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## The Black Body Radiation Shift

The environment surrounding the interrogated atoms emits thermal radiation around room temperature (a few of ten of THz). This far detuned radiation causes a frequency shift, called the blackbody radiation shift. The shift is described as a power series of T4, T6 and T8. The first static term stat is proportional to the difference between the static polarizabilities of both clock states and the mean thermal field of the environment radiation. It corresponds to 90% of the total BBR shift. The second term, dyn, is the dynamic correction to the static polarizability and it includes the frequency dependence of polarizabilities of all states through the BBR spectrum:where stat = −2.13013(6) Hz and dyn = −147.6(23) mHz are the shifts at the temperature T0 = 300 K [124]. This correction is applied in our clock.

The estimation of the BBR shift is based on the precise measurement of the thermal environment around the atoms. The temperature around the atoms is measured continuously by 6 calibrated pt100 sensors. The temperature of the science chamber is not homogeneous, mostly due to the heating of the MOT coils. Although, they are cooled, they dissipate heat and warm up the vacuum chamber. Therefore, sensors are spread in pairs in the hottest points – around each MOT coil – and coldest points – far away from coils – of the science chamber. The last pair is placed in the mid temperature points. The temperature of the chamber shows inhomogeneity as large as 0.9 K. Except the points where the temperature is measured, we do not have any other information about the temperature gradient in the chamber. We simply, but pessimistically assume that th effective temperature is not larger or smaller than the extreme measured temperatures and it is equally distributed in this range. The associated uncertainty is 1/ p12 of the total spread, so it is equal to 0.26 K [47,125]. However, our vacuum chamber has two large view ports, one on each side of the atomic ensemble. Windows have larger emissivity than the metallic part of the chamber and occupy a large solid angle. Moreover, the measured temperature does not differ from the temperatures of mid points.

We need to include in the total BBR induced shift the fact that the atoms can see the hot effusive oven. Due to the implementation of the deflection beam, atoms do not share common sight axis with the oven. The distance between the source of hot atoms and the trap of cold atoms is 700 mm. Hot atoms are transferred via 5 mm diameter aperture. As in the case of the vacuum chamber, we do not have a precise model of potential reflections and the possible distribution of this radiation. We use a simple model that assumes the worst case scenario that atoms are covered by a sphere of 700 mm radius and emissivity 0.11 at room temperature. On its surface, we consider a small BBR source which has the same characteristics as the oven. The small source emits the thermal radiation which is uniformly reflected by the big sphere. Using this model the BBR shift in the center of the sphere is 10−17, which we take as the uncertainty of the BBR shift induced by the hot oven. The model does not include that indirect emission of the BBR of the oven to the atoms is shielded. It takes into account only homogeneous emissivity of the solid angle from the atoms, which is additionally higher than assumed 0.1. Therefore this model overestimates the uncertainty of the shift.

### The Lattice Light Shifts

In the previous chapter, we shown that the differential light-shift due to the present of the trapping light is as follows:The evaluation of lattice light-shift is based on a differential measurement of the clock frequency for various trap depths from 50 Er to 1000 Er. At the magic wavelength, hyperpolarizability effects, proportional to the square of the average trapping potential, are visible and the data are fitted to a parabola f(U0) = aU2 0 + bU0 + c. The correction and the uncertainty of the light-shift at the normal clock operation (70 Er) are taken from the parameters of the fit. However, based on only the single measurement the uncertainty of the coefficient a is rather large, which leads to a large uncertainty on the extrapolation to zero trap depth. Over years, repeated lattice light-shift measurements show that the value of a is fully reproducible for different lattice sources, atomic densities and the lattice polarizations. Therefore, the value of a = (0.457 ± 0.066)μHz/E2 r taken to extrapolate the lattice light-shift at zero trap depth (figure 4.3.1), is fixed at the average value, which allows us to decrease the associated uncertainty. The typical value of the correction and uncertainty are at low 10−17 level (the accuracy budget is at the end of the chapter). During the last campaign in June 2017, the Sr2 clock operated with the trap depth at 25 Er and the associated uncertainty goes down to 3×10−18. We observed also the additional light-shift due to the incoherence of the background spectrum of the semiconductor sources. This shift depends on the given tapered amplifier chip and it changes with the aging of the chip. To avoid this problem, we resigned from using semiconductor sources, and for the measurements described in the next chapters we use a Titanium Sapphire (TiSa) laser. Nevertheless, we investigated the problem, and the separated chapter 7 is dedicated to describe the light-shift that arises from the spectrum background.

#### The Scalar and Tensor Shifts

In the previous chapter, we have shown that the dipole polarizability (!) can be split in three terms: scalar s, vector v and tensor t. The polarizability depends on the trap polarization, therefore, the evaluation of the clock accuracy needs to include this dependence. To be more adapted to the experimental conditions, we define the trap depth U0 = s|E/2|2 and introduce the differential s,t coefficients: where the s,t = s,t(3P0)−s,t(3S0). Now, we consider both scalar s and tensor t coefficients, because they depend linearly on the average trap potential: E1 = (s + t)U0.

**Table of contents :**

**1 Introduction **

1.1 Characterization of Optical Frequency Standards

1.2 State-of-the-Art Optical Clocks

1.2.1 87Strontium Optical Lattice Clocks

1.2.2 88Strontium Optical Lattice Clocks

1.3 Thesis At-a-Glance

**2 Atoms in the Dipole Trap Potential **

2.1 Dipole Trap

2.2 Dipole Polarizability

2.3 The Magic-Wavelength

2.4 1D Optical Lattice

2.5 Lamb-Dicke Regime

2.6 Ultrahigh Resolution Spectroscopy

**3 Clock Operation **

3.1 Cooling and Trapping Sr Atoms

3.2 Lattice Laser Systems

3.3 Loading Atoms in the Dipole Trap

3.4 State Preparation

3.5 Stabilization of the Clock Light

3.6 Clock Spectroscopy and Detection

**4 Evaluation of Systematic Effects **

4.1 The Zeeman Shifts

4.2 The Black Body Radiation Shift

4.3 The Lattice Light Shifts

4.3.1 The Scalar and Tensor Shifts

4.3.2 The Vector Shift

4.3.3 Hyperpolarizability

4.4 The Density Shift

4.5 The Probe Light AC Shift

4.6 The Line Pulling Shift

4.7 The Background Gas Collision Shift

4.8 AOM Phase Chirp

4.9 The DC Stark Shift

4.10 Servo Error

4.11 The Accuracy Budget

**5 Comparisons of Optical Clocks **

5.1 Applications of Clock Comparisons

5.1.1 Confirmation of an Accuracy Budget

5.1.2 Variation of Fundamental Constants

5.1.3 Dark Matter

5.1.4 Astronomy

5.1.5 Special Relativity Theory

5.1.6 General Relativity Theory

5.1.7 Time Scales

5.2 Methods for Frequency Transfer

5.2.1 Local Comparisons

5.2.2 Comparisons by Satellite

5.2.3 Comparisons by Fiber Link

5.3 Gravitational Redshift Corrections

5.4 Sr-Sr Comparison

5.5 Sr-Microwave Standards Comparisons

5.6 Sr-Hg Comparison

5.7 SYRTE-PTB

5.8 SYRTE-NPL

5.9 Test of Special Relativity

**6 Temps Atomique International **

6.1 A Timescale with Optical Clocks

6.2 Reliability of Clocks

6.3 Estimation of Uncertainty of Dead Times

6.4 Contribution to TAI

6.5 SYRTE-NICT

**7 A Lattice with Semiconductor Sources **

7.1 Methods

7.1.1 Coherent Light in the Cavity

7.1.2 Incoherent Light in the Cavity

7.1.3 Filtering System

7.1.4 Theory

7.1.5 Experiment

7.2 Titanium-Sapphire Laser

7.3 Slave Diodes

7.4 Tapered Amplifier

7.4.1 Model of the ASE Background

7.5 Results

7.5.1 Dependence on Temperature

7.5.2 Filtering of the Slave Spectrum

7.5.3 Filtering by Band Filter

7.5.4 Filtering of the TA Spectrum

**8 Summary **

**Appendices **

**A List of publications **

**B Résumé **

B.1 Principe de fonctionnement

B.2 Évaluation des effets systématiques

B.3 Comparaisons d’horloges

B.3.1 Comparaison locale de deux horloges au strontium

B.3.2 Comparaisons locales d’horloges d’espèces différentes

B.3.3 Comparaisons intracontinentales

B.4 Vers une redéfinition de la seconde fondée sur une transition optique

B.4.1 Horloges au strontium opérationnelles

B.4.2 Contribution au TAI

B.4.3 Comparaisons intercontinentales

B.5 Un réseau optique avec des sources semi-conducteurs

B.6 Conclusion

**Bibliography**