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Dissipative cat qubits are a promising physical platform for quantum computing, since their large noise bias can enable more hardware-efficient quantum error correction. In this work we theoretically study the long-term prospects of a hybrid cat-transmon quantum computing architecture where dissipative cat qubits play the role of data qubits, and error syndromes are measured using ancillary transmon qubits. The cat qubits' noise bias enables more hardware-efficient quantum error correction, and the use of transmons allows for practical, high-fidelity syndrome measurement. While correction of the dominant cat Z errors with a repetition code has recently been demonstrated in experiment, here we show how the architecture can be scaled beyond a repetition code. In particular, we propose a cat-transmon entangling gate that enables the correction of residual cat X errors in a thin rectangular surface code, so that logical error can be arbitrarily suppressed by increasing code distance. We numerically estimate logical memory performance, finding significant overhead reductions in comparison to architectures without biased noise. For example, with current state-of-the-art coherence, physical error rates of $10^{-3}$ and noise biases in the range $10^{3} - 10^{4}$ are achievable. With this level of performance, the qubit overhead required to reach algorithmically-relevant logical error rates with the cat-transmon architecture matches that of an unbiased-noise architecture with physical error rates in the range $10^{-5} - 10^{-4}$., Comment: 13+14 pages, 8+9 figures

Cat qubits, a type of bosonic qubit encoded in a harmonic oscillator, can exhibit an exponential noise bias against bit-flip errors with increasing mean photon number. Here, we focus on cat qubits stabilized by two-photon dissipation, where pairs of photons are added and removed from a harmonic oscillator by an auxiliary, lossy buffer mode. This process requires a large loss rate and strong nonlinearities of the buffer mode that must not degrade the coherence and linearity of the oscillator. In this work, we show how to overcome this challenge by coloring the loss environment of the buffer mode with a multi-pole filter and optimizing the circuit to take into account additional inductances in the buffer mode. Using these techniques, we achieve near-ideal enhancement of cat-qubit bit-flip times with increasing photon number, reaching over $0.1$ seconds with a mean photon number of only $4$. Concurrently, our cat qubit remains highly phase coherent, with phase-flip times corresponding to an effective lifetime of $T_{1,\text{eff}} \simeq 70$ $\mu$s, comparable with the bare oscillator lifetime. We achieve this performance even in the presence of an ancilla transmon, used for reading out the cat qubit states, by engineering a tunable oscillator-ancilla dispersive coupling. Furthermore, the low nonlinearity of the harmonic oscillator mode allows us to perform pulsed cat-qubit stabilization, an important control primitive, where the stabilization can remain off for a significant fraction (e.g., two thirds) of a $3~\mathrm{\mu s}$ cycle without degrading bit-flip times. These advances are important for the realization of scalable error-correction with cat qubits, where large noise bias and low phase-flip error rate enable the use of hardware-efficient outer error-correcting codes., Comment: Comments welcome!

In order to solve problems of practical importance, quantum computers will likely need to incorporate quantum error correction, where a logical qubit is redundantly encoded in many noisy physical qubits. The large physical-qubit overhead typically associated with error correction motivates the search for more hardware-efficient approaches. Here, using a microfabricated superconducting quantum circuit, we realize a logical qubit memory formed from the concatenation of encoded bosonic cat qubits with an outer repetition code of distance $d=5$. The bosonic cat qubits are passively protected against bit flips using a stabilizing circuit. Cat-qubit phase-flip errors are corrected by the repetition code which uses ancilla transmons for syndrome measurement. We realize a noise-biased CX gate which ensures bit-flip error suppression is maintained during error correction. We study the performance and scaling of the logical qubit memory, finding that the phase-flip correcting repetition code operates below threshold, with logical phase-flip error decreasing with code distance from $d=3$ to $d=5$. Concurrently, the logical bit-flip error is suppressed with increasing cat-qubit mean photon number. The minimum measured logical error per cycle is on average $1.75(2)\%$ for the distance-3 code sections, and $1.65(3)\%$ for the longer distance-5 code, demonstrating the effectiveness of bit-flip error suppression throughout the error correction cycle. These results, where the intrinsic error suppression of the bosonic encodings allows us to use a hardware-efficient outer error correcting code, indicate that concatenated bosonic codes are a compelling paradigm for reaching fault-tolerant quantum computation., Comment: Comments on the manuscript welcome!

Entangling microwave and optical photons is one of the promising ways to realize quantum transduction through quantum teleportation. This paper investigates the entanglement of microwave-optical photon pairs generated from an electro-optomechanical system driven by a blue-detuned pulsed Gaussian pump. The photon pairs are obtained through weak parametric-down-conversion, and their temporal correlation is revealed by the second-order correlation function. We then study the discrete variable entanglement encoded in the time bin degree of freedom, where entanglement is identified by Bell inequality violation. Furthermore, we estimate the laser-induced heating and show that the pulse-pumped system features lower heating effects while maintaining a reasonable coincidence photon counting rate.

Optomechanical crystals (OMCs) enable coherent interactions between optical photons and microwave acoustic phonons, and represent a platform for implementing quantum transduction between microwave and optical signals. Optical absorption-induced thermal noise at cryogenic (millikelvin) temperatures is one of the primary limitations of performance for OMC-based quantum transducers. Here, we address this challenge with a two-dimensional silicon OMC resonator that is side-coupled to a mechanically detached optical waveguide, realizing a six-fold reduction in the heating rate of the acoustic resonator compared to prior state-of-the-art, while operating in a regime of high optomechanical-backaction and millikelvin base temperature. This reduced heating translates into a demonstrated phonon-to-photon conversion efficiency of 93.1 $\pm$ 0.8% at an added noise of 0.25 $\pm$ 0.01 quanta, representing a significant advance toward quantum-limited microwave-optical frequency conversion and optically-controlled quantum acoustic memories.

Entanglement is an extraordinary feature of quantum mechanics. Sources of entangled optical photons were essential to test the foundations of quantum physics through violations of Bell's inequalities. More recently, entangled many-body states have been realized via strong non-linear interactions in microwave circuits with superconducting qubits. Here we demonstrate a chip-scale source of entangled optical and microwave photonic qubits. Our device platform integrates a piezo-optomechanical transducer with a superconducting resonator which is robust under optical illumination. We drive a photon-pair generation process and employ a dual-rail encoding intrinsic to our system to prepare entangled states of microwave and optical photons. We place a lower bound on the fidelity of the entangled state by measuring microwave and optical photons in two orthogonal bases. This entanglement source can directly interface telecom wavelength time-bin qubits and GHz frequency superconducting qubits, two well-established platforms for quantum communication and computation, respectively., Comment: 14 pages, 12 figures

Optomechanical crystals are a promising device platform for quantum transduction and sensing. Precise targeting of the optical and acoustic resonance frequencies of these devices is crucial for future advances on these fronts. However, fabrication disorder in these wavelength-scale nanoscale devices typically leads to inhomogeneous resonance frequencies. Here we achieve in-situ, selective frequency tuning of optical and acoustic resonances in silicon optomechanical crystals via electric field-induced nano-oxidation using an atomic-force microscope. Our method can achieve a tuning range >2 nm (0.13%) for the optical resonance wavelength in the telecom C-band, and >60 MHz (1.2%) for the acoustic resonance frequency at 5 GHz. The tuning resolution of 1.1 nm for the optical wavelength, and $150$ kHz for the acoustic frequency allows us to spectrally align multiple optomechanical crystal resonators using optimal oxidation patterns determined via an inverse design protocol. Our results establish a method for precise post-fabrication tuning of optomechancical crystals. This technique can enable coupled optomechanical resonator arrays, scalable resonant optomechanical circuits, and frequency matching of microwave-optical quantum transducers.

Noise within solid-state systems at low temperatures, where many of the degrees of freedom of the host material are frozen out, can typically be traced back to material defects that support low-energy excitations. These defects can take a wide variety of microscopic forms, and for amorphous materials are broadly described using generic models such as the tunneling two-level systems (TLS) model. Although the details of TLS, and their impact on the low-temperature behavior of materials have been studied since the 1970s, these states have recently taken on further relevance in the field of quantum computing, where the limits to the coherence of superconducting microwave quantum circuits are dominated by TLS. Efforts to mitigate the impact of TLS have thus far focused on circuit design, material selection, and material surface treatment. In this work, we take a new approach that seeks to directly modify the properties of TLS through nanoscale-engineering. This is achieved by periodically structuring the host material, forming an acoustic bandgap that suppresses all microwave-frequency phonons in a GHz-wide frequency band around the operating frequency of a transmon qubit superconducting quantum circuit. For embedded TLS that are strongly coupled to the electric qubit, we measure a pronounced increase in relaxation time by two orders of magnitude when the TLS transition frequency lies within the acoustic bandgap, with the longest $T_1$ time exceeding $5$ milliseconds. Our work paves the way for in-depth investigation and coherent control of TLS, which is essential for deepening our understanding of noise in amorphous materials and advancing solid-state quantum devices., Comment: 11 + 25 pages, 4 + 22 figures, 6 tables; comments welcome!

Quantum error correction with erasure qubits promises significant advantages over standard error correction due to favorable thresholds for erasure errors. To realize this advantage in practice requires a qubit for which nearly all errors are such erasure errors, and the ability to check for erasure errors without dephasing the qubit. We demonstrate that a "dual-rail qubit" consisting of a pair of resonantly coupled transmons can form a highly coherent erasure qubit, where transmon $T_1$ errors are converted into erasure errors and residual dephasing is strongly suppressed, leading to millisecond-scale coherence within the qubit subspace. We show that single-qubit gates are limited primarily by erasure errors, with erasure probability $p_\text{erasure} = 2.19(2)\times 10^{-3}$ per gate while the residual errors are $\sim 40$ times lower. We further demonstrate mid-circuit detection of erasure errors while introducing $< 0.1\%$ dephasing error per check. Finally, we show that the suppression of transmon noise allows this dual-rail qubit to preserve high coherence over a broad tunable operating range, offering an improved capacity to avoid frequency collisions. This work establishes transmon-based dual-rail qubits as an attractive building block for hardware-efficient quantum error correction., Comment: 9+13 pages, 16 figures

Modern computing and communication technologies such as supercomputers and the internet are based on optically connected networks of microwave frequency information processors. In recent years, an analogous architecture has emerged for quantum networks with optically distributed entanglement between remote superconducting quantum processors, a leading platform for quantum computing. Here we report an important milestone towards such networks by observing non-classical correlations between photons in an optical link and a superconducting electrical circuit. We generate such states of light through a spontaneous parametric down-conversion (SPDC) process in a chip-scale piezo-optomechanical transducer. The non-classical nature of the emitted light is verified by observing anti-bunching in the microwave state conditioned on detection of an optical photon. Such a transducer can be readily connected to a superconducting quantum processor, and serve as a key building block for optical quantum networks of microwave frequency qubits., Comment: 22 pages, 14 figures

Coherent transduction of quantum states from the microwave to the optical domain can play a key role in quantum networking and distributed quantum computing. We present the design of a piezo-optomechanical device formed in a hybrid lithium niobate on silicon platform, that is suitable for microwave-to-optical quantum transduction. Our design is based on acoustic hybridization of an ultra-low mode volume piezoacoustic cavity with an optomechanical crystal cavity. The strong piezoelectric nature of lithium niobate allows us to mediate transduction via an acoustic mode which only minimally interacts with the lithium niobate, and is predominantly silicon-like, with very low electrical and acoustic loss. We estimate that this transducer can realize an intrinsic conversion efficiency of up to 35% with <0.5 added noise quanta when resonantly coupled to a superconducting transmon qubit and operated in pulsed mode at 10 kHz repetition rate. The performance improvement gained in such hybrid lithium niobate-silicon transducers make them suitable for heralded entanglement of qubits between superconducting quantum processors connected by optical fiber links.

Synthesis of many-body quantum systems in the laboratory can help provide further insight into the emergent behavior of quantum materials, whose properties may provide improved methods for energy conversion, signal transport, or information processing. While the majority of engineerable many-body systems, or quantum simulators, consist of particles on a lattice with local interactions, quantum systems featuring long-range interactions are particularly challenging to model and interesting to study due to the rapid spatio-temporal growth of quantum entanglement and correlations. Here, we present a scalable quantum simulator architecture based on a linear array of superconducting qubits locally connected to an extensible photonic-bandgap metamaterial. The metamaterial acts both as a quantum bus mediating qubit-qubit interactions, and as a readout channel for multiplexed qubit-state measurement. As an initial demonstration, we realize a 10-qubit simulator of the one-dimensional Bose-Hubbard model with in situ tunability of both the hopping range and the on-site interaction. We characterize the Hamiltonian of the system using a measurement-efficient protocol based on quantum many-body chaos. Further, we study the many-body quench dynamics of the system, revealing through global bit-string statistics the predicted crossover from integrability to ergodicity as the hopping range increases. The metamaterial quantum bus architecture presented here can be extended to two-dimensional lattice systems and used to generate a wide range of qubit interactions, expanding the accessible Hamiltonians for analog quantum simulation and increasing the flexibility in implementing quantum circuits for gate-based computations, Comment: 34 pages, 18 figures, 1 Table

Multidimensional photonic graph states, such as cluster states, have prospective applications in quantum metrology, secure quantum communication, and measurement-based quantum computation. However, to date, generation of multidimensional cluster states of photonic qubits has relied on probabilistic methods that limit the scalability of typical generation schemes in optical systems. Here we present an experimental implementation in the microwave domain of a resource-efficient scheme for the deterministic generation of 2D photonic cluster states. By utilizing a coupled resonator array as a slow-light waveguide, a single flux-tunable transmon qubit as a quantum emitter, and a second auxiliary transmon as a switchable mirror, we achieve rapid, shaped emission of entangled photon wavepackets, and selective time-delayed feedback of photon wavepackets to the emitter qubit. We leverage these capabilities to generate a 2D cluster state of four photons with 70\% fidelity, as verified by tomographic reconstruction of the quantum state. We discuss how our scheme could be straightforwardly extended to the generation of even larger cluster states, of even higher dimension, thereby expanding the scope and practical utility of such states for quantum information processing tasks., Comment: 31 pages, 17 figures

Protected qubits such as the 0-$\pi$ qubit, and bosonic qubits including cat qubits and GKP qubits offer advantages for fault-tolerance. Some of these protected qubits (e.g., 0-$\pi$ qubit and Kerr cat qubit) are stabilized by Hamiltonians which have (near-)degenerate ground state manifolds with large energy-gaps to the excited state manifolds. Without dissipative stabilization mechanisms the performance of such energy-gap-protected qubits can be limited by leakage to excited states. Here, we propose a scheme for dissipatively stabilizing an energy-gap-protected qubit using colored (i.e., frequency-selective) dissipation without inducing errors in the ground state manifold. Concretely we apply our colored dissipation technique to Kerr cat qubits and propose colored Kerr cat qubits which are protected by an engineered colored single-photon loss. When applied to the Kerr cat qubits our scheme significantly suppresses leakage-induced bit-flip errors (which we show are a limiting error mechanism) while only using linear interactions. Beyond the benefits to the Kerr cat qubit we also show that our frequency-selective loss technique can be applied to a broader class of protected qubits., Comment: 19 pages, 10 figures, comments welcome!

We present a comprehensive architectural analysis for a proposed fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout. Using estimated physical parameters for the hardware, we perform a detailed error analysis of measurements and gates, including CNOT and Toffoli gates. Having built a realistic noise model, we numerically simulate quantum error correction when the outer code is either a repetition code or a thin rectangular surface code. Our next step toward universal fault-tolerant quantum computation is a protocol for fault-tolerant Toffoli magic state preparation that significantly improves upon the fidelity of physical Toffoli gates at very low qubit cost. To achieve even lower overheads, we devise a new magic-state distillation protocol for Toffoli states. Combining these results together, we obtain realistic full-resource estimates of the physical error rates and overheads needed to run useful fault-tolerant quantum algorithms. We find that with around 1,000 superconducting circuit components, one could construct a fault-tolerant quantum computer that can run circuits which are currently intractable for classical computers. Hardware with 18,000 superconducting circuit components, in turn, could simulate the Hubbard model in a regime beyond the reach of classical computing., Comment: 117 pages (main text 32 pages), 62 figures, 15 tables. Comments welcome! V2 adds additional appendices and conforms to journal specifications

Recent advances in cavity-optomechanics have now made it possible to use light not just as a passive measuring device of mechanical motion, but also to manipulate the motion of mechanical objects down to the level of individual quanta of vibrations (phonons). At the same time, microfabrication techniques have enabled small-scale optomechanical circuits capable of on-chip manipulation of mechanical and optical signals. Building on these developments, theoretical proposals have shown that larger scale optomechanical arrays can be used to modify the propagation of phonons, realizing a form of topologically protected phonon transport. Here, we report the observation of topological phonon transport within a multiscale optomechanical crystal structure consisting of an array of over $800$ cavity-optomechanical elements. Using sensitive, spatially resolved optical read-out we detect thermal phonons in a $0.325-0.34$GHz band traveling along a topological edge channel, with substantial reduction in backscattering. This represents an important step from the pioneering macroscopic mechanical systems work towards topological phononic systems at the nanoscale, where hypersonic frequency ($\gtrsim$GHz) acoustic wave circuits consisting of robust delay lines and non-reciprocal elements may be implemented. Owing to the broadband character of the topological channels, the control of the flow of heat-carrying phonons, albeit at cryogenic temperatures, may also be envisioned., Comment: 20 pages, 9 figures

While designing the energy-momentum relation of photons is key to many linear, non-linear, and quantum optical phenomena, a new set of light-matter properties may be realized by employing the topology of the photonic bath itself. In this work we investigate the properties of superconducting qubits coupled to a metamaterial waveguide based on a photonic analog of the Su-Schrieffer-Heeger model. We explore topologically-induced properties of qubits coupled to such a waveguide, ranging from the formation of directional qubit-photon bound states to topology-dependent cooperative radiation effects. Addition of qubits to this waveguide system also enables direct quantum control over topological edge states that form in finite waveguide systems, useful for instance in constructing a topologically protected quantum communication channel. More broadly, our work demonstrates the opportunity that topological waveguide-QED systems offer in the synthesis and study of many-body states with exotic long-range quantum correlations., Comment: 28 pages 17 figures

Bidirectional conversion of electrical and optical signals lies at the foundation of the global internet. Such converters are employed at repeater stations to extend the reach of long-haul fiber optic communication systems and within data centers to exchange high-speed optical signals between computers. Likewise, coherent microwave-to-optical conversion of single photons would enable the exchange of quantum states between remotely connected superconducting quantum processors, a promising quantum computing hardware platform. Despite the prospects of quantum networking, maintaining the fragile quantum state in such a conversion process with superconducting qubits has remained elusive. Here we demonstrate the conversion of a microwave-frequency excitation of a superconducting transmon qubit into an optical photon. We achieve this using an intermediary nanomechanical resonator which converts the electrical excitation of the qubit into a single phonon by means of a piezoelectric interaction, and subsequently converts the phonon to an optical photon via radiation pressure. We demonstrate optical photon generation from the qubit with a signal-to-noise greater than unity by recording quantum Rabi oscillations of the qubit through single-photon detection of the emitted light over an optical fiber. With proposed improvements in the device and external measurement set-up, such quantum transducers may lead to practical devices capable of realizing new hybrid quantum networks, and ultimately, distributed quantum computers., Comment: 17 pages, 11 figures

A structured electromagnetic reservoir can result in novel dynamics of quantum emitters. In particular, the reservoir can be tailored to have a memory of past interactions with emitters, in contrast to memory-less Markovian dynamics of typical open systems. In this Article, we investigate the non-Markovian dynamics of a superconducting qubit strongly coupled to a superconducting slow-light waveguide reservoir. Tuning the qubit into the spectral vicinity of the passband of this waveguide, we find non-exponential energy relaxation as well as substantial changes to the qubit emission rate. Further, upon addition of a reflective boundary to one end of the waveguide, we observe revivals in the qubit population on a timescale 30 times longer than the inverse of the qubit's emission rate, corresponding to the round-trip travel time of an emitted photon. By tuning of the qubit-waveguide interaction strength, we probe a crossover between Markovian and non-Markovian qubit emission dynamics. These attributes allow for future studies of multi-qubit circuits coupled to structured reservoirs, in addition to constituting the necessary resources for generation of multiphoton highly entangled states., Comment: 19 pages, 11 figures with appendices

Optomechanical systems offer new opportunities in quantum information processing and quantum sensing. Many solid-state quantum devices operate at millikelvin temperatures -- however, it has proven challenging to operate nanoscale optomechanical devices at these ultralow temperatures due to their limited thermal conductance and parasitic optical absorption. Here, we demonstrate a two-dimensional optomechanical crystal resonator capable of achieving large cooperativity $C$ and small effective bath occupancy $n_b$, resulting in a quantum cooperativity $C_{\text{eff}}\equiv C/n_b \approx 1.3 > 1$ under continuous-wave optical driving. This is realized using a two-dimensional phononic bandgap structure to host the optomechanical cavity, simultaneously isolating the acoustic mode of interest in the bandgap while allowing heat to be removed by phonon modes outside of the bandgap. This achievement paves the way for a variety of applications requiring quantum-coherent optomechanical interactions, such as transducers capable of bi-directional conversion of quantum states between microwave frequency superconducting quantum circuits and optical photons in a fiber optic network., Comment: 21 pages, 12 figures, 7 appendices

We present measurements at millikelvin temperatures of the microwave-frequency acoustic properties of a crystalline silicon nanobeam cavity incorporating a phononic bandgap clamping structure for acoustic confinement. Utilizing pulsed laser light to excite a co-localized optical mode of the nanobeam cavity, we measure the dynamics of cavity acoustic modes with single-phonon sensitivity. Energy ringdown measurements for the fundamental $5$~GHz acoustic mode of the cavity shows an exponential increase in phonon lifetime versus number of periods in the phononic bandgap shield, increasing up to $\tau \approx 1.5$~seconds. This ultralong lifetime, corresponding to an effective phonon propagation length of several kilometers, is found to be consistent with damping from non-resonant two-level system defects on the surface of the silicon device. Potential applications of these ultra-coherent nanoscale mechanical resonators range from tests of various collapse models of quantum mechanics to miniature quantum memory elements in hybrid superconducting quantum circuits., Comment: 43 pages, 22 figures

Wavelengths in the telecommunication window (~1.25-1.65 microns) are ideal for quantum communication due to low transmission loss in fiber networks. To realize quantum networks operating at these wavelengths, long-lived quantum memories that couple to telecom-band photons with high efficiency need to be developed. We propose coupling neutral ytterbium atoms, which have a strong telecom-wavelength transition, to a silicon photonic crystal cavity. Specifically, we consider the 3P0-3D1 transition in neutral 171Yb to interface its long-lived nuclear spin in the metastable 3P0 'clock' state with a telecom-band photon at 1.4 microns. We show that Yb atoms can be trapped using a short wavelength (~470 nm) tweezer at a distance of 350 nm from the silicon photonic crystal cavity. At this distance, due to the slowly decaying evanescent cavity field at a longer wavelength, we obtain a single-photon Rabi frequency of g/(2pi)~100 MHz and a cooperativity of C~47 while maintaining a high photon collection efficiency into a single mode fiber. The combination of high system efficiency, telecom-band operation, and long coherence times makes this platform well suited for quantum optics on a silicon chip and long-distance quantum communication., Comment: 12 pages, 7 figures

Waveguide quantum electrodynamics studies photon-mediated interactions of quantum emitters in a one-dimensional radiation channel. Although signatures of such interactions have been observed previously in a variety of physical systems, observation of coherent cooperative dynamics has been obscured by radiative decay of atoms into the waveguide. Employing transmon qubits as artificial atoms coupled to a microwave coplanar waveguide, here we observe dynamical oscillations in an open system where a designated probe qubit interacts with an entangled dark state of an array of qubits which effectively traps radiation as an atomic cavity. The qubit-cavity system is shown to achieve a large cooperativity of $\mathcal{C}=172$ due to collective enhancement of photon-mediated interactions, entering the strong coupling regime. The quantum coherence of the dark state cavity is also explored through its nonlinear response at the single-excitation level. With realistic refinements, this system is suitable for studying the many-body dynamics of large ($N>10$) quantum spin chains, synthesizing highly non-classical radiation fields on demand, and implementing universal quantum logic operations with high fidelity on information encoded within decoherence-free subspaces., Comment: 19 pages, 8 figures, 5 appendices

Radiation pressure within engineered structures has recently been used to couple the motion of nanomechanical objects with high sensitivity to optical and microwave electromagnetic fields. Here, we demonstrate a form of electromechanical crystal for coupling microwave photons and hypersonic phonons by embedding the vacuum-gap capacitor of a superconducting resonator within a phononic crystal acoustic cavity. Utilizing a two-photon resonance condition for efficient microwave pumping and a phononic bandgap shield to eliminate acoustic radiation, we demonstrate large cooperative coupling ($C \approx 30$) between a pair of electrical resonances at $10$GHz and an acoustic resonance at $0.425$GHz. Electrical read-out of the phonon occupancy shows that the hypersonic acoustic mode has an intrinsic energy decay time of $2.3$ms and thermalizes close to its quantum ground-state of motion (occupancy $1.5$) at a fridge temperature of $10$mK. Such an electromechanical transducer is envisioned as part of a hybrid quantum circuit architecture, capable of interfacing to both superconducting qubits and optical photons., Comment: 16 pages, 12 figures, 8 appendices

The properties of coupled emitters can differ dramatically from those of their individual constituents. Canonical examples include sub- and super-radiance, wherein the decay rate of a collective excitation is reduced or enhanced due to correlated interactions with the environment. Here, we systematically study the properties of collective excitations for regularly spaced arrays of quantum emitters coupled to a one-dimensional (1D) waveguide. We find that, for low excitation numbers, the modal properties are well-characterized by spin waves with a definite wavevector. Moreover, the decay rate of the most subradiant modes obeys a universal scaling with a cubic suppression in the number of emitters. Multi-excitation subradiant eigenstates can be built from fermionic combinations of single excitation eigenstates; such "fermionization" results in multiple excitations that spatially repel one another. We put forward a method to efficiently create and measure such subradiant states, which can be realized with superconducting qubits. These measurement protocols probe both real-space correlations (using on-site dispersive readout) and temporal correlations in the emitted field (using photon correlation techniques)., Comment: 21 pages, 9 figures

The embedding of tunable quantum emitters in a photonic bandgap structure enables the control of dissipative and dispersive interactions between emitters and their photonic bath. Operation in the transmission band, outside the gap, allows for studying waveguide quantum electrodynamics in the slow-light regime. Alternatively, tuning the emitter into the bandgap results in finite range emitter-emitter interactions via bound photonic states. Here we couple a transmon qubit to a superconducting metamaterial with a deep sub-wavelength lattice constant ($\lambda/60$). The metamaterial is formed by periodically loading a transmission line with compact, low loss, low disorder lumped element microwave resonators. We probe the coherent and dissipative dynamics of the system by measuring the Lamb shift and the change in the lifetime of the transmon qubit. Tuning the qubit frequency in the vicinity of a band-edge with a group index of $n_g = 450$, we observe an anomalous Lamb shift of 10 MHz accompanied by a 24-fold enhancement in the qubit lifetime. In addition, we demonstrate selective enhancement and inhibition of spontaneous emission of different transmon transitions, which provide simultaneous access to long-lived metastable qubit states and states strongly coupled to propagating waveguide modes., Comment: 13 pages, 7 figures

We present the fabrication and characterization of transmon qubits formed from aluminum Josephson junctions on two different silicon-based substrates: (i) high-resistivity silicon (Si) and (ii) silicon-on-insulator (SOI). Key to the qubit fabrication process is the use of an anhydrous hydrofluoric vapor process which removes silicon surface oxides without attacking aluminum, and in the case of SOI substrates, selectively removes the lossy buried oxide underneath the qubit region. For qubits with a transition frequency of approximately $5$GHz we find qubit lifetimes and coherence times comparable to those attainable on sapphire substrates ($T_{1,\text{Si}} = 27\mu$s, $T_{2,\text{Si}} = 6.6\mu$s; $T_{1,\text{SOI}} = 3.5\mu$s, $T_{2,\text{SOI}} = 2.2\mu$s). This qubit fabrication process in principle permits co-fabrication of silicon photonic and mechanical elements, providing a route towards chip-scale integration of electro-opto-mechanical transducers for quantum networking of superconducting microwave quantum circuits., Comment: 6 pages, 4 figures

We show how the snowflake phononic crystal structure, which has been realized experimentally recently, can be turned into a topological insulator for sound waves. This idea, based purely on simple geometrical modifications, could be readily implemented on the nanoscale., Comment: 5 pages, 4 figures

Synthetic magnetism has been used to control charge neutral excitations for applications ranging from classical beam steering to quantum simulation. In optomechanics, radiation-pressure-induced parametric coupling between optical (photon) and mechanical (phonon) excitations may be used to break time-reversal symmetry, providing the prerequisite for synthetic magnetism. Here we design and fabricate a silicon optomechanical circuit with both optical and mechanical connectivity between two optomechanical cavities. Driving the two cavities with phase-correlated laser light results in a synthetic magnetic flux, which in combination with dissipative coupling to the mechanical bath, leads to nonreciprocal transport of photons with 35dB of isolation. Additionally, optical pumping with blue-detuned light manifests as a particle non-conserving interaction between photons and phonons, resulting in directional optical amplification of 12dB in the isolator through direction. These results indicate the feasibility of utilizing optomechanical circuits to create a more general class of nonreciprocal optical devices, and further, to enable novel topological phases for both light and sound on a microchip., Comment: 18 pages, 8 figures, 4 appendices

There is a growing effort in creating chiral transport of sound waves. However, most approaches so far are confined to the macroscopic scale. Here, we propose a new approach suitable to the nanoscale which is based on pseudomagnetic fields. These fields are the analogon for sound of the pseudomagnetic field for electrons in strained graphene. In our proposal, they are created by simple geometrical modifications of an existing and experimentally proven phononic crystal design, the snowflake crystal. This platform is robust, scalable, and well-suited for a variety of excitation and readout mechanisms, among them optomechanical approaches.

We present the optical and mechanical design of a mechanically compliant quasi-two-dimensional photonic crystal cavity formed from thin-film silicon in which a pair of linear nanoscale slots are used to create two coupled high-$Q$ optical resonances. The optical cavity supermodes, whose frequencies are designed to lie in the $1500$~nm wavelength band, are shown to interact strongly with mechanical resonances of the structure whose frequencies range from a few MHz to a few GHz. Depending upon the symmetry of the mechanical modes and the symmetry of the slot sizes, we show that the optomechanical coupling between the optical supermodes can be either linear or quadratic in the mechanical displacement amplitude. Tuning of the nanoscale slot size is also shown to adjust the magnitude and sign of the cavity supermode splitting $2J$, enabling near-resonant motional scattering between the two optical supermodes and greatly enhancing the $x^2$-coupling strength. Specifically, for the fundamental flexural mode of the central nanobeam of the structure at $10$~MHz the per-phonon linear cross-mode coupling rate is calculated to be $\tilde{g}_{+-}/2\pi = 1$~MHz, corresponding to a per-phonon $x^2$-coupling rate of $\tilde{g}'/2\pi=1$~kHz for a mode splitting $2J/2\pi = 1$~GHz which is greater than the radiation-limited supermode linewidths., Comment: 17 pages, 10 figures

We present a silicon optomechanical nanobeam design with a dynamically tunable acoustic mode at 10.2 GHz. The resonance frequency can be shifted by 90 kHz/V^2 with an on-chip capacitor that was optimized to exert forces up to 1 $\mu$N at 10 V operation voltage. Optical resonance frequencies around 190 THz with Q factors up to $2.2 \times 10^6$ place the structure in the well-resolved sideband regime with vacuum optomechanical coupling rates up to $g_0/2\pi = 353$ kHz. Tuning can be used, for instance, to overcome variation in the device-to-device acoustic resonance frequency due to fabrication errors, paving the way for optomechanical circuits consisting of arrays of optomechanical cavities., Comment: 13 pages, 5 figures

Linking classical microwave electrical circuits to the optical telecommunication band is at the core of modern communication. Future quantum information networks will require coherent microwave-to-optical conversion to link electronic quantum processors and memories via low-loss optical telecommunication networks. Efficient conversion can be achieved with electro-optical modulators operating at the single microwave photon level. In the standard electro-optic modulation scheme this is impossible because both, up- and downconverted, sidebands are necessarily present. Here we demonstrate true single sideband up- or downconversion in a triply resonant whispering gallery mode resonator by explicitly addressing modes with asymmetric free spectral range. Compared to previous experiments, we show a three orders of magnitude improvement of the electro-optical conversion efficiency reaching 0.1% photon number conversion for a 10GHz microwave tone at 0.42mW of optical pump power. The presented scheme is fully compatible with existing superconducting 3D circuit quantum electrodynamics technology and can be used for non-classical state conversion and communication. Our conversion bandwidth is larger than 1MHz and not fundamentally limited., Comment: 8 pages, 5 figures; comments welcome!

Fabrication processes involving anhydrous hydrofluoric vapor etching are developed to create high-$Q$ aluminum superconducting microwave resonators on free-standing silicon membranes formed from a silicon-on-insulator wafer. Using this fabrication process, a high-impedance $8.9$GHz coil resonator is coupled capacitively with large participation ratio to a $9.7$MHz micromechanical resonator. Two-tone microwave spectroscopy and radiation pressure back-action are used to characterize the coupled system in a dilution refrigerator down to temperatures of $T_f = 11$~mK, yielding a measured electromechanical vacuum coupling rate of $g_{0}/2\pi \approx 24.6$~Hz and a mechanical resonator $Q$-factor of $Q_{m}=1.7\times 10^7$. Microwave back-action cooling of the mechanical resonator is also studied, with a minimum phonon occupancy of $n_{m} \approx 16$ phonons being realized at an elevated fridge temperature of $T_f = 211$~mK., Comment: 8 pages, 5 figures

We present a platform based upon silicon nitride nanomembranes for integrating superconducting microwave circuits with planar acoustic and optical devices such as phononic and photonic crystals. Utilizing tensile stress and lithographic patterning of a silicon nitride nanomembrane we are able to reliably realize planar capacitors with vacuum gap sizes down to $s \approx 80$nm. In combination with spiral inductor coils of micron pitch, this yields microwave ($\approx 8$GHz) resonant circuits of high impedance ($Z_{0} \approx 3.4$k$\Omega$) suitable for efficient electromechanical coupling to nanoscale acoustic structures. We measure an electromechanical vacuum coupling rate of $g_{0}/2\pi = 41.5$~Hz to the low frequency ($4.48$MHz) global beam motion of a patterned phononic crystal nanobeam, and through parametric microwave driving reach a backaction cooled mechanical mode occupancy as low as $n_{m} = 0.58$., Comment: 21 pages, 9 figures

Cavity-optomechanical systems realized in single-crystal diamond are poised to benefit from its extraordinary material properties, including low mechanical dissipation and a wide optical transparency window. Diamond is also rich in optically active defects, such as the nitrogen-vacancy (NV) and silicon-vacancy (SiV) centers, which behave as atom-like systems in the solid state. Predictions and observations of coherent coupling of the NV electronic spin to phonons via lattice strain has motivated the development of diamond nanomechanical devices aimed at realization of hybrid quantum systems, in which phonons provide an interface with diamond spins. In this work, we demonstrate diamond optomechanical crystals (OMCs), a device platform to enable such applications, wherein the co-localization of ~ 200 THz photons and few to 10 GHz phonons in a quasi-periodic diamond nanostructure leads to coupling of an optical cavity field to a mechanical mode via radiation pressure. In contrast to other material systems, diamond OMCs operating in the resolved-sideband regime possess large intracavity photon capacity (> 10$^5$) and sufficient optomechanical coupling rates to reach a cooperativity of ~ 20 at room temperature, allowing for the observation of optomechanically induced transparency and the realization of large amplitude optomechanical self-oscillations.

The mechanical properties of light have found widespread use in the manipulation of gas-phase atoms and ions, helping create new states of matter and realize complex quantum interactions. The field of cavity-optomechanics strives to scale this interaction to much larger, even human-sized mechanical objects. Going beyond the canonical Fabry-Perot cavity with a movable mirror, here we explore a new paradigm in which multiple cavity-optomechanical elements are wired together to form optomechanical circuits. Using a pair of optomechanical cavities coupled together via a phonon waveguide we demonstrate a tunable delay and filter for microwave-over-optical signal processing. In addition, we realize a tight-binding form of mechanical coupling between distant optomechanical cavities, leading to direct phonon exchange without dissipation in the waveguide. These measurements indicate the feasibility of phonon-routing based information processing in optomechanical crystal circuitry, and further, to the possibility of realizing topological phases of photons and phonons in optomechanical cavity lattices., Comment: 16 pages, 7 figures

Optical measurement of the motion of a 940 kHz mechanical resonance of a silicon nitride nanostring resonator is demonstrated with a read out noise imprecision reaching 37 dB below that of the resonator's zero-point fluctuations. Via intensity modulation of the optical probe laser, radiation pressure feedback is used to cool and damp the mechanical mode from an initial room temperature occupancy of $\bar{n}_{b} = 6.5 \times 10^6$ ($T_{b}=295$K) down to a phonon occupation of $\langle n \rangle = 66 \pm 10$, representing a mode temperature of $T_{m} \approx 3$mK. The five decades of cooling is enabled by the system's large single-photon cooperativity $(C_{1} = 4)$ and high quantum efficiency of optical motion detection ($\eta_{t} = 0.27$)., Comment: 13 pages, 13 figures

We present the design, fabrication, and characterization of a planar silicon photonic crystal cavity in which large position-squared optomechanical coupling is realized. The device consists of a double-slotted photonic crystal structure in which motion of a central beam mode couples to two high-Q optical modes localized around each slot. Electrostatic tuning of the structure is used to controllably hybridize the optical modes into supermodes which couple in a quadratic fashion to the motion of the beam. From independent measurements of the anti-crossing of the optical modes and of the optical spring effect, the position-squared vacuum coupling rate is measured to be as large as 245 Hz to the fundamental in-plane mechanical resonance of the structure at 8.7MHz, which in displacement units corresponds to a coupling coefficient of 1 THz/nm$^2$. This level of position-squared coupling is approximately five orders of magnitude larger than in conventional Fabry-Perot cavity systems., Comment: 11 pages, 6 figures

Utilizing a silicon nanobeam optomechanical crystal, we investigate the attractor diagram arising from the radiation pressure interaction between a localized optical cavity at $\lambda = 1552$nm and a mechanical resonance at $\omega/2\pi = 3.72$GHz. At a temperature of $T \approx 10$K, highly nonlinear driving of mechanical motion is observed via continuous wave optical pumping. Introduction of a time-dependent (modulated) optical pump is used to steer the system towards an otherwise inaccessible dynamically stable attractor in which mechanical self-oscillation occurs for an optical pump red-detuned from the cavity resonance. An analytical model incorporating thermo-optic effects due to optical absorption heating is developed, and found to accurately predict the measured device behavior., Comment: 5 pages, 3 figures

Using pulsed optical excitation and read-out along with single phonon counting techniques, we measure the transient back-action, heating, and damping dynamics of a nanoscale silicon optomechanical crystal cavity mounted in a dilution refrigerator at a base temperature of 11mK. In addition to observing a slow (~740ns) turn-on time for the optical-absorption-induced hot phonon bath, we measure for the 5.6GHz `breathing' acoustic mode of the cavity an initial phonon occupancy as low as 0.021 +- 0.007 (mode temperature = 70mK) and an intrinsic mechanical decay rate of 328 +- 14 Hz (mechanical Q-factor = 1.7x10^7). These measurements demonstrate the feasibility of using short pulsed measurements for a variety of quantum optomechanical applications despite the presence of steady-state optical heating., Comment: 16 pages, 6 figures

Using an optical probe along with single photon detection we have performed effective phonon counting measurements of the acoustic emission and absorption processes in a nanomechanical resonator. Applying these measurements in a Hanbury Brown and Twiss set-up, phonon correlations of the nanomechanical resonator are explored from below to above threshold of a parametric instability leading to self-oscillation of the resonator. Discussion of the results in terms of a "phonon laser", and analysis of the sensitivity of the phonon counting technique are presented., Comment: 10 pages, 8 figures

We fabricate and characterize a microscale silicon electro-opto-mechanical system whose mechanical motion is coupled capacitively to an electrical circuit and optically via radiation pressure to a photonic crystal cavity. To achieve large electromechanical interaction strength, we implement an inverse shadow mask fabrication scheme which obtains capacitor gaps as small as 30 nm while maintaining a silicon surface quality necessary for minimizing optical loss. Using the sensitive optical read-out of the photonic crystal cavity, we characterize the linear and nonlinear capacitive coupling to the fundamental 63 MHz in-plane flexural motion of the structure, showing that the large electromechanical coupling in such devices may be suitable for realizing efficient microwave-to-optical signal conversion., Comment: 8 papers, 4 figures

Optical measurements of a nanoscale silicon optomechanical crystal cavity with a mechanical resonance frequency of 3.6GHz are performed at sub-kelvin temperatures. We infer optical-absorption-induced heating and damping of the mechanical resonator from measurements of phonon occupancy and motional sideband asymmetry. At the lowest probe power and lowest fridge temperature (10mK), the localized mechanical resonance is found to couple at a rate of 400Hz (Q=9x10^6) to a thermal bath of temperature 270mK. These measurements indicate that silicon optomechanical crystals cooled to millikelvin temperatures should be suitable for a variety of experiments involving coherent coupling between photons and phonons at the single quanta level., Comment: 11 pages, 9 figures

We present a comprehensive study of dispersion-engineered nanowire photonic crystal waveguides suitable for experiments in quantum optics and atomic physics with optically trapped atoms. Detailed design methodology and specifications are provided, as are the processing steps used to create silicon nitride waveguides of low optical loss in the near-IR. Measurements of the waveguide optical properties and power-handling capability are also presented.