Perspectives on quantum transduction

Cold, optically or magnetically trapped neutral atoms offer a pristine system in which transitions in both the microwave and optical regimes can be driven with high fidelity. As such they are a natural setting for the generation of the nonlinearities required for single-photon microwave-to-optical transduction [38]. In addition the availability of atomic states with long coherence times makes it possible to combine transduction of quantum optical fields with quantum memories that will form the basis for quantum network nodes.

While electric dipole atomic transitions at optical frequencies are relatively strong, at microwave frequencies the transitions that couple to atomic ground states are of magnetic dipole character, and are much weaker. Consequently, a pronounced challenge with this platform is the ability to engineer a sufficiently high vacuum Rabi frequency of the microwave transition in a physical system that is simultaneously compatible with laser cooling and trapping and a cryogenic environment. To meet this challenge either the atom has to be positioned extremely close to the microwave source or resonant cavity, or an ensemble of N atoms should be used to gain a $\sqrt{N}$ collective enhancement of the coupling.

The alternative is to excite atoms to Rydberg states which have microwave frequency electric dipole allowed transitions with very large dipole moments, such that strong interactions of single atoms with single microwave photons in free space are possible. This characteristic of Rydberg atoms was recognized long ago and exploited for detection of single quanta [39, 40]. Since the atom-microwave coupling via Rydberg states is approximately a factor of 10⁶ larger [41], than the coupling via a hyperfine transition in the ground state manifold, a single Rydberg-excited atom can provide the same interaction strength as 10¹² ground state atoms.

We proceed by highlighting the two primary level schemes employed in transduction experiments with cold, neutral atoms. The first and simplest approach is based on three-wave mixing [26]. It employs a microwave transition between a pair of hyperfine ground states in an alkali atom; both of which are optically coupled to the same electronically-excited state (see figure 1(a)). The microwave field $\hat{b}$ (${\hat{b}}^{\dagger }$) annihilates (creates) an excitation of the hyperfine transition $| 1\rangle -| 2\rangle$, classical pump field with the Rabi frequency Ω₂₃ drives the $| 2\rangle -| 3\rangle$ transition and the optical field $\hat{a}$ (${\hat{a}}^{\dagger }$) annihilates (creates) an excitation of the adjacent optical transition $| 1\rangle -| 3\rangle$. Generically, there will be a finite detuning between these fields and their corresponding atomic transitions which we call δ₂ and δ₃, respectively. This system is amenable to pulsed or continuous-wave (CW) operation. Note that the atomic population is predominantly in $| 1\rangle$, which is directly attached to both the microwave and optical transitions.

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Crucially, the magnetic dipole moment of the hyperfine transition $| 1\rangle -| 2\rangle$ is only μ ≈ 1 μ_B (Bohr magneton). Accordingly, even if the atom is ≈5 μm from the surface of a superconducting resonator, the vacuum coupling strength for a single atom would only be g_μ/2π ≈ 50 Hz [26, 42, 43]. Since superconducting quantum circuits operate with ≈MHz single-photon bandwidths [44], an ensemble of N ≈ 10⁹ atoms would be required to obtain a sufficiently large $\sqrt{N}$ collective enhancement [25]. While such values of N can easily be reached with atomic impurities in solid-state crystals [45, 46], this requirement is very daunting for the cold, trapped atom-based platform.

The second and more commonly pursued approach to transduction with cold, trapped atoms relies on microwave transitions between two highly-excited Rydberg states [41, 47–52]. (Note that some of these references focus primarily on the microwave-coupling step, which is certainly the most challenging.) Such a transition is employed in a four-wave-mixing scheme (see figure 1(b)), or in some cases even six- or seven-wave-mixing. Rydberg–Rydberg transitions have extremely large electric dipole moments that scale as n²e a₀ with n the principal quantum number, e the electronic charge, and a₀ the Bohr radius. The microwave frequency of the S−P transition for a given n scales as 1/n³, so the desired frequency can be selected by choosing the principle quantum number n of the Rydberg levels appropriately. For a microwave frequency of f_μ ≈ 5 (17) GHz in cesium (Cs), n ≈ 90 (60) would be selected. Typical electric dipole moments of such transitions are d ≳ 1000 e a₀, such that the coupling strength of a single atom at a similar distance of ≈5 μm is g_μ/2π ≈ 1 MHz [41, 50]. Hence, an ensemble of atoms may not be necessary, and several efforts focus on the use of a single atom.

However, magneto-optical cooling and optical trapping of atoms—one or many—within several μm of a superconducting waveguide is highly non-trivial. Recall that the superconducting state is easily destroyed upon the absorption of excess photons or magnetic fields [44]. Moreover, the use of Rydberg atoms near surfaces introduces additional challenges associated with their large DC polarizabilities from residual electric fields [53]. While schemes have been devised to reduce the sensitivity of the Rydberg states to these fields [54], alternative approaches in which the atoms can remain far from any surfaces are highly desirable.

Of course as the atom(s) are moved further from a surface, the coupling strength of the microwave transition decreases rapidly. Naively, one might assume that this decrease in the coupling strength can be compensated by a collective enhancement via the use of an atomic ensemble. However, here in lies a subtle yet crucial point. In contrast to three-wave mixing schemes considered above, in the Rydberg-atom-based scheme operating in the single-photon regime there is no steady-state atomic polarization in either of the states coupled to the microwave field (i.e. $| 2\rangle$ and $| 3\rangle$). This is precluded by the relatively short lifetimes, scaling approximately as n³ (≈1 ms at n = 90 in a 4 K cryostat [55]). Thus, there is no collective enhancement of the microwave transition in the single-photon regime. Reference [56] provides a detailed analysis of this effect, and proposes a compromise between coupling strength and distance from surfaces. Note, however, that a collective enhancement could be engineered in a resonant, pulsed regime [57]. Even so, it is difficult to continuously excite a dense sample of Rydberg atoms in the multi-photon regime due to the blockade effect [53].

It is desirable that the wavelength of the optical photon lie in the telecommunications window (∼1.25 μm to ∼1.65 μm) for more efficient photon transfer over long fiber links. Generally, most atom and atom-like emitters have strong optical transitions in the visible band where the wavelengths are much shorter, and frequency conversion into the telecom window is often required. Erbium ions are a notable exception to this trend with transitions at 1.54 μm; however, these transitions are very weak and must be Purcell- or collectively-enhanced with an optical resonator to achieved desired bandwidths compatible with superconducting circuits. In alkali atoms telecom-band transitions are only available between high-lying states, so complex schemes involving six or seven internal levels are required to make use of them for transduction [41, 49–52]. Recently, a CW four-wave-mixing scheme based on an ensemble of alkaline-earth(-like) ytterbium (Yb) atoms was proposed [56] in which a strong transition in the telecommunication E-band at 1389 nm is employed [58].

Experimental progress on cold atom-based transduction lags substantially behind leading approaches in this field, primarily because of its relative complexity. Early work in this field focused on atomic beams [48, 59–61] rather than cold, trapped atoms; however, a number of efforts based on the latter are now in progress. References [26, 42, 43, 62–64] provide an overview of approaches based on three-wave mixing using a ground-state hyperfine transition. References [41, 47–52] provide an overview of ideas and recent experimental efforts with Rydberg states, but we emphasize that this list is not exhaustive.

The first demonstration of the microwave-to-optical conversion with Rydberg atoms to our knowledge was performed in 2018 in [51]. However, the efficiency was low (η ∼ 0.003) and the conversion was performed in the classical regime with many extra photons. Nevertheless, a respectable conversion bandwidth of $\bar{{\rm{\Gamma }}}\approx 4\,\mathrm{MHz}$ was observed. In an improvement upon this first result, the same group demonstrated a higher efficiency of η ∼ 0.05 [52], albeit still in the classical regime with many excess photons. Using numerical simulations of their system the authors conclude that the conversion efficiency could be increased up to η ≈ 0.7 with corresponding conversion bandwidth of $\bar{{\rm{\Gamma }}}\approx 15\,\mathrm{MHz}$. While there have been few demonstrations of microwave to optical transduction with Rydberg atoms to date, more effort has focused on the related problem of coupling atoms to microwave fields—specifically superconducting resonators. Reference [25], for instance, demonstrated the strong coupling of an ultracold gas to a superconducting waveguide cavity using a hyperfine transition in the ground state (not Rydberg). They observed g_eff/2π ∼ 40 kHz, which is large compared to the cavity linewidth κ/2π ∼ 7 kHz.

Ensembles of REIs doped into transparent crystals are an appealing platform to devise transducer interconnects. They are well known for their long optical coherence times and form the basis for solid state implementations of optical quantum memories. More recently their spin coherence times were measured and analyzed in more detail, and coherent ensemble coupling to microwave cavities was demonstrated [45] allowing the development of transducer proposals in this medium. First transducer proposals for REI ensembles were made a few years ago [27, 28]. Both proposal suggest to use Er³⁺ ions doped into yttrium orthosilicate (YSO) crystal due to its prominent optical transition in the telecom wavelength region at 1536 nm. With half-integer total spin Er³⁺ belongs to the so called Kramers ions and as such has a doubly degenerate ground and optical excited state. This degeneracy could be lifted by applying an external static magnetic field allowing for a magnetic dipole transition in the microwave range. Another advantage of Er³⁺ is its relatively high magnetic dipole moment, up to 15 μ_B in YSO host crystal [65]. However, magnetic dipole transitions are in general considerably weaker than electric dipole transitions and a high Q microwave resonator as well as large number ensembles have to be used to enhance the coupling to microwave fields.

The system can be considered as an ensemble of three level atoms in Λ-type configuration, with two lower lying spin levels and one common optically excited state, similar to the three-wave mixing scheme for the trapped atoms (se figure 2). But unlike the trapped atom ensembles, the transitions in REI ensembles have large inhomogeneous broadening due to slightly different local environment in the host crystal. One way to mitigate the detrimental effects of large inhomogeneous broadening is to use optical and microwave cavities that are far detuned from the resonant transitions (as suggested in [28]), with detunings δ₃ and δ₂ being larger than the inhomogeneously broadened linewidths of corresponding transitions. In this case the matter part can be adiabatically eliminated from the equations of motion and one is left with an effective nonlinear interaction between the classical optical field, quantum optical cavity field and quantum microwave cavity field with the interaction Hamiltonian (1), where Ω is the Rabi frequency of a classical field driving the $| 2\rangle -| 3\rangle$ transition, and $\hat{a}$ (${\hat{a}}^{\dagger }$) and $\hat{b}$ (${\hat{b}}^{\dagger }$) are the annihilation (creation) bosonic operators for the optical and microwave cavity respectively. The effective coupling strength g_eff depends on the collective coupling strength of the optical and microwave transitions and is inversely proportional to the detunings. The fact that the detunings should be large results in the usually weak effective coupling strength g_eff and efficient conversion requires large cooperativity factors for both cavities.

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The scheme above, with classical field being constantly present, is best suited for a conversion of CW fields. But pulsed operations could be implemented in the same medium as well, as was shown in [27]. In this proposal the conversion of an optical photon to a microwave photon is achieved by first mapping an incoming single photon pulse onto an optical matter excitation and subsequently transferring this excitation to a spin excitation by means of a series of classical laser pulses. Afterwards the spin excitation is resonantly coupled to a microwave cavity which leads to a coherent mapping of the spin excitation to a microwave photon. To store the incoming photon into a collective matter excitation the authors propose to use gradient echo quantum memory protocol, a type of a controlled reversible inhomogeneous broadening (CRIB) quantum memory protocol [66], which together with atomic frequency comb protocol [67] belongs to echo type quantum memories and was especially designed for systems with long optical coherence times and large inhomogeneous broadening, such as rare-earth doped crystals. The idea of the gradient echo memory protocol is to induce controllable inhomogeneous broadening in the system by applying a field gradient. Large inhomogeneous broadening leads to a fast dephasing (much faster than the excited state lifetime) of the collective excitation that couples to the electromagnetic field resulting in the effective storage of the photon. In order to transport this optical excitation to a rephased spin excitation a series of π-pluses is applied between the spin and excited levels. The timing of the pulses is such that at the end of the rephasing procedure the system is left in the collective spin state that interacts resonantly with the microwave cavity. The overall efficiency of the protocol is given by η = η_Sη_T, where η_S is the storage efficiency of the optical photon into the symmetric spin excitation and η_T describes the transport efficiency of this spin excitation to a microwave photon inside the cavity. The storage efficiency is bounded by the spatial overlap of the spin excitation with the microwave cavity mode. Using gradient echo scheme where the field gradient is applied along the propagation direction of the optical photon this overlap can be maximized by spectral tailoring of the incoming photon, since in this configuration the spatial profile of the spin excitation is given by the frequency spectrum of the optical photon.

A more recent proposal [68] suggest modification of the proposal by O'Brien et al [27] by using the Zeeman levels of the optically excited state instead of the ground state. The modest ratio between the coupling strength and the decoherence rate limited the conversion efficiency in the original proposal [27] to ∼90%. Using the sublevels in the optically excited state manifold have advantage of longer spin coherence times due to reduced spin-spin interaction with the neighboring ions and could potentially improve the overall conversion efficiency near unity.

Most experimental investigations for light–matter coupling using rare-earth crystals have focused on demonstrations of quantum memories for light or microwaves. There are several reviews of quantum memories for light [69, 70], and many works on quantum-oriented microwave or radio-frequency coupling using bulk and stripline resonators [45]. Nevertheless, there has been little experimental work using enesmbles of rare earth ions for microwave to optical transduction.

The first efforts came from the Longdell group of Otago that followed a route based on off-resonant fields and three-wave mixing [28, 29]. In one of their experiments a cylindrical sample of erbium doped YSO crystal sat inside a shielded loop gap resonator and optical Fabry–Perot resonator in a superconducting magnet at 4.6 K under 146 mT field. Using this system it was possible to demonstrate microwave to optical telecom-wavelength conversion for classical input fields with an efficiency of order 10⁻⁵ [71]. The optical cavity in the system has provided an enhancement in the conversion efficiency of nearly 10⁴ compared to its counterpart without an optical cavity [29]. The authors predict that by matching the impedance of the microwave and optical cavities as well as by lowering the temperature to mK one should obtain near unit conversion efficiency. A more recent proposal from the group suggests to use fully concentrated crystals where rare-earth ions are part of the crystal structure, rather than being dopants [33, 34]. Using such crystals allows one to increase the concentration of REI without significantly broadening the optical and the spin transition, as is the case in the crystals with doped REI. At high concentrations the strong interaction between the REI can lead to a magnetic ordering of the system at low temperatures, and the interaction with microwave photons can be described in terms of collective excitations (magnons) of the system (see section 5 for more details on the magnon based transduction).

Another experimental effort comes from the Faraon group at Caltech that has pioneered approaches using nanophotonic waveguide and cavities used focused ion beam milling of yttrium vanadate (YVO) crystals. For transduction, their approach involves using Yb ions doped into YVO under weak magnetic field. Yb ions have a non-zero nuclear spin, which results in a simple hyperfine structure (characterized by a nuclear spin of 1/2) that also features a long coherence lifetime due to zero first order interaction of the spin with its surrounding magnetic field bath. In their design the nanophotonic components are positioned within microwave coplanar waveguides and cavities that allow the necessary microwave coupling [72]. In this proof-of-principle experiment coherent microwave-to-optical conversion with an efficiency of the order of 10⁻¹³ was demonstrated and a strategy to improve this efficiency up to 30% was discussed.

Before presenting the recent advances of the optomechanical based photon conversion we theoretically explain how the mechanical resonator facilitates the photon conversion between microwave and optical domains. Figure 3(a) shows the modes coupling diagram of a double-cavity optomechanical system in which a microwave resonator mode C₁ with resonance frequency ω_c,1 and an optical cavity mode C₂ with resonance frequency ω_c,2, are simultaneously coupled to the vibrational mode of a mechanical resonator M with frequency ω_m. In figure 3(c) we schematically show a circuit representing this mode coupling in which a mechanical resonator forms one of the mirrors of the optical cavity while it capacitively coupled to a superconducting microwave resonator. The Hamiltonian describing this tripartite interaction is given by [73]

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}\hat{H} & = & \hslash {\omega }_{m}{\hat{b}}^{\dagger }\hat{b}+\hslash \displaystyle \sum _{j=1,2}\left[{\omega }_{c,j}{\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}+{g}_{0,j}({\hat{b}}^{\dagger }+\hat{b}){\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}\right.\\ & & +\left.\,{\rm{i}}\,{{ \mathcal E }}_{j}({\hat{a}}_{j}^{\dagger }{{\rm{e}}}^{-{\rm{i}}{\omega }_{d,j}t}-{\hat{a}}_{j}{{\rm{e}}}^{{\rm{i}}{\omega }_{d,j}t})\right],\end{array}\end{array}\end{eqnarray} \tag{ 2 }$$

where $\hat{b}$ is the annihilation operator of the mechanical resonator, ${\hat{a}}_{j}$ is the annihilation operator for resonator j whose coupling rate to the mechanical resonator is g_{0, j}. As shown in figure 3(b) the optical cavity and microwave resonator are driven by external coherent pumps with amplitude ${{ \mathcal E }}_{j}$ and frequencies ${\omega }_{{d},j}$ [74].

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Moving in the interaction picture with respect to ${\hslash }{\omega }_{{d},1}{\hat{a}}_{1}^{\dagger }{\hat{a}}_{1}+{\hslash }{\omega }_{{d},2}{\hat{a}}_{2}^{\dagger }{\hat{a}}_{2}$, neglecting terms oscillating at $\pm 2{\omega }_{{d},{j}}$, and linearized the Hamiltonian by expanding the resonator modes around their steady-state field amplitudes, ${\hat{c}}_{j}={\hat{a}}_{j}-\sqrt{{n}_{j}}$, where n_j ≫ 1 is the mean number of intracavity photons induced by the cavity pumps [74, 75], result in the linearized system Hamiltonian

$$\begin{eqnarray}&&\hat{H}={\hslash }\sum _{j=1,2}{G}_{j}(\hat{b}{{\rm{e}}}^{-{\rm{i}}{\omega }_{m}t}+{\hat{b}}^{\dagger }{{\rm{e}}}^{{\rm{i}}{\omega }_{m}t})({\hat{c}}_{j}^{\dagger }{{\rm{e}}}^{{\rm{i}}{{\rm{\Delta }}}_{j}t}+{\hat{c}}_{j}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{j}t}),\end{eqnarray} \tag{ 3 }$$

where ${{\rm{\Delta }}}_{j}={\omega }_{c,j}-{\omega }_{{d},j}$ is the cavity/resonator-pump detuning and ${G}_{j}={g}_{0,j}\sqrt{{n}_{j}}$ is the multiphoton optomechanical cavity rate. By selecting the detuning parameter we can either choose the beam splitter or parametric like interaction in the optomechanical system. By setting the effective resonator detunings so that Δ₁ = −Δ₂ = −ω_m and neglecting the terms rotating at ±2ω_m, the Hamiltonian (3) reduces to

$$\begin{eqnarray}&&{\hat{H}}_{\mathrm{TMS}}={\hslash }{G}_{1}({\hat{c}}_{1}\hat{b}+{\hat{b}}^{\dagger }{\hat{c}}_{1}^{\dagger })+{\hslash }{G}_{2}({\hat{c}}_{2}{\hat{b}}^{\dagger }+\hat{b}{\hat{c}}_{2}^{\dagger }),\end{eqnarray} \tag{ 4 }$$

The first two terms of the above Hamiltonian are responsible for generating entanglement between photonic excitation of the microwave mode C₁ and mechanical mode M while the last two terms represent a beam splitter like interaction which exchanges the excitation between the optical mode C₂ and mechanical mode M. This specific form of the interaction can be used to generate entanglement or two mode squeezing between the output radiation of the cavities [76–78]. This type of interaction can be used for high-fidelity quantum states transfer between optical and microwave fields in forms of continuous variable quantum teleportation [18].

On the other hand, by selecting the effective resonator detunings Δ₁ = Δ₂ = ω_m and neglecting the terms rotating at ±2ω_m, the Hamiltonian (3) reduces to [20]

$$\begin{eqnarray}&&{\hat{H}}_{\mathrm{BS}}={\hslash }{G}_{1}({\hat{c}}_{1}{\hat{b}}^{\dagger }+\hat{b}{\hat{c}}_{1}^{\dagger })+{\hslash }{G}_{2}({\hat{c}}_{2}{\hat{b}}^{\dagger }+\hat{b}{\hat{c}}_{2}^{\dagger }),\end{eqnarray} \tag{ 5 }$$

indicates beam splitter-like interaction between mechanical degree of freedom and microwave resonator (optical cavity) mode, appropriates for the photon conversion between microwave resonator and optical cavity. In the photon transduction process, the microwave photons indicated by ${\hat{a}}_{1}$ are down-converted into the mechanical mode at frequency, i.e. ${\hat{a}}_{1}\mathop{\longrightarrow }\limits^{\hat{H}\propto {\hat{a}}_{1}{\hat{b}}^{\dagger }}\hat{b}$. Next, during an up-conversion process the mechanical mode transfers its energy to the optical cavity mode ${\hat{a}}_{2}$, i.e. $\hat{b}\mathop{\longrightarrow }\limits^{\hat{H}\propto \hat{b}{\hat{a}}_{2}^{\dagger }}{\hat{a}}_{2}$. This process is bidirectional which means the photons of the optical mode can be converted to the microwave mode by reversing the conversion process.

The photon conversion efficiency between the outputs of the microwave resonator and the optical cavity in the steady state and in the weak coupling regime is given by [13, 20]

$$\begin{eqnarray}&&\zeta =\displaystyle \frac{4{C}_{1}{C}_{2}}{{(1+{C}_{1}+{C}_{2})}^{2}},\end{eqnarray} \tag{ 6 }$$

where ${C}_{i}=\tfrac{4{G}_{i}^{2}}{{\kappa }_{i}{\gamma }_{m}}$ is the optomechanical cooperativity for cavity i = 1, 2 in which κ_i is the total damping rate of the microwave and optical cavities, and γ_m is the damping rate of the mechanical resonator. Note that in equation (6) we ignore the internal losses of the optical cavity and microwave resonator. In the limit of equal and large cooperativity C₁ = C₂ and C_i/n_m ≫ 1, the unity coherent photon conversion can be achieved ζ = 1 where n_m is the thermal occupation of the mechanical mode. The bandwidth of the conversion is set by the total mechanical damping Γ = γ_m(1+C₁+C₂) which is the total back-action-damped linewidth of the mechanical resonator.

Among the initial experiments, Bagci et al [17] demonstrated a strongly-coupled opto-electro-mechanical transducer using an electrostatic nanomembrane that is displaced out-of-plane using a radio-frequency resonance circuit and is simultaneously coupled to light reflected off its surface. The mechanical resonator used in this experiment was a 500 μm square SiN membrane coated with Al. The radio-frequency signals are detected as an optical phase shift with quantum-limited sensitivity. Thermal noise fluctuation and the quantum noise of the light are the two major sources of the noise which both dominated the Johnson noise of the input.

Andrews et al [16] have shown bidirectional transduction with 10% efficiency overall albeit with about 1700 noise quanta at cryogenic temperatures. Their system employs a thin SiN membrane that acts as one of the mirrors of an optical Fabry–Perot cavity while being capacitively coupled to a superconducting microwave resonator. The vibration of the membrane simultaneously modulates both optical cavity and microwave resonator which consequently transfers the excitation between microwave and optical modes. This measurement has been done in near resolved-sideband regime in which the mechanical frequency ${\omega }_{m}$ exceeds the damping rates of the microwave resonator κ₁ and the optical cavity κ₂ set by 4ω_m > {κ₁, κ₂}. The conversion efficiency in this experiment was limited by the loss of the microwave resonator and the imperfect optical mode matching. The thermal vibrational noise at 4 K temperature as well as the spurious mechanical modes in the membrane were the two main sources of the noise in this experiment. These issues have been resolved in the recent experiment from the same group [79]. Improving the sample design to remove the unwanted mechanical modes, having low-loss optical cavity and microwave resonator, larger opto and electromechanical coupling rates, and operating the sample below 40 mK temperature result in considerable improvement of the photon conversion to 48% with only 38 added noise quanta.

Another avenue involves piezoelectricity to achieve the electro-mechanical coupling, which does not involve defining an electro-mechanical capacitor, but instead using the traveling phonons.

Bochmann et al [80], demonstrated bidirectional microwave-to-optical conversion of strong fields using a piezoelectric aluminum nitride optomechanical photonic crystal cavity. The piezeoelectric coupling allowed mechanical strain to couple to microwave fields via an interdigital transducer while a one-dimensional optomechanical cavity hosted high quality mechanical and optical modes. In their experiment, coherent mechanical modes are driven through the interdigital transducer and optical read out is provided by an evanescently coupled waveguide. Internal conversion efficiencies are only at the few percent level.

Another experimental effort using piezoelectrics optomechanics [81] comes from the Srinivasan group at NIST that coherently coupled radio frequency, optical, and acoustic waves in an integrated chip. In this experiment an optomechanical cavity with photonics wavelength 1550 nm and localized phononics mode 2.5 GHz are placed between two inter-digitated transducers (IDTs). The strong optomechanical coupling rate in the order of 1 MHz allowed efficient coupling between the optical mode and the localized mechanical breathing mode. The RF excitation is first converted to surface acoustic wave using the IDTs and then routed to the optomechanical cavity using phononic crystal waveguides which ultimately excited the mechanical mode of the optomechanical cavity and therefore facilitated the energy conversion between the optical and radio frequency modes.

In a similar effort Forsch et al [82] have implemented the quantum groundstate microwave-to-optical photon conversion. In this device a one-dimensional optomechanical crystal is coupled to an IDT. The optomechanical cavity supports a breathing mechachanical mode at 2.7 GHz while its photonics mode is in the telecome band. The piezoelectric effect, which creates the electromechanical coupling, links the microwave excitation to the optomechanical cavity, allows quantum noise limited bidirectional conversion with efficiency in the order of 5.5 × 10⁻¹².

Similarly, the Safavi-Naeini's group have demonstrated a low-noise on-chip lithium niobate piezo-optomechanical transducers using acousto-optic modulation [83]. This system provides conversion efficiency of 10⁻⁵ with red-detuned optical pump and 5.5% with blue-detuned pump.