Tests of Lorentz Invariance with atomic clocks and optical cavities

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. Wolf1, F. Chapelet1, S. Bize1, A. Clairon1 1LNE-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 k 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/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: - bw, dw, kw, gw, lw 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, mF>  |F=4, mF> transition in Cs is: SME part Classical part: Z(1)B≈ mF 1400 Hz Ke≈ 10-5 ; Kp≈ 10-2 (Schmidt nuclear model). • Direct measurement limited by first order Zeeman shift (fluctuations of B). •  measure “simultaneously” n3, n-3, and n0: • cancellation of first order Zeeman • second order Zeeman ≈ -2 mHz

8. Nat~2109; r~1.5-3 mm; T ~1mK Vlaunch ~ 4m.s-1; H ~1m; T ~500ms Tc ~0.8-2s; B ~ 200 nT Stability (mF=0): 1.5 x 10-4 Hz/√t 1.5 x 10-6 Hz observed Accuracy (mF=0): 6.0 x 10-6 Hz Atomic fountains: Principle of operation B Quantization field transition probability

9. Experimental strategy • Alternate mF = 3 and mF = -3 measurement every second (interleaved servo-loops). • Measure mF = 0 clock transition every 400 s (reference). • Limited by stability of magnetic field at t < 4 s. • Reduce launching height to optimize stability of observable. • Transforming to sun-frame SME parameters: • A, Ci, Si, 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

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 mF=+3 and mF=-3 atoms can lead to incomplete cancellation of Z(1). • Confirmed by TOF difference ≈ 158 ms ( 623 mm). • Variation of B with launching height ≈ 0.02 pT/mm (at apogee).  MC simulation gives offset of only ≈ 6 mHz. • Contrast as function of mF: 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 in GeV • Sensitivity to cTJ reduced by a factor v/c (≈ 10-4). • Assuming no cancellation between cTJ 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 cQ.

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 cTJ parameters. • Ultimate limitation will probably come from residual first order Zeeman effect (magnetic field gradient coupled to asymmetries in the  mF 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.