Sideband-resolved resonator electromechanics based on a nonlinear Josephson inductance probed on the single-photon level

Light-matter interaction in optomechanical systems is the foundation for ultra-sensitive detection schemes as well as the generation of phononic and photonic quantum states. Electromechanical systems realize this optomechanical interaction in the microwave regime. In this context, capacitive coupling arrangements demonstrated interaction rates of up to 280 Hz. Complementary, early proposals and experiments suggest that inductive coupling schemes are tunable and have the potential to reach the single-photon strong-coupling regime. Here, we follow the latter approach by integrating a partly suspended superconducting quantum interference device (SQUID) into a microwave resonator. The mechanical displacement translates into a time varying flux in the SQUID loop, thereby providing an inductive electromechanical coupling. We demonstrate a sideband-resolved electromechanical system with a tunable vacuum coupling rate of up to 1.62 kHz, realizing sub-aN Hz−1/2 force sensitivities. The presented inductive coupling scheme shows the high potential of SQUID-based electromechanics for targeting the full wealth of the intrinsically nonlinear optomechanics Hamiltonian. Recently, inductively-coupled optomechanical systems have been realized. They represent an important step forward towards achieving strong light-matter interaction, offer extreme sensitivity to mechanical displacement, and allow to study quantum phenomena on a single quantum level. In this work, a superconducting device is inductively coupled to a microwave resonator forming an electromechanical system operating at the single-photon level.

D esigning, investigating, and understanding the optomechanical interaction plays a key role in tailoring the light-matter interaction and hence for testing quantum mechanics [1][2][3][4] . In addition, optomechanics is also the basis for detection schemes with extreme sensitivity used in gravitational wave detection 5 , mechanical sensing 6 as well as the creation of mechanical quantum states 7,8 . This potential triggered a multitude of realizations of optomechanical systems 4,9 including systems based on superconducting circuits, which define the field of nanoelectromechanics [10][11][12][13][14][15][16] . Here, the photonic cavity is implemented as a microwave resonator and the electromechanical interaction is typically realized using a mechanically compliant capacitance, which transfers a mechanical displacement into a change of the resonance frequency of the microwave circuit [10][11][12][13][14]17 . It was proposed early on that inductive coupling schemes can allow for higher coupling rates than capacitive ones [18][19][20][21] .
SQUID tunable microwave resonators can be designed to realize qubits or effective two-level systems 22,23 . In this context, electromechanical interactions have also been studied 15,[24][25][26][27] enabling large optomechanical couplings with the potential to realize quantum interference of massive objects' trajectories 28 . In addition, they provide a route to implement even more complex coupled quantum systems, which, e.g., allow to create photon-phonon entanglement, phonon Fock state generation, and three partite entanglement [29][30][31] . However, these systems employ the effective two-level character and hence go beyond the traditional optomechanical system.
Only recently, an electromechanical coupling on-par with capacitive approaches was demonstrated by using an inductive coupling scheme based on a lumped-element microwave resonator and a mechanically compliant Josephson inductance 32 . By design, the linear inductance of the resonator was chosen much larger than the Josephson inductance, limiting the coupling strength in favor of tolerating large photon numbers. Complementary, it was shown that a distributed resonator with an embedded Josephson inductance allows to design arbitrarily large nonlinearities and can be used to realize two-level systems 33 .
Here, we present such an inductive coupling scheme based on a Josephson nonlinearity integrated into a coplanar waveguide resonator (CPW). For this device, we find a magnetic field tunable electromechanical vacuum coupling of up to 1.62 kHz.
Moreover, we find a proportionality factor of 3.1 MHz T −1 , between the coupling rate and the applied magnetic field, exceeding the one reported in ref. 32 by a factor of 120. In addition, the high coupling rate allows us to detect the mechanical motion with probe powers of 2.7 fW corresponding to an occupation of the microwave resonator with only a few photons. Hence, our device concept is naturally compatible with quantum microwave sources, which typically offer only a limited output power, and quantum-limited amplifiers with low saturation powers. These benefits make the device class ideally suited to act as a building block for the realization of inductively coupled devices integrated in more complex quantum circuits. Our results represent a significant step toward reaching the single-photon strong-coupling regime in electromechanics, allowing the investigation of quantum mechanical effects such as single-photon single-phonon blockade 4,34 , the creation of mechanical quantum states 35 , and the generation of non-classical light 36 .

Results and discussion
Circuit concept and fabrication. The nano-electromechanical device discussed here is based on a λ/4 superconducting CPW resonator which is shunted to ground at one of its ends via a direct-current superconducting quantum interference device (dc-SQUID) (cf. Fig. 1). As the SQUID acts as a flux-dependent inductance, the resonance frequency of the microwave circuit becomes flux sensitive. In addition, the SQUID loop is partly suspended and contains two nanomechanical string oscillators. Any displacement x of the strings translates into a change of the magnetic flux Φ threading the SQUID loop and, in turn, into the microwave resonance frequency ω c . The mechanically induced frequency shift can be described by the electromechanical inter- scales with the length l of the string, the zero-point displacement , and the mode-shape γ of the mechanical resonator. Note, that g 0 is tunable as the applied magnetic field A dc-SQUID with two freely suspended arms, which are separated by w, shunts the resonator to ground on the other end. The resonance frequency is tunable by varying the flux through the SQUID loop, e.g., by adjusting the external magnetic field B ext . In addition, the freely suspended SQUID arms (strings) modulate the SQUID inductance with the mechanical frequency via a change in the area of the SQUID loop and hereby realize the electromechanical coupling. In total, we expect the presence of four mechanical modes (one in-and one out-of-plane for each of the two strings) in our device. In our experiments, we focus only on one of them as sketched in panel (a). Panel b shows a microscope image of the aluminum resonator with the SQUID located at one end fabricated using a lift-off process. Panel c shows a tilted scanning electron microscope (SEM) image of a similar partly suspended SQUID structure. Note that the strings investigated in this work have dimensions of 20 μm × 110 nm × 200 nm.
B ext and the flux to resonance frequency transfer function ∂ω c / ∂Φ, which we call in the following flux responsivity, can be controlled in situ.
We fabricate the electromechanical system as an all-aluminum superconducting circuit using electron beam lithography, doublelayer shadow evaporation, and reactive ion etching (cf. Fig. 1 and Supplementary Note 3 for details). The SQUID contains two freely suspended mechanical string oscillators as well as the two Josephson junctions. Each string has a length l = 20 μm, a width w = 200 nm, and a thickness t = 110 nm, corresponding to an effective mass of m eff = 0.6 pg (all parameters of the device are summarized in Table 1). During the device fabrication, we anneal the entire chip at 350 ∘ C resulting in tensile stressed aluminum strings with mechanical resonance frequencies Ω m ≃ 6.3 MHz at millikelvin temperatures. Thus, we obtain a zero-point fluctuation of x zpm = 47 fm. The microwave resonator is coupled capacitively at one of its ends to a microwave CPW feed-line, enabling the microwave spectroscopy of the device. The other end, is shunted to ground via the SQUID. In this way, we obtain a flux-tunable resonator with large flux responsivity ∂ω c /∂Φ, which is essential for the realization of a large electromechanical coupling strength.
Microwave resonator characterization. We start the characterization of the microwave circuit by performing microwave transmission measurements as shown in Fig. 2 (see Supplementary Note 1 B for details). Figure 2a shows the data as a function of the applied flux Φ, where the resonator appears as an absorption signature in dark blue. Figure 2b, c shows transmission data for two selected flux bias. For a quantitative analysis of the evolution of the microwave resonance frequency ω c , we locate the transmission minimum for each flux bias and plot ω c as well as ∂ω c /∂Φ versus the applied flux in Fig. 2d, e. We find that ∂ω c / ∂Φ reaches values of up to 10 GHz/Φ 0 , underlining the performance of this coupling scheme. In addition, the ω c (Φ) dependence allows us to extract the single-junction critical current of the SQUID, I c = 0.44 μA, and the minimum Josephson inductance, L J = 0.36 nH, at ∂ω c /∂Φ = 0 (see Supplementary Note 2). Recording transmission data for selected flux bias points with higher frequency resolution (as shown in Fig. 2b, c) allows to determine the external coupling rate κ ext between the resonator and the feed-line as well as the internal loss rate κ int of the microwave circuit. At ∂ω c /∂Φ = 0, we find a total linewidth of κ/ 2π = (κ int + κ ext )/2π = 2.5 MHz. Figure 2f shows κ ext and κ int as a function of the flux responsivity. While κ ext remains nearly constant, κ int increases for large ∂ω c /∂Φ, which is attributed to the increased sensitivity of the circuit to flux noise. However, even at the bias point K, the device remains in the resolved sideband regime (Ω m > κ).
Quantification of the electromechanical readout performance. Next, we investigate the mechanical properties including the electromechanical coupling rate. For this experiment, we probe the microwave sideband fluctuations originating from the interaction of the incident probe tone with the mechanics. The probe tone is set to a blue sideband configuration (ω p = ω c + Ω m ) to enable a sensitive detection of the scattered photons 4 . The resulting signal is down-converted with ω p . Figure 3a shows the voltage power spectral density S UU (Ω) detected using a spectrum analyzer (cf. Supplementary Note 1) for a temperature of 185 mK, when the microwave resonator is biased to working point K at ω c /2π = 6.887 GHz with B ext = −470 μT. We find a mechanical signature with a resonance frequency Ω m /2π = 6.34311 MHz and a full-width at half-maximum linewidth of Γ m /2π = 33.6 Hz. In addition, we modulate the frequency of the incident microwave probe tone resulting in a calibration peak visible in panel a at −2 kHz. Similar to refs. 12,38 , the comparison of the calibration tone amplitude S UU ðΩ mod Þ with the mechanical response S UU (Ω m )Γ m /2 allows us to quantify the electromechanical coupling rate g 0 . Due to the specific detection scheme used for this experiment, we need to account for an additional factor Y, which relates the transfer function of the mechanical motion to the transmission function of the calibration tone (refer to Supplementary Note 4 for details). From this data, we obtain g 0 /2π = (1.62 ± 0.12) kHz. This value exceeds the highest coupling rate of 280 Hz 17 achieved for capacitive coupling by a factor of 5.8 and corresponds to a single-photon cooperativity C 0 of about 0.1. This large coupling strength allows us to use ultra-low probe powers for the detection of the mechanical sidebands. In fact, we use a   At Ω m /2π = 6.34311 MHz, we observe the mechanical signature with a peak amplitude of S UU (Ω m ) = 1.26 μV 2 Hz −1 and a linewidth of Γ m /(2π) = 33.6 Hz. In this experiment, we configure the microwave spectroscopy tone to the blue sideband configuration (ω p = ω c + Ω m ) to efficiently detect the Stokes field within the bandwidth of the resonator. However, we suppress back-action induced heating by using an ultra-weak probe tone corresponding to an average photon number of n r ¼ 1:6 in the microwave resonator. The sharp peak at (Ω − Ω m )/2π ≈ −2 kHz with an amplitude of S UU ðΩ mod Þ ¼ 2:56 μV 2 Hz À1 stems from the phase modulation of the microwave spectroscopy tone (ϕ 0 = 3.94 × 10 −4 ). Combining the information of the mechanical signature and the calibration peak, we find an electromechanical vacuum coupling of g 0 /2π = (1.62 ± 0.12) kHz. We further show the mechanical displacement density S det xx in panel (b) for T = 126 mK (blue), 186 mK (green), and 232 mK (red dots) including the Lorentzian fits to the data (solid lines). The peak area scales as expected with temperature via the mechanical linewidth. All spectra are recorded with the same averaging. Panel c shows the spectral force sensitivity S det FF at T = 126 mK for the microwave probe powers as indicated. On-resonance, we reach sub-attonewton force sensitivities even for these ultra-low microwave probe powers.
probe power corresponding to an average photon number of n r ¼ 1:6 in the microwave resonator for the spectrum shown in Fig. 3a. At this power level, we suppress back-action induced heating of the mechanical mode. In particular, for the parameters used in our experiment, we estimate an electromechanically induced damping rate of |Γ em = −2.6 Hz| ≪ Γ m , resulting in an increase in the thermal phonon occupation by 4% 4 , as we probe with a cooperativity of C ≃ C 0 ≃ 0.1. This contribution is already accounted for in the stated value of g 0 (for details see Supplementary Note 5). The background of S UU (Ω) shows the experimental imprecision noise S imp UU of the experiment, which is presently limited by the performance of the cryogenic microwave amplifier used as the first amplification stage.
Our calibration technique allows us to express S UU (Ω) as the mechanical displacement spectrum S xx (Ω) (see Supplementary Note 4). Figure 3b displays these spectra for various temperatures. We observe the expected increase in the phonon number with temperature in the form of an enlarged peak area. This manifests itself as an enhancement of the linewidth Γ m with temperature (cf. Supplementary Information, Fig. S10a). This behavior has been observed for aluminum nano-strings and is attributed to the reduced dimensionality (1D) of the phonon mode 39 . In addition, we find a non-monotonic dependence of the imprecision noise floor on the environmental temperature. This behavior correlates with the temperature-dependent quality factor of the microwave resonator, which decreases above 185 mK. We speculate that this observation is connected to its population with thermal microwave photons, the presence of quasiparticles, and the motion of flux vortices (for details see Supplementary Note 4B).
The detected displacement power spectral density S xx directly relates to the force sensitivity S det FF via S det FF ¼ 2S xx ðΩÞ=jχj 2 40 with the mechanical susceptibility χ ¼ ½m eff ðΩ 2 À Ω 2 m À iΓ m ΩÞ Tuning of the vacuum coupling strength. Next, we investigate the scaling of the electromechanical coupling strength with the applied magnetic field B ext and the flux responsivity ∂ω c /∂Φ, cf. Eq. (1). To this end, we excite the mechanical motion using an oscillatory mechanical force provided by piezo actuators resulting in a controlled oscillating displacement of the nano-strings. Under these conditions, the displacement amplitude is much larger compared to the thermal noise-driven measurements and hence much easier to detect (details are found in Supplementary Note 7). Figure 4a shows the measured resonance frequency shift of the microwave resonator δω c for various flux responsivities using a coherent excitation force at an approximately constant magnetic field of B ext ≈ −440 μT. As the frequency shift δω c is proportional to the product (∂ω c /∂Φ)δΦ, we observe a large δω c for large flux responsivity ∂ω c /∂Φ. Figure 4b summarizes the coupling rates derived from the peak amplitudes presented in Fig. 4a (dark green triangles) as well as the results of further experiments with a lower piezo actuator drive power (light green triangles) and links them to displacement noise measurements of g 0 (red cross). The data corroborate the predicted linear scaling of the electromechanical coupling with the flux responsivity of up to 10 GHz/Φ 0 .
In a similar fashion, we measure the electromechanical response as a function of the applied magnetic field for a fixed flux responsivity of ∂(ω c /2π)/∂Φ = 6.6 GHz/Φ 0 (bias point K) and present the data in Fig. 4c. Again, we link these data points to the thermal displacement measurements (red circles) and corroborate the linear scaling with B ext .
From Fig. 4c, we extract a scaling factor of (3.13 ± 0.20) MHz T −1 for the electromechanical coupling with respect to the applied magnetic field B ext . This value exceeds those of previous reports by two orders of magnitude 32 and represents an important figure of merit as it allows one to estimate g 0 at the given physical parameters of the device. For example, for the 70 nm thin aluminum film, we expect an in-plane critical field of 130 mT 41 which extrapolates to g 0 /2π = 410 kHz. This coupling rate does not yet satisfy the singlephoton strong coupling condition. However, improved microwave decoherence rates of up to 400 kHz as reported for superconducting quantum circuits 42 in combination with an optimized critical current I c of the SQUID will allow to reach the strong-coupling regime (see Supplementary Note 9). Nevertheless, ground-state cooling for n r ¼ 1 is expected to be within reach for this device using moderate fields of B ext = 20 mT applied in parallel to the superconducting layer. Overall, we find an improvement of the vacuum coupling strength by a factor of 7 in comparison to previous inductive coupling schemes 32 .
Finally, Eq. (1) includes a mode-shape factor γ. With our experimental parameters outlined so far, we find γ = 0.99, which is in good agreement with previous findings 16,32 .
Besides electromechanically induced transparency (EMIT), which can be used to measure g 0 given quantitative knowledge of the photon number 12,32,[43][44][45] , one can utilize the signature of a modification of the microwave resonator absorption lineshape, using the strong binding condition δω c > Ω m 46 to confirm g 0 . Given our parameters (see Table 1), we use the second strategy. Here, a large amplitude coherent oscillation of the displacement is excited by a coherent mechanical drive causing the characteristic pattern of the microwave resonator absorption lineshape. When combined with the nonlinear response of the string-oscillator, we can quantify the displacement and hence g 0 . Corresponding data (shown in Supplementary Information, Fig. S12) corroborates the coupling strengths determined via thermal displacement noise (data analysis is presented in Supplementary Note 8).
Conclusions. In summary, we present a device implementing an inductive electromechanical coupling scheme based on a superconducting coplanar waveguide resonator and a dc-SQUID operating in the resolved sideband regime. We demonstrate an electromechanical coupling rate of 1.62 kHz, exceeding values of capacitive coupling schemes by almost an order of magnitude. This high coupling rate enables an ultra-high force sensitivity of 0.70 aN Hz −1/2 at an ultra-low microwave readout power of 5.4 fW, making this device a very promising low-power force sensor. In addition, the maximum electromechanical coupling of 1.62 kHz in combination with the investigated tuning of the electromechanical coupling indicates that this coupling strategy has the potential to reach the single-photon strong-coupling regime of electromechanics. Moreover, the tunability of the coupling provides the access to new features of electromechanical systems such as state amplification and readout techniques using the resonator as a parametric amplifier.
During the review process, we became aware of similar works by refs. 47,48 .

Methods
Calibration of the electromechanical coupling via thermal motion. For the determination of the electromechanical coupling constant, we use a scheme similar to the calibration tone technique discussed by Gorodetsky et al. in ref. 38 . In particular, we use the microwave detection scheme where the frequency modulates the microwave source to generate the calibration tone. We down-convert the signal COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-00501-3 ARTICLE COMMUNICATIONS PHYSICS | (2020) 3:233 | https://doi.org/10.1038/s42005-020-00501-3 | www.nature.com/commsphys coming from the dilution fridge with an unmodulated local oscillator and perform a spectrum analysis of the down-converted signal. This results in the spectra shown in Fig. 3a. Via this scheme, the vacuum coupling strength is determined by This is derived in full detail in Supplementary Note 4 Determination of the resonator transfer function. For a quantitative analysis of Eq. (2), we need to obtain information about the transfer functions Y involved as well. For our case, we find the transfer function depending in general on the microwave resonator characteristics and the employed probe tone configuration as we derive in more detail in Supplementary Note 4A.
Scaling of the electromechanical coupling. Due to the nature of the inductive coupling, the electromechanical coupling rate is flux tunable (see Eq. (1)). Thus, we expect a linear scaling of the electromechanical coupling rate with ∂ω c /∂Φ and B ext . As discussed in the main text, one option to explore the scaling of g 0 are thermal displacement noise experiments. Alternatively, and more time efficiently, one can measure the frequency shift of the microwave resonator for a fixed mechanical excitation amplitude. In this experiment, the signal amplitudes are significantly larger resulting in a faster data acquisition. In detail, we use a piezo actuator mounted on the outside of the sample enclosure to excite the nanostring. The piezo actuator is addressed using a fixed excitation voltage. As the displacement strength is independent of ∂ω c /∂Φ and B ext , the piezo actuator induces a displacement of constant amplitude. Thus, the recorded frequency shift δω c is determined only by the coupling strength, without the need for a calibration of the piezo susceptibility. If a quantification of the vacuum coupling strength is desired, the extracted frequency shift has to be calibrated independently. For that purpose, we use the spectroscopy of the nanostring's thermal motion.

Data availability
Experimental data are available upon reasonable request at the corresponding author.
Received: 30 August 2020; Accepted: 18 November 2020;  Fig. 4 Scaling of the electromechanical coupling rate at T = 126 mK. Panel a shows the mechanically driven resonator shift δω c close to Ω m for a coherent mechanical drive at angular frequency Ω generated by a piezo actuator for the various operation points labeled in Fig. 2. The external static magnetic field is set to B ext ≈ −440 μT. At magnetic field bias points (e.g., C, D) with low flux responsivity, almost no mechanical signature is observed, in contrast to those points (E to H) with large flux responsivity. Panel b displays the electromechanical coupling rate g 0 deduced from the response in panel (a). The derived values are plotted versus the flux responsivity for low (light green triangles) and high piezo drive amplitudes (dark green triangles), and are linked to the thermal displacement measurements (red cross). This data confirms the linear scaling of the electromechanical coupling strength with ∂ω c /∂Φ. In panel c, we plot the coupling rate g 0 obtained both from driven (green) and thermal displacement measurements (red) versus the applied magnetic field B ext for a fixed flux responsivity of ∂(ω c /2π)/∂Φ = 6.6 GHz/Φ 0 . In the given experimental configuration, the external field sets the working point of the microwave resonator. As flux jumps occurred during the experiments the recorded driven measurements could not be performed at identical field bias as the thermal ones. The error bars on the data in panels (b) and (c) include the standard error originating from the model-fit to the data and errors originating from the calibration. If not shown, error bars are smaller than the symbol size.