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  • vrije universiteit amsterdam

    Towards Doppler-free two-photon spectroscopy of trapped and cooled

    HD+ ions

    academisch proefschrift

    ter verkrijging van de graad Doctor of Philosophy aan de Vrije Universiteit Amsterdam,

    op gezag van de rector magnificus prof.dr. V. Subramaniam,

    in het openbaar te verdedigen ten overstaan van de promotiecommissie van de Faculteit der Bètawetenschappen op donderdag 9 mei 2019 om 11.45 uur

    in de aula van de universiteit, De Boelelaan 1105


    Sayan Patra geboren te Burdwan, India

  • promotoren: prof.dr. W.M.G. Ubachs prof.dr. K.S.E. Eikema

    copromotor: dr. J.C.J. Koelemeij

  • i

    To family and friends

    A handful of elegant designs support Nature’s exuberant construction, from simple building blocks, of the material world.

    - Wilczek, Frank. A Beautiful Question. New York: Penguin Press, 2015.

  • ii

    This thesis was approved by the members of the reviewing committee:

    dr. R. Gerritsma, Universiteit van Amsterdam, the Netherlands prof.dr. P. Gori-Giorgi, Vrije Universiteit Amsterdam, the Netherlands prof.dr. D. Iannuzzi, Vrije Universiteit Amsterdam, the Netherlands dr. D. Leibfried, NIST Boulder, the United States of America prof.dr. P.O. Schmidt, Leibniz Universität Hannover and PTB Braunschweig, Germany dr. S. Sturm, Johannes Gutenberg-Universität Mainz and Max-Planck-Institut für Kernphysik, Heidelberg, Germany

    This work is done as a part of the FOM-projectruimte “Time will tell; leads to new physics from a molecular optical clock” (13PR3109), which was financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The research was performed at the LaserLaB, Vrije Universiteit Am- sterdam.

  • Contents

    1 Introduction 1 1.1 Prologue 1 1.2 Determination of the proton-electron mass ratio µpe 3 1.3 The hydrogen molecular ion HD+: Benchmark system for testing

    molecular theory 4 1.4 Fundamental constants from HD+ 6 1.5 Test of the Standard Model of particle physics and search for

    physics beyond it 8 1.6 Vibrational spectroscopy of the HD+ ion 10 1.7 Outline and summary of this thesis 11

    2 Necessary background information 15 2.1 Advances in ab initio calculations of the energy levels of HD+ 15

    2.1.1 Summary of recent advances in the calculations of the ro-vibrational transition frequencies of HD+ 15

    2.1.2 Hyperfine structure calculation 16 2.2 Summary of spectroscopy of the (v, L) : (0, 2)→ (8, 3) overtone

    in HD+ 17 2.3 Doppler-free two-photon spectroscopy of ro-vibrational transi-

    tions of HD+ 23 2.3.1 Feasibility study 23 2.3.2 Limitations of the OBE model 30

    3 Experimental setup 33 3.1 Laser-cooling of Be+ ions and the ion trap setup 33

    3.1.1 Ion trap setup 33 3.1.2 Laser-cooling setup for Be+ ions 35 3.1.3 Experimental control and automation 39 3.1.4 Performance of the ion trap and laser-cooling setup 43

    3.2 Frequency stabilisation of the spectroscopy lasers 44


  • Contents

    3.2.1 The optical frequency comb (OFC) and the ultra-stable laser 45

    3.2.2 The transfer-oscillator locking scheme 46 Principle and implementation 46 Performance of the frequency stabilisation scheme 54 Effect of intra-office optical fibre on the spectra purity of

    the lasers 58 Selection of frep of the OFC for multiple lasers 60

    4 A priori determination of required experimental parameters 63 4.1 Introduction 63 4.2 Choice of hyperfine components 64

    4.2.1 Zeeman shifts 64 4.2.2 Suppression of Doppler-broadened background 66

    4.3 Modelling the Doppler-free two-photon spectrum: a priori de- termination of required experimental parameters 68 4.3.1 Brief review of the rate equation model 69 4.3.2 Calculation of the (v, L) : (0, 3)→ (4, 2)→ (9, 3) Doppler-

    free two-photon spectrum: results 72 4.3.3 Estimate of frequency shift due to underlying Doppler-

    broadened background 77 4.4 Zeeman and AC Stark shifts 79

    4.4.1 Zeeman shift 79 4.4.2 AC Stark shift due to the cooling laser at 313 nm and

    the dissociation laser at 532 nm 81

    Appendices 84 4.A AC Stark shift due to the cooling laser and the dissociation laser 84

    5 Ghost features in Doppler-broadened spectra of ro-vibrational transitions in trapped HD+ ions 87 5.1 Introduction 87 5.2 Brief review of the rate equation model 88 5.3 Calculation of the (v, L) : (0, 3)→ (4, 2) spectrum 90 5.4 Ghost features in (v, L) : (0, 3)→ (4, 2) single-photon spectrum 92 5.5 Significance of ghost features for spectroscopy of trapped ions 99 5.6 Summary and conclusion 100

    6 Proton-electron mass ratio from HD+ revisited 103 6.1 Introduction 103 6.2 Improvement in theory 104 6.3 Comparison between experiment and theory 107


  • Contents

    6.4 Determination of the proton-electron mass ratio µpe 110 6.5 Outlook and conclusion 115 6.6 Acknowledgments 116

    7 Observation of the (v, L) : (0, 3)→ (9, 3) two-photon transition in HD+ 117 7.1 Doppler-broadened (v, L) : (0, 3) → (9, 3) ro-vibrational transi-

    tion in HD+ 117 7.2 Doppler-free two-photon (v, L) : (0, 3)→ (9, 3) transition in HD+ 119

    7.2.1 Experimental protocol and measured spectrum 119 7.2.2 Statistical test to distinguish “signal” from “noise” 123

    The statistical test 123 Welch’s t-test 125 Statistical analysis of the measured spectrum 126

    Appendices 130 7.A Choice of optimum REMPD duration for the Doppler-free two-

    photon spectroscopy experiment 130

    8 Conclusion and Outlook 135

    Bibliography 137

    Acknowledgement 153


  • Chapter 1


    1.1 Prologue

    A theory describing the interaction between electromagnetic radiation and mat- ter, treating both the radiation field and matter quantum mechanically, was first formulated by Dirac in the 1920s. This theory was successfully employed to calculate the coefficient of spontaneous emission of an atom with a single electron [1]. In subsequent years, Dirac proposed a quantum theory of elec- trons consistent with the Special Theory of Relativity [2]. Dirac’s theory also predicted the existence of positrons [3], which were experimentally detected by Anderson in 1932 [4]. However, it was found that Dirac’s theory led to serious divergences in the calculation of the electron’s self-energy when it is expanded in powers of α, the fine structure constant [5–7]. Of particular interest to the development of the modern day formulation of Quantum Electrodynam- ics (QED) are further two predictions of Dirac’s theory, namely, the magnetic moment of the electron and the degeneracy of the 2S1/2 and 2P1/2 levels in atomic hydrogen. The magnetic dipole moment of the electron is expressed in terms of the g-factor. The Dirac equation predicts the value of the g-factor for the electron ge = 2. In 1947, Kusch and co-workers measured ge to be 2(1.00119± 0.00005), which is slightly different from 2 as predicted by Dirac [8–10]. The same year, in their classic experiment of measuring the fine structure of hydrogen atom using mi- crowaves, Lamb and Retherford [11] discovered that the 2S1/2 and the 2P1/2 levels are not degenerate (Lamb shift), contrary to the prediction of Dirac’s theory. Moreover, the hyperfine splitting in the ground states of atomic hydro- gen and deuterium were also measured to be different from their theoretically predicted values, which assumed ge = 2 [12]. These experimental results led to the present-day formalism of Quantum Electrodynamics, the relativistic quan-


  • 1. Introduction

    tum field theory (QFT) of electrodynamics successfully combining the Special Theory of Relativity and quantum mechanics. QED was developed by Bethe, Feynman, Schwinger, Tomonaga and Dyson in the 1940s [13–16]. In their for- mulation, the divergences encountered were avoided by writing the equations in terms of measurable, finite quantities such as charge and mass of the elec- tron (renormalisation). Thorough and mathematically rigorous discussions on Quantum electrodynamics can be found in standard textbooks and is beyond the scope of this thesis (also beyond the expertise of the author). A primary consequence of this renormalisation essential to this thesis is that certain mea- surable quantities in terms of which the equations of QED are written, cannot be calculated ab initio, but have to be measured (this is true not only for QED, but for any renormalisable Quantum Field Theory (QFT)). These quantities in most cases, appear in most basic equations of physics and are called “fun- damental constants”. Examples of fundamental constants are the velocity of light in vacuum c, the mass of the electron me, the charge of the electron e, the fine structure constant α, the Planck’s constant h, the Rydberg constant R∞ etc. A precise and accurate knowledge of the values of these fundamental constants is essential to the scientific community to provide accurate quan- titative descriptions of the physical world. The Task Group on Fundamental Constants (TGFC) of the Committee on Data for Science and Technology (CO- DATA) was established in 1969 to provide the scientific community with a set of internationally accepted values of the fundamental constants. The first such compilation of fundamental constants was published in 1973 [17] and has since been periodically updated to take into account recent measurements of the fundamental constants. These constants can have dimensions (e.g. charge of an electron e) or also can be dimensionless (e.g. fine structure constant α). In this thesis, the