Hydrogen spectroscopy and the proton radius puzzle

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Ti:Sa and Verdi frequency stabilization

Figure 2.5 offers a general view of the stabilization scheme. The Ti:Sa and Verdi laser frequencies are both stabilized in a similar manner.
First, to correct rapid frequency fluctuations (jitter), the laser cavity length is locked on a resonance of an auxiliary Fabry-Perot cavity. There are two such cavities, one for each laser, that we will call FPATi:Sa and FPAV6.
Both cavities are composed of a 25-cm-long invar bar and two mirrors, one of which is piezo-mounted. They are placed inside heavy cylinders to reduce acoustic perturbations. The finesse of the FPATi:Sa cavity is on the order of 100, that of the FPAV6 cavity is about 400 [Galtier2014a]. The stabilization is done using a Pound-Drever-Hall scheme [Drever1983]. This locking method uses an electro-optic modulator (EOM) to apply a frequency modulation to the light beam before the entrance of the cavity. The two sidebands thus created are not resonant with the cavity, but are reflected in phase if the laser is resonant. We monitor the reflected light using a photodiode and, by mixing its signal with the modulation driving the EOM, we obtain an error signal which presents a very steep dispersion shape.
For the Ti:Sa laser, this error signal is used to perform a retroaction on the piezo-mounted M4 mirror and the electro-optic modulator in the Ti:Sa cavity. For the Verdi laser, it acts on the two piezo-actuators in the Verdi laser cavity. In the case of the Verdi laser, the error signal is also used to modulate the frequency applied to the double-pass acousto-optic modulator (AOM) placed at the output of the laser. This additional correction, having a greater bandwidth than the piezo actuators, allows to reduce the rapid frequency fluctuations and the spectral width of the laser.

Frequency measurement

The frequencies of our cw lasers are measured by means of an optical frequency comb, which is a mode-locked pulsed laser, emitting a coherent train of short pulses. The spectrum of such a laser is composed of many regularly spaced modes, which can be used as a sort of “optical ruler” to determine the frequency of cw lasers.
The comb spectrum can be characterized by two frequencies: the repetition rate frep, which is the spacing between two adjacent modes, and the carrier-envelope offset (CEO) frequency f0, which is an overall offset due to the phase difference between two successive pulses. The frequency of a given mode of the comb is given by fn = nfrep ± f0, where n is a large ( 106) positive integer. The ± sign is due to the fact that the sign of the CEO is not known a priori; by convention, the quantity f0 is positive. A beat between this mode and a cw laser of frequency fcw will have the frequency fbeat = ±(fn − fcw) = ±(nfrep ± f0 − fcw).

3S–2P fluorescence detection

The Balmer- photons at 656 nm, resulting from the decay of the excited atoms to the 2P level, are collected by an imaging system as shown on Fig. 2.11, and detected by a photomultiplier (R943-02 model from Hamamatsu).
A first condenser guides the photons towards an interference filter at 656 nm.
They are then focused by another condenser onto a slit, to reduce stray light. A third condenser images this slit, which is placed parallel to the atomic beam, on the photocathode of the photomultiplier. Furthermore, a spherical metallic mirror situated below the cavity redirects photons emitted downwards, thus increasing the total opening angle of the detection system. Since the 656-nm interference filter has an acceptance angle of 11, the detection region is a 12-mm-long segment of the atomic beam, at the center of the build-up cavity. A close-up view of the detection region will be given later (Fig. 3.5 in Chapter 3).

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Magnetic field production

In order to estimate the atomic velocity distribution and thus the second order Doppler shift, we apply a vertical magnetic field in the excitation region, following the method described in Chapter 1.

The Helmholtz coils

This magnetic field is produced by two coils, placed in Helmholtz configuration on either side of the detection region [Hagel2001]. The coils, of mean diameter 34.2 cm, are placed 11.6 cm apart. Each coil is formed of 23 turns of copper tube in which circulates a DC current on the order of a hundred amperes. To prevent overheating, closed-circuit cooling water flows inside the copper tubes.
In order to compensate the terrestrial magnetic field, smaller compensation coils are placed in the other two directions.

Table of contents :

Introduction
1 Context and principle 
1.1 Theory of hydrogen energy levels
1.1.1 From the Bohr model to the Dirac equation
1.1.2 The Lamb shift
1.1.3 Hyperfine structure
1.2 Hydrogen spectroscopy and the proton radius puzzle
1.2.1 High-resolution spectroscopy
1.2.2 Determining the Rydberg constant
1.2.3 The proton radius puzzle
1.3 Our 1S−3S experiment
1.3.1 The second-order Doppler effect
1.3.2 Experimental improvements
1.3.3 Previous results and perspectives
2 The 1S−3S experimental setup 
2.1 The 205-nm laser source
2.1.1 The titanium-sapphire laser
2.1.2 The frequency-doubled Verdi laser
2.1.3 Sum frequency generation
2.2 Frequency stabilization and scanning
2.2.1 The rubidium-stabilized standard laser
2.2.2 Ti:Sa and Verdi frequency stabilization
2.2.3 Frequency scanning
2.3 Frequency measurement
2.3.1 General principle
2.3.2 The frequency comb
2.3.3 The frequency beat notes
2.4 Excitation and detection
2.4.1 The atomic beam
2.4.2 The power build-up cavity
2.4.3 3S–2P fluorescence detection
2.5 Magnetic field production
2.5.1 The Helmholtz coils
2.5.2 Calibration of the magnetic field
2.6 Data acquisition and signals
2.6.1 Data acquisition
2.6.2 Observed signals
3 Systematic effects 
3.1 The theoretical line profile
3.1.1 Fluorescence calculation
3.1.2 The velocity distribution
3.1.3 The complete fitting function
3.2 Shifting effects
3.2.1 Light shift
3.2.2 Pressure shift
3.3 Broadening effects
3.3.1 Saturation broadening
3.3.2 Transit-time broadening
3.3.3 Collisional broadening
3.3.4 Observed broadening
3.4 Cross-damping effect
3.4.1 Method
3.4.2 Details of the calculation
3.4.3 Results
4 Data analysis and results 
4.1 Experimental data
4.1.1 Recordings
4.1.2 Fit with theoretical line profile
4.2 Determination of the velocity distribution
4.2.1 Chi-square minimization
4.2.2 Uncertainties
4.3 Correction of systematic effects
4.3.1 Light shift
4.3.2 Pressure shift
4.4 Final result
4.4.1 1S–3S transition frequency
4.4.2 Rydberg constant, Lamb shift and proton charge radius .
4.4.3 New analysis of Sandrine Galtier’s recordings
Conclusion
A Estimate of cross-damping shift 
B Integration of the fluorescence over the detection region 
C Least-squares method and uncertainties 
Résumé en français 
Bibliography 

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