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Superconducting Nanoelectromechanical Transducer Resilient to Magnetic Fields

Nanoscale electromechanical coupling provides a unique route towards control of mechanical motions and microwave fields in superconducting cavity electromechanical devices. Though their successes in utilizing the optomechanical or electromechanical back-action effects for various purposes, aluminum imposes severe constraints on their operating conditions with its low superconducting critical temperature (1.2 K) and magnetic field (0.01 T). To extend the potential of the devices, here we fabricate a superconducting electromechanical device employing niobium and demonstrate a set of cavity electromechanical dynamics including back-action cooling and amplification, and electromechanically induced reflection at 4.2 K and in strong magnetic fields up to 0.8 T. This device could be used to realize electromechanical microwave components for quantum technologies by integrating amplifiers, converters, and circulators on a single chip that can be installed at the 4K stage of dilution refrigerators. Moreover, with its ability to control and readout nanomechanical motions simultaneously, this niobium electromechanical transducer could provide powerful nanomechanical sensing platforms.


Introduction
The motion of a nanoscale mechanical object implemented in an electromagnetic cavity at microwave 1,2 (~ GHz) or optical frequencies [3][4][5] (> 100 THz) can be detected by measuring the motion-induced intensity modulation of the electromagnetic field coming out from the cavity.
This interferometric detection method is essential in measuring nanomechanical motions as it enables picometer-level sensitivity. Thanks to this powerful method, nanomechanical devices are considered promising for sensing small perturbations like mass-loading by single molecules [6][7][8] . More sensitive detection requires the enhancement of the quality factor, Q, of the optical cavity, which ultimately leads to strong interactions of mechanical and optical eigenmodes (i.e. phonons and photons) via the radiation pressure, enabling energy transfer between different physical domains 9 . This dynamic process is, in general, described by a linearized optomechanical Hamiltonian 10 , ̂= −ℏ △̂ †̂+ ℏ̂ †̂+̂ in the rotating frame at d, where  = d -c is the detuning frequency, d is the driving frequency, c is the resonant frequency of an optical cavity, m is the resonant frequency of a mechanical resonator.
̂(̂ †) and ̂(̂ † ) are the annihilation (creation) operators for photon and phonon modes, respectively, and ̂ is the photon-phonon interaction Hamiltonian. If the system is driven in the red-detuned regime ( = −m), ̂= ℏ (̂ †̂+̂̂ †) , which describes coherent exchange of the two bosonic modes, resulting in the cooling of mechanical motions. In contrast, a blue-detuned drive ( = m) sets ̂= ℏ (̂ †̂ † +̂̂), which leads to parametric downconversion of the drive photons, increasing phonons in the mechanical resonator and photons in the optical cavity. In both cases, = 0 √ denotes the optomechanical coupling rate when the number of the drive photons in the cavity is . = √ ℏ 2 Ω is the single photon coupling rate, where meff is the effective mass of the mechanical resonator.
Cavity optomechanics have been explored in various experimental platforms such as nanomembranes in the middle of high Q optical cavities 11 , toroidal optical cavities 12,13 , optomechanical crystals [14][15][16]  Extending the operating conditions of the devices is strongly desired to utilize their versatility as nanomechanical sensors and microwave components for quantum technologies.
Here we describe the realization of a superconducting nanoelectromechanical transducer with niobium and demonstrate their operations at 4.2 K and in strong magnetic fields (up to 0.8 T).
We show that the micro-fabricated niobium microwave resonator exhibits relatively high quality factor (~ 4,000) at 4.2 K and preserves its microwave resonance even in external magnetic fields. We demonstrate efficient back-action cooling and amplification of nanomechanical motions, and electromechanically induced reflection of microwave signals.
We also experimentally show the detection of nanomechanical motions in 0.8 T magnetic field, manifesting the potential for exploring spin-phonon interactions 28-31 using the niobium device.

Results
Niobium superconducting electromechanical transducer. We explore the optomechanical interaction of microwave photons and nanomechanical phonons by fabricating a LC microwave resonator circuit made of superconducting niobium (Fig. 1a) where a free-standing, 130-nm thick nanomechanical resonator forms a vacuum-gap capacitor with a 100-nm thick bottom electrode (Fig. 1b). This membrane resonator is connected to a spiral inductor and the bottom electrode is connected to the circuit ground. The diameter of the membrane is 12.6 m and the bottom electrode is 10 m in diameter. The effective mass, meff, of the membrane is ~ 140 pg based on its dimensions. Our device is a typical set-up for studying microwave cavity electromechanics 19-21,23,24,32 via the dispersive electromechanical coupling, which induces the shift of the microwave resonant frequency by the motion of the membrane. The nanomechanical resonator is suspended over the bottom electrode with a 130-nm vacuum gap by releasing a sacrificial layer of Poly-methyl-methacrylate (PMMA) ( Fig. 1c and Methods).
To characterize microwave and mechanical responses of our device, we perform all the measurements at 4.2 K, the liquid helium temperature (Fig. 1d). The microwave resonator is capacitively coupled to the input and the output ports to drive and measure the responses of the device. From a microwave transmission measured with a network analyzer, we obtain c ( 2  3. 777 GHz) and the total decay rate of the microwave resonator,  (  + ext = 2  960 kHz) with intrinsic loss  ( 2  330 kHz) and external loss ext ( 2  640 kHz) (Fig. 1e).
We characterize the mechanical responses by measuring a noise spectrum around c. This spectrum results from the frequency up-conversion of the thermomechanical noise around m when the device is excited by a red-detuned microwave drive at d = c − m. This microwave drive is weak enough to induce negligible optomechanical back-action effects. We also obtain m ( 2  8.4 MHz) and the intrinsic mechanical damping rate, m ( 2  810 kHz) (Fig.   1f). Note that this m corresponds to the fundamental flexural mode of the membrane as the bottom electrode provide an isotropic electrostatic force distribution. The ratio of m/ ~ 9 implies that our system is in the sideband resolved regime, meaning that only one of the two sidebands (Stokes or anti-Stokes) is dominant depending on .  Fig. 1). We observe the consequent optomechanical spring and damping effects which originates from the retardation effect in the cavity field 10 . The optomechanical damping clearly shows its maximum at  = −m. We also characterize the back-action cooling and amplification performance of our device at  = −m (Fig. 2a) and  = m (Fig. 2b) for different drive powers, nd. For  = −m, Sx () of the up-converted sidebands, which exhibit Lorentzian resonance curves, show significant linewidth broadening, decreasing the amplitude of the resonance peaks as we increase nd (Fig. 2a). The signal-to-noise ratio (SNR), Sx mech (m)/Sx bg (m) also changes with increasing nd, manifesting the interplay of the thermomechanical noise, the measurement imprecision and the dynamic back-action ( Supplementary Fig. 2a). We obtain the best force sensitivity of 400 aN/Hz 1/2 at nd ≈ 4 × 10 6 , which is limited by the thermomechanical noise ( Supplementary Fig. 2b). On the other hand, Here, p = p−d is the probe detuning. R and L denote the input and output couplings and equal to ext/2. We learn from this equation that the linewidth and depth of the reflection window strongly depends on the drive power, nd. To systematically study the EMIR effect in our device, we measure the transmission spectrum of a weak probe signal at p with the reddetuned drive ( = −m) while we vary nd from 2 × 10 4 to 6.5 10 8 (Fig. 3b). When p ≈c, the depth of the reflection window, AEMIR, increases as nd increases and reaches −32 dB at the maximum drive power (nd ≈ 6.510 8 ). The linewidth of the reflection window, EMIR, corresponds to the damping rate of our nanomechanical resonator and broadens up to 32 kHz at the maximum nd (Fig. 3c). In the resolved sideband regime, the window linewidth at  = −m is given by EMIR = m(1+C), where C = 4g0 2 nd /(m) represents the electromechanical cooperativity. In our device, we obtain the maximum cooperativity of C ≈ 40. Note that C≥ 1 leads to notable electromechanical back-action effects 12 . Based on the currently available g0 (≈ 2 150 Hz) demonstrated in aluminum-based devices 24 , it should be possible to engineer our niobium device to achieve C ≈ 10 5 at 4 K.
Electromechanical transduction in strong external magnetic fields. Niobium is also expected to enable reliable operation of superconducting electromechanical devices in magnetic fields, which could enable studying and harnessing, for example, spin-phonon interactions 28-31 in hybrid quantum devices. To support this idea, we study the responses of our electromechanical device to an external magnetic field, B. We apply the field in the direction parallel to the device and measure microwave transmission while varying B from 0 to 0.89 T, with 0.089 T steps (Fig. 4a). We find that c decreases from 3.778 GHz (B = 0 T) to 3.744 GHz (B = 0.8 T) and  increases from 950 kHz (B = 0 T) to 31 MHz (B = 0.8 T) (Fig. 4b). This increase of  mainly originates from quasiparticle generation in niobium 35 .  becomes comparable to m around 0.5 T and starts to exceed m. In this regime, the resolved sideband approximation becomes irrelevant as the contribution of the Stokes sideband is not negligible.
To characterize electromechanical responses, we measure the EMIR spectra for different B by driving the device in the red-detuned regime ( = −m). The highest B at which we observe the EMIR effect is 0.8 T with AEMIR of ~ 0.5 dB at nd ≈ 3.3510 7 (Fig. 4c). Even at B = 0.8 T, we still observe non-negligible electromechanical back-action damping effect with C ~ 0.46.
EMIR exhibits reduced electromechanical effects as B increases (Fig. 4d). This mainly originates from the decrease of the single-photon electromechanical cooperativity, C0 = 4g0 2 /(m ), due to the increase of  To check this, we estimate g0 from C0 for different B and it remains almost constant up to 0.62 T and starts to decrease as the resolved sideband approximation becomes invalid (Fig. 4d inset).

Discussion
Utilizing the cavity-electromechanical transduction scheme could be a promising approach for   The blue and red dots in (c) and (d) correspond to cooling and amplification cases, respectively.  Here, S21,0( = c) is the transmission signal with the thru reference and  is the total decay rate. The intrinsic decay rate, o is o =  − ext.
Relation of the sideband power and the displacement of the mechanical resonator. We derive the relation of the sideband power and the displacement of the nanomechanical resonator by modeling our device using circuit elements (Fig. 1d). To simplify the analysis, we covert the circuit in Figure 1d into an equivalent circuit consisting of parallel components only (Supplementary Fig. 3). We solve for the voltage in the circuit, V(t). The Here, Psideband is the sideband power and Pd is the driving power, both measured at the spectrum analyzer. Note we use the following relation: Here, is the power of the microwave drive measured at the spectrum analyzer.  Calculation of displacement and force noise spectra from noise power spectrum.

Calibration of phonon and drive photon numbers.
The total displacement noise spectrum, Sx (), near the microwave resonance can be expressed as the sum of the mechanical sideband noise spectrum, Sx mech (), and the background noise spectrum, Sx bg (). This background noise is usually a white noise originates from the measurement circuits and thermal fluctuations in the microwave resonator. To begin with the derivation, we consider the normalized sideband power, which is given by The blue, orange, and red solid circles correspond to Sf mech (m), Sf bg (m), and Sf mech (m) + Sf bg (m), respectively. The procedure for the conversion between the displacement noise and the force noise is described in Methods.