Abstract
Cavityelectromechanical systems are extensively used for sensing and controlling the vibrations of mechanical resonators down to their quantum limit. The nonlinear radiationpressure interaction in these systems could result in an unstable response of the mechanical resonator showing features such as frequencycombs, perioddoubling bifurcations and chaos. However, due to weak lightmatter interaction, typically these effects appear at very high driving strengths. By using polariton modes formed by a strongly coupled fluxtunable transmon and a microwave cavity, here we demonstrate an electromechanical device and achieve a singlephoton coupling rate \(\left({g}_{0}/2\pi \right)\) of 160 kHz, which is nearly 4% of the mechanical frequency ω_{m}. Due to large g_{0}/ω_{m} ratio, the device shows an unstable mechanical response resulting in frequency combs in subsingle photon limit. We systematically investigate the boundary of the unstable response and identify two important regimes governed by the optomechanical backaction and the nonlinearity of the electromagnetic mode. Such an improvement in the singlephoton coupling rate and the observations of microwave frequency combs at singlephoton levels may have applications in the quantum control of the motional states and critical parametric sensing. Our experiments strongly suggest the requirement of newer approaches to understand instabilities.
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Introduction
Light carries momentum, and it can be used to control and manipulate the motion of a mechanical resonator down the quantum regime^{1}. Such control over the motional states is essential for technological advancement as well as to probe the fundamental physics^{2,3}. In cavityelectromechanical devices, two linear modes, namely an electromagnetic (EM) mode and a mechanical mode are coupled with the nonlinear radiationpressure interaction^{1,4}. In the microwave domain, the vibrations of the mechanical resonators are typically integrated into the EM mode using charge modulation^{5,6,7,8,9,10,11,12,13}. Recently, cavity electromechanical devices utilizing the magneticflux modulation of Josephson inductance have shown interesting results such as large electromechanical coupling rates^{14,15,16,17}, the nearground state cooling of the mechanical resonator by fourwavemixing^{18}, and Kerrenhanced techniques^{19}. Further, such devices have been proposed to reach the singlephoton strong coupling regime using its linear scaling with the magnetic field^{20,21,22}.
Apart from providing the electromechanical coupling, the nonlinear nature of the Josephson inductance in fluxcoupled electromechanical systems can be a valuable resource^{18,19}. It allows us to control the nature of the electromagnetic mode to change from the weakKerr to a single photon strongKerr oscillator where the response remains nonlinear down to a single excitation. Further, under certain control parameters, the electromagnetic (EM) mode can made to undergo a dissipative phase transition with strong fluctuations in the photon number^{23}. Therefore, coupling a mechanical resonator to such an EM mode using radiationpressure interaction offers a unique platform to study and discern the effects stemming from the nonlinear nature of the radiationpressure interaction^{13,24}, as well as the ones from the nonlinear nature of the electromagnetic mode^{23}.
Here, we demonstrate an electromechanical device consisting of a linear cavity, a frequency tunable transmon qubit, and a mechanical resonator^{15}. First, the fluxtunability of the transmon qubit is used to implement the electromechanical coupling. Second, tuning of the transmon frequency in resonance with the cavity forms the new eigenstates due to the designed strong coupling. The anharmonicity of the resultant polariton modes can be controlled by transmoncavity detuning^{25,26}. Such a tripartite system thus allows us to control the electromechanical coupling and the nonlinearity of the EM mode in a single device. Therefore, by controlling the strength of the nonlinearity, the electromechanical effects can be explored as the EM mode undergoes various regimes such as supersplitting, multiphoton transitions^{25,27}, critical slowing down^{28}, photon blockade breakdown^{29,30}, and transmonionization^{31}.
Results
Device concept
The frequency tunable transmon qubit is enabled by a SQUID loop and a capacitor. Figure 1a shows the schematic design of the device. The coaxial cavity is placed inside a 2axis vector magnet, allowing us to control the axial and the normal components of the magnetic field independently. The cavity is machined from oxygenfree high conductivity (OHFC) copper. The transmon qubit is fabricated on an intrinsicSi chip using electron beam lithography and shadow evaporation of aluminum. To achieve a larger critical inplane magnetic field, we use 28 nm thin Al film to fabricate the device^{32}. The mechanical resonator is realized by suspending one of the arms of the SQUID loop by selective isotropic dry etching of the silicon substrate. The transmon frequency can be tuned by varying the magnetic flux through the SQUID loop.
A false color scanning electron microscope image of the SQUID loop with the suspended mechanical resonator is shown in Fig. 1b. The patterned chip is placed inside the coaxial cavity and cooled down to 20 mK in a dilution refrigerator. The details of device fabrication, and measurement setup are given in Supplementary Note 1. Since superconducting films show more resilience to the inplane magnetic field^{32}, we focus on the electromechanical coupling of the outofplane vibrational mode of the mechanical resonator. Further, by using the control over the normal component of the magnetic field, we can cancel any outofplane component of the magnetic field arising from the misalignment between the axial field and SQUID plane.
Measurements
We report measurements from two similar devices but with substantially different transmoncavity coupling rates. Detailed parameters for both the devices are listed in Supplementary Note 2. We first begin by measuring the voltage transmission ∣S_{21}(ω)∣ of Device1 through the cavity while varying the magnetic flux through the SQUID loop. Depending on the flux passing through the SQUID loop, the transmon frequency can vary from its maximum value to a minimum value. Figure 1c shows the color plot of the cavity transmission. As the transmon is brought in resonance with the cavity, the measured transmission splits into two wellseparated dressed modes as shown in Fig. 1d, demonstrating the strong coupling. Provided low strength of the input power, the dressed cavity mode for Φ/Φ_{0} = 0.5, and the vacuum Rabisplit peaks (VRS) for Φ/Φ_{0} = 0.27 have the characteristic Lorentzian lineshape. From the separation of VRS, we infer the strength of the transmoncavity coupling to be 72 MHz. The reduction in the peak height of VRS in resonant conditions suggests that energy dissipation via the transmon qubit dominates over the cavity dissipation rate.
In the resonant limit of transmoncavity, it is suitable to describe the system in terms of new eigenstates, denoted by upper and lower polariton modes with frequency ω_{+} and ω_{−}, respectively. Away from the resonant limit, these modes exhibit a cavitylike or a transmonlike behavior with distinct anharmonicity as represented by the gradual change of color in Fig. 1e. These polariton modes retain the flux tunability due to underlying SQUID inductance, allowing them to be utilized for implementing electromechanical interaction. Further, due to the large spectral separation between the two modes, we can analyze and describe each mode’s optomechanical effect separately. Considering the upper mode, the Hamiltonian of the tripartite system can be reduced to
where ω_{+}, K_{+}, ω_{m}, and g_{+} are the upper mode frequency, Kerr nonlinearity, mechanical mode frequency, and singlephoton electromechanical coupling rate, respectively. The ladder operators of the upper polariton mode and the mechanical mode are denoted by \({\hat{a}}_{+}({\hat{a}}_{+}^{{{\dagger}} })\) and \(\hat{b}({\hat{b}}^{{{\dagger}} })\), respectively. The singlephoton electromechanical coupling rate of the upper polariton mode can be expressed as g_{+} ≃ ξG_{+}B^{∥}l x_{zp}, where G_{+} = dω_{+}/dΦ is the flux responsivity of the upper polariton mode, B^{∥} is the axial component of the applied magnetic field, l is the length of the mechanical resonator, x_{zp} is the quantum zeropoint fluctuations of the mechanical resonator, and ξ is a geometrical factor of order unity and depends on the mechanical modeshape. It is evident that a higher singlephoton electromechanical coupling rate can be achieved by increasing the fluxresponsivity G_{+} and magnetic field B^{∥}. However, the increased fluxresponsivity of the dressed mode comes at the cost of reduced transmission. For the experiments discussed in the next section, we choose an operating frequency ω_{+} in Device1 that is 20–80 MHz detuned from the bare cavity frequency.
Cavityenabled qubitphonon absorption
To probe the electromechanical coupling, we use a pump–probe scheme, similar to the electromagnetically induced transparency technique^{33,34}. Figure 2a shows the schematic of the different transitions involved in the measurement. Two coherent signals are sent to the device, i.e., a pump signal at a red detuned frequency (ω_{+}–ω_{m}) which drives the \( \vert 0 , \, m + 1 \rangle \leftrightarrow \vert +, \, m \rangle \) transition, and a weak probe signal at ω_{p} (near ω_{+}) to measure the transmission. We emphasize that due to the tunable nature of the polariton modes in our device, we can carry out such experiments in both low and strong anharmonicity limits, as shown in the lower panel of Fig. 2a.
We first perform the experiment in Device1 with a cavitylike mode. Without the pump, the voltage transmission ∣S_{21}(ω_{p})∣ shows the characteristic Lorentzian lineshape, as shown in Fig. 2b. In the presence of the pump, an absorption feature appears in the transmission spectrum, as shown in Fig. 2c. It arises from the destructive interference of the probe field from two different pathways^{35}. The shape of the absorption feature is determined by the response of the mechanical resonator and can be used to extract its resonant frequency and effective dissipation rate. From these measurements, we obtain the mechanical resonant frequency and linewidth to be ω_{m}/2π ~3.97 MHz and ~13 Hz, respectively. In addition, we determine the singlephoton coupling strength g_{+} by fitting the absorption feature with an analytical expression calculated by treating the electromagnetic mode as an anharmonic oscillator (Supplementary Note 3). We repeat the experiment for various pump strengths and extract the electromechanical coupling strength g_{+}/2π ~ 40.0 ± 2.7 kHz, with uncertainty arising from the statistical measurements. Since the singlephoton electromechanical coupling rate is linearly proportional to the axial magnetic field B^{∥}, we carry out similar absorption experiments at different magnetic fields and calculate the singlephoton electromechanical coupling rates. Figure 2d shows the plot of experimentally determined g_{+} for different values of the magnetic field.
Next, we utilize the transmonlike mode which exhibits a significantly larger anharmonicity compared to its dissipation rate. These measurements are done on Device2, which is designed to have a transmoncavity coupling of J/2π ~ 193 MHz. For the threemode system discussed here, measurement of the transmon–phonon interaction is enabled by the cavity, we call this process cavityenabled qubitphonon absorption (CEQA). In the large anharmonicity limit, the absorption feature in probe transmission is shown in Fig. 2e. Due to its large anharmonicity, the CEQA feature effectively arises dominantly from the participation of ground and first excited states of the transmonlike mode. Consequently, we can approximate the electromechanical interaction as an effective twolevel system longitudinally coupled to a mechanical resonator. Keeping this in mind, we derive an analytical expression for the absorption feature and fit the experimental data, resulting in the black line shown in Fig. 2e. Additional details pertaining to the calculations are given in Supplementary Note 3. We emphasize that due to large g_{+} in the system, the CEQA feature appears in both weak and strong nonlinearity regimes down to the meanphoton occupation of ~10^{−1} and ~10^{−3}, respectively.
Optomechanical backaction and instability
A high singlephoton coupling rate results in significant optomechanical backaction at very low pump strengths. To investigate the effects of dynamical backaction on the mechanical resonator, we send a pump signal and measure the power spectral density (PSD) of the output microwave signal while varying the pump power and detuning. Depending on the pump detuning, an imbalance between the up and downscattering rates of microwave photons by the mechanical resonator results in its cooling or heating. We first operate at a lower magnetic field of B^{∥} ~ 18 mT, and at a mode frequency of ω_{+}/2π ~ 5.873 GHz, which corresponds to an estimated Kerr nonlinearity of K_{+}/2π ~ 5.1 MHz/photon (see Supplementary Note 2). At such operating frequency, the upper polariton mode has an estimated flux responsivity of G_{+}/2π ~ 1.16 GHz/Φ_{0}, and expected g_{+}/2π ~ 13.4 ± 0.8 kHz (see Supplementary Note 2). Figure 3a shows the measurements of the effective mechanical linewidth and the shift in the mechanical frequency, extracted from the PSD measurements. As expected, we observe a broadening of the mechanical linewidth for negative detunings and heating and an unstable response for the positive detunings. With the increase in the strength of the pump signal, the backaction effects become enhanced, and the onset of unstable response shifts towards negative detuning due to the Kerr nonlinearity.
To theoretically understand the experimental observation, we model the system as a weak Kerr oscillator with an energy decay rate κ/2π ~ 14 MHz, estimated from the transmission spectrum. We incorporate the pump by adding a driving term in the effective Hamiltonian given by Eq. (1). To compute the backaction effects on the mechanical resonator, we solve the equations of motion for the coupled modes and obtain the expressions of the effective linewidth Γ_{m} and the frequency shift Δω_{m} of the mechanical resonator (see Supplementary Note 4). The solid black lines in Fig. 3a are the theoretically calculated results.
We next focus on the unstable response of the mechanical resonator. It is defined when the first and second mechanical sideband amplitudes show an abrupt change with the pump power (as shown in Supplementary Fig. 6). Such large vibrational motion of the mechanical resonator produces a prominent frequency comb structure in the PSD with multiple peaks separated by ω_{m}, as shown in Fig. 3b. In linear cavity optomechanics, such features have been studied extensively, both theoretically and experimentally^{1}. We investigate the boundary of the unstable response by varying the pump power and its detuning. Figure 3c shows the results from Device2, while operating at ω_{+}/2π ~ 5.82 GHz, corresponding to an energy decay rate of κ/2π ~ 9 MHz and an electromechanical coupling rate of g_{+}/2π ~ 45.0 ± 1.9 kHz, which is determined from a separate CEQA experiment. It is important to note that due to such a large coupling, the onset of mechanical instability occurs at a mean photon occupation of 0.01 in the polariton mode. To theoretically understand the boundary of the instability, we use a weakKerr model as mentioned earlier (Supplementary Note 4). Mechanical instability is defined when the effective mechanical damping rate becomes nonpositive. The results from these calculations are shown by the solidblack line in Fig. 3c. By comparing the two theoretical results, we observe that due to the nonlinearity present in the system, the onset of instability shifts toward lower frequencies as the pump power is increased.
Mechanical Instabilities in singlephoton strong Kerr limit
We first note that the response of the circuitQED system can be substantially different at high driving powers. Figure 4a, b shows the cavity transmission ∣S_{21}∣ of Device2 while varying B^{⊥}. Apart from the dominating vacuumRabi splitting, additional transitions arising from higher levels can also be seen. A schematic of the new (polariton) eigenstates is shown in the right panel of Fig. 4c. The left panel depicts the uncoupled states as \(\vert {n}_{c},\, {n}_{q}\rangle\), where n_{c} and n_{q} are the number of excitation in cavity and transmon, respectively. The symmetric and antisymmetric combination of single excitation states \(\left\vert 1,\, 0\right\rangle\) and \(\left\vert 0,\, 1\right\rangle\) are denoted by \(\left\vert+\right\rangle\) and \(\left\vert \right\rangle\), respectively. Similarly, the new eigenstates in the twoexcitation manifold are labeled as \(\left\vert \alpha \right\rangle\), \(\left\vert \beta \right\rangle\), and \(\left\vert \gamma \right\rangle\). The additional peaks in the transmission spectrum of Fig. 4b are arising from the higherlevel transitions, namely \(\{\left\vert+\right\rangle,\, \left\vert \right\rangle \}\leftrightarrow \{\left\vert \alpha \right\rangle,\, \left\vert \beta \right\rangle,\, \left\vert \gamma \right\rangle \}\). Such transitions are possible with a single frequency drive due to nonzero thermal occupation of \(\left\vert+\right\rangle\) and \(\left\vert \right\rangle\) states.
To identify these transitions, we treat the transmon as a nonlinear oscillator and model the system using an extended Jaynes–Cummings Hamiltonian added with a coherent drive. We numerically solve the Markovian master equation to compute the voltage transmission. The results are plotted as a solid black line in Fig. 4b. It can be seen that the transition frequencies corresponding to the peaks in the transmission spectrum match closely with the numerical results. Clearly, these transitions arise from the singlephoton excitation and are distinct from the multiphoton transitions^{25}, which usually occur at stronger drive strengths.
We next activate the electromechanical coupling by applying the parallel magnetic field B^{∥} ~ 9 mT and investigate the boundary of the unstable response of the mechanical resonator in a wider span of pump power and frequency. Figure 5a, c shows the region of unstable response from Device2 at two different values of fluxoperating point, and Fig. 5b, d shows the corresponding transmission spectra at B^{∥} = 0. For the operating flux in Fig. 5a, the transmon qubit becomes near resonant to the cavity, and the electromechanical coupling rate of the higher frequency polariton mode is estimated to reach g_{+}/2π ~ 160 kHz, which is nearly 4% of the mechanical mode frequency ω_{m}. As a consequence, the onset of mechanical instability takes place at a very low mean photon occupation of n_{d} ~ 3 × 10^{−4}. The onset of mechanical instability coincides with the bluedetuned region of the transition frequency ω_{+} and ω_{−}, similar to the situation in the linear cavity optomechanics. As pump power is increased, the mechanical instability emerges near the higher transition frequencies around ω_{−β}, and possibly around ω_{−α} or ω_{+β}. With further increases in the pump power, we observe instabilities near the bare cavity frequency. Surprisingly, we do not see any instability branch surrounding the transition frequency ω_{+γ}. We also note that unstable mechanical response resulting in frequency comb features in PSD persist even beyond −5 dBm pump power. At these powers, we expect transmon to be ionized, and SQUID would operate in a nonzero voltage regime^{31}. Due to the decoupling of the transmon mode, the transmission shows the bare cavity response, as shown in the lower panel of Fig. 5b.
Figure 5c shows similar measurements but at a different fluxoperating point, such that the transmon qubit is approximately 240 MHz detuned above the cavity frequency. An instability region corresponding to singlephoton transitions with frequency ω_{−β} and ω_{+γ} emerges, along with the instability branches surrounding the frequencies ω_{+} and ω_{−}. At moderately higher power, the instability region near the twophoton transition corresponding to frequency ω_{γ}/2 can be seen. We also observe the supersplitting of the ω_{−} peak, resulting in an abrupt widening of the instability region.
Semiclassical analysis and a model based on a quantum twolevel system
To gain insight into these observations, we plot the transmission measurements ∣S_{21}∣ over a large frequency at B^{∥} = 0 and the boundary of instability, together as shown in Fig. 6a. The instability parameterspace can be divided into three regions—(i) the lowpower region, where the singlephoton instabilities stem from the lower transitions, (ii) the midpower region, where higher energy single and multiphoton transitions, supersplitting, bistability of the electromagnetic mode are important, and (iii) the ionization region, where the frequencycomb persists despite the SQUID operating in the nonzero voltage regime.
As the EM mode emerges from the strong coupling between the transmon and the cavity, we model it by expanding it using polariton eigenstates to capture the lowpower behavior. Such an approach effectively breaks down the EM mode into independent twolevel systems, which are longitudinally coupled to a mechanical resonator. Details of the model are included in the Supplementary Note 6. Results from these calculations are plotted in Fig. 6b. As expected, the theoretical model is able to capture the behavior at low pump powers, however, it does not capture the experimental observations at intermediate or high pump powers.
For intermediate powers, we expect the system to behave more classically. Therefore, we also attempt a model based on the semiclassical analysis while treating the transmon as an anharmonic oscillator (details are included in Supplementary Note 5). We first note that a driven transmoncavity system alone (no optomechanical coupling) can have bistable and unstable solutions of the intracavity field^{36}. We use a semiclassical approach and perform the linear stability test^{37}. We identify the regions with one unstable fixedpoint (FP), and regions with one unstable and two stable FP, as shown in Fig. 6c. In these regions, the intracavity field is expected to show either an unstable or a bistable behavior resulting in large photonfluctuations.
Next, we include the optomechanical interaction, and obtain the fixedpoints. The regions of the mechanical instability is shown by the dashed lines in Fig. 6c. At low powers, the semiclassical analysis could produce the mechanical instability region near the ω_{±} polaritons, arising from the optomechanical backaction. In addition, we note that in the optically bistable region, one of the stable solutions also give rise to mechanical instability.
Clearly, the quantum model based on the polaritonbasis, and the threemode semiclassical analysis do not capture the experimentally observed behavior completely. While the model based on the polaritoneigenstates captures the lowpower behavior, it is not as effective in the midpower range. Particularly, in the middle part of the region (ii), the transmoncavity system undergoes a dissipative phase transition and can take a long time to reach equilibrium^{23,28,30}. In this region, the presence of an unstable response in the experimental data, and the absence of it in the modeled results strongly suggest the role of fluctuating photon pressure on the mechanical resonator during the transition. These observations would require theoretical investigations beyond the standard semiclassical or twolevel approach^{38}.
Discussion
To summarize our findings, our current experiment reaches a singlephoton coupling rate which is nearly 4% of the mechanical resonator frequency. It highlights that instabilities arising from the residual thermal occupations of the mechanical and EM modes would become important as one reaches the singlephoton strong coupling regime g_{0} ≳ {κ, ω_{m}} unless the mechanical oscillator is cooled to its quantum ground state. Singlephoton strong coupling regimes seem experimentally feasible as transmon qubits have been shown to operate in higher magnetic fields^{32}.
By changing the dimensions of the mechanical resonator and operating it in a higher magnetic field, it is feasible to achieve groundstate cooling via sideband driving below the level of a single photon. This also suggests ways to prepare nonclassical mechanical states, including Schrödingercat states^{21,22,39}. Such methods and techniques can be extended to the lowfrequency fluxfamily superconducting qubits to realize transverse electromechanical couplings, thus extending the toolbox available with the fluxcoupled devices^{40}. The generation of microwave frequency combs at the singlephoton level could have applications in quantum sensing^{41,42}. Our experimental observations further demand theoretical investigation into the parametric instabilities in single photon limit.
Methods
The coaxial cavity is machined from oxygenfree highconductivity copper. The central post (solid cylinder in the lower half) has a length of 11.5 mm and a diameter of 2.5 mm. The inner diameter of the outer cylinder is 5.5 mm, and the total height of the cavity is 20.5 mm. The transmon qubit is fabricated on an intrinsic silicon(100) substrate. Using EBL and shadow evaporation of aluminum, we fabricate the device using a single step of lithography. The evaporated aluminum film is annealed under ambient conditions to transform the compressive stress into tensile stress. Selective etching of silicon is then carried out to release the nanowire (mechanical oscillator) from the substrate. To nullify the effects of annealing and etching on Josephson junction inductance, the oxidation parameter is tuned accordingly at the time of deposition so that we get the desired junction resistance at the end of all processes. Finally, the silicon chip is mounted inside the coaxial cavity and the cavity is placed inside a homebuilt vector magnet setup. This assembly is placed inside a double layer of infrared and magnetic field shields and mounted to the mixing chamber plate for cooling down to 20 mK for measurements.
Data availability
All the datasets used in this study are available in the Zenodo.
Code availability
The codes of this study are available from the corresponding authors upon request.
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Acknowledgements
T.B. thanks Bhoomika Bhat, Harsh Vardhan Upadhyay and S. Majumder for assisting during the device fabrication and helpful discussions. This work is supported by the Air Force Office of Scientific Research under Award No. FA23862014003. G.S.A. thanks the Infosys Foundation Chair of the Department of Physics, IISc Bangalore, which made this collaboration possible, and the AFOSR award no FA95502010366 for the support. The authors acknowledge the support under the CoEQT by MEITY and the QuEST program by DST, Govt. of India. The authors acknowledge device fabrication facilities at CeNSE, IISc Bangalore, and central facilities at the Department of Physics funded by DST (Govt. of India).
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V.S. conceived and supervised the experiment. T.B. fabricated the device and performed the measurements. T.B. and V.S. have done the data analysis. T.B., M.K., and G.S.A. carried out the theoretical calculations. All the authors have contributed to preparing the paper.
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Bera, T., Kandpal, M., Agarwal, G.S. et al. Singlephoton induced instabilities in a cavity electromechanical device. Nat Commun 15, 7115 (2024). https://doi.org/10.1038/s4146702451499z
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DOI: https://doi.org/10.1038/s4146702451499z
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