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Creation and control of multi-phonon Fock states in a bulk acoustic wave resonator

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, which can act as long-lived 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. 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 multi-phonon Fock states in a macroscopic bulk-acoustic wave resonator. We also perform Wigner tomography and state reconstruction to highlight the quantum nature of the prepared states. 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 acousto-dynamics (circuit QAD) enables sophisticated quantum control of macroscopic mechanical objects and opens the door to using acoustic modes as novel quantum resources.

onators. 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 many-body quantum systems 3,13 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 high-overtone bulk acoustic wave resonator (HBAR). These demonstrations are enabled by a robust new flip-chip 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 flip-chip 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 plano-convex 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 plano-convex resonator design, which allows us to control the frequency spacing between transverse modes. To further increase the mode selectivity of the qubit-phonon 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 electro-mechanical 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 l = l 1 and m, n = 0 mode of the BAR, which reveals a single distinct anticrossing feature (Figure 1b). Here l is the longitudinal mode number, and m, n are the mode numbers of the Hermite-Gaussian-like transverse modes.  Supplementary Information). However, the closest observable higher order mode is ∼1 MHz away from the m, n = 0 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 m, n = 0 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 T 1 of (64 ± 2) µs, a Ramsey T 2 of (38 ± 2) µs, and an echo T 2 of (45 ± 2) µs. On other devices, we measured that the phonon T 1 can be as long as (113 ± 4) µs.
These coherence times are now comparable to that of state-of-the art superconducting qubits and indicate that the plano-convex resonator design does indeed support much longer lived phonons.
The qubit in this device has a T 1 of (7 ± 1) µs, 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 ν 0 that is δ = −5 MHz detuned from the target l 1 phonon mode, which has frequency ν 1 . The qubit ideally starts out in the ground state |g , but in reality has a finite thermal population ofn ∼ 4 − 8% in the excited state |e . 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 l 2 mode with frequency ν 2 . This procedure effectively uses an additional mode of the phonon resonator to cool the qubit ton ∼ 2%. The qubit is then excited with a π pulse and subsequently brought into resonance with the l 1 mode to swap the energy into the acoustic resonator. This is then repeated N times to climb up the Fock state ladder, ideally resulting in a state of N phonons, which is then where p n,N is then the population in |g, n after performing a N phonon preparation. The fit for each N is subject to the constraints p n,N ≤ 1 ∀n and nmax n=1 p n,N ≤ 1. Finally, the population in the zero phonon state is calculated as p 0,N = 1 − nmax n=1 p n,N . Ideally, p n,N = δ n,N . As shown in Figure 3d, we observe that the resulting distribution of populations for each experiment is indeed peaked at n = N . However, the population in the nominally prepared state decreases with increasing N . For N = 1, we find that 86% of the population is in the n = 1 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 |g . For larger N 's, the state preparation may be affected by additional effects such as off-resonant driving of the phonon mode during the qubit π pulses, which could lead to excess population in the n > N states. We also found that the largest source of potential error in extracting p n,N comes from uncertainty in the system parameters that are used in simulating p e,n (t). In particular, slight drifts of the qubit frequency can result in a mismatch between the value of 2g 0 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 Here P (α) and W (α) are the values of the displaced parity and Wigner functions at a phase space amplitude α, respectively, ρ is the prepared state, andP is the parity operator. From now on we will plot the values of P (α) for clarity, but use the terms displaced parity and Wigner function interchangeably. In our case, the cavity displacementD(α) is implemented by a microwave pulse at the phonon frequency while the qubit is detuned at ν 0 . 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 α with a 1 µs RMS width and truncated to 4 µs total length. We then measure the subsequent Fock state populations p n,|0 (α) 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 |0 using P |0 (α) = n (−1) n p n,|0 (α). Similarly, we can measure the displaced parity P ρ (α) for an arbitrary state ρ 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 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 T 1 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

Fabrication procedures
The transmon qubit used in our device is fabricated on sapphire using a standard e-beam lithography and Dolan bridge process 1 . The fabrication procedures for the acoustic resonator chip is shown in Figure S1a. 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 (AZ-EBR) on a hotplate at 60 • C.
After 2-3 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  Figure S1b. However, the simulated mode radius is only ∼20 µm (see Section 2), which means that the mode does not extend beyond the AlN part of the curved surface.
After making the resonators, we use photolithography, e-beam 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 home-built alignment and flip-chip 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 three-axis 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.

Acoustic mode simulations
Detailed descriptions of the design and simulation of high-Q HBAR resonators are given in [2][3][4] . 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 well-defined 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 v l = 11100 m/s and v t = 8540 m/s, which is similar to typical measured values and the values used in our previous work 4 . 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 2 × 10 5 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 4 . Figure S2c shows the resulting predicted coupling rate g 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 flip-chip device geometry in the future.

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 µs RMS width and 4 µs total width. We find that they agree well with the expected Poisson distributions for coherent states whose amplitudes α 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 off-resonant 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.

State reconstruction
State reconstruction was done using the formalism described in 5 . We use a maximum likelihood method that finds the most probable density matrix ρ that results in the Wigner functions we measured, subject to the constraints that ρ is physical (positive semi-definite, Hermitian, and unit trace).
We used Fock states up to n = 14 for extracting the populations p n,ρ (α) 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 n = 9 for the reconstruction. Figure S4 shows the reconstructed density matrices that were used to plot the Wigner functions in