Light shift in Ba

Light shift in Ba ⁺ and Ra ⁺ Ions

Jo¨ el Hussels June 30, 2014

Figure: trapped barium ions

Abstract

In the radium ion experiment of the Van Swinderen Institute we search for physics beyond the Standard Model. To do this we attempt to show that the value of the Weinberg angle given by the Standard Model is not found experimentally. Atomic parity violation in a single radium ion provides for a measurement of the Weinberg angle (sin²(θ_w)). The light shift of the energy levels is needed to extract this value for sin²(θ_w). This thesis describes light shifts in atoms. Light shifts caused by different laser beams in barium and radium ions are calculated with a two-level approximation.

Actually, barium and radium ions are not two-level systems. More research is needed to find the details of the influence of all transitions on each other. For a laser beam at wavelength 589 nm and intensity 6.0(6)*10^{8 W}_m2 we have calculated the light shift for different states in the Ba⁺ ions in a two-level approximation. For a laser beam at wavelength 802 nm and intensity 3.0(3)*10^{7 W}_m2 we have calculated the light shift for different states in the Ra⁺ ions in the same two-level approximation. The calculations are accurate to order 1%. However, the radius of the laser beam can only be measured at 5 % accuracy at present. Therefore we can compare experiments and theory not better then at the 10% level. But if the radius of the laser beam would be known more precise, we would get an accuracy of about 1%. To improve the calculations to sub-1%

level, the Einstein co¨efficients need to be known at better accuracy.

Contents

1 Light shifts in atoms 1

2 Atomic Parity Violation 1

3 Radium ion to measure Atomic Parity Violation 3

4 Light shift in a two-level system 5

4.1 The Barium Ion . . . 9 4.2 The Radium Ion . . . 16

5 More than two levels 20

6 Discussion of the results 21

7 Conclusion 21

8 Appendices 22

8.1 Deriving and solving the Bloch equations for a two level system . . . . 22 8.2 Deriving the Bloch equations for a three level system . . . 25

9 Literature 27

10 Acknowledgements 28

1 Light shifts in atoms

Atoms, basic units of matter, consist of a dense, positively charged, central nucleus surrounded by a cloud of negatively charged electrons. In the quantum mechanical model of the atom, the electrons can only be in certain states (Bohr model). Each electron-state corresponds to a certain energy level. Sometimes different states have the same energy level. This is known as degeneracy. The state with the lowest energy is called the ground state, each other state is an excited state. If an atom is put in an electric field the energy levels shift. This is called Stark shift. If the shift is due to the oscillating electric field of a laser beam, this is called light shift.

Physical systems aim to be at the lowest possible state of total energy. According to the Pauli principle, 2 electrons can never be in the same state. If an atom contains multiple electrons, they can not all be in the ground state. The electron configuration of an atom describes the states of the electron of that atom. The total energy of this configuration is the sum of the energies of all states. The configuration which has the lowest total energy is called the ground state of the atom. In the case of barium and radium ions, there is one “valence” electron, the electron in the highest state. This va- lence electron is mostly in the outer regions of the space the atom occupies. Typically, when a photon is absorbed, it is almost always by the valence electron, bringing the electron to an excited state. When this electron falls back to its lowest possible state, it sends out a photon again. The valence electron can be excited to many different states by different photon energies. And thus it can send out light of different wavelengths.

All these wavelengths together define the atomic spectrum. If the energy levels are shifted by an external AC-fiels (i.e. a light shift is induced), the atomic lines in the spectrum move by amounts that depends on the intesity and frequency of the incoming light [1,2,3,5,7].

2 Atomic Parity Violation

Today, the most accurate model that discribes all of particle physics and in particular atoms and interactions between atoms is the Standard Model. This Standard Model includes as one of its impotant parameters of the Weinberg angle (θ_w). This is the so-called “mixing-angle” between photons and Z⁰-bosons. It’s value can be determined experimentally at different momentum scales rather precisely. Their relative size with respect to eachother can be predicted by the Standard Model. The measured values at intermediate and low energies come from parity violation measurements. Deviation of one value from the prediction based value would be a sign of new physics beyond the Standard Model. And thus, the Standard Model would not be complete. For an accu- rate measurement of θw (or actually sin²(θw)) in a Ra⁺ ion, the light shift of different

transitions between the energy levels needs to be determined accurately [1]. For tests, Ba⁺ is used, to determine the agreement of the calculations and experiments.

The electromagnetic (EM) interaction between the electron cloud and the quarks in the nucleus is most important to describe most properties of an atom. The EM interac- tion is the exchange of a photon between an electron and a quark in the nucleus. This process is described by quantum electrodynamics (QED). A parity transformation can be seen as a transformation that flips the sign of al coordinates in a system, according

_x

yz

→−x

−y

−z



. (1)

QED is invariant under parity transformations, so the atomic spectrum must also be.

However, there are also weak interactions between quarks and electrons. Though their influence is very small compared to EM interactions, they can not be neglected in some cases. The weak interaction is the exchange of W^± and Z⁰ bosons. In atoms the weak interaction between an electron and a quark in the nucleus is by Z⁰ bosons only, because W^± bosons transfer charge. The main difference between the weak and the EM interaction is that the photon is massless and Z⁰ bosons have a significant mass.

Thus, compared to an EM interaction, a weak interaction is very short-ranged and the exchange of Z⁰ bosons is less probable than photon exchange. Nevertheless, weak interactions occur. This causes the states of the ion to change a little. The states mix, though for most purposes the mixing is small enough to neglect. If the states are mixed, dipole transitions between states can occur, which were otherwise forbidden. And when a decay occurs between 2 states that are light shifted, we can calculate sin²(θ_w), in par- ticular if the transition is forbidden by QED. For more information, see e.g. [1].

The Weinberg angle (sin²(θ_w)) depends on the energy scale that is used. The experiment of Atomic Parity Violation (APV) is on a low energy scale, but it has high precision. Figure 1 shows sin²(θw) as predicted by the Standard Model (solid line), as a function of energy. If the Standard Model is violated, this could be explained by e.g.

“dark” Z-bosons. Dark matter might then be explained. The dotted lines in Figure 1 show different possibilities for the mass of the dark Z-boson [1].

sin2 θW

Momentum scale [GeV]

0.225 0.230 0.235 0.240 0.245 0.250

10⁻³ 10⁻² 10⁻¹ 1 10¹ 10² 10³ Møller

ν-DIS E158

Planned experiments

LEP APV(Cs)

SoLID APV(Ra⁺) MESA Qweak

SLD Qweak

(first 4%)

200 MeV 100 MeV 50 MeV

mdark Z

Figure 1: sin²(θ_w) as a function of momentum transfer for different experiments. The solid is the value that follows from the Standard Model. The dotted lines give sin²(θ_w) for different values of the mass of a dark Z-boson [1].

3 Radium ion to measure Atomic Parity Violation

There are various reasons to use a single trapped radium ion in such an experiment.

For a single trapped ion, there is good systematics and a long coherence time of the system [1]. Furthermore, only really small quantities of the radioactive element are needed. Especially for the rare radium, this is an advantage. A heavy atomic system with a single valence electron is preferred, because the wavefunction are better known for such atoms. All the earth-alkali atoms have this. It is proposed that the nS¹

2-(n- 1)D³

2 transitions in the alkali-like Ba⁺ (n=6), Sr⁺ (n=5) and Ca⁺ (n=4) ions are good transitions for a light shift measurement [4]. Ra⁺ (n=7), is even a better candidate.

Radium has a big advantage over the lighter earth-alkalis, and that is because the dependence of the APV effect on the proton number of the nucleus of the ion.

The matrix element describing APV is [1]

< ns¹

2|H_{AP V}|n⁰p¹

2 >∝ Z²Q_WK_rel(Z, R) (2) where ns¹

2 and n’p¹

2 are quantum states.

This scaling is called the ”faster than Z³law”. This is because Q_W (the weak charge)

is proportional to Z, and the relativistic factor K_rel also has its dependence on Z: [1]

K_rel(Z, r) '

"

Γ(3) Γ(2γ + 1)

 2Zr a₀

γ−1#2

(3)

with

γ =p

1 − (Zα)² (4)

These equations contain many constants like α, the fine strucure constant and a₀, the Bohr radius. The most important is that K_rel increases with increasing Z. This scaling of the APV effect is really a lot faster than Z³ (see Figure 2).

Figure 2: The scaling of the APV matrix element for alkali-earth ions. The scaling is stronger than Z³ [6].

There are more advantages for using radium. All relevant wavelengths are in or near the visible regime. There are different isotopes of radium that are have a suf- ficiently large lifetime against radioactive decay. Thus it is possible to do an inves- tigation of the APV effect in different isotopes. This adds the possibility of doing

”ratio-measurements”. By taking the ratio of an APV measurement in two different isotopes, the uncertainty associated with the atomic wavefunction calculation cancels.

This is a big advantage [1].

4 Light shift in a two-level system

A fictional ion that has only two energy levels, can be in a ground state or in an excited state. When this ion is placed in a laser beam, it oscillates between these two energy levels. The change that it is in the excited state changes with the Rabi-frequency (Ω) and the detuning (δ). When the laserlight has a frequency that is excatly the frequency needed for the transition, it is called “on-resonance”. The on-resonance Rabi-frequency can be found by [2]

Ω² = E²D₁₂²

~² , (5)

where E is the amplitude of the electric part of the electromagnetic wave and D₁₂ is the dipole-matrix element that corresponds to this transition. ~ is the Planck constant devided by 2π This D₁₂ can be found by [9]

A₂₁= g₁ g₂

8π²D₁₂² 3₀~λ³0

, (6)

where λ₀ is the wavelength of the transition and A₂₁ is the Einstein coefficient of the transition. ₀ is the vacuum permittivity, a physical constant. This Einstein coefficient is determined the decay time of the transition [2,3].

The electric field can easily be found by the intensity of the (Gaussian) beam, [3]

I = 2P

πr² = c₀E²

2 , (7)

where r is the radius and P the power of the laser beam. c is the speed of light in vacuum, which is defined to be constant.

Combining the last three equations gives Ω² = g₂

g₁

3λ³₀P A₂₁

2π³r²c~ . (8)

If the laser is a off-resonance, the detuning of the laserlight has to be taken into account. The detuning is given by [2]

δ = ω_L− ω₀ = 2πcλ₀− λ_l

λ²₀ , (9)

where ω_L and ω₀ are the frequency of the laserlight and the transition respectively.

λ_l is the wavelength of the laserlight.

The off-resonance (generalized) Rabi-frequency is given by [2]

Ω⁰ =√

Ω² + δ². (10)

To calculate the change that the ion is in the excited state, the Bloch equations can be exploited. In these Bloch equations the ion is described by the density matrix

operator,[5]

ˆ ρ =X

i,j

ρ_ij|iihj|, (11)

where i and j represent different states of the ion. For now, we only have two states in our ion: the ground state (g) and the excited state (e)

The time evolution of ˆρ is given by the Liouville equation,[5]

d ˆρ dt = −i

~

[ ˆH, ˆρ] + L_decay. (12)

But since we have a two level system, there can be no spontaneous decays to other levels: L_decay = 0 The Hamiltonian ( ˆH) of the system consists of two parts: the Hamil- tonian ˆH₀ of the ion, with a small pertubation ˆH_int, the interaction term [5].

Hˆ₀ = ~−^ω₂⁰ 0 0 ^ω₂⁰



. (13)

In the rotating wave approximation (RWA) the interaction Hamiltoninian is [5]

Hˆ_int = ~ 2

 0 Ωe^iωLt Ωe^−iωLt 0



. (14)

Note: the “real” interaction Hamiltonian also contains a term that is negligible for our purpose: the so-called “counter-rotating” term. In [5], this “Rotating Wave Approximation” is explained. This time dependence in ˆH_int may be a serious issue in further calculations that involve several significantly different timescales. Therefore, this time dependence is eliminated by moving to another frame: the rotating wave frame. This is not an approximation and can be done by an well chosen unitary matrix U .

Then the operators transform like [5]

ˆ˜

ρ = U ˆρU^† (15)

and

H = U ˆˆ HU^†− i~UdU^†

dt . (16)

The transformation causes the resulting eigenstates to have energy differences that are shifted by ω_L compared to the laboratorium frame. The observable quantities (the diagonal elements of the density matrix) are not changed by this transformation. The ion is either in the ground state or in the excited state, that should be the same in all

frames. For the two level system, the appropriate unitary matrix is

U = e^−iωLt2 0 0 e^iωLt2

!

. (17)

The resulting Hamiltonian is then H = ~

2

−δ Ω

Ω δ



. (18)

The elements of the density matrix in the rotating wave frame are described by a set of Equations 31-34, which are given in Appendix 8.1. In the laboratorium frame the equations for ˜ρ_gg and ˜ρ_ee are exactly the same because of the unitary transformation.

The equations for ˜ρge and ˜ρeg will be different in the lab frame, but in the two-level approximation, these off diagonal elements don’t play a role. The sum of ˜ρ_gg and ˜ρ_ee is 1, since the ion is either in a ground state or in an excited state. As we can see in Figure 3, the amplitude of the oscilations decreases with increasing δ. The frequency with wich it oscillates increases with increasing δ.

2 4 6 8 10 12 tK s

WO

0.2 0.4 0.6 0.8 1.0

Ρ_ee

∆=0

∆=W

∆=3W

∆=15W

Figure 3: The Rabi oscillations for a two-level system at different detunings δ in units of the Rabi frequency Ω.

The eigenvalues of this Hamiltonian are given by Equation 26 in Appendix 8.1. In the lab frame, the eigenvalues of the Hamiltonian are shifted by ±^~₂ω_L. Then the energy of the levels are

E_g,e = ∓~

2ω_L± ~

2Ω⁰. (19)

If we assume that the detuning is much larger than the Rabi frequency (|δ| >> Ω, the far off-resonance limit)

Ω⁰ =√

δ²+ Ω² ≈ δ(1 + Ω²

2δ² + ...) (20)

If we fill this in in Equation 19 we get E_g,e= ∓~

2ω₀± ~Ω²

4δ . (21)

The shifts of the energy of the eigenstates are [2]

∆Eg,e= ±~Ω²

4δ . (22)

To know whether there can be an accurate measurement, actual light shift is needed.

The frequency that corresponds to this light shift is [2]

∆ω_g,e= ∆Eg,e

~ = ±Ω²

4δ. (23)

This is a the angular frequency, corresponding to the energy. The actual frequency change that can be measured is [7]

∆ν_g,e = ∆ω_g,e

2π . (24)

If the ion falls back from an excited state to a ground state, it sends out a different photon then normal. Because the frequency of this photon is changed by

∆ν_tot = |∆ν_g| + ∆ν_e|. (25)

4.1 The Barium Ion

This two-level system is an appropriate way to look at all energy levels of an ion seperately. For a ¹³⁸barium ion, the most important energy levels are shown in Figure 4.

Figure 4: Grotrian diagram for the barium ion. The zero point of the energy axis in such diagrams are set at the energy of the ground state [6].

Let’s look at a two-level system in a laser beam. In the following we shall focus on the dependence of the light shift on the wavelength and intensity of the laserlight.

For this survey two systems are used. They correspond to the 6s ²S¹

2 - 6p ²P¹

2 and the 5d²D³

2 - 6p ²P¹

2 transitions in barium ions. For the laser beam the power and the wavelength are varied. Furthermore, we use that r = 46.0(2.3)µm. Note that the error of the radius of the laser beam (that is assumed to be Gaussian) is assumed very large, 5 %. We take here the accuracy quoted by the manifacturer of a commercial device (Thorlabs beam profiler BP209-VIS). This value can be magnificently improved in a dedicated effort, if needed, to . 1% [7]. This evenually causes a correspondingly large error in the light shift (around 10 % ), since it goes with _r¹2. Using Equations 8,9,23 and 24, we are able to calculate the light shift of the lower states caused by the laserlight.

Table 1: The detuning δ, Rabi-frequency Ω and light shift of the lower state ∆ν_g due to different laser beams for the 6s²S¹

2 - 6p ²P¹

2 transition in the Ba⁺-ion.

6s²S¹

2 - 6p ²P¹

2

λ₀ = 493.54538(3) nm [13,7]

A = 9.53(12) ∗ 10⁷ rad s⁻¹ [13,7]

λ_l (nm) P (W) Ω (rad s⁻¹) δ (rad s⁻¹) ∆ν_g (Hz) 390.000000(1)

10⁻⁴ 9.1(9) ∗ 10⁸ 8.0071343(2) ∗ 10¹⁴ 41(4) 10⁻² 9.1(9) ∗ 10⁹ 8.0071343(2) ∗ 10¹⁴ 4.1(4) ∗ 10³

1 9.1(9) ∗ 10¹⁰ 8.0071343(2) ∗ 10¹⁴ 4.1(4) ∗ 10⁵ 490.000000(1)

10⁻⁴ 9.1(9) ∗ 10⁸ 2.741632(2) ∗ 10¹³ 1.2(1) ∗ 10³ 10⁻² 9.1(9) ∗ 10⁹ 2.741632(2) ∗ 10¹³ 1.2(1) ∗ 10⁵ 1 9.1(9) ∗ 10¹⁰ 2.741632(2) ∗ 10¹³ 1.2(1) ∗ 10⁷ 590.000000(1)

10⁻⁴ 9.1(9) ∗ 10⁸ −7.4588079(2) ∗ 10¹⁴ −44.2(4.5) 10⁻² 9.1(9) ∗ 10⁹ −7.4588079(2) ∗ 10¹⁴ −4.42(45) ∗ 10³

1 9.1(9) ∗ 10¹⁰ −7.4588079(2) ∗ 10¹⁴ −4.42(45) ∗ 10⁵

Table 2: The detuning δ, Rabi-frequency Ω and light shift of the lower state ∆ν_g due to different laser beams for the 5d²D³

2 - 6p ²P¹

2 transition in the Ba⁺-ion.

5d²D³

2 - 6p ²P¹

2

λ₀ = 649.86936(3) nm [13, 7]

A = 3.10(4) ∗ 10⁷ rad s⁻¹ [13, 7]

λ_l (nm) P (W) Ω (rad s⁻¹) δ (rad s⁻¹) ∆ν_g (Hz) 390.000000(1)

10⁻⁴ 5.5(6) ∗ 10⁸ 1.15905361(3) ∗ 10¹⁵ 11(1) 10⁻² 5.5(6) ∗ 10⁹ 1.15905361(3) ∗ 10¹⁵ 1.1(1) ∗ 10³

1 5.5(6) ∗ 10¹⁰ 1.15905361(3) ∗ 10¹⁵ 1.1(1) ∗ 10⁵ 490.000000(1)

10⁻⁴ 5.5(6) ∗ 10⁸ 7.1303966(3) ∗ 10¹⁴ 17(2) 10⁻² 5.5(6) ∗ 10⁹ 7.1303966(3) ∗ 10¹⁴ 1.7(2) ∗ 10³

1 5.5(6) ∗ 10¹⁰ 7.1303966(3) ∗ 10¹⁴ 1.7(2) ∗ 10⁵ 590.000000(1)

10⁻⁴ 5.5(6) ∗ 10⁸ 2.6702570(3) ∗ 10¹⁴ 45.9(4.6) 10⁻² 5.5(6) ∗ 10⁹ 2.6702570(3) ∗ 10¹⁴ 4.59(46) ∗ 10³

1 5.5(6) ∗ 10¹⁰ 2.6702570(3) ∗ 10¹⁴ 4.59(46) ∗ 10⁵ From this table follows:

- Ω⁰ ≈ |δ| since |δ| >> Ω, and the approximation used in Equation 20 is justified.

- If the wavelength of the laser comes closer to the wavelength of the transition, the light shift increases but the Rabi frequency stays constant. The detuning becomes smaller, so the generalized Rabi frequency becomes smaller.

- If the power of the laser increases, the light shift grows faster than the (generalized) Rabi frequency.

- The sign of the light shift depends on the sign of the detuning.

0.001 0.01 0.1 1

P HWL

-50 0 50 100 150

DΝ

HkHzL

390 nm 490 nm 590 nm

Figure 5: Light shift for different wavelengths of the laserlight vs power of the laser for the 6s²S¹

2 - 6p ²P¹

2 transition in the barium ion.

0.001 0.01 0.1 1

P HWL

5 10 15 20 25 30 35

DΝ

HkHzL

390 nm 490 nm 590 nm

Figure 6: Light shift for different wavelengths of the laserlight vs power of the laser for the 5d²D³

2 - 6p ²P¹

2 transition in barium ion.

For all the transitions in Figure 4, we can now calculate the light shift of the energy levels. For the transitions we use the constants in Table 3 [7,13].

Table 3: The Einstein coefficients A₂₁ and transition wavelengths λ₀ of different tran- sitions in the Ba⁺-ion.

transition A₂₁ (rad s⁻¹) λ₀ (nm) 6s²S¹

2 - 6p ²P¹

2 9.53(12) ∗ 10⁷ 493.54538(3) 6s²S¹

2 - 6p ²P³

2 1.11(3) ∗ 10⁸ 455.53098(3) 5d ²D³

2 - 6p ²P¹

2 3.10(4) ∗ 10⁷ 649.86932(4) 5d ²D³

2 - 6p ²P³

2 6.00(16) ∗ 10⁶ 585.52973(4) 5d ²D⁵

2 - 6p ²P³

2 4.12(10) ∗ 10⁷ 614.34129(4)

Table 4: The wavelength λ_L, power P, radius r and intensity I of the laser beam used in this survey.

λ_L 589.000000(1) nm

P 2 W

r 46.0 (2.3) µm I 6.0(6)*10^{8 W}_m2

Here we used Equation 7 to calculate the intensity. The laser of this wavelength is used, because in the measurement the light shift in the 5d ²D³

2 state is the most important for the experiment at the Van Swinderen Institute. By this we can calculate δ, Ω and ∆ν_g, with Equations 8,9,23 and 24 (see Table 5).

Table 5: The detuning δ, Rabi-frequency Ω and light shift of the lower state ∆ν_g for different transitions in the Ba⁺-ion.

transition δ (rad s⁻¹) Ω (rad s⁻¹) ∆ν_g (MHz) 6s ²S¹

2 - 6p ²P¹

2 −7.3814761(8) ∗ 10¹⁴ 1.28(7) ∗ 10¹⁰ -0.89(9) 6s ²S¹

2 - 6p ²P³

2 −1.2115605(2) ∗ 10¹⁵ 1.74(9) ∗ 10¹⁰ -1.0(1) 5d²D³

2 - 6p ²P¹

2 2.7148562(4) ∗ 10¹⁴ 7.8(4) ∗ 10¹⁰ 0.90(9) 5d²D³

2 - 6p ²P³

2 −1.9065889(3) ∗ 10¹³ 4.2(2) ∗ 10¹⁰ -3.6(4) 5d²D⁵

2 - 6p ²P³

2 1.2647656(2) ∗ 10¹⁴ 9.6(5) ∗ 10¹⁰ 2.9(3)

∆ν_g corresponds to the energy shift of the lower state. ∆ν_e, that corresponds to the energy shift of the higher states, is, according to Equations 23 and 24: ∆ν_e= −∆ν_g The total ∆ν of every state is just the sum of the seperate ∆ν⁰s it has from different transitions (see Table 6).

Table 6: Light shift ∆ν_tot of the energy levels of the Ba⁺-ion, using the parameters in Table 4. Note for the 6p ²P¹

2 state, the contribution of both transitions to the total light shift cancels.

State ∆ν_tot (MHz) 6s ²S¹

2 -1.9(2)

6p²P¹

2 −8.5(9) ∗ 10⁻³ 6p²P³

2 1.7(2)

5d²D³

2 -2.7(3) 5d²D⁵

2 2.9(3)

Note: ∆ν dependends linearly on P, while δ is constant for changing P. Furthermore, the states that are looked at can also interact with higher states, like the 7s²S¹

2 state.

This will also give a light shift, though it will be a lot smaller than what we have now, because these states are further away.

The light shift of the 6p ²P¹

2 state is very small. This is because the two transitions that cause this light shift (almost) cancel eachother.

550 600 650 700 Λ_lHnmL

-5 5

DΝ_gHMHzL

5d-D3 2 5d-D_{5 2} 6s-S1 2 6p-P_{1 2} 6p-P_{3 2}

585 590 595 600 Λ_lHnmL

-10 -5 5 10

DΝ_gHMHzL

5d-D3 2 5d-D_{5 2} 6s-S1 2 6p-P_{1 2} 6p-P_{3 2}

Figure 8: Calculated light shift of different energy levels of the barium ion versus wavelenght of the laserlight. The laserlight has an intensity of 6.0(6)*10^{8 W}_m2. Zoomed in around 589 nm.

The graphs above shows that de dependence of the light shift on the laserlight is different for every energy level. Note that the graph does not make sense when the wavelength of the laser comes close to the wavelength of a transition. Because then the detuning will be very small compared to the Rabi frequency, and the approximation we made in Equation 20 is not valid anymore.

4.2 The Radium Ion

Analogue to what is done in in section 2.1 for the ¹³⁸barium ion, it is also possible to calculate the light shift of the energy levels in the²¹⁴radium ion.

Figure 9: Grotrian diagram for the radium ion. The zero point of the energy axis in such diagrams are set at the energy of the ground state [6].

First, we need the wavelengths and the Einstein coefficients of the relevant transi- tions (see Table 7) [7,13].

Table 7: The Einstein coefficients A₂₁ and transition wavelengths λ₀ of different tran- sitions in the Ra⁺-ion.

transition A₂₁ (rad s⁻¹) λ₀ (nm) 7s ²S¹

2 - 7p ²P¹

2 9.3(1) ∗ 10⁷ 468.31266(2) 7s ²S¹

2 - 7p ²P³

2 1.19(2) ∗ 10⁸ 381.52027(1) 6d ²D³

2 - 7p ²P¹

2 3.34(4) ∗ 10⁷ 1079.14454(12) 6d ²D³

2 - 7p ²P³

2 4.70(9) ∗ 10⁶ 708.00115(5) 6d ²D⁵

2 - 7p ²P³

2 3.53(7) ∗ 10⁷ 802.1980(6)

Table 8: The wavelength λ_L, power P, radius r and intensity I of the laser beam used in this survey.

λ_L 802.000000(1) nm

P 0.1 W

r 46.0 (2.3) µm I 3.0(3)*10^{7 W}_m2

By this we can calculate δ, Ω and ∆ω_g (see Table 9).

Table 9: The detuning δ, Rabi-frequency Ω and light shift of the lower state ∆ν_g for different transitions in the Ra⁺-ion.

transition δ (rad s⁻¹) Ω (rad s⁻¹) ∆ν_g (kHz) 7s ²S¹

2 - 7p ²P¹

2 −2.865949309(8) ∗ 10¹⁵ 2.63(13) ∗ 10¹⁰ -9.6(9) 7s ²S¹

2 - 7p ²P³

2 −5.441396869(15) ∗ 10¹⁵ 3.09(16) ∗ 10¹⁰ -7.0(7) 6d²D³

2 - 7p ²P¹

2 4.48278245(1) ∗ 10¹⁴ 3.90(20) ∗ 10¹⁰ 1.35(14) ∗ 10² 6d²D³

2 - 7p ²P³

2 −3.532280513(9) ∗ 10¹⁴ 1.10(6) ∗ 10¹⁰ -13.6(1.4) 6d²D⁵

2 - 7p ²P³

2 5.79565616(2) ∗ 10¹¹ 2.96(15) ∗ 10¹⁰ 6.0(6) ∗ 10⁴

And again, with this we can calculate the total light shift of each level (see Table 10).

Table 10: Light shift ∆νtot of the energylevels of the Ra⁺-ion, using the parameters in Table 8.

State ∆ν_tot (kHz) 7s ²S¹

2 -17(2)

7p²P¹

2 −1.3(1) ∗ 10² 7p²P³

2 -6.0(6) ∗ 10⁴ 6d²D³

2 1.2(1) ∗ 10² 6d²D⁵

2 6.0(6) ∗ 10⁴

The light shift in the Ra⁺ ion is very different from the light shift in the Ba⁺ ion.

This is because the light shift in the Ra⁺ ion is caused by a laser beam with a lower power, but with a lower detuning from the most important transition.

750 800 850 900 Λ_lHnmL

-0.6 -0.4 -0.2 0.2 0.4 0.6

DΝ_gHMHzL

5d-D3 2

5d-D_{5 2} 6s-S_{1 2} 6p-P1 2

6p-P3 2

Figure 10: Calculated light shift of different energy levels of the radium ion versus wavelenght of the laserlight. The laserlight has an intensity of 3.0(3)*10^{7 W}_m2.

801.8 802.0 802.2 802.4 Λ_lHnmL

-50 50

DΝ_gHMHzL

5d-D3 2

5d-D5 2

6s-S1 2

6p-P1 2

6p-P3 2

Figure 11: Calculated light shift of different energy levels of the radium ion versus wavelenght of the laserlight. The laserlight has an intensity of 3.0(3)*10^{8 W}_m2. Zoomed in around 802 nm.

The graphs above shows that also for the radium ion the dependence of the light shift on the laserlight is different for every energy level. Note that the graph does not make sense when the wavelength of the laser comes too close to the wavelength of a transition. Because then the detuning will be very small compared to the Rabi frequency, and the approximation we made in Equation 20 is not valid anymore. For the 6d²D⁵

2 - 7p²P³

2 transition, there is a very low detuning, but it is still much bigger than the Rabi frequency.

5 More than two levels

In Chapter 4 a two-level approximation was used. The real ion has more levels and the question rises what their influence is. We have added up all light shifts, treating the energy levels as 5 independent two-level systems, which is an incomplete description. If we consider e.g. the 5 energy levels of the barium ion, used in Chapter 4, coherences and spontaneous decay to other levels need to be considered for a more precise description [7,10,15].

Figure 12: The (Three-level) Λ-system. The coherences between the transitions influ- ence the light shift.

First consider the so-called Λ-system, a three level system, which is aligned like the lowest ²S¹

2, ²P¹

2 and ²D³

2 in the barium- or radium ion (see Figure 12). The Bloch equations for the Λ-system (see Equations 35-43 in Appendix 8.2) show that the off- diagonal terms of the density matrix play an important role. The ion does not have to populate the |c > state when it is excited from state |a > to state |b >.

In Appendix 8.2, the spontaneous decay to other levels is not taken into account This should be done for a three (or more) level system. The resulting Bloch equations can not be solved analytically. This is mainly because the rotating wave frame is not as helpfull as it was in a two-level system. Since we have two transitions, the time dependence is on two different scales. Numerical solutions are required, or more approximations have to be done. This will not be further disscussed in this thesis. Furthermore, the three-level system is not sufficient to describe the problem. We need at least a five-level system, which also describes the lowest ²P³

2 and ²D⁵

2 state of the ions. More research is needed to find the details of the influence of all transitions on each other.

6 Discussion of the results

The light shift of the energy levels of the barium and radium ion is calculated, using the two level approximation. The results can be found in Table 6 and 10 respectively.

In an experiment at the Van Swinderen Institute, the actual light shift will soon be measured. The distribution of the photons coming from the trapped ion, has a width of 15 MHz [7]. A light shift of -2.7 MHz (barium, 5d ²D³

2) or 2.9 MHz (barium, 5d

2D⁵

2) will be measurable. And especially a light shift of 60 MHz (radium, 7p ²P³

2 and 6d²D⁵

2) are measurable. However, for the expected input of the radius, the predicted light shift is accurate to about 10 %. The biggest cause of this large error is the error in the radius of the laser beam. This is 5 %, and since the light shift goes with the radius squared, this error causes most of the error in the light shift. Other quantities, like the wavelength of the transitions, the wavelength of the laser and the power of the laser are known very precise. The only other quantity that has some influence in the error of the light shift, is the Einstein coefficient. It is known to somewhat more than 1%, if we want to know the light shift to sub-1% level, we should know the Einstein coefficients more precise. This can be achieved by measuring the lifetime of the states more precise.

7 Conclusion

For a laser beam with a wavelength of 589 nm we have calculated the light shift of different states in the Ba⁺ ions in a two-level approximation. For a laser beam with a wavelength of 802 nm we have calculated the light shift of different states in the Ra⁺ ions in a two-level approximation. This is done accurate to order 1%. However, the input of the radius of the laser beam has an error of 5 %. Therefore we get a light shift with an error of 10 %, for now. If we want to calculate the light shift to sub-1% level, the Einstein coefficients should be measured more precise. The light shifts calculated can be measured at the Van Swinderen Institute. More research is needed to find the details of the influence of all transitions on each other.

8 Appendices

8.1 Deriving and solving the Bloch equations for a two level system

For deriving the Bloch equations and the eigenvalue of the Hamiltonian, we used the following in Mathematica:

Clear[“Global`*”]

matr2 = {{ρ_gg[t], ρ_ge[t]}, {ρ_eg[t], ρ_ee[t]}};

(*The density matrix in the rotating wave frame*)

mathi2 = (~/2) ∗ {{−ω0, Ω ∗ E^∧(i ∗ ωl ∗ t)}, {Ω ∗ E^∧(−i ∗ ωl ∗ t), ω0}};

(* the initial hamiltonian in the lab frame *)

matu2 = {{E^∧(−i ∗ ωl ∗ t/2), 0}, {0, E^∧(i ∗ ωl ∗ t/2)}};

(*the unitary matrix used for transformation to the rotating wave frame*) matu2c = {{E^∧(i ∗ ωl ∗ t/2), 0}, {0, E^∧(−i ∗ ωl ∗ t/2)}};

(*the complex conjugate of the unitary matrix*)

dmatu2c = i ∗ ωl ∗ 0.5 ∗ {{E^∧(i ∗ ωl ∗ t/2), 0}, {0, −1 ∗ E^∧(−i ∗ ωl ∗ t/2)}};

(*the derivative of the complex conjugate above*)

dmath2 = (matu2.(mathi2.matu2c)) − i ∗ ~ ∗ (matu2.dmatu2c);

(* the hamiltonian in the rotating wave frame, see Equation 16*) dmatr2 = −(i/~) ∗ ((matr2.dmath2) − (dmath2.matr2));

(* the Liouville equation, see Equation 12*) dmatr2x = FullSimplify[dmatr2];

dmatr2x//MatrixForm Eigenvalues[dmath2]



1iΩ(ρ [t] − ρ [t]) −i(ω0 − ωl)ρ [t] +¹iΩ(ρ [t] − ρ [t])



n−~p

0.25Ω²+ 0.25ω0²− 0.5ω0ωl + 0.25ωl², ~p

0.25Ω²+ 0.25ω0²− 0.5ω0ωl + 0.25ωl²o The last answer are the eigenvalues of the Hamiltonian in the rotating wave frame.

Rewriting them gives:

E_g,e = ±~

2Ω⁰ (26)

The first matrix is equal to the time derivative of the density matrix in the rotating wave frame. This gives the Bloch equations:

d ˜ρ_gg

dt (t) = i

2Ω[ ˜ρeg(t) − ˜ρge(t)] (27) d ˜ρ_ge

dt (t) = i

2Ω[ ˜ρ_ee(t) − ˜ρ_gg(t)] + iδ ˜ρ_ge(t) (28) d ˜ρeg

dt (t) = i

2Ω[ ˜ρ_gg(t) − ˜ρ_ee(t)] − iδ ˜ρ_eg(t) (29) d ˜ρ_ee

dt (t) = i

2Ω[ ˜ρ_ge(t) − ˜ρ_eg(t)] (30) Assumed that the ion is in the ground state on t = 0, these equations are solvable.

For solving these, we used the following in mathematica:

ClearAll[ ˜ρgg, ˜ρge, ˜ρeg, ˜ρee, δ, Ω, sol, deqns]

deqns = { ˜ρ⁰_gg[t]==I ∗ Ω ∗ ( ˜ρ_eg[t] − ˜ρ_ge[t])/2,

˜

ρ⁰_ge[t]==(I ∗ Ω ∗ ( ˜ρ_ee[t] − ˜ρgg[t])/2)+I∗δ∗ ˜ρge[t],

˜

ρ⁰_eg[t]==(I ∗ Ω ∗ ( ˜ρ_gg[t] − ˜ρ_ee[t])/2) − I ∗ δ ∗ ˜ρ_eg[t],

˜

ρ⁰_ee[t]==I ∗ Ω ∗ ( ˜ρ_ge[t] − ˜ρ_eg[t])/2

˜

ρ_gg[0]==1, ˜ρ_ge[0]==0, ˜ρ_eg[0]==0, ˜ρ_ee[0]==0};

sol = DSolve[deqns, { ˜ρ_gg[t], ˜ρ_ee[t], ˜ρ_ge[t], ˜ρ_eg[t]}, {t}]

˜

ρ_gg[t] → ^e

−t

√

−δ2−Ω2 4e^t

√

−δ2−Ω2δ²+Ω²+2e^t

√

−δ2−Ω2Ω²+e^2t

√

−δ2−Ω2Ω²



4(δ²+Ω²) ,

˜

ρge[t] → −^e

−t

√

−δ2−Ω2

−1+e^t

√

−δ2−Ω2 Ω

iδ²+ie^t

√

−δ2−Ω2δ²+iΩ²+ie^t

√

−δ2−Ω2Ω²−δ√

−δ²−Ω²+e^t

√

−δ2−Ω2δ√

−δ²−Ω² 4√

−δ²−Ω²(δ²+Ω²) ,

˜

ρ_eg[t] → −^e

−t√

−δ2−Ω2

−1+e^t

√

−δ2−Ω2 Ω

−iδ²−ie^t

√

−δ2−Ω2δ²−iΩ²−ie^t

√

−δ2−Ω2Ω²−δ√

−δ²−Ω²+e^t

√

−δ2−Ω2δ√

−δ²−Ω² 4√

−δ²−Ω²(δ²+Ω²) ,

˜

ρ_ee[t] → −^e

−t

√

−δ2−Ω2

−1+e^t

√

−δ2−Ω22

Ω² 4(δ²+Ω²)

These 4 answers seem complicated, but can easily be rewritten in:

˜

ρ_gg= 1 − Ω²

Ω⁰²sin²Ω⁰t

2 (31)

˜

ρge = −iΩ

2Ω⁰sinΩ⁰t − Ωδ

Ω⁰²sin²Ω⁰t

2 (32)

˜

ρ_eg = iΩ

2Ω⁰sinΩ⁰t − Ωδ

Ω⁰²sin²Ω⁰t

2 (33)

˜

ρ_ee= Ω²

Ω⁰²sin²Ω⁰t

2 (34)

8.2 Deriving the Bloch equations for a three level system

Clear[“Global`*”];

matr3 = {{ρ_aa[t], ρ_ab[t], ρ_ac[t]}, ρ_ba[t], ρ_bb[t], ρ_bc[t]}, ρ_ca[t], ρ_cb[t], ρ_cc[t]}};

(*The density matrix in the rotating wave frame*)

mathi3 = hbar ∗ {{ω_a, 0, 0.5 ∗ Ω_ac∗ e^∧(I ∗ ω_l∗ t)}, {0, ω_b, 0.5 ∗ Ω_bc∗ e^∧(I ∗ ω_l∗ t)}

{0.5 ∗ Ω_ac∗ e^∧(−I ∗ ω_l∗ t), 0.5 ∗ Ω_bc∗ e^∧(−I ∗ ω_l∗ t), ω_c}}

(* the initial hamiltonian in the lab frame *)

matu3 = {{e^∧(−I ∗ ω_l∗ t/2), 0, 0}, {0, e^∧(−I ∗ ω_l∗ t/2), 0}, {0, 0, e^∧(I ∗ ω_l∗ t/2)}};

(*the unitary matrix used for transformation to the rotating wave frame*)

matu3c = {{e^∧(I ∗ ω_l∗ t/2), 0, 0}, {0, e^∧(I ∗ ω_l∗ t/2), 0}, {0, 0, e^∧(−I ∗ ω_l∗ t/2)}};

(*the complex conjugate of the unitary matrix*)

dmatu3c = I ∗ ω_l∗ 0.5 ∗ {{e^∧(I ∗ ω_l∗ t/2), 0, 0}, {0, e^∧(I ∗ ω_l∗ t/2), 0}, {0, 0, −1 ∗ e^∧(−I ∗ ω_l∗ t/2)}};

(*the derivative of the complex conjugate above*)

math3 = (matu3.(mathi3.matu3c)) − I ∗ hbar ∗ (matu3.dmatu3c);

(* the hamiltonian in the rotating wave frame, see Equation 16*) L = (I/(hbar)) ∗ ((matr3.math3) − (math3.matr3));

(* the Liouville equation, see Equation 12*) bloch = FullSimplify[L];

bloch//MatrixForm













0.5iΩ_acρ_ac[t] − 0.5iΩ_acρ_ca[t]

i((ω_a− ω_b)ρ_ba[t] + 0.5Ω_acρ_bc[t] − 0.5Ω_bcρ_ca[t])

−0.5iΩ_acρ_aa[t] − 0.5iΩ_bcρ_ba[t] + i(ω_a− ω_c+ ω_l)ρ_ca[t] + 0.5iΩ_acρ_cc[t]

























−i((ω_a− ω_b)ρ_ab[t] − 0.5Ω_bcρ_ac[t] + 0.5Ω_acρ_cb[t]) 0.5iΩ_bcρ_bc[t] − 0.5iΩ_bcρ_cb[t]

−0.5iΩ_acρ_ab[t] − 0.5iΩ_bcρ_bb[t] + i(ω_b− ω_c+ ω_l)ρ_cb[t] + 0.5iΩ_bcρ_cc[t]

























0.5iΩ_acρ_aa[t] + 0.5iΩ_bcρ_ab[t] − i(ω_a− ω_c+ ω_l)ρ_ac[t] − 0.5iΩ_acρ_cc[t]

0.5iΩ_acρ_ba[t] + (0.5i)Ω_bcρ_bb[t] − i(ω_b− ω_c+ ω_l)ρ_bc[t] − 0.5iΩ_bcρ_cc[t]

i(−0.5Ω_acρ_ac[t] − 0.5Ω_bcρ_bc[t] + 0.5Ω_acρ_ca[t] + 0.5Ω_bcρ_cb[t])













The last 3 matrices together are actually together one matrix, which represent the Bloch equations. The Bloch equations are:

d ˜ρaa

dt (t) = 0.5iΩ_acρ_ac[t] − 0.5iΩ_acρ_ca[t] (35) d ˜ρ_ab

dt (t) = −i((ω_a− ω_b)ρ_ab[t] − 0.5Ω_bcρ_ac[t] + 0.5Ω_acρ_cb[t]) (36) d ˜ρ_ac

dt (t) = 0.5iΩ_acρ_aa[t] + 0.5iΩ_bcρ_ab[t] − i(ω_a− ω_c+ ω_l)ρ_ac[t] − 0.5iΩ_acρ_cc[t] (37) d ˜ρ_ba

dt (t) = i((ωa − ωb)ρba[t] + 0.5Ωacρbc[t] − 0.5Ωbcρca[t]) (38) d ˜ρ_bb

dt (t) = 0.5iΩ_bcρ_bc[t] − 0.5iΩ_bcρ_cb[t] (39) d ˜ρ_bc

dt (t) = 0.5iΩ_acρ_ba[t] + (0.5i)Ω_bcρ_bb[t] − i(ω_b− ω_c+ ω_l)ρ_bc[t] − 0.5iΩ_bcρ_cc[t] (40)

d ˜ρ_ca

dt (t) = −0.5iΩ_acρ_aa[t] − 0.5iΩ_bcρ_ba[t] + i(ω_a− ω_c+ ω_l)ρ_ca[t] + 0.5iΩ_acρ_cc[t] (41)

d ˜ρ_cb

dt (t) = −0.5iΩacρaa[t] − 0.5iΩbcρba[t] + i(ωa− ωc+ ωl)ρca[t] + 0.5iΩacρcc[t] (42) d ˜ρ_cc

dt (t) = i(−0.5Ω_acρ_ac[t] − 0.5Ω_bcρ_bc[t] + 0.5Ω_acρ_ca[t] + 0.5Ω_bcρ_cb[t]) (43)

9 Literature

[1] L. Wansbeek, Atomic Parity Violation in a Single Radium Ion, Thesis, Rijksuniver- siteit Groningen, 2011.

[2] J.A. Sherman, arXiv:0907.0459v1 [physics.atom-ph], 2009.

[3] W. Demtr¨oder, Laser spectroscopy, third edition, 2003, ISBN 9788181282057.

[4] N. Fortson, Phys.Rev.Lett. 70,2383 (1993).

[5] T. W. Koerber, Measurements of light shift ratios with a single trapped ¹³⁸Ba⁺ ion, and prospects for a parity violation experiment, PhD thesis, University of Wash- ington, Seattle, 2003.

[6] wiki.kvi.nl, Triµp group.

[7] Private meetings with people from the radium ion experiment at the Van Swinderen Institute.(2014)

[8] D. Budker, D. F. Kimball, D. P. DeMille, Atomic Physics, An exploration through problems and solutions, First Edition, 2004, ISBN 9780198509509.

[9] C. J. Foot, Atomic Physics, 2013 Edition, ISBN 9780198506966.

[10] Y.R. Shen, The principles of Nonlinear Optics, 1984, ISBN 0471889989.

[11] S. Kumar, W. Marciano, Annu. Rev. of Nucl. Part. Sci. 63, 237 (2013).

[12] H. Davoudiasl, Hye-Sung Lee, W. Marciano, arxiv. 1402.3620 (2014).

[13] NIST database, www.nist.gov.

[14] R.G. Brewer, E.L. Hahn, Phys. Rev. A 11, 5 (1975)

[15] R.G. Brewer, Coherent optical transients, Phys. Today, May 1977.

[16] J.A. Sherman, A. Andalkar, W. Nagourny, E.N. Fortson, Phys. Rev. A 78, 052514 (2008)

[17] T. Zanon-Wilette, E. de Clercq, E. Arimondo, Phys. Rev. A 84, 062502 (2011) [18] M. Crescimanno, M. Hohensee, J. Opt. Soc. Am. B, Vol 25, No. 12, Dec 2008.

10 Acknowledgements

I would like to thank prof. dr. Klaus Jungmann and dr. Lorenz Willmann for super- vising my bachelor thesis. I highly appreciate their support, and the help from all the people of the radium ion experiment at the Van Swinderen Institute. Especially my discussions with prof. dr. Klaus Jungmann helped a lot for expanding my knowledge about light shifts. I would like to thank my officemate Mayerlin Nu˜nez Portela for all the help, advice, music and coffee which really helped me. Advice given by Elwin Dijck has been a great help for understanding all the computer programs I used. It was highly appreciated. At last I would like to thank all the people of the FIS Group for listening to my talk and for the feedback they gave.