Abstract
The experimental investigation of quantum devices incorporating mechanical resonators has opened up new frontiers in the study of quantum mechanics at a macroscopic level. It has recently been shown that surface acoustic waves (SAWs) can be piezoelectrically coupled to superconducting qubits, and confined in highquality Fabry–Perot cavities in the quantum regime. Here we present measurements of a device in which a superconducting qubit is coupled to a SAW cavity, realising a surface acoustic version of cavity quantum electrodynamics. We use measurements of the AC Stark shift between the two systems to determine the coupling strength, which is in agreement with a theoretical model. This quantum acoustodynamics architecture may be used to develop new quantum acoustic devices in which quantum information is stored in trapped onchip acoustic wavepackets, and manipulated in ways that are impossible with purely electromagnetic signals, due to the 10^{5} times slower mechanical waves.
Introduction
The study of the quantum nature of mechanical systems has rapidly increased in the last decade^{1,2,3}. The primary goal of these experiments has been the demonstration of the quantum behaviour of macroscopic objects when suitably isolated from their environment, with the intent to corroborate the validity of quantum mechanics at macroscopic scales. Pioneering work has now experimentally proved the possibility to prepare mechanical objects close to their quantum ground state^{4, 5} and to coherently manipulate their state^{6}. These results have encouraged new lines of research utilising mechanical quantum devices, including the development of microwaveoptical converters^{7}, mechanical quantum memories^{8} and quantumlimited amplifiers^{9}, the generation of squeezed vacuum states of mechanical objects^{10, 11}, and the detection of nonclassical correlations of photon–phonon pairs^{12}.
A highly successful architecture for the exchange of single quanta between coupled quantum systems is the solidstate version of cavity quantum electrodynamics (QED), known as circuit QED^{13}, in which the electrical interaction between a qubit and a highquality microwave resonator offers the possibility to reliably control, store, and read out quantum bits of information on a chip. Although many quantum experiments involving mechanical objects have been reported that use an optomechanical coupling between an electric field and a mechanical system^{2}, a parallel series of investigations have employed such a circuitQED type of interaction between mechanical resonators and superconducting qubits^{6, 14,15,16}, which in principle enables full control of the quantum state of the mechanical mode via the qubit.
In contrast to these experiments involving localised mechanical modes, studies have also recently emerged on the coupling of superconducting qubits to travelling surface acoustic waves (SAWs)^{17, 18}, which are mechanical perturbations that propagate on the surface of a crystal^{19}, and are naturally coupled to superconducting circuits using the piezoelectric effect. As well as being of fundamental interest to study such acoustic waves at the quantum level, they may find uses in quantum signal processing, since their slow speed of travel (five orders of magnitude slower than light) means manywavelength signals can be manipulated on a mmscale chip^{18, 20, 21}. It has been demonstrated that Fabry–Perot SAW cavities formed using superconducting surface Bragg mirrors can reach quality factors in the 10^{5} range at microwave frequencies^{22, 23}, opening up the possibility of realising surface acoustic cavity QED, either with superconducting qubits, or with other solidstate quantum systems^{24}. SAW cavities have also been proposed as a potential quantum acoustooptic transducer between superconducting qubits and optical photons by exploiting stimulated Brillouin scattering^{25}.
In this work, we present measurements of a device in which a tuneable transmon qubit^{26} is piezoelectrically coupled to a SAW cavity, displaying a surface acoustic version of cavity QED which we call circuit quantum acoustodynamics (QAD). We characterise the dispersive interaction between the two systems in several ways. First, we measure the frequency shift of the acoustic mode as the qubit is flux tuned. Secondly, we measure the acoustic Stark shift of the qubit due to the population of the mechanical resonator and we observe a preferential coupling of the qubit to one longitudinal mode of the acoustic cavity. We extract the coupling and we show that it is in agreement with theoretical expectations. In order to demonstrate the possibility to control the device in the time domain, we show a timedelayed Stark shift made possible by the slow travel of the wave. We also present spectroscopic measurements of the qubit via the Stark shift of the acoustic cavity, indicating that SAWRs can in principle be adopted as an alternative qubit readout scheme in quantum information processors.
Results
Description of the experiment
Our measured device (Fig. 1) is fabricated on STX quartz, on which the free SAW speed is v _{f} ≈ 3158 m/s at room temperature^{19}. This travelling mode is excited by applying an oscillating voltage to the electrodes of an interdigitated transducer (IDT) patterned on the surface of the substrate. The propagating SAW is confined to a small region of the chip between two Bragg mirrors facing each other forming a Fabry–Perot acoustic cavity; each mirror consists of a regular array of shorted metallic strips. A tuneable transmon qubit is situated in the middle of the SAW cavity and consists of a SQUID shunted by an interdigitated capacitance with periodicity λ _{0} matching the SAW IDTs (see Supplementary Note 1 for further details on device parameters). The transmon is also coupled to an auxiliary coplanar waveguide resonator (CPWR) employed for independent dispersive qubit readout^{27}. All measurements presented hereafter have been performed at the base temperature T ≈ 10 mK of a dilution refrigerator. Microwave ports 1 and 2 are connected to room temperature via low (≈16 dB) attenuation lines in order to easily populate and measure the SAW cavity modes, while port 3 is highly attenuated (≈70 dB) such that the CPWR and qubit are close to their quantummechanical ground states.
SAW cavity response in frequency and time domain
The SAW cavity contains two transducers for the excitation and detection of the acoustic wave. Figure 2a shows the transmission coefficient S _{21} of the cavity, measured via the two IDTs. The frequency of the central mechanical mode is f _{m2} = 523.435 MHz, while side peaks seen at f _{m1} = 522.83 MHz and f _{m3} = 524.58 MHz are likely to be additional mechanical modes. The quality factors of these modes, Q _{m1,m2,m3} = {4820, 6980, 7580}, are obtained from additional measurements of S _{11} (see Supplementary Note 1 for more information). Since the periodicity of the IDTs is set to λ _{0} = 6 in fabrication, the central mode frequency is consistent with a speed of sound of v _{e} = f _{m2} λ _{0} = 3140.6 m/s, assuming a symmetric device. The slight difference between v _{e} and the textbook room temperature value v _{f} may be due to slight device asymmetry and/or stiffness tensor changes or crystal contraction at millikelvin temperatures. Note that the SAW cavity modes are not in their ground state, due to the mode frequencies f _{mi } ≲ k _{B} T/h and lowattenuation connections to room temperature.
The acoustic nature of the observed resonant modes can be further tested by measuring the device response to a short coherent drive pulse (Fig. 2b). An exponentially decaying train of pulses is clearly observed in the response, consistent with the applied 800 ns pulse reflecting back and forth between the mirrors of the cavity. The decay time of the pulses, the lifetime of phonons in the SAW cavity, is τ ≈ 2.4 μs. The pulses measured in the response are separated by Δt = 870 ns, consistent with a cavity length of L _{c} = v _{e}Δt/2 ≈ 1365 μm. As expected, the cavity length is slightly longer than the distance between the two gratings d = 1260 μm. This is consistent with the fact that SAWs slightly penetrate into the gratings by an amount L _{p} = (L _{c} − d)/2 = 55 μm before being efficiently reflected. The frequency spacing of adjacent modes (the free spectral range) of the cavity are also consistent with the same cavity length, \({L_{\rm{c}}} = {v_{\rm{e}}}{\rm{/}}2\left {{f_{{\rm{m2}}}}  {f_{{\rm{m3}}}}} \right \approx 1365\) μm, where we have used the two higherquality modes f _{m2} and f _{m3}. Note, however, that f _{m1} is slightly closer in frequency to f _{m2}. This asymmetric behaviour in the frequency domain may be due to the fact that the grating stopband does not perfectly coincide with the resonant frequency of the IDTs^{28}.
Interaction between a SAW cavity and a superconducting qubit
Having characterised the SAW cavity, we now proceed to examine its interaction with the superconducting qubit, a fluxtuneable transmon. An appropriate quantummechanical description of this system (including the readout CPWR) is the generalised Jaynes–Cummings Hamiltonian of two resonators both coupled to the same transmon qubit (for reasons that will become clear, we will only consider the central acoustic mode at f _{m2}, which we hereafter refer to as the SAW resonator, SAWR):
where f _{ j }(Φ) are the flux dependent transmon transition frequencies, f _{r} = 5.83 GHz is the CPWR frequency, \(\hat a\) \(\left( {{{\hat a}^\dag }} \right)\) and \(\hat b\) \(( {{{\hat b}^\dag }} )\) are the annihilation (creation) operators of the microwave cavity and of the mechanical resonator, respectively and g _{ ij }(Φ) [λ _{m2,ij }(Φ)] is the coupling strength between the qubit and the CPWR (SAWR). Hereafter, we will denote the coupling strength between the CPWR (SAWR) and the first energy level of the qubit simply by g (λ _{m2}). The transition frequency f _{q} between the first two energy levels of the qubit is tuned by applying an external magnetic flux Φ to its superconducting loop, and its value is given by:
where Φ_{0} = h/2e is the magnetic flux quantum, and E _{C} = h × 0.31 GHz, E _{J0} = h × 10.7 GHz are the Coulomb and maximum Josephson energy of the qubit (obtained from standard qubit spectroscopy). The electrical coupling between the CPWR and the qubit mainly originates from the capacitance C _{2} shown in Fig. 1b. The acoustic coupling is instead due to the potential difference generated by the acoustic wave on the electrodes of the qubit and is given by:
where e is the electron charge, C _{q} and C _{Σ} are the qubit capacitance and the total capacitance seen by the qubit respectively, e _{pz} is the piezoelectric coupling coefficient, ε is the substrate permittivity, ρ is the substrate mass density, A _{c} is the acoustic cavity area and A(f) is a normalised array factor (see Supplementary Note 2 for a derivation of Eq. (3)). From values related to our experiment, this formula predicts λ _{m2} = 6.0 MHz (for a qubit frequency of f _{q} = 2.52 GHz, for comparison with later measurements).
SAW cavity response as a function of flux
As a first probe of the interaction between the qubit and the SAW mode, we measure both the frequency of the qubit (via the CPWR) and the acoustic mode f _{m2} as a function of magnetic flux Φ (Fig. 3). The qubit frequency (Fig. 3a) fits well to Eq. (2), while the SAW mode frequency (Fig. 3b) also shows a flux dependence with the same periodicity. By fitting the experimental curve of Fig. 3b with a QuTiP numerical model^{29} based on Eq. (1), we can extract the value of the acoustic coupling and we find λ _{m2} = 5.7 ± 0.5 MHz at f _{q} = 2.52 GHz. In this model, the transmon coupling strengths and level spacings are calculated by diagonalising the Hamiltonian for a Cooperpair box including many charge states^{26}. The free parameters of the model are the coupling strength λ _{m2}, the asymmetry of the critical currents of the two Josephson junctions [d _{sym} = (I _{c1} − I _{c2})/(I _{c1} + I _{c2}) = 0.09] and the effective temperature of the device (T = 85 mK). The additional SAW modes at f _{m1} and f _{m3} do not show any detectable flux dependence. This is in agreement with the expectation that these modes are antisymmetric with respect to the centre of the cavity, whereas the central mode f _{m2} and the qubit transducer geometry are both symmetric.
Acoustic Stark shift of the qubit
A second method to investigate the acoustic coupling between the qubit and the SAW is to measure the AC Stark shift between the two systems in the fardetuned limit (Fig. 4), in which the Hamiltonian of the SAWqubit system becomes:
where \({\hat \sigma _z}\) is a Pauli operator and we have approximated the transmon as a twolevel system for simplicity. We first set the magnetic flux such that the qubit frequency is f _{q} = 3.29 GHz, and the qubit–SAW detuning is \(\Delta = 2.77\,{\rm{GHz}} \gg {\lambda _{{\rm{m2}}}}\). In Fig. 4a, we show the qubit frequency shift as a function of SAW drive frequency close to the acoustic modes at two different drive powers. At the lower power of P _{in} = −74 dBm, we clearly observe a qubit frequency shift only at the frequency of the central SAW mode f _{m2}. The shift fits well to a Lorentzian centred at f _{m2}, and has a FWHM of 60 ± 5 kHz, close to that obtained for the SAW mode measured via S _{21}. No such frequency shift is observed at the other SAW mode frequencies f _{m1} and f _{m3} until higher power. From a second measurement at P _{in} = −64 dBm, high enough to observe small shifts at f _{m1} and f _{m3}, we can estimate the coupling of the qubit to these additional modes. Assuming that λ _{m2} = 5.7 MHz from the fit to the flux dependence of f _{m2}, and taking into account the different drive powers of the two experiments, we can estimate the coupling to the two side modes to be λ _{m1} ≈ 380 kHz and λ _{m3} ≈ 340 kHz, more than an order of magnitude lower than the coupling to the central mode. Figure 4b illustrates the qubit frequency shift as a function of drive power at frequency f _{m2}. The shift is observed to be linear, in agreement with the AC Stark effect.
Acoustic dispersive readout of the qubit
In a complementary experiment, we measure the Stark shift of the SAW mode frequency when the qubit excited state is populated (Fig. 4c), an acoustic equivalent of circuit QED dispersive qubit readout^{27}. This measurement can be used to extract an independent estimate of the acoustic coupling λ _{m2}. We carry out such a measurement at two different values of the magnetic flux, in both cases measuring the phase shift Δϕ of a probe drive at f _{m2} under strong drive of the qubit. We have \(\Delta \phi = 2\,{\rm{arctan}}\left( {\left \chi \right{\rm{/}}{\kappa _{{\rm{m2}}}}} \right)\), where κ _{m2} is the mechanical mode linewidth, \(\chi =  \lambda _{{\rm{m2}}}^2{E_{\rm{C}}}{\rm{/}}\Delta \left( {\Delta  {E_{\rm{C}}}} \right)\) is the dispersive shift for a transmon and Δ = f _{q} − f _{m2} is the frequency detuning. From Fig. 4c, we have Δ = 2.00 GHz and Δϕ = 4.9° ± 0.3° and since κ _{m2} = 75 kHz, we find \(\left \chi \right\) = 3.2 kHz and λ _{m2} = 5.9 ± 0.2 MHz, in close agreement with our earlier estimate obtained by fitting the flux dependence of f _{m2}. We can now use this measurement of χ to obtain an estimate for the average coherent phonon population \(\bar n = \langle {{{\hat b}^\dag }\hat b} \rangle\) of the SAW mode in the experiment shown in Fig. 4a, obtaining \(\bar n \approx {10^4}\left( {{{10}^5}} \right)\) for P _{in} = −74(−64) dBm. These values are within the limit in which the dispersive Hamiltonian remains valid^{30}, \({\lambda _{{\rm{m2}}}}\sqrt {\bar n} < \Delta\).
Timedelayed acoustic Stark shift of the qubit
We finally report an experiment in which we use the slow travel of the acoustic wave to apply a timedelayed Stark shift to the qubit, occurring as a SAW pulse passes the qubit inside the SAW cavity (Fig. 5). A short 100 ns pulse is first applied to one SAW IDT, then a timedelayed pulsed measurement of the qubit is subsequently carried out via the CPWR (with a 100 ns measurement pulse). A continuous drive is applied to the qubit throughout the experiment, the frequency of which is varied to determine the qubit frequency. The qubit is observed to shift lower in frequency initially at a time 170 ns after the SAW pulse is applied, exactly consistent with the timeofflight of the SAW pulse between the IDT and qubit. Several further frequency dips are then observed, spaced by ∼430 ns, again consistent with the SAW timeofflight from qubit to one Bragg mirror and back again. The qubit frequency is observed to decay back to its undisturbed value over a timescale of ≈3 μs, similar to the phonon lifetime of the SAW cavity. The multiple reflections between qubit, IDTs and Bragg mirrors likely serve to smooth out the response beyond a time delay of around 1 μs. As well as demonstrating the unique slow propagation of SAWs in a quantum device, this experiment also serves to further prove that the Stark shifts that we observe are indeed due to the acoustic field of the SAW mode, rather than a crosstalk of the electromagnetic signal applied to the IDT directly to the qubit.
Superconducting qubit coherence
Although, as detailed above, we have been able to observe clear signatures of the coherent coupling between a transmon and SAW resonator in our device, the coherence of our transmon (T _{1(2)} = 46(67) ns at f _{q} = 2.6 GHz (see Supplementary Note 1 for more details)) is significantly lower than the current state of the art. There are several likely explanations for this short coherence. Firstly, the lowtemperature dielectric loss tangent of our quartz substrate may be significantly higher than the lowloss sapphire and silicon substrates typically used in current superconducting circuits. Secondly, our transmon geometry necessarily includes a finepitched interdigitated capacitor to couple to the SAWs, which increases the dielectric participation ratio with respect to current highcoherence designs^{31}. Lastly, since quartz is a piezoelectric material, there may be a noticeable contribution to qubit energy relaxation from bulk (as opposed to surface) phonon emission.
Discussion
The prototype quantum acoustic device that we have presented here may be improved, opening up the possibility of using cavitytrapped SAWs for quantum memories, time delays and quantum signal filtering applications. In particular, we have used a relatively weak piezoelectric substrate for our experiment, nevertheless achieving a qubit–SAWR coupling strength of 5.7 MHz. Stronger piezoelectrics such as lithium niobate or zinc oxide could dramatically increase this coupling strength. This could have the additional benefit of enabling the qubit coherence to be improved, as the electric field of the qubit could be designed to only partially rather than fully reside in the piezoelectric substrate (which in the present case likely limits coherence due to undesired bulk acoustic emission). The 10^{5} times reduced speed of travel of SAWs compared to electromagnetic signals also makes our device a miniaturised mechanical implementation of traditional cavity QED, and an ideal engineered platform to push the boundaries of cavity QED physics, opening up the possibility to explore, for instance, strong coupling multimode cavity QED with mechanical devices.
Data availablity
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Poot, M. & van der Zant, H. S. Mechanical systems in the quantum regime. Phys. Rep. 511, 273–335 (2012).
 2.
Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014).
 3.
Treutlein, P., Genes, C., Hammerer, K., Poggio, M. & Rabl, P. Hybrid Mechanical Systems 327–351 (Springer, 2014).
 4.
Teufel, J. D. et al. Sideband cooling of micromechanical motion to the quantum ground state. Nature 475, 359–363 (2011).
 5.
Chan, J. et al. Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 478, 89–92 (2011).
 6.
O’Connell, A. D. et al. Quantum ground state and singlephonon control of a mechanical resonator. Nature 464, 697–703 (2010).
 7.
Andrews, R. W. et al. Bidirectional and efficient conversion between microwave and optical light. Nat. Phys. 10, 321–326 (2014).
 8.
Palomaki, T. A., Harlow, J. W., Teufel, J. D., Simmonds, R. W. & Lehnert, K. W. Coherent state transfer between itinerant microwave fields and a mechanical oscillator. Nature 495, 210–214 (2013).
 9.
Massel, F. et al. Microwave amplification with nanomechanical resonators. Nature 480, 351–354 (2011).
 10.
Wollman, E. E. et al. Quantum squeezing of motion in a mechanical resonator. Science 349, 952–955 (2015).
 11.
Pirkkalainen, J.M., Damskägg, E., Brandt, M., Massel, F. & Sillanpää, M. A. Squeezing of quantum noise of motion in a micromechanical resonator. Phys. Rev. Lett. 115, 243601 (2015).
 12.
Riedinger, R. et al. Nonclassical correlations between single photons and phonons from a mechanical oscillator. Nature 530, 313–316 (2016).
 13.
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
 14.
LaHaye, M. D., Suh, J., Echternach, P. M., Schwab, K. C. & Roukes, M. L. Nanomechanical measurements of a superconducting qubit. Nature 459, 960–964 (2009).
 15.
Pirkkalainen, J.M. et al. Hybrid circuit cavity quantum electrodynamics with a micromechanical resonator. Nature 494, 211–215 (2013).
 16.
Rouxinol, F. et al. Measurements of nanoresonatorqubit interactions in a hybrid quantum electromechanical system. Nanotechnology 27, 364003 (2016).
 17.
Gustafsson, M. V. et al. Propagating phonons coupled to an artificial atom. Science 346, 207–211 (2014).
 18.
Aref, T. et al. Quantum Acoustics with Surface Acoustic Waves, 217–244 (Springer International Publishing, 2016).
 19.
Morgan, D. Surface Acoustic Wave Filters (Academic Press, 2007).
 20.
Kockum, F. A., Delsing, P. & Johansson, G. Designing frequencydependent relaxation rates and Lamb shifts for a giant artificial atom. Phys. Rev. A 90, 013837 (2014).
 21.
Guo, L., Grimsmo, A., Kockum, A. F., Pletyukhov, M. & Johansson, G. Giant acoustic atom: a single quantum system with a deterministic time delay. Phys. Rev. A 95, 053821 (2017).
 22.
Manenti, R. et al. Surface acoustic wave resonators in the quantum regime. Phys. Rev. B 93, 041411 (2016).
 23.
Magnusson, E. B. et al. Surface acoustic wave devices on bulk ZnO at low temperature. Appl. Phys. Lett. 106, 063509 (2015).
 24.
Schuetz, M. J. A. et al. Universal quantum transducers based on surface acoustic waves. Phys. Rev. X 5, 031031 (2015).
 25.
Shumeiko, V. S. Quantum acoustooptic transducer for superconducting qubits. Phys. Rev. A 93, 023838 (2016).
 26.
Koch, J. et al. Chargeinsensitive qubit design derived from the cooper pair box. Phys. Rev. A 76, 042319 (2007).
 27.
Wallraff, A. et al. Approaching unit visibility for control of a superconducting qubit with dispersive readout. Phys. Rev. Lett. 95, 060501 (2005).
 28.
Uno, T. & Jumonji, H. Optimization of quartz SAW resonator structure with groove gratings. IEEE Trans. Sonics Ultrason. 29, 299–309 (1982).
 29.
Johansson, J. R., Nation, P. D. & Nori, F. QuTiP 2: a Python framework for the dynamics of open quantum systems. Comput. Phys. Commun. 184, 1234–1240 (2013).
 30.
Boissonneault, M., Gambetta, J. M. & Blais, A. Dispersive regime of circuit QED: photondependent qubit dephasing and relaxation rates. Phys. Rev. A 79, 013819 (2009).
 31.
Wang, C. et al. Surface participation and dielectric loss in superconducting qubits. Appl. Phys. Lett. 107, 162601 (2015).
Acknowledgements
This work has received funding from the UK Engineering and Physical Sciences Research Council (under Grant Nos. EP/J001821/1 and EP/J013501/1), CREST, the John Templeton Foundation, and the Japan Society for the Promotion of Science.
Author information
Affiliations
Clarendon Laboratory, Department of Physics, University of Oxford, OX1 3PU, Oxford, UK
 Riccardo Manenti
 , Andrew Patterson
 , Tanja Behrle
 , Joseph Rahamim
 , Giovanna Tancredi
 & Peter J. Leek
Center for Emergent Matter Science, RIKEN, Saitama, 3510198, Japan
 Anton F. Kockum
 & Franco Nori
Physics Department, The University of Michigan, Ann Arbor, MI, 481091040, USA
 Franco Nori
Authors
Search for Riccardo Manenti in:
Search for Anton F. Kockum in:
Search for Andrew Patterson in:
Search for Tanja Behrle in:
Search for Joseph Rahamim in:
Search for Giovanna Tancredi in:
Search for Franco Nori in:
Search for Peter J. Leek in:
Contributions
R.M. designed, fabricated and measured the device, and carried out all of the data analysis. A.F.K. carried out the QuTip simulations and derived the formula for the coupling strength. A.P., T.B., J.R. and G.T. contributed to the experimental setup. R.M. and P.J.L. wrote the paper with contributions from A.F.K. and F.N.; P.J.L. conceived and supervised the project.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Riccardo Manenti or Peter J. Leek.
Electronic supplementary material
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
Further reading

1.
Onchip temporal focusing of elastic waves in a phononic crystal waveguide
Nature Communications (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.