Abstract
Control over the quantum states of a massive oscillator is important for several technological applications and to test the fundamental limits of quantum mechanics. Addition of an internal degree of freedom to the oscillator could be a valuable resource for such control. Recently, hybrid electromechanical systems using superconducting qubits, based on electriccharge mediated coupling, have been quite successful. Here, we show a hybrid device, consisting of a superconducting transmon qubit and a mechanical resonator coupled using the magneticflux. The coupling stems from the quantuminterference of the superconducting phase across the tunnel junctions. We demonstrate a vacuum electromechanical coupling rate up to 4 kHz by making the transmon qubit resonant with the readout cavity. Consequently, thermalmotion of the mechanical resonator is detected by driving the hybridizedmode with meanoccupancy well below one photon. By tuning qubit away from the cavity, electromechanical coupling can be enhanced to 40 kHz. In this limit, a small coherent drive on the mechanical resonator results in the splitting of qubit spectrum, and we observe interference signature arising from the LandauZenerStückelberg effect. With improvements in qubit coherence, this system offers a platform to realize rich interactions and could potentially provide full control over the quantum motional states.
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Introduction
Cavity optomechanical systems, where a mechanical mode parametrically modulates the resonant frequency of an electromagnetic (EM) mode, have been very successful in controlling the motional states of massive oscillators^{1}. Starting from the earlier demonstration of the motional quantum ground state by the sideband cooling technique^{2,3}, these experiments have reached several milestones related to the displacementdetection^{4} and the preparation of the nonclassical states of mechanical motion^{5,6}. Beyond the traditional twomode systems, consisting of one EM and one mechanical mode, cavity optomechanical systems with an auxiliary mode provides a wide range of interactions. Such systems have been used to realize nonreciprocal devices^{7,8,9}, and to demonstrate quantum entanglement between two mechanical resonators^{10,11}.
Among the twomode cavity optomechanical devices, preparation of the quantum states of motion appears to be technologically challenging. One successful strategy in the microwave domain, to circumvent this, is to introduce an auxiliary nonlinear mode such as a qubit. The qubit can be used as a singlephoton source^{12}, photoncounter^{13}, or directly coupled to a mechanical mode using its charge dispersion^{14,15,16,17} or the piezoelectric effect^{18,19,20}. In such devices, the qubit mode “acts” like an additional degree of freedom which couples to mechanical mode via an intermediate mode or directly using the “charge” dispersion.
While systems utilizing the chargebased coupling have been studied extensively, the experimental progress of the hybrid systems based on magneticflux has been very limited. Here we design and study the performance of a hybrid electromechanical device based on a fundamentally different coupling scheme based on the magnetic flux. We engineer the device parameters such that in addition to the fluxbased electromechanical coupling, one mode maintains sufficient anharmonicity to be qualified as a qubit. This approach results in an electromechanical system with an internal spinhalf degree of freedom. By recording the thermomechanical motion of the resonator, we demonstrate a magnetic field tunable electromechanical coupling rate up to 4 kHz when qubit is tuned in resonance with the cavity. On one hand, the strong and tunable nonlinearity of the qubit mode improves the displacement sensitivity. On the other hand, the large electromechanical coupling modifies the qubit spectrum in the dispersive limit. As a consequence, we observe the Landau–Zener–Stückelberg (LZS) interference in the qubit spectrum when the qubit is detuned in the dispersive limit. Similar to vacuumelectromechanical coupling rate’s scaling with total charge in chargedispersionbased schemes^{14,15,16}, the coupling rate here scales linearly with the magnetic field. Therefore, such an approach has the potential to reach the elusive singlephoton strong coupling regime with suitable choice of materials^{21}.
Results and discussion
Concept
The hybrid device consists of a transmon qubit coupled to a mechanical resonator and a readout cavity, as shown in Fig. 1a. The transmon qubit couples to the cavity via a dipole coupling, commonly referred to as transverse coupling, as it connects the ground and the first excited state of the qubit^{22}. The mechanical resonator couples to the qubit via a fluxmediated coupling. Such coupling is achieved by embedding a mechanical resonator into one of the arms of a SQUID (Superconducting Quantum Interference Device) loop, which provides the necessary Josephson inductance to form a transmon qubit. Due to the quantum interference of the superconducting phase, the Josephson inductance of the SQUID depends on the magnetic flux threading the loop as schematically shown in Fig. 1b. In the presence of a magnetic field, applied normal to the plane of the SQUID, it acts like a displacementdependent nonlinear inductor. By shunting the SQUID “inductor” to a suitable capacitance, a transmon qubit mode can be designed. As the motion of the mechanical resonator directly affects the qubit transition frequency, this coupling is referred to as longitudinal coupling. A fluxcoupled hybrid system formed this way can be thought of a dual to the “charge” coupling approach realized with the Cooperpair box qubit^{16}.
Theoretically, the fluxmediated electromechanical coupling has been considered in the context of fluxqubits^{23}, cavityelectromechanical devices^{24}, and more recently with the transmon qubit^{21,25}. On the experimental side, the scheme has been used for large bandwidth displacement detection^{26}, and to demonstrate the cavityelectromechanical system by embedding Josephson elements in the microwave circuitry^{27,28}.
In comparison to existing fluxcoupling approaches^{27,28}, our design methodology circumvents several issues by using a tunable transmon mode. First, the requirement of large Josephson inductance for transmon design helps in suppressing hysteretic effects with magnetic flux arising from geometrical and kinetic inductance. Second, our approach here is to implement a longitudinal coupling between transmon qubit and a mechanical resonator through the modulation of Josephson inductance. This enables the interaction of the mechanical mode with the qubit in two distinct ways. First, due to strong coupling between the qubit and cavity in the resonant limit, the mechanical motion directly couples to the hybridized states. Second, in the dispersive limit when the qubit is detuned far away from the cavity, a sufficiently large coupling between the qubit and the mechanical mode can be maintained. It thus provides a mean to use the qubit as an internal degree of freedom to the mechanical mode and further paves ways for measurementbased cooling and control protocols^{25,29,30}.
Device design
We use a threedimensional (3D) cavity to implement the transmon design as shown schematically in Fig. 1c. Unlike the conventional 3Dtransmon qubit, which couples differentially to the cavity mode, we design a singleended qubit mode by grounding one end of the SQUID loop to the cavity wall using a small wirebond^{31}. The other end of the SQUID loop extends toward the center of the cavity and provides the necessary qubit capacitance and coupling with the fundamental cavity mode. The rectangular cavity (35 × 4 × 35 mm^{3}) is machined using copper with the fundamental resonant mode TE_{101} at ω_{c} ≈ 2π × 6 GHz. A falsecolor scanning electron microscope (SEM) image of the SQUID loop is shown in Fig. 1d. The nanobeamshaped mechanical resonator, formed by a 100nm highly stressed SiN film coated with 50 nm of aluminum, and suspended part of the Josephson junctions can be clearly seen. Important device parameters are listed in the Supplementary Note 1. The offset in the SQUID position (away from the center of the cavity) in transmon design allows us to bring a RF drive line for the electrostatic actuation of the mechanical resonator. Results from design simulations are included in the Supplementary Note 3.
The transmon qubit frequency ω_{q} is given by \(\hslash {\omega }_{{\mathrm{q}}}\approx \sqrt{8{E}_{{\mathrm{C}}}{E}_{{\mathrm{J}}}^{0}\left\cos \left(\pi {{\Phi }}/{{{\Phi }}}_{0}\right)\right}{E}_{{\mathrm{C}}}\), where \({E}_{{\mathrm{J}}}^{0}\) is the maximum Josephson energy, E_{C} is the charging energy, Φ is the total flux threading the SQUID loop, and Φ_{0} = h/2e is the magnetic flux quanta. The tunability of qubit frequency with flux allows access to its dispersive or resonant interaction with the cavity. The interaction between the qubit and the cavity mode can be expressed as \(\hslash J(\hat{a}{\hat{\sigma }}^{+}+{\hat{a}}^{\dagger }{\hat{\sigma }}^{})\), where \(\hat{a}\)(\({\hat{\sigma }}^{}\)) is the ladder operator for the cavity(qubit) mode and J is the dipole coupling rate. The electromechanical coupling arises from the modulation of qubit frequency caused by mechanical displacement. As the qubit frequency can be tuned over a large range, it is convenient to define the vacuum electromechanical coupling rate between the qubitcavity hybridized states \(\left({\omega }_{\pm }\right)\) and the mechanical resonator as,
where \({G}_{{{\Phi }}}^{\pm }=\partial {\omega }_{\pm }({{\Phi }})/\partial {{\Phi }}\) is the fluxresponsivity, x_{zp} is the quantum zeropoint fluctuations of the mechanical resonator, and w is the effective width of the SQUID loop. Equation (1) defines the coupling rate over the entire range of qubit frequencies. The hybridized modes \({\omega }_{\pm }=\bar{{{\Delta }}}\pm \sqrt{{({{\Delta }}/2)}^{2}+{J}^{2}}\) with Δ = ω_{q} − ω_{c}, \(\bar{{{\Delta }}}=({\omega }_{{\mathrm{q}}}+{\omega }_{{\mathrm{c}}})/2\) approach the uncoupled qubit and cavity frequencies in the dispersive limit. Restricting the coupled qubit–cavity system to single excitation subspace, in the resonant limit Δ ≪ J, the hybridized modes essentially act like independent cavity optomechanical systems. However, it is worth pointing out here that the interaction between the hybridized modes and the mechanical motion can be used to enhance the quantum nonlinearity by designing J comparable to the mechanical frequency^{32,33}.
Qubit spectroscopy and fluxresponsivity
We use spectroscopic measurements to characterize the qubit. Figure 2a shows the transmission (∣S_{21}(ω)∣) through the cavity as applied magnetic flux is varied. When the qubit becomes resonant with the cavity, the vacuum Rabi splitting is observed which signifies the strong coupling between the qubit and cavity mode. We determine a dipole coupling rate J = 2π × 85 MHz, the bare cavity frequency ω_{c} = 2π × 5.993 GHz, maximum qubit frequency \({\omega }_{{\mathrm{q}}}^{0}=2\pi \times\)7.982 GHz, and an anharmonicity of −132 MHz (see the Supplementary Note 4 for details). Due to the fluxperiodicity of qubit frequency, the vacuum Rabi splitting pattern repeats with every new fluxquanta added. The extension panel of Fig. 2a shows the transmission measurement at a higher magnetic field. Apart from a small reduction (~15 MHz) in the maximum qubit frequency and an increase in the dressedcavity mode linewidth, we do not observe any significant change in the device parameters up to a field of ~3.7 mT (310 Φ_{0}).
To understand the fluxtransduction of hybridized modes, we compute the fluxresponsivity \({G}_{{{\Phi }}}^{\pm }\) using the measured qubit and cavity parameters and assuming identical junctions. Figure 2b shows the plot of \({G}_{{{\Phi }}}^{\pm }\) with respect to the hybridizedmode frequencies. The fluxresponsivity of the hybridized modes increases as their relative detuning \(\left( {\omega }_{\pm }{\omega }_{{\mathrm{c}}} \right)\) increases. However, reduced transmission at frequencies far away from ω_{c} hinders their use for the mechanical transduction. We choose an optimum operating point of 6.025 GHz, corresponding to \({G}_{{{\Phi }}}^{+}/2\pi \sim\) 1.8 GHz/Φ_{0}, for the mechanical resonator characterization. This fluxresponsivity is significantly larger than the values reported with the SQUID cavity^{27}. In addition, the fluxresponsivity of qubit \({G}_{{{\Phi }}}^{{\mathrm{q}}}=\partial {\omega }_{{\mathrm{q}}}/\partial {{\Phi }}\) can be much larger near the halfinteger flux quantum as shown in Fig. 2c. In the dispersive limit, while the effective coupling between the dressed cavity and mechanical resonator degrades by a factor of \({\left(J/{{\Delta }}\right)}^{2}\), a large coupling between the qubit and the mechanical resonator can be maintained.
Detection of mechanical mode and vacuum electromechanical rate
We first focus on the driven response of the mechanical resonator. For electrostatic actuation, a weak ac signal and a dc voltage V_{dc} are applied at the mechanical drive port. We finetune the magnetic flux near 190 Φ_{0} to operate the hybridized mode ω_{+}/2π at 6.025 GHz. We inject a microwave tone at ω_{+} creating a mean photon occupation of ≈1, calibrated independently using acStark shift. Details are provided in the Supplementary Note 5. The signal that emerges from the cavity is then mixeddown and recorded by a network analyzer. Figure 3a shows the amplitude of the signal in a color plot as the mechanical drive frequency and V_{dc} are varied. The change in color over the background signifies the mechanical resonance. We measure the inplane vibrational mode at ω_{m} ~ 2π × 6.585 MHz with a characteristic capacitive frequency softening with V_{dc}.
Next, we focus on the thermal motion of the mechanical resonator. We operate the hybridized mode at ω_{+} = 2π × 6.025 GHz and drive the system with a microwave tone tuned to lower sideband (ω_{+} − ω_{m}), creating a mean photon occupation of ~0.1. The power spectral density (PSD) of the output signal is then recorded with a spectrum analyzer. The average PSD, along with the fitted Lorentzian, is shown in Fig. 3b. We measure a mechanical linewidth of γ_{m} = 2π × 6 Hz, corresponding to a quality factor of ~1.1 × 10^{6}.
For a drive at the lower sideband, the ratio of integrated power at the upconverted frequency (P_{m}) near ω_{+} to the power of transmitted carrier signal (P_{d}) at ω_{+} − ω_{m} can be conveniently expressed as \({P}_{{\mathrm{m}}}/{P}_{{\mathrm{d}}}={\left(2{g}_{+}/\kappa \right)}^{2}{n}_{{\mathrm{m}}}^{{\mathrm{th}}}\), where \({n}_{{\mathrm{m}}}^{{\mathrm{th}}}\) is the mean thermal occupation of the mechanical mode, and κ is the hybridizedmode linewidth. To estimate the thermal occupation of the mechanical mode, we first measure \((\kappa /2)\sqrt{({P}_{{\mathrm{m}}}/{P}_{{\mathrm{d}}})}\) for different values of applied flux. We fit the expected linear dependence of this quantity on magnetic flux to obtain the slope \(\sqrt{{n}_{{\mathrm{m}}}^{{\mathrm{th}}}}({g}_{+}/{{\Phi }})\). By using the estimated value of g_{+}/Φ from the device parameters (Eq. (1)), we then obtain the thermal phonon occupancy of 169. This corresponds to a mode temperature of 53 ± 4 mK, which is consistent with the independent estimate obtained by varying the fridge temperature (additional details are given in the Supplementary Note 6). Using \({n}_{{\mathrm{m}}}^{{\mathrm{th}}}\) = 169, we then calculate the g_{+} as shown in Fig. 3c.
We emphasize that the vacuum electromechanical coupling rate of g_{+} ~ 2π × 4 kHz is limited by choice of ω_{+}, and the magnetic field range available in our measurement setup. By operating at ω_{c} ± J, one can achieve the \({G}_{{{\Phi }}}^{+}/{G}_{{{\Phi }}}^{{\mathrm{q}}}=1/2\), resulting in g_{+} ~ 2π × 15 kHz. In addition, thin films of Al can withstand a larger magnetic field than the maximum field used here (3.7 mT). As the inplane critical magnetic field is much larger than the perpendicular critical magnetic field for thin Al films, a configuration with field applied inplane to the SQUID loop would result in significantly higher coupling rates for the outofplane mechanical mode.
LZS interference in the dispersive limit
Next, we investigate the system by tuning the qubit away from ω_{c}. In the dispersive limit ∣Δ∣ ≫ J, the mechanics essentially decouples from the cavity mode. While the qubit–cavity interaction is given by \(\left({J}^{2}/{{\Delta }}\right){\hat{a}}^{\dagger }\hat{a}{\hat{\sigma }}_{z}\), the longitudinal interaction between the uncoupled qubit and the mechanical resonator is given by \({g}_{{\mathrm{qm}}}{\hat{\sigma }}_{z}(\hat{b}+{\hat{b}}^{\dagger })\), where \({g}_{{\mathrm{qm}}}=(\partial {\omega }_{{\mathrm{q}}}/\partial x){x}_{{\mathrm{zp}}}\) is the qubitelectromechanical coupling rate and \(\hat{b}\)\(({\hat{b}}^{\dagger })\) is the lowering (raising) operator for the mechanical mode. With a superconducting qubit device, timedependent longitudinal coupling scheme has been used to perform highfidelity qubit measurements^{34}. In the present device, a static g_{qm} would instead result in a small qubitstatedependent displacement \((\sim {g}_{{\mathrm{qm}}}{x}_{{\mathrm{zp}}}/{\omega }_{{\mathrm{m}}})\)^{29}. Here, we focus on the qubit dynamics while driving the mechanical resonator. The qubit is detuned to 4.9 GHz to enhance g_{qm} to 40 kHz, and its spectrum is probed using the twotone spectroscopy technique. The mechanical resonator is coherently actuated at its resonant frequency. It is equivalent to the fluxmodulation of the qubit frequency at ω_{m}, and a frequency deviation set by g_{qm} and the mechanical amplitude.
Figure 4a shows the qubit spectrum as the strength of mechanical drive is varied. We observe a splitting in the qubit spectrum with a weak modulation inbetween. The separation between the primary splitting varies linearly with the mechanical amplitude. The primary splitting can be understood by considering the passage of the system across the region of avoided crossing with separation set by the strength of the spectroscopic tone (the Rabiflop rate Ω_{R})^{35}. As the system moves across the avoided crossing at a rate set by ω_{m}, the transition during multiple passages mix the two states, eventually resulting in almost equal population of the two energy levels. Hence, it results in the splitting of the qubit spectrum. This splitting of qubit spectrum can be captured in semiclassical calculation as shown by the dotted lines in Fig. 4b. Calculation details are included in the Supplementary Note 7.
At large mechanical drive power, the system crosses avoidedcrossing region with higher speed. In this regime, one would expect to see the interference fringes, arising from multiple Landau–Zener transitions, at a separation close to ω_{m}^{35}. In our experiment, as the qubit linewidth is comparable to the modulation frequency ω_{m}, the fringes are not well resolved. Their signatures are visible as weak modulation between the primary splitting. We have performed numerical calculations based on the Lindblad master equation. The solid line in Fig. 4b shows the result from such calculations. Apart from capturing the linear amplitude dependence of the primary splitting, the calculated results show the weak modulation in the experimental data, whereas the semiclassical calculation only shows the primary splitting.
Conclusion
In summary, we have developed a hybrid electromechanical device by integrating a modified 3Dtransmon qubit and extremely lowloss mechanical resonator of SiN/Al. The detection of thermomechanical motion by driving the system with less than one photon highlights the large underlying coupling rate. Accessibility to different regimes of interaction is further demonstrated by the observation of the LZS interference. Looking ahead, by accessing inplane vibration mode through changes in the design geometry and in combination with higher magnetic field, the flux coupling rate can be increased. With further improvements in the coupling strength, the device in consideration can reach resolved sideband regime and strong coupling regime. This could enable experiments in the regime of the singlephoton cooperativity exceeding one, and conditional cooling of the mechanical resonator to the quantum ground state.
Methods
Device fabrication and measurement setup
For device fabrication, we use an intrinsic Si (100) substrate coated with a 100nmthick highstress SiN layer grown using the lowpressure chemical vapor deposition method. Using standard lithography and shadow evaporation techniques, the transmon design is patterned in a single lithography step. To release the mechanical resonator, a combination of dry and wet etching processes is used. First, the exposed SiN is vertically etched by the reactive ion etching using SF_{6} and CHF_{3} plasma. The aluminum film naturally acts as a mask layer and thus protects the SiN underneath it. In the second step of etching, a modifiedTMAHbased etchant is used to remove the exposed silicon, while providing excellent selectivity against Al and SiN. Additional details are given in the Supplementary Note 2. After the wet etch process, the samples are blowdried gently with N_{2}, requiring no critical point drying. The (111)facets of Si resulted from the wet etch process can also be seen in Fig. 1d. The sample placed inside a copper cavity, along with a small solenoid, is kept inside a cryopermshield to protect it from the ambient magnetic field fluctuations.
Data availability
All raw data and processed data as well as supporting code for processing and figure generation are available in Zenodo with the DOI: https://doi.org/10.5281/zenodo.4314415.
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Acknowledgements
V.S. acknowledges the support received from Infosys Science Foundation, and under the “Early Career Research Award” by SERB, Department of Science and Technology (DST), Govt. of India. T.B. and S.M. thank Tata Trust for providing the travel support. The authors acknowledge device fabrication facilities at CeNSE, IISc, Bangalore, and central facilities at the Department of Physics funded by DST. Authors thank R. Vijayraghavan, Manas Kulkarni, Diptiman Sen, and G. S. Agarwal for valuable discussions.
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T.B. and S.M. contributed equally to this work. V.S. conceived the experiments. T.B. and S.M. fabricated the devices. T.B., S.M., and S.K.S. performed the measurements. T.B., S.M., and V.S. performed the data analysis. All the authors contributed to writing of the manuscript and discussing the results.
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Bera, T., Majumder, S., Sahu, S.K. et al. Large fluxmediated coupling in hybrid electromechanical system with a transmon qubit. Commun Phys 4, 12 (2021). https://doi.org/10.1038/s4200502000514y
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DOI: https://doi.org/10.1038/s4200502000514y
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