TJ reduced by a factor v  /c (≈ 10 -4 ). Assuming no cancellation between c TJ and others. First measurements of four components. Improvement by 11 and 13 orders of magnitude on previous limits (re-analysis of IS experiment, [Lane C., PRD 2005]). Dominated by statistical uncertainty (factor 2) except for c Q. in GeV

13. Outlook and Conclusion New test of LI, with first measurements of four proton parameters and large improvements on four others. Exploring a qualitatively new region of the SME. Still no evidence for Lorentz violation. Repeating the experiment over a year or more allows reduction of statistical uncertainty (gain of about a factor 2), and resolution of annual sidebands  decorrelation of c TJ parameters. Ultimate limitation will probably come from residual first order Zeeman effect (magnetic field gradient coupled to asymmetries in the  m F trajectories). Could be resolved by using simultaneously Rb as a magnetic field probe. That may also allow precise measurements of the magnetic field dependent transitions in a Rb – Cs combination  access to additional parameters. Similar experiment could be carried out onboard ACES (2009). Faster integration due to 90 min orbital period.