Calculated 𝛼 Parameter

REI TRANSDUCTION THEORY

ER 3 YVO 4 SPECTROSCOPY

3.5 Calculated 𝛼 Parameter

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.

Table 3.6: Er³⁺:YVO₄𝛼parameters

𝑍₁-𝑌₁ 𝑍₁-𝑌₂ Density, 𝜌[cm⁻³] 5.4·10¹⁸ 5.4·10¹⁸ Optical Inhomogeneity,Δ𝑜[MHz] 260 225

Spin Inhomogeneity,Δ𝜇[MHz] 65 65

Optical Dipole Type 𝜎_{𝑀 𝐷} 𝜋_{𝐸 𝐷}

Optical Dipole Strength [C·m] 3.26·10⁻³² 2.08·10⁻³² Spin Dipole Strength [𝜇_𝐵] 3.55 (2.29) 3.55 (3.37) 𝛼[s⁻¹] 1.54·10⁻¹⁰ 1.14·10⁻¹⁰

The 𝛼 parameter for 𝑍₁-𝑌₁ and 𝑍₁-𝑌₂ are very similar, with 𝑍₁-𝑌₁ being slightly larger. It should be noted again that there is another optical dipole strength term in the optical Rabi frequency term, which makes 𝑍₁-𝑌₂ slightly more favorable.

Another difference is the dipole type. 𝑍₁-𝑌₁ is best with a 𝜎_{𝑀 𝐷} optical dipole, while𝑍₁-𝑌₂is best with a𝜋_{𝐸 𝐷} optical dipole. Therefore, the coupling to the optical resonator will require different designs, which will be discussed more in Chapter 4.

Comparison to Other Materials

It is useful to compare the 𝛼 value we get for Er³⁺:YVO₄ with other rare-earth ion materials. Unfortunately, it is not necessarily very straight forward to find the optimal 𝛼 for a given material. As shown above, several parameters have strong dependence on the magnetic field direction used for the experiment. Also,𝛼∝ _Δ^𝜌

𝑜Δ𝜇, which means it is important to find the highest density possible that also has most narrow linewidths. This complete type of analysis has not really been done on many (if any) materials, so we can only compare with experimental data we are aware about. In other words, there are possibly (or likely) other rare-earth ion/host materials that perform even better and using them will only improve things, but additional spectroscopy is needed to identify them.

In terms of known materials, we can compare to ¹⁷¹Yb:YVO₄ [57, 108] and Er³⁺:YSO [54, 98]. For ¹⁷¹Yb:YVO4, 𝛼 = 1.6 · 10⁻⁸ using the spectroscopic parameters provided in the references [57, 108]. This shows that ¹⁷¹Yb:YVO₄ is another promising material. However, it should be noted that most of the improve- ment is due to the narrow spin inhomogeneity of¹⁷¹Yb:YVO₄(0.1 MHz compare to 30 MHz). When coupling with an on-chip superconducting microwave resonator, the spin inhomogeneity tends to increase due to magnetic field inhomogeneity from the gaps in the superconducting film. ¹⁷¹Yb:YVO₄ can operate at small magnetic fields if a small microwave frequency is acceptable, so potentially there are regimes where this is not a problem.

For Er³⁺:YSO,𝛼 =1.4·10⁻¹⁰ using the spectroscopic parameters assumed in the reference [98] or𝛼 = 5.4· 10⁻¹³ using the spectroscopic parameters in [54]. The difference in these values largely arises from the difference in the spin linewidth, which is a factor of 25. We should also note that these Er³⁺:YSO parameters assume a much smaller Er density (10 ppm), which is a factor of 56 smaller than our Er³⁺:YVO₄calculations.

3.6 ¹⁶⁷Er:YVO₄Hyperfine Transitions

Although we do not intend to use¹⁶⁷Er isotopes for transduction, their presence in natural abundance Er³⁺:YVO₄crystals can result in their transitions overlapping or being near transitions of interest. Therefore, it is useful to determine their transition frequencies and see how close they are to the even isotope spin transitions that we intend to use. ¹⁶⁷Er has a nuclear spin I =7/2, which increases the number of states from two (in the case of zero nuclear spin isotopes) to 16 states and makes it much more complicated.

The ground state¹⁶⁷Er can be modelled using the following Hamiltonian [66]:

𝐻_{𝑒 𝑓 𝑓} =𝜇_𝐵B·g·S+I·A·S+I·P·I (3.30) where the first term is the electron spin term, the second term represents the hyperfine interaction between the electron and nuclear spin, and the last term describes the nuclear quadrupole interaction.

Due to the axial site symmetry, the spin Hamiltonian can be simplified to [66, 105]:

𝐻_{𝑒 𝑓 𝑓} =𝜇_𝐵

𝑔_∥𝐵_𝑧𝑆_𝑧+𝑔_⊥(𝐵_𝑥𝑆_𝑥 +𝐵_𝑦𝑆_𝑦) +𝐴_∥𝑆_𝑧𝐼_𝑧+ 𝐴_⊥(𝑆_𝑥𝐼_𝑥+𝑆_𝑦𝐼_𝑦) +𝑃_∥

𝐼²

𝑧 − 1

3𝐼(𝐼+1)

(3.31)

where𝑔_∥ =3.544,𝑔_⊥ =7.085, |𝐴_∥| =1.226·10⁶[cm⁻¹], |𝐴_⊥|=2.491·10⁶[cm⁻¹] and|𝑃_∥|=1.39·10⁵[cm⁻¹]for¹⁶⁷Er³⁺:YVO4using values from Ranon et al. [105].

Figure 3.8: ¹⁶⁷Er³⁺:YVO₄ hyperfine transitions for 𝐵_𝑑𝑐 50° from the c-axis. The blue line is the 𝑍₁ electron spin transition and the red line is the𝑌₁ electron spin transition.

Solving the Hamiltonian, we can determine the energy of all the states. The hyperfine transitions for𝐵_𝑑𝑐being 50°from the c-axis are shown in Figure 3.8. The transition strengths was evaluated by taking the inner product of the initial and final states mediated by both a𝜎_𝑥⊗𝐼_𝑁 (and𝜎_𝑧⊗𝐼_𝑁) operator to account for the selection rules for 𝐵_{𝑎 𝑐} ∥a (and 𝐵_{𝑎 𝑐} ∥c) [66, 103]. In our on-chip resonator, we expect 𝐵_{𝑎 𝑐} to be along both crystal directions, so we consider both direction here.

The blue line is the𝑍₁electron spin transition and the red line is the𝑌₁electron spin transition to show the proximity of these transitions to the hyperfine transitions. It is evident that at particular magnetic fields and particular microwave frequencies, we have overlap between the even isotope spin transitions and hyperfine transitions.

This will be important later when we do the experiment, so this will be referred to again in that section and does suggest that there can be improvements for an isotopically purified zero spin Er³⁺:YVO₄material.