Large flux-mediated coupling in hybrid electromechanical system with a transmonqubit

Tanmoy Bera,1 Sourav Majumder,1 Sudhir Kumar Sahu,1 and Vibhor Singh1, ∗

1Department of Physics, Indian Institute of Science, Bangalore-560012 (India)(Dated: January 17, 2020)

Control over the quantum states of a massiveoscillator is important for several technologicalapplications and to test the fundamental limitsof quantum mechanics [1–10]. Addition of an in-ternal degree of freedom to the oscillator couldbe a valuable resource for such control [11, 12].Recently, hybrid electromechanical systems usingsuperconducting qubits, based on electric-chargemediated coupling, have been quite successful[13–17]. Here, we realize a hybrid device, con-sisting of a superconducting transmon qubit and amechanical resonator coupled using the magnetic-flux. The coupling stems from the quantum-interference of the superconducting phase acrossthe tunnel junctions. We demonstrate a vacuumelectromechanical coupling rate up to 4 kHz bymaking the transmon qubit resonant with thereadout cavity. Consequently, thermal-motion ofthe mechanical resonator is detectable by driv-ing the dressed-mode with mean-occupancy wellbelow one photon. By tuning the qubit awayfrom the cavity, electromechanical coupling be-tween qubit and mechanical mode can be furtherenhanced to 40 kHz. In this limit, a small co-herent drive of the mechanical resonator resultsinto the splitting of qubit spectrum and we ob-serve interference signature arising from the Lan-dau–Zener–Stuckelberg effect [18]. With furtherimprovements in the qubit coherence, this systemoffers a novel platform to realize rich interactionsand could potentially provide full control over thequantum motional states [19].

Cavity optomechanical systems, where a mechanicalmode parametrically modulate the resonant frequency ofan electromagnetic (EM) mode, have been very success-ful in controlling the motional states of massive oscilla-tors [2]. Starting from the earlier demonstration of themotional quantum ground state by the sideband coolingtechnique [3, 4], these experiments have reached severalmilestones related to the displacement-detection [5] andthe preparation of the non-classical states of mechani-cal motion [6, 7]. Beyond the traditional two-mode sys-tems, consisting of one EM and one mechanical mode,cavity optomechanical systems with an auxiliary modeprovides a wide range of interactions. Such systems havebeen used to realize nonreciprocal devices [8–10], andto demonstrate quantum entanglement between two me-

chanical resonators [20, 21].

Among the two-mode cavity optomechanical devices,preparation of the motional states with non-Gaussianstatistics appears to be technologically challenging. Toaddress this, a strategy in the microwave domain is toutilize the high nonlinearity of a superconducting qubit.One approach is to use the superconducting qubit as asingle-photon source [22] or as a photon detector [23].In such devices, propagation loss and smaller qubit co-herence time limit the state-transfer fidelity between theEM and the mechanical mode. Alternative techniquesbased on the direct coupling of superconducting qubitto the mechanical mode using piezoelectricity, or thecharge-dispersion of the cooper-pair box (CPB) havebeen quite successful, resulting in the observation ofthe multi-phonon Fock states [14], and nearly phonon-resolved splitting of the qubit spectrum [15–17].

Here, we realize a hybrid device consisting of a mechan-ical resonator and a transmon qubit transversely coupledto a readout cavity as shown in Fig. 1(a). The mechan-ical resonator couples to the qubit via a flux-mediatedlongitudinal coupling. Such coupling is achieved by em-bedding a mechanical resonator into one of the arms of aSQUID (Superconducting Quantum Interference Device)loop, which provides the necessary Josephson inductanceto form a transmon qubit. Due to the quantum interfer-ence of the superconducting phase, the Josephson induc-tance of the SQUID depends on the magnetic flux thread-ing the loop as schematically shown in Fig. 1(b). In thepresence of a magnetic field applied normal to the planeof the SQUID, it acts like a displacement-dependent non-linear inductor. By shunting the SQUID “inductor” toa suitable capacitance, a transmon qubit mode can bedesigned. A flux-coupled hybrid system formed this waycan be thought of a dual to the “charge” coupling ap-proach realized with the CPB qubit [15].

Theoretically, the flux-mediated electromechanicalcoupling has been considered in the context of flux-qubits[24], cavity-electromechanical devices [25], and more re-cently with the transmon qubit [19, 26]. On the exper-imental side, the scheme has been used for large band-width displacement detection [27], and to demonstratecavity-electromechanical system by embedding a SQUIDloop in the microwave circuitry [28]. Our approach hereis to implement a longitudinal coupling between trans-mon qubit and a mechanical resonator through the mod-ulation of Josephson inductance. The requirement of

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FIG. 1. Device concept and design: (a) A schematic showing various components of the hybrid electromechanical system.The transmon qubit couples to an electromagnetic (EM) cavity via transverse coupling. A low-frequency mechanical resonatorcouples to the transmon qubit via the longitudinal coupling. (b) A schematic of the SQUID loop with suspended arm. Due tothe magnetic flux Φ(x) dependence of Josephson inductance, it forms a displacement-dependent inductor. (c) A cross-sectionalview of a 3D-cavity based transmon device. Input-output ports for microwaves and a third port added for mechanical actuationis shown. The SQUID loop is placed inside a small recess of the cavity (outside the cavity chamber). (d) A false color SEMimage of the SQUID loop, showing the suspended portion of the Josephson junctions and the nanobeam. The mechanicalresonator has a length and width of 45 µm and 300 nm, respectively. It consists of a 50 nm coating of aluminum over 100 nmthick highly-stressed SiN film. The T -shaped electrode in the lower-half of image is used to actuate the mechanical resonator.

large Josephson inductance for transmon design helps insuppressing hysteretic effects with magnetic flux arisingfrom geometrical and kinetic inductance. The qubit read-out can be performed by coupling it to a linear cavity.The frequency tunable qubit thus allows interaction withthe mechanical mode in two distinct ways. In the reso-nant limit, the mechanical motion directly couples to thehybridized-states of qubit and the cavity. In the disper-sive case, a sufficiently large coupling between qubit andthe mechanical mode can be maintained, thus providinga mean to use the qubit as an internal degree of freedomto the mechanical mode [19, 29].

We use a three-dimensional (3D) cavity to implementthe transmon design as shown schematically in Fig. 1(c).Unlike the conventional 3D-transmon qubit, which cou-ples differentially to the cavity mode, we design a single-ended qubit mode by grounding one end of the SQUIDloop to the cavity wall using a small wirebond [30]. Theother end of the SQUID loop extends towards the centerof the cavity and provides the necessary qubit capaci-tance and coupling with the fundamental cavity mode.The rectangular cavity (35×4×35 mm3) is machined us-ing OFHC copper with the fundamental resonant modeTE101 at ωc ≈ 2π× 6 GHz. A false color SEM image ofthe SQUID loop is shown in Fig. 1(d). The nanobeam-shaped mechanical resonator, formed by 100 nm highly-

stressed SiN film coated with 60 nm of Aluminum, andsuspended part of the Josephson junctions can be clearlyseen. See methods section for details. The offset in theSQUID position (away from the center of cavity) in trans-mon design allows to bring a RF drive line for the elec-trostatic actuation of the mechanical resonator (see sup-plementary information (SI) for design simulations).

The transmon qubit frequency ωq is given by hωq ≈√8ECEJ(Φ) − EC , where EJ(Φ) = E0

J |cos (πΦ/Φ0)| isthe Josephson energy, Ec is the charging energy, Φ is thetotal flux threading the SQUID loop, and Φ0 = h/2e isthe magnetic flux quanta. The tunability of qubit fre-quency with flux allows access to its dispersive or res-onant interaction with the cavity. On one hand, theinteraction between qubit and the cavity mode can beexpressed as hJ(aσ+ + a†σ−), where the dipole couplingrate J can easily be engineered to reach the strong cou-pling limit. On the other hand, the electromechanicalcoupling arises from the modulation of qubit frequencycaused by the mechanical displacement. Due to the tun-ability of qubit over a large frequency range, it is con-venient to define the vacuum electromechanical couplingrate between the dressed states (ω±) and the mechanicalresonator as,

g±(Φ) =∂ω±(Φ)

∂xxzp = ΦG±Φ

xzpw, (1)

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FIG. 2. Qubit spectroscopy and flux-responsivity at 25 mK: (a) Color scale plot of transmission |S21| through the cavityas the magnetic flux through the SQUID loop is varied. The strong qubit-cavity coupling (J) manifests as the avoided-crossing,yielding J ∼ 2π × 85 MHz. The extension panel shows the trend of avoided crossing at larger values of the magnetic flux.(b) The flux-responsivity G±

Φ = ∂ω±/∂Φ, computed using the measured device parameters, plotted as a function of dressedfrequencies. Panel (c) shows the flux-responsivity of uncoupled qubit Gq

Φ = ∂ωq/∂Φ with the magnetic flux.

where G±Φ = ∂ω±(Φ)/∂Φ is the flux-responsivity, xzp isthe quantum zero-point fluctuations of the mechanicalresonator, and w is the effective width of the SQUIDloop. The dressed frequencies ω± = ∆ ±

√(∆/2)2 + J2

with ∆ = ωq − ωc and ∆ = (ωq + ωc) /2, approaches theuncoupled qubit and cavity frequencies in the dispersivelimit. Restricting the coupled qubit-cavity system to sin-gle excitation subspace, in the resonant limit ∆� J , thedressed-modes essentially act like independent cavity op-tomechanical systems. However, it is worth pointing outhere that in a suitable parameter regime (J ∼ ωm), in-teraction between the dressed modes and the mechanicalmotion can be used to enhance of the quantum nonlinear-ity [31]. Further, the independent cavity optomechanicalmodel breaks down under the strong cavity drive [32].

We use spectroscopic measurements to characterizethe qubit. Fig. 2(a) shows the transmission (|S21(ω)|)through the cavity as applied magnetic flux is varied(see SI for details of the measurement setup). Whenqubit becomes resonant with the cavity, the vacuum Rabisplitting is observed which signifies the strong couplingbetween the qubit and cavity mode. We determine adipole coupling rate J = 2π×85 MHz, the bare cav-ity frequency ωb

c = 2π×5.993 GHz, maximum qubit fre-quency ω0

q = 2π×7.982 GHz, and an anharmonicity of-132 MHz (see SI for details). Due to the flux-periodicityof qubit frequency, the vacuum-Rabi splitting pattern re-

peats with every new flux-quanta added. The extensionpanel of Fig. 2(a) shows the transmission measurementat higher magnetic field. Apart from a small reduction(∼15 MHz) in the maximum qubit frequency and an in-crease in the dressed-cavity mode linewidth, we do notobserve any significant change in the device parametersup to a field of ∼ 3.7 mT (310 Φ0).

To understand the flux-transduction of dressed-modes,we compute the flux-responsivity G±Φ using the mea-sured qubit and cavity parameters. Fig. 2(b) shows theplot of G±Φ with respect to the dressed-mode frequen-cies. The flux-responsivity of the dressed-mode increasesas their detuning (|ω± − ωc|) approaches J . However,reduced transmission at these frequencies hinders theiruse for the mechanical transduction. We choose an op-timum operating point of 6.025 GHz, corresponding toG+

Φ ∼1.8 GHz/Φ0, for the mechanical resonator charac-terization. This flux-responsivity is significantly largerthan the values reported with SQUID cavity [28]. Inaddition, the flux-responsivity of qubit Gq

Φ = ∂ωq/∂Φcan be much larger near the half-integer flux quanta asshown in Fig. 2(c). In dispersive limit, while the effec-tive coupling between the dressed cavity and mechanicalresonator degrades by a factor of (J/∆)

2, a large cou-

pling between qubit and the mechanical resonator canbe maintained.

We first focus on the driven response of the mechani-

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FIG. 3. Detection of mechanical mode and vacuum electromechanical rate: (a) Color plot of down-mixed signal asmechanical actuation frequency and dc voltage is varied. The mechanical resonance appears as a sharp change in the color.Blue(red) color represents low(high) values of the signal. To reduce the total measurement time, the mechanical actuationfrequency range is automatically adjusted to follow the mechanical mode. (b) A 2D-histogram of 200 power-spectral density(PSD) traces measured near ωc, while driving at the lower-sideband with ∼ 0.1 photons. The cumulative color density bringsout the thermal motion of the wire peaked at ωm/2π = 6.5822 MHz. (c) Average PSD along with a fitted curve yielding amechanical linewidth γm ∼ 2π×6 Hz corresponding to a quality factor of ∼ 1.1×106. (d) Plot of the vacuum electromechanicalcoupling rate between dressed-mode and the mechanical resonator as the magnetic flux through the SQUID loop is increased,while ω+ = 2π × 6.025 GHz is kept fixed. The maximum flux applied corresponds to a field of 3.7 mT. The blue-dotted lineshows the expected coupling rate calculated from the device parameters.

cal resonator. For electrostatic actuation, a weak ac sig-nal and a dc voltage Vdc are applied at the mechanicaldrive port (see SI for details). We fine-tune the mag-netic flux near 190 Φ0 to operate the dressed mode ω+

at 6.025 GHz. We inject a microwave tone at ω+ creat-ing a mean photon occupation of ≈ 1, calibrated inde-pendently using ac-Stark shift. The signal emerges fromthe cavity is then mixed-down and recorded by a networkanalyzer. Fig. 3(a) shows the amplitude of the signal in acolor plot as the mechanical drive frequency and Vdc arevaried. The change in color over the background signi-fies the mechanical resonance. We measure the in-planevibrational mode at ωm ∼ 2π×6.585 MHz with a char-acteristic frequency softening with Vdc.

Next, we focus on the thermal motion of the mechan-ical resonator. We operate the dressed mode at ω+ =2π× 6.025 GHz and drive the system with a microwavetone tuned to lower sideband (ω+−ωm), creating a meanphoton occupation of ∼ 0.1 photons. The power spectraldensity (PSD) of the output signal is then recorded witha spectrum analyzer. We record individual traces of the

PSD from the spectrum analyzer to filter out any flux-jump events. This is validated by measuring the trans-mission at ω+ before and after each trace is recorded.Fig. 3(b) shows a 2D-histogram of the PSD measurednear ω+, where the cumulative color density brings outthe thermomechanical motion. The average PSD alongwith the fitted Lorentzian is shown in Fig. 3(c). We mea-sure a mechanical linewidth of γm = 2π×6 Hz, corre-sponding to a quality factor of ∼ 1.1×106.

For a drive at the lower sideband, the ratio of in-tegrated power at the up-converted frequency (Pm)near ω+ to the power of transmitted carrier signal(Pd) at ω+ − ωm can be conveniently expressed as

Pm/Pd = (2g+/κ)2nthm , where nthm is the mean ther-

mal occupation of the mechanical mode, and κ is thedressed-mode linewidth. By varying the fridge temper-ature, we estimated the mechanical mode temperatureto be 50 mK, corresponding to a thermal phonon occu-pancy of 157. Fig. 3(d) shows the variation in g+ asthe magnetic flux through the SQUID loop is varied,

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FIG. 4. Onset of the Landau-Zener-Stuckelberg in-terference: (a) Measurement of the qubit spectrum, usingtwo-tone spectroscopy, as the amplitude of mechanical drivesignal εm is varied. The top panel shows a line cut from theposition marked with dotted line. A weak modulation be-tween the dips can be seen. (b) Qubit spectrum calculatedusing the master equation, which captures both, the primarymechanical drive amplitude dependent splitting and the weakmodulation.

while the dressed-mode frequency is maintained fixed atω+/2π = 6.025 GHz. The dotted line shows the expectedelectromechanical coupling rate estimated from the mea-sured device parameters.

We emphasize that the vacuum electromechanical cou-pling rate of g+ ∼ 2π×4 kHz is limited by the choice ofω+, and the magnetic field range available in our mea-surement setup. By operating closer to ωc ± J , one canreach the theoretical limit of G+

Φ/GqΦ = 1/2, resulting in

g+ ∼ 2π×15 kHz. In addition, thin film of Al can with-stand a larger magnetic field than the maximum fieldused here (3.7 mT). As in-plane critical magnetic fieldis much larger than the perpendicular critical magneticfield for thin Al films, a configuration with field appliedin-plane to the SQUID loop would result into signifi-cantly higher coupling rates for the out-of-plane mechan-ical mode.

Next, we investigate the system by tuning the qubitaway from ωc. In the dispersive limit |∆| � J ,the mechanics essentially decouples from the cavitymode. While the qubit-cavity interaction is given by(J2/∆

)a†aσz, the longitudinal interaction between qubit

and the mechanical resonator is given by gqmσz(b + b†),where gqm = (∂ωq/∂x)xzp is the qubit-electromechanical

coupling rate and b (b†) is the lowering (raising) op-erator for the mechanical mode. With superconduct-ing qubit device, time-dependent longitudinal couplingscheme has been used to perform high-fidelity qubit mea-surements [33]. In the present device, a static gqm wouldrather result in a small qubit-state dependent displace-ment (∼ gqmxzp/ωm) [29]. Here, we focus on the qubitdynamics while driving the mechanical resonator. The

qubit is detuned to 4.9 GHz to enhance gqm to 40 kHz,and its spectrum is probed using two-tone spectroscopytechnique. The mechanical resonator is coherently ac-tuated at its resonant frequency. It is equivalent to theflux-modulation of the qubit frequency at a rate set byωm, and a frequency deviation set by the mechanicalamplitude. Fig. 4(a) shows the qubit spectrum as thestrength of mechanical drive εm is varied. We observe asplitting with a weak modulating pattern and the sepa-ration between the primary splitting varies linearly withthe mechanical amplitude.

Intuitively, the splitting can be understood by con-sidering an adiabatic modulation of the qubit frequencyby the mechanical resonator. Therefore, during one pe-riod of the mechanical oscillation, the qubit essentiallyspends longer time at each turning point leading to anincrease in the probability of transition from ground tothe excited state. The weak modulation observed be-tween the primary splitting is the onset of the Landau-Zener-Stuckelberg (LZS) interference [18]. The qubit iscoherently driven in transverse direction by the photonfield, and in longitudinal direction by a phonon field(ωm). The contrast of interference pattern is lower asthe qubit linewidth κq is larger than the modulation rate(κq/ωm ∼ 2). We have performed numerical calcula-tions based on the Lindblad master equation (details areincluded in SI). Fig. 4(b) shows the result from such cal-culations. Apart from capturing the linear amplitudedependence of the primary splitting, we are able to cap-ture the weak modulation observed between the primarydips.

In summary, we develop a hybrid electromechanical de-vice by integrating a mechanical resonator of SiN/Al witha modified 3D-transmon qubit. The detection of thermo-mechanical motion by driving the system with less thanone photon highlights the large underlying coupling rate.Accessibility to different regimes of interaction is furtherdemonstrate by the observation of the LZS interference.Looking ahead, the device discussed here can be pushedinto the sideband-resolved limit by improving the qubitcoherence. This could enable experiments in the regimeof the single-photon cooperativity exceeding one, and aconditional cooling of the mechanical resonator to thequantum ground state.

Methods

For device fabrication, we use an intrinsic Si (100) sub-strate coated with 100 nm thick high-stress SiN layergrown using LPCVD method. Using standard lithogra-phy and shadow evaporation techniques, the transmondesign is patterned in a single lithography step. To re-lease the mechanical resonator, a combination of dry andwet etching processes is used. First, the exposed SiN isvertically etched by the reactive ion etching using SF6

6

and CHF3 plasma. The aluminum film naturally acts asa mask layer and thus protects the SiN underneath it.In the second step of etching, a modified-TMAH basedetchant is used to remove the exposed silicon, while pro-viding excellent selectivity against Al and SiN (see SI foradditional details). After the wet etch process, the sam-ples are blow-dried gently with N2, requiring no criticalpoint drying. The (111)-facets of Si resulted from the wetetch process can also be seen in Fig. 1(d). The sampleplaced inside a copper cavity, along with a small solenoidis kept inside a cryoperm-shield to protect it from theambient magnetic field fluctuations.

Acknowledgments

V.S. acknowledges the support received from InfosysScience Foundation, and under the “Early Career Re-search Award” by SERB, Department of Science andTechnology (DST), Govt. of India. T. B. and S. M.thank Tata Trust for providing travel support. The au-thors acknowledge device fabrication facilities at CeNSE,IISc, Bangalore, and central facilities at the Departmentof Physics funded by DST. Authors thank R. Vijayragha-van, Manas Kulkarni, Diptiman Sen, and G. S. Agarwalfor valuable discussions.

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