Measurement of the Phase Chirp Introduced by the Pulsing of the Clock Acousto-Optics Modulator

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Current and prospective applications of optical fre-quency standards

I would now like to emphasize several new envisioned applications for optical frequency standards, which provide the motivation for world-wide research efforts towards increasingly high levels of clock frequency accuracy and stability, and for this thesis in particular.
The first one is chronometric geodesy, namely that a measurement of the gravitational redshift of the clock transition with respect to the known bare-transition frequency can allow a local probing of the grav-itational field at the position of the atoms [109]. On earth, the gravi-tational redshift scales as 1 10 16 per meter of elevation. This means that a clock working with an accuracy of 10 18 can probe the gravi-tational field with a resolution of 1 cm, making it a very useful tool for testing geoid models and for geophysics applications. To that end, transportable optical clocks are being developed in several laboratories around the world [49].
Another topic of high interest to the optical clock community is the possible redefinition of the SI second in terms of (an) optical transi-tion(s), and the dissemination of optical time and frequency references via optical fibers networks [94]. Such a network is currently being built on the French side in a collaboration between our group and the LPL (Laboratoire de Physique des Laser) group in Villetaneuse, linking SYRTE (SYstemes` de Ref´erences´ Temps-Espace) in Paris with other national metrology institutes in Europe, namely NPL (National Physical Laboratory) in the UK, PTB (Physikalisch-Technische Bunde sanstalt) in Germany and INRIM (Istituto Nazionale di Ricerca Metro-logica) in Italy [56] (see Figure 2). These fiber links also allow distant comparisons of optical clocks for fundamental physics tests [20]. Finally, another application in fundamental physics is the probing and monitoring of the variation of fundamental constants of nature, such as the fine structure constant, by looking at the time evolution of frequency ratios between clocks based on different atomic species over a few years [114] (see Section 0.2 for a more detailed discus-sion). These variations are predicted by many theories aiming at uni-fying electroweak and strong interactions (Standard Model) with grav-ity (currently described by General Relativity), a longstanding goal of modern physics.

Context and objectives of my PhD work

My PhD work, performed in the optical frequency group of SYRTE in Paris Observatory, is part of a global effort to improve the perfor-mances of optical clocks to a level compatible with applications men-tioned above.
In our lab, two Strontium optical lattice clocks are currently under development, which have reached an accuracy of 4.1 10 17 [57] and have shown nice agreement when compared with each other, as well as great reproducibility when comparing with microwave clocks [52], [57].
On an international scale, a single clock instability of 1.4 10 15 at one second has been demonstrated for a single Al+ ion optical clock [13], optical lattice clocks have shown instabilities as low as 1.4 10 16 at one second [102], reaching 10 18 measurement precision af-ter a few thousand seconds. As far as accuracy is concerned, single ion clocks have long been the reference for optical clocks, with a sin-gle ion Al+ clock demonstrating a total frequency uncertainty of 8.6 10 18 [13]. Recently, a single ion Yb+ clock has improved upon this remarkable accuracy with a record 3 10 18 total frequency uncer-tainty [39]. Optical lattice clocks are now reaching similar accuracies at the 10 18 level [75], [113].
However, I want to emphasize the fact that in order to be useful tool for precision measurements, clocks have to be compared with each other, and most of the applications detailed in Section 0.1.3 rely upon comparison of optical clocks over long distances which has been demonstrated by [109] with a resolution of 5.9 10 18, and/or com-parison of optical clocks based on different atomic elements for which the best published accuracy is the ratio reported in [73] with an accu-racy of 4 10 17.
At the beginning of this thesis, the mercury clock had demonstrated a short term stability of 5.7 10 15 at one second averaging time [63], and the overall accuracy of the clock was 5.4 10 15 [65]. The objective of my PhD work was to lower the instability and improve the accuracy of the mercury clock below the performances of state of the art microwave frequency standards, and perform clock compar-isons with the mercury clock once the accuracy was established. This manuscript details the technical improvements which allowed to push the short term stability to 1.2 10 15 at one second, to subsequently lower the uncertainty to 9.6 10 17, and finally to measure three fre-quency ratios involving the mercury clock.

Mercury Level Structure: the Key to a Highly Accurate Frequency Standard

Following Klechkowski’s rule, the electronic configuration for mer-cury can be written [Xe]4f145d106s2. It is an alkaline-earth-like atom with 2 valence electrons, giving rise to two categories of electronic states with spin singlet and spin triplet states exhibiting long lived metastable states, making it an ideal candidate for a high accuracy frequency standard. The structure of the energy levels of 199Hg relevant for our work is shown on Figure 3. The ground state is 1S0, and in our experiment, the relevant excited states are those of the triplet P state (3P1 and 3P0). In commonly used alkaline-earth-like atoms like Yb or Sr, the atoms are first cooled using the broad 1S0 ! 1P1 transition. However, in the particular case of mercury, this transition has a wavelength of 185 nm, making it highly impractical for laser cooling, and it has a 119 MHz natural linewidth, corresponding to a Doppler temperature of 2.8 mK, too high for directly loading an optical lattice.
Instead, we use single-stage cooling on the 1S0 ! 3P1 inter-combi-nation line at a wavelength of 254 nm (more details on the mercury laser-cooling can be found in Chapter 1). This transition is in principle not electric-dipole-coupled, due to the dipole selection rule S = 0, and is only weakly allowed due to the breakdown of LS coupling (A = 8.17 106 s 1 [35]). This weak but closed dipole transition permits single-stage cooling below 100 K temperatures and straightforward loading in the optical lattice.
Similarly, the 1S0 ! 3P0 transition is in principle strictly forbidden, but for the fermionic isotope, the presence of a non zero nuclear spin allows the hyperfine interaction to weakly mix the 3P0 state with the 1P1 and 3P1 states, giving rise to a very weak dipole coupling (A = 0.76 s 1 for 199Hg [4], [87]). This very weak transition has a lifetime of 1.3 s (199Hg) and a natural linewidth of = A/2 = 121 mHz, making it a very good narrow “clock” transition. However, the wavelength of 265.6 nm presents a challenge for high resolution spectroscopy, as we will discuss in Chapter 1. Furthermore, the frequency of this clock transi-tion is highly immune to many environmental perturbations which are the source of frequency offsets and fluctuations of atomic frequency standards. Of particular interest is the weak sensitivity of mercury to the blackbody radiation shift, which stems from the low polarizability of neutral mercury. This frequency shift is one of the limiting factor of many neutral-atom-based optical clocks (see Section 5.5 for an in depth discussion of this effect). We can therefore argue that the mercury atom possess key prop-erties that make it a potential highly accurate frequency standard at room temperature, provided that the experimental challenges related to the complex laser systems needed to manipulate it can be overcome.

Pre-cooling with a 2D-MOT

In the very near future, we plan to enhance the MOT loading rate by using a 2D-MOT. The 2D-MOT creates a bright beam of pre-cooled atoms [95], and therefore its main effect is to greatly enhance the load-ing rate of the 3D-MOT.
A foreseeable benefit of this enhancement lies in the prospect of lowering the pressure in the 3D-MOT chamber. The lifetime of the atoms in the optical lattice is approximately 300 ms (see inset of Fig-ure 1.3 (a)). Based on our study of the MOT loading time as a function of the mercury atoms background pressure (1.3 (b)), we have good reason to believe that this lifetime, and therefore the lattice lifetime, is limited by collisions with the hot background gas of mercury atoms, whose pressure is set relatively high (several 10 9 mbar) to allow effi-cient loading in the 3D-MOT. A more efficient loading with the 2D-MOT would therefore allow us decrease the vapor pressure of mercury in the chamber, resulting in an increased lifetime of the atoms in the lattice. With longer lifetimes, we could interrogate the atoms for several hun-dreds of ms while keeping a good signal to noise ratio, and therefore obtain narrower linewidths.
Finally, the 2D-MOT will allow us to trap more atoms (potentially up to a factor of 10 [84], [85]), which will provide a great increase in signal to noise ratio, as well as a much needed increase in lever arm for study of collisional properties of mercury (see Chapter 5), which has been very little studied in the cold and ultra-cold regimes.

Trapping in a 1D “Magic” Optical Lattice

We will now present the experimental setup for trapping mercury atoms at the magic wavelength. As we have briefly mentioned in the introduction, the chosen magic wavelength for neutral mercury is ex-pected to lie close to 362.5 nm. At this wavelength, the polarizability of the 2 clock states is equal to first order, and rather small (’20 a. u., or 5.7 kHz/(kW/cm2)) [80], [44]. This implies that a given lattice intensity will create a magic trap of relatively small depth compared for example to the case of Sr, for which the polarizability at the magic wavelength is close to 9 times higher.
The low polarizability at the magic wavelength and the need for ac-curate and stable control of the lattice light frequency put stringent constraints on the laser system needed to trap the atoms. Our ex-perimental effort to meet these constraints will be the subject of the following chapter.

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The trapping laser system

Several requirements must be met by the trapping laser system in order to reach clock performances below the 10 17 level. We need a highly tunable laser, over several hundreds of MHz to study lattice light shifts and accurately pinpoint the magic wavelength. Moreover, we also want a laser which is able to generate intensities on the atoms of several Watts, both for normal clock operation, which supposes a trap depth between 50 and 100 recoil energies (corre-sponding to 20 W incident power on the atoms), and for lattice light-shift studies.
Finally, we want to operate the laser at the magic wavelength, whose value was measured to be close to 362.6 nm [120]. A natural choice to meet all those demands is a doubled Ti:Sa laser seeding a build-up cavity. We have installed and tested during the course of this thesis a new commercial Ti:Sa laser from Msquared (SolsTiS model, see Figure 1.4). This laser is pumped by 16 W of green light at 532 nm (Verdi V18) and can output up to 5.5 W of light at 725 nm. Moreover, it is highly tunable in frequency, without mode-hop over several GHz. This laser allowed a gain of a factor 5 on the available trap depth for our exper-iment, greatly improving the lever-arm for the measurement of lattice related frequency shifts (see Section 5.3 for more details) as well as the number of atoms transferred from the MOT to the lattice.
This laser is fed into a home-built doubling bow-tie cavity contain-ing a 15 mm LBO crystal and locked at resonance using the Hansch¨-Couillaud locking technique. We have produced up to 800 mW of dou-bled light at 362.6 nm with this setup, but usually operate it with 200 mW of UV being produced out of the doubling cavity. The SolsTiS laser has two frequency modulation inputs whose char-acteristics are detailed in Table 1.1.

Absolute frequency calibration with a frequency comb

In the near future, our express purpose is to control the frequency of the magic trap to the 10 18 level accuracy and below. With this goal in mind, we will need a better control over the lattice light-shift, and since this light-shift is proportional to the detuning of the trapping laser from the magic frequency (see section 5.3), we therefore need to control and measure this detuning better than the MHz level. As we have seen in Section 1.3.2, the commercial wavemeter which we used so far to reference the frequency of the lattice light has a frequency accuracy of 5 MHz if properly calibrated (at least every hour), and is therefore not suited for this purpose.
To resolve this issue, we have decided to reference our laser to the operational optical-frequency-comb of SYRTE. Unfortunately, the wavelength range in which the comb has a good enough signal to noise ratio to allow for a beatnote between the Ti:Sa and the comb to be realized doesn’t reach below 800 nm, whereas as we have seen above, the Ti:Sa is operating at twice the magic wavelength, close to 725 nm. Therefore, in order to reference the laser to the comb, we have to divise a slightly more complicated scheme, which is shown schematically on We use a transfer oscillator scheme [110] to bridge the frequency gap using a ECDL laser outputting 100 mW (To check) at 1450 nm (Toptica DL100pro). A small fraction (’ 1 mW) of the ECDL light is sent to the comb lab using an uncompensated optical fiber, and beat with a comb tooth close to 1450 nm. The resulting optical beatnote is mixed with a DDS at 34 MHz. In order to realize a digital frequency modulation lock, the DDS is frequency modulated at 1 MHz, to probe the mid-points of higher slope at the half-maxima of the beatnote. The electronics beatnote is then filtered in a 1 MHz bandwidth and sub-sequently demodulated providing and error signal to react on the PZT of the ECDL and lock the ECDL to the comb. In this scheme, the micro-controller is used as a digital lock-in amplifier.
The remainder of the ECDL light is frequency doubled in a single-pass configuration in a PPLN crystal to yield about 100 W of light at 725 nm. We beat the Ti:Sa to the ECDL using the same digital lock-in modulation technique as the ECDL to the comb, using a second DDS at 70 MHz. The whole locking system and the laser fit in a standard 3U rack unit, and can stay locked for several hours without interruption.

Table of contents :

Acknowledgements
Introduction 
0.1 Optical Atomic Clocks
0.1.1 Ion-based optical clocks
0.1.2 Optical lattice clocks
0.1.3 Current and prospective applications of optical frequency standards
0.1.4 Context and objectives of my PhD work
0.2 The Mercury Atom: a Short Overview
0.3 Mercury Level Structure: the Key to a Highly Accurate
Frequency Standard
0.4 Thesis Overview
1 A Mercury Optical Lattice Clock 
1.1 Overview of the Experimental Setup
1.2 Cooling of Mercury Atoms in a Magneto-Optical Trap
1.2.1 The cooling-laser system
1.2.2 3D-MOT of 199Hg
1.2.3 Vapor pressure and MOT lifetime
1.2.4 Pre-cooling with a 2D-MOT
1.3 Trapping in a 1D “Magic” Optical Lattice
1.3.1 The trapping laser system
1.3.2 Locking scheme for the lattice light
1.3.3 A build-up cavity for a deeper trap
1.3.4 Absolute frequency calibration with a frequency comb
1.3.5 Lifetime of the atoms in the lattice
1.4 An Ultra-Stable Laser System for Coherent Atomic Interrogation
1.4.1 Fabry-Perot cavity for laser stabilization
1.4.2 Ultra-stable laser setup
1.4.3 Laser noise and frequency doubling
1.5 Fluorescence Detection
2 A New Laser System for Cooling Mercury Atoms 
2.1 Requirements
2.1.1 Spectral purity
2.1.2 Laser power
2.2 Architecture of the Cooling Laser
2.2.1 External-Cavity Diode Laser
2.2.2 Laser amplifier and single stage doubling to 507 nm
2.2.3 Frequency doubling to 254 nm
2.2.4 Locking to the cooling transition via saturated-absorption spectroscopy
2.3 A Second System for the 2D-MOT
2.3.1 Frequency locking of the two seed lasers for 2DMOT operation
3 High-Resolution Spectroscopy in an Optical Lattice Trap 
3.1 Theory: Spectroscopy in a 1D Optical Lattice
3.1.1 Clock spectroscopy in the Lamb-Dicke regime .
3.1.2 Structure of the clock transition
3.1.3 Rabi and Ramsey spectroscopy
3.2 Experimental Spectroscopic Signals and Their Interpretation
3.2.1 Magnetic field zeroing using clock spectroscopy measurements
3.2.2 Carrier spectroscopy of the two Zeeman sublevels
3.2.3 Control of atomic noise: implementing a normalized detection
3.2.4 Towards improved stability: high-resolution Rabi and Ramsey spectroscopy
3.2.5 Rabi flopping and excitation inhomogeneities
3.3 Estimation of the Trap Depth with Transverse Sideband Spectroscopy
3.3.1 Spectroscopy with a misaligned probe beam
3.3.2 Estimation of the trap depth
3.4 Studies of Parametric Excitation in the Trap
3.4.1 Trap depth estimation
3.4.2 Atomic temperature filtering
4 Clock Operation and Short-Term Stability Optimization 
4.1 Locking to the Atomic Resonance
4.2 Allan Deviation and Clock Stability
4.3 Fundamental Sources of Noise
4.3.1 Quantum projection noise
4.3.2 The Dick effect
4.3.3 Optimization of clock stability
4.4 Study of the Detection Noise
4.5 Estimating the Mercury Clock Stability Without Referencing to a Second Optical Clock
4.5.1 The atoms against the ultrastable cavity
4.5.2 Stability for systematics evaluation
4.6 Stability of a Two-Clocks Comparison: Correlated Interrogation
4.6.1 Principle of correlated interrogation
4.6.2 Transfer of spectral purity via the optical frequency comb
4.6.3 Correlated interrogation – experiments
5 Ascertaining the Mercury Clock Uncertainty Beyond the SI Second Accuracy 
5.1 Clock Accuracy
5.1.1 Digital lock-in technique for studying systematics
5.2 Collisional Shift
5.2.1 Theoretical introduction
5.2.2 Experimental results
5.3 Lattice AC Stark-Shift
5.3.1 Linear shift
5.3.2 Vector shift
5.3.3 Higher order terms
5.4 Zeeman Shift
5.4.1 Linear Zeeman effect
5.4.2 Quadratic Zeeman effect
5.5 Blackbody Radiation Shift
5.6 Measurement of the Phase Chirp Introduced by the Pulsing of the Clock Acousto-Optics Modulator
5.6.1 Digital I/Q demodulation for phase estimation
5.6.2 Results and shift estimation
5.7 Other Shifts
5.7.1 Background gas collisions
5.8 Final Uncertainty Budget
6 Frequency Ratio Measurements for Fundamental Physics and Metrology 
6.1 Purpose of Frequency Ratios Measurements
6.1.1 Redefinition of the SI second
6.1.2 Time variation of fundamental constants
6.2 Detailed Experimental Scheme
6.3 Comparison With Microwave Frequency Standards
6.3.1 Hg/Cs frequency ratio
6.3.2 Hg/Rb frequency ratio
6.3.3 Gravitational redshift estimation and correction
6.4 Comparison With a Strontium Optical Lattice Clock .
6.5 Measurement of Frequency Ratios via European Fiber Network
6.6 Long-Term Monitoring and Fundamental Constants .
Conclusion – Perspectives
Bibliography

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