Abstract
In recent years, important progress has been made towards encoding and processing quantum information in the large Hilbert space of bosonic modes. Mechanical resonators have several practical advantages for this purpose, because they confine many highqualityfactor modes into a small volume and can be easily integrated with different quantum systems. However, it is challenging to create direct interactions between different mechanical modes that can be used to emulate quantum gates. Here we demonstrate an in situ tunable beamsplittertype interaction between several mechanical modes of a highovertone bulk acousticwave resonator. The engineered interaction is mediated by a parametrically driven superconducting transmon qubit, and we show that it can be tailored to couple pairs or triplets of phononic modes. Furthermore, we use this interaction to demonstrate the Hong–Ou–Mandel effect between phonons. Our results lay the foundations for using phononic systems as quantum memories and platforms for quantum simulations.
Similar content being viewed by others
Main
Mechanical degrees of freedom are a particularly interesting quantum platform, as they involve the collective motion of massive particles, can have long coherence times and can be combined with many other quantum systems^{1}. Circuit quantum acoustodynamics (cQAD) systems, where a superconducting qubit is coupled to gigahertzfrequency acoustic modes, have recently been engineered^{2,3,4} and used to demonstrate the generation and measurement of nontrivial quantum states^{4,5,6,7,8} and entanglement between mechanical modes^{9}. Due to the small mode volumes, low crosstalk and high coherence times of acoustic modes, cQAD devices have become the target platform of recent proposals for the realization of a quantum randomaccess memory^{10} as well as faulttolerant quantum computing architectures^{11,12}. In particular, cQAD devices that incorporate highovertone bulk acousticwave resonators (HBARs) can take advantage of the HBAR’s large effective mass and multimode properties, making them excellent platforms for the implementation of bosonic quantum simulations^{13,14,15}, bosonic encodings^{16,17}, quantum metrology applications^{18} and fundamental studies of quantum mechanical interference phenomena between phonons^{19,20,21}.
An important yet currently missing tool for the realization of these applications is the generation of a phononic iSWAP gate, which is an operation that allows for a direct exchange of quanta between mechanical modes. This can be engineered via a beamsplitter interaction, a coupling mechanism that has already been studied between photonic modes^{22,23}, in optomechanical systems^{24}, in trapped ions^{25}, between mechanical resonators in the classical regime^{19,26} and between travelling mechanical waves^{21}. When brought to the quantum regime, this phononic beamsplitter interaction will not only become a building block of quantum computing architectures^{10,11,12} but will also offer new possibilities for the simulation of complex quantum systems and the phononic realization of quantumopticstype experiments that have so far been mostly explored with photonic systems.
In this work, we demonstrate a beamsplitter interaction between multiple phonon modes of an HBAR coupled to a superconducting transmon qubit. We create this interaction by applying two offresonant drives on the qubit^{27} such that it acts as a nonlinear mixing element. We first study the effects of this bichromatic driving through qubit spectroscopy, observe the generation of multiple sidebands and show how these sidebands mediate the desired beamsplitter coupling. Having realized this interaction, we then perform timedomain experiments to demonstrate both iSWAP and \(\sqrt{{\rm{i}}{{{\rm{SWAP}}}}}\) gates, subsequently using the latter to demonstrate entanglement between two acoustic overtone modes of our HBAR. Furthermore, by choosing another parameter regime, we create an interference between three phononic modes and explore the multimode dynamics governing the system. Finally, we utilize the beamsplitter interaction to exchange multiple excitations between the modes and observe the Hong–Ou–Mandel interference^{21,22,25,28,29,30} between macroscopic mechanical modes.
The device used in this work is a cQAD system where a superconducting qubit is flipchip bonded to an HBAR^{7}. The qubit is a threedimensional transmon with a frequency of ω_{q} = 2π × 5.97 GHz, an energy relaxation time of T_{1} = 9.5 µs, a Ramsey decoherence time of \({T}_{2}^{\,* }\) = 7.2 µs and an anharmonicity α = 2π × 218 MHz. The longitudinal free spectral range (FSR) of the HBAR is approximately 2π × 12.63 MHz, and the two subsystems are coupled through a piezoelectric transducer that mediates a Jaynes–Cummings (JC) interaction with a coupling strength of g_{m} = 2π × 257 kHz. The device is housed in a threedimensional aluminium cavity, which we use to both shield the qubit from its environment and read its state via the dispersive interaction between the qubit and the cavity. Supplementary Table I provides a full list of system parameters.
Although the cQAD device used in this work has been previously studied in both dispersive^{5} and resonant coupling regimes^{6}, here we focus on direct multimode interactions that arise when two parametric drives are applied to the qubit. The Hamiltonian of our system in the presence of these drives is given by
where we assume g_{m} to be real. Here the first two terms describe the qubit as an anharmonic mode with lowering operator q. The sum over phonon modes m = a, b, c… with frequencies ω_{m} and lowering operators m includes their energies as well as their JC interaction with the qubit. The last term, given by \({H}_{{{{\rm{qd}}}}}=\left({\varOmega }_{1}{{\rm{e}}}^{{\rm{i}}{\omega }_{1}t}+{\varOmega }_{2}{{\rm{e}}}^{{\rm{i}}{\omega }_{2}t}\right){q}^{{\dagger} }+{{{\rm{h.c.}}}}\), describes two offresonant microwave drives applied to the qubit with frequencies ω_{1} = ω_{q} + 2π × 492.5 MHz and ω_{2} ≈ ω_{1} + FSR. The drives, together with two modes a and b, can participate in a fourwave mixing process mediated by the Josephson nonlinearity of the superconducting qubit^{10,27,31}. In particular, when the resonance condition Δ_{21} ≡ ω_{2} − ω_{1} = ω_{b} − ω_{a} is satisfied, equation (1) leads to a bilinear coupling between the phonon modes. Even though this picture is quantitatively accurate for large phonon–phonon detunings and small drive strengths, we now present a framework that extends this picture to address the case of large drive strengths and small phonon–phonon detunings. Furthermore, our analysis readily lends itself to systems with many bosonic modes by explicitly considering processes involving multiple drive photons.
We first consider only the effect of the drives on the qubit itself. Due to transmon anharmonicity, going into the displaced frame of the drives results in a modulated a.c. Stark shift of the qubit frequency given by (Supplementary Section V)
with the dimensionless drive strengths ξ_{j} = Ω_{j}/Δ_{j}, where Δ_{j} = ω_{j} − ω_{q} for j ∈ {1, 2}. This shift has a timeindependent as well as a timedependent contribution, the latter arising from the beating between the two drives, which modulates the qubit frequency with Δ_{21}. As usual for a frequencymodulated system^{32,33,34} (Fig. 1a), this gives rise to the appearance of multiple qubit sidebands separated by Δ_{21}, whose amplitudes are given by \({J}_{n}\left(\frac{\Lambda }{{\Delta }_{21}}\right)\). Here J_{n}(x) is the Bessel function of the first kind for a given sideband number n, and Λ = −4αξ_{1}ξ_{2}. We note that due to the interplay of the parametric drives with the third energy level of the qubit, H_{Stark} acquires a correction, which we derive using timeindependent perturbation theory (Supplementary Section III). In the following, we use the corrected value for the modulation depth, which we label as Λ′. Furthermore, we will use the shorthand \({J}_{n}={J}_{n}\left(\frac{{\Lambda }^{{\prime} }}{{\Delta }_{21}}\right)\).
We experimentally confirm these effects via twotone spectroscopy. Specifically, we sweep a weak probe signal across the qubit frequency with the offresonant drives turned on and subsequently measure the resulting qubit population using dispersive readout. As expected, we find multiple resonances separated by Δ_{21} with different peak heights, which are the qubit sidebands described above (Fig. 1b). The measured steadystate population of the qubit is quantitatively described in the same way as in a regular qubit spectroscopy experiment^{35}, with the probe strength adjusted by the sideband amplitude (Fig. 1b, continuous black line). After repeating the measurement for a range of parametric drive strengths ξ_{1}ξ_{2} (ξ_{1} = ξ_{2}), we find the result shown in Fig. 1c, where we observe multiple diagonal lines spaced in frequency by Δ_{21} and with varying intensities. These qubit sidebands shift to lower frequencies with increasing drive power, as expected from the Stark shift described by the first term in equation (2).
The JC interaction between the driven qubit and phonon modes results in anticrossings where the frequency of a sideband matches that of a phonon mode (Fig. 1c,d). However, the effective qubit–phonon coupling strength is scaled by the amplitude of the sideband closest to the phonon mode. Therefore, the gap of the anticrossing will be reduced from 2g_{m} to 2J_{n}g_{m}, as indicated for n = 0 (Fig. 1d).
In the dispersive regime, where all the qubit sidebands and phonon modes are far detuned, it is useful to enter the interaction picture of the sidebandmediated qubit–phonon coupling via the Schrieffer–Wolff transformation^{36}. After applying the rotatingwave approximation, we can identify two effects in the resulting effective Hamiltonian. First, there is a frequency shift in the phonon modes, due to their hybridization with the qubit^{37}, such that the phonon frequency in the presence of the driven qubit is ω_{m} + δ_{m} with
where \({\tilde{\Delta }}_{m}={\omega }_{m}{\tilde{\omega }}_{q}\) is the detuning between phonon mode m and the Starkshifted qubit. We see that a phonon mode’s frequency shift is dominated by the sideband for which the denominator in equation (3) is the smallest. Second, although the Schrieffer–Wolff transformation typically eliminates the JC coupling term between the qubit and phonons, in our case, it also gives rise to phonon–phonon coupling terms. For example, the coupling between two neighbouring phonon modes b and c is given by g_{bc}(b^{†}c + bc^{†}), with
when Δ_{21} = ω_{c} − ω_{b} + δ_{c} − δ_{b}, such that this term remains after the rotatingwave approximation. Here δ_{b,c} refer to the frequency shift of phonons b and c as described by equation (3). Similarly, the nextnearestneighbouring phonon modes a and c experience a coupling of g_{ac}(a^{†}c + ac^{†}), with
when 2Δ_{21} = ω_{c} − ω_{a} + δ_{c} − δ_{a}.
The numerator of equation (4), which contains the product of two successive Bessel functions, represents the physical process of the qubit converting one photon between the parametric drives. The frequency conversion of the drive photons compensates for the energy difference between the phonon modes, making the beamsplitter interaction resonant. Interestingly, the effective coupling strength for this process does not become larger monotonically with increasing drive strengths ξ_{1}ξ_{2}. Instead, the speed of the singlephoton conversion is reduced in favour of multiphoton processes, for example, converting two drive photons to bridge the energy gap between the phonon modes with a frequency difference of 2Δ_{21} (equation (5)). Supplementary Sections II and IV provide a more detailed derivation of the different transformations and their effects on the system Hamiltonian.
The dependence of the qubit sidebands on the Bessel functions is what allows us to choose different combinations of coupling strengths between the phonon modes and frequency shifts throughout this work. Naively, it might seem that due to the equal frequency spacing of the phonon modes, one cannot choose interactions between only a subset to be resonant. However, this is not the case. For instance, by choosing an appropriate modulation depth Λ′/Δ_{21}, we can choose the amplitude of J_{0} to be larger than those of the neighbouring sidebands, namely, J_{1} and J_{−1}. According to equation (3), the phonon mode closest to the zeroth sideband will shift by a larger amount \((\propto {J}_{0}^{2})\) than the adjacent phonon modes \((\propto {J}_{1}^{2},\,{J}_{1}^{2})\), giving rise to a unique frequency spacing between the two phonon modes equal to Δ_{21} and promoting a beamsplitter interaction between them (Fig. 2a). If, on the other hand, we choose a regime where J_{0} = J_{1} = − J_{−1}, the three phonon modes a, b and c adjacent in frequency to the n = −1, 0 and 1 sidebands, respectively, will be equally shifted, promoting beamsplitter interactions between these three modes. Note that in the latter case, the nextnearestneighbour modes a and c are coupled via a twophoton conversion described by equation (5).
We now experimentally investigate the first case of coupling between only the two modes b and c (Fig. 2a). By choosing appropriate drive strengths ξ_{1,2}, we set the modulation depth to Λ′/Δ_{21} = 0.610 ± 0.001 such that J_{0} = 0.91 ± 0.01 and J_{1} = 0.29 ± 0.01. Here the errors are propagated from uncertainties in the independent measurement of system parameters (Supplementary Table I). Our experimental protocol starts with swapping an excitation from the qubit into mode c using the resonant JC interaction. Note that we use a third microwave drive, far detuned from the parametric drives, to independently adjust the frequency of the qubit for this swap operation and to compensate the Stark shift of the qubit from the parametric drives during the beamsplitter interaction to set \({\tilde{\varDelta }}_{{b}}\) = 2π × 1.0 MHz ± 17 kHz. We then turn on the parametric drives for a variable time τ_{BS} (Fig. 2b). Afterwards, the qubit has a finite excitedstate population due to the offresonant drives. We reset the qubit to its ground state by swapping its residual population to an ancillary phonon mode detuned by several FSRs from the modes of interest^{6}. Finally, we swap the excitation from mode b or c into the qubit and measure its excitedstate population.
Repeating this experiment for different values of Δ_{21}, we observe the expected chevron pattern produced by a beamsplittertype interaction between the two modes (Fig. 2c). Here we vary Δ_{21} by only about ±1%, such that we can treat the modulation depth as constant. When Δ_{21} matches the unique detuning between the two modes Δ, we satisfy the resonance condition for the fourwave mixing process, and the exchange of quanta between the modes becomes most efficient. This occurs for a modulation frequency of (Δ_{21} − FSR) = −2π × 44 kHz, which matches our prediction from equation (3). We plot the phononmode populations for Δ_{21} = Δ (Fig. 2d) and fit them each to a decaying oscillation, yielding a beamsplitter coupling rate of g_{bc} = 2π × 15.6 ± 0.1 kHz. Note that the contrast for the oscillation in phonon mode b is slightly lower than that for phonon mode c. This is a result of the different decay rates between the two phonon modes, as well as a small but finite leakage to the next phonon mode, namely, m_{−1} (Fig. 2a). The microscopic origin of the different decay rates for different HBAR modes is a subject of ongoing research^{38}.
At the time τ_{BS} = π/4g_{BS} = 8.0 μs (Fig. 2d, dashed line), the interaction becomes a 50:50 beamsplitter or \(\sqrt{{\rm{i}}{{{\rm{SWAP}}}}}\) gate, which creates an entangled state between the two phonon modes. We experimentally confirm this by performing twoqubit state tomography on the resulting state (Fig. 2e). Here, in contrast to the data shown in Fig. 2c,d, we measure the observables of both phonon modes in the same sequence, thereby accessing joint twomode observables necessary for fullstate tomography. To quantify the created entanglement, we compute an overlap of the reconstructed density matrix with the maximally entangled state \(\left\vert bc\right\rangle =(\left\vert 01\right\rangle +{{\rm{e}}}^{{\rm{i}}\phi }\left\vert 10\right\rangle )/\sqrt{2}\) of F_{Bell} = 0.69 ± 0.01, with ϕ chosen to optimize F_{Bell}. This confirms the presence of entanglement between the two phonon modes. We attribute the difference between the reconstructed density matrix and the maximally entangled state to phonon decay during the \(\sqrt{{\rm{i}}{{{\rm{SWAP}}}}}\) gate and an imperfect state preparation of the initial Fock state in mode c. Supplementary Section VI provides details on the tomography procedure.
Having demonstrated a beamsplitter interaction between the two phonon modes, we now move on to create simultaneous interactions between three modes. To that end, we tune the modulation depth to Λ′/Δ_{21} = 1.430 ± 0.003 such that J_{0} = J_{1} = −J_{−1} = 0.55 ± 0.01. In this regime, phonon modes a, b and c are equally shifted such that Δ_{cb} = Δ_{ba} ≡ Δ. This is schematically shown in Fig. 3a. In this case, phononmode pairs (b, c) and (a, b) are coupled via equation (4), whereas the mode pair (a, c) is coupled via equation (5), with ∣g_{ab}∣ ≈ ∣g_{bc}∣ ≈ ∣g_{ac}∣.
To explore the dynamics of this threemode coupling scheme, we extend the experiment presented in Fig. 2. Specifically, we load an excitation into phonon mode b and turn on the parametric drives, thereby activating beamsplitter interactions between all the three modes, and finally measure their population. As before, we sweep the interaction time τ_{BS} and the modulation frequency Δ_{21}, with \({\tilde{\varDelta }}_{b}\) = 2π × 1.0 MHz ± 17 kHz. The results are shown in Fig. 3b,c. Although they show the expected qualitative aspects of the excitation swapping between all the three modes, we observe two interesting features. First, when Δ_{21} = Δ, the initial excitation in mode b flows to modes a and c with approximately equal rates (Fig. 3d). However, the excitation does not fully swap to modes a and c, which is visible from the reduced oscillation contrast (Fig. 3d, greyshaded area). Although counterintuitive at first, this is the expected behaviour of a threemode system with coupling between all the mode pairs. The coupling between modes a and c hybridizes them into new normal modes with frequencies shifted by the coupling strength. As a result, the coupling between these normal modes and mode b is no longer resonant, resulting in the reduced oscillation contrast we observe. We note that the frequency of the population exchange observed in Fig. 3d, namely, 2π × 64 ± 1.5 kHz, is in good agreement with theoretical calculations.
The second observation is that the data in Fig. 3c for mode a are approximately the mirror image of mode c with respect to Δ_{21} − Δ = 0. For instance, when Δ_{21} − Δ > 0 (Δ_{21} − Δ < 0), the initial excitation in mode b predominantly flows to mode a (c). Although the roles of modes a and c are symmetric when Δ_{21} = Δ, this symmetry is broken away from the resonance condition due to the coupling between modes a and c and the resulting normalmode splitting. Supplementary Section VIII presents a detailed explanation for both these effects.
Although we present experimental details on two interesting values of modulation depth, we note that we can tune from one regime to the other by changing the drive powers, thereby observing a gradual change in both coupling strength and relative detuning (Fig. 3e). To acquire the effective interaction strengths between the three modes as well as their respective phonon frequency shifts, we perform the experiment shown in Fig. 3b,c for different values of ξ_{1}ξ_{2}, thereby varying Λ′/Δ_{21}. We then fit the measured phonon populations to a set of coupled equations of motion with beamsplitter couplings g_{mk} and relative phonon detunings δ_{mk} as free parameters (m, k ∈ {a, b, c}). Supplementary Section VII provide details on the fitting procedure. The fit results are plotted alongside equations (3)–(5) with no free parameters (Fig. 3e) and show good agreement between experiment and theory. The observed difference between ∣g_{ab}∣ and ∣g_{bc}∣ is a result of the different relative contributions from the sidebands in equation (4) depending on the position of the phonon modes involved. In particular, the observed reduction in ∣g_{ab}∣ and ∣g_{bc}∣ for larger modulation depths, as well as the accompanying increase in ∣g_{ac}∣, are well captured by theory. We emphasize that previous works have only investigated a much smaller range of modulation depths; therefore, these effects were not evident^{16,27,31,39}.
So far, we have studied the two and threemode coupling regimes for the particular case where a single phononic quantum is shared between all of the participating modes. We now investigate the interplay of two quanta during a beamsplitter operation. We first create a cb〉 = 11〉 state in modes b and c by repeatedly exciting the qubit and swapping its excitation into each mode^{7}. We then turn on the twomode beamsplitter interaction and subsequently measure the resulting phonon Fockstate distributions of either mode by monitoring the qubit population during a resonant qubit–phonon JC interaction, as shown in previous work^{7} (Fig. 4a). As an example, the results for a beamsplitter time of τ_{BS} = 6.7 μs are shown in Fig. 4b. Here, to optimize the coupling strength and reduce the residual JC interaction with the qubit, we use a slightly larger qubit–phonon detuning of \({\tilde{\Delta }}_{b}\) = 2π × 1.2 MHz ± 17 kHz and modulation depth of Λ′/Δ_{21} = 0.850 ± 0.002, resulting in g_{bc} = 2π × 18.5 ± 0.8 kHz.
The Hong–Ou–Mandel effect predicts that the outcome of this experiment should depend on whether or not the two phonons are distinguishable. If they are, no interference between them will occur and the excitations will be equally shared between the two phonon modes. On the other hand, if they are indistinguishable, both excitations will bunch in one of the two phonon modes after the beamsplitter. To experimentally confirm this, we compare the probability of the bunched (P_{20} + P_{02}) with that of the antibunched outcome (P_{11}). We extract the bunched outcome probability from the individual Fock distributions by assigning P_{02} + P_{20} to \({P}_{2}^{c}+{P}_{2}^{b}\), where \({P}_{2}^{c(b)}\) is the probability of finding two quanta in mode c (b). Doing so relies on the assumption that our system contains a maximum of two excitations at the start of the beamsplitter interaction and that no additional quanta are added during the sequence. This assumption is justified because the residual thermal population of the phonon modes is less than 1.6% (ref. ^{40}). Under the same assumption, we can put an upper bound on the antibunched probability, namely, \({\bar{P}}_{11}=\min ({P}_{1}^{b},{P}_{1}^{\,c})\ge {P}_{11}\). Nevertheless, we still take into account the possibility for leakage into higher Fock states by fitting the qubit–phonon Rabi oscillations for the first five energy levels. The population contribution of these higher levels is 0.01 on average and is then included in the error bars (Fig. 4c).
In Fig. 4c, we show both \({\bar{P}}_{11}\) and P_{20} + P_{02} for various beamsplitter interaction times τ_{BS}, normalized by the entire twoexcitation subspace \({P}_{\Sigma }={P}_{20}+{P}_{02}+{\bar{P}}_{11}\). As expected, the twoexcitation manifold of the phonon state in the beginning of the interaction is dominated by 11〉. After τ_{BS} = 6.7 µs, which corresponds to a 50:50 beamsplitter (Fig. 4c, vertical dashed line), the joint state is more probably bunched with (P_{20} + P_{02})/P_{Σ} = 0.622 ± 0.028.
Although we cannot straightforwardly access the joint Fock distributions of the two phonon modes in our experiment, we can do so in a master equation simulation of our system using independently measured system parameters. The results are plotted as continuous lines in Fig. 4c, showing good agreement between data and theory. The fast oscillations that can be seen for lower interaction times in both theory and experiment arise due to an offresonant JC interaction with the qubit. This result demonstrates how two apriori distinguishable phononic quanta in modes at different frequencies are made indistinguishable by a frequencyconverting coupling, which compensates for the energy difference between the two modes, thereby confirming that the lattice vibrations constituting our phonons display behaviour that cannot be classically described.
In conclusion, we have engineered a direct beamsplitter coupling between two and three distinct mechanical modes of an HBAR. We have used the twomode interaction to create a phononic \(\sqrt{{\rm{i}}{{{\rm{SWAP}}}}}\) gate, allowing us to generate entanglement between the modes and observe the Hong–Ou–Mandel effect between two phonons. In addition to our experimental data, we have also presented a theoretical model that is in good agreement with our findings. Parametrically driven beamsplitters are being actively studied for the purpose of bosonic quantum computing^{16,21,39,41}. Our work explores a new regime of this interaction, where sidebands generated by a large frequency modulation depth and the conversion of more than one drive photon plays an important role. We find our beamsplitter operation to be limited in speed by the qubit–phonon coupling strength and in fidelity by the phonon lifetimes. Larger values for both these parameters have been observed^{42,43}, though combining both remains a challenge. Nevertheless, on the basis of these recent developments, we expect to be able to improve our device quality in the near future.
Our results provide a fundamental building block for performing quantumopticstype experiments with massive mechanical excitations^{6}. They also address a key challenge towards realizing a mechanical quantum randomaccess memory by providing one of two required operations^{10}, the other one being a conditional phase operation^{31}. Furthermore, our technique, in principle, allows for alltoall coupling between a large number of phononic modes, all compactly hosted within a single physical resonator. This makes our device a hardwareefficient platform for future studies of nonreciprocal interactions^{19,44} and quantum simulations with bosonic modes^{13,14,45}. Finally, our current system and the concepts discussed here can potentially be extended to single and twomode squeezing interactions, enabling Gaussian quantum information processing using mechanical resonators^{46}.
Data availability
All data that support the plots within this paper and other findings of this study are available from the corresponding authors on reasonable request. Source data are provided with this paper.
Code availability
The analysis and simulation codes that support the findings of this study are available from the corresponding authors on request.
References
Chu, Y. & Gröblacher, S. A perspective on hybrid quantum opto and electromechanical systems. Appl. Phys. Lett. 117, 150503 (2020).
Chu, Y. et al. Quantum acoustics with superconducting qubits. Science 358, 199–202 (2017).
Satzinger, K. J. et al. Quantum control of surface acousticwave phonons. Nature 563, 661–665 (2018).
ArrangoizArriola, P. et al. Resolving the energy levels of a nanomechanical oscillator. Nature 571, 537–540 (2019).
von Lüpke, U. et al. Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics. Nat. Phys. 18, 794–799 (2022).
Bild, M. et al. Schrödinger cat states of a 16microgram mechanical oscillator. Science 380, 274–278 (2023).
Chu, Y. et al. Creation and control of multiphonon Fock states in a bulk acousticwave resonator. Nature 563, 666–670 (2018).
Sletten, L., Moores, B., Viennot, J. & Lehnert, K. Resolving phonon Fock states in a multimode cavity with a doubleslit qubit. Phys. Rev. X 9, 021056 (2019).
Wollack, E. A. et al. Quantum state preparation and tomography of entangled mechanical resonators. Nature 604, 463–467 (2022).
Hann, C. T. et al. Hardwareefficient quantum random access memory with hybrid quantum acoustic systems. Phys. Rev. Lett. 123, 250501 (2019).
Pechal, M., ArrangoizArriola, P. & SafaviNaeini, A. H. Superconducting circuit quantum computing with nanomechanical resonators as storage. Quantum Sci. Technol. 4, 15006 (2019).
Chamberland, C. et al. Building a faulttolerant quantum computer using concatenated cat codes. PRX Quantum 3, 010329 (2022).
Wang, C. S. et al. Efficient multiphoton sampling of molecular vibronic spectra on a superconducting bosonic processor. Phys. Rev. X 10, 021060 (2020).
Huh, J., Guerreschi, G. G., Peropadre, B., McClean, J. R. & AspuruGuzik, A. Boson sampling for molecular vibronic spectra. Nat. Photon. 9, 615–620 (2015).
Sparrow, C. et al. Simulating the vibrational quantum dynamics of molecules using photonics. Nature 557, 660–667 (2018).
Teoh, J. D. et al. Dualrail encoding with superconducting cavities. Proc. Natl. Acad. Sci. 120, e2221736120 (2023).
Lau, H. & Plenio, M. B. Universal quantum computing with arbitrary continuousvariable encoding. Phys. Rev. Lett. 117, 100501 (2016).
Munro, W. J., Nemoto, K., Milburn, G. J. & Braunstein, S. L. Weakforce detection with superposed coherent states. Phys. Rev. A 66, 023819 (2002).
del Pino, J., Slim, J. J. & Verhagen, E. NonHermitian chiral phononics through optomechanically induced squeezing. Nature 606, 82–87 (2022).
Wanjura, C. C. et al. Quadrature nonreciprocity in bosonic networks without breaking timereversal symmetry. Nat. Phys. 19, 1429–1436 (2023).
Qiao, H. et al. Splitting phonons: Building a platform for linear mechanical quantum computing. Science 380, 1030–1033 (2023).
Gao, Y. Y. et al. Programmable interference between two microwave quantum memories. Phys. Rev. X 8, 021073 (2018).
Rodrigues, I. C., Bothner, D. & Steele, G. A. Cooling photonpressure circuits into the quantum regime. Sci. Adv. 7, eabg6653 (2021).
Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391 (2014).
Toyoda, K., Hiji, R., Noguchi, A. & Urabe, S. Hong–Ou–Mandel interference of two phonons in trapped ions. Nature 527, 74–77 (2015).
Hälg, D. et al. Strong parametric coupling between two ultracoherent membrane modes. Phys. Rev. Lett. 128, 094301 (2022).
Zhang, Y. et al. Engineering bilinear mode coupling in circuit QED: theory and experiment. Phys. Rev. A 99, 012314 (2019).
Kobayashi, T. et al. Frequency–domain Hong–Ou–Mandel interference. Nat. Photon. 10, 441–444 (2016).
Lopes, R. et al. Atomic Hong–Ou–Mandel experiment. Nature 520, 66–68 (2015).
Lang, C. et al. Correlations, indistinguishability and entanglement in Hong–Ou–Mandel experiments at microwave frequencies. Nat. Phys. 9, 345–348 (2013).
Gao, Y. Y. et al. Entanglement of bosonic modes through an engineered exchange interaction. Nature 566, 509–512 (2019).
Strand, J. D. et al. Firstorder sideband transitions with fluxdriven asymmetric transmon qubits. Phys. Rev. B 87, 220505(R) (2013).
Naik, R. K. et al. Random access quantum information processors using multimode circuit quantum electrodynamics. Nat. Commun. 8, 1904 (2017).
Kervinen, M., RamírezMuñoz, J. E., Välimaa, A. & Sillanpää, M. A. LandauZenerStückelberg interference in a multimode electromechanical system in the quantum regime. Phys. Rev. Lett. 123, 240401 (2019).
Schuster, D. et al. a.c. Stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field. Phys. Rev. Lett. 94, 123602 (2005).
Schrieffer, J. R. & Wolff, P. A. Relation between the Anderson and Kondo Hamiltonians. Phys. Rev. 149, 491 (1966).
Gely, M. F. & Steele, G. A. Superconducting electromechanics to test Diósi–Penrose effects of general relativity in massive superpositions. AVS Quantum Sci. 3, 035601 (2021).
Cleland, A. Y., Wollack, E. A. & SafaviNaeini, A. H. Studying phonon coherence with a quantum sensor. Preprint at https://arxiv.org/abs/2302.00221 (2023).
Chapman, B. J. et al. HighOnOffRatio BeamSplitter Interaction for Gates on Bosonically Encoded Qubits. PRX Quantum 4, 020355 (2023).
Schrinski, B. et al. Macroscopic quantum test with bulk acoustic wave resonators. Phys. Rev. Lett. 130, 133604 (2023).
Lu, Y. et al. Highfidelity parametric beamsplitting with a parityprotected converter. Nat. Commun. 14, 5767 (2023).
Kervinen, M., Rissanen, I. & Sillanpää, M. Interfacing planar superconducting qubits with high overtone bulk acoustic phonons. Phys. Rev. B 97, 205443 (2018).
Gokhale, V. J. et al. Epitaxial bulk acoustic wave resonators as highly coherent multiphonon sources for quantum acoustodynamics. Nat. Commun. 11, 2314 (2020).
Koch, J., Houck, A. A., Le Hur, K. & Girvin, S. Timereversalsymmetry breaking in circuitQEDbased photon lattices. Phys. Rev. A 82, 043811 (2010).
Hartmann, M. J. Quantum simulation with interacting photons. J. Opt. 18, 104005 (2016).
Weedbrook, C. et al. Gaussian quantum information. Rev. Mod. Phys. 84, 621 (2012).
Acknowledgements
We thank E. Verhagen, Y. Zhang and M. Bild for useful discussions. The fabrication of the device was performed at the FIRST cleanroom of ETH Zürich and the BRNC cleanroom of IBM Zürich. We acknowledge support from the Swiss National Science Foundation under grant 200021_204073. M.F. was supported by The Branco Weiss Fellowship—Society in Science, administered by the ETH Zürich.
Funding
Open access funding provided by Swiss Federal Institute of Technology Zurich.
Author information
Authors and Affiliations
Contributions
U.v.L. designed and fabricated the device. U.v.L., I.C.R. and Y.Y. performed the experiments and analysed the data. U.v.L. developed the theoretical model and performed the QuTiP simulations of the experiments. M.F. provided the theory support. Y.C. supervised the work. U.v.L., I.C.R. and Y.C. wrote the paper with input from all authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Physics thanks Audrey Bienfait and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Supplementary Information
Supplementary Sections I–VIII, Figs. 1–7 and Tables I and II.
Source data
Source Data Fig. 1
Source data for Fig. 1.
Source Data Fig. 2
Source data for Fig. 2.
Source Data Fig. 3
Source data for Fig. 3.
Source Data Fig. 4
Source data for Fig. 4.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
von Lüpke, U., Rodrigues, I.C., Yang, Y. et al. Engineering multimode interactions in circuit quantum acoustodynamics. Nat. Phys. 20, 564–570 (2024). https://doi.org/10.1038/s4156702302377w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s4156702302377w
This article is cited by

Sound interactions across multiple modes
Nature Physics (2024)