Magnetic Field Orientation

REI TRANSDUCTION THEORY

ER 3 YVO 4 SPECTROSCOPY

3.4 Magnetic Field Orientation

0 10 20 30 40 50 60 70 80 90 B-field angle from c-axis

2 3 4 5 6 7 8

g dc

Z1

Y1

Y2

Figure 3.5: 𝑔_𝑑𝑐for 𝑍₁, 𝑌₁and𝑌₂as a function of the magnetic field angle from the c-axis.

Lastly, magnetic field orientation for the branching ratio of the optical transitions dipoles can be determined. We will use an effective spin Hamiltonian approach as previously shown in the literature that has matched quite well to their experiments [111–113]. We can use the effective spin Hamiltonian for the ground and excited state (Equation 3.1) to determine the states at a given magnetic field orientation.

First we can write down the states:

|Ψ_{+, 𝑍1}⟩=−𝛽∗ |Φ_−3/2, 𝑍₁⟩ +𝛼^∗|Φ_+3/2, 𝑍₁⟩ (3.9)

|Ψ₋, 𝑍₁⟩=−𝛼^∗|Φ_−3/2, 𝑍₁⟩ +𝛽|Φ_+3/2, 𝑍₁⟩ (3.10)

|Ψ_+,𝑌1⟩=−𝛿^∗

𝑌₁|Φ_−3/2,𝑌₁⟩ +𝛾^∗

𝑌₁|Φ_+3/2,𝑌₁⟩ (3.11)

|Ψ_−,𝑌1⟩=−𝛿_𝑌

1|Φ_−3/2,𝑌₁⟩ +𝛾_𝑌

1|Φ_+3/2,𝑌₁⟩ (3.12)

|Ψ₊,𝑌₂⟩=−𝛿^∗

𝑌₂|Φ_−1/2,𝑌₂⟩ +𝛾^∗

𝑌₂|Φ_+1/2,𝑌₂⟩ (3.13)

|Ψ_−,𝑌2⟩=−𝛿_𝑌

2|Φ_−1/2,𝑌₂⟩ +𝛾_𝑌

2|Φ_+1/2,𝑌₂⟩ (3.14) where |Φ𝑖, 𝑗⟩are the labels of the states with a magnetic field along the c-axis and 𝑖 is the crystal field quantum number. |Ψ𝑖, 𝑗⟩is the label of the state when a static magnetic field is applied in the a-c plane. 𝛼, 𝛽, 𝛿_𝑌

1, 𝛾_𝑌

1, 𝛿_𝑌

2, and𝛾_𝑌

2 are the state overlap coefficients.

We can then apply the transition selection rules (see Table 3.1) and transition oper- ator time-reversal symmetry properties [66, 112] to estimate the relative transition

strengths for the electric and magnetic dipole transitions for𝑍₁-𝑌₁: 𝑅_𝑌

1, 𝐵∥𝑐,⊥ = | ⟨Ψ_{+, 𝑍1}|𝑃ˆ_𝐵∥𝑐|Ψ_−,𝑌1⟩ |²=|𝛼𝛿+𝛾 𝛽|² (3.15) 𝑅_𝑌

1, 𝐵∥𝑐,∥ = | ⟨Ψ₊, 𝑍₁|𝑃ˆ_𝐵∥𝑐|Ψ₊,𝑌₁⟩ |²= |𝛼𝛿^∗−𝛽𝛾^∗|² (3.16) 𝑅_𝑌

1,𝐸∥𝑎,⊥ =| ⟨Ψ_{+, 𝑍1}|𝑃ˆ_𝐸∥𝑎|Ψ_−,𝑌1⟩ |²=|𝛼𝛿+𝛽𝛾|² (3.17) 𝑅_𝑌

1,𝐸∥𝑎,∥ =| ⟨Ψ_{+, 𝑍1}|𝑃ˆ_𝐸∥𝑎|Ψ_+,𝑌1⟩ |²= |𝛼𝛾^∗− 𝛽𝛿^∗|² (3.18) 𝑅_𝑌

1, 𝐵∥𝑎,⊥ =| ⟨Ψ₊, 𝑍₁|𝑃ˆ_𝐵∥𝑎|Ψ₋,𝑌₁⟩ |² =|𝛼𝛿−𝛽𝛾|² (3.19) 𝑅_𝑌

1, 𝐵∥𝑎,∥ = | ⟨Ψ_{+, 𝑍1}|𝑃ˆ_𝐵∥𝑎|Ψ_+,𝑌1⟩ |²=|𝛼𝛾^∗+𝛽𝛿^∗|² (3.20) where𝐸/𝐵correspond to magnetic and electric dipoles, ∥ 𝑎/𝑐refers to the polar- ization of the dipole, ∥ /⊥ correspond to transitions between sign-preserving (i.e.

|Ψ₊, 𝑍₁⟩ ↔ |Ψ₊,𝑌₁⟩) or sign-flipping transitions (i.e. |Ψ₊, 𝑍₁⟩ ↔ |Ψ₋,𝑌₁⟩), and ˆ𝑃is the transition dipole operator.

We can do the same thing for𝑍₁-𝑌₂: 𝑅_𝑌

2,𝐸∥𝑐,⊥ =| ⟨Ψ₊, 𝑍₁|𝑃ˆ_𝐸∥𝑐|Ψ₋,𝑌₂⟩ |² =|𝛽𝛿+𝛼𝛾|² (3.21) 𝑅_𝑌

2,𝐸∥𝑐,∥ = | ⟨Ψ_{+, 𝑍1}|𝑃ˆ_𝐸∥𝑐|Ψ_+,𝑌2⟩ |²=|𝛼𝛿^∗−𝛽𝛾^∗|² (3.22) 𝑅_𝑌

2,𝐸∥𝑎,⊥ =| ⟨Ψ₊, 𝑍₁|𝑃ˆ_𝐸∥𝑎|Ψ₋,𝑌₂⟩ |²= |𝛼^∗𝛿^∗−𝛽^∗𝛾^∗|² (3.23) 𝑅_𝑌

2,𝐸∥𝑎,∥ = | ⟨Ψ₊, 𝑍₁|𝑃ˆ_𝐸∥𝑎|Ψ₊,𝑌₂⟩ |² =|𝛼^∗𝛾+𝛽^∗𝛿|² (3.24) 𝑅_𝑌

2, 𝐵∥𝑎,⊥ = | ⟨Ψ_{+, 𝑍1}|𝑃ˆ_𝐵∥𝑎|Ψ_−,𝑌2⟩ |²=|𝛼^∗𝛿^∗+𝛽^∗𝛾^∗|² (3.25) 𝑅_𝑌

2, 𝐵∥𝑎,∥ =| ⟨Ψ₊, 𝑍₁|𝑃ˆ_𝐵∥𝑎|Ψ₊,𝑌₂⟩ |²=|𝛼^∗𝛾− 𝛽^∗𝛿|². (3.26) We can then define the branching ratio for each dipole transition as:

𝐵_{𝑖, 𝑗} =

𝑅_{𝑖, 𝑗 ,⊥} 𝑅_{𝑖, 𝑗 ,⊥}+𝑅_{𝑖, 𝑗 ,∥}

(3.27) for transition 𝑖 (i.e. 𝑍₁-𝑌₁ or 𝑍₁-𝑌₂) and 𝑗 dipole type and polarization. The branching ratio is defined in such a way that an ideal (or even) branching ratio gives a value of 1/2 which results in even mixing between the states, while a branching ratio of 0 or 1 corresponds to no mixing.

The branching ratio for the three different possible transition dipoles for 𝑍₁-𝑌₁are shown in Figure 3.6a. When the DC magnetic field is along either the c-axis or the a-axis, the branching ratio for all transitions dipoles is either 0 or 1. Only the𝐸 ∥ 𝑎 (𝜎_{𝐸 𝐷}) transition dipole can reach an even branching ratio at a magnetic field angle of 35°from the c-axis. 𝐵||𝑎 (𝜋_{𝑀 𝐷}) and 𝐵||𝑐(𝜎_{𝑀 𝐷}) can reach non-zero branching ratios, but the mixing is far from even (97% and 3%, respectively).

Figure 3.6: Branching ratio for a) 𝑍₁-𝑌₁ and b) 𝑍₁-𝑌₂ optical transitions for the different dipole operators (i.e. electric or magnetic) and orientations (i.e. parallel to the a-axis or c-axis).

Similarly, we can look at the branching ratios for 𝑍₁-𝑌₂ in Figure 3.6b. Both orientation of the electric dipole can achieve an even branching ratio at a magnetic field angle 24° from the c-axis. The branching ratio for 𝐵 ∥ 𝑎 (𝜋_{𝑀 𝐷}) reaches a maximum value of 3·10⁻³which is very small.

For transduction, we care about the branching ratio, but we also need to consider the overall strength of the transition dipole. In other words, if the branching ratio is not even, but the total strength is much larger, that can still be the better transition dipole to use. We can calculate the transition dipole moments as:

𝑑²

𝑖, 𝑗 ,⊥ =𝑑²

𝑖, 𝑗

𝑅_{𝑖, 𝑗 ,⊥}

𝑅_{𝑖, 𝑗 ,⊥} +𝑅_{𝑖, 𝑗 ,∥}

(3.28) 𝑑²

𝑖, 𝑗 ,∥ =𝑑²

𝑖, 𝑗

𝑅_{𝑖, 𝑗 ,∥}

𝑅_{𝑖, 𝑗 ,⊥} +𝑅_{𝑖, 𝑗 ,∥} (3.29)

where we are using the same notation as the branching ratio. The transduction efficiency scales as𝜂 ∝ |𝑑_{𝑖, 𝑗 ,⊥}𝑑_{𝑖, 𝑗 ,∥}|² if we assume that we are only able to use one of the transition dipole moments (i.e. only electric or only magnetic dipole) for the transduction process. This is the case for us in our standing wave optical cavities and will be mentioned again in the optical cavity design section in Chapter 4.

Figure 3.7: Dipole moment product for a)𝑍₁-𝑌₁and b)𝑍₁-𝑌₂for different transition dipole operators (i.e. electric or magnetic) and orientations (i.e. parallel to the a-axis or c-axis).

The dipole product factor for𝑍₁-𝑌₁is shown in Figure 3.7a. The strongest transition dipole product for𝑍₁-𝑌₁is the magnetic dipole along the c-axis (𝜎_{𝑀 𝐷}) even though the branching ratio is far from even. Similarly, the dipole product for 𝑍₁-𝑌₂ is shown in Figure 3.7b. Here, the strongest transition dipole for 𝑍₁-𝑌₂is the electric dipole along the c-axis (𝜋_{𝐸 𝐷}), which has both a strong dipole strength and an even branching ratio. Comparing the two different transitions, we would expect ∼7x larger transduction signal for𝑍₁-𝑌₂relative to𝑍₁-𝑌₁based on the optical transition dipole strengths and assuming all other parameters are constant.

Initial devices we made were designed to couple to 𝑍₁-𝑌₂ optical transitions, but after some worry about the optical coherence due to the close proximity of𝑌₁and 𝑌₂, we switched to devices coupled to 𝑍₁-𝑌₁ optical transitions. Both transitions should be usable for transduction, but our 𝑍₁-𝑌₁ devices had better performance.

This is likely most attributed to more device iterations and improvements, while our 𝑍₁-𝑌₂device was made in the early days of our fabrication process development and did not have the luxury of improvements that were made over the years.

Table 3.5: 𝑍₁-𝑌₁spectroscopic parameters at𝐵_𝜃 =50° Parameter Value

𝑔_𝑑𝑐, 𝑍₁ 5.89 𝑔_𝑑𝑐,𝑌₁ 4.55 𝑑_{𝐵||𝑐,⊥}[10⁻³²C·m] 0.48 𝑑_{𝐵||𝑐,∥} [10⁻³² C·m] 3.26 𝑑_{𝐵||𝑎,⊥} [10⁻³²C·m] 1.58 𝑑_{𝐵||𝑎,∥} [10⁻³²C·m] 0.23 𝑑_{𝐸||𝑎,⊥}[10⁻³² C·m] 0.52 𝑑_{𝐸||𝑎,∥} [10⁻³²C·m] 0.86 Best Magnetic Field Orientation for 𝑍₁-𝑌₁

Based on the dipole moment product, for 𝑍₁-𝑌₁ the best magnetic field is at an angle of 35° from the c-axis. However, it is also beneficial to have the 𝑔_𝑑𝑐 for the ground and excited state to differ as much as possible which occurs at larger magnetic field angles (see Figure 3.5). Therefore, we decide to use a magnetic field angle of 50°as a balance of these two effects. At an angle of 50° from the c-axis, the dipole moment product is still near its maximum, so we only expect a decrease in transduction efficiency of∼25%, which is a small factor.

A summary of the magnetic field dependent 𝑍₁-𝑌₁ spectroscopic parameters for a magnetic field angle of 50°are summarized in Table 3.5. The𝑌₁label for the dipole moments was dropped, since we are only including the𝑍₁-𝑌₁transitions here.