Optomechanical measurement of a millimeter-sized mechanical oscillator near the quantum limit

Cavity optomechanics is a tool to study the interaction between light and micromechanical motion. Here we observe near-quantum limited optomechanical physics in a truly macroscopic oscillator. As the mechanical system, we use a mm-sized piezoelectric quartz disk oscillator. Its motion is coupled to a charge qubit which translates the piezo-induced charge into an effective radiation-pressure interaction between the disk and a microwave cavity. We measure the thermal motion of the lowest mechanical shear mode at 7MHz down to 35 mK, corresponding to roughly 100 quanta in a 20mg oscillator. The work opens up opportunities for macroscopic quantum experiments.


Introduction
Small but nevertheless macroscopic systems have been operated in the limit where they exhibit quantum mechanical behavior in some of their degrees of freedom. One of the most successful systems have been the superconducting quantum bits (qubits) [1,2], where the most complicated quantum states have been constructed [3,4]. Observing mechanical oscillators at the quantum limit of their motion, where the excess phonon number n m approaches zero, was a long-standing goal. The ground state was reached some years ago [5][6][7]. In earlier experiments, n 10 m 2 < was reached with nanostrings already a decade ago [8][9][10]. In the cavity optomechanical scheme, where optical and mechanical resonances are coupled [11], micro mirrors up to 0.2 mg weight [12][13][14], cantilevers [15][16][17] up to 0.5 mm long, or nitride membranes [18] have been used. Ground-state cooling [6], entanglement [19], and squeezed mechanical states [20][21][22] have been reported in the realization involving superconducting radio frequency cavities together with drum oscillators. A major motivation is to study the fundamentals of quantum mechanics.
Here we propose and demonstrate a new cavity optomechanical scheme which involves a genuinely macroscopic mechanical oscillator relatively near the ground state (see figure 1). We use a 6 mm diameter quartz disk oscillator with a total mass of 20 mg, and the vibrating effective mass of the order 0.1 mg, whose vibrations at the lowest shear mode frequency 2 7MHz m w p  we observe at a temperature of 30 mK. The quartz disk is used both as the mechanical oscillator, and as a substrate for fabricating a superconducting micro circuit. The latter includes a charge qubit and a transmission line resonator. With the notable exception of LIGO [23], it is among the most macroscopic system in such studies. Our system is three orders of magnitude more massive than earlier [24], and in terms of mass over measured phonon number, it is two decades more macroscopically quantum.
The work is based on a generic idea where a quantum two-level system, a qubit, mediates and enhances an interaction between a linear mechanical oscillator and a linear cavity. For example, the energy of a charge qubit depends sensitively on the charge in its vicinity, and the frequency of an electrical system consisting of the qubit and a cavity coupled to the qubit becomes charge-sensitive. A voltage-biased conductive mechanical oscillator was proposed to this end in [25,26], and the scheme was experimentally demonstrated in [27]. An analogous setup has been proposed using a flux-biased qubit [28]. The main motivation for introducing a qubit is that the qubit can dramatically enhance the effective optomechanical interaction, possibly allowing for entering the elusive single-photon strong coupling regime.
In the current work, the charge sensed by a charge qubit arises due to the piezoelectricity of the quartz oscillator. The deformation corresponding to the mechanical vibrations induces charge on the chip surface.
Here, the role of the qubit is particularly critical. The piezo oscillations and the on-chip microwave cavity as such do couple to each other, however, that is a small linear coupling of two linear, highly detuned oscillators. That coupling as such is of very limited use. In a recent theory proposal [29], a related system was introduced where a coupling between a transmon qubit and piezo vibrations was mediated via a low-frequency cavity, but the optomechanical aspect was not considered.
2. The optomechanical system 2.1. Radiation-pressure interaction As illustrated in figure 1(a), the traditional cavity optomechanics setup involves two optical mirrors facing each other, assembled such that one of the mirrors is free to oscillate. The cavity and the mechanical oscillator have the resonance frequencies c w and m c w w  , and decay rates κ and g k  , respectively. The coupled system is described by the radiation-pressure Hamiltonian, which couples the cavity and the mechanical oscillator, and reads Here, a † and b † are the creation operators of the cavity and the mechanical oscillator, respectively. The singlequantum coupling energy g 0 k  is the small parameter in the system. Experimentally, it is often the most important figure in the sense that it sets limits for how strong mechanics-cavity interaction one can obtain. One usually strongly pumps the cavity up to a photon number n 1 P  . As a result, the relevant coupling is the effective coupling G g n 0 P = . The significance of the cavity interaction on the mechanical oscillator dynamics is described by the cooperativity C G 4 2 = kg , or alternatively by a decoherence parameter )is the equilibrium phonon number set by coupling to a bath of temperature T. The η quantity describes what is the importance of the cavity back-action on the oscillator decoherence due to the bath, and it is often the single most relevant parameter in the quantum experiments. The basic optomechanical phenomenon is the dynamic backaction from the cavity. It appears as an optically induced change in damping to eff , leading either to sideband cooling or lasing. The effective phonon number becomes n n T m m eff g g = .
2.2. Qubit-mediated optomechanics As already mentioned, systems described by equation (1) can be physically built from very different ingredients. In qubit-mediated cavity optomechanics [25][26][27][28]34], see figure 1(b), the qubit can drastically enhance the intrinsically small radiation-pressure coupling g 0 between the mechanics and an microwave cavity. This is ) , and x s and z s are the Pauli spin matrices. The coupling of the charge qubit to mechanical oscillators is described by , where g m is the coupling energy. In the case of capacitive coupling to vibrating metallic beams or membranes [35,36], g m is proportional to a dc voltage bias, and can be a sizable fraction of the oscillator frequency.
In order to create the optomechanical interaction starting from the qubit-oscillator coupling, we include in the circuit design a transmission line resonator, which shunts the qubit, as presented in figure 2(a). We label the qubit-cavity coupling energy as g qc . The radiation-pressure interaction can be derived by considering the microwave cavity having the Hamiltonian H aa The cavity frequency c w is Lamb shifted from the intrinsic value c0 w as g , where Δ is the detuning between qubit and cavity frequencies.
Supposing the qubit stays in the ground state, the detuning becomes . The cavity Hamiltonian now assumes the form H g a a b b ) .

Coupling to piezo motion
Let us return to description of the piezo oscillator. We consider a piezoelectric disk having thickness t, shear mode stress coefficient e s of the material, shear modulus Y s , and relative permittivity r  . One further defines the dimensionless piezoelectric coupling coefficient K e Y   [30][31][32][33]. The lowest shear mode at 7 MHz couples to the microwave circuit. Blue denotes aluminum metallization, and white is bare quartz. The bonding pads are labeled as: A is the input/output for the reflection measurement, B is the gate bias, C is a test junction, D is the flux bias, E is the on-chip inductor L, F is the junctions and island region, and H is used for actuating the mechanical vibrations in order to test the system before the actual experiments. (b) Equivalent electrical resonator of the piezoelectric quartz oscillator. equivalent circuit (see appendix A), the qubit-piezo interaction can be derived as g The effective radiation-pressure interaction between the piezo motion and a cavity frequency arises then as discussed in section 2.2. Notice that in contrast to previous work where the qubit acts as to enhance the radiation-pressure interaction, in the present case the qubit is needed in order to get any coupling whatsoever.

Optimizing device parameters
Next we will work out some numbers using the parameters of the experiment discussed below. We begin with the interaction of the qubit and cavity. Typically [25,27], g 0.1 qc c w » , thus the qubit and cavity are close to the ultra-strong coupling regime [37]. The analysis presented in the end of section 2.2 holds only if g qc D  , so it is not immediately clear how large a ratio g g 0 m one can achieve. In order to make quantitative estimates of the radiation-pressure coupling, we treat the effective cavity as a charge-tunable resonator. The radiation-pressure coupling is x n g  [27], the modulation of cavity frequency as a function of charge is of the order 2 50MHz , and we estimate g 2 1 Hz . This is two orders of magnitude smaller than typical of aluminum drum oscillators [6], and is mostly limited by the small island A A 1 qb  .
Instead of enlarging the qubit island in order to increase g n A 0 zp 0 q qb s µ = , our primary strategy is to focus most of the piezo charge nearby the island. This is achieved through manipulation of the mechanical mode shape, thereby locally increasing q s . We use Comsol Multiphysics simulations with an actuation electrode to drive the mode, and adjust the drive such that the strain energy equals the mode zero-point energy. Then we integrate the piezo charge around the island to obtain n zp 0 . The basic circuit layout used in earlier related work [27] is shown in figure 2(a), and the corresponding normalized charge density in the top row of figure 3(b), where we observe an approximately uniform density all over the chip. For the final design shown in figure 3(a), we add grounded spikes which extend from the ground planes surrounding the chip towards the qubit island. The spikes strongly enhance the nearby electric field and act as anchors, focusing most of the piezoelectric strain/charge to the center of the disk (bottom row of figure 3(b)). Comparing the realizations in figure 3(b), the integrated charge around the qubit island differs by two orders of magnitude in favor of the charge focusing design, translating into a dramatic increase in the radiation-pressure coupling. The other property contributing to the coupling is the charge dispersion n c g w ¶ ¶ which we obtain by numerically diagonalizing the charge qubit Hamiltonian. This quantity depends on both the qubit and cavity properties and on the gate and flux bias, and it is sensitive to the E E J C ratio, strongly enhanced in the deep charge qubit limit E E 1 J C  , suggesting that g 0 is maximized in the said limit. Nonetheless, as explained below, nearly the opposite holds true.
In the simulations we consider three potentially realizable devices which substantially differ in the island size as written in the second column of table 1. In figure 4 we plot the predicted coupling for the three devices as a function of gate charge. A device with a small island and consequently small capacitance (figure 4(a)), benefits from large maximum values n c g w ¶ ¶ that causes the predicted g 0 to strongly peak near n 0.5 g =  as seen in figure 4(a). By increasing the island size ( figure 4(b)), the charge modulation is decreased, but this is overwhelmed by a substantially larger integrated charge. At small island size, g 0 grows roughly proportional to the island size. Before reaching a very large island as in device 3 (figure 4(c)), the dependence levels off, on one hand because charge focusing loses its effect at large size, on the other hand because charge dispersion is reduced. Device 3 represents a design which is expected to roughly maximize the radiation-pressure coupling. Notice that the island can be made much larger than that of a typical charge sensitive device, owing to the low dielectric constant 4 r  » of quartz which keeps the capacitance moderate. As seen in figure 4, in the optimum bias points, g 0 can attain values up to several kHz. These values are an order of magnitude higher than those of mesoscopic aluminum drum oscillators, showing a great promise for the proposed scheme. We note that in spite that large values are predicted in the charge qubit limit (figures 4(a) and (b)), these values may not be reachable in practice. Namely, large n c g w ¶ ¶ implies a high sensitivity to background charge fluctuations. This entails a strong dephasing of the effective cavity, visible as an inhomogeneously broadened resonance, or a fast cavity decay. The latter have been prohibiting reaching the single-photon strong coupling limit in [27], and as we see later in section 4, these effects are present in the current experiment as well.
We now briefly discuss the effective mass related to the quartz vibrations. In the case that the shear mode is spread all over the disk as roughly in figure 3(b) (top), the effective mass of the mode is the disk mass times a geometric factor of the order but smaller than one. Since charge focusing also focuses mechanical strain in the same ratio as charge is focused, also the vibrating effective mass is clearly reduced as a result. A rough estimate is that the effective mass is of the order the disk mass divided by the focusing ratio, in the present experiment we obtain ∼0.1 mg. We note, finally, that in spite of the focusing, non-negligible part of the vibrations are delocalized all over the 6 mm chip, hence substantiating the claims of true macroscopity. Table 1. Parameters and simulation results of the devices 1 and 2 measured in this work, and for a hypothetical device 3.

Experimental results
In the experiments, we use devices made according to the layout shown in figures 3(a) and (b) (bottom row). They are fabricated by shadow mask evaporation of 30 and 60 nm of aluminum, and oxidizing the first layer in order to create the Josephson junctions. We investigate two devices, labeled 1 and 2. The devices are otherwise nominally similar, but differ in the size of the island as displayed in table 1. In the following, unless noted otherwise, we discuss device 1. We also label image panels according to the device in question.

Basic characterization
The measurements were carried out in a dilution refrigerator with a base temperature of 30 mK. The microwave tones are applied to the port A of figure 3(a), and we record the scattered signal from the same port. Regarding basic characterization, both devices showed very similar behavior. The cavity linewidth below 200 mK was 2 7MHz k p » . In figures 5(a) and (b) we show examples of the cavity resonance absorption and phase shift. The cavity frequency is sensitive both to flux and charge as seen in figures 5(c) and (d), respectively. The solid lines represent a theoretical fit produced by numerical diagonalization of the coupled qubit-resonator Hamiltonian, displaying an excellent agreement.
Around the gate bias points n 1 2 g = , where the maximum n c g w ¶ ¶ takes place, the cavity resonance becomes smeared out due to enhanced decay or dephasing due to background charge fluctuations. The maximum radiation-pressure couplings, expected to appear in this region by figures 4(a) and (b) in the charge qubit limit, are hence not available. We thus have selected to operate at n 0.4 g  where the cavity resonance is stable, although somewhat broadened compared to the value at n 0.0 g  . We now turn the discussion on observing the quartz vibrations by cavity optomechanical means. We first studied the driven motion by exciting the quartz through the actuation pad (labeled H in figure 3(a)), and probed possible nonlinear changes under an intense driving. A piezo charge which is a substantial fraction of one electron will change the time averaged cavity response (see figure 5(d)), causing a clear signature in the S 11 response to a weak probe tone. Figure 6 shows this type of the detection, revealing the lowest shear mode resonance at the expected frequency. From the simulations, we obtain an independent estimate of the driven piezo charge e 0.1 , which is in the correct range.

Detection of piezo thermal motion
The benchmark of cavity optomechanics is the measurement of the motional sidebands due to thermally excited vibrations. Here, a pump tone p w is applied near the cavity resonance, and the sidebands appear in the spectrum at the frequencies p m w w  . The spectral density S out w ( )in units of quanta (power/bandwidth) divided by system gain , when the pump is applied at the red or blue sideband, is S n 4 , 2 which is a Lorentzian centered at the mechanical frequency with respect to the pump frequency. Our cavities reach a maximum n 10 20 P  -limited by the Josephson nonlinearity [27,38]. In figure 7(a) we show the results of such measurement for different cryostat temperatures. The pump power is optimized for maximizing the signal. We use equation (2) to plot the theory curves in figure 7(a). As adjustable parameters, we use  which is common to all curves, and the conversion factor A from the generator power setting P 0 to the effective coupling, viz.G AP 0 = . We note that this way we in principle cannot separate the contributions from g 0 and n P to G, however, we know that cavity nonlinearity limits n 10 20 P~-, giving a calibration point. The range of estimated n P here is large because we do not know exactly what amount of nonlinearity corresponds to the maximum signal. We fit  such that the intermediate-temperature data point (106 mK) in figure 7(a) is fitted properly. This leads to the fact that at lower temperatures, the data points fall above the fit curve. This is visible  also by the fact that the blue points indicate slightly imperfect thermalization. The value of  is benchmarked against a roughly known fixed point given by the effective noise temperature of the system, setting the noise floor in the spectrum. Here we obtain a reasonable value of ∼5 K for the noise temperature. For device 1, we extracted mechanical Q-value of 6.4e5, g 2 140 30 Hz 0 p =  with a maximum n 15 P » , reasonably agreeing with the simulated g 0 for our charge focusing design for device 1 when n 0.4 g  as in the measurement. The area under the power spectrum in equation (2) is expected to be proportional to the mode temperature, which behavior is qualitatively seen in figure 7(a). To get the points in figure 7(b), we fit a Lorentzian (not limited to equation (2)) to each curve individually. We follow the usual practice and calibrate the equilibrium phonon number n T m by relying on n k T . Based on the data and a linear fit shown in figure 7(b) for both devices 1 and 2, we conclude that the shear mode thermalizes down to roughly T 30 mK » , which corresponds to only n 90 15 T m =  phonons in the mm-sized vibrating disk.

Experimental prospects
In this section we discuss the possibilities to further enhance the coupling or cooperativity in piezo optomechanics in order to control the mechanics via cavity back action. We first review the prospects to increase the coupling in standard microwave optomechanics with linear cavities, where one goal is to approach the single-photon strong coupling condition g 0 k  .

Parametric coupling
Let us now study a mechanical oscillator which directly couples to the cavity frequency. This is the parametric coupling usually encountered in cavity optomechanics. In the following we discuss drumhead or membrane oscillators used in many microwave-frequency experiments [6, 19-22, 24, 39]. Typical parameters for drum oscillator experiments are 2 10MHz m w p » , 2 500 kHz k p » , 2 100 Hz g p » . For state-of-the art experiments such as squeezing [20][21][22], values of cooperativity in the range C 10 10 3 4 -, and decoherence parameter 10 10 2 h~have been reached.
In the microwave regime, the motion changes the capacitance of a microwave cavity having the capacitance C. The total capacitance is the sum of C and a small x-dependent part C x g ( ). The strength of the coupling becomes The last form used the plate capacitor model, with the vacuum gap d separating the oscillator from the cavity electrode. In order to maximize the coupling, one clearly benefits from a small vacuum gap. In the smalloscillator limit C C g  , the coupling g d 0 2 µis a strong function of the gap, and a linear function of the drum surface area. The current experiments typically have roughly C C g , and exhibit coupling of the order g 2 100 Hz 0 p~. When trying to increase the coupling by reducing the gap, or when making the oscillator larger, one rather soon is faced with the limit C C g  , leading to the fact that the coupling at a fixed cavity frequency does not benefit from increasing the membrane area, but depends only on the vacuum gap inversely. Let us consider a membrane size of 200 mm, with d=100 nm. The membrane metallization will roughly be optimized to cover only the center region of 50 mm. With this values, we obtain g 2 50Hz 0 p » that is lower than state-of-the art. Hence it looks unlikely that the coupling could be much increased in the drum or membrane approach.

Qubit-mediated effective parametric coupling
In the following we discuss some promising applications for quartz optomechanics, and compare the typical performance indicators obtainable to the state-of-the-art in microwave optomechanics evaluated above in section 5.1. For the current device 2, we obtain C 0. 3 , and decoherence parameter 10 3 h~which clearly are far from those needed for quantum limited work. The corresponding values at the single-quantum level (with n 1 P = ), however, are already the same as the drum oscillators exhibit.
As discussed in section 3, the optimal island size for maximizing g 0 is a balance between the piezo charge sensed by the qubit, and increased qubit capacitance which reduces the charge dispersion. Device 3 (see table 1) with a very large 380 m 2 m( ) sized island is expected to be relatively close to an optimum, yielding g 2 18kHz 0 p~. In stark contrast to true parametrical coupling, the qubit-mediated coupling thus scales very favorably in the large-size limit. The large size, indeed, is what cavity optomechanics usually is striving for in the quantum-regime investigations. Besides, large size usually implies enhanced mechanical Q values.
In order to spell out numbers for prospective experiments with optimized devices, we note that now we are limited to about n 10 P = because of cavity nonlinearity. Device 3 in the current setup would be sideband cooled down to the ground state, n 1 m  quanta with the current mechanical Q. We now introduce a somewhat optimized but feasible setup consisting of device 3, and 2 1MHz k p » , 2 1Hz g p » (mechanical Q-value of 10 7 ). With such a device we reach C 10 3 and 10 h~similar to state-of-the-art drum resonators, but now with a drastically smaller energy pumping. The single-quantum values are, on the other hand, around six orders of magnitude higher than state-of-the-art. We remark that it is generally beneficial to reach a given performance at low pumping, because non-idealities such as heating that the pumping likely introduces.
The experiments that are foreseen in the mentioned optimized setup include new investigations on squeezing, entanglement [40], in particular in potential time-resolved investigations [41], all for the case of a tangible object. With the above values we obtain quantum squeezing of ∼5 dB, and entanglement Duan quantity ∼0.6 below vacuum between two quartz oscillators (the actual realization still being unspecified).
Another intriguing opportunity offered by our approach is the test of possible extensions of quantum mechanics, in particular that of hypothetical collapse models. These show up as a spurious decoherence channel, and can become visible in a cold oscillator with a long ringdown. [42][43][44]. With the high Q-values 10 9 [45,46], according to [44] we obtain a spontaneous heating in the DP model of the order 1 mK, which is relevant already in the temperatures encountered in the present experiment.
One can also consider coupling of the piezo motion to the qubit-like mode in the microwave circuit. In the limit of high E E 1 J C  corresponding to the transmon qubit, the qubit-motion coupling energy becomes (see appendix A) g

Conclusions
To conclude, monolithic quartz oscillators are a promising platform for physical experiments. They can show exceedingly high quality factors at tens of MHz frequencies, and they present a tangible object in contrast to many other important types of mechanical oscillators. However, creating an intrinsically nonlinear coupling, needed for quantum experiments, from the quartz mechanical vibrations to electrical systems is difficult. In this work, we solve the challenge, and demonstrate cavity optomechanics on such quartz oscillator near the quantum ground state. Focusing of piezo charge allowed coupling of the mechanical oscillations of a mm-sized quartz disk, via a charge qubit, to a microwave cavity. We obtained single-photon coupling energies similar or larger than in state-of-the-art microwave optomechanical setups. As the best future prospect we foresee a design involving a qubit with nearly a mm-size island, allowing for further increasing the coupling by two orders of magnitude. The approach is suitable for such experiments in cavity optomechanics which do not require strongcoupling conditions, but which benefit from a sizable single-phonon coupling and low pumping powers, and long ringdown times. These include macroscopic squeezing and entanglement, as well as tests of extensions of quantum mechanics in unequivocally macroscopic moving bodies.
) which depends on the flux bias. The junction capacitance is C J , the (small) gate capacitance is C g , and the island selfcapacitance is C 0 . The total island capacitance is C C C C J g 0 = + + S . The dynamical variables are the node flux Θ that corresponds to the qubit degree of freedom, and m Q which is that of the mechanical mode.
The Lagrangian of the circuit in figure 8 is In the charge qubit limit, we express p q in the basis of charge states of the island. The island charge Q is the sum of the charges on all capacitors connected to the island, and is related to the momenta as follows: In the high-capacitance limit relevant for the transmon qubit or phase qubit, one expresses the qubit momentum p q in the harmonic oscillator basis: L C 1 0 J w = S is the frequency of small oscillations of the qubit mode. We get the transmon-mechanics coupling