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The rapid development of superconducting quantum hardware is expected to run into significant I/O restrictions due to the need for large-scale error correction in a cryogenic environment. Classical data centers rely on fiber-optic interconnects to remove similar networking bottlenecks and to allow for reconfigurable, software-defined infrastructures. In the same spirit, ultra-cold electro-optic links have been proposed and used to generate qubit control signals, or to replace cryogenic readout electronics. So far, the latter suffered from either low efficiency, low bandwidth and the need for additional microwave drives, or breaking of Cooper pairs and qubit states. In this work we realize electro-optic microwave photonics at millikelvin temperatures to implement a radio-over-fiber qubit readout that does not require any active or passive cryogenic microwave equipment. We demonstrate all-optical single-shot-readout by means of the Jaynes-Cummings nonlinearity in a circulator-free readout scheme. Importantly, we do not observe any direct radiation impact on the qubit state as verified with high-fidelity quantum-non-demolition measurements despite the absence of shielding elements. This compatibility between superconducting circuits and telecom wavelength light is not only a prerequisite to establish modular quantum networks, it is also relevant for multiplexed readout of superconducting photon detectors and classical superconducting logic. Moreover, this experiment showcases the potential of electro-optic radiometry in harsh environments - an electronics-free sensing principle that extends into the THz regime with applications in radio astronomy, planetary missions and earth observation.

Recent advancements in superconducting circuits have enabled the experimental study of collective behavior of precisely controlled intermediate-scale ensembles of qubits. In this work, we demonstrate an atomic frequency comb formed by individual artificial atoms strongly coupled to a single resonator mode. We observe periodic microwave pulses that originate from a single coherent excitation dynamically interacting with the multi-qubit ensemble. We show that this revival dynamics emerges as a consequence of the constructive and periodic rephasing of the five superconducting qubits forming the vacuum Rabi split comb. In the future, similar devices could be used as a memory with in-situ tunable storage time or as an on-chip periodic pulse generator with non-classical photon statistics.

The number of ordered factorizations and the number of recursive divisors are two related arithmetic functions that are recursively defined. But it is hard to construct explicit representations of these functions. Taking advantage of their recursive definition and a geometric interpretation, we derive three closed-form expressions for them both. These expressions shed light on the structure of these functions and their number-theoretic properties. Surprisingly, both functions can be expressed as simple generalized hypergeometric functions.

We recently introduced the recursive divisor function $\kappa_x(n)$, a recursive analogue of the usual divisor function. Here we calculate its Dirichlet series, which is ${\zeta(s-x)}/(2 - \zeta(s))$. We show that $\kappa_x(n)$ is related to the ordinary divisor function by $\kappa_x * \sigma_y = \kappa_y * \sigma_x$, where * denotes the Dirichlet convolution. Using this, we derive several identities relating $\kappa_x$ and some standard arithmetic functions. We also clarify the relation between $\kappa_0$ and the much-studied number of ordered factorizations $K(n)$, namely, $\kappa_0 = {\bf 1} * K$.

The Kauffman model is the archetypal model of genetic computation. It highlights the importance of criticality, at which many biological systems seem poised. In a series of advances, researchers have honed in on how the number of attractors in the critical regime grows with network size. But a definitive answer has proved elusive. We prove that, for the critical Kauffman model with connectivity one, the number of attractors grows at least, and at most, as $(2/\!\sqrt{e})^N$. This is the first proof that the number of attractors in a critical Kauffman model grows exponentially., Comment: 5 pages, 3 figures

The Kauffman model of genetic computation highlights the importance of criticality at the border of order and chaos. The model with connectivity one is of special interest because it is exactly solvable. But our understanding of its behavior is incomplete, and much of what we do know relies on heuristic arguments. Here, we show that the key quantities in the model are intimately related to aspects of number theory. Using these links, we derive improved bounds for the number of attractors as well as the mean attractor length, which is harder to compute. Our work suggests that number theory is the natural language for deducing many properties of the critical Kauffman model with connectivity one, and opens the door to further insight into this deceptively simple model., Comment: 15 pages, 3 figures

The critical Kauffman model with connectivity one is the simplest class of critical Boolean networks. Nevertheless, it exhibits intricate behavior at the boundary of order and chaos. We introduce a formalism for expressing the dynamics of multiple loops as a product of the dynamics of individual loops. Using it, we prove that the number of attractors scales as $2^m$, where $m$ is the number of nodes in loops - as fast as possible, and much faster than previously believed.

Entanglement is a genuine quantum mechanical property and the key resource in currently developed quantum technologies. Sharing this fragile property between superconducting microwave circuits and optical or atomic systems would enable new functionalities but has been hindered by the tremendous energy mismatch of $\sim10^5$ and the resulting mutually imposed loss and noise. In this work we create and verify entanglement between microwave and optical fields in a millikelvin environment. Using an optically pulsed superconducting electro-optical device, we deterministically prepare an itinerant microwave-optical state that is squeezed by $0.72^{+0.31}_{-0.25}$\,dB and violates the Duan-Simon separability criterion by $>5$ standard deviations. This establishes the long-sought non-classical correlations between superconducting circuits and telecom wavelength light with wide-ranging implications for hybrid quantum networks in the context of modularization, scaling, sensing and cross-platform verification., Comment: Main text 6 pages with 3 figures. Supplement 22 pages with 12 figures

The photon blockade breakdown in a continuously driven cavity QED system has been proposed as a prime example for a first-order driven-dissipative quantum phase transition. But the predicted scaling from a microscopic system - dominated by quantum fluctuations - to a macroscopic one - characterized by stable phases - and the associated exponents and phase diagram have not been observed so far. In this work we couple a single transmon qubit with a fixed coupling strength $g$ to an in-situ bandwidth $\kappa$ tuneable superconducting cavity to controllably approach this thermodynamic limit. Even though the system remains microscopic, we observe its behavior to become more and more macroscopic as a function of $g/\kappa$. For the highest realized $g/\kappa \approx 287$ the system switches with a characteristic dwell time as high as 6 seconds between a bright coherent state with $\approx 8 \times 10^3$ intra-cavity photons and the vacuum state with equal probability. This exceeds the microscopic time scales by six orders of magnitude and approaches the near perfect hysteresis expected between two macroscopic attractors in the thermodynamic limit. These findings and interpretation are qualitatively supported by semi-classical theory and large-scale Quantum-Jump Monte Carlo simulations. Besides shedding more light on driven-dissipative physics in the limit of strong light-matter coupling, this system might also find applications in quantum sensing and metrology., Comment: 15 pages, 6 figures

Recent quantum technologies have established precise quantum control of various microscopic systems using electromagnetic waves. Interfaces based on cryogenic cavity electro-optic systems are particularly promising, due to the direct interaction between microwave and optical fields in the quantum regime. Quantum optical control of superconducting microwave circuits has been precluded so far due to the weak electro-optical coupling as well as quasi-particles induced by the pump laser. Here we report the coherent control of a superconducting microwave cavity using laser pulses in a multimode electro-optical device at millikelvin temperature with near-unity cooperativity. Both the stationary and instantaneous responses of the microwave and optical modes comply with the coherent electro-optical interaction, and reveal only minuscule amount of excess back-action with an unanticipated time delay. Our demonstration enables wide ranges of applications beyond quantum transductions, from squeezing and quantum non-demolition measurements of microwave fields, to entanglement generation and hybrid quantum networks.

Developing and maintaining life requires a lot of computation. This is done by gene regulatory networks. But we have little understanding of how this computation is organized. I show that there is a direct correspondence between the structural and functional building blocks of regulatory networks, which I call regulatory motifs. I derive a simple bound on the range of function that these motifs can perform, in terms of the local network structure. I prove that this range is a small fraction of all possible functions, which severely constrains global network behavior. Part of this restriction is due to redundancy in the function that regulatory motifs can achieve - there are many ways to perform the same task. Regulatory motifs help us understanding how genetic computation is organized and what it can achieve.

State-of-the-art transmon qubits rely on large capacitors which systematically improves their coherence due to reduced surface loss participation. However, this approach increases both the footprint and the parasitic cross-coupling and is ultimately limited by radiation losses - a potential roadblock for scaling up quantum processors to millions of qubits. In this work we present transmon qubits with sizes as low as 36$ \times $39$ \mu$m$^2$ with $\gtrsim$100 nm wide vacuum gap capacitors that are micro-machined from commercial silicon-on-insulator wafers and shadow evaporated with aluminum. After the release in HF vapor we achieve a vacuum participation ratio up to 99.6\% in an in-plane design that is compatible with standard coplanar circuits. Qubit relaxation time measurements for small gaps with high vacuum electric fields of up to 22 V/m reveal a double exponential decay indicating comparably strong coupling to long-lived two-level-systems (TLS). The exceptionally high selectivity of $>$20 dB to the superconductor-vacuum surface allows to precisely back out the sub-single-photon dielectric loss tangent of aluminum oxide exposed to ambient conditions. In terms of future scaling potential we achieve a qubit quality factor by footprint area of $20 \mu \mathrm{s}^{-2}$, which is on par with the highest $T_1$ devices relying on larger geometries and expected to improve substantially for lower loss superconductors like NbTiN, TiN or Ta.

The ability to control the direction of scattered light in integrated devices is crucial to provide the flexibility and scalability for a wide range of on-chip applications, such as integrated photonics, quantum information processing and nonlinear optics. In the optical and microwave frequency ranges tunable directionality can be achieved by applying external magnetic fields, that modify optical selection rules, by using nonlinear effects, or interactions with vibrations. However, these approaches are less suitable to control propagation of microwave photons inside integrated superconducting quantum devices, that is highly desirable. Here, we demonstrate tunable directional scattering with just two transmon qubits coupled to a transmission line based on periodically modulated transition frequency. By changing the symmetry of the modulation, governed by the relative phase between the local modulation tones, we achieve directional forward or backward photon scattering.

We investigate the deterministic generation and distribution of entanglement in large quantum networks by driving distant qubits with the output fields of a non-degenerate parametric amplifier. In this setting, the amplifier produces a continuous Gaussian two-mode squeezed state, which acts as a quantum-correlated reservoir for the qubits and relaxes them into a highly entangled steady state. Here we are interested in the maximal amount of entanglement and the optimal entanglement generation rates that can be achieved with this scheme under realistic conditions taking, in particular, the finite amplifier bandwidth, waveguide losses and propagation delays into account. By combining exact numerical simulations of the full network with approximate analytic results, we predict the optimal working point for the amplifier and the corresponding qubit-qubit entanglement under various conditions. Our findings show that this passive conversion of Gaussian- into discrete- variable entanglement offers a robust and experimentally very attractive approach for operating large optical, microwave or hybrid quantum networks, for which efficient parametric amplifiers are currently developed., Comment: 15 pages, 6 figures

The inductively shunted transmon (IST) is a superconducting qubit with exponentially suppressed fluxon transitions and a plasmon spectrum approximating that of the transmon. It shares many characteristics with the transmon but offers charge offset insensitivity for all levels and precise flux tunability with quadratic flux noise suppression. In this work we propose and realize IST qubits deep in the transmon limit where the large geometric inductance acts as a mere perturbation. With a flux dispersion of only 5.1 MHz we reach the 'sweet-spot everywhere' regime of a SQUID device with a stable coherence time over a full flux quantum. Close to the flux degeneracy point the device reveals tunneling physics between the two quasi-degenerate ground states with typical observed lifetimes on the order of minutes. In the future, this qubit regime could be used to avoid leakage to unconfined transmon states in high-power read-out or driven-dissipative bosonic qubit realizations. Moreover, the combination of well controllable plasmon transitions together with stable fluxon states in a single device might offer a way forward towards improved qubit encoding schemes., Comment: 13 pages, 8 figures

Magnetic-field-resilient superconducting circuits enable sensing applications and hybrid quantum-computing architectures involving spin or topological qubits and electro-mechanical elements, as well as studying flux noise and quasiparticle loss. We investigate the effect of in-plane magnetic fields up to 1 T on the spectrum and coherence times of thin-film 3D aluminum transmons. Using a copper cavity, unaffected by strong magnetic fields, we can solely probe the magnetic-field effect on the transmons. We present data on a single-junction and a SQUID transmon, that were cooled down in the same cavity. As expected, transmon frequencies decrease with increasing fields, due to a suppression of the superconducting gap and a geometric Fraunhofer-like contribution. Nevertheless, the thin-film transmons show strong magnetic-field resilience: both transmons display microsecond coherence up to at least 0.65 T, and $T_1$ remains above 1 $\mathrm{\mu}$s over the entire measurable range. SQUID spectroscopy is feasible up to 1 T, the limit of our magnet. We conclude that thin-film aluminum Josephson junctions are a suitable hardware for superconducting circuits in the high-magnetic-field regime.

Networks of gene regulation govern morphogenesis, determine cell identity and regulate cell function. But we have little understanding, at the local level, of which logics are biologically preferred or even permitted. To solve this puzzle, we studied the consequences of a fundamental aspect of gene regulatory networks: genes and transcription factors talk to each other but not themselves. Remarkably, this bipartite structure severely restricts the number of logical dependencies that a gene can have on other genes. We developed a theory for the number of permitted logics for different regulatory building blocks of genes and transcription factors. We tested our predictions against a simulation of the 19 simplest building blocks, and found complete agreement. The restricted range of biological logics is a key insight into how information is processed at the genetic level. It constraints global network function and makes it easier to reverse engineer regulatory networks from observed behavior., Comment: 6 pages

The quantum bits (qubits) on which superconducting quantum computers are based have energy scales corresponding to photons with GHz frequencies. The energy of photons in the gigahertz domain is too low to allow transmission through the noisy room-temperature environment, where the signal would be lost in thermal noise. Optical photons, on the other hand, have much higher energies, and signals can be detected using highly efficient single-photon detectors. Transduction from microwave to optical frequencies is therefore a potential enabling technology for quantum devices. However, in such a device the optical pump can be a source of thermal noise and thus degrade the fidelity; the similarity of input microwave state to the output optical state. In order to investigate the magnitude of this effect we model the sub-Kelvin thermal behavior of an electro-optic transducer based on a lithium niobate whispering gallery mode resonator. We find that there is an optimum power level for a continuous pump, whilst pulsed operation of the pump increases the fidelity of the conversion., Comment: 12 pages, 4 figures

The superconducting circuit community has recently discovered the promising potential of superinductors. These circuit elements have a characteristic impedance exceeding the resistance quantum $R_\text{Q} \approx 6.45~\text{k}\Omega$ which leads to a suppression of ground state charge fluctuations. Applications include the realization of hardware protected qubits for fault tolerant quantum computing, improved coupling to small dipole moment objects and defining a new quantum metrology standard for the ampere. In this work we refute the widespread notion that superinductors can only be implemented based on kinetic inductance, i.e. using disordered superconductors or Josephson junction arrays. We present modeling, fabrication and characterization of 104 planar aluminum coil resonators with a characteristic impedance up to 30.9 $\text{k}\Omega$ at 5.6 GHz and a capacitance down to $\leq1$ fF, with low-loss and a power handling reaching $10^8$ intra-cavity photons. Geometric superinductors are free of uncontrolled tunneling events and offer high reproducibility, linearity and the ability to couple magnetically - properties that significantly broaden the scope of future quantum circuits., Comment: 8 pages, 5 figures, 1 table

Fiber optic communication is the backbone of our modern information society, offering high bandwidth, low loss, weight, size and cost, as well as an immunity to electromagnetic interference. Microwave photonics lends these advantages to electronic sensing and communication systems, but - unlike the field of nonlinear optics - electro-optic devices so far require classical modulation fields whose variance is dominated by electronic or thermal noise rather than quantum fluctuations. Here we present a cavity electro-optic transceiver operating in a millikelvin environment with a mode occupancy as low as 0.025 $\pm$ 0.005 noise photons. Our system is based on a lithium niobate whispering gallery mode resonator, resonantly coupled to a superconducting microwave cavity via the Pockels effect. For the highest continuous wave pump power of 1.48 mW we demonstrate bidirectional single-sideband conversion of X band microwave to C band telecom light with a total (internal) efficiency of 0.03 % (0.7 %) and an added output conversion noise of 5.5 photons. The high bandwidth of 10.7 MHz combined with the observed very slow heating rate of 1.1 noise photons s$^{-1}$ puts quantum limited pulsed microwave-optics conversion within reach. The presented device is versatile and compatible with superconducting qubits, which might open the way for fast and deterministic entanglement distribution between microwave and optical fields, for optically mediated remote entanglement of superconducting qubits, and for new multiplexed cryogenic circuit control and readout strategies., Comment: 16 pages, 11 figures

Practical quantum networks require low-loss and noise-resilient optical interconnects as well as non-Gaussian resources for entanglement distillation and distributed quantum computation. The latter could be provided by superconducting circuits but - despite growing efforts and rapid progress - existing solutions to interface the microwave and optical domains lack either scalability or efficiency, and in most cases the conversion noise is not known. In this work we utilize the unique opportunities of silicon photonics, cavity optomechanics and superconducting circuits to demonstrate a fully integrated, coherent transducer connecting the microwave X and the telecom S bands with a total (internal) bidirectional transduction efficiency of 1.2% (135 %) at millikelvin temperatures. The coupling relies solely on the radiation pressure interaction mediated by the femtometer-scale motion of two silicon nanobeams and includes an optomechanical gain of about 20 dB. The chip-scale device is fabricated from CMOS compatible materials and achieves a V$_\pi$ as low as 16 $\mu$V for sub-nanowatt pump powers. Such power-efficient, ultra-sensitive and highly integrated hybrid interconnects might find applications ranging from quantum communication and RF receivers to magnetic resonance imaging., Comment: 26 pages, 13 figures

Despite our familiarity with specific technologies, the origin of new technologies remains mysterious. Are new technologies made from scratch, or are they built up recursively from new combinations of existing technologies? To answer this, we introduce a simple model of recursive innovation in which technologies are made up of components and combinations of components can be turned into new components---a process we call technological recursion. We derive a formula for the extent to which technological recursion increases or decreases the likelihood of making new technologies. We test our predictions on historical data from three domains and find that technologies are not built up from scratch, but are the result of new combinations of existing technologies. This suggests a dynamical process by which known technologies were made and a strategy for accelerating the discovery of new ones., Comment: 6 pages, 4 figures

Microelectromechanical systems and integrated photonics provide the basis for many reliable and compact circuit elements in modern communication systems. Electro-opto-mechanical devices are currently one of the leading approaches to realize ultra-sensitive, low-loss transducers for an emerging quantum information technology. Here we present an on-chip microwave frequency converter based on a planar aluminum on silicon nitride platform that is compatible with slot-mode coupled photonic crystal cavities. We show efficient frequency conversion between two propagating microwave modes mediated by the radiation pressure interaction with a metalized dielectric nanobeam oscillator. We achieve bidirectional coherent conversion with a total device efficiency of up to ~ 60 %, a dynamic range of $2\times10^9$ photons/s and an instantaneous bandwidth of up to 1.7 kHz. A high fidelity quantum state transfer would be possible if the drive dependent output noise of currently $\sim14$ photons$\ \cdot\ $s$^{-1}\ \cdot\ $Hz$^{-1}$ is further reduced. Such a silicon nitride based transducer is in-situ reconfigurable and could be used for on-chip classical and quantum signal routing and filtering, both for microwave and hybrid microwave-optical applications., Comment: 9 pages, 3 figures

We propose an efficient microwave-photonic modulator as a resource for stationary entangled microwave-optical fields and develop the theory for deterministic entanglement generation and quantum state transfer in multi-resonant electro-optic systems. The device is based on a single crystal whispering gallery mode resonator integrated into a 3D microwave cavity. The specific design relies on a new combination of thin-film technology and conventional machining that is optimized for the lowest dissipation rates in the microwave, optical and mechanical domains. We extract important device properties from finite element simulations and predict continuous variable entanglement generation rates on the order of a Mebit/s for optical pump powers of only a few tens of microwatt. We compare the quantum state transfer fidelities of coherent, squeezed and non-Gaussian cat-states for both teleportation and direct conversion protocols under realistic conditions. Combining the unique capabilities of circuit quantum electrodynamics with the resilience of fiber optic communication could facilitate long distance solid-state qubit networks, new methods for quantum signal synthesis, quantum key distribution, and quantum enhanced detection, as well as more power-efficient classical sensing and modulation.

Quantum illumination is a powerful sensing technique that employs entangled signal-idler photon pairs to boost the detection efficiency of low-reflectivity objects in environments with bright thermal noise. The promised advantage over classical strategies is particularly evident at low signal powers, a feature which could make the protocol an ideal prototype for non-invasive biomedical scanning or low-power short-range radar. In this work we experimentally investigate the concept of quantum illumination at microwave frequencies. We generate entangled fields using a Josephson parametric converter to illuminate a room-temperature object at a distance of 1 meter in a free-space detection setup. We implement a digital phase conjugate receiver based on linear quadrature measurements that outperforms a symmetric classical noise radar in the same conditions despite the entanglement-breaking signal path. Starting from experimental data, we also simulate the case of perfect idler photon number detection, which results in a quantum advantage compared to the relative classical benchmark. Our results highlight the opportunities and challenges on the way towards a first room-temperature application of microwave quantum circuits., Comment: 12 pages, 7 figures

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may tame the infinities that arise from quantizing gravity, and dispense with the machinery of the real numbers, which has no direct observational support. Yet despite sophisticated attempts at formulating discrete space, researchers have failed to construct even the simplest geometries. We investigate graphs as the most elementary discrete models of two-dimensional space. We show that if space is discrete, it must be disordered, by proving that all planar lattice graphs exhibit the same taxicab metric as square grids. We give an explicit recipe for growing disordered discrete space by sampling a Boltzmann distribution of graphs at low temperature. We then propose three conditions which any discrete model of Euclidean space must meet: have a Hausdorff dimension of two, support unique straight lines and obey Pythagoras' theorem. Our model satisfies all three, making it the first discrete model in which continuum-like behavior is recovered at large lengths., Comment: 6 pages. 7 figures

Laws and regulations across the United States require that all students, including those in the custody of the juvenile courts, probation services, or other child-serving agencies, receive education services. Too often, because of short lengths of stay and logistical challenges associated with student mobility, educators and juvenile justice administrators do not provide education services to which youth are entitled. Providing high-quality services in juvenile detention is essential. The recommendations offered in this brief are intended to "raise the bar" and do not suggest that "one size fits all" or that all juvenile detention facilities will develop identical programs. Rather, this brief suggests an approach that includes an intake classroom designed with principles of engagement, encouragement, and transition planning followed by a more traditional detention center program is the most logical way to assist youth in reconnecting to school and to ensure that they receive the services to which they are entitled.

We prove that the observable telegraph signal accompanying the bistability in the photon-blockade-breakdown regime of the driven and lossy Jaynes--Cummings model is the finite-size precursor of what in the thermodynamic limit is a genuine first-order phase transition. We construct a finite-size scaling of the system parameters to a well-defined thermodynamic limit, in which the system remains the same microscopic system, but the telegraph signal becomes macroscopic both in its timescale and intensity. The existence of such a finite-size scaling completes and justifies the classification of the photon-blockade-breakdown effect as a first-order dissipative quantum phase transition., Comment: 14 pages, 9 figures

Mechanical systems facilitate the development of a new generation of hybrid quantum technology comprising electrical, optical, atomic and acoustic degrees of freedom. Entanglement is the essential resource that defines this new paradigm of quantum enabled devices. Continuous variable (CV) entangled fields, known as Einstein-Podolsky-Rosen (EPR) states, are spatially separated two-mode squeezed states that can be used to implement quantum teleportation and quantum communication. In the optical domain, EPR states are typically generated using nondegenerate optical amplifiers and at microwave frequencies Josephson circuits can serve as a nonlinear medium. It is an outstanding goal to deterministically generate and distribute entangled states with a mechanical oscillator. Here we observe stationary emission of path-entangled microwave radiation from a parametrically driven 30 micrometer long silicon nanostring oscillator, squeezing the joint field operators of two thermal modes by 3.40(37) dB below the vacuum level. This mechanical system correlates up to 50 photons/s/Hz giving rise to a quantum discord that is robust with respect to microwave noise. Such generalized quantum correlations of separable states are important for quantum enhanced detection and provide direct evidence for the non-classical nature of the mechanical oscillator without directly measuring its state. This noninvasive measurement scheme allows to infer information about otherwise inaccessible objects with potential implications in sensing, open system dynamics and fundamental tests of quantum gravity. In the near future, similar on-chip devices can be used to entangle subsystems on vastly different energy scales such as microwave and optical photons., Comment: 13 pages, 5 figures

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