Quantum communication with itinerant surface acoustic wave phonons

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 single-phonon 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 high-fidelity node-to-node transfer of quantum states as well as the generation of a two-node 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 quantum-optics-style experiments with itinerant phonons.

The 2-mm-long 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 general-purpose design, with physically separate and distinct phonon emitter and receiver.
We use this device to demonstrate two-node quantum state transfers as well as the phonon-mediated 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 single-phonon 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 1 the roles of the qubits in the previous experiment and demonstrating the versatility of this architecture.

Phonon-mediated 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 Q1 e at times that are multiples of the phonon round-trip time τ RT~1 μs, each revival corresponding to the traveling phonon reflecting off the other transducer before re-exciting 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 ω uni Q1;2 =2π ¼ 3.976 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 ω bi Q1;2 =2π ¼ 4.102 GHz . These frequencies are marked by the dashed white and red lines, respectively, in Fig. 2a.
To maximize the efficiency of phonon-mediated quantum state transfers, we need to carefully shape the emission and absorption of the phonon wave packet, which is done by time-dependent 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 input-output 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~100-m-long coaxial cable, illustrating the very long delays achievable with phonon-based quantum channels.
In Fig. 2c, we show quantum process tomography for both regimes. For the unidirectional process, we find a process fidelity of F uni ¼ ð82:0 ± 0:3Þ%, while for the bidirectional regime, the process fidelity is limited to F bi ¼ ð39:0 ± 0:3Þ%. We compare these experimental process fidelities with predictions and find ¼ 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 phonon-mediated remote entanglement We further explore the capabilities of itinerant phonon communication by performing a phonon-mediated 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 ð e0g j iþ g1g j iÞ= ffiffi ffi 2 p (writing the state Q 1 γ Q 2 j iwhere γ represents the itinerant phonon). During the time the emitted "half-phonon" 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 half-phonon is then captured by Q 2 , generating a Bell state ψ j i ¼ ð eg j i þ e iφ ge j iÞ= ffiffi ffi 2 p between the two qubits, with φ a relative phase. Figure 3a shows the time-dependent qubit state populations P e for each qubit, which agrees well with a master equation simulation. Following the capture of the half-phonon, we perform quantum state tomography at time t m = 750 ns; these measurements are used to reconstruct the two-qubit density matrix ρ shown in Fig. 3b. We find a Bell state fidelity F Bell ¼ Tr ðρ ideal Á ρÞ ¼ 72% and a concurrence C ¼ 0.53 , close to the master equation simulation results, with a trace distance Each node comprises a superconducting qubit, a tunable coupler that allows shaping phonon release and capture, and a unidirectional phonon transducer. Details in the Methods section. b Circuit diagram of the assembled device. Each qubit Q j is excited through its dedicated Q xyj microwave line and frequency-controlled through a separate Q z j flux-bias line, and each tunable coupler G k is controlled via its associated G z k flux-bias line.

Phonon-qubit 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 two-level system, shown in Fig. 4a. We use qubit Q 1 as the emitter and receiver of a "half-phonon" 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 stand-in 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 half-phonon 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 half-phonon, 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 Δφ exp ¼ 0:40π attributed to the dispersive interaction between Q 2 and the traveling half-phonon, close to our fit-free simulated value of Δφ sim ¼ 0:41π (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 ð g j i þ e iθ e j iÞ= ffiffi ffi 2 p by the initial π/2 rotation, performed about an axis rotated in the xy plane of the Bloch sphere by θ, and the θ-dependent 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 Δθ exp ¼ 0:95 π close to our simulation Δθ sim ¼ 0:99 π.

DISCUSSION
In conclusion, we demonstrate controlled phonon-mediated quantum state transfer and remote entanglement between two quantum nodes, each node comprising a superconducting qubit with a time-variable coupler, using individual itinerant SAW phonons traveling in an acoustic transmission line after a controlled, on-demand 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 2-mm-long 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 quantumoptics-like experiments realized with individual phonons instead of photons.

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 flip-chip assembled 30 . The full circuit schematic is shown in Fig. 1b.
The acoustic die is fabricated using a single layer of~25-nm-thick aluminum patterned by PMMA liftoff on a LiNbO 3 wafer, 500 μm thick. The central part of the acoustic device is the ℓ = 2-mm-long 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 equal-amplitude 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 IDT-mirror 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 430-μm-thick sapphire using standard lithographic processing 15 . The qubits Q 1,2 are tunable xmon-style 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 Ramsey 2 = 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 flux-insensitive point). These times demonstrate the potential for excellent qubit coherence when using a flip-chip assembly 30  , showing a dependence of the phonon phase on Q 2 's state, with a shift of~0.4π rad. c Qubit pulse sequences (see text for details). d Schematic roles played by each element for the dispersive phasedependent interferometric probe. e Ramsey interference in Q 2 (blue circles) reveals Q 2 's dependence on the relative phase with the phonon. When repeating the same protocol with no initial π pulse on Q 1 , we measure the Ramsey interference of Q 2 (salmon squares), shifted by π compared to the first measurement. Blue (salmon) points show the φ dependence of Q 1 's final P e when Q 2 is in g j i ( e j i), showing a dependence of the phonon phase on Q 2 's state, with a shift of~0.4π rad. f Qubit pulse sequences, similar to c except Q 2 is always placed in ð g j i þ e iθ e j iÞ= ffiffi ffi 2 p and the θ-dependent P Q2 e is measured.