Resolving Phonon Fock States in a Multimode Cavity with a Double-Slit Qubit

We resolve phonon number states in the spectrum of a superconducting qubit coupled to a multimode acoustic cavity. Crucial to this resolution is the sharp frequency dependence in the qubit-phonon interaction engineered by coupling the qubit to surface acoustic waves in two locations separated by ∼40 acoustic wavelengths. In analogy to double-slit diffraction, the resulting interference generates high-contrast frequency structure in the qubit-phonon interaction. We observe this frequency structure both in the coupling rate to multiple cavity modes and in the qubit spontaneous emission rate into unconfined modes. We use this sharp frequency structure to resolve single phonons by tuning the qubit to a frequency of destructive interference where all acoustic interactions are dispersive. By exciting several detuned yet strongly coupled phononic modes and measuring the resulting qubit spectrum, we observe that, for two modes, the device enters the strong dispersive regime where single phonons are spectrally resolved. Quantum control over mechanical degrees of freedom promises insight into fundamental physics as well as the development of innovative quantum technologies. As mechanical resonators are massive and macroscopic, they can probe quantum theories at large scales [1-3], while the ability of mechanical motion to couple to a variety of quantum systems has inspired numerous mechanics-based transduc-tion schemes [4-10]. Additionally, mechanical elements are compact compared to their electromagnetic counterparts, enabling the on-chip fabrication of many wavelength microwave structures such as high-performance filters and multi-mode resonators [11-13]. High-fidelity control over the large number of modes achievable in acoustic platforms would be a powerful resource for quantum information processing [14]. The field of circuit quantum electrodynamics (cQED) has provided both guidance and tools for achieving quantum control over mechanical excitations. In cQED, the state of a photonic mode is measured and manipulated using super-conducting qubits. These qubits can also interact with mechanical systems using piezoelectric materials. Two seminal works leveraged this fact to couple a qubit to a dilatational resonator [15] and to propagating surface acoustic waves (SAWs) [16]. Both surface and bulk acoustic waves can be confined to form high-overtone resonators [17,18], leading to demonstrations of qubit-phonon coupling in multimode cavities [19-23]. Most recently, a pair of experiments used resonant interactions to create number and superposition states of an acoustic cavity mode, thereby demonstrating basic quantum control of acoustic phonons [3,24]. Following the example of cQED, achieving strong dispersive interactions in acoustic systems would lead to improved quantum control through quantum nondemolition phonon measurement [25,26] and qubit-mediated phonon-phonon interactions [27,28]. Realizing these techniques in acoustic systems would enable the exploration of a multi-mode analogy of cQED [29]. But coupling a single qubit with uniform strength to multiple modes of an acoustic cavity reduces the number of coherent interactions achievable with a given mode. Consider that for any qubit frequency inside the cavity bandwidth, there exists some nearest mode k with detuning Δ k less than half the cavity's free spectral range f s. To be in the dispersive limit for all modes, the qubit must have coupling g ≪ Δ k < f s =2. This limited coupling then bounds the number of operations possible within the qubit's coherence time ð2πγÞ −1 at approximately g 2 =ðΔ k γÞ. The number of interactions achieved for mode k is further reduced by a large factor ≈jn − kjf s =Δ k for the nth cavity mode. Eschewing the dispersive limit by choosing g ≈ f s yields strong, resonant interactions between the qubit and multiple cavity modes. The resulting hybrid modes are composed predominantly of linear cavity modes, effectively diluting the qubit's nonlinearity and thereby increasing the time required for coherent operations [21]. This reduction in coherent operations can be overcome by engineering a *