The real effects of changes in public transport networks are complex and often estimated on the basis of incomplete data, which increases the risk of poor decisions

triply

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Appendix A: Theory 1. Coupled mode theory for optical modes hybridization Using a pair of optical resonances increases the intracavity photon number for a given input optical power, resulting in an enhanced optomechanical interaction. Such an optical doublet is implemented here by evanescently coupling two micro-rings, forming a photonic molecule. Consider two bare optical modes with resonance frequencies ωl and ωr that correspond to the left and right micro-rings, respectively. The two modes couple with a rate J, which yields the Hamiltonian [1] and the equations of motion With Laplace transform, they can be rewritten in frequency domain as 2. Transducer Hamiltonian and input-output relations In terms of bare micro-ring modes, the transducer Hamiltonian is given by Note that the piezo-coupled acoustic mode interacts with only one of the micro-rings. To compute quantities directly related to the experiments, we rewrite the Hamiltonian in terms of the supermodes (Appendix A 1) as Finally, by defining effective optomechanical coupling rates and going into a rotating frame defined by the operator we have in the interaction picture. The time evolution of the bosonic modes can be computed using the Heisenberg-Langevin equations given by We can move to the frequency domain by using the Fourier transform along with the properties We then have 3. Conversion efficiency and added noise For the following analysis, we consider the triply resonant configuration where ω+ −ω− = ωm and the optical pump is on resonance with one of the optical supermodes. First, we consider the anti-Stokes configuration where ∆− = 0 and ∆+ = −ωm. From Eq. A24, we have the input-output relation The equations of motion is given by 4. Estimation of correlated microwave-optical pair generation rates Equations A40 and A43 suggest that entangled pairs of microwave and optical photons can be generated in the Stokes configuration since it gives rise to an effective two-mode squeezing Hamiltonian [3–5]. In order to verify that the HBAR transducer can generate entangled pairs of microwave and optical photons, we compute the on-chip pair generation rate, expected to be equal to the output photon flux from the lower frequency optical mode as in standard spontaneous parametric down-conversion (SPDC) scheme. Integrating over the transduction bandwidth, we have In practice, the pump laser should be filtered for the optical output to be dominated by heralding photons, and other losses from the measurement setup can further reduce the effective rate. For η oc = 7.9×10−5 , κ− = 2π×300 MHz and κm = 2π × 10 MHz, this would corresponds to an on-chip pair generation rate of 190 Hz. Losses in the measurement chain will have to be accounted for to estimate the count rate at the detector. In order to measure entanglement between the microwave and optical fields, the on-chip rate must be higher than the thermal decoherence rate of the acoustic mode κmnth,m(ωm), where the thermal occupancy nth,m(ωm) is given by the Bose-Einstein distribution. For ωm = 2π×3.5 GHz and κm = 2π×10 MHz, the decoherence rate at 800 mK is 43 MHz and 0.5 Hz at 10 mK. Therefore, the transducer needs to be operated in the mixing chamber of the dilution refrigerator, with a pulsed pump to reduce heat load. The total rate of optical heralding is hence gated by the pump duty cycle.

Bidirectional Microwave-optical Transduction Based on Integration: Appendix A: Theory

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency III. DISCUSSION Non-classically correlated microwave-optical photon pairs can be generated through spontaneous parametric down-conversion in our transducer. These correlated photon pairs constitute a key ingredient toward entangling distant quantum processors via the Duan-LukinCirac-Zoller protocol [4, 45]. With the current device parameters, the pair generation rate on chip amounts to 190 Hz, which is well above the thermal decoherence rate for 3.5 GHz at 10 mK as explained in Appendix A 4. However, losses in the measurement setup specific to such an experiment, as well as the gating of the optical pump required to alleviate the effective heat load, are expected to limit further the final heralding rate. Quasi-free-standing structures, such as 1D OMCs, do not readily thermalize and are thus more susceptible to heating effects. Even though our transducer also utilizes a suspended acoustic resonator, it may be feasible to operate at a higher duty cycle than what is presented in Fig. 4. It has been shown that a buffer gas environment can facilitate thermalization of a 1D OMC, at the cost of increased damping of the mechanical breathing mode [46]. On the contrary, the HBAR mode of interest here should be relatively insensitive to such viscous damping, as the acoustic wave propagates predominantly along the longitudinal direction inside the cladding. Therefore, a sample cell affixed to the mixing chamber flange and filled with buffer superfluid helium—an inviscid fluid that is both an excellent thermal conductor and electrical insulator—could potentially improve the heralding rate without hampering the conversion efficiency [47]. The present design is also suited for tasks beyond quantum state transfer. The transducer can be a key component of photonic interconnects for superconducting qubits [7, 8], where laser lights routed through optical fibers are used to encode microwave signals directly inside the dilution refrigerator during qubit readout. It may be challenging for traveling-wave EOMs to attain the requisite half-wave voltage to be competitive in noise performance against conventional all-electrical readout scheme utilizing high-electron-mobility transistor amplifiers [8]. Our triply resonant approach affords a path towards a scalable optically controlled cryogenic waveform generator for both qubit readout and control. The multimode nature of the transducer is particularly suited for frequency-multiplexed dispersive readout. Transduction leveraging multiple HBAR modes is already demonstrated in Fig. 3e. One can additionally utilize line-type instead of point-type dimer coupling for the photonic molecule, which gives rise to a dispersion in the supermode splitting [48]. In combination with the multitude of acoustic overtones, a single transducer can therefore support several spectrally distinguishable triply resonant systems that serve as qubit multiplex channels. Finally, since the transducer requires no superconducting element to function, it may be pertinent for applications in classical microwave photonics. Optimization of design and fabrication should engender further improvement on device performance. The conversion efficiency, proportional to microwave extraction efficiency (Eq. 2), can be improved by replacing AlN with scandium-doped AlN [49], which features a piezoelectric coefficient d33 about five times larger than AlN [50, 51] while preserving CMOS compatibility. There is likewise additional upside on the photonics (Appendix C 1). Bending radiation loss can be significantly reduced by increasing the micro-ring radius r. In particular, intrinsic quality factor exceeding 107 has been demonstrated with an r approximately ten times larger using the same fabrication process [52]. It may be anticipated that the efficiency gain here may be chiefly compensated by the reduction in optomechanical coupling rate g0 ∼ 1/ √ r. Nevertheless, by thinning the waveguide to 200 nm that approaches half of the acoustic wavelength, g0 is expected to remain unchanged. As in the thick-core waveguide fabricated using the photonic Damascene process, such a thin-core waveguide on a subtractive manufacturing platform is expected to exhibit negligible bending loss [53]. Assuming critical coupling conditions (Appendix F), we therefore expect another 1.5- time gain in η tot. Finally, optical insertion loss can be nearly eliminated by employing a spot-size converter to facilitate mode-matching with lensed fiber [54–56]. Combining these improvements, our transducer could achieve a total efficiency η tot = 6 × 10−3 . In conclusion, we have designed, fabricated, and characterized a compact piezo-optomechanical microwaveoptical transducer that integrates wafer-scale, CMOScompatible HBAR and Si3N4 photonic technologies. Free of any superconducting components, this triply resonant transducer attains a bidirectional off-chip photon number conversion efficiency of 1.6 × 10−5 and a bandwidth of 25 MHz, with an input pump power of 21 dBm at room temperature. Multimode transduction leveraging distinct HBAR modes have been demonstrated, suggesting prospects for designing frequency-multiplexed photonic interconnects for superconducting qubits. Realistic design and fabrication improvements may even allow access to experiments in the quantum regime.

Bidirectional Microwave-optical Transduction Based on Integration: Discussion

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency C. Bidirectional microwave-optical transduction We measure the off-chip efficiency η tot of the microwave-to-optical (up-conversion) and optical-tomicrowave (down-conversion) processes as a function of off-chip CW input optical pump power Pin. Compared to the on-chip efficiency η oc, this efficiency also accounts for the loss channels of the microwave and optical ports. First, we study the transducer in the triply resonant configuration. For up-conversion, the optical sideband generated from the microwave input, detuned by ωin = ωm from the pump frequency, is measured via a self-calibrated heterodyne detection method. The off-chip optical output containing both the pump (ωL) and the sideband (ωs) is combined with a local oscillator (LO; ωLO) and detected. By placing the LO at a frequency ωLO = (ωL + ωs) /2 + δ such that the photodetector response is constant over a frequency span of δ, we determine the sideband power relative to the pump. Picking off a fraction of the optical output then enables the determination of its absolute power and, by extension, that of the sideband with a power meter. The measured optical sideband power is compared to that of the microwave input to yield η tot. In the case of down-conversion, the optical input is generated by modulating the phase of the pump with an electrooptic modulator (EOM) driven by a microwave source of frequency ωEOM = ωm. Only one of the resulting sidebands is admitted into the photonic molecule and transduced, as the other is far off-resonance. The converted microwave power is directly probed with an electrical spectrum analyzer (ESA). Summarized in Figs. 3a and c, bidirectional transduction processes corresponding to both the effective beam-splitter (ωL = ω−; anti-Stokes) and two-mode-squeezing (ωL = ω+; Stokes) interactions are investigated. We reach a maximal η tot of −48 dB at 21 dBm pump for each configuration. Knowing the port losses, we estimate an on-chip efficiency η oc = −41 dB (Appendix E). Accounting for the extraction efficiency η ext, we further obtain an internal conversion efficiency η int = 2 × 10−3 from Eq. 2. Furthermore, we deviate from triple resonance to map out the transduction bandwidth. First, for a pump still resonant with one of the optical supermodes, the input microwave frequency ωin and input optical detuning (controlled by EOM drive frequency ωEOM) are varied for up- and down-conversion, respectively. Shown in Fig. 3b and d, transduction leveraging the main transduction mode exhibits a full width at half maximum (FWHM) of 25 MHz. The multimode nature of the transducer is further manifest in Fig. 3e, where the pump is in the beam-splitter configuration but slightly detuned from aˆ−. An additional transduction peak in η tot with a FWHM of 10 MHz is observed around ωin = 2π × 3.165 GHz, which corresponds to another HBAR one FSR away from the main transduction mode ωm = 2π × 3.480 GHz. The engineering degrees of freedom such as cladding and actuator thickness (HBAR FSR), and optical dispersion (supermode splitting as a function of optical wavelength) in the present system offer possibilities for frequencymultiplexed transduction.

Bidirectional Microwave-optical Transduction Based on Integration: C. Bidirectional

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency B. Device characterization We perform metrological characterization to affirm the integrity of the stack. Shown in Fig. 2a, the region of suspended cladding is first identified through optical micrography; the released substrate below leads to a contrast in the image. To further access the layer structure, we use focused ion beam milling to create an opening on the device surface at the suspension site. This opening provides sufficient clearance to directly image the cladding acoustic resonator, pictured in Fig. 2b, confirming the removal of silicon. We then characterize the transducer as an optoelectronic network with one microwave port, two optical ports, and one auxiliary electrical DC port. Each optical port comprises an edge-coupled lensed fiber and on-chip bus waveguide terminated by 1D nanotapers designed for TE-polarized light, yielding a fiber-to-fiber coupling efficiency of −8 dB. For adjusting the frequency splitting between optical supermodes, a bias voltage is applied to the DC port to drive either the piezoelectric actuator [44] or an integrated thermo-optic heater placed in the vicinity of the micro-rings (Appendix C 2). The former affords no additional heat load and is hence cryogenic-compatible, whereas the latter provides an easy alternative for fast room-temperature characterization. The avoided mode crossing characteristic of a photonic molecule is exemplified in Fig. 2c. We access the HBARs through the microwave port. Calibrated high-frequency probes are used to contact the top and bottom electrode pads of the piezoelectric actuator, effectively linking the acoustic resonator with a transmission line. The reflection spectrum (Fig. 2d) reveals predominantly one single series of HBARs with a free spectral range (FSR) of 320 MHz, which corresponds to the acoustic length of the cladding. The transduction acoustic mode at 3.48 GHz exhibits a typical total linewidth of 13 MHz with a microwave extraction efficiency of κex,m/κm ≈ 11%. The microwave response is distinct from that of an identical stack composition with unreleased substrate. There, a periodic envelope corresponding to the cladding modes that are more strongly coupled to the microwave is superimposed over the full-stack HBAR response (Appendix D and Ref. [40]). The absence of these full-stack modes provide another piece of evidence of successful cladding suspension. Finally, we study the device acousto-optic response in the triply resonant configuration, where the transducer effectively operates as a resonant single-sideband modulator. The optical output containing both the pump at the symmetric-mode frequency and generated sideband at the antisymmetric-mode frequency are mixed by a photodetector. Such a beat-note spectrum shown in Fig. 2e displays response peaks aligned with the HBAR frequencies in Fig. 2d, demonstrating three-wave mixing.

Bidirectional Microwave-optical Transduction Based on Integration: B. Device Characterization

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency II. RESULTS A. Physics and design Figure 1 delineates the principle of operation of the present transducer, while the theoretical formalism is detailed in Appendix A. The requisite nonlinear interaction for microwave-optical transduction is a parametric threewave mixing process described by standard optomechanical Hamiltonian Through the bilinear piezoelectric interaction, the acoustic resonance (ˆb) is coupled to the itinerant transmission line mode (cˆin) of the same frequency ωin = ωm ≈ 3.5 GHz. This frequency matches the detuning of the optical pump from the optical resonance (aˆ). Specifically, an optical pump addressing the red (blue) side of the resonance induces an effective beam-splitter (two-modesqueezing) interaction between aˆ and ˆb, as illustrated by the signal flow graph in Fig. 1a (1b). The primary figure of merit for a transducer is its efficiency, defined as the number of output photon number for each input photon. An efficient device necessitates strong nonlinear interaction between the modes of interest, as well as ease of coupling to these modes internal to the device. Our design addresses these two aspects. First, the internal efficiency, depends solely on the three-wave mixing cooperativity C = 4g 2 0n/¯ (κoκm). Here, n¯, κo, and κm denote the photon number in the optical cavity, and the total optical and acoustic linewidths of the transduction modes respectively. The single photon optomechanical coupling rate g0 = − (∂ωa/∂x) xZPF, a product of the optical cavity frequency-pull parameter and zero-point displacement of the acoustic wave. In Eq. 1, the plus and minus signs in the denominator correspond to the scenario where the pump is red- and blue-detuned (“anti-Stokes” and “Stokes” processes), respectively. Using a triply resonant configuration as in Refs. [26–29] enhances the intracavity photon number n¯ for a given off-chip input pump power Pin, or corresponding photon flux n˙ in = Pin/ (ℏωL). The improved photon pumping efficiency can be explicitly seen by considering a “hot” cavity with coupling rate κex,o driven at a detuning ∆ = ±ωm, yielding

Bidirectional Microwave-optical Transduction Based on Integration: A. Physics and Design

This paper is available on arxiv under CC 4.0 license. Authors: (1) Terence Blésin, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (2) Wil Kao, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (3) Anat Siddharth, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL); (4) Alaina Attanasio, OxideMEMS lab, Purdue University; (5) Hao Tian, OxideMEMS lab, Purdue University; (6) Sunil A. Bhave, OxideMEMS lab, Purdue University; (7) Tobias J. Kippenberg, Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL) & Center of Quantum Science and Engineering (EPFL). Table of Links Introduction Results A. Physics and design B. Device characterization C. Bidirectional microwave-optical transduction D. Pulsed transduction Discussion Acknowledgments Appendix A: Theory Appendix B: Fabrication process flow Appendix C: Optical characterization Appendix D: Characterization of acoustic resonances Appendix E: Characterization of bidirectional transduction Appendix F: Optimization of optical extraction efficiency Coherent interconversion between microwave and optical frequencies can serve as both classical and quantum interfaces for computing, communication, and sensing. Here, we present a compact microwave-optical transducer based on monolithic integration of piezoelectric actuators atop silicon nitride photonic circuits. Such an actuator directly couples microwave signals to a high-overtone bulk acoustic resonator defined by the suspended silica cladding of the optical waveguide core, which leads to enhanced electromechanical and optomechanical couplings. At room temperature, this triply resonant piezo-optomechanical transducer achieves an off-chip photon number conversion efficiency of 1.6 × 10−5 over a bandwidth of 25 MHz at an input pump power of 21 dBm. The approach is scalable in manufacturing and, unlike existing electro-optic transducers, does not rely on superconducting resonators. As the transduction process is bidirectional, we further demonstrate synthesis of microwave pulses from a purely optical input. Combined with the capability of leveraging multiple acoustic modes for transduction, the present platform offers prospects for building frequency-multiplexed qubit interconnects and for microwave photonics at large. I. INTRODUCTION Modern data centers have seen rapidly increasing traffic that motivates an overhaul of existing network infrastructures. As optical fibers support nearly lossless transport and high bandwidths, twisted-pair copper deployment has shrunk in favor of optical interconnects. Energy-efficient optical transceivers [1] and optical network architectures [2] are being explored in parallel to accommodate emerging data- and resource-intensive applications. In an analogous fashion, processing quantum information in superconducting circuits and networking via photonic interconnects has been envisioned as an effective strategy to address the scalability challenges in advancing quantum technologies [3–5]. The scheme, featuring the full universal set of microwave quantum gates [6] with vanishing thermal occupancy and loss of the optical channels, calls for the development of microwaveoptical transducers to bridge the two energy scales that differ by more than four orders of magnitude. Aside from facilitating the networking of remote quantum processors, these transducers may enable fully optical control and readout of microwave qubits [7, 8]. The subsequent replacement of coaxial lines bridging room temperature and cryogenic environments by optical fibers is expected to significantly ease the space and heat load constraints in dilution refrigerators, opening up a path toward upscaling processor units housed in a single fridge. Efficient frequency conversion requires a nonlinear interaction stronger than the coupling to loss channels. The highest conversion efficiency to date has been achieved using an electro-optomechanical approach [9]. There, a silicon nitride membrane interacts with an optical mode of a free-space Fabry-Pérot cavity via radiation pressure and simultaneously serves as the top plate of a capacitor, parametrically coupling the mechanics to the microwave resonator. By virtue of the high resonator quality factors and pump power handling capability, 47% of input photons can be interconverted between optical and microwave domains [10], approaching the 50% efficiency required for attaining finite quantum capacity [11]. This highly efficient transducer has enabled optical dispersive qubit readout with negligible excess backaction [12]. The low-frequency (MHz) mechanical intermediary nevertheless limits the transduction bandwidth and leads to appreciable added noise even at dilution refrigerator temperature. To address these drawbacks, piezo-optomechanical transducers based on optomechanical crystals (OMC) have been developed [13–22], where a tightly confined high-frequency (GHz) mechanical mode and a co-localized optical mode can interact at a vacuum optomechanical coupling rate g0 ∼ 2π × 500 kHz. The trade-off lies instead in the sophistication required for microwave-phonon wave matching that significantly increases design complexity, as well as thermo-optic instability that constrains the intra-cavity photon number. An on-chip efficiency of 5% has recently been reported on such a platform [20]. However, low thermal conductance of these suspended quasi-one-dimensional structures hinders correlated microwave-optical photon pair generation at a practical rate. Cavity electro-optic modulators constitute a conceptually simpler approach where Pockels effect directly mediates microwave-optical interaction [23–25]. On-chip realizations employing planar superconducting microwave resonators, benefited from the deep sub-wavelength mode volume of the vacuum electric field, to reach single photon electro-optic coupling rates g0 ∼ 2π × 1 kHz [26– 29]. However, the material science of χ (2) crystals poses additional challenges. Photorefractive effects—observed, for instance, in LiNbO3 thin films [30]—hamper optical power handling, while piezoelectric loss and scattered optical photons degrade the quality factor of superconducting resonators [31, 32]. The difficulty of producing smooth sidewall surfaces in the workhorse Pockels material, LiNbO3 , through dry etching results in propagation losses an order of magnitude above the absorption limit [33]. As a result, the maximum on-chip efficiency achieved with integrated electro-optic transducers exceeds just 2% [26]. In the spirit of Ref. [9], a bulk transducer comprising a mm-size mechanically polished LiNbO3 whispering gallery mode (WGM) resonator coupled to a 3D superconducting microwave cavity has proved competitive, trading g0 for improved power handling and quality factors [34, 35]. Taking one step further with pulsed optical pumping, the system has demonstrated not only a hallmark 14.4% total efficiency but also electro-optic dynamical backaction [36, 37] and microwave-optical quadrature entanglement [38]. With the aforementioned design trade-offs in mind, we present a new piezo-optomechanical transducer based solely on wafer-scale, CMOS-compatible fabrication processes. We utilize a low-loss silicon nitride (Si3N4 ) photonic molecule, as well as multiple GHz high-overtone bulk acoustic resonances (HBAR) parametrically coupled to the optical modes. Endowed with power handling capabilities superior to existing integrated transducers, the device fits compactly within a 100 µm-by-50 µm footprint, in contrast to state-of-the-art bulk designs [10, 36]. In addition, the transduction HBAR modes are more readily coupled to the microwave signal, unlike OMC transducers. We demonstrate bidirectional microwave-optical transduction with a bandwidth of 25 MHz and total efficiency up to 1.6 × 10−5 by pumping with 21 dBm of off-chip optical power in continuous-wave (CW) operation. The device is also characterized with a pulsed optical pump, which constitutes the first step toward optical control and readout of qubits, as well as heralded microwave-optical photon pair generation. The simple design, ease of fabrication, robust operation, and compact form factor anticipate wide applicability in quantum technologies and microwave photonics at large.

Bidirectional Microwave-optical Transduction Based on Integration: Introduction

Christian Science by Mark Twain, is part of the HackerNoon Books Series. You can jump to any chapter in this book . THE CHRISTIAN SCIENCE PASTOR-UNIVERSAL here THE CHRISTIAN SCIENCE PASTOR-UNIVERSAL When Mrs. Eddy turned the pastors out of all the Christian Science churches and abolished the office for all time as far as human occupancy is concerned—she appointed the Holy Ghost to fill their place. If this language be blasphemous, I did not invent the blasphemy, I am merely stating a fact. I will quote from page 227 of Science and Health (edition 1899), as a first step towards an explanation of this startling matter—a passage which sets forth and classifies the Christian Science Trinity: “Life, Truth, and Love constitute the triune God, or triply divine Principle. They represent a trinity in unity, three in one—the same in essence, though multiform in office: God the Father; Christ the type of Sonship; Divine Science, or the Holy Comforter.... “The Holy Ghost, or Spirit, reveals this triune Principle, and (the Holy Ghost) is expressed in Divine Science, which is the Comforter, leading into all Truth, and revealing the divine Principle of the universe—universal and perpetual harmony.” I will cite another passage. Speaking of Jesus— “His students then received the Holy Ghost. By this is meant, that by all they had witnessed and suffered they were roused to an enlarged understanding of Divine Science, even to the spiritual interpretation.. ... of His teachings,” etc. Also, page 579, in the chapter called the Glossary: “HOLY GHOST. Divine Science; the developments of Life, Truth, and Love.” The Holy Ghost reveals the massed spirit of the fused trinity; this massed spirit is expressed in Divine Science, and is the Comforter; Divine Science conveys to men the “spiritual interpretation” of the Saviour's teachings. That seems to be the meaning of the quoted passages. Divine Science is Christian Science; the book “Science and Health” is a “revelation” of the whole spirit of the Trinity, and is therefore “The Holy Ghost”; it conveys to men the “spiritual interpretation” of the Bible's teachings and therefore is “the Comforter.” I do not find this analyzing work easy, I would rather saw wood; and a person can never tell whether he has added up a Science and Health sum right or not, anyway, after all his trouble. Neither can he easily find out whether the texts are still on the market or have been discarded from the Book; for two hundred and fifty-eight editions of it have been issued, and no two editions seem to be alike. The annual changes—in technical terminology; in matter and wording; in transpositions of chapters and verses; in leaving out old chapters and verses and putting in new ones—seem to be next to innumerable, and as there is no index, there is no way to find a thing one wants without reading the book through. If ever I inspire a Bible-Annex I will not rush at it in a half-digested, helter-skelter way and have to put in thirty-eight years trying to get some of it the way I want it, I will sit down and think it out and know what it is I want to say before I begin. An inspirer cannot inspire for Mrs. Eddy and keep his reputation. I have never seen such slipshod work, bar the ten that interpreted for the home market the “sell all thou hast.” I have quoted one “spiritual” rendering of the Lord's Prayer, I have seen one other one, and am told there are five more. Yet the inspirer of Mrs. Eddy the new Infallible casts a complacent critical stone at the other Infallible for being unable to make up its mind about such things. Science and Health, edition 1899, page 33: “The decisions, by vote of Church Councils, as to what should and should not be considered Holy Writ, the manifest mistakes in the ancient versions: the thirty thousand different readings in the Old Testament and the three hundred thousand in the New—these facts show how a mortal and material sense stole into the divine record, darkening, to some extent, the inspired pages with its own hue.” To some extent, yes—speaking cautiously. But it is nothing, really nothing; Mrs. Eddy is only a little way behind, and if her inspirer lives to get her Annex to suit him that Catholic record will have to “go 'way back and set down,” as the ballad says. Listen to the boastful song of Mrs. Eddy's organ, the Christian Science Journal for March, 1902, about that year's revamping and half-soling of Science and Health, whose official name is the Holy Ghost, the Comforter, and who is now the Official Pastor and Infallible and Unerring Guide of every Christian Science church in the two hemispheres, hear Simple Simon that met the pieman brag of the Infallible's fallibility: “Throughout the entire book the verbal changes are so numerous as to indicate the vast amount of time and labor Mrs. Eddy has devoted to this revision. The time and labor thus bestowed is relatively as great as that of—the committee who revised the Bible.... Thus we have additional evidence of the herculean efforts our beloved Leader has made and is constantly making for the promulgation of Truth and the furtherance of her divinely bestowed mission,” etc. It is a steady job. I could help inspire if desired; I am not doing much now, and would work for half-price, and should not object to the country. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Mark Twain (2004). Christian Science. Urbana, Illinois: Project Gutenberg. Retrieved October 2022 https://www.gutenberg.org/cache/epub/3187/pg3187-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at , located at . www.gutenberg.org https://www.gutenberg.org/policy/license.html

THE CHRISTIAN SCIENCE PASTOR-UNIVERSAL

Tarzan and the Jewels of Opar by Edgar Rice Burroughs, is part of the HackerNoon Books Series. You can jump to any chapter in this book here . La Seeks Vengeance La Seeks Vengeance Swinging back through the jungle in a wide circle the ape-man came to the river at another point, drank and took to the trees again and while he hunted, all oblivious of his past and careless of his future, there came through the dark jungles and the open, parklike places and across the wide meadows, where grazed the countless herbivora of the mysterious continent, a weird and terrible caravan in search of him. There were fifty frightful men with hairy bodies and gnarled and crooked legs. They were armed with knives and great bludgeons and at their head marched an almost naked woman, beautiful beyond compare. It was La of Opar, High Priestess of the Flaming God, and fifty of her horrid priests searching for the purloiner of the sacred sacrificial knife. Never before had La passed beyond the crumbling outer walls of Opar; but never before had need been so insistent. The sacred knife was gone! Handed down through countless ages it had come to her as a heritage and an insignia of her religious office and regal authority from some long-dead progenitor of lost and forgotten Atlantis. The loss of the crown jewels or the Great Seal of England could have brought no greater consternation to a British king than did the pilfering of the sacred knife bring to La, the Oparian, Queen and High Priestess of the degraded remnants of the oldest civilization upon earth. When Atlantis, with all her mighty cities and her cultivated fields and her great commerce and culture and riches sank into the sea long ages since, she took with her all but a handful of her colonists working the vast gold mines of Central Africa. From these and their degraded slaves and a later intermixture of the blood of the anthropoids sprung the gnarled men of Opar; but by some queer freak of fate, aided by natural selection, the old Atlantean strain had remained pure and undegraded in the females descended from a single princess of the royal house of Atlantis who had been in Opar at the time of the great catastrophe. Such was La. Burning with white-hot anger was the High Priestess, her heart a seething, molten mass of hatred for Tarzan of the Apes. The zeal of the religious fanatic whose altar has been desecrated was triply enhanced by the rage of a woman scorned. Twice had she thrown her heart at the feet of the godlike ape-man and twice had she been repulsed. La knew that she was beautiful—and she was beautiful, not by the standards of prehistoric Atlantis alone, but by those of modern times was La physically a creature of perfection. Before Tarzan came that first time to Opar, La had never seen a human male other than the grotesque and knotted men of her clan. With one of these she must mate sooner or later that the direct line of high priestesses might not be broken, unless Fate should bring other men to Opar. Before Tarzan came upon his first visit, La had had no thought that such men as he existed, for she knew only her hideous little priests and the bulls of the tribe of great anthropoids that had dwelt from time immemorial in and about Opar, until they had come to be looked upon almost as equals by the Oparians. Among the legends of Opar were tales of godlike men of the olden time and of black men who had come more recently; but these latter had been enemies who killed and robbed. And, too, these legends always held forth the hope that some day that nameless continent from which their race had sprung, would rise once more out of the sea and with slaves at the long sweeps would send her carven, gold-picked galleys forth to succor the long-exiled colonists. The coming of Tarzan had aroused within La’s breast the wild hope that at last the fulfillment of this ancient prophecy was at hand; but more strongly still had it aroused the hot fires of love in a heart that never otherwise would have known the meaning of that all-consuming passion, for such a wondrous creature as La could never have felt love for any of the repulsive priests of Opar. Custom, duty and religious zeal might have commanded the union; but there could have been no love on La’s part. She had grown to young womanhood a cold and heartless creature, daughter of a thousand other cold, heartless, beautiful women who had never known love. And so when love came to her it liberated all the pent passions of a thousand generations, transforming La into a pulsing, throbbing volcano of desire, and with desire thwarted this great force of love and gentleness and sacrifice was transmuted by its own fires into one of hatred and revenge. It was in a state of mind superinduced by these conditions that La led forth her jabbering company to retrieve the sacred emblem of her high office and wreak vengeance upon the author of her wrongs. To Werper she gave little thought. The fact that the knife had been in his hand when it departed from Opar brought down no thoughts of vengeance upon his head. Of course, he should be slain when captured; but his death would give La no pleasure—she looked for that in the contemplated death agonies of Tarzan. He should be tortured. His should be a slow and frightful death. His punishment should be adequate to the immensity of his crime. He had wrested the sacred knife from La; he had lain sacrilegious hands upon the High Priestess of the Flaming God; he had desecrated the altar and the temple. For these things he should die; but he had scorned the love of La, the woman, and for this he should die horribly with great anguish. The march of La and her priests was not without its adventures. Unused were these to the ways of the jungle, since seldom did any venture forth from behind Opar’s crumbling walls, yet their very numbers protected them and so they came without fatalities far along the trail of Tarzan and Werper. Three great apes accompanied them and to these was delegated the business of tracking the quarry, a feat beyond the senses of the Oparians. La commanded. She arranged the order of march, she selected the camps, she set the hour for halting and the hour for resuming and though she was inexperienced in such matters, her native intelligence was so far above that of the men or the apes that she did better than they could have done. She was a hard taskmaster, too, for she looked down with loathing and contempt upon the misshapen creatures amongst which cruel Fate had thrown her and to some extent vented upon them her dissatisfaction and her thwarted love. She made them build her a strong protection and shelter each night and keep a great fire burning before it from dusk to dawn. When she tired of walking they were forced to carry her upon an improvised litter, nor did one dare to question her authority or her right to such services. In fact they did not question either. To them she was a goddess and each loved her and each hoped that he would be chosen as her mate, so they slaved for her and bore the stinging lash of her displeasure and the habitually haughty disdain of her manner without a murmur. For many days they marched, the apes following the trail easily and going a little distance ahead of the body of the caravan that they might warn the others of impending danger. It was during a noonday halt while all were lying resting after a tiresome march that one of the apes rose suddenly and sniffed the breeze. In a low guttural he cautioned the others to silence and a moment later was swinging quietly up wind into the jungle. La and the priests gathered silently together, the hideous little men fingering their knives and bludgeons, and awaited the return of the shaggy anthropoid. Nor had they long to wait before they saw him emerge from a leafy thicket and approach them. Straight to La he came and in the language of the great apes which was also the language of decadent Opar he addressed her. “The great Tarmangani lies asleep there,” he said, pointing in the direction from which he had just come. “Come and we can kill him.” “Do not kill him,” commanded La in cold tones. “Bring the great Tarmangani to me alive and unhurt. The vengeance is La’s. Go; but make no sound!” and she waved her hands to include all her followers. Cautiously the weird party crept through the jungle in the wake of the great ape until at last he halted them with a raised hand and pointed upward and a little ahead. There they saw the giant form of the ape-man stretched along a low bough and even in sleep one hand grasped a stout limb and one strong, brown leg reached out and overlapped another. At ease lay Tarzan of the Apes, sleeping heavily upon a full stomach and dreaming of Numa, the lion, and Horta, the boar, and other creatures of the jungle. No intimation of danger assailed the dormant faculties of the ape-man—he saw no crouching hairy figures upon the ground beneath him nor the three apes that swung quietly into the tree beside him. The first intimation of danger that came to Tarzan was the impact of three bodies as the three apes leaped upon him and hurled him to the ground, where he alighted half stunned beneath their combined weight and was immediately set upon by the fifty hairy men or as many of them as could swarm upon his person. Instantly the ape-man became the center of a whirling, striking, biting maelstrom of horror. He fought nobly but the odds against him were too great. Slowly they overcame him though there was scarce one of them that did not feel the weight of his mighty fist or the rending of his fangs. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Edgar Rice Burroughs (1993). Tarzan and the Jewels of Opar. Urbana, Illinois: Project Gutenberg. Retrieved October 2022 https://www.gutenberg.org/cache/epub/92/pg92-images.html This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org , located at https://www.gutenberg.org/policy/license.html.

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