Phonon-mediated quantum state transfer and remote qubit entanglement

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Science  26 Apr 2019:
Vol. 364, Issue 6438, pp. 368-371
DOI: 10.1126/science.aaw8415

Good vibrations in quantum communication

Quantum information processing platforms typically require qubits to talk to each other. To date, photons (either optical or microwave) have been the carrier of choice to transfer quantum states between the qubits. For some solid-state systems, however, the vibrational properties of the materials themselves, phonons, could be advantageous. Bienfait et al. describe the deterministic emission and capture of itinerant phonons through an acoustic communication channel, enabling the phonon-based coherent transfer of quantum states from one superconducting qubit to another and the quantum entanglement of the two qubits over the acoustic channel. The results provide a route to couple hybrid quantum solid-state systems using surface acoustic waves.

Science, this issue p. 368


Phonons, and in particular surface acoustic wave phonons, have been proposed as a means to coherently couple distant solid-state quantum systems. Individual phonons in a resonant structure can be controlled and detected by superconducting qubits, enabling the coherent generation and measurement of complex stationary phonon states. We report the deterministic emission and capture of itinerant surface acoustic wave phonons, enabling the quantum entanglement of two superconducting qubits. Using a 2-millimeter-long acoustic quantum communication channel, equivalent to a 500-nanosecond delay line, we demonstrate the emission and recapture of a phonon by one superconducting qubit, quantum state transfer between two superconducting qubits with a 67% efficiency, and, by partial transfer of a phonon, generation of an entangled Bell pair with a fidelity of 84%.

Electromagnetic waves, whether at optical or at microwave frequencies, have played a singular role as carriers of quantum information between distant quantum nodes, providing the principal bus for distributed quantum information processing. Several recent experiments have used microwave photons to demonstrate deterministic as well as probabilistic remote entanglement generation between superconducting qubits, with entanglement fidelities in the range of 60 to 95% (15). However, for some solid-state quantum systems, such as electrostatically defined quantum dots or electronic spins, the strong interactions with the host material make acoustic vibrations, or phonons, an alternative and potentially superior candidate to photons. In particular, surface-acoustic wave (SAW) (6) phonons have been proposed as a universal medium for coupling remote quantum systems (7, 8). These phonons also have the potential for efficient conversion between microwave and optical frequencies (9, 10), linking microwave qubits to optical photons. These proposals followed experiments showing the coherent emission and detection of traveling SAW phonons by a superconducting qubit (11, 12). Traveling SAW phonons have also been used to transfer electrons between quantum dots (13, 14) and couple to nitrogen-vacancy centers (15), as well as drive silicon carbide spins (16). When localized in Fabry-Pérot resonators, standing-wave SAW phonons have been coherently coupled to superconducting qubits (1721), allowing the on-demand creation, detection, and control of quantum acoustic states (21). Related experiments have demonstrated similar control of localized bulk acoustic phonons (2224).

Here we report the use of itinerant SAW phonons to realize the coherent transfer of quantum states between two superconducting qubits. We show that a single superconducting qubit can launch an itinerant phonon into a SAW resonator when operating near the strong multimode coupling regime where the coupling between the qubit and one Fabry-Pérot mode exceeds the resonator free spectral range. This allows the phonon to be completely injected into the acoustic channel before any reexcitation of the emitting qubit. Using techniques developed for microwave photon transfer (5, 25), the emitting qubit can recapture the phonon at a later time, with a 67% efficiency. Using the same acoustic channel, two-qubit quantum-state transfer is performed as well as remote qubit entanglement with a fidelity exceeding 80%.

The acoustic part of the device is a SAW resonator (Fig. 1), with an effective Fabry-Pérot mirror spacing of 2 mm, corresponding to a single-pass itinerant phonon travel time of about 0.5 μs. The resonator is coupled to two frequency-tunable superconducting Xmon qubits (26), Q1 and Q2; their coupling is electrically controlled using two tunable couplers, G1 and G2 (27). The tunable couplers, the qubits, and their respective control and readout lines are fabricated on a sapphire substrate, whereas the SAW resonator is fabricated on a separate lithium niobate substrate. The qubits have native relaxation times T1,int of 22 and 26 μs and coherence times T2,Ramsey of 2.1 and 0.6 μs.

Fig. 1 Experimental device.

(A to C) Micrograph of flip-chip assembled device (A), with two superconducting qubits (Q1 and Q2, blue), connected to two tunable couplers (G1 and G2, purple), fabricated on sapphire (B). These are connected via two overlaid inductors (green) to a SAW resonator (C), fabricated on lithium niobate. The SAW resonator comprises two Bragg mirrors (orange), spaced by 2 mm, defining a Fabry-Pérot acoustic cavity probed by an interdigitated transducer (red). The red and blue outlines in (A) represent the locations of (B) and (C), respectively. (D) Simplified circuit diagram, with the gray box indicating elements on the flipped lithium niobate chip. (E) Excited-state population Pe for qubit Q1, with coupler G1 set to maximum and G2 turned off. Q1 is prepared in |e using a π pulse, its frequency set to ωQ1 (vertical scale) for a time t (horizontal scale), before dispersive readout of its excited population Pe (28). Q1 relaxes owing to phonon emission via the IDT, and if its frequency is within the mirror stop band from 3.91 to 4.03 GHz, the emitted phonon is reflected and generates qubit excitation revivals at times τ (orange line) and 2τ. The inset shows the pulse sequence. (F) Measured qubit energy decay time T1 for ωQ,i/2π=3.95 GHz as a function of the coupler Josephson junction phase δi, showing the qubit emission can be considerably faster than the phonon transit time (orange line), for both Q1 (circles) and Q2 (squares).

The SAW resonator is defined by two acoustic mirrors, which are arrays of 30-nm-thick aluminum lines, spaced by 0.5 μm, with 400 lines in each array, defining two Bragg mirrors on each side of the central acoustic emitter-receiver. The acoustic emitter is an interdigitated transducer (IDT), comprising forty 30-nm-thick lines with alternate lines connected to a common electrical port. An electrical pulse applied to the IDT results in two symmetric SAW pulses traveling in opposite directions, reflecting off the mirrors, and completing a round trip in a time τ = 508 ns (28). The mirrors support a 125-MHz-wide stop band, centered at 3.97 GHz, localizing about 60 Fabry-Pérot standing modes with a free spectral range νFSR=1/τ=1.97 MHz. Each port of the IDT is inductively grounded, with each inductor forming a mutual inductance through free space with a similarly grounded inductance in the couplers G1 and G2 on the sapphire chip. The coupled inductors are precisely aligned to one another in a flip-chip assembly (21, 29). Each coupler comprises a π inductive network, with a Josephson junction bridging two fixed inductors to ground, one of the fixed inductors coupled to one port of the SAW transducer and the other contributing to the qubit inductance. A flux line controls the phase δi across the coupling Josephson junction, thus controlling its effective inductance and the qubit-IDT coupling. Each coupler can be switched from maximum coupling to off in a few nanoseconds (27), isolating the qubits (21).

The qubits’ controlled coupling to the IDT enables the time domain–shaped emission of itinerant phonons into the resonator (21). We characterize this emission by exciting the qubit and then monitoring its excited-state population (Fig. 1E), with the excited state decaying as a result of phonon emission. As expected, the frequency dependence of the decay rate closely follows the frequency response of the IDT’s electrical admittance [see (21, 28)]. When near maximum admittance, the qubit relaxes in T1<15 ns, a value roughly 3% of the phonon’s resonator transit time τ and less than 0.1% of the qubit’s intrinsic relaxation time T1,int. The relaxation rate of the qubit also sets the extent of the emitted phonon wave packet: The phonon can truly be considered as an itinerant excitation, with each oppositely traveling packet having a spatial extent of about 60 μm, compared with the 2-mm length of the resonator. For phonon frequencies within the mirror stop band, the emitted itinerant phonon is strongly reflected by the mirrors and can reexcite the qubit at integer multiples of the phonon transit time t=nτ (Fig. 1E). The transit time is unusually long for superconducting qubit experiments: The equivalent microwave cable length would be ~100 m, much larger than the typical ~1-m connections used for quantum communication experiments (15). We fix the qubits’ operating frequencies to ωQ,i/2π=3.95 GHz, well within the mirror stop band.

We first explore the emission and recapture of phonons by one qubit. The highest efficiencies are achieved by controlling the qubit-acoustic channel coupling rates κi so as to match the phonon envelope during the emission and capture of the phonon (13, 5, 25). Each qubit’s coupling rate κi is controlled by the coupler Gi and is calibrated by measuring the qubit energy decay rate T11=κi+T1,int1 as a function of the coupler flux bias and then subtracting the contribution from the intrinsic qubit lifetime T1,int (Fig. 1F). At maximum coupling, we measure 1/κ1=7.6 ns and 1/κ2=10.6 ns, the difference being due to a 5% mismatch in the couplers’ Josephson junction inductances. At this maximum point, the qubit couplings gi to an individual Fabry-Pérot mode are g1/2π=2.57±0.1 MHz and g2/2π=2.16±0.1 MHz (28). The system is thus at the threshold of “strong” multimode coupling (18, 28), where the coupling rate gi is larger than the free spectral range of the resonator, the qubit linewidth, and the SAW energy decay rate [respectively, νFSR=1.97 MHz, 1/T2,Ramsey=2π×76 kHz, and 1/T1,SAW2π×133 kHz (28)].

We demonstrate a one-qubit single-phonon “ping-pong” experiment using qubit Q1 (Fig. 2). With G2 off, we apply calibrated control pulses to G1 and Q1 to obtain a symmetric phonon envelope of characteristic bandwidth 1/κc=10 ns, and monitor Q1’s excited-state population Pe (Fig. 2A). The emission takes about 150 ns, after which Pe remains near zero during the phonon transit. After ~0.5 μs, the phonon returns and is recaptured, with Pe increasing and leveling off for times t>tf=0.65 μs. The capture efficiency is η=Pe(tf)/Pe(t=0)=0.67±0.01.

Fig. 2 Emission and capture of a shaped itinerant phonon.

(A) Calibrated control pulses (inset) ensure the release of a time-symmetric phonon and its efficient capture. Circles represent the measured excited-state population of Q1 when interrupting the sequence after a time t. (B) Measured excited-state population of Q1 while sweeping the delay between the emission and capture control pulses, evidencing a population geometrically decreasing with the number of transits (gray line). (C) Quantum process tomography at the maximum efficiency point of (B), with a process fidelity F1=0.83±0.002. I stands for the identity operator and X, Y, and Z for the Pauli operators. In (A) to (C), dashed lines indicate the results of a master equation simulation including a finite transfer efficiency and qubit imperfections [see (28)].

Successive transits show a geometric decrease in capture efficiency, ηn=ηn with transit number n (Fig. 2B). We attribute the decrease to losses in the acoustic channel. An independent measurement yields a Fabry-Pérot mode energy decay rate T1,SAW=1.2 μs (28), close to similar resonators in the literature (17, 18). This sets an upper limit ηeτ/T1,SAW0.65 (28), close to our measurement.

Quantum process tomography of the one-qubit release-and-catch operation is performed by preparing four independent qubit states and reconstructing the process matrix χ at time tf (Fig. 2C), yielding a process fidelity F1=Tr(χχideal)=0.83±0.002. A master equation simulation, taking into account an acoustic channel loss of η=0.67 and the finite qubit coherence T2R=2.1 μs, yields a process matrix whose trace distance to χ is Tr[(χχsim)2]=0.07.

To demonstrate the interferometric nature of the one-qubit phonon emission and capture (Fig. 3), we prepare Q1 in |e, emit a half-phonon, and then capture it with Q1 after one transit. As capture is the time reversal of emission, it would seem that the photonic half-excitation left in Q1 would be emitted during the capture process. Depending on the relative phase Δϕ between the stored half-photon and the reflected half-phonon, the two half-quanta will interfere destructively or constructively, giving results ranging from reexcitation of the qubit to total emission. When Δϕ=π, the reflected half-phonon interferes constructively with the emitted half-photon stored in Q1, and all the energy is transferred to the SAW resonator, but when Δϕ=0, destructive interference results in qubit reexcitation. Figure 3A shows the qubit population for Δϕ=0 and Δϕ=π, with the final population Pe(tf) reaching 0.77 and 0.08, respectively. Figure 3B shows that the final qubit population oscillates as expected between these two limiting values as a function of Δϕ. A simulation including channel loss and qubit dephasing accounts partially for the reduction in interferometric amplitudes, with the remaining mismatch being attributed to control-pulse imperfections (28).

Fig. 3 Phonon interferometry.

With Q1 initially prepared in |e, a control signal on G1 releases and subsequently recaptures half a phonon to the resonator. Simultaneously, a 20-MHz detuning pulse of varying duration is applied to Q1 to change its phase by ∆ϕ. (A) Measured Q1 excited-state population when interrupting the sequence after a time t, with a phase difference ∆ϕ = 0 (squares) or π (circles). The inset shows the control sequence. (B) Q1 final state Pe(t=tf) for tf=0.65 μs as a function of the phase difference ∆ϕ between the half-photon and half-phonon. Circles are experimental points. Dashed lines are simulations based on an input-output theory model [see (28)].

We can use the acoustic communication channel to transfer quantum states as well as generate remote entanglement between the two qubits. In Fig. 4A, we demonstrate a quantum swap between Q1 and Q2. As the phonon transit time is substantially larger than the phonon temporal extent, up to three itinerant phonons can be stored sequentially in the SAW resonator. Thus, a double swap can be performed by having each qubit successively emit an excitation and then, at a later time, capture the phonon emitted by the other qubit. This process has a fidelity F2=0.63±0.01 (28), in agreement with the one-qubit state transfer fidelity F2F12 and with infidelity dominated by acoustic losses.

Fig. 4 Quantum state transfer and remote entanglement.

(A) Qubit state swap via the acoustic channel, with control pulses shown on the left. (B) Acoustic entanglement. With Q1 initially in |e, a control signal applied to G1 releases half a phonon to the channel, captured later by Q2. In (A) and (B), circles and squares are Q1 and Q2 excited-state populations measured simultaneously after a time t. (C and D) Expectation values of two-qubit Pauli operators (C) for the reconstructed Bell state density matrix (D) at t = 0.65 μs. In (C) and (D), solid lines indicate values expected for the ideal Bell state |Ψ=(|eg+|ge)/2. In (A) to (D), dashed lines are simulation results including a finite transfer efficiency and qubit imperfections (28).

We also use the acoustic channel to share half of an itinerant phonon, generating remote entanglement between Q1 and Q2. We first prepare Q1 in |e and then apply a calibrated pulse to coupler G1 to release half of Q1’s excitation to the acoustic channel. This half-phonon is then captured, after one transit, by Q2, by using the pulse sequence shown in Fig. 4B, creating a Bell state |Ψ=(|eg+eiα|ge)/2, the phase α resulting from the relative detunings of the qubits during the sequence. The excited-state population of each qubit is shown as a function of time in Fig. 4B, and the Pauli matrices and reconstructed density matrix ρ, at time t = 0.65 μs, are shown in Fig. 4, C and D. We find a state fidelity, referenced to the ideal Bell state, of FB=Tr(ρρ|Ψ)=0.84±0.01 and a concurrence C=0.61±0.04. A master equation simulation yields a density matrix ρsim, which has a small trace distance Tr[(ρρsim)2]=0.06 to ρ, with errors dominated by acoustic losses.

These results comprise clear and compelling demonstrations of the controlled release and capture of itinerant phonons into a confined Fabry-Pérot resonator and are limited primarily by acoustic losses. As the phonons have a spatial extent much smaller than the resonator length, the emission and capture processes are “blind” to the length of the resonator, so the same processes would work in a nonresonant acoustic device. We demonstrate that these processes can generate fidelity entanglement between two qubits that is quite high. These results constitute a step toward realizing fundamental quantum communication protocols (30, 31) with phonons.

Supplementary Materials

Supplementary Text

Figs. S1 to S3

Tables S1 and S2

References (3247)

References and Notes

  1. See supplementary materials.
Acknowledgments: We thank P. J. Duda for helpful discussions and W. D. Oliver and G. Calusine at Lincoln Laboratories for the provision of a traveling-wave parametric amplifier (TWPA). Funding: Devices and experiments were supported by the U.S. Air Force Office of Scientific Research MURI program and by the U.S. Army Research Laboratory under contract W911NF-15-2-0058. K.J.S. was supported by NSF GRFP (NSF DGE-1144085), É.D. was supported by LDRD funds from Argonne National Laboratory, and A.N.C. was supported by the U.S. Department of Energy, Office of Basic Energy Sciences. This work was partially supported by the UChicago MRSEC (NSF DMR-1420709) 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-1542205). Author contributions: A.B. fabricated the devices with assistance from K.J.S., C.R.C., and É.D.; K.J.S., H.-S.C., M.-H.C., and J.G. developed the fabrication processes; K.J.S., É.D., G.A.P., and A.N.C. contributed to device design; A.B. performed the experiments and analyzed the data with assistance from Y.P.Z. and É.D.; and A.N.C. advised on all efforts. All authors contributed to discussions and production of the manuscript. Competing interests: The authors declare no competing financial interests. Data and materials availability: All data are available in the manuscript or the supplementary materials. Correspondence and requests for materials should be addressed to corresponding author A.N.C.

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