Production and Polarization of Radioactive Rubidium for Testing Lorentz Invariance

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Production and Polarization of Radioactive Rubidium for Testing Lorentz Invariance

Master’s Thesis in Experimental Physics

Elwin Dijck

Supervisor: 2^nd Reader:

Dr. ir. C. J. G. Onderwater (kvi) Dr. ir. J. P. M. Beijers (kvi)

Abstract

Lorentz symmetry is a fundamental property of the Standard Model and no compelling evidence of its violation has been found to date. In the triµp programme at kvi, the rotational-dependence of the weak decay rate of spin-polarized alkali nuclei will be tested.

Two of the challenges in performing this experiment have been investigated for this thesis:

the polarization and production of the radioactive nuclei.

The polarization is achieved using optical pumping. Data sheets with the relevant optical parameters have been composed for isotopes of interest⁸⁰Rb and⁸²Rb. Numerical calculations have been performed to investigate the experimental requirements for creating highly polarized rubidium in krypton gas, showing that a laser flux of 5 mW/cm²is sufficient to achieve > 90% nuclear polarization when pumping the rubidium D1line in an environment of up to 10²mbar krypton gas.

For the production of radioactive rubidium using a proton beam on a krypton target, production cross sections and several aspects of the target design have been explored, including the production purity, yield and beam losses due to scattering. Finally an analysis was performed of the isotopes produced during an experiment in preparation of the Lorentz invariance measurement.

Kernfysisch Versneller Instituut, Rijksuniversiteit Groningen

July, 2012

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Introduction

Lorentz Invariance

Lorentz symmetry refers to the assumed invariance of physical laws under rotations and velocity boosts. This principle of relativity implies that all inertial frames of reference are physically equivalent and that no preferred frame exists [1]. Lorentz covariance is postulated as a key property physical laws in special relativity and thereby underlies the Standard Model of particle physics. In general relativity, Lorentz symmetry still holds as a local symmetry. Additionally, Lorentz symmetry is closely related to cpt symmetry and breaking of the latter would imply a breakdown of Lorentz covariance as well, although the converse is not necessarily true [2].

Despite over a century of testing, to date no compelling observation of Lorentz symmetry breaking (lsb) has been made. However, speculative quantum gravity theories operating at the Planck scale such as string theory may incorporate (spontaneous) Lorentz symmetry breaking in a natural way, but their connection to experimental observables is at present unclear. Nevertheless, the signals of these theories might appear as small radiative corrections to known processes at low energy. High-precision low-energy experiments could therefore possibly discern these traces of physics beyond the Standard Model.

Standard-Model Extension

As an intermediate step in joining theory to experiment, Kostelecký et al. used a phenomenological approach in developing the Standard-Model Extension (sme) [3]. In this model-independent framework, the Standard Model Lagrangian is extended with a complete set of Lorentz non- covariant terms, both cpt invariant as well as cpt non-invariant. All Lorentz symmetry violating processes can thus be represented by nonzero amplitudes for some of the sme terms. The values of the amplitudes should then be explained by more fundamental theories.

The sme allows experimental results to be translated into values for its coefficients and be compared. Many experiments have been converted into bounds on these coefficients, including low-energy high-precision qed measurements, mass comparison experiments and astrophysical observations [4].

To date, none of these experiments have shown clear evidence of Lorentz symmetry breaking, with many sme coefficients already stringently constrained in a way consistent with an exact Lorentz symmetry. However, the coefficients related to Lorentz symmetry breaking in the weak interaction are still largely unexplored and provide a unique discovery potential for new physics.

Lorentz Invariance Experiment at KVI

A test of Lorentz symmetry in weak decays is being pursued at kvi as part of the triµp¹ programme supported by the Foundation for Fundamental Research on Matter (fom). This

1Trapped Radioactive Ions: microlaboratories for Fundamental Physics [5]

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effort incorporates experimental as well as theoretical physics in performing an experiment and developing the theory necessary to translate the experimental results into constraints on the sme parameters.

The approach of the Lorentz invariance experiment is to search for a rotational dependence of the weak-decay rate of spin-polarized nuclei. Assuming Lorentz covariance to be broken in the weak interaction, a preferred frame of reference could be established, related to some kind of hidden background field. The weak decay rate could then depend on the spin direction of the decaying nucleus with respect to this hypothetical field. The observation of such an effect would indicate the breakdown of rotational invariance and thereby Lorentz covariance. Note that this change of the decay rate would be induced by a particle Lorentz transformation, i.e. physically reorienting the setup in space, while a Lorentz transformation of the observer would still leave the laws of physics unaffected as this merely changes the coordinate system used to described the experiment [3].

Assuming that the Standard-Model decay rate should be modified, Γ → ΓSM(1 + δLSB),

in a purely phenomenological fashion, there are several ways in which such an effect could depend on the vector quantities present in the system of a nucleus undergoing weak decay (disregarding the neutrino that is difficult to measure): the nuclear polarization I, the momentum p and polarization σ of the β-particle, and some assumed preferred-frame direction ˆn. The modification includes terms of the form

δLSB ≈ ξ₁p ·ˆn + ξ2I · ˆn + ξ3σ ·ˆn,

among terms describing more complicated interactions and terms assuming the background field to be a tensor field rather than the vector field n.

The experiment at kvi is set to probe ξ2 by polarizing weak-decaying nuclei and measuring their decay rate. If an effect of the proposed type would exist, it would manifest itself as a modification of the decay rate depending on the direction of the polarization with respect to the background field. Assuming the preferred frame to be fixed on a cosmological scale, the Sun-centered frame can be taken as reference [4]. The decay rate would then show a possible sidereal and annual variation as the experimental setup rotates with the Earth and circles around the Sun. The polarization direction could also be changed deliberately during the experiment to probe the effect.

Experiment concept

A major decision for performing the experiment is choosing the isotope to perform the measurement on. In absence of theoretical results indicating the expected sensitivity of various nuclei to the proposed effects, it was chosen to work with alkali metals as these can be efficiently polarized using the process of optical pumping. This constitutes transferring angular momentum to the nuclei by irradiating them with circularly-polarized laser light. Using this process it is feasible to achieve nuclear polarization degrees of over 90% as will be shown in this work.

Evidently the chosen isotope has to decay via the weak decay. In order to be able to obtain sufficient statistics in a reasonable amount of time, the lifetime of the isotope should not exceed several minutes and a lifetime much shorter than a second would complicate the experimental setup. Ideally the isotope should have a pure Gamow–Teller decay with a large β-asymmetry that could be used to monitor the degree of nuclear polarization during the experiment. At the start of the work for this thesis, the isotopes envisioned for the experiment were rubidium-80 and rubidium-82, which will be investigated in the present work in more detail.

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Irrespective of the chosen isotope, the Lorentz invariance experiment can conceptually be described by four steps to be achieved in the setup:

Production Producing the radioactive isotopes for the experiment using the agor cyclotron at kvi and a suitable target.

Purification Remove contaminations by selecting only the isotopes of interest and bringing them into a suitable environment for polarization, to the degree necessary.

Polarization Using polarized laser light to optically pump the nuclei and polarize their nuclear spin parallel or anti-parallel to an applied magnetic field.

Probing Measuring the decay rate of the nuclei and measuring its rotational dependence, e.g.

by detecting decay radiation, the disappearance of fluorescence, or some other detection scheme.

The focus of this thesis lies on the optimization of the first and third of these steps.

Thesis outline

The sensitivity of the proposed Lorentz invariance experiment increases with the number of measured atoms and the degree of nuclear polarization that can be reached:

∆ ∝ 1 P ·√1

ηN,

where P is the nuclear polarization degree, N is the number of measured nuclei and η accounts for detector efficiencies.

This thesis will be divided in two parts. The first part will describe the polarization of alkali atoms by optical pumping for optimizing parameter P . This part will start with the investigation and calculation of the required optical parameters of the atoms to be polarized (Chapter 1). Then a framework for calculating the effect of polarized laser light on the spins of atoms will be described (Chapter 2), concluded by presenting the results of such calculations for the rubidium isotopes of interest for the Lorentz invariance experiment.

The second part of this thesis will examine the production of radioactive rubidium isotopes for optimizing the parameter N. A calculation package will be used to perform production cross sections and compare different production reaction for rubidium-80 (Chapter 3). Based on these results, some of the issues in designing a target for the Lorentz invariance experiment will be treated (Chapter 4). Finally, an analysis of the isotopes produced during a beamtime used for testing the partial setup for the experiment will be presented (Chapter 5).

Elwin Dijck – e.a.dijck@student.rug.nl

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Part I

Polarization

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Chapter 1

Rubidium Atomic Structure

The first part of this thesis will describe the requirements for achieving nuclear polarization of alkali atoms using optical pumping. The absorption of circularly-polarized photons will convey angular momentum to the alkali atoms, producing atomic polarization. This process ideally brings the atoms into a maximally-polarized stretched state, where the electronic and nuclear spins are both aligned with the magnetic field, thus producing maximal nuclear polarization as well.

This chapter will deal with the literature search and calculation of various physical properties of the rubidium isotopes of interest for the Lorentz invariance experiment at kvi. To investigate the requirements on the experimental setup and laser system, detailed optical pumping calculations were performed, as will be described in the next chapter. These calculations require knowledge of the physical and optical properties of the rubidium isotopes. The relevant properties will be described and tabulated in this chapter, along with some of the expressions used in their calculation.

Comprehensive reviews of the physical and optical properties relevant to quantum optics experiments for the stable isotopes of sodium, cesium and rubidium have been compiled by Daniel Steck and are available on his website [6]. The content and organization of his work was very useful and similar data sheets were prepared for the radioactive isotopes selected as candidates for the Lorentz invariance experiment: ⁸⁰Rb and⁸²Rb. The content of these data sheets is included as Appendix A.

1.1 Physical Properties

Rubidium is the fifth period alkali metal in the periodic table of elements and the ground-state electronic configuration of atomic rubidium is therefore [Kr] 5s¹. The chemical and optical properties of atomic rubidium are mostly determined by the single unpaired electron in the valence shell. ⁸⁵Rb is the only truly stable isotope, but⁸⁷Rb with a half-life of 5 × 10¹⁰ year also occurs naturally [7].

For performing the Lorentz invariance measurement of the β-decay rate, the radioactive isotopes⁸⁰Rb and⁸²Rb were selected as candidates. Both are β⁺-emitters with a ground-state half-life of 33 and 76 seconds, respectively [7]. Both have pure 1⁺→0⁺ Gamov–Teller decays with a β-asymmetry that can be used to measure their nuclear polarization. Some of the relevant physical properties of⁸⁰Rb and⁸²Rb are given in Table 1.1.

80Rb decays to the stable isotope⁸⁰Kr, mostly to the ground state and the first excited state at 617 keV. The decay scheme is depicted in Figure 1.1. The decay of ground-state ⁸²Rb is similar, decaying primarily to the ground state of ⁸²Kr and its first excited state at 777 keV.

However,⁸²Rb also has a metastable state with an excitation energy of 69 keV and a spin-parity

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80Rb ⁸²Rb

Atomic Number Z 37 37

Total Nucleons Z+ N 80 82

Atomic Mass m 79.923 u 81.918 u

Nuclear Spin and Parity I^π 1⁺ 1⁺

Nuclear Lifetime τN 48.2 s 110.2 s

Nuclear Magnetic Moment µI −0.084 · µN 0.554 · µN

Electronic Configuration [Kr] 5s¹ [Kr] 5s¹ Table 1.1– Selected physical properties of ⁸⁰Rb and ⁸²Rb, see Appendix A for details and source references.

of I^π = 5⁻. The large spin severely hinders its decay and results in a half-life of 6.5 hours, making this metastable state problematic in the production of ⁸²Rb. The metastable state is denoted ^82mRb and also decays via β⁺-emission, with no internal conversion taking place [8].

Figure 1.1– Decay schemes of⁸⁰Rb and⁸²Rb (ground state), showing only the most important daughter isotope levels [8].

1.2 Optical Properties

Since the proposed Lorentz invariance experiment involves polarizing rubidium atoms with laser light, their optical properties are important. Of most interest is the so-called D line in the near-infrared, the lowest-energy transition of the valence electron. Inexpensive diode lasers at the required transition frequency are readily available.

Figure 1.2– D line level scheme for⁸⁰Rb and⁸²Rb (splittings not to scale).

Detailed level schemes with all frequency splittings can be found in Appendix A.

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The D line is split into two components by the fine-structure coupling shown in Figure 1.2 (described in more detail in the next section), resulting in the D1 line (the 5²S1/2 → 5²P1/2

transition) and the D2 line (the 5²S1/2 →5²P3/2transition). Relevant optical properties of these transitions are given in Tables 1.2 and 1.3.

80Rb ⁸²Rb

Frequency ω0 2π · 384.230 132 THz 2π · 384.230 251 THz Transition Energy ~ω0 1.589 048 00 eV 1.589 048 50 eV

Wavelength (Vacuum) λ 780.241 926 nm 780.241 684 nm

Isotope Shift ω0− ω₀(⁸⁷Rb) 2π · −353 MHz 2π · −234 MHz

Excited State Lifetime τ 26.235 ns 26.235 ns

Decay Rate/ Γ 38.117 × 10⁶ s⁻¹ 38.117 × 10⁶ s⁻¹

Natural Line Width 2π · 6.067 MHz 2π · 6.067 MHz

Oscillator Strength f 0.696 0.696

Table 1.2 – Selected D2 (5²S1/2 → 5²P3/2) transition optical properties of

80Rb and ⁸²Rb, see Appendix A for details and source references.

80Rb ⁸²Rb

Frequency ω₀ 2π · 377.107 113 THz 2π · 377.107 235 THz Transition Energy ~ω0 1.559 589 57 eV 1.559 590 07 eV

Wavelength (Vacuum) λ 794.979 590 nm 794.979 333 nm

Isotope Shift ω₀− ω₀(⁸⁷Rb) 2π · −350 MHz 2π · −231 MHz

Excited State Lifetime τ 27.679 ns 27.679 ns

Decay Rate/ Γ 36.128 × 10⁶ s⁻¹ 36.128 × 10⁶ s⁻¹

Natural Line Width 2π · 5.750 MHz 2π · 5.750 MHz

Oscillator Strength f 0.342 0.342

Table 1.3 – Selected D1 (5²S1/2 → 5²P1/2) transition optical properties of

80Rb and ⁸²Rb, see Appendix A for details and source references.

The different nuclear masses and volumes of the isotopes cause their transition frequencies to differ slightly. The resulting shift with respect to ⁸⁷Rb is reported as the isotope shift. The lifetimes of the excited states will also differ slightly for the various isotopes, but this difference is not visible at the present level of experimental uncertainty.

The absorption oscillator strength f can be used to express the relative intensity of a transition and is related to the excited-state lifetime τ or transition probability Γ as [6, 9]

Γ = 1

τ = e²ω²₀ 2π0m_ec³

2J + 1

2J⁰+ 1f, (1.1)

for a J → J⁰ fine structure transition.

1.3 Energy Level Splittings

1.3.1 Without external fields

The optical pumping calculations take into account the energy level structure of the atoms. The broad features of the valence electron energy levels of an alkali metal atom are determined by the

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atomic orbitals with principal quantum number n and the orbital angular momentum quantum number l. With only a single valence electron, each orbital corresponds to a single term with L= l and S = s. The rubidium D line is the transition L = 0 → L⁰ = 1 for n = 5 [9, 10].

On closer inspection the D line is split into two lines due to the fine structure splitting:

the electron moving through the electric field of the nucleus experiences a magnetic field that interacts with its intrinsic magnetic moment, known as the spin-orbit interaction [6, 9, 10]. The energy shifts caused by this effect can be described in terms of the total angular momentum of the electron, defined as the sum of the orbital angular momentum L and the spin angular momentum S:

J= L + S.

Denoting the eigenvalue of J² as J(J + 1)~² and the eigenvalue of Jz as mJ~ (and for other angular momentum operators likewise), the quantum number J lies in the range

|L − S| ≤ J ≤ L+ S.

Since S = 1/2 for a single electron, this will split all terms, except the L = 0 ground state, into two fine structure levels. For the rubidium D line this splitting is approximately 15 nm, large enough to be resolved by most lasers and therefore the two transitions are generally treated separately [6].

In addition to the fine structure splitting of the D line into a doublet, the transitions also have a hyperfine structure caused by the interaction of the electrons with the magnetic moment of the nucleus [6, 9, 10]. This coupling between the total electron angular momentum J and the total nuclear angular momentum I can be described in terms of the total atomic angular momentum, defined as

F= J + I.

Analogously to J, the quantum number F is bound by

|J − I| ≤ F ≤ J+ I.

For ⁸⁰Rb and⁸²Rb with a nuclear spin of I = 1, the ²S1/2 ground state and ²P1/2 excited state (both with J = 1/2) are split into two hyperfine levels and the²P3/2 excited state (with J = 3/2) is split into three hyperfine levels. Since the magnetic moment of the nucleus is much smaller than the magnetic moment of the electron, the hyperfine energy splittings are also much smaller than the fine structure, being of the order of several hundred megahertz for⁸⁰Rb to a few gigahertz for ⁸²Rb, depending on their respective magnetic moments.

The hyperfine energy shifts are described by the following Hamiltonian that includes contributions up to quadrupole [6, 9]:

H_hfs= AhfsI · J+ Bhfs

3(I · J)²+³₂I · J − I(I + 1)J(J + 1)

2I(2I − 1)J(2J − 1) , (1.2)

which results in energy shifts relative to the fine-structure level of Ehfs= 1

2AhfsK+ Bhfs 3

2K(K + 1) − 2I(I + 1)J(J + 1)

4I(2I − 1)J(2J − 1) , (1.3)

where K = F (F + 1) − I(I + 1) − J(J + 1). Note that the electric quadrupole contribution is only present for states with J > ¹₂ and I > ¹₂. The eigenstates of this Hamiltonian may be labeled as |IJF mFi, since the operators I², J², F² and Fz all commute with each other and with the Hamiltonian in the absence of external fields. The magnitude of the energy shifts and the ordering of the levels is determined by the constants Ahfs and Bhfs which are given for the relevant levels in Table 1.4.

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80Rb ⁸²Rb 5²S1/2 Magnetic Dipole Constant A₅2S_1/2 h · −156 MHz h · 1 032 MHz 5²P1/2 Magnetic Dipole Constant A₅2P_1/2 h · −19 MHz h ·123 MHz 5²P3/2 Magnetic Dipole Constant A₅2P_3/2 h · −2 MHz h ·27 MHz 5²P3/2 Electric Quadrupole Constant B₅2P_3/2 h ·33 MHz h ·18 MHz

Table 1.4– D transition hyperfine structure constants of⁸⁰Rb and ⁸²Rb, see Appendix A for details and source references.

1.3.2 With static magnetic fields

Each of the hyperfine energy levels is degenerate with 2F +1 magnetic sublevels in the absence of external fields. The application of a magnetic field will break this degeneracy by the interaction of the various magnetic moments with the magnetic field, described by the Hamiltonian [6, 9, 10]

H_B= −µ · B = µ_B

~ (gSS+ gLL+ gII) · B. (1.4) If the magnetic field is taken as the atomic quantization axis (conventionally the z-axis), the Hamiltonian becomes

H_B= µ_B

~ (gSS_z+ gLL_z+ gII_z)Bz. (1.5) The g-factors relate the angular momentum quantum numbers to the corresponding magnetic moments and their values are listed in Table 1.5.

For a nonzero magnetic field, the angular momentum operators F² and J² no longer commute with the Hamiltonian, but for sufficiently weak fields their quantum numbers can still be considered good quantum numbers. If the energy shifts due to the magnetic field are small compared to the fine structure splitting, but large compared to the hyperfine structure, both J and Jz are good quantum numbers and the Hamiltonian can be approximated as

HB = µB

~ (gJJz+ gIIz)Bz, (1.6)

where the Landé g-factor is given by [6, 9]

g_J = gL

J(J +1) − S(S +1) + L(L+1)

2J(J +1) + gS

J(J +1) + S(S +1) − L(L+1) 2J(J +1)

≈ 3

2 +S(S +1) − L(L+1)

2J(J +1) . (1.7)

Table 1.5 lists the values of gJ for the different fine structure states, where available these are experimental measurements.

If the energy splittings due to the magnetic field are small compared to the hyperfine structure, F and mF are good quantum numbers and the interaction Hamiltonian can be simplified to

H_B= µ_B

~ g_FF_zB_z, (1.8)

where the hyperfine g-factor is defined analogously to (1.7) [6, 9]:

gF = gJ

F(F +1) − I(I +1) + J(J +1)

2F (F +1) + gI

F(F +1) + I(I +1) − J(J +1) 2F (F +1)

≈ g_JF(F +1) − I(I +1) + J(J +1)

2F (F +1) . (1.9)

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80Rb ⁸²Rb Electron Orbital g-factor g_L 0.999 99 0.999 99 Fine Structure Landé g-factor

gJ(5²S1/2) 2.002 33 2.002 33 g_J(5²P1/2) 0.666 0.666 gJ(5²P3/2) 1.336 2 1.336 2

Nuclear g-factor g_I 0.000 045 −0.000 301

Electron Spin g-factor gS 2.002 32

Table 1.5 – D transition magnetic field interaction parameters of⁸⁰Rb and

82Rb, see Appendix A for details and source references.

In the low-field regime where expression (1.8) is valid, the magnetic field perturbs the hyperfine energy levels, resulting in a lowest order shift of

∆E|F m_Fi = µBgFmFBz, (1.10) which is linear in the magnetic field strength and known as the Zeeman effect.

1.4 Electric Dipole Operator

To first order, the interaction of light with an atom can be described using the electric dipole operator D = er, whose matrix elements couple the various states and determine their transition probabilities depending on the polarization of the electric field. This description will be used for the optical pumping calculations.

The electric dipole operator is a spherical tensor operator and therefore its matrix elements in the basis of angular momentum eigenstates can be reduced using the Wigner–Eckart theorem² [6, 9, 11]

DαJ m_JT_q^kα⁰J⁰m⁰_J^E= (−1)^2kC_J^{J m}0m^J⁰_Jk q

DαJT^kα⁰J⁰^E, (1.11) whereby the matrix element factorizes into a Clebsch–Gordan coefficient that contains all the information about the angular dependence, and a reduced matrix element that is independent of q, m_J and m⁰_J.

The Clebsch–Gordan coefficient is zero unless the following conditions are satisfied:

|J − k| ≤ J⁰ ≤ J + k and mJ = m⁰J+ q.

These conditions essentially express the conservation of angular momentum and define the selection rules for electric dipole transitions. The electric dipole operator has order unity, k = 1, giving

∆J = 0, ±1 and ∆mJ = −q = 0, ±1, (1.12)

where q is related to the polarization with respect to the atomic quantization axis of the light needed to drive the transition.

Applying equation (1.11) to component q in the spherical basis of the electric dipole matrix element between hyperfine states gives [6]

hF m_F|D_q|F⁰m⁰_Fi= C_F^{F m}0m^F⁰_F1 qhF kDkF⁰i, (1.13)

2The normalization convention used in [6] will be taken, which differs from [11] by a factor√

2J + 1 in the reduced matrix element

αJ T^k

α⁰J⁰ .

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This expression can be reduced further by using a Wigner 6-j symbol that factors out the F and F⁰ dependence [6]

hF kDkF⁰i= (−1)^F⁰^{+J +1+I}^q(2F⁰+ 1)(2J + 1)

( J J⁰ 1 F⁰ F I

)

hJ kDkJ⁰i. (1.14) The numerical value of the reduced matrix element in this expression can be calculated from the lifetime of the J⁰ excited state as [6]

1

τ = ω³₀ 3π0~c³

2J + 1 2J⁰+ 1

hJ kDkJ⁰i

2 (1.15)

and is listed in Table 1.6.

D2 (5²S1/2 → 5²P3/2) Transition

Dipole Matrix Element hJ = 1/2 k D k J⁰ = 3/2i 4.228 ea0

3.584 × 10⁻²⁹ C m D1 (5²S1/2 → 5²P1/2) Transition

Dipole Matrix Element hJ = 1/2 k D k J⁰ = 1/2i 2.993 ea0

2.538 × 10⁻²⁹ C m Table 1.6 – D transition dipole matrix elements of ⁸⁰Rb and ⁸²Rb, see Ap- pendix A for details and source references.

The parameters introduced in this chapter will be used in the optical pumping calculations in the following chapter. A compilation of the optical properties of ⁸⁰Rb and ⁸²Rb is included as Appendix A.

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Chapter 2

Optical Pumping

To investigate the requirements on the laser and the environment of the rubidium atoms to be polarized, optical pumping calculations were performed based on the textbook by Happer et al.[11]. This chapter will provide a summary of the calculation method described in this book and then present the results when applying this method to the Lorentz invariance experiment with rubidium in krypton buffer gas. The book also contains code snippets to perform the various calculations, written for matlab [12]. These were adapted to the present experiment and the resulting code can be found in Appendix C.

2.1 Calculation Method

The process of optical pumping is often represented with models including only a few spin sublevels, but real atoms can have a large numbers of sublevels and especially at low buffer-gas pressures, optical interactions depend in a detailed way on the interplay between the laser light and the various transitions. Modern scientific computing software makes it practical to analyze the full multilevel atomic systems to obtain realistic results [11].

2.1.1 General

The density matrix formalism will be used to describe the sublevel distribution and coherences of an ensemble of optically-pumped atoms:

ρ=^X

µν

ρ_µν|µihν|. (2.1)

The time evolution of the pure states |ii is given by Schrödinger equation and therefore the evolution of a quantum mechanical system in the density matrix formalism is determined by the commutator with the Hamiltonian:

d dtρ= 1

i~[H, ρ] . (2.2)

In the case of optical pumping, the Hamiltonian contains a static part that includes the energy level splittings and a second part describing the electromagnetic interaction with the atom’s dipole moment:

H= H0− D · E. (2.3)

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Liouville space

The calculations are performed in Liouville space, where a density matrix ρ that is a Hermitian matrix with g rows and columns in the more customary Schrödinger (Hilbert) space is transformed into a column vector |ρ) with g² elements by placing each successive column of the matrix below the one to its left:

|ρ) =^X

µν

|µν)(µν|ρ), (2.4)

where the Liouville basis vectors are

|µν) = |µihν|

and the amplitudes are

(µν|ρ) = ρµν.

The ‘state vector’ |ρ) allows the time-derivative of the density matrix to be described by matrix exponentiation, a calculation that can be easily and efficiently performed by a desktop computer running, e.g. matlab, rather than requiring the integration of higher-order tensors.

The full density matrix in the Liouville space of a two-level atom is given by:

|ρ) =







|ρ^{ee})

|ρ^{ge})

|ρ^{eg})

|ρ^{gg})







, (2.5)

where following the notation of [11], the g and e superscripts label the J ground state and J⁰ excited states, respectively. Note that only the excited fine-structure state that is resonant with the laser light will be included in the calculation.

Any matrix in Schrödinger space can be transformed into a vector in Liouville space in analogy to (2.4), with row vectors being defined as the Hermitian conjugates of column vectors:

(X| = |X)^† with elements (X|µν) = (µν|X)^∗. The unit operator in the full Liouville space of the atom is given by

|1) =







|1^{e})

|0^{ge})

|0^{eg})

|1^{g})







. (2.6)

Expectation values of operators are calculated as

hXi= Tr[Xρ] = (X^†|ρ), e.g. h1i = Tr[ρ] = (1|ρ) = 1.

In Liouville space, any linear transformation of the density matrix can be represented by matrix multiplication of |ρ) and special notation is introduced for the Liouville transformations of some often encountered operator expressions. Suppose that ρ = ρ^{mn} is a block of density matrix (2.5) with g^{m} rows and g^{n} columns, where m and n each label either the ground or excited state, then the following superoperators are introduced for operations on ρ:

flat superoperators: |Aρ) = A^[|ρ), A^[= A^[{n}= 1^{n}⊗ A; sharp superoperators: |ρA) = A^]|ρ), A^]= A^]{m}= A^T ⊗1^{m};

o-dot superoperators: |A .∗ ρ) = A|ρ), (µν|A|µ⁰ν⁰) = δµµ⁰δ_νν⁰hµ|A|νi; The .∗ operator is an element-by-element multiplication of two matrices of the same dimension, as in matlab: hµ|A .∗ B|νi = hµ|A|νihµ|B|νi. These superoperators will be used in describing the time-dependence of the density matrix.

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2.1.2 Static Hamiltonian

The static part of the Hamiltonian can be written as

H₀ = ~ω0Q+ H^{e}+ H^{g} with Q = 1 2

1^{e}−1^{g}, (2.7) where ω0 is the nominal resonance frequency between the ground and excited states picked out by Q, including the fine structure shift as listed in Section 1.2. The hyperfine shifts and magnetic field interaction are as given by equations (1.2) and (1.4):

H^{g} = A^{g}I · J − µ^{g}· B with µ^{g}= −gJµ_BJ+µI

I I and H^{e}= A^{e}I · J⁰+ B^{e}3(I · J⁰)²+ ³₂I · J⁰− I(I + 1)J⁰(J⁰+ 1)

2I(2I − 1)J⁰(2J⁰−1) − µ^{e}· B, with µ^{e}= −gJ⁰µBJ⁰+µ_I

I I.

The values of the various atomic constants were listed in Section 1.3. Note that the Hamiltonian contains no kinetic part since the movement of atoms will not be included directly.

2.1.3 Interaction Hamiltonian

The electric dipole moment operator D can be split into a conjugate pair of dimensionless complex vector operators ∆ and ∆^† and an amplitude D:

D= D + D^†= D∆ + D^∗∆^†, (2.8)

D= ^X

m_Jm⁰_J

|J m_JihJ m_J|D|J⁰m⁰_JihJ⁰m⁰_J|.

The matrix elements of the complex dipole operator can be calculated by decomposing it along the spherical basis vectors ξq and using the Wigner–Eckart theorem (1.11), where the direction of the magnetic field defines the z-axis:

∆=^X

q

(−1)^q∆qξ_−q with ξ₀ = z, ξ±1 = ∓x ± iy

√2 , (2.9)

where ∆_q = ∆ · ξ_q =

s 3

2J + 1 X

m_Jm⁰_J

|J m_JihJ⁰m⁰_J|C_J^{J m}0m^J⁰_J1 q.

The amplitude of the dipole operator is related to the matrix element from equation (1.15):

|D|² = (2J + 1)~rec²f

2ω0 = 2J + 1 3

hJ kDkJ⁰i

2. (2.10)

The interaction component of the Hamiltonian is then written as

−D · E ≈ V+ V^† with V(t) = −D^†· E(t) = −D^∗∆^†· E(t), (2.11) where only the terms of the inner product corresponding to an atom being excited by absorbing a photon from the pumping beam V, and an atom being de-excited by emitting a photon V^†, are retained. This facilitates solving the problem by allowing a description in a rotating frame where the Hamiltonian will be time-independent.

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The (classically modeled) electric field is also split into two parts:

E= E + E^∗, E = ˜Ee^{i(k·r−ωt)}, (2.12) where r = r0+ vt gives the position of the atom at time t and ˜E = θ ˜E_θ+ φ ˜E_φ is the transverse projection of the electric field along the propagation direction k and determines the polarization state of the light. Typically, r0 = 0 and v = 0 will be used for ease of calculation; the effect of thermal movement in an ensemble of atoms can be taken into account in a different way. The mean energy flux is given by

S= c

2π| ˜E|². (2.13)

2.1.4 Spontaneous emission

The spontaneous decay of the excited state of the atom will result in a damping effect of the density matrix that is not included in the Hamiltonian:

˙ρ^{ee}s = −γ1ρ^{ee}, (2.14)

where γ1 = Γ^{ge}s = _τ¹ is the natural radiative decay rate of the excited state as given in Section 1.2. This will populate the ground state at a rate of

˙ρ^{gg}_s = 4ω³₀

3~c³D · ρ^{ee}D^†= Γ^{ge}_s 2J⁰+ 1

3 ∆ · ρ^{ee}∆^†. (2.15) In Liouville space this can be written as

|˙ρ^{gg}_s ) = Γ^{ge}_s A_s^{ge}|ρ^{ee}) with A^{ge}_s = 2J⁰+ 1 3

X

j

∆^∗_j⊗∆_j, (2.16)

where the sum extends over the three Cartesian projections of the dimensionless dipole operator:

∆j = ∆ · xj.

The damping of the coherences between the ground and excited state are similarly given by

˙ρ^{ge}s = −γ2^∗ρ^{ge} and ˙ρ^{eg}s = −γ2ρ^{eg}, (2.17) with γ2 = ¹₂Γ^{ge}s + Γcα, where the last term models the collisional broadening and shift of the transition for atoms in a buffer gas, treated in the next section.

Note that the light emitted by spontaneous decays is not included in the electric field. This corresponds to assuming that the number density of the atoms is sufficiently low for the medium to remain optically transparent and emitted photons unlikely to be reabsorbed. If instead the density of atoms is high, the (randomly polarized) spontaneous emission can be reabsorbed by other atoms leading to radiation trapping, an additional depolarizing effect. To prevent radiation trapping, experiments performing optical pumping in high-pressure buffer gas typically add a quenching gas that will cause most atoms to de-excite before they decay radiatively [11, 13]. For these scenarios the electric field as modeled here would correspond to perfect quenching without any radiation trapping effects.

2.1.5 Spin relaxation in a buffer gas

The polarization of an atom may be altered by interacting with other atoms. Optical pumping is normally performed in a buffer gas and therefore several types of interactions causing spin relaxation are possible, including binary collisions between alkali metal atoms, binary collisions

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between alkali metal and buffer-gas atoms, and the formation of long-lived Van der Waals molecules. In addition, the spin relaxation due to the diffusion of alkali metal atoms to the walls of the experimental cell can be modeled in a similar way. Based on the involved number densities and cross sections, the most important contributions for the Lorentz invariance experiment are likely to be due to binary collisions with buffer-gas atoms, and spatial diffusion at low pressure.

The various processes will now be described in some more detail.

Modeling spin relaxation

The effect of collisions that do not involve excitation or de-excitation on the density matrix can be modeled by including additional terms in the time-evolution of the density matrix of the form

|˙ρ^{gg}c ) = −ΓcA^{gg}_c |ρ^{gg}) (2.18) for the ground-state part of the density matrix and an analogous term for the excited state.

Here the rate of collisions is given by Γcand the matrices Ac determine the effect on the density matrix.

A simple collisional model is ‘uniform relaxation’, where every collision destroys the entire polarization, i.e. coherences are destroyed and populations are evenly redistributed in the chosen basis:

A^{gg}_u = 1^{gg}− 1

g^{g}|1^{g})(1^{g}|. (2.19) This expression can be used to model long-lasting collisions (longer than several picoseconds) where both the electronic and nuclear polarization of an atom are destroyed, for instance collisions of atoms with (uncoated) walls of an experimental cell, where the atoms may be stuck for nanoseconds before evaporating into the gas again [11].

For sudden collisions (with interaction times on the order of picoseconds), a more realistic relaxation model is ‘J-damping’ which takes into account that the electronic polarization may be destroyed in a such a collision, while the nuclear polarization is preserved because the interaction time is too short to change the weakly-interacting nuclear spin. This can be modeled by operating on the density matrix with projection operators Π^{JJ_l ⁰^}, which decompose it into its different electronic spin polarization components [11]:

ρ=^X

l

Π^{JJ}_l ρ=^X

l

ρ_l. For electronic spin relaxation of ground-state atoms, this gives

A^{gg}_c = α1Π^{JJ}1 , (2.20)

where Π^{JJ}₁ is the projection operator that projects out the part of the density function with electronic polarization and the dimensionless constant α₁ determines the collision rate.

For the ground state and D1 excited state with J = 1/2, the complete set of operators that project out the different J spin-components of the density matrix are given in Liouville space by [11]

Π^{0¹²¹²^} = 1

4+ J^[· J^], Π^{₁¹²¹²^} = 3

4− J^[· J^]

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and for the D2 excited state with J = 3/2 [11], Π^{₀³²³²^}= 33

128− 31

96J^[· J^]− 5 72

J^[· J^]²+ 1 18

J^[· J^]³, Π^{1³²³²^}=− 81

128+117

160J^[· J^]+ 9 40

J^[· J^]²− 1 10

J^[· J^]³, Π^{₂³²³²^}= 165

128− 23

96J^[· J^]− 17 72

J^[· J^]²+ 1 18

J^[· J^]³, Π^{₃³²³²^}= 11

128− 27

160J^[· J^]+ 29 360

J^[· J^]²− 1 90

J^[· J^]³.

Note that similar projection operators could be used to decompose the density matrix into components with different F or I polarization. For example, for an I = 1 atom, Π^{II}₀ gives the unpolarized component of ρ, Π^{II}₁ gives the nuclear vector-polarized component, and Π^{II}₂ gives the nuclear l = 2 tensor-polarized component.

The uniform relaxation matrix introduced in equation (2.19) corresponds to the simultaneous relaxation of all polarization components if it is evaluated in the eigenspace of an angular momentum operator, i.e. when evaluated in J-eigenspace, Au= 1 − Π^{JJ}₀ .

Binary collisions with buffer-gas atoms

Binary collisions of alkali metal atoms with buffer-gas atoms can cause two distinct effects on the spins of the involved particles. The first effect is spin relaxation due to an interaction between the spin J of the alkali metal atom and the relative angular momentum of the two colliding particles N, the so-called spin rotation interaction³ [11, 13] with a potential of the form:

V = γN · J,

where γ is the coupling strength. The interaction time of binary collisions is typically on the order of picoseconds, which is sudden with respect to the hyperfine interaction with the nuclear spin [11, 13, 14]. Therefore the nuclear polarization of the alkali metal atom is not affected by these collisions and the effect will be the relaxation of the electronic polarization at a rate of

Γsr = Γcα1 = nhvσsri, (2.21)

where n is the number density of the buffer gas, v is the relative velocity of the colliding atoms, and σ is the cross section of the interaction. The cross sections⁴ are given in Table 2.1, simply taking the mean velocity for v. The buffer-gas number density can be calculated from the ideal gas law,

n= P N_A RT

and the mean relative velocity can be obtained by adding the (uncorrelated) mean velocities of the atoms taken from their corresponding Maxwell–Boltzmann distributions,

v=^qv²_gas+ v²_alkali with vi =

s8RT πM_i.

3The spin rotation interaction is also referred to as spin–axis interaction or spin–orbit coupling.

4Note that cross sections obtained in older experiments from relaxation times such as those listed in [9] may have to be multiplied by a nuclear slowing-down factor to account for mixing with the angular momentum stored in the nuclear polarization; for natural rubidium this factor is 10.8 [13, 15].

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Kr atomic mass m Rb–Kr spin rotation σsr Rb–Kr spin exchange σse

83.798 u 2.7 × 10⁻²⁰ cm² 5 × 10⁻²³ cm² Table 2.1 – Cross sections of spin relaxation and spin exchange collisions between rubidium and krypton atoms [11]. The spin rotation interaction cross section is given at 27 ℃ and is typically strongly temperature-dependent. Also the atomic mass of natural krypton is given.

Additionally, collisions with buffer-gas atoms that have a nonzero nuclear spin can cause spin exchange through the Fermi contact interaction [11, 13]:

V = αJ · K,

where K is the nuclear spin of the buffer-gas atom. This interaction causes a transfer of polarization from the alkali metal atoms to the buffer gas, which can be used to obtain polarized noble gasses [13, 15]. However, the cross section of this process, also given in Table 2.1, is typically small compared to the spin rotation interaction and properly modeling its effect would require to also keep track of the buffer-gas polarization K, so this effect will be neglected.

Van der Waals molecule formation

When using a heavy noble gas such as krypton as buffer gas, the formation of weakly bound Van der Waals molecules will occur in three-body collisions (where the third body is needed for momentum and energy conservation) [11, 13, 14]. Compared to the binary collision interaction times, Van der Waals molecules are relatively long-lived (10⁻⁷ s for Rb–Kr at low pressure) before breaking up in a subsequent collision. The formation rate of these weakly-bound molecules is proportional to n² due to their three-body nature, but since the molecules will be destroyed by subsequent two-body collisions, the molecular lifetime is proportional to n⁻¹. This causes the depolarization rate of the Van der Waals interaction to increase with pressure until saturating around 10 mbar at about 10²s⁻¹for the Rb–Kr system [14] and at high pressure binary collisions dominate.

Although the effect of the Van der Waals interactions becomes of comparable size to the binary collisions at a narrow pressure range between about 1–10 mbar, as diffusion is most important at lower pressure and binary collisions dominate at higher pressure, the effect of the formation of Van der Waals molecules is assumed to be small and is not taken into account in the present calculation. Additionally, the presence of a static magnetic field of several Gauss will further limit the formation of Van der Waals molecules [14].

Binary collisions between alkali metal atoms

To first order, alkali–alkali collisions conserve spin, resulting in spin exchange between alkali atoms. In addition, there is spin–axis coupling that causes damping of the electronic polarization.

The cross sections of these effects for rubidium are listed in Table 2.2 and are orders of magnitude larger than the cross sections for interactions with buffer-gas atoms (Table 2.1). However, to calculate the rate of these effects, the number density of the alkali metal atoms has to be substituted in equation (2.21) and for the radioactive isotopes of the Lorentz invariance experiment this is typically many orders of magnitude smaller than the buffer gas number density, so these effects will be negligible.

It is also possible to make use of these spin exchange reactions for polarization, e.g. by using a vapor of stable alkali atoms that will become polarized and subsequently transferring

Production and Polarization of Radioactive Rubidium for Testing Lorentz Invariance