Efficient microwave frequency conversion mediated by the vibrational motion of a silicon nitride nanobeam oscillator

Microelectromechanical systems and integrated photonics provide the basis for many reliable and compact circuit elements in modern communication systems. Electro-opto-mechanical devices are currently one of the leading approaches to realize ultra-sensitive, low-loss transducers for an emerging quantum information technology. Here we present an on-chip microwave frequency converter based on a planar aluminum on silicon nitride platform that is compatible with slot-mode coupled photonic crystal cavities. We show efficient frequency conversion between two propagating microwave modes mediated by the radiation pressure interaction with a metalized dielectric nanobeam oscillator. We achieve bidirectional coherent conversion with a total device efficiency of up to ~ 60 %, a dynamic range of $2\times10^9$ photons/s and an instantaneous bandwidth of up to 1.7 kHz. A high fidelity quantum state transfer would be possible if the drive dependent output noise of currently $\sim14$ photons$\ \cdot\ $s$^{-1}\ \cdot\ $Hz$^{-1}$ is further reduced. Such a silicon nitride based transducer is in-situ reconfigurable and could be used for on-chip classical and quantum signal routing and filtering, both for microwave and hybrid microwave-optical applications.


Introduction
Silicon nitride (Si 3 N 4 ) thin films show exceptional optical and mechanical properties [1], and are used in many microelectromechanical and photonic devices. The material's large bandgap [2], high power handling due to the absence of two-photon absorption in the telecom band [3] and the low absorption losses in Si 3 N 4 thin films [4] make it an ideal candidate for many photonics applications, ranging from nonlinear optics [5,6], to atom trapping [7,8] and tests of quantum gravity [9]. The structural stability and high mechanical quality factor [10] of high tensile stress Si 3 N 4 thin films grown by low-pressure chemical vapor deposition enables nanostructures to be patterned with extreme aspect ratios [11] and form up to centimeter scale patterned membranes [12,13] with high reflectivity [14,15]. New soft clamping techniques make use of the tensile stress to maximize the mechanical quality factor [16,17,18], allowing for an unprecedented regime in which quantum coherence can be reached for micromechanical systems even in a room temperature environment [19]. Additionally, slot mode 1D photonic crystal cavities have been developed using Si 3 N 4 thin films to realize strong optomechanical interactions in a small mode volume and fully integrated on-chip [20,21,22,23].
In the microwave domain, Si 3 N 4 is widely used for wiring capacitors and cross-overs. Early work focussed on the study of Si 3 N 4 as a low loss dielectric to realize compact capacitive circuit elements operated in the quantum regime [24]; however, the amorphous material and its surface are known to host two-level defects [25], such as hydrogen impurities with sizable dipole moments and life-times, which led to the observation of strong coupling between a single two-level system and a superconducting resonator [26,27,28]. Nonetheless, due to its unique mechanical properties, high quality factor membranes [29,30] as well as micro-machined Si 3 N 4 nanobeams [31] have been coupled capacitively to superconducting resonators [32,33] in the context of cavity electromechanics. In the latter experiments the achievable coupling strength was fundamentally limited by the small participation ratio of the motional capacitance. We recently presented a new platform that uses the membrane itself as a low loss substrate for the microwave resonator, drastically lowering the parasitic circuit capacitance and maximizing the electromechanical coupling between a metalized silicon nitride nanobeam and a high impedance superconducting coil resonator.
This allowed for high cooperativities and successfully demonstrating sideband cooling of the low MHz frequency nanobeam to the motional ground state [27].
Silicon nitride membrane-based devices are currently the leading approach to couple optical and mi-crowave systems [34,35]. Realizing noiseless conversion with a mechanical oscillator [36,37,38,39,40,41,42] would allow one to build transducers for quantum networks of superconducting processors connected via resilient and low loss optical fiber networks. Efficient wavelength conversion has been realized between optical wavelengths using silicon optomechanical crystals [43], between microwave frequencies using metallic drum resonators [44,45,46] and silicon nanobeams [47], and also between microwave and optical wavelength using silicon nitride membrane based Fabry-Perot cavities [34,35]. Alternative approaches include the use of Josephson circuits for conversion in the microwave domain [48,49,50], Bragg scattering in silicon nitride rings [6] and dispersion engineering of silicon nitride waveguides [2] in the optical domain. Coupling RF and microwave fields to optics has been achieved with membranes [51,52], via a mechanical intermediary in combination with the piezoelectric effect and optomechanical interactions [53,54,55,56], and microwave to optics conversion has been proposed [57,58,59] and realized with high bandwidth via the electro-optic effect [60,61].
It is an outstanding challenge to realize an onchip integrated microwave to optics converter based on the radiation pressure (optomechanical) interaction alone (i.e., on both the microwave and optical side of the converter). Mechanical systems offer the potential to fully separate the sensitive optical modes (superconductors cause optical loss) from the equally sensitive superconducting circuits (optical light generates quasi-particles in superconductors); for example using phononic waveguides. Here we present a coherent microwave frequency converter on the aluminum-on-Si 3 N 4 platform [27] that is compatible with on-chip optomechanics [21,22]. In the future this approach could be used to realize conversion between microwave and optical fields, or to implement low voltage modulation and fully electrical tunability in Si 3 N 4 -based photonic devices.

Physics
We realize a system where one mechanical oscillator mode with frequency ω m and damping rate γ m is coupled to two electromagnetic resonator modes with resonance frequencies ω i and linewidths κ i (i = {1, 2}) via the optomechanical radiation pressure interaction as proposed in Refs. [37,38,39]. In the presence of two red detuned classical drive fields α d,i near the red sideband of the respective microwave mode at ω d,i = ω i − ω m the parametric interaction can be linearized and described by the sum of two beam splitter type interactions that allow to swap excitations between the mechanical and the two electromagnetic modes, see Fig. 1(a). In the resolved-sideband limit (ω m κ i , γ m ) the linearized electromechanical Hamiltonian in the rotating frames and the rotating wave approximation is given by whereâ i is the annihilation operator for the microwave field mode,b is the annihilation operator of the mechanical mode, ∆ i = ω i − ω d,i = ω m is the detuning between the external driving field and the relevant resonator resonance, and g i = g 0,i √ n i is the electromechanical coupling strength between the mechanical mode and resonator i with the number of intra-resonator drive photons for the microwave input power with P in,i .
The interaction terms of the Hamiltonian in Eq. (1) have two closely related effects. Optomechanical damping cools the mechanical motion with the rate Γ i = 4g 2 i κi . At the same time this leads to the desired bidirectional photon conversion between two distinct electromagnetic frequencies. Using input-output theory, we can relate the itinerant input and output modes to the intra-cavity modes asâ out,i = √ κ ex,iâi −â in,i .
In the photon conversion process, an input microwave signal at frequency ω s,1 with amplitudeâ in,1 is downconverted to the mechanical mode at frequency ω m , which corresponds tob †â 1 in Eq. (1). Next, during an up-conversion process the mechanical mode transfers its energy to the output of the other microwave resonator at frequency ω 2 and amplitudeâ out,2 , which corresponds toâ † 2b in Eq. (1). In this process the mechanical resonance is virtually populated, in the sense that the input signal is rapidly converted to the output signal, leaving little time for the population of the intermediate mechanics. Likewise, an input microwave signal at frequency ω 2 can be converted to frequency ω 1 by reversing the conversion process, see Fig. 1(a). The Hermitian aspect of the Hamiltonian (1) makes this process bidirectional, without any unwanted loss, gain or noise.
We define the photon conversion efficiency via the transmission scattering parameter, i.e. as the ratio of the output-signal photon flux over the inputsignal photon flux, |S ij | 2 = âout,î ain,j 2 . By solving the linearized Langevin equations we find that for signals on resonance with the microwave resonator, in the steady state the bidirectional conversion efficiency is given as [62] with i, j = {1, 2} the indices of the two modes. C i = Γi γm is the electromechanical cooperativity for resonator 2 a out,1 S12 a in,1 a out,1 S11 a in,2 a out,2 S22 (a), Schematic presentation of the frequency conversion.
The spectral density of the two microwave resonators at frequencies ω i (black lines), the strong drive tones at frequencies ω d,i = ω i − ωm (long red and blue arrows) and the signal tones at the optimal frequencies ω s,i = ω i (short red and blue arrows) for i = {1, 2}, as well as the conversion scattering parameters S 21 and S 12 are indicated. (b), Circuit diagram of the converter. The silicon nitride nanobeam in-plane fundamental mode displacement (color indicates displacement amplitude) is coupled capacitively via its two modulated capacitances C m,i to two parallel inductance-capacitance resonators realized with high characteristic impedance planar spiral inductors with the inductances L i and the stray capacitances C s,i . The two resonant circuits are coupled inductively to a transmission line to couple in and out the propagating microwave modesâ in,i andâ out,i . (c), False color scanning electron micrograph of the converter device with thin-film aluminum (white) on suspended silicon nitride membrane (blue). Mechanical beam, cross-over and capacitor region are shown enlarged. (d), Experimental setup. Three microwave sources and one vector network analyzer (VNA) output are combined at room temperature, attenuated by α i and coupled to the device at about 12 mK using semirigid coaxial cables, a low loss printed circuit board and an onchip coplanar waveguide. The reflected signals at the two frequencies of interest are routed to the output path using a cryogenic circulator and after passing another isolator (not shown) are amplified by β i at the 4 Kelvin stage and also at room temperature before detection with either a spectrum analyzer (SA) or the VNA input.
i and η i = κex,i κi is the waveguide-to-resonator coupling ratio with κ i = κ in,i + κ ex,i the total damping rate, κ ex,i the decay rate into the waveguide and κ in,i the decay rate to any other mode. We also obtain a simple equation for the two reflection coefficients, which are given by For lossless microwave cavities η i = 1 and in the limit C 1 = C 2 1 near unity photon conversion efficiency with |T | 2 = 1 and |S 11 | 2 = |S 22 | 2 = 0 (zero reflection) can be achieved. The former condition (C 1 = C 2 ) balances the photon-phonon conversion rates Γ i , while the latter condition (C i 1) guarantees the mechanical damping rate is much smaller than the conversion rates γ m Γ i . In this limit the photon-to-photon conversion rate exceeds the mechanical damping rate -the rate at which phonons are exchanged with the noisy environment. This conversion process is coherent with the bandwidth given by Γ = γ m + Γ 1 + Γ 2 , which is the total backaction-damped linewidth of the mechanical resonator in the presence of the two microwave drive fields.

Circuit
We implement bidirectional frequency conversion in a circuit as shown in Fig. 1(b). The two microwave resonators with resonance frequencies ω 1 = 7.444 GHz and ω 2 = 9.308 GHz are realized using two lumped element inductor-capacitor (LC) circuits formed from a planar spiral inductor of high impedance. The capacitance of these lumped element resonant circuits is defined by the sum of the stray capacitance of the circuit, which is dominated by the inductor stray capacitance, and the mechanically modulated capacitance.
The two resonators are inductively coupled to a single physical port -a 50 Ω coplanar waveguide that is shorted to ground using a thin superconducting wire close the the two inductors.

Device
The described circuit is fabricated on the aluminumon-Si 3 N 4 platform similar to Ref. [27]. Here the entire aluminum circuit, which is shown in Fig. 1(c), is suspended on a fully under-etched high-stress Si 3 N 4 membrane on a high resistivity silicon chip. The inductors are realized as planar spiral inductors with a pitch of 1 µm which maximizes the obtained geometric inductance per unit length, and together with the small effective permittivity of the 60 nm thin membrane, minimizes the stray capacitance of the circuit. This in turn maximizes the obtained electromechnical couplings yielding measured values of g 0,1 /2π = 33 Hz and g 0,2 /2π = 44 Hz for the fundamental in-plane mechanical mode of the patterned silicon nitride nanobeam with an intrinsic damping rate of γ m /2π = 7 Hz at a resonance frequency of ω m /2π = 4.118 MHz. This is in good agreement with calculations based on perturbation theory and electromagnetic modeling of the electric field strength at the dielectric and metallic boundaries of the vacuum gap capacitor [63]. While not measured in this work, the Si 3 N 4 nanobeam has been designed as a phononic bandgap crystal that also localizes a high frequency acoustic defect mode [27]. Very recently, quantum-level transduction of hypersonic mechanical motion could be demonstrated with a similar device [64].

Setup
The experiment is performed at 12 mK inside a dilution refrigerator. At room temperature we apply the drive and signal tones with low noise microwave sources and detect both reflection scattering parameters with a vector network analyzer (VNA) and the two transmission scattering parameters with a spectrum analyzer (SA), as shown in Fig. 1(d). Inside the dilution refrigerator we distribute 50 Ω attenuators at various temperature stages to thermalize the electromagnetic mode temperature with the refrigerator temperature. We use one circulator to couple to the single physical port of the device in a reflection geometry. A second isolator is used to isolate the device from noise at 4 Kelvin where a commercial low noise HEMT amplifier is positioned. The four scattering parameters S ii between the two mode frequencies ω i presented in this work refer to the ratios of the 4 propagating modes in a single on-chip waveguide as schematically shown in Fig. 1(d).

Resonators
As a first step we characterize the resonator properties using a VNA. The magnitudes of the measured complex reflection coefficients S ii are normalized with α i β i → 1, which now corresponds to the scattering parameter at the position of the on-chip waveguide. Then the measured in-phase and quadrature phase components are fitted to the real and imaginary components of where φ is a global phase offset and τ ≈ 50 ns is the delay of the signal in our setup. The result for both resonator modes are shown in Fig. 2(a). Here we plot the magnitude and phase of the measurement (blue points) and the fit (red lines) with excellent agreement. One can see that the resonators are both over-coupled, i.e. κ ex,i > κ in,i as indicated by the full phase shift of ∼ 2π. The comparably very high quality factors of Q in,i = ωi κin,i = {2.2 × 10 5 , 5.5 × 10 4 } enable the large waveguide coupling constants of up to η i = {0.92, 0.68} that are essential for the efficient conversion process (c.f. Eq. (2)). In general, we find the intrinsic losses to be drive power dependent, likely due to saturation of two-level system absorption in the amorphous Si 3 N 4 [24]. For the powers studied in this manuscript we determined coupling efficiencies in the range of η i = {0.80 − 0.92, 0.54 − 0.68}.

Two-mode EIT
As a second step we study the reflection scattering parameters measured with a weak probe tone using the VNA in the presence of two strong reddetuned drive tones. Here we observe a variant of optomechanically induced transparency [65,66,67], an analog to electromagnetically induced transparency (EIT), where the mechanical sideband generated from the drives by the optomechanical interaction interfere with the weak probe tone from the VNA to modify the coherent resonator spectrum. In the case of two resonators and drive tones interacting with one mechanical mode we can model [68] and fit the measurements with with i, j = {1, 2} the indices of the resonator modes, the resonator susceptibilities and χ −1 m = γ m /2 + i(ω m − ω) the mechanical susceptibility. Figure 2(b) shows a measurement (blue points) of the reflection scattering parameter in the vicinity of the two resonator modes together with a fit to Eq. (5) (red lines).
Similar to the case of standard EIT-type measurements, the formation of the peak or dip due to the mechanical mode with total linewidth Γ (as indicated in the insets of Fig. 2(b)) depends on the level of cooperativity C and the degree of coupling η. However, in the present case the interpretation is more complicated because there are two terms that cause optomechanical damping. They affect each other and the double-EIT spectrum as a whole. For the chosen pump power and cooperativity combination, which is indicated in Fig. 2(c) and (d) by red data points, we observe a full suppression of the reflected probe signal for resonator mode 1 Fig. 2(b). This is a necessary condition for high efficiency conversion. In the case of resonator mode 2 on the other hand, which is only critically coupled to the waveguide (η ∼ 0.5), we observe a peak in the center of the resonator, indicating a finite reflection that results in a limited conversion efficiency.

Cooperativity
We performed two-mode EIT measurements and analyses as presented above for all of the following  (mK) pump power combinations: P d,1 from -14 dBm to 0 dBm and P d,2 from -10 dBm to 2 dBm, both in steps of 2 dB. Here the input power at the device P in,i = P d,i − α i (on a log scale) is related to the shown drive powers via the attenuations α i = {69.0, 70.4} dB for the two mode frequencies of interest. For each power combination we fit both spectra to a single set of parameters (specifically g i ) and summarize the results in Fig. 2(c) and (d). Shown are the mean and the statistical error of the cooperativities C i = Γi γm . Using γ m /2π = 7 Hz obtained from cooling measurements discussed below, we also back out the optomechanical damping Γ i = 4g 2 i κi for each of the drive powers P d,i , and the intra-resonator drive photon numbers n d,i . We find that the dependence on drive power follows the expected behavior (dashed lines) based on the applied drive powers, attenuations and the calibrated g 0,i . The small error bars confirm that C 1 is not significantly affected by changing P d,2 and vice versa. The maximum C 1,2 ∼ 10 2 that were obtained suggest that internal conversion efficiencies |Sij | 2 η1η2 ∼ 1 can be achieved.

Sideband cooling
To estimate the noise generated by this device we perform motional sideband cooling [69,70,71,72,27] using each resonator mode independently, i.e. with only one cooling tone at a time. The data is obtained by measuring the electronic noise spectrum due to the thermal Brownian motion of the nanobeam at the resonance frequency ω m with the linewidth Γ = Γ i +γ m as a function of the cooling drive tone power P d,i applied at the optimal detuning ∆ i = ω i − ω m . The data shown in Fig. 2(e) and f for modes 1 and 2 has been calibrated and analyzed as described in Ref. [27]. For small drive photon numbers we see a decrease of the phonon occupancies n m,i (blue squares) in line with the expectations (blue lines), i.e. n m,i = n b Ci+1 with n b = 60 the mechanical bath corresponding to the refrigerator temperature of ∼ 12 mK and the intrinsic mechanical linewidth γ m (blue squares). However, we also observe a power dependent increase of the noise floor with the bandwidth of the resonator mode [33]. Using a theoretical model [73,27] we fit the resonator occupations n r,i , which we show (red circles) together with a power law fit (dashed red line) in Fig. 2(e) and (f). The power dependent resonator noise limits the minimum phonon occupation to n m ∼ 5 at drive powers of about -5 dBm, independent of which resonator mode is used for cooling.
The residual thermal population of the mechanical oscillator (n m ∼ 5) and the two microwave resonators (n r,i ∼ 4) leads to incoherent added noise when the device is used as a transducer. In the limit C 1 ∼ C 2 1 and the realistic assumption that the waveguide modes are well thermalized and unpopulated, the noise added to any converted signal at the output of resonator port i is given as [45] n add,i = η i (n r,1 + n r,2 + 2n m ), which results in n add,i ∼ {16, 12} photons · s −1 · Hz −1 .
For this estimate we have cautiously assumed the same mechanical population as measured with only a single drive tone, due to the cooling limitations imposed by the drive dependent resonator noise. In addition, finite sideband resolution could lead to gain and amplified vacuum noise. With sideband resolution factors of κi 4ωm ≤ 10 −3 this is expected to be negligible in the presented devices.

Scattering parameters
In order to measure and quantify the efficiency of the wavelength converter we obtain the full set of scattering parameters for all 56 reported cooperativity combinations. The two reflection coefficients |S ii | 2 are extracted from the center region of the twomode EIT response shown in Fig. 2(b). For the transmission we apply a coherent signal on resonance with one of the two microwave resonators and measure the mechanically transduced signal appearing in the center of the other resonator using a spectrum analyzer. Measuring this in both directions allows us to calibrate the product |S 21 ||S 12 | = |T | 2 , i.e. the total bidirectional conversion efficiency [34], using the product of both measured off-resonant reflection coefficients √ α 1 β 1 √ α 2 β 2 → 1, which were already used to normalize the reflection parameters in Fig. 2(a) and (b). In practice, we sweep the signal frequency in a small range δ with a span on the order of the damped mechanical linewidth and extract the maximum value of the conversion efficiency. The result is shown in Fig. 3(a) as a function of both cooperativities where red color indicates high and blue color indicates low values of the three scattering parameters. Qualitatively we see that matching the cooperativities leads to higher transmission and minimizes the reflection.
In panel (b) of Fig. 3 we show the quantitative result for the three cooperativity combinations indicated with white lines in panel (a). In the first case we show all three S parameters as a function of C 1 for C 2 ≈ 30. As expected we observe the maximum conversion efficiency of |S ij | 2 and the minimum reflection |S ii | 2 for C 1 ∼ C 2 . In the second plot we fix a higher value (C 1 ≈ 95) where we find that the optimal matching condition is relaxed, i.e. the high conversion efficiency is achieved for a larger range of C 2 . Finally, keeping the product constant C 1 · C 2 ≈ 660 and plotting the scattering parameters as a function of the ratio C 1 /C 2 , the matching condition C 1 = C 2 is very clear. Although the cooperativities are highest in the second case, the maximum conversion efficiency of |S ij | 2 ≈ 0.6 is the same due to the predicted limitations imposed by the finite waveguide coupling efficiencies of η 1 η 2 ≈ 0.63. The estimated internal photon to photon conversion efficiency is 10 7 10 8 10 9 10 10 and Eq. (3) shown as solid lines in Fig. 3(b).

Bandwidth and dynamic range
We have seen that the conversion efficiency saturates for C 1 = C 2 1. The conversion bandwidth Γ on the other hand is scaling with the cooperativity. Figure 3(c) shows multiple signal frequency sweeps for different drive powers that match the cooperativity on both resonators. Here we achieve a maximum conversion bandwidth at the maximum efficiency of Γ/2π = 1.72 kHz. The dynamic range is tested by extracting the maximum conversion efficiency as a function of the converted signal power for C 1 ≈ C 2 ≈ 35 and the result is shown in Figure 3(d). We see no significant compression up to signal levels of 2 × 10 9 photons/s.

Conclusion and Outlook
In summary, we have presented a very versatile new dielectric nano-mechanical system that is suitable for efficient and near quantum limited wavelength conversion in the microwave frequency band. The unique properties of silicon nitride make it a natural platform for the realization of microwave-to-optical transducers and heralded entanglers [74]. Nevertheless, to achieve close to noise-free operation will require a deeper understanding of the electromagnetic and mechanical loss mechanisms, [29] and the thermal conductivity of low dimensional nano-beams [75] made from glassy materials like Si 3 N 4 at temperatures in the range of only a few millikelvin [76]. This could lead to the observation of new and surprising physics along the way to fully operational quantum transducers and networks.