Quantum jumps of sound

AI summary

Overview

Research area: Quantum acoustics / quantum electromechanics — the quantum control and measurement of gigahertz-frequency mechanical resonators coupled to superconducting qubits.

Technical level: Advanced (assumes familiarity with circuit QED, dispersive coupling, Ramsey interferometry, and Bayesian state estimation).

Scope (one sentence): The paper reports repeated quantum nondemolition (QND) measurements of phonon number in a lithium niobate (LN) nanomechanical resonator coupled to a transmon qubit, culminating in the direct observation of discontinuous quantum jumps between the resonator's first excited state and its ground state.

What This Paper Is About

Any measurement that couples to a mechanical resonator's displacement yields a smooth exponential decay, exactly as a classical oscillator would — the discrete energy levels predicted by quantum mechanics remain hidden. To see the quantization of sound, one must instead measure the resonator's energy directly, in a way that does not disturb it, which requires coupling the oscillator to an intrinsically nonlinear system such as a qubit. This paper builds such a device — a piezoelectric phononic-crystal resonator integrated with a superconducting transmon — and uses it to track phonon number in real time, heralding single-phonon states and observing individual quantum jumps of a massive vibrating object.

Key Contributions

1. A fabrication platform that preserves both subsystems. The authors fabricate the LN mechanical resonator and the tantalum transmon qubit separately, then integrate them with an aligned transfer-print process using a submicron die bonder, suspending the phononic crystal just 75 nm above a 500-nm qubit island gap. This avoids degrading either the piezoelectric or superconducting material.

2. Simultaneous long lifetime, strong coupling, and qubit coherence. The device achieves a mechanical lifetime T₁ = 2.1 ms, a dispersive shift 2χ/2π = 328 kHz per phonon, and qubit coherence T₂* = 3.2 μs with T₁q = 31 μs — a combination the paper states has not previously been realized, enabling roughly 170 parity checks within a single phonon lifetime.

3. Heralded mechanical Fock states via repeated QND parity measurements. Using a Ramsey sequence of duration τ = π/2χ = 1.524 μs that maps phonon parity onto qubit state, six consecutive measurements of |e⟩ herald an n = 1 state with P(n=1) = 0.85.

4. Real-time observation of quantum jumps of a mechanical resonator. Three individual trajectories of 294 consecutive parity checks each show abrupt transitions from predominantly |e⟩ to predominantly |g⟩, analyzed with forward and forward-backward Bayesian estimation.

Main Findings

• Long mechanical lifetime: T₁ = 2.1 ms (fit value 2070 ± 240 μs from qubit spectroscopy of an n̄ = 1 coherent state), described as exceptionally long for any piezoelectric resonator at gigahertz frequencies.

• Strong dispersive shift: 2χ/2π = 328 kHz per phonon, with χ/2π = 164 kHz reported in Table S1. The mechanical mode is detuned by roughly 21 MHz from the qubit e→f transition, placing it in the straddling regime.

• Qubit performance after integration: T₂* = 3.2 μs and T₁q = 31 μs. A control qubit on the same chip that saw no transfer printing had T₁ = 61 μs and T₂* = 8.7 μs, comparable to three other mechanics-integrated qubits (T₁ = 31, 81, 60 μs; T₂* = 3.2, 10.5, 13.4 μs), suggesting qubit coherence is limited by qubit fabrication rather than the integration.

• Parity measurement fidelity: Single-shot qubit readout fidelity of 0.91 (0.912 in the appendix, using a 1792 ns square pulse), but a lower single-shot parity assignment fidelity of [P(e|odd) + 1 − P(e|even)]/2 = 0.68, limited primarily by qubit dephasing during the Ramsey sequence.

• Spectroscopic resolution of phonon number: Individual phonon peaks separated by 2χ are resolved in postselected qubit spectra, with peak areas giving P(n). Weight in the n = 1 peak grows monotonically across k = 1 to 6 consecutive |e⟩ outcomes.

• Bayesian model agreement: Fitting the evolution of P(n=1) yields α = 0.356 ± 0.087 and P(e|even) = 0.296 ± 0.037; an independent extraction from the n = 0 and n = 1 peak areas in Fig. 2B gives α = 0.283 ± 0.032.

• Nearly QND operation: Residual backaction reduces the effective lifetime to T₁eff = 649 μs during continuous monitoring, corresponding to a demolition probability P_D = 0.013 per measurement — about 99% QND. A master-equation simulation attributes this to ~1.6% swap probability between |e,1⟩ and |f,0⟩, combined with a roughly 67.1% f-state reset fidelity, giving an estimated ~1.1% demolition.

• Jump statistics: The distribution of jump times from n = 1 to n = 0 is exponential with a time constant T₁ = 645 ± 29 μs, consistent with memoryless spontaneous phonon emission. The extracted value varies by only ±21 μs as the algorithm's input T₁ is swept from 570 to 730 μs.

• Large cooperativity: C = 4g²T₁T₂*, with g/2π = 1.72 MHz, exceeds 3 × 10⁶, which the authors describe as an order-of-magnitude improvement over other quantum acoustic platforms.

• Device parameters: ω_ge/2π = 3.389 GHz, ω_m/2π = 3.277 GHz, α/2π = 133 MHz, effective mass m_eff = 0.1995 pg, zero-point fluctuation x_zpf = 3.576 fm. The mechanical mode is a shear mode.

• Mechanical dephasing (not measured on the primary device): Measured on a nominally identical resonator on the same chip, averaging 55 μs and fluctuating substantially over roughly 24 hours. The authors state they did not measure the dephasing time of the resonator reported in the main text and did not carefully investigate its two-level system properties.

Methodology in Plain English

The team placed a tiny vibrating structure — a one-dimensional phononic crystal cavity carved from thin-film lithium niobate, a piezoelectric material — very close to a superconducting qubit. The periodic patterning acts as acoustic mirrors that trap the vibration in a micron-scale central block and shield it from leaking sound into the substrate.

The critical fabrication trick is that the two components are made separately, since the processes that optimize each are incompatible, then physically stamped together. The resonator sits on raised LN pillars so it floats 75 nm above a narrow 500-nm gap in the qubit, where the electric field is concentrated. Bonding used 10 N of force at 80 °C for 5 minutes.

Because the qubit's transition frequency shifts by 2χ for every phonon present, measuring the qubit frequency tells you the phonon number without absorbing the phonon. The measurement itself is a Ramsey sequence: a π/2 pulse puts the qubit on the equator, it precesses at a rate set by the phonon number, and after τ = π/2χ = 1.524 μs, even phonon numbers return to the start while odd ones pick up a π phase. A final −π/2 pulse and readout converts that into a g/e answer — a "parity check" denoted Π̂. Each check, including readout and conditional reset, takes about 12 μs, more than two orders of magnitude faster than the phonon lifetime.

To turn a string of noisy binary outcomes into a phonon-number trajectory, the authors use Bayesian inference: a forward pass updates the probability distribution P(n) after each outcome using the emission probabilities and a matrix accounting for decay between measurements, and a backward pass folds in later measurements to smooth the estimate and suppress noise.

Why This Matters

Impact on research. Quantum jumps were first proposed by Bohr and first directly observed in trapped ions, then in electromagnetic cavities — first a Fabry-Perot cavity probed by Rydberg atoms, later a 3D microwave cavity coupled to a superconducting qubit. This work extends that arc to a mechanical object. The authors frame the result as completing the sequence of state initialization, coherent manipulation, and QND measurement of quanta for mechanical vibrations. The demonstrated cooperativity above 3 × 10⁶ is the kind of resource that underlies bosonic quantum error correction.

Real-world applications (as motivated in the paper):

• Mass sensing and mass spectrometry: Integrated electromechanical resonators are exceptionally sensitive to small masses and forces; the paper cites subpicogram effective masses and large quality factors as what single-molecule mass sensing requires (m_eff here is 0.1995 pg).
• Probing single atomic defects in materials, using the resonator's sensitivity to local perturbations.
• Quantum transduction between physical degrees of freedom, exploiting the resonator's ability to couple to diverse systems.
• Precision measurement broadly: the paper notes kilogram-scale pendulums are used for gravitational wave detection, positioning mechanical resonators as central to metrology. The ability to prepare and nondestructively read out nonclassical mechanical states offers a route to sensing frequency changes at much lower drive power.
• Quantum information processing: compact size, long lifetimes, and low cross-talk make these resonators promising, and phonons' inability to propagate through vacuum provides natural isolation between modes.

Industry relevance. The work sits at the intersection of superconducting quantum computing hardware and MEMS/NEMS fabrication. The aligned transfer-print integration technique is a packaging-level capability that could be adopted by groups building hybrid quantum devices, and the demonstrated compatibility of piezoelectric material with a tantalum-on-silicon transmon addresses a known integration bottleneck. The experiment used commercial control electronics (Quantum Machines OPX and Octave), a Lincoln Labs traveling-wave parametric amplifier, and a SCALINQ LINQER16 package, indicating the measurement stack is built from available instrumentation.

Future Directions

1. A mechanical dual-rail qubit. The authors propose coupling two mechanical resonators to a shared qubit and performing joint parity measurements, so that phonon loss becomes a detectable erasure error. Slow frequency fluctuations could be handled either by spatially overlapping the modes to create correlated dephasing or by parametrically swapping the modes faster than the dephasing timescale. The transfer-print technique enables this because additional resonators can be stamped onto the same qubit.

2. Reducing measurement backaction. The 1.3% per-measurement demolition probability, attributed mainly to unwanted swapping between |e,1⟩ and |f,0⟩ plus imperfect f-state reset, could be lowered by more careful engineering of the qubit and resonator frequencies.

3. Improving parity assignment fidelity. At 0.68, this is limited by qubit dephasing during the Ramsey sequence rather than readout error, so better qubit coherence would directly sharpen each measurement and reduce the number of rounds needed to herald a Fock state.

4. Characterizing mechanical dephasing and two-level-system defects. The paper explicitly leaves this open, noting that the dephasing time of the reported resonator was not measured and that TLS properties were not carefully investigated — despite TLS defects being widely considered the dominant dephasing mechanism in quantum acoustic systems.

Target Audience

Specialists in quantum acoustics, circuit QED, and superconducting qubit hardware; researchers developing hybrid quantum devices and MEMS/NEMS integration processes; and physicists interested in foundational demonstrations of quantum discreteness in massive mechanical objects. The paper is written for readers comfortable with dispersive readout, Ramsey interferometry, and Bayesian state estimation, and much of its experimental detail (fabrication recipes, wiring, device parameter tables) is in appendices aimed at practitioners who might reproduce or extend the platform.

Authors’ abstract

Quantum mechanics predicts that a vibrating object's energy comes in discrete packets, yet no measurement of its position reveals this discreteness. Resolving individual energy levels requires a qualitatively different measurement, one coupling to the resonator's energy rather than its displacement. We use a superconducting qubit dispersively coupled to a nanomechanical resonator to perform repeated quantum nondemolition measurements of the phonon number. An aligned transfer-print technique integrating the qubit and resonator yields a mechanical lifetime of $T_1 = 2.1$ milliseconds and a dispersive shift of $2χ/2π= 328$ kilohertz per phonon. We heralded single-phonon states with 85% fidelity and observed quantum jumps between the resonator's first excited state and ground state. These discontinuous transitions are a striking manifestation of quantum mechanics in a massive, vibrating object.