Adiabatic Model

REI TRANSDUCTION THEORY

2.1 Adiabatic Model

In the adiabatic model (originally presented in Reference [98]), we operate in the regime where

𝛿_{𝜇, 𝑘} ≫

𝑔_{𝜇, 𝑘} ,

𝛿_{𝑜, 𝑘}

≫

𝑔_{𝑜, 𝑘} ,

𝛿_{𝜇, 𝑘}𝛿_{𝑜, 𝑘} ≫

Ω𝑜, 𝑘

2, such that we can adiabatically eliminate the excited states of the atomic system. For simplicity, we also neglect any atomic dephasing or energy dissipation and assume that T= 0 K (these assumptions will be dropped for subsequent models). Importantly, in this highly detuned regime used to satisfy the adiabatic condition, we expect to have reduced added noise from processes such as spontaneous emission and reduced absorption from parasitic ions (ions that are only coupled to one of the two cavities).

In the adiabatic limit, we have the following effective Hamiltonian:

𝐻_{𝑒 𝑓 𝑓}/ℏ =𝛿_𝑐,𝑜𝑎ˆ^†𝑎ˆ+𝛿_{𝑐, 𝜇}𝑏ˆ^†𝑏ˆ+

𝑁

∑︁

𝑘

−

𝛿_{𝜇, 𝑘}|𝑔_{𝑜, 𝑘}|²

𝛿_{𝑜, 𝑘}𝛿_{𝜇, 𝑘} − |Ω𝑜, 𝑘|²𝑎ˆ^†𝑎ˆ−

𝛿_{𝑜, 𝑘}|𝑔_{𝜇, 𝑘}|² 𝛿_{𝑜, 𝑘}𝛿_{𝜇, 𝑘} − |Ω𝑜, 𝑘|²

𝑏ˆ^†𝑏ˆ

+

Ω𝑜, 𝑘𝑔_{𝜇, 𝑘}𝑔^∗

𝑜, 𝑘

𝛿_{𝑜, 𝑘}𝛿_{𝜇, 𝑘} − |Ω𝑜, 𝑘|²𝑎ˆ^†𝑏ˆ +

Ω^∗_{𝑜, 𝑘}𝑔^∗

𝜇, 𝑘𝑔_{𝑜, 𝑘} 𝛿_{𝑜, 𝑘}𝛿_{𝜇, 𝑘} − |Ω𝑜, 𝑘|²𝑏ˆ^†𝑎ˆ

. (2.4) This effective Hamiltonian has four new terms, where the first two new terms correspond to the cavity mode-pulling from the atomic transitions and the last two terms correspond to an effective linear coupling between the two cavity modes ˆ𝑎 and ˆ𝑏, which we shall denote as 𝑆. We can further simplify the linear coupling coefficient due to the adiabatic elimination, such that:

𝑆=

𝑁

∑︁

𝑘

Ω𝑜, 𝑘𝑔_{𝜇, 𝑘}𝑔^∗

𝑜, 𝑘

𝛿_{𝑜, 𝑘}𝛿_{𝜇, 𝑘}

. (2.5)

Now we can write the effective Hamiltonian in a simplified beam splitter-like form:

𝐻_{𝑒 𝑓 𝑓}/ℏ = (𝛿_𝑐,𝑜−𝜔^′

𝑎)𝑎ˆ^†𝑎ˆ+ (𝛿_{𝑐, 𝜇}−𝜔^′

𝑏)𝑏ˆ^†𝑏ˆ +𝑆𝑎ˆ^†𝑏ˆ +𝑆^∗𝑏ˆ^†𝑎ˆ (2.6) where𝜔^′

𝑎 =Í^𝑁

𝑘

𝛿_{𝜇 , 𝑘}|𝑔_{𝑜, 𝑘}|² 𝛿𝑜, 𝑘𝛿𝜇 , 𝑘−|Ω𝑜, 𝑘|², 𝜔^′

𝑏 =Í^𝑁

𝑘

𝛿_{𝑜, 𝑘}|𝑔_{𝜇 , 𝑘}|²

𝛿𝑜, 𝑘𝛿𝜇 , 𝑘−|Ω𝑜, 𝑘|² are the cavity mode pulling frequencies for the optical and microwave cavity, respectively.

We can use the input-output formalism to relate the cavity fields to their respective input/output modes:

¤ˆ

𝑎 =−𝑖(𝛿_𝑐,𝑜−𝜔^′

𝑎)𝑎ˆ−𝑖 𝑆𝑏ˆ − 𝜅_𝑜 2 𝑎ˆ−√

𝜅_{𝑜,𝑖𝑛}𝐴_𝑖𝑛

¤ˆ

𝑏=−𝑖(𝛿_{𝑐, 𝜇} −𝜔^′

𝑏)𝑏ˆ −𝑖 𝑆^∗𝑎ˆ− 𝜅_𝜇

2 𝑏ˆ −√

𝜅_{𝜇,𝑖𝑛}𝐵_𝑖𝑛. (2.7) We can solve these equations in steady state and assume that we are only inputting a field into one cavity (i.e. 𝐴_𝑖𝑛or𝐵_𝑖𝑛 =0) and use 𝐴_𝑜𝑢𝑡 =√

𝜅_{𝑜,𝑖𝑛}𝑎for microwave to optical transduction (or𝐴_𝑜𝑢𝑡 =√

𝜅_{𝜇,𝑖𝑛}𝑏for optical to microwave transduction), such that the photon number efficiency is:

𝜂=

4𝑖 𝑆√ 𝜅_𝑜𝜅_𝜇 4𝑆²+ (2𝑖(𝛿_𝑐,𝑜−𝜔^′

𝑎) +𝜅_𝑜) (2𝑖(𝛿_{𝑐, 𝜇} −𝜔^′

𝑏) +𝜅_𝜇)

2

· 𝜅_{𝑜,𝑖𝑛} 𝜅_𝑜

· 𝜅_{𝜇,𝑖𝑛}

𝜅_𝜇 (2.8) From Equation 2.8, we are able to obtain an impedance matching condition when 𝑅 ≡ ^4|𝑆|2

𝜅_𝜇𝜅_𝑜 = 1, the input field and pump laser frequency are chosen such that 𝛿_𝑐,𝑜 = 𝜔^′

𝑎 and 𝛿_{𝑐, 𝜇} = 𝜔^′

𝑏 and the two cavities are perfectly over-coupled, which provides a theoretical path for unit transduction efficiency.

More generally, the transduction efficiency when the light is resonant with mode- pulled cavity frequencies is𝜂=

2𝑅 𝑅²+1

2

·^{𝜅𝑜,𝑖 𝑛}

𝜅𝑜

· ^{𝜅𝜇 ,𝑖 𝑛}

𝜅𝜇 , which results in for small𝑅(i.e.

𝑅 ≪ 1) the efficiency scales as𝜂 ∼ 𝑅². We can also define the internal efficiency to be𝜂_𝑖𝑛𝑡 =𝜂/_𝜅

𝑜,𝑖 𝑛

𝜅𝑜

· ^{𝜅𝜇 ,𝑖 𝑛}

𝜅𝜇

.

In Figure 2.2, the transducer internal efficiency and bandwidth are plotted using Equation 2.8 in the limit of 𝜅_𝜇 ≪ 𝜅_𝑜 and the light frequencies is at the optimal detuning. The bandwidth is the 3 dB bandwidth of the transducer efficiency. When 𝑅 is small (i.e. 𝑅 ≪ 1), the bandwidth follows the bandwidth of the narrowest cavity, which in this case is the microwave cavity. When 𝑅 is large (i.e. 𝑅 ≫ 1), the bandwidth can be increased, but at the cost of reduced efficiency. In practice, it would be more beneficial to work in a regime with larger cavity bandwidths (i.e.

larger𝜅) to reduce𝑅to unity and maximize the transducer bandwidth that way.

10^-2 10^-1 10⁰ 10¹ 10^-3

10^-2 10^-1 10⁰

10⁰ 10¹ 10²

Bandwidth/

Figure 2.2: Adiabatic model efficiency and bandwidth as a function of the impedance matching parameter, 𝑅, in the limit of𝜅_𝜇 ≪ 𝜅_𝑜.

Linear Transduction Coupling Coefficient –𝑆

In order to maximize our efficiency it is important to maximize the linear trans- duction coupling coefficient,𝑆, and have high quality factor optical and microwave cavities. 𝑆 was defined in Equation 2.5, but we can take a closer look at different contributions of the transducer affect the parameter,𝑆.

First, we can write down𝑆more explicitly:

Ω𝑜, 𝑘 =√︁

⟨𝑛_𝑜⟩

√︂ 𝜔 2ℏ𝜖₀

𝑑₃₂

√ 𝑉_𝑒

𝐸_{𝑑 ,𝑜,2}(𝑟_𝑘)

√︁(𝜖_𝑟(𝑟_𝑘) |𝐸_𝑜,2|²)𝑚 𝑎𝑥

(2.9)

𝑔_{𝜇, 𝑘} =

√︂𝜇₀𝜔 2ℏ

𝜇₂₁

√ 𝑉_𝑚

𝐵_{𝑚, 𝜇}(𝑟_𝑘)

|𝐵_{𝜇,𝑚 𝑎𝑥}| (2.10)

𝑔_{𝑜, 𝑘} =

√︂ 𝜔 2ℏ𝜖₀

𝑑₃₁

√ 𝑉_𝑒

𝐸_{𝑑 ,𝑜,1}(𝑟_𝑘)

√︁(𝜖_𝑟(𝑟_𝑘) |𝐸_𝑜,1|²)𝑚 𝑎𝑥

(2.11)

where 𝑑₃₁ is the electric dipole moment of optical transition |1⟩ ↔ |3⟩, 𝑑₃₂ is the electric dipole moment of optical transition|2⟩ ↔ |3⟩, and𝜇₂₁is the dipole moment of microwave transition|1⟩ ↔ |2⟩. The subscripts𝑚, 𝑑correspond to the field along the magnetic or electric dipole direction and the subscripts 1,2 are used to note the different electric fields between the pump field and the transduction signal field. We have assumed that an electric dipole moment for the optical transitions here, but similar expressions can be shown for a magnetic dipole transition (i.e. it is the same form as the microwave magnetic dipole transition).

If we assume no spectral and position correlations among the ion ensemble and assume the ion density is sufficiently large, then we can replace the sum with integrals and arrive at:

𝑆=√

𝜔_𝑜𝜔_𝜇𝛼 𝐹Ω𝑚 𝑎𝑥 (2.12)

where 𝛼 contains the spectroscopic parameters of the atomic transitions, 𝐹 is the mode overlap, andΩ𝑚 𝑎𝑥 is the maximum optical Rabi frequency of the pump field.

𝛼can be expressed explicitly as:

𝛼=

√︂ 𝜇₀ ℏ²𝜖₀

𝑑₃₁𝜇₂₁𝜌

∫ ∞

𝜖𝜇

𝐷_𝜇(𝛿_𝜇) 𝛿_𝜇

𝑑 𝛿_𝜇

∫ ∞

𝜖𝑜

𝐷_𝑜(𝛿_𝑜) 𝛿_𝑜

𝑑 𝛿_𝑜 (2.13) where 𝜌 is the number density of atoms and 𝐷_𝜇 and 𝐷_𝑜 are the inhomogeneous broadening distribution functions of the microwave and optical transitions, respec- tively.

𝐹can be defined as:

𝐹 = 1

√︁𝑉_𝑜𝑉_𝜇

∫ 𝐵_𝜇(𝑟)𝐸_𝑜,1(𝑟)𝐸_𝑜,2(𝑟) 𝐵_{𝜇,𝑚 𝑎𝑥}√︁

(𝜖_𝑟(𝑟_𝑘) |𝐸_𝑜,1|²)𝑚 𝑎𝑥𝐸_{𝑜,2,𝑚 𝑎𝑥𝑌 𝑉 𝑂} 𝑑𝑉

. (2.14)

Ω𝑚 𝑎𝑥 can be defined as:

Ω𝑚 𝑎𝑥 =√︁

⟨𝑛_𝑜⟩

√︂ 𝜔_𝑜 2ℏ𝜖₀

𝑑₃₂

√ 𝑉_𝑒

𝐸_{𝑜,2,𝑚 𝑎𝑥}(𝑟)

√︁(𝜖_𝑟(𝑟) |𝐸_𝑜,2|²)𝑚 𝑎𝑥

. (2.15)

We can see from Equation 2.12 what the important spectroscopic properties of the rare-earth ion ensemble are and also what mode profiles for the microwave and optical cavity should be used in order to increase𝑆.

In order to have high transduction efficiency, we want the rare-earth ion ensemble to have large optical and microwave dipole moments (i.e. 𝑑₃₁, 𝑑₃₂ & 𝜇₂₁), high density, and narrow inhomogeneity in the optical and spin transitions (as the light- atom detunings should be large compared to the transition inhomogeneities). That is, in the low efficiency limit, the efficiency scales with the spectroscopic parameters as:

𝜂 ∝ 𝜁 =

𝑑₃₁𝑑₃₂𝜇₂₁𝜌 Δ𝑜Δ𝜇

2

(2.16) whereΔ𝑜andΔ𝜇are the optical and microwave transition linewidths.

For the mode profiles of our three fields, we want to maximize the mode overlap between them. In a low quality factor optical cavity (i.e. 𝜅_𝑜 > 𝜔_𝜇), the two optical fields can couple to the same cavity mode, which simplifies this a bit. Due to the five orders of magnitude difference in wavelength between microwave and optical photons, increasing the overlap tends to require squeezing the microwave mode as much as possible and increasing the optical mode to match the size of the microwave mode.

It is also worth noting thatΩ ∼ ^√1

𝑉𝑜

, which means we can obtain larger pump Rabi frequencies when we decrease the mode volume of the optical cavity (assuming all other parameters are constant). AlthoughΩcan be increased by using more optical power, for quantum transduction it is desired to minimize the optical power in order to limit device heating or added noise, so there are practical advantages to using a smaller mode volume for the optical pump.