Climbing the phonon Fock state ladder
Abstract
Quantum states of mechanical motion can be important resources for quantum information, metrology, and studies of fundamental physics. Recent demonstrations of superconducting qubits coupled to acoustic resonators have opened up the possibility of performing quantum operations on macroscopic motional modes[1, 2, 3], which can act as longlived quantum memories or transducers. In addition, they can potentially be used to test for novel decoherence mechanisms in macroscopic objects and other modifications to standard quantum theory[4, 5]. Many of these applications call for the ability to create and characterize complex quantum states, putting demanding requirements on the speed of quantum operations and the coherence of the mechanical mode. In this work, we demonstrate the controlled generation of multiphonon Fock states in a macroscopic bulkacoustic wave resonator. We also perform Wigner tomography and state reconstruction to highlight the quantum nature of the prepared states[6]. These demonstrations are made possible by the long coherence times of our acoustic resonator and our ability to selectively couple to individual phonon modes. Our work shows that circuit quantum acoustodynamics (circuit QAD)[7] enables sophisticated quantum control of macroscopic mechanical objects and opens the door to using acoustic modes as novel quantum resources.

Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA and Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA
Light and sound are two familiar examples of wave phenomena in the classical world. By now, the field of quantum optics has extensively demonstrated the particle nature of light in quantum mechanics through the study of single photons and other nonGaussian electromagnetic states. The concept of particles of sound, or phonons, is used widely in solid state physics. However, the ability to create states of individual phonons has only been demonstrated in a few instances[1, 2], while complete quantum tomography of such states has only been achieved in a single trapped ion[6]. This disparity between the electromagnetic and acoustic degrees of freedom is largely because sound propagates inside the complex and potentially lossy environment of a massive material rather than vacuum. As a result, an open question remains: Is it feasible to control and measure complex quantum states in the motion of a macroscopic solid state object, or what we usually think of as sound, analogously to what has been done with light?
The relatively new field of quantum acoustics is attempting to answer this question[8, 9], and one particularly promising approach is circuit QAD[1, 10, 7, 3], where, in analogy to circuit quantum electrodynamics (circuit QED), superconducting quantum circuits are interfaced with mechanical resonators. Circuit QAD takes advantage of the strong interactions between mechanical and electromagnetic degrees of freedom enabled by, for example, piezoelectricity. It also incorporates the nonlinearity provided by the Josephson junction, which is a crucial ingredient for creating nonGaussian states of motion. In turn, the ability to create these states make mechanical resonators useful as resources in quantum circuits that offer capabilities beyond those of electromagnetic resonators. For example, mechanical transduction is a promising method for transferring quantum information between microwave circuits and other systems such as optical light or spin qubits[11, 12]. In addition, due to the difference between the speeds of sound and light, compared to an electromagnetic resonator, an acoustic resonator at the same frequency is much more compact and well isolated. For the same reason, a single acoustic resonator can provide a multitude of independent modes that are individually addressable by a superconducting qubit. Such an architecture is desirable for simulating manybody quantum systems[13, 3] and also provides a highly hardware efficient way of storing, protecting, and manipulating quantum information using superpositions of Fock states and coherent states[14, 15]. These examples show that, by repurposing the toolbox of circuit QED through the similarities between light and sound, circuit QAD allows us to make use of the important differences between these quantum degrees of freedom. However, in order to access this toolbox, we first need to demonstrate that a circuit QAD system can be engineered to have the necessary mode structure, strong enough interactions, and sufficient quantum coherence to create and characterize quantum states of motion.
In this work, we experimentally prepare and perform full quantum tomography on Fock states of phonons and their superpositions inside a highovertone bulk acoustic wave resonator (HBAR). These demonstrations are enabled by a robust new flipchip device geometry that couples a superconducting transmon qubit to the HBAR. This geometry allows us to separately optimize the design of the acoustic resonator and qubit to extend phonon coherence while enhancing the selectivity of the coupling to a single mode. The combination of these improvements leads to a device that is deeper in the strong coupling regime of circuit QAD, which is necessary for the generation and manipulation of more complex quantum states.
We now motivate and describe the design of our circuit QAD system in more detail. Figure 1a shows a schematic of our device, which we call the BAR from now on. The first important difference from our previous device is the use of a flipchip geometry, where the superconducting transmon qubit and acoustic resonator are now on separate sapphire chips[16]. This simplifies the fabrication procedure and increases the yield of successful devices (see Supplementary Information), while allowing for the qubits and acoustic resonators to be individually tested before assembly. Second, the BAR now incorporates a planoconvex acoustic resonator that supports stable, transversely confined acoustic modes than can have significantly extended lifetimes relative to our previous demonstration[2]. The convex surface is fabricated using a simple and robust recipe[17] that can produce a range of controllable radii of curvature (see Supplementary information). Another important requirement for robust quantum control of phonons is the ability to selectively couple the qubit to a single well defined acoustic mode. This is partly achieved in the current device by the planoconvex resonator design, which allows us to control the frequency spacing between transverse modes. To further increase the mode selectivity of the qubitphonon coupling, our third design improvement was the addition of a optimized transduction electrode to the qubit geometry. The electrode shape and size was chosen to match the strain profile of the fundamental Gaussian transverse mode of the acoustic resonator (see Supplementary information). We point out that even though the acoustic resonator is not in physical contact with the electrode, the electric field of the qubit extends across the gap between the two chips and through the AlN film, thus allowing for piezeoelectric transduction.
We now experimentally show that the new design does indeed lead to improvements in the electromechanical coupling, acoustic mode spectrum, and coherence of our device. As in our previous work, the BAR is measured using a standard circuit QED setup that allows for flux tuning of the qubit frequency. Figure 1b shows qubit spectroscopy near the and mode of the BAR, which reveals a single distinct anticrossing feature (Figure 1b). Here is the longitudinal mode number, and are the mode numbers of the HermiteGaussianlike transverse modes. corresponds to the highest frequency longitudinal mode fully within the tunable range of the qubit, as indicated in Figure 2, where we investigate the mode structure of the BAR over several longitudinal freespectral ranges. Figure 2a shows time dynamics of the qubitphonon interaction, which reveals vacuum Rabi oscillations every MHz as we tune the qubit frequency, each corresponding to an anticrossing feature similar to the one shown in Figure 1b. The Fourier transform of the data in Figure 2a is shown in Figure 2b, which allows us to determine the qubitphonon coupling rate as kHz. In addition to the dominant set of vacuum Rabi oscillations corresponding to the Gaussian modes, there are clear signatures of other acoustic modes visible in both the time domain data and its Fourier transform, which simulations indicate correspond to higher order transverse modes (see Supplementary Information). However, the closest observable higher order mode is 1 MHz away from the mode and about ten times less strongly coupled to the qubit, while all others are at least five times less strongly coupled. From now on, we use only the longitudinal mode number to represent the modes, unless otherwise specified. These results indicate that the BAR is a good approximation of a system in which the qubit can be tuned to interact with a single acoustic mode at a time.
We demonstrate improvements in the coherence of our system by performing quantum operations on the phonon mode using the qubit. Using techniques described in our previous work[2], we find that the phonon mode has a of s, a Ramsey of s, and an echo of s. On other devices, we measured that the phonon can be as long as s. These coherence times are now comparable to that of stateofthe art superconducting qubits and indicate that the planoconvex resonator design does indeed support much longer lived phonons. The qubit in this device has a of s, which is similar to our previous device. As will be discussed later, we believe these device parameters can be further improved through modifications of the materials, fabrication procedure, and device geometry.
The improvements presented above allow us to perform quantum operations on the phonon mode with a new level of sophistication, which we now illustrate by creating and measuring multiphonon Fock states. We use a procedure for Fock state preparation that has previously only been demonstrated in electromagnetic systems[18] (Figure 3a). The experiment begins with the qubit set to a frequency that is MHz detuned from the target phonon mode, which has frequency . The qubit ideally starts out in the ground state , but in reality has a finite thermal population of in the excited state . The phonon modes, on the other hand, were shown to be colder[2]. Therefore we first perform a swap operation between the qubit and the mode with frequency . This procedure effectively uses an additional mode of the phonon resonator to cool the qubit to . The qubit is then excited with a pulse and subsequently brought into resonance with the mode to swap the energy into the acoustic resonator. This is then repeated times to climb up the Fock state ladder, ideally resulting in a state of phonons, which is then probed by bringing the qubit and phonon on resonance for a variable time and measuring the final qubit population in the excited state. The resulting time traces for up to are shown in Figure 3b. In Figure 3c, we plot the Fourier transform of the data in Figure 3b. As expected, we observe oscillations with a dominant frequency of , corresponding to the rate of energy exchange between the and states.
In order to more quantitatively characterize the states we have created, we can extract the population in each phonon Fock state after performing a phonon preparation experiment. We do this by first simulating the expected time traces if the phonon mode is prepared in an ideal Fock state ranging from to . The independently measured value of , along with the qubit and phonon decay and dephasing rates, are used in the simulations. Then, the experimental data for each (Figure 3b) are fitted to a weighted sum of the form
(1) 
where is then the population in after performing a phonon preparation. The fit for each is subject to the constraints and . Finally, the population in the zero phonon state is calculated as . Ideally, . As shown in Figure 3d, we observe that the resulting distribution of populations for each experiment is indeed peaked at . However, the population in the nominally prepared state decreases with increasing . For , we find that 86 of the population is in the Fock state. This is consistent with a simple estimate that takes into account the energy decay from the one excitation manifold during a swap operation, which is dominated by the qubit decay rate, and the imperfect preparation of the qubit in . For larger ’s, the state preparation may be affected by additional effects such as offresonant driving of the phonon mode during the qubit pulses, which could lead to excess population in the states. We also found that the largest source of potential error in extracting comes from uncertainty in the system parameters that are used in simulating . In particular, slight drifts of the qubit frequency can result in a mismatch between the value of used in the simulations and the actual effective oscillation frequency of the vacuum Rabi data. An estimate of the effect of such miscalibrations are given by the errorbars in Figure 3d.
We now build upon our ability to extract the phonon number distribution to perform full Wigner tomography and explore the quantum nature of the prepared mechanical state. As in previous experiments in circuit QED and trapped ions[6, 19], we make use of the definition
(2) 
Here and are the values of the displaced parity and Wigner functions at a phase space amplitude , respectively, is the prepared state, and is the parity operator. From now on we will plot the values of for clarity, but use the terms displaced parity and Wigner function interchangeably. In our case, the cavity displacement is implemented by a microwave pulse at the phonon frequency while the qubit is detuned at . Under these conditions, the phonon mode is still coupled to the microwave drive port, in part due to its hybridization with the qubit. To verify this and calibrate our displacement amplitudes, we first apply a Gaussian phonon drive pulse of varying amplitude with a 1 s RMS width and truncated to 4 s total length. We then measure the subsequent Fock state populations and check that they agree well with the expected Poisson distributions up to an overall scaling between the amplitudes of the applied drive and the actual displacement (see Supplementary Information). We can then calculate the displaced parity for the vacuum state using . Similarly, we can measure the displaced parity for an arbitrary state by adding a phonon drive pulse between state preparation and measurement.
In Figure 4, we present the results of Wigner state tomography on the nominally prepared states , , and . The state was prepared by performing a pulse on the qubit and followed by a swap operation with the phonon mode. From the measured data shown in Figures 4a, b, and c, we can reconstruct the measured state using a maximum likelihood method[15]. The Wigner functions of the reconstructed states are presented in Figures 4d, e, and f. Figures 4g, h, and i show that the reconstructed parities agree well with the raw data. The negativity of the Wigner functions clearly demonstrate the quantum nature of the states. From the reconstructed density matricies, we find that the fidelities of the prepared states to the target states are , , and . The infidelity for all three states are dominated by excess population in the lower number Fock states, which is an expected consequence of energy decay during state preparation and measurement (see Supplementary Information).
These results show that the quantum state of motion in a macroscopic mechanical resonator can be prepared, controlled, and fully characterized in a circuit QAD device. The demonstration of even more complex quantum states should be possible with further improvements of the device performance. A crucial parameter that needs to be improved is the qubit coherence, which is currently the dominant loss mechanism during our state preparation procedure. We found that the qubit is higher when the phonon resonator chip is either not present or rotated by 180 relative to the qubit chip. This indicates that the qubit lifetime may be limited by loss due to the AlN, which could be mitigated by using a different piezoelectric material or optimizing the device geometry to minimize the electric field in the AlN that does not contribute to transduction. The current limitations on the phonon coherence also require further investigation. The energy loss is likely to be dominated by surface roughness or imperfections in the fabricated geometry, while additional dephasing could result from thermal excitations of the qubit while detuned from the phonon mode. In addition, we can more carefully characterize the final device geometry, such the spacing between the chips and the alignment of the qubit to the acoustic resonator. The flipchip assembly process can then be modified accordingly, potentially leading to further improvements of the coupling and mode selectivity.
The next generation of devices could give us access to even more sophisticated methods for quantum control of the acoustic resonator. Our current device is close to being able to reach the strong dispersive regime where circuit QED systems currently operate, which would allow for quantum nondemolition measurements of phonon numbers[20] and more sophisticated techniques for generating arbitrary quantum states of harmonic resonators[21] such as Schrödinger cat states that are used in bosonic encodings of quantum information[14]. Furthermore, our technique for cooling the qubit already takes advantage of the multimode nature of the acoustic resonator. Future experiments would, for example, demonstrate interactions between multiple modes mediated by the qubit and the creation of multipartite entangled states of mechanical motion. Recent efforts in improving the efficiency of electromechanical and optomechancial transduction with mechanical resonators could enable conversion of quantum information between the microwave and optical domains[11, 17]. Beyond the use of acoustic resonators as resources for quantum information, the creation of increasingly complex quantum states in highly coherent mechanical resonators can provide insight into the question of whether quantum superpositions of massive objects are suppressed due to mechanisms other than environmental decoherence[4, 22]. In addition, the ability to perform quantum control on our large effective mass, high frequency, and low thermal occupation mechanical system may put new bounds on modifications to quantum mechanics at small length scales[23, 24]. These examples suggest that the wide range of quantum acoustics demonstrations that may soon be possible with BAR will give rise to new quantum technologies while furthering our understanding of fundamental physics.
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We thank Michel Devoret, Steve Girvin, Yaxing Zhang, and Vijay Jain for helpful discussions. We thank Katrina Silwa for providing the Josephson parametric converter amplifier. This research was supported by the US Army Research Office (W911NF1410011), ONR YIP (N000141712514), NSF MRSEC (DMR1119826), and the Packard Fellowship for Science and Engineering. Facilities use was supported by the Yale SEAS cleanroom, the Yale West Campus Cleanroom, and the Yale Institute for Nanoscience and Quantum Engineering (YINQE).

R.J.S., and L.F. are founders and equity shareholders of Quantum Circuits, Inc.

Correspondence and requests for materials should be addressed to Y. Chu (email: yiwen.chu@yale.edu) or R. J. Schoelkopf (robert.schoelkopf@yale.edu ).
Supplementary information for:
Climbing the phonon Fock state ladder
Appendix A Fabrication procedures
The transmon qubit used in our device is fabricated on sapphire using a standard ebeam lithography and Dolan bridge process[26]. The fabrication procedures for the acoustic resonator chip is shown in Figure S1a. We begin with a commercially purchased double side polished 2” sapphire wafer with a 1m thick film of caxis oriented AlN grown on one side (Kyma Technologies, part number H.AT.U.050.1000). The convex surfaces of the device resonator and the spacer resonators at the edges of the chip are then fabricated on the side with AlN using a procedure that has previously been demonstrated for making microlenses and highQ highovertone bulk acoustic wave resonators (HBAR) in other materials[27]. The procedure uses photoresist that has been reflown into hemispheres as a mask for reactive ion etching (RIE), which transfers the hemispherical geometry into the substrate due to the finite etch selectivity.
We first pattern disks of photoresist (AZP 4620) on the wafer, which is then attached to a heated chuck at 55 C and inverted above a beaker of solvent (AZEBR) on a hotplate at 60 C. After 23 hours, the disks of resist will have reabsorbed the solvent and reflowed into hemispheres due to surface tension. The mode structure of the resonator depends on the radius of curvature of the final convex surface, which in turn depends on the radius of curvature and geometry of the reflowed resist hemisphere. The radii of each hemisphere is approximately that of the original disk, while the height and shape depend on the disk radius, the thickness of the resist, and the reflow time. Therefore, we use identical radii for the device and spacer resonators within each chip to ensure that they result in nominally identical heights of the final surface.
After reflowing the resist, we bake the wafer starting at 90 C, gradually increasing to 145 C over the course of 15 minutes to remove the solvent and harden the resist. The wafer is then etched in an Oxford 100 RIE/ICP etcher (Cl/BCl/Ar at 4/26/15 sccm, 8 mTorr, 70 W RF power, 350 W ICP power). The resist mask has an etch rate of 200 nm/min, while AlN and sapphire have etch rates of 60 nm/min and 20 nm/min, respectively. We stop the etch when the maximum thickness of the AlN is , where is the acoustic wavelength at the qubit frequency, to maximize the qubitphonon coupling strength. The different etch rates of the AlN and sapphire results in a twotiered geometry of the final surface, as shown in Figure S1b. However, the simulated mode radius is only 20 m (see Section B), which means that the mode does not extend beyond the AlN part of the curved surface.
After making the resonators, we use photolithography, ebeam evaporation, and liftoff to add Al spacers on top of the spacer resonators. The wafer is then diced into individual chips, which are then combined with the qubit chips using a homebuilt alignment and flipchip bonding setup. We first pick up the resonator chip with a small drop of PDMS at the end of a tungsten tip, which is attached to a threeaxis translation stage. We then use the stage to align the Al spacers on the resonator chip to corresponding features patterned on the qubit chip, which results in the device resonator being aligned with the transduction electrode on the qubit. It is also possible to directly align the device resonator using interference rings that are visible once the two chips are brought into contact. Once alignment is achieved, we add small drops of GE varnish on the edges to hold the chips together.
Appendix B Acoustic mode simulations
Detailed descriptions of the design and simulation of highQ HBAR resonators are given in[28, 27, 29]. The simulation propagates the acoustic wave over many roundtrips through the resonator and calculates the interferometric sum of the fields. At frequencies where a stable mode exists, the interference is constructive, and we obtain a welldefined mode profile with a large total intensity. We note that the simulated mode spectrum should not depend on the initial excitation, but certain modes can be missed if the initial excitation has particular symmetries. Therefore, we use an initial acoustic excitation that was found by first simulating the qubit electric field pattern with HFSS, then calculating the strain profile that is generated by this field through the piezoelectric transducer.
We now show that the simulated mode structure and coupling strengths are in qualitative agreement with what we observe. Figure S2a shows one free spectral range (FSR) of the measured data from Figure 2a of the main text, where we can clearly see a collection of modes that couple to the qubit. We compare this to the simulated acoustic mode spectrum shown in Figure S2b, where we plot the total mode intensity versus the frequency of the acoustic excitation. The free parameters to match the simulated mode frequencies to the experimental data are the longitudinal and transverse sound velocities of sapphire, which are not well known at low temperatures. We find them to be m/s and m/s, which is similar to typical measured values and the values used in our previous work[29]. All other geometrical parameters, including the shape of the curved surface, are known from independent measurements such as the one shown in Figure S1b. The simulated spectrum agrees reasonably well with the experimental data, especially for the first few transverse modes in each FSR. There appear to be additional modes in the measured data, which may be higher order transverse modes from the neighboring longitudinal mode number. Another possibility is the existence of acoustic modes with other polarizations in the resonator, which are not included in the simulations. The qubit may couple to these modes through the nonzero transverse components of the electric field and shear components of the AlN piezoelectric tensor.
The relative heights of the peaks in Figure S2b give some sense of how well each mode is coupled to the qubit. However, to obtain the coupling rates more accurately, we find the strain profiles at only the simulated mode frequencies after round trips. We then calculate the coupling rate, which includes the overlap of the simulated qubit electric field, the shape of the piezoelectric transducer, and the simulated acoustic mode profiles, as described in the Supplementary Materials of[29]. Figure S2c shows the resulting predicted coupling rate for each mode. We see that the relative coupling strengths qualitatively agree with what we experimentally observe, with the fundamental Gaussian mode being at least a factor of five more strongly coupled than any higher order mode. The overall strength of the coupling is higher in the simulations than what is observed in our experiment, which could be due to a variety of reasons. One is a lower piezoelectric constant for the actual material than the value used in the similation, which we have not independently measured. Misalignment and a larger gap between the two chips than expected would also lead to different overall and relative coupling strengths. To resolve these discrepancies, we hope to more systematically characterize the flipchip device geometry in the future.
Appendix C Coherent displacement calibration
In this section, we show that our procedure for doing a direct displacement on the phonon mode does indeed result in a coherent state. Figure S3 shows the extracted Fock state populations after performing displacements with six different amplitudes within the range used for the measurements of Wigner functions. As in the main text, the displacement drive is a Gaussian pulse at the phonon frequency with a 1 s RMS width and 4 s total width. We find that they agree well with the expected Poisson distributions for coherent states whose amplitudes are given by a single scale factor multiplied by the actual drive amplitudes used in the experiments. This is in practice how we calibrate the experimental drive amplitudes to the actual displacement amplitudes, and the same scale factor is used in all Wigner tomography experiments. We find that for shorter pulses and larger drive amplitudes, the population distributions deviate from that of coherent states. These effects are reproduced in simulations, and are potentially due to offresonant excitations of the qubit. In the case of significant initial excited state population of the qubit, our procedure for measuring the phonon populations is no longer valid. On the other hand, a longer pulse with smaller amplitudes results in more decoherence during the displacement pulse, thus limiting our ability to perform Wigner tomography on a prepared state. The pulse length used here and in the data shown in Figure 4 of the main text is a compromise between these two considerations. Future optimization of the qubit detuning during the displacement and the pulse shape could potentially improve the robustness and range of amplitudes of the displacement drive.
Appendix D State reconstruction
State reconstruction was done using the formalism described in[30]. We use a maximum likelihood method that finds the most probable density matrix that results in the Wigner functions we measured, subject to the constraints that is physical (positive semidefinite, Hermitian, and unit trace). We used Fock states up to for extracting the populations of both the Fock states shown in Figure 3 of the main text and for calculating the Wigner functions shown in Figure 4 of the main text. We truncate the Hilbert space at Fock state for the reconstruction. Figure S4 shows the reconstructed density matrices that were used to plot the Wigner functions in Figures 4d, e, and f of the main text.
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