Abstract
Surface acoustic waves are commonly used in classical electronics applications, and their use in quantum systems is beginning to be explored, as evidenced by recent experiments using acoustic Fabry–Pérot resonators. Here we explore their use for quantum communication, where we demonstrate a singlephonon surface acoustic wave transmission line, which links two physically separated qubit nodes. Each node comprises a microwave phonon transducer, an externally controlled superconducting variable coupler, and a superconducting qubit. Using this system, precisely shaped individual itinerant phonons are used to coherently transfer quantum information between the two physically distinct quantum nodes, enabling the highfidelity nodetonode transfer of quantum states as well as the generation of a twonode Bell state. We further explore the dispersive interactions between an itinerant phonon emitted from one node and interacting with the superconducting qubit in the remote node. The observed interactions between the phonon and the remote qubit promise future quantumopticsstyle experiments with itinerant phonons.
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Introduction
Quantum communication is of significant interest for the generation of remote entanglement and the secure transmission of information, as well as for distributed quantum computing^{1,2,3,4,5,6,7}. There are several demonstrations of longdistance quantum communication protocols using optical methods, in parallel with demonstrations of similar protocols using microwavefrequency photons, including Bell state entanglement of remote qubits as well as the transmission of multiqubit entangled states^{8,9,10,11,12,13,14,15,16}. Microwavefrequency phonons, as opposed to photons, can also be used for quantum communication as well as for coupling hybrid quantum systems^{17,18,19,20}, in the latter case taking advantage of the strong strain coupling in some optical as well as atomicscale systems. Microwavefrequency acoustic resonators may be able to serve as very longlived quantum memories^{21}. Quantum communication protocols implemented with phonons are thus of significant scientific as well as practical interest. Recent advances in the quantum control of phonons include the creation and measurement of stationary phonon quantum states^{22,23,24}, the emission and absorption of phonons in an acoustic resonator^{25}, and the generation of entangled phonons in a phononmediated quantum eraser experiment^{26}.
Here we report the experimental realization of a phononbased quantum communication channel, enabling the communication of quantum states via traveling phonons linking two physically distinct quantum nodes. The phonons are emitted in the communication channel as shortduration acoustic pulses, sufficiently brief that the extent of the acoustic pulses is significantly less than the length of the channel, such that the phonons travel in a particlelike fashion along the channel, which we term itinerant.
The experimental system is shown schematically in Fig. 1, with the physical setup in Fig. 1a and the circuit schematic in Fig. 1b. The 2mmlong phonon communication channel (500 ns singletrip time) is terminated at each end by a specially designed unidirectional interdigitated transducer (IDT), which is in turn connected to a superconducting qubit via a superconducting tunable coupler. The unidirectional transducers (UDTs) differ from conventional acoustic transducers, here emitting itinerant phonons in only one direction, as opposed to more standard bidirectional transducers, which emit excitations equally in two opposing directions (see Supplementary Note 1; a related but distinct design appears in ref. ^{27}). We note this device differs from the experimental construction in e.g. ref. ^{25}, which uses a single bidirectional transducer in a Fabry–Pérot cavity. In that experiment, a single phonon comprises acoustic excitations that travel in two opposing directions to distant acoustic mirrors, from which the excitations reflect and return to interfere constructively at the emitting transducer, where the excitation can be intercepted by one of two qubits. In the experiment here, two distinct UDTs are used to link two physically separate nodes. Each transducer is constructed to emit an acoustic excitation in only one direction, creating a significantly more flexible and generalpurpose design, with physically separate and distinct phonon emitter and receiver.
We use this device to demonstrate twonode quantum state transfers as well as the phononmediated deterministic generation of an entangled Bell state, representing a significant advance over prior work, in which a single transducer was coupled to a Fabry–Pérot acoustic cavity formed by two acoustic mirrors^{22,23,24,25,26}. We also realize a singlephonon interferometer, using one qubit to emit and detect a traveling phonon, where the phonon is used to probe the state of the second qubit, effectively demonstrating the dispersive interaction of a photon (localized in the remote qubit) and a traveling phonon. Finally, we demonstrate a Ramsey interferometer, using the second qubit to detect the presence of a traveling phonon emitted by the first qubit, thus interchanging the roles of the qubits in the previous experiment and demonstrating the versatility of this architecture.
Results
Phononmediated quantum state transfer
We first probe the interaction between the qubits and the phonon channel, as shown in Fig. 2a. We excite Q_{1} with a π pulse, then set its coupler G_{1} to an intermediate coupling, sufficient that Q_{1}’s relaxation is dominated by phonon emission. We set Q_{2}’s coupler G_{2} off during this measurement so that Q_{2} does not interact with the traveling phonon. For frequencies inside the transducer’s active band, from 3.87 to 4.01 GHz, where the emission is almost entirely unidirectional itinerant phonons, we observe a timedelayed revival of qubit Q_{1}’s excited state population \({P}_{e}^{{Q}_{1}}\) at times that are multiples of the phonon roundtrip time τ_{RT} ~ 1 μs, each revival corresponding to the traveling phonon reflecting off the other transducer before reexciting Q_{1}. Outside the unidirectional band, we see a complex structure in P_{e} as a function of frequency and interaction time, with broad swings of width ~150 MHz superposed with narrow oscillations of width ~7 MHz. The broad swings and finer details are in accordance with expectations (see Supplementary Note 1)^{28}.
The itinerant phonon experiments are performed at the working frequency \({\omega }_{{Q}_{1,2}}^{{{{{{\rm{uni}}}}}}}/2\pi =\,{{\mbox{3.976}}}\,\ \,{{\mbox{GHz}}}\,\), inside the unidirectional band. By working outside this band, we can explore the regime where the transducers are effectively bidirectional, using the second working frequency \({\omega }_{{Q}_{1,2}}^{{{{{{\rm{bi}}}}}}}/2\pi =\,{{\mbox{4.102}}}\,\ \,{{\mbox{GHz}}}\,\). These frequencies are marked by the dashed white and red lines, respectively, in Fig. 2a.
To maximize the efficiency of phononmediated quantum state transfers, we need to carefully shape the emission and absorption of the phonon wave packet, which is done by timedependent control of the coupling between the qubit and its transducer^{11,12,13,14,15,25,29}. We experimentally optimize the transfer efficiency, with results shown in Fig. 2b for both the unidirectional (left) and bidirectional (right) regimes. The transfer starts with the shaped emission of a phonon, shown by the decrease of Q_{1}’s excited state population with the expected time dependence. Both qubits then remain in their ground states until the phonon reaches Q_{2}, which absorbs the itinerant phonon, following the expected time dependence, and ultimately reaching a plateau once the transfer is complete. The total transfer takes ~700 ns, including the ~500 ns phonon travel time. The final Q_{2} population reaches a maximum of 68% for the unidirectional transfer, limited mostly by phonon loss in the channel. For the bidirectional transfer, the final Q_{2} population reaches 15%, 4.5 times less than the unidirectional population, which is 12% higher than the ideal value, demonstrating good agreement with theory and excellent unidirectionality for the transducer design. We simulate the transfer process using a cascaded quantum inputoutput model^{25} (solid green line). From this model, we estimate that phonon loss reduces the final unidirectional transfer efficiency by 27%, and the finite Q_{1} and Q_{2} coherence times reduce the fidelity by 1 and 2%, respectively. We note that an equivalent photon travel time would require a ~100mlong coaxial cable, illustrating the very long delays achievable with phononbased quantum channels.
In Fig. 2c, we show quantum process tomography for both regimes. For the unidirectional process, we find a process fidelity of \({{{{{{\mathcal{F}}}}}}}^{{{{{{\rm{uni}}}}}}}=\,({{\rm{82.0}}}\,\pm \,{\rm{0.3}}){{\%}}\), while for the bidirectional regime, the process fidelity is limited to \({{{{{{\mathcal{F}}}}}}}^{{{{{{\rm{bi}}}}}}}=\,({{\rm{39.0}}}\,\pm\,{{\rm{0.3}}}){{{\%}}}\). We compare these experimental process fidelities with predictions and find trace distances \(d=\sqrt{{{{{{\rm{Tr}}}}}}\,{({\chi }_{\exp }{\chi }_{{{{{{\rm{sim}}}}}}})}^{2}}=0.07\) and 0.3 for the unidirectional and bidirectional regimes. The contrast in fidelities and trace distances underlines the importance of the UDTs.
Traveling phononmediated remote entanglement
We further explore the capabilities of itinerant phonon communication by performing a phononmediated remote entanglement of the two qubits, shown in Fig. 3. The protocol is similar to that for the quantum state transfer, except here we calibrate the emission pulse to only emit Q_{1}’s excitation as a phonon with a probability of 1/2, meaning that immediately following the “halfemission,” with qubit Q_{2} in the ground state, the system is ideally in the state \((\lefte0g\right\rangle +\leftg1g\right\rangle )/\sqrt{2}\) (writing the state \(\left{Q}_{1}\ \gamma \ {Q}_{2}\right\rangle\) where γ represents the itinerant phonon). During the time the emitted “halfphonon” travels along the phonon channel, Q_{1}’s remaining excitation decays following Q_{1}’s intrinsic T_{1} time, with Q_{1}’s coupling to the channel set to zero. The traveling halfphonon is then captured by Q_{2}, generating a Bell state \(\left\psi \right\rangle =(\lefteg\right\rangle +{e}^{i\varphi }\leftge\right\rangle )/\sqrt{2}\) between the two qubits, with φ a relative phase.
Figure 3a shows the timedependent qubit state populations P_{e} for each qubit, which agrees well with a master equation simulation. Following the capture of the halfphonon, we perform quantum state tomography at time t_{m} = 750 ns; these measurements are used to reconstruct the twoqubit density matrix ρ shown in Fig. 3b. We find a Bell state fidelity \({{{{{{\mathcal{F}}}}}}}_{{{{{{\rm{Bell}}}}}}}={{{{{\rm{Tr}}}}}}\,({\rho }_{{{{{{\rm{ideal}}}}}}}\cdot \rho )=\,{{\rm{72}}}{{{\%}}}\,\) and a concurrence \({{{{{\mathcal{C}}}}}}=\,{{\mbox{0.53}}}\,\), close to the master equation simulation results, with a trace distance \({d}^{{{{{{\rm{Bell}}}}}}}=\sqrt{{{{{{\rm{Tr}}}}}}\,{({\rho }_{\exp }{\rho }_{{{{{{\rm{sim}}}}}}})}^{2}}=\,{{\mbox{0.13}}}\,\).
Phononqubit dispersive interaction
Sensing traveling phonons without absorbing them would provide a highly useful capability, as would being able to use a traveling phonon as a probe of a remote quantum system, which we explore in a pair of related experiments. First, we use a traveling phonon as a probe of a remote quantum twolevel system, shown in Fig. 4a. We use qubit Q_{1} as the emitter and receiver of a “halfphonon” that is detected interferometrically^{25,26} when returning to Q_{1}. This allows us to measure how the phase of the traveling phonon is affected by interacting dispersively with qubit Q_{2}, which serves as a standin for a generic quantum system.
The pulse sequence for this state detection is shown to the right in Fig. 4a. We first prepare Q_{1} in its excited state and emit a halfphonon, which reflects from the distant transducer, whose coupling to Q_{2} is turned on during the reflection process, and the halfphonon interacts with Q_{1} on its return. During the halfphonon transit, we briefly shift Q_{1}’s frequency so that Q_{1}’s excited state acquires a relative phase φ, yielding an interferometric interaction with the returning halfphonon, either interfering constructively to return Q_{1} toward its excited state, or destructively and having Q_{1} emit its remaining energy and relax to its ground state. In Fig. 4a, we show the final Q_{1} population as a function of the phase φ (blue points), showing a characteristic interference pattern with a visibility of 32%.
We repeat the experiment with Q_{2} excited by a π pulse at the beginning of the experiment, with the experiment otherwise unchanged; the results are shown in Fig. 4a (salmon points). There are three effects on the oscillation pattern: a slight increase in the oscillation minima, attributed to a decrease of the phonon coherence^{25} in its interaction with Q_{2}; a more marked reduction of visibility attributed to inadequate absorption of the phonon wave packet; and, most significantly, a phase shift of \({{\Delta }}{\varphi }_{\exp }=0.40\pi\) attributed to the dispersive interaction between Q_{2} and the traveling halfphonon, close to our fitfree simulated value of \({{\Delta }}{\varphi }_{{{{{{\rm{sim}}}}}}}=0.41\pi\) (see Supplementary Note 9). This last effect points to the interesting possibility of using phonons as dispersive probes of other quantum systems.
In a separate experiment, shown in Fig. 4b, we swap the roles of the qubits, so Q_{2} is now used as a dispersive probe for the phonon released by Q_{1}, using a Ramsey fringe measurement of Q_{2}. The pulse sequence is shown to the right in Fig. 4b, where Q_{2} is placed in the state \((\leftg\right\rangle +{e}^{{{{{{\rm{i}}}}}}\theta }\lefte\right\rangle )/\sqrt{2}\) by the initial π/2 rotation, performed about an axis rotated in the x – y plane of the Bloch sphere by θ, and the θdependent evolution of Q_{2} is compared for where Q_{1} is not excited (no probe phonon) to where Q_{1} is excited and Q_{2} interacts with the subsequently released traveling phonon. In the latter case, the Ramsey fringe visibility is reduced, which we attribute to leakage from Q_{2} into the phononic channel, but we again observe a significant phase shift, here as high as \({{\Delta }}{\theta }_{\exp }=0.95\ \pi\) close to our simulation \({{\Delta }}{\theta }_{{{{{{\rm{sim}}}}}}}=0.99\ \pi\).
Discussion
In conclusion, we demonstrate controlled phononmediated quantum state transfer and remote entanglement between two quantum nodes, each node comprising a superconducting qubit with a timevariable coupler, using individual itinerant SAW phonons traveling in an acoustic transmission line after a controlled, ondemand release, followed by capture. Using this architecture, we also demonstrate the dispersive interaction between an itinerant phonon and a superconducting qubit. These results have been made possible by the integration of broadband, highly UDT in a 2mmlong phonon communication channel, as well as the use of a quantum state protocol requiring tunable coupling to each qubit node^{2}. Achieving a quite impressive quantum state transfer fidelity of 82.0(2)%, limited by the loss in the phonon channel, this platform paves the way for quantumopticslike experiments realized with individual phonons instead of photons.
Methods
Device fabrication and characterization
The device used in these experiments comprises two dies, a sapphire die with the two superconducting qubits (Q_{1} and Q_{2}), and their associated tunable couplers (G_{1} and G_{2}, respectively), as well as control and readout wiring, and a lithium niobate die with the phononic channel and the two UDTs. The two dies are fabricated separately then flipchip assembled^{30}. The full circuit schematic is shown in Fig. 1b.
The acoustic die is fabricated using a single layer of ~25nmthick aluminum patterned by PMMA liftoff on a LiNbO_{3} wafer, 500 μm thick. The central part of the acoustic device is the ℓ = 2mmlong phononic channel, with width W = 150 μm, terminated at each end by a UDT (UDT_{1,2}).
The UDTs are described more completely in Supplementary Note 1. Briefly, the two (nominally identical) UDTs each comprise a standard bidirectional IDT combined with an acoustic mirror (a reflective grating). The IDT emits equalamplitude acoustic excitations in opposite directions, one toward and the other away from the second UDT. The acoustic mirror, placed immediately adjacent to the IDT on the side opposite the second UDT, reflects its incident excitation back toward the second UDT, such that it interferes constructively with the other excitation. Each UDT is coupled inductively to one of the two qubits.
We have separately characterized similar IDTmirror designs, wherein the frequency band from about 3.85–4 GHz, excellent directionality is achieved, with emission from the UDT almost entirely directed away from the mirror. Typical directivities are greater than 20 dB. Outside this unidirectional band, the mirrors are less effective and the devices emit more strongly in both directions^{27}.
The superconducting qubit die is fabricated on 430μmthick sapphire using standard lithographic processing^{15}. The qubits Q_{1,2} are tunable xmonstyle qubits^{31,32}, where each qubit’s frequency is controlled by a flux line Q_{z1,z2}, and excited using a capacitively coupled microwave line Q_{xy1,xy2}. Each qubit is coupled to the SAW chip through a superconducting tunable coupler G_{1,2}, whose coupling is controlled^{33} using external flux lines G_{z1,z2}. Qubit states are inferred from standard dispersive measurements using a separate readout resonator for each qubit. The readout resonators are connected to a common readout line; more details are given in Supplementary Note 1.
The qubits are characterized with their couplers turned off (see Supplementary Notes 6 and 7). At the qubit idle frequency ω_{idle}/(2π) ~4.3 GHz, we find the qubits have an energy relaxation time T_{1} = 57 μs (Q_{1}) and 38 μs (Q_{2}), with a coherence time \({T}_{2}^{{{{{{\rm{Ramsey}}}}}}}\)= 1.11 μs (Q_{1}) and 0.88 μs (Q_{2}) (most likely limited by flux noise as the qubits are tuned far away from their fluxinsensitive point). These times demonstrate the potential for excellent qubit coherence when using a flipchip assembly^{30}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. Correspondence and requests for materials should be addressed to A. N. Cleland (anc@uchicago.edu).
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Acknowledgements
Devices and experiments were supported by the Air Force Office of Scientific Research and the Army Research Laboratory, and material for this work was supported by the Department of Energy (DOE). É. D. was supported by LDRD funds from Argonne National Laboratory. K. J. S. was supported by NSF GRFP (NSF DGE1144085) and A. N. C. was supported by the DOE, Office of Basic Energy Sciences. This work was partially supported by the UChicago MRSEC (NSF DMR2011854), AFOSR under award FA95502010270, the NSF QLCI for HQAN (NSF Award 2016136), and made use of the Pritzker Nanofabrication Facility, which receives support from SHyNE, a node of the National Science Foundation’s National Nanotechnology Coordinated Infrastructure (NSF NNCI ECCS2025633).
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É. D. designed and fabricated the devices, performed the experiment, and analyzed the data. K. J. S., G. A. P., and M.H. C. participated to the design process of the unidirectional transducer. É. D., K. J. S., A. B., H.S. C., J. G., and Y. P. Z. developed the fabrication process of the superconducting circuit. É. D., K. J. S., and A. B. wrote code to model surface acoustic waves. A. N. C. advised on all efforts. All authors contributed to the discussion and production of the manuscript.
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Dumur, É., Satzinger, K.J., Peairs, G.A. et al. Quantum communication with itinerant surface acoustic wave phonons. npj Quantum Inf 7, 173 (2021). https://doi.org/10.1038/s41534021005111
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DOI: https://doi.org/10.1038/s41534021005111
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