Deterministic multi-phonon entanglement between two mechanical resonators on separate substrates

Mechanical systems have emerged as a compelling platform for applications in quantum information, leveraging recent advances in the control of phonons, the quanta of mechanical vibrations. Several experiments have demonstrated control and measurement of phonon states in mechanical resonators integrated with superconducting qubits [1, 2, 3, 4], and while entanglement of two mechanical resonators has been demonstrated in some approaches [5, 6, 7, 8, 9], a full exploitation of the bosonic nature of phonons, such as multi-phonon entanglement, remains a challenge. Here, we describe a modular platform capable of rapid multi-phonon entanglement generation and subsequent tomographic analysis, using two surface acoustic wave resonators on separate substrates, each connected to a superconducting qubit. We generate a mechanical Bell state between the two mechanical resonators, achieving a fidelity of $\mathcal{F}=0.872\pm 0.002$, and further demonstrate the creation of a multi-phonon entangled state (N=2 N00N state), shared between the two resonators, with fidelity $\mathcal{F}=0.748\pm 0.008$. This approach promises the generation and manipulation of more complex phonon states, with potential future applications in bosonic quantum computing in mechanical systems. The compactness, modularity, and scalability of our platform further promises advances in both fundamental science and advanced quantum protocols, including quantum random access memory [10, 11] and quantum error correction [12].

Mechanical systems have significantly smaller footprints than existing circuit quantum electrodynamics (cQED) systems at similar frequencies [4], potentially long lifetimes [13], and a large number of accessible microwave-frequency modes [14, 15, 2, 16]. Mechanical systems have been operated in the quantum limit [1, 17], with explorations of quantum information storage and processing [10, 18, 12, 13, 19] and quantum sensing [20, 21, 22]. Additional achievements include the quantum control of mechanical motion [1, 2, 3], entanglement between macroscopic mechanical objects [23, 5, 6, 7, 8], coupling between surface acoustic waves (SAW) and qubits [24, 25, 26, 14, 27], the deterministic emission and detection of individual SAW phonons as well as phonon-phonon entanglement [28, 29, 30], and the transmission of quantum information [28, 29, 30, 31, 32], among other demonstrations [33, 34, 35]. Mechanical systems have also been investigated as a platform for interconnecting microwave qubits with optical photons [36, 37, 38, 39, 40] and spin assemblies [41], with the potential for realizing long-distance quantum communication. Many of these advances have been enabled through the integration of superconducting qubits with mechanics, affording the quantum control of highly linear mechanical modes as well as straightforward quantum measurement.

Here, we demonstrate the deterministic generation and distribution of multi-phonon entanglement between two physically separated mechanical modes. We use a modular architecture, in which the two mechanical resonators are fabricated on separate piezoelectric substrates and electrically coupled to a pair of superconducting qubits on a third, non-piezoelectric substrate. This design supports the generation of complex entangled states as well as straightforward quantum tomography, with potential applications in quantum sensing and high-precision measurements [42, 43, 44, 45, 46].

The experimental layout is shown in Fig 1. Our device comprises two nodes, each node including a mechanical SAW resonator inductively coupled via a variable coupler [3] to a frequency-tunable superconducting Xmon qubit [48, 49]. The two qubits are capacitively coupled to one another, supporting entangling gates. The two SAW resonators ($R_{A}$ and $R_{B}$) are fabricated on separate lithium niobate (LN) substrates, while the qubits ($Q_{A}$ and $Q_{B}$), couplers ($G_{A}$ and $G_{B}$), and their associated readout resonators and control lines, are fabricated on a sapphire substrate. The two LN dies are sequentially aligned and attached to the sapphire substrate, using a non-galvanic flip-chip assembly [47]. Each mechanical resonator includes a central interdigitated transducer (IDT) and two acoustic mirrors, situated on either side of, and immediately adjacent to, the transducer. Each acoustic mirror is an array of two hundred 10 nm-thick parallel aluminium lines, forming a Bragg mirror with a $\sim$ 50 MHz-wide acoustic stop-band, centered on the respective mechanical resonator frequencies of 3.027 GHz ($R_{A}$) and 3.295 GHz ($R_{B}$). The free spectral range (FSR) of each acoustic resonator is designed to be slightly larger than mirror stop-band, confining a single acoustic mode in each resonator. By virtue of the piezoelectric response of the LN substrates, applying an electrical signal to either IDT generates symmetric, oppositely-directed surface acoustic waves, whose retro-reflection by the two mirrors forms a single Fabry-Pérot resonance, generating a sympathetic electrical response at the corresponding IDT. In Fig. 2a we show the calculated SAW resonator transmission, mirror stopband, and IDT admittance for each resonator, using their design parameters. Each superconducting qubit is coupled to its respective mechanical resonator via a variable coupler, whose coupling strength is controlled externally by magnetic flux bias of an rf-SQUID [3], with the coupler connected to the respective IDT through an air-gap inductive coupler. The three-die assembly is mounted in an aluminium box with wire-bond electrical connections and external magnetic shielding, operated in a dilution refrigerator with a base temperature of about 10 mK.

We first characterize each node by measuring the qubit-resonator interactions, measured as a function of time versus qubit frequency (see Fig. 2b). Each qubit is initially excited to its $\ket{e}$ state by a tuned microwave $\pi$ pulse, following which the coupling between the qubit and resonator is turned on. When the qubit is tuned into resonance with the corresponding acoustic mode, qubit-resonator Rabi swaps give rise to the expected chevron patterns (Fig. 2b). Outside the $\sim 50$ MHz acoustic mirror stop band, visible in Fig. 2a (dashed green line), but within the IDT emission band of $\sim 600$ MHz (shown by the larger calculated admittance, dashed purple line), the qubit decays rapidly by acoustic emission that escapes through the mirrors. When the qubit is tuned outside the IDT emission band (left and right margins of either plot in panel b), the qubit lifetime increases rapidly due to the reduced phonon emission rate, due to the smaller admittance (purple dashed line in panel a).

The system supports multiplexed Rabi swap measurements, shown in Fig. 2c. Each qubit is set to its idle frequency of 3.245 GHz (3.557 GHz), a microwave $\pi$ pulse applied, and the qubits then tuned into resonance with their corresponding mechanical resonators while both variable couplers are turned on, yielding simultaneous parallel swaps. The swap times for nodes A and B are 42 ns and 35 ns, respectively. We next use the qubits to measure the mechanical resonator lifetimes at the single-phonon level, by swapping an excitation from the qubit into the resonator and then measuring the decay of the resulting one-phonon state as a function of time, for both nodes A and B. The resonators’ energy relaxation times extracted from the measurements are $T_{1,A}^{m}=380\pm 8$ ns and $T_{1,B}^{m}=270\pm 3$ ns. The dephasing time for each resonator is then measured by exciting either qubit with a $\pi/2$ microwave pulse and swapping the qubit superposition state into the corresponding resonator, then measuring the decoherence time with a Ramsey fringe measurement [3]. We find dephasing times of $T_{2,A}^{m}=709\pm 16$ ns and $T_{2,B}^{m}=527\pm 6$ ns, approximately twice the $T_{1}$ times (see Extended Data Fig. 1). The corresponding quality factors for the resonators are $Q_{A}\approx 7200$ and $Q_{B}\approx 5600$, roughly twice the value for single-mode SAW resonator in Ref. [3]. The improvement is possibly due to a modified SAW resonator geometry as well as more thorough surface cleaning of the LN substrates (see Supplementary).

We then use the qubits to prepare entangled mechanical resonator states, distributed across the two LN dies, as illustrated in Fig. 3a. In Ref. [8], a two-resonator mechanical Bell state was prepared, then measured dispersively using a single-qubit Ramsey measurement. Here we use direct swaps between the resonators and their respective qubits for state analysis, using short pulse sequences with improved state fidelity. Following a protocol similar to Ref. [51] (pulse sequence shown in Fig. 3b), we prepare a mechanical Bell state by exciting qubit $Q_{A}$ and performing a half-swap to qubit $Q_{B}$, generating a two-qubit Bell state $(|eg\rangle+|ge\rangle)/\sqrt{2}$. We then bring each qubit into resonance with its corresponding mechanical resonator, and turn on the variable couplers to perform full qubit-resonator swaps, ideally resulting in a dual-resonator Bell state $(|10\rangle+|01\rangle)/\sqrt{2}$. Note this method is not negatively impacted by the different swap times for nodes A and B (44.8 ns and 36.4 ns, respectively). To analyze the resulting entangled resonator state, coherent displacement pulses $\hat{D}_{A}$ and $\hat{D}_{B}$ are applied to each resonator, following which the qubits interact resonantly with their corresponding resonators, followed by simultaneous two-qubit state readout  [52, 51]. By varying the interaction time $\tau$, we can map out the two-qubit state probabilities $P_{gg},P_{ge},P_{eg}$ and $P_{ee}$ as a function of time, shown for zero displacement ($\hat{D}_{A}=\hat{D}_{B}=0$) in panel c. These data show coherent swaps between the resonators and qubits while $P_{ee}$ remains zero, consistent with the expectation that only a single phonon is shared between the two resonators. Panel d shows the populations for the resonators, corresponding to the fit solid lines in c. By performing similar measurements with a total of $15\times 15$ different combinations of displacement pulses $\hat{D}_{A}$ and $\hat{D}_{B}$ (see Supplementary), we reconstruct the Bell state using convex optimization. The resulting density matrix $|\rho|$ is shown in Fig. 3e, with a fidelity $\mathcal{F}=\sqrt{\Tr(\rho_{\rm Bell}\cdot|\rho|)}=0.872\pm 0.003$ to the ideal Bell state $\rho_{\rm Bell}$, close to our simulated result, which predicts a fidelity $\mathcal{F}=0.92$ (see Supplementary). The infidelity is dominated by the resonator lifetime combined with a reduced qubit $T_{1}$ when each qubit is coupled to its resonator (see Supplementary and Extended Data Table. 1).

We finally display the capabilities of this system for generating and measuring multi-phonon entangled states, doing so for an $N=2$ N00N state shared between the two mechanical resonators. Our protocol is illustrated in Fig. 4a, with the corresponding pulse sequence in panel b. We use a similar process to Fig. 3 to prepare a two-qubit Bell state $(|eg\rangle+|ge\rangle)/\sqrt{2}$, then use a microwave $\pi$ pulse to selectively excite each qubit to its second excited state $|f\rangle$, yielding the entangled qutrit state $(|fg\rangle+|gf\rangle)/\sqrt{2}$ (stage 1 in Fig. 4a). We then tune each qubit’s $f\leftrightarrow e$ transition into resonance with the corresponding resonator, and perform a full swap, resulting in the four-fold entangled state $(|eg10\rangle+|ge01\rangle)/\sqrt{2}$ (stage 2 in Fig. 4a). We note that during the $f\leftrightarrow e$ swap with the resonator, the $e\leftrightarrow g$ transition also falls inside the active SAW transducer bandwidth, but detuned from the resonator transition and outside the mirror bandwidth of $\sim 50$ MHz; the qubit $|e\rangle$ state thus decays in parallel by emitting unwanted, non-resonant phonons via the transducer, competing with the desired $f\rightarrow e$ transition. There is thus a trade-off between the qubit-resonator coupling strength and this unwanted phonon emission. This issue could be alleviated by reducing the IDT bandwidth so that $e\leftrightarrow g$ transition is outside IDT emission bandwidth due to qubit anharmonicity, e.g. by adding more IDT finger pairs. In our experiment, we carefully control both couplers to maximize the final N00N state fidelity.

In the last step, each qubit’s $e\leftrightarrow g$ transition is brought into resonance with its corresponding resonator, swapping the remaining qubit excitation into the resonator, ideally resulting in a final state $|gg\rangle\otimes(|20\rangle+|02\rangle)/\sqrt{2}$. This protocol can be extended to $(|N0\rangle+|0N\rangle)/\sqrt{2}$ states by iterating the first two steps. Alternatively, we can generate $(|N0\rangle+|0M\rangle)/\sqrt{2}$ N00M states, if one qubit is initially excited to its $|f\rangle$ state while the other qubit remains in $|e\rangle$.

We analyze the final resonator state using Wigner tomography, similar to the Bell state analysis. The evolution of the two-qubit state probabilities for zero displacement are shown in Fig. 4c; the corresponding joint resonator population distribution is shown in panel d. We see that the $P_{eg}$ and $P_{ge}$ oscillations are approximately $\sqrt{2}$ faster than the corresponding oscillations for the Bell state in Fig. 3b, while the measured $P_{ee}$ remains zero, consistent with the expectation that detection of a phonon by one qubit precludes detection by the other qubit. Using a total of 261 different combinations of tomography displacement pulses, we use convex optimization to reconstruct the density matrix $\rho$, shown in Fig. 4e. We find a state fidelity $\mathcal{F}=\sqrt{\Tr(\rho_{\rm N00N}\cdot|\rho|)}=0.748\pm 0.008$ to the ideal $N=2$ N00N state $\rho_{\rm N00N}$, in reasonable agreement with our simulation fidelity of $\mathcal{F}=0.745$ (see Supplementary). The unwanted phonon emission mentioned above, together with the short mechanical resonator lifetimes, limit our ability to generate higher $N$ N00N states.

In conclusion, we deterministically entangle two macroscopic mechanical resonators on separate substrates, using two independently-controlled superconducting qubits to synthesize and then analyze multi-phonon states, including high fidelity phonon Bell and N00N states. This platform is scalable, supporting simultaneous entanglement of larger numbers of mechanical resonators, enabling e.g. the direct synthesis of Greenberger-Horne-Zeilinger (GHZ) and W states, as well as synthetic cat states. Using larger SAW cavities with smaller free spectral ranges, each qubit could access multiple acoustic resonances, opening the possibility for multi-mode quantum information processing with small form-factor acoustic devices (see Extended Data Fig. 4). Our architecture promises further insight into the fundamental science of entangled mechanical systems, as well as an approach to distributed quantum computing. This hybrid quantum system would clearly benefit from increased coherence lifetimes for the SAW resonators, which will be essential for implementing more complex quantum operations [53, 54, 12]. This could be achieved by better material growth [55], different device designs, or possibly lowering the frequency of the SAW resonators [56].