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Dipole Polarizability
In previous section, we modeled the atom by a simple oscillator, but a real atom is more complicated. Due to the complex sub-structure of electronic transitions, the dipole potential depends on the given atomic sub-state. In this section, a multi-level atom will be described with an irreducible operator of polarizability (!), depending on the trapping light polarization and the orientation of the quantization axis. Calculations are based on refs [60, 98, 110]. We consider an atom with a ground state |g, {n, J, F,mF }i and an exited state |e, {n0, J0, F0,m0 F }i, with unit angular momentum along the quantization axis, parallel to the bias magnetic field Bs. Atom is placed in the dipole potential with the presence of Bs. The Hamiltonian of that system is: ˆH tot = ˆHZ + ˆH, (2.2.10).
where ˆHZ is the Hamiltonian describing the Zeeman interactions: ˆH Z = gμB ~ Fˆ · Bs, (2.2.11). where g is the Landé factor and μB is the Bohr magneton. The Hamiltonian of the electric dipole interaction between atom and the off-resonance trapping light is as follows: ˆH = −dˆ· E(r, t), (2.2.12).
which, in the second-order perturbation theory can be expressed as [111]: ˆH = − 1 4 ˆ|E|2.
Ultrahigh Resolution Spectroscopy
The ultra-narrow clock transition is the ultimate goal of the atomic clocks. Thanks to the Lamb-Dicke and well-resolved sidebands regimes, the pure electronic excitation is minimally perturbed. At the low saturation limit (below unity) the amplitude of sidebands are smaller at least by factor of 10. As a result of the clock interrogation, the absorption signal is Rabi-shaped. An example of the observed resonance is shown in figure 2.6.5.
For negligible broadening effects, the linewidth of the absorption line is Fourier limited by interrogation time, itself limited by the coherence time of the probe ultra-stable laser. But the phase coherent signal, generated by ultra stable laser, should be longer than time of the single clock interrogation and it extends over several spectroscopy probings. Longer coherence times allow to obtain narrower resonances. Currently, observed resonances have linewidth at sub-Hz level by using state-of-the-art laser [119]. But recent progress in ultra-stable lasers pushed forward the linewidth of the atomic spectrum [22]. The stability of cryogenic-stabilized Fabry-Pérot cavities is thermal noise limited at 4 × 10−17 with a coherence time up to 55 s. The linewidth of the laser can be then as small as sub-mHz level, which is comparable with predicted atomic linewidth.
Cooling and Trapping Sr Atoms
To obtain ultra-cold Sr atoms, we need to heat them up to temperature 500 C – 600 C in an effusive oven, due to the low vapor pressure of Sr. The hot atomic flux (1012 atoms/s) is cooled down by using the 1S0−1P1 transition at 461 nm. In the first step, atoms are cooled in a Zeeman slower in order to be trapped in a 3D magnetooptical (MOT). The cooling transition with linewidth = 2 · 32 MHz is fully allowed by electric dipole selection rules and guarantees efficient cooling of atoms to a Doppler limit of TDopp = ~ 2kB = 720 μK, where ~ and kB are the Planck’s and Boltzmann’s constants respectively. In practical realization, the obtained temperatures are higher – a few mK due to the extra-heating mechanism, induced by transverse spatial intensity fluctuations of the optical molasses [120]. To generate the 461 nm light, a doubling ring cavity with PPKTP crystal (periodically poled potassium titanyl phosphate) is used. An infrared beam at 922 nm is amplified by a tapered amplifier (TA) before the cavity. The injection beam is provided to the TA from the SrB system. The laser system is locked to the 1S0−1P1 transition via spectroscopy in an independent Sr oven with a feedback loop to the 922 nm master laser. The cavity is digitally locked by using a RedPitaya (FPGA – field-programmable gate array). We can obtain 250-350 mW of optical power from 1 W of infrared light. Then the output beam is divided on the deflection, MOT and Zeeman beams. The general scheme of the distribution of blue beams is shown in figure 3.1.2.
Loading Atoms in the Dipole Trap
Cold atoms, collected in the MOT, must be transferred to the dipole trap. At SYRTE, the loading technique, the so-called «atomic drain» is used. It is based on pumping the atoms to the metastable states 3P0 and 3P2 via the 1S0−3P1 (689 nm) and 3P1-3S1 (688 nm) transitions. The relevant transitions are depicted in figure figure 3.3.7. The technique is in contrast to all other laboratories, which employ the second stage MOT based on the narrow 1S0-3P1 transition as a second stage cooling (the so-called «red- MOT»). The atomic drain method has several advantages. The first one, very low power (4 μW for 689 nm and 20 μW for 688 nm) is necessary to implement this method. The red-MOT needs around 20 mW of power. The 689 nm laser in the atomic drain method requires only one frequency as opposed to the red-MOT, where, in order to effectively collect atoms, trapping and stirring frequencies are necessary due to the large hyperfine splitting of the 3P1 state. Moreover, the red-MOT cooling and trapping take more time.
With the atomic drain method, the loading to the dipole trap is performed in parallel to the first stage blue cooling and only the coldest atoms of the Maxwell distribution are transferred to the dipole trap, where they are cooled by short second stage cooling (section 3.4). Finally, because the blue MOT is ten times bigger than the red-MOT, the atoms are spread in more lattice sites and therefore, the atomic density in the lattice potential is low (1-2 atoms per site), the interaction between atoms are small and the density shift of the clock transition is reduced. On the other hand, the atomic drain method employs one extra laser at 688 nm.
As mentioned before, atoms are loaded to the dipole potential during the blue MOT phase. At this time, the trap depth is set to its maximum value in order to capture as many atoms as possible. Two drain lasers are tuned to the 1S0−3P1 (689 nm) and 3P1-3S1 (688 nm) transitions, overlapped and tightly focused at the center of the dipole trap. When cold enough atoms pass through the region of the dipole potential, they are transferred by the drain lasers to the metastable 3P0 and 3P2 states via the 3S1 state. In these states, atoms can be trapped in the dipole potential and they become invisible to the blue cooling transition. The MOT dynamics does not kick out them from the dipole trap. The transfer process is repeated, as the cloud of cold atoms thermalizes and new cold atoms are produced in the MOT. To have efficient transfer, the drain lasers operate at the nominal power 4 μW for 689 nm laser and at least 20 μW for 688 nm laser. Both lasers are locked to the corresponding atomic transitions in an independent Sr oven : the 689 nm laser via saturation spectroscopy and 688 nm via absorption spectroscopy. The transfer process is finished with the end of the blue cooling stage. Then, atoms in the metastable states are repumped by using two repumper lasers : 3P0−3S1 at 679 nm and 3P2−3S1 at 707 nm during 15 ms. Atoms confined in the dipole potential take part in the further preparation process.
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 Atoms in the Dipole Trap Potential
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
