molecules. The quest behind the investigation of these constants is the search for the mechanism that defines them. Every precision measurement on fundamental constants constrains theory beyond the SM, by setting a new upper limit for a potential variation.
The pioneer work in this field brings us one step closer to understanding the universe.
1.3 Outline
This thesis describes the design of a precision laser system, that will ultimately be used in measurements to test physics beyond the SM. The first benchmark system for testing this apparatus are the A2Σ+, v0 = 0 ← X2Π3/2, v00 = 0, J00 = 3/2 electronic transitions in OH and its deuterated isotopologue, OD.
Chapter 2 introduces the theory needed to describe the rich electronic structure of these molecules. Afterwards, Chapter 3 describes the complex laser system used in the measurement of these electronic transitions, with emphasis on the optical frequency comb (OFC). The OFC is required to phase stabilize the ultraviolet spectroscopy laser to an infrared reference laser, which has higher stability. Both of these lasers, as well as the OFC, rely on nonlinear optical conversion, which is explained in Chapter 4. Furthermore, this chapter introduces the optical parametric oscillator (OPO), which will be the core of future vibrational transition measurement in OH. Although no vibrational transitions are measured in this thesis, they are the long term goal of building the described laser system in this thesis. Chapter 5 completes the discussion of the involved lasers, by highlighting the various frequency standards used to stabilize the lasers. In particular, Doppler-free sat-uration spectroscopy is used to stabilize the laser system on a short time scale, introduced in Chapter 6. Although the laser system is stable, frequency shifts can still occur when measuring the molecules in a molecular beam. Thus, Chapter 7 gives a brief overview of molecular beams, potential frequency shifts of the measured transitions that can occur due to the Doppler-effect, and discusses ways to correct for these shifts.
Frequency shifts can also occur due to interactions with the electric field of the laser.
Chapter 8 covers these shifts and introduces the theoretical model to fit the measured electronic transitions described in Chapter 9. A detailed analysis of the measured spectra and comparison to previous measurements is provided in Chapter 10. Although we suc-ceeded in precisely measuring electronic transitions in OH, Chapter 11 emphasizes certain improvements to the measurement setup to make future measurements even more precise.
Chapter 2
The Hydroxyl Radical
This chapter introduces the rich electronic structure of the hydroxyl radical (OH). After explaining the term symbols of diatomic molecules, the Hund’s case (a) basis is discussed, which is the most suitable for describing the OH electronic ground state. the ground state of OH is emphasized which is preferably described in Hund’s case (a) basis. The later introduced Hund’s case (b) basis simplifies the qualitative description of the first electronic excited state. The quantum numbers of the ground and the excited state provide selection rules for the electronic transitions between both states. Since this thesis also involves measurements on the deuterated hydroxyl radical (OD), it is convenient to clarify the differences between OH and OD.
2.1 Term Schema of Diatomic Molecules
The molecular structure of the OH depends on the motion of both nuclei and the nine elec-trons. In the Born-Oppenheimer approximation, the motion of the electrons is independent of the slow motion of the heavy nuclei[40]. This allows separating the wavefunction into an electronic and nuclear part. The description of the electronic states of a diatomic molecule is analogous to atomic energy states[41]. In an atom, the individual electronic angular momentali of all electrons inside the atom couple to a total electronic angular momentum P
ili =L, with the associated good quantum number L. A diatomic molecule also has a total angular momentumL, butLis no longer a good quantum number. In contrast to the spherical symmetry of an atom, the internuclear axis defines the symmetry of a diatomic molecule. Thus, the projection ofLalong the internuclear axis is a good quantum number.
This projection is denoted with the quantum number Λ = 0,1,2,··and the corresponding molecular states are Σ,Π,∆,··, in analogy to the atomic statesS, P, D,··. An important second degree of freedom is the total electron spin angular momentumS =P
isi, with the corresponding quantum numbersS (total electron spin angular momentum) and Σ (pro-jection ofSalong the internuclear axis). Since Σ can range from−Σ to Σ, the multiplicity 2S+ 1 describes the number of possible values for Σ. The term symbol2S+1Λ provides a
2.1. Term Schema of Diatomic Molecules compact way of summarizing the values of S and Λ that define an electronic state. For example, the state X2Π means S = 1/2 and Λ = 1 (Section 2.2.1). The label ‘X’ in front of the symbol marks the state as the electronic ground state. The only different letter from ‘X’ appearing in this thesis is ‘A’, generally denoting the first electronic excited state with the same S as the ground state. A basis of the quantum mechanical state is chosen such that the resulting eigenvectors are as diagonal as possible. Consider at the moment only Hund’s case (a) basis (Section 2.2). In this case, the projection of the total electron angular momentum along the internuclear axis Ω = Λ + Σ is a good quantum number and Ω can be added to the term symbol as a subscript, for instance, 2Σ+1Λ|Ω|. However, not all states will have nearly diagonal eigenvectors in Hund’s case (a), and for these Ω will not be a good quantum number. In many cases, these states can be better described by a Hund’s case (b) basis (Section 2.3). In such cases, Ω is left out in the term symbol. An example is the A2Σ+ state (Section 2.2.1), with the symmetry label ’+’ in the superscript. This symbol describes the potential sign change of the wave function after a reflection through a plane, containing the internuclear axis. If the sign of the wave function stays the same, then the label is ‘+’, otherwise it is ‘−’.
2.1.1 Parity
The discussion of parity is often a source of confusion since more than one kind of sym-metry operation is possible. For diatomic heteronuclear molecules, the plane through the internuclear axis is used to consider the effect of reflection on the sign of the wavefunc-tion. However, reflection across planes is not the only possible symmetry operawavefunc-tion. The inversion operation E∗ is defined as a change of the sign of the space-fixed coordinates X, Y, Z and is equivalent to a combination of the previous reflection and an additional rotation of the molecular frame by 180°[42]. If the purely electromagnetic Hamiltonian H commutes withE∗ such that [H, E] = 0, then simultaneous eigenstates ofH andE∗ exist.
Consider the inversion operator acting on the wavefunction ones[42]
E∗φ(X, Y, Z) = φ(−X,−Y,−Z) and twice E∗E∗φ(X, Y, Z) = φ(X, Y, Z). (2.1) ApplyingE∗twice on the wavefunction sets the system back into its original state, meaning the square of the eigenvalue is 1. Therefore the eigenvalue of E∗ must be±1. The sign of the wavefunction after transformation withE∗ defines the parity of the state. The doubly degenerate states of
|Λ,±i= 1
√2(|Λi ±(−1)p|−Λi) for |Λ|>0 (2.2) have different parities. While there are interactions that mix different electronic states and lift this degeneracy, the parity remains a good quantum number. The value of p in
Chapter 2. The Hydroxyl Radical
Equation (2.2) is given by J−S+s in Hund’s case (a) basis (Section 2.2) and in Hund’s case (b) (Section 2.3) by N +s. The parameter s is zero for Λ > 0 or Σ+ states and one for Σ− states. In order to more easily determine the energetic order of the levels in a nearly-degenerate doublet, it is convenient to define an alternative version formulation of parity, namely the e and f labeling scheme[43]. These labels depend solely on the parity and the total angular momentum J. For half-integral J, if (−1)J−1/2 is the sign of the parity, then the state is labeled e, otherwise it is a f level. For integral J, if the sign of the parity is (−1)J, then it is an e level, otherwise it is a f level. The lower level in each parity doublet will always have the same e/f label in a given electronic state.