A clamped transducer arrangement for quantum level transduction and method for fabrication thereof

Abstract

The disclosure relates to a transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwaves to mechanical waves, and vice versa; b) transduction from mechanical waves to optical waves, and vice versa; c) transduction from microwaves to optical waves, via mechanical waves, and vice versa; said arrangement further comprising: a transducer device (EMT; OMT; hybrid EMT-OMT) having a region which is either optomechanically or electromechanically active, or both, for said transduction; and a substrate The transducer device (EMT; OMT; hybrid EMT-OMT) is clamped to the substrate in a manner so that said active region has at least one side in full contact with said substrate.

Description

[0001] A TRANSDUCER ARRANGEMENT AND A METHOD FOR FABRICATING SAID TRANSDUCER ARRANGEMENT

[0002] FIELD OF THE INVENTION

[0003] The present disclosure relates to a transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwaves to mechanical waves, and vice versa; b) transduction from mechanical waves to optical waves, and vice versa; and c) transduction from microwaves to optical waves, via mechanical waves, and vice versa. The arrangement comprises a transducer device having a region which is either optomechanically or electromechanically active, or both, for said transduction, and a substrate.

[0004] The present disclosure also relates to a method for manufacturing a transducer arrangement.

[0005] BACKGROUND OF THE INVENTION

[0006] In the field of information processing, it is known that signals in the microwave and optical domains can be transduced both in quantum and classical applications. For this type of wave conversion, there is in particular a requirement for high transduction efficiency, low pump power, low dissipated energy per (qu)bit, and often high entanglement rate and low thermal noise.

[0007] Several types of microwave-to-optical wave transduction principles are previously known, for example electro-optomechanical conversion, in which mechanical waves can be used as an intermediate process for transduction between the microwaves and the optical waves. Various types of arrangements have been described in which conversion of waves is obtained from electrical signals at high frequencies (i.e. microwave photons) via mechanical signals (based on phonons) to optical signals (based on optical photons), and vice versa.

[0008] A number of previously known documents describe different types of the above- mentioned electromechanical and / or optomechanical transduction. In particular, the following publications can be mentioned: Zhang, J. et al. Subwavelength Control of Photons and Phonons in Release-Free Silicon Optomechanical Resonators. ACS Photonics 9, 3855-3862 (2022);

[0009] Ma, X., Shandilya, P. K. & Barclay, P. E. Semiconductor-on-diamond cavities for spin optomechanics. Preprint at https: / / doi.org / 10.48550 / arXiv.2302.04967 (2023);

[0010] Painter, O. et al. Techniques for transduction and storage of quantum level signals. US Patent (2020);

[0011] Jiang, W. et al. Optically heralded microwave photons. Preprint at https: / / doi.org / 10.48550 / arXiv.2210.10739 (2022); and

[0012] Sarabalis, C. J., Dahmani, Y. D., Patel, R. N., Hill, J. T. & Safavi-Naeini, A. H. Release- free silicon-on-insulator cavity optomechanics. Optica l, 1147-1150 (2017).

[0013] Also, certain patent documents relate to the above-mentioned field of technology, for example the documents US 2021 / 0278745 and US 9454061. However, further improvements high transduction efficiency, low pump power, low dissipated energy per (qu)bit, high entanglement rate and often low thermal noise are desired. Also, improvements regarding efficient manufacturing processes and an improved level of cointegration of optimal combinations of electromechanical and / or optomechanical materials should be obtained.

[0014] For this reason, there is a requirement for improved devices and methods in which the drawbacks of the prior can be overcome and in which certain advantages can be obtained.

[0015] SUMMARY OF THE INVENTION

[0016] In accordance with the disclosure, there is provided an improved arrangement for transducing signals, and an improved method for manufacturing said arrangement, by means of which one or more of the drawbacks of known technology within this field can be overcome.

[0017] For this reason, and in accordance with the disclosure, there is provided a transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwaves to mechanical waves, and vice versa; b) transduction from mechanical waves to optical waves, and vice versa; c) transduction from microwaves to optical waves, via mechanical waves, and vice versa. Furthermore, the arrangement comprises a transducer device (EMT; OMT; hybrid EMT- OMT) having a region which is either optomechanically or electromechanically active, or both, for said transduction; and a substrate. Furthermore, the transducer device is clamped to the substrate in a manner so that said active region has at least one side in full contact with said substrate.

[0018] According to an aspect of the disclosure, the a transducer device is either an electromechanical transducer (EMT) (for said transduction from microwaves to mechanical waves), an optomechanical transducer (OMT) (for said transduction from mechanical waves to optical waves) or an electro-optomechanical transducer, i.e. a hybrid (hybrid EMT-OMT) (for said transduction from microwaves to optical waves via mechanical waves).

[0019] In the context of the disclosure, the active region is the physical region of space where either the optomechanical interaction and / or the electromechanical interaction takes place. Generally speaking, it is the region where there is significant mechanics; often defined by a 1 / e drop in mechanical energy. The active region in the embodiment which is shown in fig. 4.1 corresponds to the region where there is non-negligible mechanical motion, which is roughly the 5-7 unit cells in the center of the structure in this case. Similar principle to find the active region applies to the OMT and the hybrid EMT-OMTs.

[0020] In some embodiments, suspended EMT active regions could be combined with clamped OMT active regions in the microwave-to-optics transduction mode. In other embodiments, clamped EMT active regions could be combined with suspended OMC active regions.

[0021] As regards relevant dimensions, the disclosure is focused on chip based devices.

[0022] Certain advantages are achieved by means of the arrangement according to the disclosure, for example high transduction efficiency, low pump power, low dissipated energy per (qu)bit, high entanglement rate and often low thermal noise. Also, the arrangement according to the disclosure provides advantages relating to a more flexible and optimal process for manufacturing. In fact, the arrangement can be configured so as to co-localize near-infrared optical waves and gigahertz mechanical waves into fully clamped structures on the surface of a chip.

[0023] In particular, the fact that the transducer device is clamped to its substrate increases their thermal contact area dramatically and improves prospects for dense co-integration of electronic, mechanical, and optical components. In particular, it can be mentioned that the arrangement is configured so as to confine mechanical modes without suspending the devices and thus realizing a much higher thermal contact area for undesired heat-carrying phonons to escape from the device. According to an aspect, the devices are fully clamped on at least their bottom side to their substrate.

[0024] According to an embodiment, there is provided a transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwaves to mechanical waves, and vice versa; b) transduction from mechanical waves to optical waves, and vice versa; c) transduction from microwaves to optical waves, via mechanical waves, and vice versa; said arrangement further comprising: a transducer device (EMT; OMT; hybrid EMT-OMT) having a region which is either optomechanically or electromechanically active, or both, for said transduction; and a substrate; wherein the transducer device (EMT; OMT; hybrid EMT-OMT) is nonsuspended from the underlying substrate and clamped to the substrate in a manner so that said active region has at least one side in full contact with said substrate, thereby defining a thermal contact area between said active region and said substrate; and that the transducer device (EMT; OMT; hybrid EMT-OMT) is based on a design in which the mechanical modes' wavevectors and frequencies are fully outside the continuum formed by surface and bulk acoustic waves in the substrate.

[0025] Alternatively, and more generally, it can be said that the above-mentioned transducer device (EMT; OMT; hybrid EMT-OMT) is based on a design in which the mechanical modes' wavevectors and frequencies are fully outside the continuum formed by acoustic excitations associated with the substrate, including surface and bulk acoustic waves.

[0026] According to an embodiment, the transducer device comprises a periodically repeated structure in the form of a plurality of unit cells, each unit cell having an internal closed volume configured for co-localizing said mechanical and / or optical waves to said active region. Regarding possible dimensions, it can be noted that there are several different types of cells with different functions which all have different dimensions. For instance, the cell in the center of the OMT is designed so it supports mechanical and optical waves and allows coupling between the two. The cell at the end of the OMT is designed so it reflects mechanical and optical waves. Similarly, the center cell of the EMT is designed so that it supports a mechanical mode and can be coupled to a microwave mode. More precise dimensions of various embodiments are given in the detailed description.

[0027] According to an embodiment, each unit cell is manufactured from either optomechanically or electromechanically active materials, having said volume extending through said unit cell.

[0028] In the context of this disclosure, optomechanically active materials are defined as generally crystalline, high refractive index materials that can be fabricated in high quality at a large scale. These must support non-zero coupling strength between optics and mechanics. In an embodiment, silicon is such a material. Similarly, electromechanically active materials are preferably also crystalline but do not need high refractive index, instead strong electromechanical effect is required. For the EMT active region, it should be manufactured from electromechanically active materials that support non-zero electromechanical coupling strength between microwaves and mechanics. In one embodiment, the electromechanically active material is lithium niobate (LN).

[0029] With regard to possible materials for the substrate, this should be materials that can be integrated well with the material of the active region. Depending on the application, it can be helpful that the material be conducting heat well. Other preferable traits are low losses for mechanical, electrical and optical fields at room temperature and / or at cryogenic temperatures depending on the application. One attractive such candidate is sapphire, other examples are diamond. There is also the possibility for more complex combinations of thin-films to form a composite substrate that has optimal properties for losses and performances of EMTs, OMTs and hybrid EMT-OMTs.

[0030] According to an embodiment, said plurality of unit cells are configured either as a 2D- confined waveguide or as a 3-D defined cavity. According to an embodiment, the structure is configured for providing a desired mechanical mode with a wavevector and frequency that differs from the wavevectors and frequencies of any mechanical mode found in the continuum of mechanical modes associated with the substrate or cladding.

[0031] Furthermore, according to an embodiment, said structure is configured for providing a desired mechanical mode with a wavevector and frequency that differs from the wavevectors and frequencies of any mechanical mode found in the continuum of mechanical modes associated with the substrate or cladding by means of a process of exploring unit cells having a relatively low cell period which results in a mechanical mode which differs from a mode corresponding to said continuum.

[0032] According to an embodiment, the unit cell period of the structure is decreased, and the surface-to-volume ratio is increased, until the lowest frequency modes associated with the substrate (often surface acoustic waves) become higher in frequency than the desired operating point for the mechanical frequency.

[0033] Furthermore, the term “continuum” is defined by all the vibrational modes in which energy is not confined to the waveguide / cavity (i.e. waves in the substrate). If the waveguide / cavity vibrates in a mode with a wavevector that differs from any in the continuum, conservation of momentum dictates that there is no way for the mechanical energy to leak out into vibrations in the substrate.

[0034] The terms “mechanical mode”, “optical mode” and “microwave mode” are generally understood by those skilled in the art.

[0035] According to an embodiment, the transducer device is an optomechanical transducer (OMT).

[0036] According to an embodiment, the the optomechanical transducer (OMT) comprises a transducer device () which is manufactured from silicon and a substrate () which is manufactured from for example sapphire or silicon dioxide. As further alternatives, the substrate can also be made from diamond, silicon-nitride, silicon, lithium niobate and GaP. According to further embodiments, other material combinations may be relevant, for example silicon-on-sapphire (SOS). This material combination is predicted to be superior to silicon-on-silicon dioxide due to the crystallinity of the sapphire as opposed to silicon dioxides amorphous structure.

[0037] According to an embodiment, the transducer device is an electromechanical transducer (EMT).

[0038] According to an embodiment, the the electromechanical transducer (EMC) comprises a transducer device which is manufactured from a metal electrode layer such as NbTiN, a mechanically active crystalline layer such as silicon, and electromechanically active layer such as piezoelectric lithium niobate (LiNbO3) or DC-biased electrostrictive materials such as silicon and a substrate which is manufactured from for example sapphire (AI2O3) or silicon dioxide (SiO2).

[0039] As further alternatives, the above-mentioned metal electrode layer may also be of aluminum, TiN, NbN, Ta or Al.

[0040] As further alternatives, the above-mentioned substrate may also be manufactured from diamond or silicon-nitride.

[0041] As further alternatives, the above-mentioned electromechanically active layer can also be formed by lithium niobate (LN), lithium tantalate, gallium nitride, GaAs, silicon, crystalline GaAs, GaP, crystalline lithium niobate, aluminum nitride (AIN), lead zirconium titanate (PZT), Lead Magnesium Niobate-lead Titanate, indium phosphide etc.

[0042] According to an embodiment, the transducer device is a hybrid electrooptical transducer (hybrid EMT-OMT), using mechanical waves as an intermediary.

[0043] According to an embodiment, the arrangement is configured for transduction at the quantum level, involving interaction between photons, corresponding to quantized units of electromagnetic radiation based on said microwave signals and optical signals; and phonons, corresponding to quantized units of mechanical vibrations based on said mechanical waves. According to a further embodiment, an arrangement is provided which is configured for either: transduction from mechanical waves to optical waves, and vice versa; or transduction from microwaves to optical waves, via mechanical waves, and vice versa; wherein the arrangement is further configured for operating in a counter-propagating or inter-modal scattering regime where the high-wavevector mechanical mode interacts with two optical waves with opposite and / or sufficiently different wavevectors such that the optomechanical three-wave-mixing satisfies both frequency- and phase-matching conditions.

[0044] In accordance with the disclosure, there is also provided a method for manufacturing a transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwave waves to acoustic waves, and vice versa; b) transduction from acoustic waves to optical waves, and vice versa; c) transduction from microwave waves to optical waves, via acoustic waves, and vice versa; said method comprising: providing a transducer device (EMT; OMT; hybrid EMT-OMT) having an active region which is either optomechanically or electromechanically active, or both, for said transduction; and providing a substrate. The method further comprises a a process for transfer of various combinations of thin-film layers of optomechanically and electromechanically active materials like silicon and lithium niobate so as to clamp said transducer (EMT ; OMT ; hybrid EMT-OMT) to the substrate in a manner so that said active region which is either optomechanically or electromechanically active, or both, has at least one side in full contact with said substrate. The method also comprises a process for assembling material layers, either the active transducers (EMT; OMT; hybrid EMT-OMT) thin-films or composite substrates and claddings, through repeated micro-transfer printing.

[0045] According to an embodiment, there is provided a method for manufacturing a transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwave waves to acoustic waves, and vice versa; b) transduction from acoustic waves to optical waves, and vice versa; c) transduction from microwave waves to optical waves, via acoustic waves, and vice versa; said method comprising: providing a transducer device (EMT; OMT) having an active region which is either optomechanically or electromechanically active, or both, for said transduction; and providing a substrate; wherein the method further comprises a process for transfer of various combinations of thin-film layers of optomechanically and electromechanically active materials so as to clamp said transducer (EMT; OMT; hybrid EMT-OMT) to the substrate in a manner so that it is non-suspended from the underlying substrate so that said active region which is either optomechanically or electromechanically active, or both, has at least one side in full contact with said substrate, thereby defining a thermal contact area between said active region and said substrate; and so that the transducer device (EMT ; OMT ; hybrid EMT-OMT) is based on a design in which the mechanical modes' wavevectors and frequencies are fully outside the continuum formed by surface and bulk acoustic waves in the substrate, comprising a process for assembling material layers, either the active EMT or OMT thin-films or composite substrates and claddings, through repeated micro-transfer printing.

[0046] Alternatively, and more generally, it can be said that the above-mentioned method is based on the fact that the mechanical modes' wavevectors and frequencies are fully outside the continuum formed by acoustic excitations associated with the substrate, including surface and bulk acoustic waves.

[0047] Certain terms and definitions are used below. The terms “mechanical mode”, “optical mode” and “microwave mode” are generally understood by those skilled in the art. With a number of distinct modes of vibration, one can mathematically construct any complex vibration the volume can exhibit. The different modes have different frequency and wavevector, where a wavevector is defined as wavevector = 2*pi / wavelength. A mechanical mode can consequently be regarded as a particular specific vibrational pattern or mode of motion in a material, for example a crystal. In this manner, it can be said to relate to the way in which mechanical energy is stored and transferred within the material. Furthermore, the mechanical mode can be confined in a device, as will be described below. Also, a wavevector of a mechanical mode can be used to describe the properties of mechanical waves in different materials.

[0048] BRIEF DESCRIPTION OF THE DRAWINGS

[0049] The disclosure will be described in greater detail below with reference to the figures shown in the appended drawings.

[0050] Figure A shows, in a schematical manner, a process for electromechanical transduction; Figure B shows, in a schematical manner, a process for optomechanical transduction;

[0051] Figure C shows, in a schematical manner, a process for electro- optomechanical transduction;

[0052] Figure 1 shows a phase-matching diagram for clamped optomechanical devices;

[0053] Figure 2 shows phase-matching for counter-propagating optomechanical interactions within standing-wave OMCs;

[0054] Figures 3.1 , 3.2 show optical and mechanical band diagrams of the unit cell for a SiO2 (3.1 ) and a sapphire (3.2, AI2O3) substrate according to the disclosure;

[0055] Figures 4.1 , 4.2 show a short OMC with SiO2 substrate (4.1) and a long OMC with sapphire (4.2, AI2O3) substrate along with the Fourier spectra of their optical and mechanical modes; Also shows the cross-section of the devices;

[0056] Figure 5 shows the physical performance of an example clamped OMC;

[0057] Figure 6 shows a clamped EMC example using electromechanically active lithium niobate on silicon with a crystalline sapphire substrate along with its band diagrams and evolution of bands along the EMC;

[0058] Figure 7.1 shows a partial mirror example for a clamped OMC on sapphire substrate such that mechanical energy can be guided out of the OMC while keeping optical energy inside the OMC along with the band diagrams;

[0059] Figure 7.2 shows group velocities for clamped defect and partial mirror cells as function of mechanical wavevector; Figure 8 shows logarithmic mechanical and optical field in an example hybrid

[0060] EMC-OMC using lithium niobate and silicon on a crystalline sapphire (AI2O3) substrate where microwave-to-optical transduction takes place; Also shows the associated electromechanical and optomechanical coupling rates;

[0061] Figure 9 shows process steps for fabrication of an EMC and / or OMC device; in this case using foundry-ready wafer for two of the thin-films involved where one of the thin-films is transferred to the other stack;

[0062] Figure 10 shows process steps for fabrication of an EMC and / or OMC device, in this case using combined transfer of several electromechanically and / or optomechanically active thin-films from native substrates to a new substrate such as crystalline sapphire or diamond;

[0063] Figure 11 shows a diagram for clamped and counter-propagating optomechanical interactions;

[0064] Figure 12 shows band diagrams for an OMC cell;

[0065] Figure 13 shows optical mode profiles for a clamped OMC;

[0066] Figure 14 shows principles for fabrication and measurement of an OMC;

[0067] Figure 15 shows an experimental setup; and

[0068] Figure 16 shows a power spectrum diagram.

[0069] DETAILED DESCRIPTION OF PARTICULAR EMBODIMENTS

[0070] Different aspects of the present disclosure will be described more fully hereinafter. The disclosure can be realized in many different forms and should not be construed as being limited to the embodiments below.

[0071] Here follows the detailed disclosure texts “Detailed disclosure (part A): Clamped gigahertz mechanics for opto- and / or electromechanical transduction at the quantum level” as well as “Detailed disclosure (part B): Clamped and sideband-resolved silicon optomechanical crystals with high single-photon cooperativity” and “Detailed disclosure (part B): Clamped and sideband-resolved silicon optomechanical crystals with high single-photon cooperativity: Supplementary information”. Part A concerns the entire concept, and part B goes in more depth on the OMT.

[0072] Detailed disclosure (part A):

[0073] General

[0074] Mechanical devices interact well with most if not all other types of excitations and systems and can be seen as a universal bus. Gigahertz mechanical waves are particularly attractive as they operate in a mature microwave band, have similar wavelengths and therefore couple well to optical waves, and are naturally in their quantum ground state at accessible cryogenic temperatures. We invent a new class of clamped micro- and nanoscale devices to confine gigahertz1mechanical modes to the surface of common semiconductor chips which support quantum-level2transduction and storage channels between the mechanical modes and near-infrared optical and gigahertz microwave systems. Crucially, the devices are clamped and anchored to their substrate, i.e. they are not suspended from their underlying substrate. This increases their thermal contact area dramatically and improves prospects for dense co-integration of electronic, mechanical, and optical components. Strong zero-point opto- and electromechanical coupling rates of, for example, at least 450 kHz and 20 MHz are available in these devices which are designed to be manufacturable at scale. This paves the way for new information transduction tasks3through fast and high-fidelity optomechanical and electromechanical interactions. This includes, but is not limited to, quantum and classical conversion between microwaves and optics, quantum acoustic processors and memory, and sensors.

[0075] The transducers have the following alternative modes of operation:

[0076] A) Electromechanical transduction (EMT) as shown in the appended figure referred to as Fig. A.

[0077] B) Optomechanical transduction (OMT) as shown in the appended Fig. B.

[0078] C) Electro-optomechanical transduction (hybrid EMT-OMT) where information can be exchanged between both microwaves, mechanics and optics in one device, and shown in the appended Fig. C. Such device can be used for microwave-to-optics transduction.

[0079] In said figures A, B and C, the electromechanical (EMT) and optomechanical transducer (OMT) could be operated separately or simultaneously (hybrid EMT-OMT or EOMT). We also invent devices that are purely electromechanical (EMTs) or purely optomechanical (OMTs)4. The OMT requires an optomechanically active region5and material. These are preferably high-refractive-index materials with strong photoelasticity like silicon (see variations). The EMT requires an electromechanically active region. These are preferably based on strong piezoelectric materials like lithium niobate (see variations). The EMT could also be based on DC-biasing of inversion-symmetric materials with strong electrostriction like silicon. We often use piezomaterials as examples while the concepts remain general. In other words, the inventors of the present disclosure have invented a new approach and class of devices to simultaneously satisfy all of the below properties (or a subset, when so desired) in electro- and / or optomechanical structures:

[0080] 1. The devices are fully attached, i.e. clamped, to their substrate. This applies to EMTs, OMTs and hybrid EMT-OMTs. a. The clamped nature of the devices dramatically increases their thermal contact area to the underlying substrate. This is essential for reaching low- noise device operation for quantum-level6applications involving optical driving at low temperatures. Various combinations of materials underneath and / or above and / or inside the device layer can be envisaged to improve the thermal properties further. For example, holes in the device layer may be filled with other materials with good thermal conduction, and the devices could be operated in He-3 or similar gaseous environment for improved thermal anchoring from above. The increased thermal contact area improves the time window during which high-fidelity operation is possible, and thereby increases repetition rates and fidelities for many tasks involving optical driving at low temperature. Heating of the mechanical system is a key limiting factor in many situations currently, and the concept according to the disclosure improves upon drawbacks of current approaches. b. The clamped nature of the devices enables gigahertz mechanics to be treated on similar footing manufacturing-wise as photonic, electronic, and superconducting devices. It means that no release steps need to be introduced in foundry processes. Any steps not commonly associated with foundries can take place before or after the foundry process without impacting the mass-manufacturability. Thus, it improves prospects for dense co-integration of gigahertz mechanics with optical and microwave components. c. Needed for electromechanics and optomechanics and electro- optomechanics

[0081] 2. Despite the clamped nature of the devices, the mechanical modes’ wavevectors and frequencies can be fully outside the continuum formed by surface and bulk acoustic waves in the unit cell design. This robustly enables dramatically higher mechanical quality factors than modes that are inside the continuum in the dispersion diagram. This applies to EMTs7, OMTs and hybrid EMT-OMTs. a. The finite length of resonators introduces small mechanical wavevector components in the continuum. These components are kept under control and clamping-limited mechanical quality factors of at least 1e6 are in reach. b. In contrast to suspended 2D OMCs, the approach does not require perfectly aligned in-plane bandgaps to function robustly. The mechanical guidance operates simply and robustly thanks to the mechanics’ high wavevector, i.e. the mechanical modes are closer to X-point than to Gamma-point in the dispersion diagram. i. The 2D suspended approach to making optomechanical crystals suffers from reproducibility issues: getting all the bands to line up needs a way more precisely defined geometry. One-dimensional OMCs and EMCs in contrast do not require in-plane bandgaps to be aligned with their frequencies. This increases fabrication tolerances compared to the suspended approach. Electro- and optomechanical transducers are complex systems with many parameters which need to be designed and fabricated within a narrow acceptable range. Improving both the design simplicity and the fabrication tolerances is key to realizing such devices with high yield. c. Needed for electromechanics and optomechanics and electro- optomechanics The optomechanical system operates in the so-called resolved-sideband regime where the mechanical frequency exceeds the optical loss rates. This is an essential condition for quantum operations such as transduction of information with low-noise. This applies to OMTs and hybrid EMT-OMTs. a. Only needed for the optomechanics and for the electro-optomechanics The mechanical modes have quantum-level channels to near-infrared optical modes with zero-point coupling rates of, for example, at least 450 kHz for optomechanical resonators with thermal contact area of at least 2 umA2. When condition (5) is met simultaneously, the zero-point coupling rate to the hybridized mode can be at least 220 kHz for optomechanical resonators with thermal contact area of at least 2 umA2. The hybridized electro-optomechanical structures have increased mechanical mode volume which reduces the heating and noise which is experienced mostly in the optical region. This applies to OMTs and hybrid EMT- OMTs. a. Only needed for the optomechanics and for the electro-optomechanics The mechanical modes have quantum-level channels to gigahertz microwave systems such as superconducting microwave resonators and qubits with zeropoint coupling rates of, for example, at least 20 MHz. When condition (4) is met simultaneously, the zero-point coupling rate to the hybridized mode can be at least 6 MHz. This applies to EMTs and hybrid EMT-OMTs. a. Only needed for electromechanics and for electro-optomechanics The devices can be designed and fabricated such that they consist only of crystalline materials. The devices can be made in a thin-film made of, for example, crystalline silicon. The substrate layer can consist of crystalline materials such as sapphire or diamond. Both the device and substrate layer can also consist of combinations of several crystalline or non-crystalline materials. This applies to OMTs, EMTs and hybrid EMT-OMTs. a. The fully crystalline nature of the devices enables dramatically higher mechanical quality factors at cryogenic temperatures than if there were non-crystalline materials present close to the mechanical mode. Noncrystalline materials such as SiO2 may be chosen in applications where mechanical quality factors above roughly 5000 to 10 000 are not desired for device operation. b. Nanometer-sized non-crystalline layers can grow in an atmospheric environment on certain materials such as silicon even though all main constituents are crystalline. This can reduce the mechanical quality factors drastically. Such oxides can be removed by e.g. hydrofluoric acid dips taking place right before loading into the cryogenic environment where the device may be operated. Alternatively, devices can be encapsulated in top claddings such as silicon nitride, gaseous or liquid environments such as nitrogen which prevent such growth. c. Needed for electromechanics, optomechanics, and for electro- optomechanics

[0082] 7. The device designs and fabrication steps are compatible with common semiconductor foundry manufacturing processes. Thus the devices are manufacturable at a commercial volume at minimum similar to the field of silicon photonics. This applies to OMTs, EMTs and hybrid EMT-OMTs. a. Needed for electromechanics, optomechanics, and for electro- optomechanics

[0083] 8. The electro- and optomechanical interactions can take place in either spatially 3D- confined resonators, in 2D-confined waveguides, or in regions of free non-confined wave propagation. This applies to OMTs, EMTs and hybrid EMT-OMTs.

[0084] 9. The gigahertz mechanical modes can either be localized or converted into sections with finite mechanical group velocity ranging from 0 m / s up to at least 4 km / s and the traveling mechanical modes can realize connectivity between other electronic, mechanical, optical, or microwave devices. Resonators for the gigahertz mechanical modes can be formed in analogy to common approaches in photonics, e.g. in the form of ring or phononic crystal resonators.

[0085] 10. When desired, suspended or released structures can locally be combined with the clamped structures. This applies to OMTs, EMTs and hybrid EMT-OMTs.

[0086] 11 . The mechanical modes are highly tailorable in terms of frequency and wavevector. They can be exploited to couple two optical waves on the chip, or couple an optical wave on the chip to optical waves off the chip. In the latter case, we speak of beam-steering. In the situation where optical waves on the chip are coupled, these optical waves have a frequency difference roughly similar to the frequency of the mechanical mode. The optical waves on the chip can be counter- or copropagating. When they are co-propagating, different spatial optical modes are desired to keep the mechanical wavevectors outside the continuum. This applies to OMTs, EMTs and hybrid EMT-OMTs. a. We focus mainly on describing the individual opto- and electromechanical elements and how they realize strong opto- and electromechanical coupling rates as well as high coherence, and how they can be manufactured at scale for both quantum and classical applications. More complex systems can be built using such elements, consisting of many such elements which could be made tunable using for example capacitive driving or second-order nonlinearities of materials like lithium niobate. One example of a more complex system is an acousto-optic beam-steering device. Such devices can consist of a linear array of adjacent and parallel electromechanical sections as presented above, each of them can for example inject the mechanical waves into long adjacent, closely spaced, and parallel optomechanical waveguides similar to the unit cell designs with periodic symmetry present above. In these waveguides coupling between on-chip optical modes and off-chip optical modes could take place via the same kind of optomechanical interactions we have described above, i.e. via photoelastic and moving boundary effects. Our devices can also improve upon key drawbacks of previous approaches also in such more complex systems and arrays. For example they support strong electro- and optomechanical coupling rate and are clamped so they do not suffer from film inhomogeneity and internal stress the same way a suspended system does. Three-wave-mixing interaction rates between one on-chip optical mode, one radiating optical mode, and the on-chip mechanical mode can be strong as the mechanical mode can be pumped very efficiently as described above. By varying the acoustic frequency, optical beam-steering and sensing can be achieved even when radiating optical fields are involved such as in autonomous systems. Coupling to other quantum information carriers such as spins in the substrate becomes more straightforward in the clamped configuration. This applies to OMTs, EMTs and hybrid EMT-OMTs. The disclosure enables systems that satisfy various subsets of the above properties. For example, properties (4) and (5) can be realized simultaneously, however when only property (4) or only property (5) need to be satisfied in separate devices the respective optomechanical or electromechanical coupling rates in those separate devices can be higher than when both of these properties need to be satisfied in one and the same device.

[0087] Variations

[0088] The concept according to the disclosure includes following variations on the same theme:

[0089] 1. Various combinations of materials underneath and / or above and / or inside the device layer can be envisaged to improve the thermal properties further. For example, holes in the device layer may be filled with other materials with good thermal conduction.

[0090] 2. The devices can be designed and fabricated such that they consist of crystalline materials. The devices can be made in a thin-film made of, for example, silicon, GaAs, GaP, InGaP, lithium niobate, AIGaAs, lithium tantalate, silicon carbide, aluminum nitride (AIN), lead zirconium titanate (PZT), Germanium, indium phosphide (InP), barium titanate (BTO), Lead Magnesium Niobate-lead Titanate (PMN-PT), silicon nitride, etc. For materials without intrinsic piezoelectricity, the electromechanical effect could be created using applied DC voltages, or from surface effects breaking inversion symmetry. For optomechanical interactions materials with high refractive index and strong photoelasticity or roto-optic effects are preferred. For electromechanical interactions materials with strong piezoelectricity, or with strong electrostriction, or with strong capability to handle DC voltages are preferred. The substrate layer can consist of crystalline materials such as sapphire or diamond or consist of amorphous or polycrystalline materials such as SiO2 or silicon nitride. Also materials like silicon carbide are possible. They are chosen such that they have low losses for mechanics and optics (OMT), low losses for mechanics and microwaves (EMT) and low losses for optics, microwaves, and mechanics (hybrid EMT-OMT) and such that are guiding heat well (OMT) and can be processed at large scale. They also need to support refractive indices below that of the thin-film device layer to confine optics. Both the device and substrate and top cladding layer may also consist of combinations of several crystalline or non-crystalline materials. For example several combinations of thin films could be used to create composite materials which may help mechanical confinement or heat guiding. Crystalline materials are preferred for many cryogenic applications where small mechanical and microwave losses are preferred.

[0091] 3. The resonators can be either standing-wave or traveling-wave type. For example the optical and mechanical waves can be confined to either traveling-wave ring resonators or standing-wave photonic crystal cavities based on quasi-bandgaps. The substrate could be patterned as well to help suppress coupling from the guided mechanical modes to the continuum modes. Fabrication processes do not necessarily need to be conducted precisely in the order depicted here. Heterogeneous material integration can take place at the beginning or at the end of the process. Microwave devices such as superconducting resonators could be integrated on the same chip as the mechanical systems, or on a different chip which could be wirebonded to or flipchipped on top of the piezo-mechanics or piezo-optomechanics chip. We have illustrated the case of counter-propagating optical waves coupling to high-wavevector mechanics. Many variations are possible where the counterpropagating waves need not be in the same spatial mode, for example the forward-propagating wave could be in a quasi-TE optical mode and the backward optical wave in a quasi-TM optical mode. In another example, one of the two optical waves could be an optical mode radiating away from the chip as for beamsteering applications. Using two different on-chip optical modes in e.g. a photonic crystal cavity can reduce optical pump power requirements by engineering both optical pump and sideband to be resonant. The mechanical structures need not be linear, i.e. straight, they can be bent in order to form e.g. spiral waveguides or ring resonators. This may require compensating for material anisotropy through geometrical changes throughout the bend. The electro- and optomechanical interactions can take place in either spatially 3D- confined resonators, in 2D-confined waveguides, or in regions of free non-confined wave propagation. The waveguides can be slow-light structures where interaction rates could be improved. The mechanical modes are highly tailorable in terms of frequency, wavevector and quality factor, for example higher frequencies of 10 GHz are accessible. In another example, lower mechanical quality factors can be engineered for applications requiring higher bandwidth. This tunability also applies to group velocity and mode shape. The electrodes for the electromechanical devices can use materials such as aluminum, NbTiN, TiN, NbN, Ta, Al, etc. Materials with high kinetic inductance are preferred for enhancing the electromechanical interactions rates. Materials with low quasiparticle lifetime are preferred for reducing effects of optical absorption on the superconducting circuit. Materials that can withstand aggressive silicon cleaning procedures are preferred to achieve long coherence. For example NbTiN satisfies these requirements.

[0092] 11 . Microwave modes can be coupled on-chip to a mechanical mode either through the use of a piezoelectric material or by capacitive driving. The piezoelectric material can be, but is not limited to, lithium niobate (LN), lithium tantalate, gallium nitride, GaAs, silicon, crystalline GaAs, GaP, crystalline lithium niobate, aluminum nitride (AIN), lead zirconium titanate (PZT), Lead Magnesium Niobate-lead Titanate, indium phosphide etc. Capacitive coupling is versatile and can work with many material systems through DC electrical pumping.

[0093] Broader references and background

[0094] The described concept relates to, at minimum but not limited to, the fields of opto- and electromechanics, Brillouin scattering, microwave-to-optics transduction and electro-optic modulation, quantum acoustics and memory, quantum computing, quantum communications, classical communications, integrated photonics, and sensing. Several of these fields are very broad and connect to other fields themselves, so this list should be seen as non-exhaustive. We foresee use-cases for the concept both in directly and indirectly related fields. Specifically, it describes a new class of techniques and devices where information can be exchanged efficiently and with low noise between gigahertz mechanical motion, near-infrared optical waves, and gigahertz microwaves. We provide a few non-exhaustive example references for some of the related fields and describe them very briefly.

[0095] Optomechanics, electromechanics & Brillouin scattering.

[0096] These fields study interactions between photons and phonons for both fundamental science and a host of applications. As the present concept describes ultra-efficient interactions between gigahertz phonons and both optical and microwave photons, it can be applied in many applications pursued in these fields including but not limited to quantum and classical electro-optic conversion, sensing, amplifiers and lasers, delay lines, microwave photonics and signal processing, frequency comb generation, and others.

[0097] Example references:

[0098] • Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391-1452 (2014). • Midolo, L., Schliesser, A. & Fiore, A. Nano-opto-electro-mechanical systems. Nature Nanotech 13, 11-18 (2018).

[0099] • Cleland, A. N. Foundations of Nanomechanics: From Solid-State Theory to Device Applications. (Springer Science & Business Media, 2002).

[0100] • Safavi-Naeini, A. H., Van Thourhout, D., Baets, R. & Van Laer, R. Controlling phonons and photons at the wavelength scale: integrated photonics meets integrated phononics. Optica, 6, 213-232 (2019).

[0101] • Eggleton, B. J., Poulton, C. G., Rakich, P. T., Steel, Michael. J. & Bahl, G. Brillouin integrated photonics. Nature Photonics 2019 13:10 13, 664-677 (2019).

[0102] Microwave-to-optics transduction and electro-optic modulation.

[0103] These fields study transduction of both quantum and classical information between microwave and optical frequencies by any means: second-order optical nonlinearities such as the Pockels effect, intermediate mechanical systems, free-carrier modulation, etc. As the present disclosure describes a new class of devices with high-fidelity and high transduction rate between microwaves and optics at extremely low pumping levels, it can be applied in many applications pursued in these fields including but not limited to quantum state transduction, classical optical read-out and control of microwave systems and qubits, low-energy electro-optic modulation in power-constrained situations, switching of optical signals in data centers or in optical Al engines, connecting small quantum processors together into larger distributed quantum computers, autonomous systems, etc. Any application that requires quantum or classical data to be moved around quickly with low power consumption can benefit.

[0104] Example references:

[0105] • Han, X., Fu, W., Zou, C.-L., Jiang, L. & Tang, H. X. Microwave-optical quantum frequency conversion. Optica, 8, 1050-1064 (2021).

[0106] • Sinatkas, G., Christopoulos, T., Tsilipakos, O. & Kriezis, E. E. Electro-optic modulation in integrated photonics. Journal of Applied Physics 130, 010901 (2021).

[0107] • Safavi-Naeini, A. H., Van Thourhout, D., Baets, R. & Van Laer, R. Controlling phonons and photons at the wavelength scale: integrated photonics meets integrated phononics. Optica, 6, 213-232 (2019).

[0108] • Miller, D. A. B. Attojoule Optoelectronics for Low-Energy Information Processing and Communications. Journal of Lightwave Technology 35, 346-396 (2017). • Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: A vision for the road ahead. Science 362, eaam9288 (2018).

[0109] Quantum acoustics and memory.

[0110] This field studies the unique properties of acoustics for quantum information processing. The acoustic wavelength is about five orders of magnitude smaller than the microwave wavelength at the same frequency - providing far more compact devices with reduced crosstalk. In addition, acoustic systems have superior coherence levels with lifetimes as high as seconds observed even in micron-scale silicon phononic crystals. This is conducive to realizing large-scale quantum computers. As the disclosure describes a new class of devices with high-fidelity and high transduction rate between microwaves and mechanics (also called acoustics), it can be applied in many applications pursued in this field, including but not limited to quantum sensors, memory, and storage.

[0111] Example references:

[0112] • Hann, C. T. et al. Hardware-Efficient Quantum Random Access Memory with Hybrid Quantum Acoustic Systems. Phys. Rev. Lett. 123, 250501 (2019).

[0113] • Chu, Y. et al. Quantum acoustics with superconducting qubits. Science 358, 199— 202 (2017).

[0114] • Chamberland, C. et al. Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes. PRX Quantum 3, 010329 (2022).

[0115] • Arrangoiz-Arriola, P. et al. Resolving the energy levels of a nanomechanical oscillator. Nature 571, 537-540 (2019).

[0116] Integrated photonics.

[0117] This field studies and exploits the flow of light in circuits integrated on a chip in materials such as lithium niobate, silicon, silicon nitride, gallium arsenide, etc. These photonic integrated circuits already have sizable markets. They currently lack proper programmability both on the chip and in their channels off the chip. This requires active and sometimes non-reciprocal components. Light and sound interact strongly and thus sound can process and structure electromagnetic signals non-reciprocally and with low power consumption. These are both features available in our new class of devices.

[0118] Example references: • Zhu, D. et al. Integrated photonics on thin-film lithium niobate. Adv. Opt. Photon., AOP 13, 242-352 (2021).

[0119] • Kaur, P. et al. Hybrid and heterogeneous photonic integration. APL Photonics 6, 061102 (2021).

[0120] • Miller, D. A. B. Attojoule Optoelectronics for Low-Energy Information Processing and Communications. Journal of Lightwave Technology 35, 346-396 (2017).

[0121] • Rahim, A. et al. Expanding the Silicon Photonics Portfolio With Silicon Nitride Photonic Integrated Circuits. J. Lightwave Technol., JLT 35, 639-649 (2017).

[0122] • Bogaerts, W. et al. Programmable photonic circuits. Nature 586, 207-216 (2020).

[0123] Sensing.

[0124] This field is broad and ranges from measuring chemical events or molecular species to exploiting gigahertz mechanics to spatially steer optical signals around and gather information on the surroundings of e.g. an autonomous system. As several embodiments of the disclosure feature both ultra-small mechanical mode volumes (which are very sensitive to the gentle perturbations one is trying to sense) and high-wavevector gigahertz mechanical waves (which can steer light around), it can be used in many applications pursued in this field.

[0125] Example references:

[0126] • Sarabalis, C. J., Van Laer, R. & Safavi-Naeini, A. H. Optomechanical antennas for on-chip beam-steering. Opt. Express, 26, 22075-22099 (2018).

[0127] • Gil-Santos, E. et al. High-frequency nano-optomechanical disk resonators in liquids. Nature Nanotech 10, 810-816 (2015).

[0128] • Carrascosa, L. G., Moreno, M., Alvarez, M. & Lechuga, L. M. Nanomechanical biosensors: a new sensing tool. TrAC Trends in Analytical Chemistry 25, 196-206 (2006).

[0129] How the concept according to the disclosure differs from certain relevant prior art references:

[0130] We select a number of the closer references and describe in each case with short keywords how the concept according to the disclosure differs from them. Here is the non- exhaustive list: • Ren, H. et al. Two-dimensional optomechanical crystal cavity with high quantum cooperativity. Nat Commun 11 , 3373 (2020). o This is suspended and has a smaller thermal contact area. Our class of devices is not suspended. o No electromechanics.

[0131] • Painter, O. et al. Techniques for transduction and storage of quantum level signals. US Patent (2020). o This is suspended and has a much smaller thermal contact area. Our class of devices is not suspended. o Does not use high-wavevector mechanics

[0132] • Mirhosseini, M., Sipahigil, A., Kalaee, M. & Painter, O. Superconducting qubit to optical photon transduction. Nature 588, 599-603 (2020). o This is suspended and has a much smaller thermal contact area. Our class of devices is not suspended. o Does not use high-wavevector mechanics

[0133] • Qiu, L., Shomroni, I., Seidler, P. & Kippenberg, T. J. Laser Cooling of a Nanomechanical Oscillator to Its Zero-Point Energy. Phys. Rev. Lett. 124, 173601 (2020). o This is suspended and has a much smaller thermal contact area. Our class of devices is not suspended. o Uses a different approach involving He-3 to try to improve thermal anchoring. Does not use high-wavevector mechanics. o No electromechanics.

[0134] • Zhang, J. et al. Subwavelength Control of Photons and Phonons in Release-Free Silicon Optomechanical Resonators. ACS Photonics 9, 3855-3862 (2022). o Not quantum-level. Much worse figures of merit: not sideband-resolved, mechanical frequencies well below 1 GHz, optomechanical coupling rates below 80 kHz, non-crystalline substrate. o Does not exploit high-wavevector mechanics in counter-propagating optomechanical interactions. o No electromechanics.

[0135] • Ma, X., Shandilya, P. K. & Barclay, P. E. Semiconductor-on-diamond cavities for spin optomechanics. Preprint at https: / / doi.org / 10.48550 / arXiv.2302.04967 (2023). o Not quantum-level. Much worse figures of merit: not sideband-resolved, mechanical frequencies too low, optomechanical coupling rates below 6 kHz for the high-wavevector mechanical modes. o No electromechanics.

[0136] • Sarabalis, C. J., Dahmani, Y. D., Patel, R. N., Hill, J. T. & Safavi-Naeini, A. H. Release-free silicon-on-insulator cavity optomechanics. Optica 4, 1147-1150 (2017). o Not quantum-level. Much worse figures of merit: not sideband-resolved, mechanical wavevector low and in continuum, mechanical quality factors only -100, non-crystalline substrate [Main text], o Much worse figures of merit: not sideband-resolved, optomechanical coupling rate 22 kHz, non-crystalline substrate [Suppl. Info.].

[0137] ■ Side note: part of the counter-propagating approach mentioned but not fully described and has low coupling rate anyway. o No electromechanics.

[0138] • Liu, S., Tong, H. & Fang, K. Optomechanical crystal with bound states in the continuum. Nat Commun 13, 3187 (2022). o Not quantum-level. Much worse figures of merit: not sideband-resolved, mechanical quality factors only -500, optomechanical coupling rate of only 100 kHz, non-crystalline substrate. [Note: the manuscript mentions higher couplings, these are unit-cell couplings that must be divided by the number of unit cells]. o Does not exploit high-wavevector mechanics in counter-propagating optomechanical interactions. This uses a different approach for confinement: bound states in the continuum at zero wavevector. Potential for much higher mechanical coherence is small. o No electromechanics.

[0139] • Weaver, M. J. et al. An integrated microwave-to-optics interface for scalable quantum computing. Preprint at https: / / doi.org / 10.48550 / arXv.2210,15702 (2022). o This is suspended and has a much smaller thermal contact area. Our class of devices is not suspended.

[0140] • Jiang, W. et al. Optically heralded microwave photons. Preprint at https: / / d0i.0rg / l 0.48550 / arXiv.2210.10739 (2022). o This is suspended and has a much smaller thermal contact area. Our class of devices is not suspended. • SCHMEING, K. & Seidler, P. F. Microwave-to-optical transducer. Patent (2020). o This is suspended and has a smaller thermal contact area. Our class of devices is not suspended. o Electromechanics is extremely weak (not quantified).

[0141] • Simon, G., Moritz, F. & Robert, H. J. S. Quantum wavelength converter between a microwave signal and an optical signal. Patent (2020). o This is suspended and has a smaller thermal contact area. Our class of devices is not suspended. o Electromechanics is extremely weak (not quantified).

[0142] • Sahu, R. et al. Entangling microwaves with optical light. Preprint at https: / / doi.org / 10.48550 / arXiv,2301.03315 (2023). o Example of a device that does not use mechanics for quantum transduction. Our approach uses mechanics on a chip, requires dramatically lower pumping powers, and offers much higher transduction rates. This example uses the Pockels effect.

[0143] • Witmer, J. D., Arrangoiz-Arriola, P., Hill, J. T., Safavi-Naeini, A. H. & McKenna, T. P. Doubly-resonant electro-optic conversion using a superconducting microwave resonator. US Patent (2020). o Example of a device that does not use mechanics for quantum transduction. Our approach uses mechanics on a chip, requires dramatically lower pumping powers, and offers much higher transduction rates. This example uses the Pockels effect.

[0144] • SAFAVI-NAEINI, A.-H., SARABALIS, C. J., WITMER, J. D. ,RRIOLA, P. A. & LAER, R. F. V. Acousto-optic beam steering system. US Patent (2021 ). o This is suspended and requires release steps. Our class of devices is not suspended.

[0145] • Arrangoiz-Arriola, P. et al. Resolving the energy levels of a nanomechanical oscillator. Nature 571, 537-540 (2019). o This is suspended and requires release steps. Our class of devices is not suspended. Principles and workings

[0146] Clamped mechanics

[0147] Gigahertz8mechanical waves have often been confined by suspending devices, i.e. physically disconnecting them from their substrate. This is often combined with mechanical radiation shields around the suspended device to eliminate leakage of mechanical energy. This simultaneously suppresses heat conduction, which is also carried by phonons, away from the mechanical device to the substrate. In contrast, here we confine mechanical modes without suspending the devices and thus realizing a much higher thermal contact area for undesired heat-carrying phonons to escape from the device. The devices are clamped on at least their bottom side to their substrate.

[0148] In our clamped class of devices, there is physical contact between at minimum the bottom side of the device and a substrate. Avoiding mechanical leakage requires careful consideration in this situation. Indeed, there is now a continuum of mechanical modes that the confined mechanical mode must not be allowed to leak into. We describe a novel class of clamped opto- and electromechanical systems which operate below the continuum. In other words, they confine mechanical energy associated with the desired mode even though there is a continuum of undesired modes in the substrate. Thus the mechanical modes of interest are confined properly to the device, while we simultaneously allow heat-carrying phonons to leak away. Our approach relies critically on using mechanical modes that do not easily leak into any mechanical mode in the continuum of undesired modes. We do so by making sure the desired mechanical modes are phase-mismatched from the continuum modes. In other words, their wavevectors and spatial profiles are such that they have vanishing overlap and thus vanishing coupling to the undesired continuum modes. It is challenging to do so while simultaneously achieving quantum-level interactions to optical and microwave systems. Below, we go into depth on how the concept according to the disclosure enables this for both optomechanical and electromechanical interactions.

[0149] Clamped optomechanics (OMT)

[0150] 8"Gigahertz": typically in the range 0.5 - 20 GHz but not necessarily limited to this. Often in the microwave C and X band from 4 - 12 GHz but not necessarily limited to this. We begin with the optomechanical interaction. We realize the interactions between mechanical and optical modes in optomechanically active materials such as silicon. We require high refractive-index and photoelasticity as these properties support strong moving-boundary and photoelastic coupling between optical and mechanical waves. The devices are patterned in thin-films which confine both optical and mechanical modes to a small region. The devices may be fabricated on a substrate made of non-crystalline materials like silicon dioxide or crystalline materials like sapphire. We invent a new method and class of devices that can co-localize near-infrared optical waves and gigahertz mechanical waves into clamped structures on the surface of a chip. The approach starts from developing a unit cell design which realizes such optomechanical interaction in a structure that has periodic symmetry, effectively a long waveguide that confines mechanics and optics in two dimensions but not in the third. Next, we use the unit cell design as a basis for designing structures that are resonant for both optics and mechanics - in other words, they confine and co-localize the optics and mechanics to a certain volume in three spatial dimensions. These structures are often referred to as optomechanical crystals (OMCs).

[0151] Clamped optomechanical unit cell design - 2D confinement

[0152] To design the optomechanical interaction, we first develop a unit cell design where all waves are confined to a volume that is repeated periodically. In this unit cell design, we make sure the desired mechanical mode has a wavevector that differs from the wavevectors of any mechanical mode found in the continuum at a given frequency (Fig.1). At the same time, this mechanical mode interacts with the optical mode in a three- wave-mixing process well-known from the fields of cavity optomechanics and Brillouin scattering. In this three-wave-mixing process, two optical waves of different frequencies interact and exchange information with the mechanical mode. This interaction is strongest when the two optical modes have a wavevector and frequency difference that approximately equal the wavevector and frequency of the desired mechanical mode. The two optical waves are often called the “pump” and “sideband”. The pump is usually excited strongly with an external laser, whereas the sideband is much weaker and could contain either classical or quantum information. The sideband could have either lower or higher frequency than the pump. Denoting angular frequency and wavevector by and ki respectively - where for pump, sideband and mechanics - we express the energy and phase-matching conditions as

[0153] In these equations, we assume that the sideband is red-detuned from the pump without loss of generality. When the sideband is blue-detuned from the pump, the indices p and s must be interchanged. Crucially, the need for a mechanical wavevector that lies outside the continuum sets stringent requirements on the optical waves. The difference between the optical wavevectors must now be such that it approximately equals this specific mechanical wavevector. There are at least two ways to realize this situation:

[0154] • The two optical waves propagate in opposite directions, i.e. they counterpropagate, such that their wavevector difference is large. They can either have the same spatial mode, e.g. both quasi-TE mode or both quasi-TM mode, or they can have a different spatial mode, e.g. one is quasi-TE and the other is a quasi-TM mode. Either way, their wavevector difference is large.

[0155] • The two optical waves propagate in the same direction, i.e. they co-propagate, and they are in a different spatial mode such that their wavevector difference is large.

[0156] We illustrate the situation of two counter-propagation optical waves in the same spatial mode in the rest of the optomechanics discussion. The principles, however, remain general. The principle of the coupling is illustrated in Fig.1a . In the figure below we use without loss of generality that due to the significant difference in optical and mechanical frequencies. This illustrates the case of intra-model coupling i.e. when optical and sideband wave are e.g. both quasi-TE or both quasi-TM polarized.

[0157] The periodic nature of the unit cell design implies a lower bound on the set of unique wavelengths that can propagate along the crystal's periodic symmetry axis. In wavevector space, this bound is known as the first Brillouin zone and limits unique wavevectors to is the period of the unit cell. This implies that any wavevector (either optical or mechanical) is defined up to multiples of the reciprocal lattice vector, i.e. one can map where n is an integer and is the reciprocal lattice vector. In other words, any state with a wavevector outside the first Brillouin zone is equivalent to a state with a wavevector inside the first Brillouin zone. As an example, this implies that X-point optical modes are not good candidates for clamped optomechanics as these would be phase-matched to T-point mechanical modes inside the mechanical continuum. Viable optical modes therefore have to interact with high-wavevector mechanical modes within the first Brillouin zone. We visualize the operating conditions implied by the discussions above for clamped optomechanics in Fig.1 (b,c). Again, we assume that the wavevector of pump and sideband optical modes to be approximately equal in magnitude without loss of generality. The figure depicts Surface Acoustic Wave (SAW) frequencies at the stimulated mechanical wavevector versus optical pump effective index and unit cell period a. The diagram is shown for both sapphire and silica substrates. These SAW frequencies are considered to belong to the slowest mechanical waves in the surrounding substrate, so they determine where the edge of the continuum lies in the dispersion diagram. Clamped optomechanics with high mechanical and optical quality factors is therefore possible for all combinations and the SAW frequency exceeds that of the desired mechanical mode. The star in the figure illustrates the operating point for a version of the proposed unit cell. We go in greater depth on the actual geometry and dispersion diagrams in following sections, here we illustrate the basic principles.

[0158] The above discussion assumes that SAWs are the slowest waves in the surrounding substrate, which need not always be the case. In more general terms, it is the slowest undesired mechanical mode in the surrounding substrate that must be used to determine the edge of the continuum.

[0159] Figure 1 (appended), (a) Phase-matching diagram for clamped optomechanics, here illustrated for the case of counter-propagating optical modes in the same spatial mode. Optical modes with angular frequency near couple to mechanics at This three- wave-mixing process can strongly couple the optical and mechanical modes in the clamped device, (b) and (c), The minimal surface acoustic wave frequencies as color maps for sapphire (b) and silica (c) substrates. The guided optical mode with vacuum wavelength of 1550 nm and effective index interacts with the mechanics at The star indicates the working point for the proposed optomechanical unit cell with mechanical frequency of ~5.5 GHz. We go in greater depth on the actual geometry and dispersion diagrams in the following sections.

[0160] Clamped optomechanical resonator design - 3D confinement and phase-matching A unit cell design can be exploited for opto- and electromechanical interactions in structures with periodic symmetry such as long waveguides. These structures have large bandwidth but require larger pumping powers to enable three-wave-mixing. To reduce these pumping powers, the next step in many applications is to develop a resonator design based on the unit cell design. The resonator design must confine the two optical waves and the mechanical mode in three spatial dimensions to a certain volume, forming a resonator along the spatial dimension that previously had periodic symmetry in the unit cell design. Since the resonators often feature periodic perturbations like in a crystal, and they simultaneously confine optics and mechanics, they are called optomechanical crystals (OMCs) in the following.

[0161] The resonators can for instance be of standing-wave or traveling-wave type:

[0162] • In the traveling-wave situation, the unit cell design is repeated such that it eventually loops back in on itself such as in a ring resonator. The interference of the waves over multiple round-trips along such a resonator discretizes the unit cell dispersion diagrams such that only certain frequencies are allowed. Reflections of optical and mechanical waves are not needed to create a traveling-wave resonance.

[0163] • In the standing-wave situation, two mirrors are added around the unit cell design such that the optical and mechanical waves bounce back and forth and only certain discretized frequencies from the unit cell dispersion diagram can exist. The unit cell design is perturbed slowly to create mirror regions where full or quasibandgaps exist.

[0164] Next, we illustrate how the standing-wave clamped OMC situation works for clamped optomechanics. The principles remain generally applicable. The concept of phasematching is an explicit result of the overlap integral between the optical and mechanical modes over a volume. In case of infinitely long waveguides with continuous symmetry along their propagation axis, the phase-matching condition must be satisfied exactly for optomechanical interaction to occur. In case of finite-length resonators with periodic symmetry, the phase-matching condition is an approximate concept that need only be satisfied up to reciprocal lattice vectors and up to the ambiguity in the wavevector set by the finite length L of the resonator. While phase-matching is an approximate concept in resonators like standing- and traveling-wave clamped OMCs, it nevertheless strongly selects and helps understand which type of optical and mechanical modes can interact well with one another.

[0165] In a standing-wave clamped OMC, both the pump, sideband, and mechanical modes can be considered as consisting of a forward and backward propagating part of the field. Together, the forward and backward propagating components form a standing wave. In

[0166] Fig.2(a) we illustrate such a standing-wave resonator in which pump and sideband optical waves propagate in forward and backward directions. These forward and backward optical components form a standing optical wave within the resonator that interacts with the standing mechanical wave. In other words, the optomechanical coupling rate scales as is the pump field, isthe sideband field and is the mechanical field. The optomechanical coupling rate thus adds together 8 terms each of which contains a pump component, sideband component, and mechanical component which are either propagating forwards or backwards. We stress that the components of the form and their conjugates are associated with counter-propagating optical pump and sideband optical fields interacting with a high-wavevector mechanical component (Fig.2). It is important to note that there are now two counter-propagating optomechanical interactions taking place at once in the standing-wave resonator: (1 ) the interaction between the forward pump component, backward sideband component, and forward mechanical component, and (2) the interaction between the backward pump component, the forward sideband component, and the backward mechanical component. In traveling-wave resonators without roughness-induced backscattering, only one of these would take place. In standing-wave resonators, the sum of both these interaction terms must be large for there to be strong optomechanical coupling.

[0167] An equivalent way of understanding this is that the optomechanical coupling rate in essence takes the overlap integral between the optical pump standing wave, the optical sideband standing wave, and the mechanical standing wave. Multiplying the optical pump and the optical sideband standing wave yields a beat note with both fast-varying and slowly-varying spatial components. The fast-varying part has a wavevector that equals the vectorial difference of the optical pump and sideband wavevectors (Fig.2). The mechanical mode can have a large wavevector that cancels this fast-varying part out and thus yields strong overlap. Put differently, realizing the strongest possible optomechanical coupling rate requires the optical beat-note to line up spatially with the standing-wave mechanical field within the resonator both in its period, shape, and in its phase. Due to the particular spatial overlap of the phase-matched modes, this integral and thus the optomechanical coupling rate scale rather slowly with resonator length as approximately l / 1^ with L the resonator length. This is the same scaling as in the more conventional case of utilizing low- wave vector mechanical modes in OMCs. It differs starkly from the strong suppression that would take place if the high-wavevector mechanical mode were to be phase-mismatched from the optical fields. We illustrate this by designing long resonators in below sections, where the optomechanical coupling rate stays high despite their longer resonator length. Due to the lack of destructive interference across neighboring unit cells, strong optomechanical coupling based on counter-propagating coupling can thus be realized in clamped OMCs. This is in stark contrast to previous OMC work based on low-wavevector mechanics. A similar argument applies to traveling-wave resonators such as ring resonators.

[0168] Figure 2 (appended). Phase-matching within standing wave OMCs (a) Pump (p) and Stokes (s) optical waves propagate in forward (f) and backward (b) directions to form standing waves. Mechanical waves with wavevector fc» are stimulated through a three- wave mixing process (energy diagram in inset). Phase-matching dictates that counter-propagating pump and Stokes waves leads to high-wavevector mechanical waves.

[0169] Clamped optomechanical resonator design - band structure and tapering

[0170] The optical and mechanical modes are co-localized in an optomechanical resonator consisting of several unit cells. Such a quasi-periodic9sequence of unit cells is called an optomechanical crystal (OMC) as described in the section above. To enable confinement and phase-matching of optical and mechanical modes, the band structures of the unit cells are tuned by patterning the cells, e.g. with a hole, allowing the dimensions of each unit cell to slightly change along the structure. For instance, in this example design increasing the lattice parameter a decreases the frequencies of the optical and mechanical bands. On the other hand, in this example design, increasing the length of the

[0171] 9"Quasi-periodic" or "semi-periodic" in the context of this disclosure means that there is a sequence of unit cells that are repeated but where there parameters can vary as they are repeated in the structure. So the structure has a quasi- or semi-periodic nature where analyses in terms of periodic boundary conditions could be made. However this is not strictly necessary and the inventive concept also applies to nonperiodic EMTs, OMTs and EMT-OMTs. ellipsis in the direction perpendicular to the beam increases optical frequencies but decreases mechanical frequencies. This is an example of how the bands can be tailored to have the desired frequency at a certain wavevector. Finite-element simulations are used to calculate the optical and mechanical band structures of the unit cells clamped to a substrate. In contrast to a suspended waveguide, the band structure of such a crystal features a continuum line not only for optical but also for the mechanical waves.

[0172] The unit cell at the center of the cavity is referred to as the defect unit cell. Its parameters, e.g. width, height, period, device layer orientation and hole size, are chosen such that

[0173] • the unit cell supports a mechanical mode at reasonable GHz frequency as well as an optical mode in the C-band

[0174] • the optical wavevector is approximately half the mechanical wavevector (Fig.3.1), as explained above, to enable good optomechanical coupling.

[0175] • the mechanical and optical fields spatially overlap well with regards to optomechanical coupling such that they yield quantum-level (see above for definition) interaction strengths and sideband-resolved operation where the mechanical frequency exceeds the optical linewidth.

[0176] • these modes lie outside of their respective continuum and support substantial optomechanical coupling.

[0177] The precise parameters of the unit cells depend on the materials of the substrate layer and the device layer.

[0178] As an example we show the band diagrams of an defect unit cell for a silicon device layer and a silica substrate layer in Fig.3.1 in a vacuum environment. The realm of continuum modes is shown as a shaded area. Dark gray lines denote modes with symmetry with respect to the xz plane, light gray lines denote modes of other symmetries. The bands of the modes of interest are black and the horizontal dashed lines indicate the approximate operating point. Fig.3a also shows the unit cell geometry in its left inset. The optical and mechanical fields overlap such that they give rise to very substantial radiation pressure at the Silicon I air interface of the hole. Using finite element simulations, we calculate the optomechanical coupling rate per unit cell in an infinite waveguide. In several embodiments of the unit cell designs they support optomechanical coupling rates of at least These unit cell coupling rates are competitive with those of the best non-clamped structures. They typically but not exclusively operate with small lattice constants a, for example around 180 to 250 nm, such that the continuum frequencies exceed the mechanical frequencies of the desired gigahertz confined mechanical mode.

[0179] To confine this mode in a standing-wave resonator a second type of cell, the mirror cell, is designed.

[0180] Its parameters, e.g. the width, height, period and hole size, are chosen such that:

[0181] • it has a quasi-bandgap at the operating frequencies of the defect cell (Fig.3.1 ) for mechanics and optics.

[0182] • a transformation from defect cell into mirror cell with several additional mirror cells will act as a low loss mirror to optical and mechanical waves.

[0183] This design scheme is applicable to any substrate. Substrates with high sound velocity, such as sapphire and diamond, can give more design freedom because the continuum is further away from the operating regime.

[0184] We show two example designs for the case of silicon on silicon dioxide (SOI: Si on SiO2; Fig.3.1) and silicon on sapphire (SOS: Si on AI2O3; Fig.3.2). Fig.3.2 shows the band structure of a unit cell on a sapphire substrate. Other material combinations are possible. For applications requiring the best possible mechanical and microwave coherence, SOS is a very attractive example.

[0185] Figure 3.1 (appended): SOI

[0186] Figure 3.2 (appended): IOMC SOS

[0187] Figure 3. Optical (a) and mechanical (b) band diagram of the unit cell. The inset in (b) shows the unit cell design. The right plot in each shows the X-point frequencies as a function of the perturbation from defect to mirror cell. The shaded area denotes the realm of continuum modes. Blue lines denote modes with symmetry with respect to the xz plane, red lines denote modes of other symmetries. The modes of interest are solid. The horizontal dashed lines indicate the approximate operating point. From the unit cells a cavity can be formed by transforming the unit cells into one another resulting in modes localized in the defect region of the optomechanical crystal. The parameters, e.g. the width, height, period and hole size, of the cells facilitating this transformation are chosen such that:

[0188] • the variation of the parameters from one unit cell to the next one is small enough (near-adiabatically) to avoid scattering of the waves.

[0189] For example, the right insets in Fig.3.1, 3.2. show the frequencies at as a function of the perturbation from defect to mirror cell, where we used a cubic function to parametrize the transition. This is to show that the perturbation opens quasi bandgaps that strongly reflect the fields. Fig.4.1a, b show the mechanical and optical mode profile respectively, which have frequencies The inset shows the cross section of the device. Fourier transformation of the cavity fields reveals that the phase matching condition is met (Fig.4.1 c). Black and gray lines indicate the Fourier transformations of the mechanical and optical field respectively. Finite element simulations are used to estimate the properties of the cavities. These cavities can have optomechanical coupling rates of at least The clamping-limited quality factors can exceed at least 1e6 (optical) and 1e5 (mechanical).

[0190] The thermal contact area for a cavity clamped to the substrate is much larger than for conventional optomechanical crystals. Because this area is proportional to the length of the defect region, it is viable to construct longer cavities. In longer cavities, the mode fields approach their unit cell equivalent more closely. The loss in coupling due to the larger mode volume is compensated for by the increase in thermal contact area which allows for more cavity photons with the same thermal load. In long cavities on silica, for instance for 31 defect cells cells, we estimate and measure optomechanical coupling rates of at least without adverse effects on the mechanical quality factor. This optomechanical coupling rate significantly exceeds the one of previous clamped approaches. Simultaneously the thermal contact area is around ten times larger compared to a clamped optomechanical crystal with only one defect cell. Applications where low loss is essential motivate the use of crystalline, high sound velocity substrates like sapphire. Such a cavity with multiple defect cells is shown in Fig.4.2. Due to the increased length of the cavity, the wavevectors of the modes are more sharply defined than for the short optomechanical crystal (Fig.4.2 c). While the principle of confinement of a clamped mechanical mode was illustrated here for a standing wave resonator, it can also be applied to a traveling wave resonator. This could for instance be a ring resonator consisting only of defect cells. Additional measures to reduce mechanical loss or increase thermal contact area could be taken, e.g. by patterning the substrate or by introducing more cladding to the device.

[0191] Figure 4.1 (appended): short OMC with SOI

[0192] Figure 4.2 (appended: long OMC with sapphire

[0193] Figures 4.1 and 4.2. (a) Defect region of clamped OMCs with u denoting normalized mechanical displacement, (b) Defect region of the same OMC showing the normalized electric field of the optical mode. Together, (a) and (b) define the active region of the optomechanical transducer, (c) Fourier transformation of the coupled cavity fieldsEv and (blue and red line respectively), showing counter propagating waves in the cavity

[0194] Clamped optomechanics: measurement

[0195] We demonstrate a version of the optomechanical part of the system, in this case realized with a 220 nm thin-film silicon OMC clamped to a silicon dioxide substrate. The example fabrication process involves patterning a mask with electron-beam lithography, followed by a reactive-ion etch that defines structures in the silicon. We measure the optomechanical properties of the devices at room temperature and atmospheric pressure.

[0196] Fig.4a shows a scanning electron micrograph of such an optomechanical crystal with 31 defect unit cells. Light enters through the center waveguide at the top of the figure and couples evanescently to the OMCs on either side. One of the two crystals has an optical resonance THz with a total linewidth GHz and external coupling rate This results in loaded and intrinsic quality factors of 140 • 103and 320 • 103 respectively. Fig.5b shows the response of the OMC to a modulated optical sideband which is swept over the resonance with a vector network analyzer. We note an electromagnetically induced transparency feature at the mechanical frequency.

[0197] A thermomechanical fingerprint is carried by light that reflects off the resonator, which we measure with a high-speed photoreceiver. Data is analyzed with a microwave spectrum analyzer and shown in Fig.5(c). The spectrum shows three optomechanically active modes with monotonically decreasing coupling. Spectral spacing and relative optomechanical coupling of the mechanical modes partially agree with simulation (Fig.3(c) inset), although absolute frequencies are shifted by -150 MHz. Discrepancies are likely due to fabrication inaccuracies. The fundamental mechanical mode is measured at Thus the optomechanical system achieves the resolved-sideband regime, improving upon key drawbacks from previous clamped approaches. The resolved-sideband regime requires the mechanical frequency to exceed the optical losses and is an essential precondition for many applications. Optomechanical coupling and the unloaded mechanical quality factor is measured by studying the mechanical linewidth at varying optical pump powers, Fig.3(d). Measurements are performed with the pump laser both blue- (gray) and red-detuned (black) from the optical resonance. The zero-point coupling rate and room temperature mechanical quality factor are measured to be 500 kHz and 850 respectively.

[0198] Figure 5 (appended). Physical performance of the optomechanical subsystem, (a) Top down scanning electron micrograph of a two silicon optomechanical crystals clamped to an underlying silica substrate. Light enters through the center waveguide at the top of the figure and couples evanescently to the OMCs. (b) Optical sideband response of the OMC. We modulate a blue-detuned pump laser by a VNA-driven electro-optic modulator. The response is measured in reflection. An electromagnetically induced transparency feature is visible at the mechanical frequency, (c) Measured thermomechanical spectrum. A strongly coupled fundamental mode along with two higher order modes are visible. The slightly asymmetric shape of the fundamental mode may be caused by inhomogeneous broadening, (d) Mechanical linewidth of the fundamental mode in (c) as a function of optical intracavity photons. We carry out the experiment with both red and blue detuned pumps. A linear fit yields zero-point optomechanical coupling

[0199] Clamped electromechanics (EMT)

[0200] As explained in overview and background, many use-cases require exchange of information between the mechanics and microwaves either separately from or combined with the exchange between the mechanics and optics. In this case, the interaction takes place via for example piezoelectricity or capacitive coupling. We illustrate the case of piezoelectricity as this usually leads to stronger electromechanical coupling rates and thus faster operations. In the case of electromechanical interactions the interaction is a form of two-wave-mixing so there is a direct relationship between the period of the microwave electrodes and the mechanical wavevector. Similar to the clamped OMC, the mechanical wave vector can be chosen to lie outside the continuum in the electromechanical section.

[0201] Similar design principles as explained above can be applied to confine mechanical modes in conjunction with an electric field in a clamped device. Such an electromechanical resonator is called an electromechanical crystal (EMC) when there is periodic patterning and again consists of defect and mirror cells.

[0202] Just as for the optomechanical crystal, the parameters of the defect cell, e.g. width, height, period, device layer orientation and hole size, are chosen such that:

[0203] • a mechanical mode at microwave frequencies exists outside of the continuum.

[0204] • the mechanical mode can be well actuated with an electric microwave field with regards to electromechanical coupling such that they yield quantum-level (see above for definition) interaction strengths.

[0205] Analogously, the parameters of the mirror cell, e.g. width, height, period and hole size, are chosen such that

[0206] • it has a quasi-band gap at the operating frequencies of the defect cell.

[0207] • a transformation from defect cell into mirror cell with several additional mirror cells will act as a low loss mirror to mechanical waves.

[0208] In addition, the number of fingers and length of the electromechanically active material (like lithium niobate) is chosen such that, in case of quantum transduction, the attached microwave circuit does not suffer excessive losses resulting from intrinsically lower microwave quality factor in this material nor from a too large mode density of electromechanically active modes which are not coherently participating in the transduction process.

[0209] As an example, we show a device featuring a piezoelectric Lithium Niobate layer above a Silicon layer clamped to a Sapphire substrate (Fig.6). Fig.6b shows the band diagram of the defect cell while the right insets show the frequencies at as a function of the perturbation from defect to mirror cell. Again, the area associated with the continuum is shaded gray and the black line denotes the mode of interest. Furthermore, the orientation of the device layer is chosen such that it is beneficial for the piezoelectric coupling. In this example, the device is designed for X-cut Lithium Niobate, oriented 45 degrees off the crystal z-axis. Thin electrodes of a metal such as NbTiN, NbN, or TiN can be deposited on the top of the device such that they overlap with the potential associated with the mechanical mode. Such an electromechanical crystal is shown in Fig.6a with its mechanical mode profile at when an alternating potential is applied, the electrical and mechanical fields overlap and couple through the piezoelectric effect. For efficient piezoelectric coupling one can choose materials such as Lithium Niobate (LiNbO3, LN). Finite element simulations are used to estimate the coupling for such a configuration, where we assume the electromechanical resonator to be coupled to a microwave resonator with around 450 Ohms impedance. Also, we assume the lithium niobate device layer to have a 15 degree sidewall angle, a realistic value in practical situations. We design EMCs with zero-point coupling rates of at least Such electromechanical coupling rates have not previously been described in clamped electromechanics. For co-integration with silicon devices it can furthermore be helpful to add a Silicon layer beneath (Fig.6a). Then, the mechanical mode can be made to leak into the silicon layer of a separate, pure silicon structure. This case is shown in the section “clamped piezo-optomechanical resonator”. When used in conjunction with optical photons, electrodes made of superconductors with short quasi-particle lifetime, such as NbTiN or NbN are preferred. In practice, the electrodes can be connected to a variety of circuitry. This could for instance be a microwave signal generator, a microwave resonator or a qubit. The latter may be either integrated on the same chip, or fabricated on a different chip in which case they could be connected via a wirebond or via capacitive / galvanic / inductive coupling using a flip-chip technique. In cases where the impedance mismatch between the EMC and the circuitry is problematic, electrodes made of superconductors with high kinetic inductance, such as NbTiN or NbN, can be used. In cases where aggressive cleaning of the device, e.g. with piranha solution or hydrofluoric acid, is desired, superconducting metals like Tantalum or NbTiN, may be appropriate.

[0210] Figure 6 (appended), (a) The active region of a clamped EMT with u denoting normalized mechanical displacement, (b) Mechanical band diagram of the unit cell. The right plot shows the X-point frequencies as a function of the perturbation from defect to mirror cell. The shaded area denotes the realm of continuum modes. The mode of interest is solid. The Lithium Niobate is assumed to have a sidewall angle of 15 degrees.

[0211] Clamped partial mirrors For a variety of applications it is advantageous to couple the mechanical mode of the optomechanical region with the mechanical mode of a different, spatially separated device. This is enabled by a third cell, the partial mirror cell.

[0212] Its parameters, e.g. width, height, period and hole size are chosen such that:

[0213] • it has a quasi-bandgap at the optical mode frequency of the optomechanical crystal.

[0214] • it simultaneously does support a mechanical mode at the mechanical mode frequency of the OMC defect cell.

[0215] • the mechanical mode does not lie inside the continuum.

[0216] In the following, we present one example design for such a clamped partial mirror cell with the same material platform as the OMC unit cells. The band structure needed for this kind of cell is for example obtained by increasing the lattice parameter a while simultaneously decreasing the ellipses size perpendicular to the beam with respect to the defect cell. The former perturbation decreases the frequencies of the optical band opening a quasibandgap while the latter increases the frequencies of the mechanical band ensuring that the partial mirror continues to support a mechanical mode while tapering.

[0217] As a consequence, the mechanical mode can propagate in the partial mirror section while the optical mode remains confined in the defect region. Fig.7.1 a, b show the optical and mechanical mode profile of such a structure. The right insets show the mode frequencies at along the perturbation from defect cell to partial mirror cell, showing that there is a quasi-bandgap for optics but not for mechanics.

[0218] Furthermore, the partial mirror region has an increased group velocity which is expected to be beneficial for using it as a phonon waveguide. In Fig.7.2 the mechanical group velocity for defect cell and partial mirror cell is presented for all available wavevectors below their respective continuum.

[0219] Figure 7.1 (appended) (a) normalized electric field E_y of the optical mode which is localized in the defect region, (b) Normalized mechanical displacement u showing that the mode is localized in both defect and partial mirror region. The insets of (a) and (b) show the X point frequencies along the perturbation from defect- to partial mirror cell for optics and mechanics respectively. In the clamped partial mirror region there is a quasi-bandgap for optics but not for mechanics. Figure 7.2 (appended) Group velocities for clamped gigahertz mechanics.

[0220] In the partial mirror unit cell, group velocities ranging from 0 to at least 4 km / s are achievable. This means such gigahertz mechanical modes can find use-cases in connectivity across a chip as well as for compact delay lines.

[0221] Very similar principles as above can be applied to yield partially leaky mirrors for pure EMTs or for the EMT component in a hybrid EMT-OMT.

[0222] The EMT can be connected via partially leaky mirrors or evanescent couplers to either mechanical waveguides or cavities for storage and delays of classical and quantum signals. Arbitrary mechanical quantum states can be generated if qubits are connected to the EMT as well.

[0223] Clamped piezo-optomechanical resonator (hybrid EMT-OMT; microwave-to-optics transduction)

[0224] The individual clamped devices, optomechanical crystal, electromechanical crystal and partial mirror can be co-integrated in a clamped piezo-optomechanical cavity (Fig.8). This device consists of an optomechanical region, where a mechanical and optical mode are co-localized, and an electromechanical region, where a mechanical and electrical mode are co-localized. The two sections are connected by a phonon waveguide with a partial mirror as described above. The partial mirror shields the electromechanical section from the optical mode. As described before, the unit cells of two adjacent sections are connected by transforming their parameters into one another.

[0225] The parameters associated with the unit cells of the EMC include e.g. period, width, height and hole size. The configuration can also differ between the thin-films if several different thin-films are involved for either electromechanical and / or optomechanical purposes.

[0226] In addition to the functions associated with the pure OMC and EMC, described before, the parameters of the defect cells of EMC and OMC are chosen such that: • the mechanical modes in the optomechanical and electromechanical regions hybridize. This means that a single mechanical mode is localized simultaneously in both the electromechanical region and the optomechanical region.

[0227] Often, the EMC region features a piezoelectric material (piezoelectric layer) on top of another material (optomechanical layer) in which the optomechanical interaction takes place. It can be advantageous to keep the piezoelectric layer spatially separated from the optomechanically active region, i.e. the piezoelectric layer only partially covers the optomechanical layer.

[0228] For the transformation from piezoelectric on optomechanical layer on substrate to only optomechanical layer on substrate the parameters of the cells are chosen such that:

[0229] • the mechanical waves are leaking from the electromechanically active region into the optomechanically active region and the other way around without too much scattering

[0230] • the variation of the parameters from one cell to the next one is small enough (near-adiabatically) to not cause scattering of the waves.

[0231] • at the end of the transformation the piezoelectric layer is removed.

[0232] • the cell continuously supports a mechanical mode at the mechanical operating point frequency throughout the transition.

[0233] This transformation as well as the individual OMC and EMC sections do not have to be based on a periodic hole structure but can be optimized with more general patterning, for instance using inverse design techniques.

[0234] As an example, this principle for realizing clamped piezo-optomechanical devices, i.e. hybridized EMC-OMC devices, is illustrated here for design featuring a silicon beam (optomechanical layer) with a lithium niobate top layer (piezoelectric layer) on the electromechanical section. The device is clamped to a sapphire substrate. The Lithium Niobate (LN, LiNbO3) layer on top of the silicon beam is patterned such that it fulfills the functions listed above. In the example shown in Fig.8 the LN is slowly removed over the course of several unit cells in order to minimize the scattering loss. This is achieved by removing the LN layer close to the holes first while simultaneously widening the ellipses in the direction perpendicular to the beam. Additionally, the choice of the unit cell parameters in this region between the two cavities can be used to tune the coupling strength between the two mechanical modes. Fig.8a, b show the mechanical and optical mode profiles respectively, where the mechanical mode is at ^W^ — u.47GHz Depending on the exact device parameters the mechanical modes in the two sections can be brought to hybridize and a single mechanical mode can exhibit substantial optomechanical- and electromechanical coupling (Fig.8c). Analogously to the example presented here, a longer optomechanical cavity can be used in order to increase the thermal contact area, as discussed before, while retaining strong electromechanical and optomechanical coupling.

[0235] For example, a full piezo-optomechanical transducer featuring LN on Silicon on Sapphire may have the following parameters, where each parameter is understood to have a range of minimum around 1% associated with it. For the OMC defect region this is around 31 unit cells, 190nm period, 700nm width, 220nm thickness, 95nm ellipse x-axis, 390nm ellipse y-axis. For the OMC mirror region: 8 unit cells, 380nm period, 700nm width, 220nm thickness, 166nm ellipse x-axis, 505nm ellipse y-axis. The transition between the two regions is 8 unit cells long.

[0236] For the OMC partial mirror region: this is around 8 unit cells, 380nm period, 700nm width, 220nm thickness, 166nm ellipse x-axis, 392nm ellipse y-axis.

[0237] For the EMC defect region: for example around 1 unit cells, 300nm period, 550nm width, 150nm thickness, 217nm ellipse x-axis, 240nm ellipse y-axis. For the EMC mirror region: 8 unit cells, 420nm period, 550nm width, 150nm thickness, 225nm ellipse x-axis, 247nm ellipse y-axis. The transition between the two regions is 8 unit cells long. Here the parameters are understood to describe the LN geometry, where a 20 degree sidewall was assumed. As a buffer, the holes in the Silicon layer are 50nm smaller. A total of 2-7 electrodes could be used for example.

[0238] As for the transition between EMC and OMC, the unit cell is tapered to a transition cell with: 190nm period, 700nm width, 220nm thickness, 330nm ellipse x-axis, 525nm ellipse y-axis,

[0239] Figure 8 (appended) (a) Logarithmic mechanical displacement in the piezo- optomechanical crystal at 5.47GHz. (b) Normalized electric field E_y of the optical mode. Part of the mechanical mode is co-localized with the optical mode in the defect region to the left. Another part is localized in the electromechanical region to the right. Together, (a) and (b) defines the active region of the hybrid EMT-OMT. (c) Example optomechanical and electromechanical coupling rates of modes supported by this simulated device.

[0240] Clamped mechanical bends

[0241] The mechanical modes in our clamped approach can be very well confined in both resonators and waveguides. Mechanical waveguides can be made with bending radii down to 3 urn while keeping the clamping-limited mechanical quality factor above 1 e10. This enables compact gigahertz and clamped mechanical ring resonators in analogy to those commonly pursued in for example the field of silicon photonics. While the mechanical properties are highly tailorable through design, we for example simulate mechanical group velocities of 1000 m / s with clamping-loss-limited propagation length around 40 mm with a bending radius of 11 micrometers. Thus a mechanical waveguide 90 degree bend can have clamping-limited loss of below 0.1 % enabling quantum-level signals to pass through such bends. Thus the approach allows for extremely compact bent mechanical waveguides, which are conducive to realizing large-scale gigahertz mechanical circuits where connectivity can be provided between many different systems by the propagating mechanical waves. The propagating mechanical waves could also be used for e.g. sensing. Mechanical material anisotropy can affect such bends and may need to be mitigated for certain devices through for example geometrical changes of the structure along the bend.

[0242] Clamped opto- and electromechanics: manufacturing process

[0243] To fabricate the new class of devices different material systems need to be integrated together into one stack. In the designs we have illustrated above based on piezoelectricity, four materials may be needed.

[0244] Material systems:

[0245] 1 . A thin-film layer with a high optical refractive index such as silicon. This layer should support high mechanical quality factors. With structures where optomechanical transduction is performed materials which support large optomechanical coupling rates are preferred.

[0246] 2. A substrate layer which supports the thin-film devices on at least their bottom side, such that the thin-film devices are clamped. And such that heat can escape well through the substrate. Said substrate layer may consist of a complex combination of different materials which may help 3. A metal layer which connects the mechanical device to the microwave circuit via electromechanical coupling mechanisms such as capacitive electromechanics or piezoelectricity.

[0247] 4. In the case of electromechanical coupling via piezoelectricity: also a piezomaterial (PM) to couple microwave signals to gigahertz mechanics. Layer may be the same as the thin-film layer with optical refractive index as described in material system number (1 ). This typically yields the strongest electromechanical coupling rates and is the case we illustrate here. Without a piezomaterial similar designs can be realized with capacitive electromechanical through DC-biased electrostrictive effect.

[0248] Through a technique known as micro-transfer printing (uTP) a variety of material systems can be combined which might not otherwise be available commercially. Moreover, process steps of one material layer that would be harmful to one or more of the other material systems can be performed prior to the uTP step. One example is that lithium niobate can be etched with argon milling prior to uTP transfer; a process which could otherwise lead to roughness in the silicon which is harmful to optical resonator quality. The uTP can be done either unaligned or aligned. In case of unaligned uTP each coupon must contain an alignment mark which can be used in the subsequent nanofabrication step for alignment.

[0249] The order of fabrication process steps can be done in a large variety of possible combinations, with a minimum need for the following steps:

[0250] 1. Silicon etch

[0251] 2. Metal deposition

[0252] 3. Piezomaterial etch

[0253] 4. Micro-transfer printing of coupons

[0254] In the case where piezoelectric effect is not utilized and capacitive electromechanics through DC-biased electrostrictive effect is used instead, step (3) can be omitted and step (4) is not strictly necessary. A non-exhaustive list is provided below to illustrate the variety of ways in which the new class of opto- and electromechanical devices can be manufactured where a PM is utilized:

[0255] 1. PM etch, 2. Si etch, 3. Metal deposition, 4. Aligned uTP

[0256] 1. PM etch, 2. Si etch, 3. Aligned uTP, 4. Metal deposition 1. PM etch, 2. Metal deposition, 3. Si etch, 4. Aligned uTP, 5. Optional 2nd Metal deposition

[0257] 1. PM etch, 2. Unaligned uTP, 3. Si etch, 4. Metal deposition

[0258] 1. Si etch, 2. Metal deposition, 3. PM etch, 4. Aligned uTP

[0259] Depending on the device design, different fabrication steps can be added into the scheme. Examples of added steps occur in case of coupon designs which require additional Si or PM etching steps after uTP. With uTP the fabrication process is versatile and common nanofabrication processes not listed here can be added, depending on precise device and material requirements.

[0260] Integrating the design with a microwave circuit, which can be superconducting at cryogenic temperatures, could be done either on the same chip, or on a separate chip. Connecting the microwave circuit can be done by wire bonding or through capacitive, galvanic, or inductive microwave coupling by the use of for example flip-chip technology. In our example designs the thin-film layer consisted of silicon (for optomechanical interaction) and lithium niobate (for electromechanical interaction). Many other material combinations are possible, and the thin-films could be manufactured out of a combination of many different crystalline or non-crystalline thin-films to optimize device properties such as confinement or thermal anchoring. Between the crystalline layer (silicon in this example) and substrate one can add an additional thin layer in between, which can function as a bonding layer and / or as a sacrificial layer to underetch some regions of the design. A bonding layer can improve contact and reduce stress in the interface of two crystalline materials which can make wafer fabrication easier. One embodiment of the invention could feature several wafers bonded together and then sequentially etched.

[0261] One may also locally release part or all of the device for improved performance. Such a layer could be fabricated by defining a device region on top of a sacrificial substrate. By selectively only creating through-holes for an etchant in those regions, the device may be only locally suspended.

[0262] In Fig.9 the fabrication process of a device utilizing piezoelectric coupling onto a clamped silicon device is outlined. To create the coupon one starts with a PM on a sacrificial layer such as, but not limited to, lithium niobate (LN) or aluminum nitride (AIN) on a sacrificial layer such as silicon dioxide. The PM is first patterned using an appropriate etching technique for the PM, such as ion beam etching for LN. The pattern outlines the region that will be transferred and is attached to the surrounding thin film though thin tethers in multiple locations. The pattern, called a “coupon”, is suspended by etching away the sacrificial layer underneath with a process that does not damage the PM layer. For silicon dioxide sacrificial layers hydrofluoric acid (HF) or buffered oxide etchant (BOE) is an effective method to underetch. Next, for example a PDMS stamp is used to break off the tethers and pick up the coupon. The picked-up coupon is then transferred onto a receiver substrate by bringing the stamp into contact. With PDMS stamps, adhesion levels can be controlled through the speed of the stamp during pick-up / release. Through slow separation it is therefore possible to transfer coupons onto a receiver substrate with high yield (> 95%). The process can realize fully crystalline material combinations such as LN on silicon on sapphire. The silicon on sapphire could be epitaxially grown or bonded with an intermediate bonding layer such as silicon dioxide. The transfer can be done either with or without alignment. If an aligned uTP is performed alignment marks have to be patterned into the source and receiving substrate beforehand. With unaligned uTP an alignment mark has to be incorporated into the coupon design. Metal is deposited onto the structure to create the finished opto- and electromechanical device. The metal is often evaporated at an angle to the thin film in order to also cover the sides of the pattern and ensure electrical contact. The chosen metal can be, but is not limited to, Al, NbTiN or Ta. Further microwave circuit fabrication steps can be added to connect for example a superconducting circuit to the opto- and electromechanical device.

[0263] Figure 9 (appended). Example process steps for fabrication of a PMSi device in this case via unaligned uTP. Transfer of PM onto Si on substrate wafer could be fabricated in large scale and for further processing in established foundries. Alternatively, the PM could be transferred with aligned uTP after a foundry process has already taken place.

[0264] The manufacturing process is compatible with established foundry10processes. From the point after uTP no additional release step of the structure is necessary and remaining

[0265] 10"Foundry" process: fabrication steps taking place inside semiconductor foundries such as those built and operated by GlobalFoundries, TSMC and others. Capable of supplying massive commercial markets e.g. associated with mobile phone and laptop fabrication steps can be performed using established foundry processes. Additional cladding depositions could be added to the device afterwards as well; for example to prevent native oxide growth or to improve thermal anchoring further. Said cladding can be solid, gas or liquid and may consist of a complex combination of different material regions.

[0266] The process of uTP can be repeated to combine the different material systems, to combine novel material combinations. With this method, we can fabricate high quality electro- and optomechanical devices which might not be possible with today’s commercially available wafers. In Fig.10 a process flow to fabricate such a device is shown using two uTP steps. First a PM coupon is prepared following the same fabrication described earlier. The coupon is transferred onto silicon with a sacrificial layer underneath. The silicon is then patterned using ICP etching to produce another coupon out of the PM and Si stack. The coupon is suspended by underetching the sacrificial layer and transferred onto a substrate wafer or chip using uTP. The silicon is then patterned once more using ICP and finally metal is deposited onto the pattern to create the finished device. The metal is evaporated at an angle to the thin film in order to also cover the sides of the pattern and ensure electrical contact.

[0267] Figure 10 (appended) Fabrication process using multiple uTP steps for fabrication of an electromechanical and / or optomechanical transducer device.

[0268] We note that such repeated transfer-printing as above can be cascaded and be used to create arbitrary composite structures that can have better optomechanical and / or electromechanical properties. For example mechanical and / or optical Bragg mirrors could be created in the substrate and / or cladding. Also composite layers made by for example repeated transfer-printing or repeated growth steps could result in better heat conduction and / or better confinement properties.

[0269] Instead of via transfer-printing, the right combination of electromechanically and optomechanically active materials can also be manufactured by wafer bonding and then chips. Our devices are at minimum manufacturable at the scale associated with the field of silicon photonics which also aims to leverage such foundries. subsequent etching. For example wafers of lithium niobate can be bonded to wafers of silicon on sapphire potentially with a sacrificial SiN or SiO2 layer in between. This could allow etching of the electromechanically active layer without damaging the optomechanically active layer (or vice versa). We note that the silicon on sapphire wafers themselves could also be based on bonding technology where a thin film of SiO2 may be in between the thin-film optomechanical Si layer and the sapphire substrate. In such a case generating a completely crystalline structure could be realized by removing the thin bonding layer and partially collapsing the thin-film Si layer on the substrate sapphire layer. Then subsequent processing to electromechanically active layers like lithium niobate could proceed as above.

[0270] Clamped opto- and electromechanical arrays and more complex systems using the building blocks

[0271] In the above, we focused on describing the individual opto- and electromechanical elements and how they realize strong opto- and electromechanical coupling rates as well as high coherence, and how they can be manufactured at scale for both quantum and classical applications. More complex systems can be built using such elements, consisting of many such elements which could be made tunable using for example capacitive driving or second-order nonlinearities of materials like lithium niobate. One example of a more complex system is an acousto-optic beam-steering device. Such devices can consist of a linear array of adjacent and parallel electromechanical sections as presented above, each of them can for example inject the mechanical waves into long adjacent, closely spaced, and parallel optomechanical waveguides similar to the unit cell designs with periodic symmetry present above. In these waveguides, coupling between on-chip optical modes and off-chip optical modes could take place via the same kind of optomechanical interactions we have described above, i.e. via photoelastic and moving boundary effects. Our devices can also improve upon key drawbacks of previous approaches also in such more complex systems and arrays. For example they support strong electro- and optomechanical coupling rate and are clamped so they do not suffer from film inhomogeneity and internal stress the same way a suspended system does. Three-wave-mixing interaction rates between one on-chip optical mode, one radiating optical mode, and the on-chip mechanical mode can be strong as the mechanical mode can be pumped very efficiently as described above. By varying the mechanical frequency, optical beam-steering and sensing can be achieved even when radiating optical fields are involved such as in autonomous systems. In the following, further aspects of the concept will be described with reference to clamped and sideband-resolved silicon optomechanical crystals with high single-photon cooperativity.

[0272] In the following, the texts “Clamped and sideband-resolved silicon optomechanical crystals with high single-photon cooperativity” and “Clamped and sideband-resolved silicon optomechanical crystals with high single-photon cooperativity: Supplementary information” will follow.

[0273] Clamped and sideband-resolved silicon optomechanical crystals with high singlephoton cooperativity

[0274] Compiled March 25, 2023

[0275] Optomechanical crystals (OMCs) are a promising and versatile platform for transduction between mechanical and optical fields. State-of-the-art suspended OMCs confine gigahertz mechanics in a way that also prevents heat-carrying noise phonons from leaking away. Thermal anchoring may instead be provided by attaching the OMCs directly to the substrate. Previous work towards such clamped OMCs suffers from weak interaction rates and insufficient lifetimes. Here, we present a new class of clamped OMCs realizing, for the first time, optomechanical interactions in the resolved-sideband regime required for quantum transduction. Our approach leverages high-wavevector mechanical modes outside the continuum. We observe record zero-point coupling rates of 9O / (2TT) = 0.50 MHz in clamped OMCs along with a single-photon cooperativity exceeding that of previous clamped OMCs by about an order of magnitude. This opens a new avenue exploiting clamped OMCs in both classical and quantum communications, sensing, and computation. © 2023 Optica Publishing Group

[0276] Confinement of electromagnetic fields and mechanical motion lies at the heart of optomechanics. Wavelength-scale optomechanical devices are of interest for instance for sensing and microwave-to-optics transduction [1-3]. A leading class of optomechanical devices is the optomechanical crystal (OMC) [4], In the state-of-the-art suspended OMCs, the low-wavevector GHz mechanical modes are confined partly by suspending the device layer. Since GHz phonons cannot propagate in free space, this eliminates direct leakage into the substrate [4], However, suspension also comes at the cost of losing a channel through which heat-carrying noise phonons created by optical absorption can dissipate [5].

[0277] Confining coherent GHz phonons while letting heat-carrying phonons leak away fast is an understudied and challenging problem. One approach is to laterally connect a suspended OMC region with the rest of the device layer. While these two-dimensional OMCs have displayed impressive results, they also require in-plane bandgaps and the associated fine- tuning of geometrical parameters [6]. Another approach to provide thermal anchoring is to attach the OMC directly to a substrate. We call this class of OMCs clamped. Besides reducing fabrication complexity, clamped devices could ease co-integration between phononic, electronic, and photonic devices, all commonly fabricated in the silicon-on- insulator (SOI) platform. Demonstrations along these lines have been made by using e.g. bound states in the continuum [7] and geometrical softening [8, 9]. These approaches have not yet been able to compete with the conventional suspended systems because of weaker interactions and worse coherence times.

[0278] Here, we propose and demonstrate a new class of clamped, i.e. non-suspended, OMCs. In our SOI-based OMCs the optomechanical three-wave-mixing interaction takes places between two counter-propagating optical modes and a high-wavevector mechanical mode. We demonstrate the first clamped OMCs with mechanical frequencies exceeding their optical loss rates. This resolved-sideband condition is essential for low-noise quantum transduction between optical and mechanical fields. The new clamped OMCs have zero-point optomechanical coupling rates of at mechanical frequencies of In addition, our clamped OMCs can have a thermal contact area exceeding that of their suspended counterparts. Our results provide a new path for clamped OMCs to become a competitive platform for both classical and quantum optomechanical circuits. In the following, we first outline the design process of the clamped OMCs and move on to show finite-element simulations of both periodically symmetric and finite structures. Finally, we present fabrication and measurement results of our devices at room temperature.

[0279] Design. A key challenge in designing clamped OMCs is avoiding excessive mechanical leakage into the substrate while keeping a high optomechanical interaction rate. In contrast to its suspended counterpart, a clamped OMC supports a continuum of states for both optical and mechanical waves. Here, we limit leakage from the OMC mechanical mode to substrate bulk- and surface-acoustic-waves (SAWs) by exploiting large mechanical wavevectors. This enables the mechanical mode to be phase-protected from the acoustic continuum, similar to optical waveguides based on total internal reflection. Realizing strong optomechanical interactions with such high-wavevector mechanical modes requires that is the mechanical (optical) wavevector. This phase-matching condition is familiar from the realm of counter-propagating Brillouin interactions

[0010] .

[0280] We illustrate the principles of confinement for our clamped OMCs in Fig. 11 (a) and (b).

[0281] Figure 11 (a) shows a phase-matching diagram for clamped and counter-propagating optomechanical interactions. The optical mode (blue) with frequency couples to a mechanical mode at cum. Counter-propagating optical modes interact with a phase- protected mechanical mode at whose wavevector lies outside the continuum of mechanical modes in the substrate. Figure 11 (b) shows SAW frequencies in silicon dioxide at the mechanical wave vector The optical pump has a vacuum wavelength nm and effective index and interacts with mechanics in an OMC with unit cell with period a. Phase-protection from both optical and mechanical continuea is achieved for all The operating point for the presented OMC defect cell is marked by a star.

[0282] For a mechanical mode with frequency to fall below the continuum, we require that the wavevector satisfies where is the substrate SAW phase-velocity. In addition, the first Brillouin zone associated with the periodicity a sets an upper bound to the set of unique wavevectors. To visualize this operating window, we plot SAW frequencies for different operating points in Fig. 11 (b). This upper bound for guided mechanical mode frequencies is calculated as Optical wavevectors stem from a pump with assumed effective index a vacuum wavelength In addition, we use for silicon dioxide

[0011] . Clamped OMC operation with low mechanical and optical radiation losses is accessible for all such that This analysis leads us to explore units cells with smaller periods a as they provide larger operating windows.

[0283] Exploring unit cells with periods around a = 190 nm, we find a guided X-point mechanical mode around At this mechanical wavevector, we calculate the SAW frequency to be This places our mode firmly below the acoustic continuum line. The mode profile resembles the “pinch mode” first reported in

[0012] for suspended OMCs (see Fig. 13a) The mechanical motion is primarily longitudinal along the OMC, in contrast to recently proposed modes

[0013] . The mode is guided despite the silicon device layer having faster speed of sound than the substrate. We attribute this to the unit cell's relatively large surface-to-volume ratio. This has been observed to reduce the effective stiffness of the structure, allowing for modes with guidable phase-velocities [9, 14], We design a defect unit cell around this mechanical mode and calculate mechanical and optical dispersion diagrams for periodic symmetry Fig. 12. We choose the parameters of the unit cell such that the waveguide supports an optical C-band mode at half the mechanical wavevector such that Considering counter-propagating optomechanical interactions, we calculate a unit cell zero-point optomechanical coupling rate of This large coupling is mostly mediated by the moving boundary effect and is comparable with those of conventional suspended OMC defect cells [4], We further investigate the implications of high-wavevector mechanics in optomechanical interactions in Supplement A.

[0284] To confine the mode described above in a standing-wave OMC, we design a second type of unit cell: the mirror cell. By modifying the parameters of the defect cell, the optical and mechanical bands can be pulled below the localized mode frequencies, opening a quasibandgap (Fig. 12)

[0015] . The OMC is thus assembled by adiabatically transforming the defect cell in the center into the mirror cell at the cavity perimeters [4], In this case, the mechanical quasi-bandgap only exists for the first half of the perturbation before the mirror cell becomes a host to the continuum modes. Yet, a mechanical quasi-band gap lasting for only a few unit cells suffices to reflect a vast majority of the mechanical field.

[0285] A consequence of the optical mode not being at the X-point is that it is not the cavity’s fundamental- but a higher order mode. Additionally, the optical mode decays slower in the mirror transition region compared to modes. Both of these effects reduce the spatial overlap between optical and mechanical modes. We keep most of the unit cell’s interaction strength in the full OMC by adding additional defect unit cells before starting the mirror region. Indeed, the field profiles of mechanics and optics approach the defect unit cell fields for a clamped OMC as the number of defect unit cells N increases. While this reduces the zero-point coupling rates as the increased interface area between a longer clamped OMC and the substrate is also expected to improve thermal anchoring.

[0286] At this stage, we optimize our design using a Nelder-Mead algorithm. For an optimized clamped OMC with N = 31 defect cells, we simulate an optical mode with THz and a mechanical mode with with radiation-limited quality factors respectively (Fig. 13a). Fourier transformation of the cavity fields indicates that the phase-matching condition km~ 2kois met for counterpropagating optomechanical interaction (Fig. 13b). The simulated zero-point optomechanical coupling of the resulting new clamped OMC design is 9O / (2TT) = 0.50 MHz.

[0287] Fig. 13a shows optical (top) and mechanical (bottom) mode profiles of the clamped OMC with Ey the normalised electric field of the optical mode and u the normalised mechanical displacement. Fig. 13b shows a Fourier transform of the coupled optical field Ey (blue) and mechanical field ux(red).

[0288] Experimental results. We transfer the device pattern to a 220 nm silicon device layer using electron beam lithography (Raith EBPG 5200) followed by an HBr / Cb-based reactive ion etch (STS ICP MPX). Next, we clean the samples in 3:1 piranha solution before measurement. A top-down scanning electron micrograph of the finished device is shown in Fig. 14a which features two clamped SOI OMCs (false color). We measure the properties of the OMC at room temperature and atmospheric pressure. A pump laser (Santec TSLS70) at 1550 nm is injected into an on-chip bus waveguide through focusing grating couplers

[0016] with a coupling efficiency of 18%. The bus waveguide couples to the OMC evanescently. Light reflected off the OMCs is amplified and subsequently detected on a high-speed photoreceiver (see SI for details).

[0289] By sweeping the laser wavelength, we detect the optical resonance at THz with a total linewidth and external coupling rate MHz. Next, we modulate the pump intensity and demodulate the reflected signal with a vector network analyzer (Fig. 14b). We thus perform an S11 measurement which is used to extract the pump detuning coo where L is the frequency of the pump laser

[0017] , This data is also used to confirm external optical coupling rates. In addition, at an on-chip power level of 375 μW, we observe an electomagnetically induced transparency window at the mechanical frequency [18, 19].

[0290] Fig. 14 shows fabrication and measurement of the OMC. Fig 14(a) shows top-down scanning electron micrograph of two silicon clamped overseas next to an optical bus waveguide. We measure the device in reflection. Scale bar indicates 1 μm. Light enters through the centre waveguide at the top of the figure and couples, evanescently to the OMCs. Fig. 14(b) shows how to extract the pump detuning A, we perform optical sideband spectrum spectroscopy of one of the OMCs measured with a vector network analyzer and a blue-detuned pump laser. The inset shows a zoomed in view of the dashed region where we observe in an electromagnetically induced transparency feature at the mechanical frequency. Fig 14(c) shows the measured thermal spectrum where the model fit (orange) shows a strongly coupled fundamental mechanical mode at along with two higher order modes. Fig 14(d) shows the mechanical line width of the fundamental mechanical mode as a function of the optical intracavity photos. We consequently consecutively carry out the experiment with first blue- and then red-detuned pumps. Fig. 14(e) how we observe cooperativity of unity and mechanical lasing for on- chip pump power of 375 pW.

[0291] Placing the pump blue-detuned from the optical resonance at roughly we measure the mechanical spectrum in the reflected light with a microwave spectrum analyzer (Fig. 14c) This reveals three mechanical modes. The spectral spacing and relative optomechanical coupling of the three mechanical modes qualitatively agree with simulation, and the absolute frequencies agreed to within 150 MHz. The fundamental mechanical mode frequency is measured to be Crucially, this puts the device operation in the resolved-sideband regime with To the best of our knowledge, this is the first demonstration of a clamped OMC in this regime. We note a slight asymmetric feature in the fundamental mode (Fig. 14c) which we suspect is related to geometrical disorder. This feature will be subject to future studies. We move on to measure the zero-point optomechanical coupling rate by measuring the mechanic linewidth at varying optical pump powers

[0020] . For both blue and red detuned pumps etc. We find a strong zero point optomechanical coupling rate by measuring the mechanical line with at varying optical pump powers for both blue and red detuned pumps. We find a strong zero-point optomechanical coupling rate of 9O / (2TT) = 0.50 MHz and a mechanical linewidth of The zero-point coupling rate is in excellent agreement with simulations. Therefore, the single photon cooperativity of the device is This cooperativity exceeds that of previously measured clamped devices by about a factor of 7 [7-9]. We suspect higher single-photon cooperativities are enriched at cryogenic temperature temperatures where phonon phonon scattering mechanisms decrease insignificance. To highlight our device performance in relation to state-of-the-art, we present a summary of important parameters for previous clamped work in Table 1.

[0292] Table 1. Table comparing clamped optomechanical structures on key parameters.

[0293] Increasing the pump power in the bus waveguide to 0.375 mW with a blue-detuned pump we reached a cooperativity of unity. This is demonstrated in Figure 14(e) where where we show mechanical power spectra as the pump detuning in swept closer to A = When the detuning approaches the mechanical frequency, we observe self-induced oscillations in the fundamental mechanical mode.

[0294] In conclusion, we designed and demonstrated a new class of clamped OMC's in SOI leveraging high-wavevector mechanical modes at gigahertz frequencies and counterpropagating optomechanical interactions. To the best of our knowledge, they are the first clamped OMC's in the resolved-sideband regime - a key requirement for low-noise quantum transduction between optics and mechanics. We observe a zero point optomechanical coupling rate of in excellent agreement with simulation. Their single-photon cooperativity exceeds that of previous clamped OMCs by about an order of magnitude. We suspect that further improvements of the optomechanical overlap are in reach. In addition, clamped OMCs can have significantly larger thermal contact area than suspended structures so they may suffer less from pump-induced mechanical heating in the cryogenic environments [5, 6]. Their operation does not require in-plane band gaps and relies on robust confinement of mechanical modes with frequencies and wave vectors outside the mechanical continuum. Our approach is not restricted to SOI; it is applicable to a wide range of materials and substrates. The clamped OMC's can be combined with e.g. spins or superconducting qubits. This opens a new avenue for scalable classical and quantum optomechanical circuits for applications in transduction, sensing and acoustic processing of electromagnetic signals [3]. Mechanical systems are often seen as a universal bus. Having them clamped on the substrate while copying strongly to light unlocks new opportunities in communication and computation.

[0295] Funding. We gratefully acknowledge support from the Wallenberg Centre for Quantum Technology and from the European Research Council via starting grant 948265.

[0296] Acknowledgement. We acknowledge Trond Hjerpekjon Haug and Per Delsing for helpful discussions. J.K. led the nanofabrication and measurement and assisted with design. P.B. led the design and assisted with nanofabrication and measurement. J.F. assisted with nanofabrication and measurement. J.K., P.B., and R.V.L. wrote the manuscript. R.V.L. provided experimental and theoretical support and conceived as well as supervised the project new.

[0297] Data availability. The datasets generated and analysed for the current study are available from the corresponding author on reasonable request.

[0298] Supplemental document. See Supplement 1 for supporting information.

[0299] References

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[0319] A. Phase-matching for counter-propagating interactions in a standing-wave OMC

[0320] As is motivated in the main text (see above), mechanical modes used for optomechanical coupling in clamped OMCs require high wavevectors to be phase-protected from the acoustic continuum. Using mechanical modes with non-zero wavevector implies a varying mechanical phase along the length of the OMC. This is in stark contrast to the T -point mechanical modes commonly adopted in suspended OMCs [1-3]. In this section, we explore how this affects the spatial phase of the local optomechanical coupling. Crucially, we require that there is no significant phase-induced cancellation of coupling when considering a cavity consisting of several unit cells.

[0321] We start by looking at the source of the optomechanical coupling. For the mechanical modes presented in the main text, the moving boundary effect is the most prominent contribution to the coupling. The interaction rate associated with this coupling is [3]

[0322] We restrict the following discussion to the contribution of this term; however, the same analysis is valid for the contribution of photoelasticity. Next, we turn our attention to the participating electrical (mechanical) fields E and (u). Crucially, we analyze the coupling between fields in a standing-wave cavity. This implies that the fields consist of both forwards (f) and backwards (b) propagating mechanical and optical components, i.e.,

[0323] Here, we assume that the longitudinal direction of the cavity is oriented in the x-direction. The fields E~ (u ) are unit cell Bloch functions and fo (x) ( frn (x)) are envelope functions for the optical (mechanical) fields. For the sake of brevity, we have without loss of generality omitted the presence of pump and sideband optical fields with differing wavevectors. The analysis remains applicable to the full three-wave-mixing scenario involving interactions between a counter-propagating pump and sideband of slightly differing frequencies.

[0324] We can now use Eq. (S1) to calculate the total coupling. First, we assume that the envelopes varies slowly, i.e. the cavity consists of many unit cells, we can therefore take the envelope functions to be effectively uniform along the OMC to illustrate the principles at work. Due to the squared optical fields, the result can be expanded into six terms with differently varying phase along the x-direction. The terms will have factors on the following forms

[0325] In the case that the mechanics has vanishing wavevector like in most suspended OMCs we have such that the terms ( / ) and ( / v) will lose their spatial phase, preventing cancellations in the total overlap integral. For the remaining terms to generate finite coupling, we find that in this scenario with small mechanical wavevector. This however leads to a contradiction that standing waves requires counter-propagating fields. Therefore, in a sufficiently long cavity, only two terms will contribute to the overall coupling and other terms are strongly suppressed. In essence, we see that low- wavevector mechanics couples well to co-propagating optical pump and sideband as is familiar from forward intra-modal Brillouin interactions [4],

[0326] On the other hand, in our clamped OMCs we have m,f = - m,b = km / = 0 such that the integrands of ( / ) and ( / v) tend to be strongly suppressed due to cancellations arising from different parts of the OMC. The phase-matching condition therefore becomes km = ±( o,f - o,b ), where the plus (minus) sign satisfies coupling in terms ( / / / ) and (v) (( / / ) and (v / )). As an example, when o,f = - o,b = ko we have that km = 2ko which is the principle upon which coupling is generated in the OMCs presented in the main text. This condition is familiar from backward intra-modal Brillouin interactions [4], Generally, integration over terms in Eq. (S3) with phase mismatch Ak leads to suppression by a factor sin(A / _) / A / _. As expected, we see that the implications of phasematching are most prominent for long cavities where wavevectors are well-defined. For the OMC presented in the main text, co-propagating terms involving our high-wavevector mechanical mode are suppressed by a factor kL = 2ko L = nN - 102 compared to counter-propagating terms. Here we used ko » rr / (2a) and L = Na with N = 31 the number of unit cells.

[0327] In summary, this analysis shows that phase-matched counter-propagating optomechanical interactions are possible in OMCs when using high-wavevector mechanical modes. Therefore, these interactions can be as strong and scale similarly to the more common approach based on co-propagating optomechanical interactions and low-wavevector mechanical modes.

[0328] B. Experimental setup

[0329] Fig. 15 (appended). Simplified measurement setup used for optical characterization of clamped OMCs. For further info, see Sec. 1 B. Legend: FPC: Fiber polarization controller. EOM: (Intensity) Electro- optic modulator. VOA: Variable optical attenuator. VNA: Vectror network analyzer. EDFA: Erbium doped fiber amplifier. ESA: Elecrical spectrum analyzer. DUT: Device under test.

[0330] A simplified diagram of the measurement setup is presented in Fig. 15. We carry out the optical characterization with a fiber-coupled continuously tunable laser in the C-band (Santec TSL570). An electro-optic intensity modulator (IXblue MX-LN-20) generates sideband tones at the mechanical frequency. To control the power sent to the device under test, we use digitally controlled variable optical attenuators (Sercalo VP1-9N-12- 16). Light polarization in the fiber network is managed through fiber polarization controllers.

[0331] After interacting with the device under test, the reflected light is circulated and amplified with an erbium doped fiber amplifier (Amonics AED FA- PA-35- B). A high-speed photoreceiver (Newport 1544-B) downconverts the GHz signals contained in the reflected light. We detect and display the thermal mechanical power spectrum with an electrical spectrum analyzer (R&S FSW26). Finally, we use a vector network analyzer (R&S ZNB20) to drive and demodulate the GHz optical modulation.

[0332] C. Wide mechanical spectrum

[0333] The mechanical spectrum shown in Fig. 14c in the main text features three prominent mechanical modes. Searching in a wider span of frequencies reveals four additional modes. We present the spectrum along with the measured frequencies in Fig. 16 (appended). To reduce the risk of misinterpreting modes as laser-induced phase noise, we perform the measurement with two independent laser sources (Santec TSL570 and Toptica CTL1550), yielding the same result.

[0334] Fig. 16 (appended). Wide range mechanical power spectrum desnisty in units of signal-to-noise ratio (SNR). Apart from the three modes presented in the main text, four additional modes are detected.

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Claims

CLAIMS1. A transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwaves to mechanical waves, and vice versa; b) transduction from mechanical waves to optical waves, and vice versa; c) transduction from microwaves to optical waves, via mechanical waves, and vice versa; said arrangement further comprising:- a transducer device (EMT; OMT; hybrid EMT-OMT) having a region which is either optomechanically or electromechanically active, or both, for said transduction; and- a substrate; characterized in that the transducer device (EMT; OMTShybrid EMT-OMT) is clamped to the substrate in a manner so that said active region has at least one side in full contact with said substrate.

2. An arrangement according to claim 1 , wherein said transducer device comprises a quasi-periodically repeated structure in the form of a plurality of unit cells (), each unit cell () having an internal closed volume () configured for co-localizing said mechanical and / or optical waves to said active region.

3. An arrangement according to claim 2, wherein each unit cell is manufactured from either optomechanically or electromechanically active materials having said volume extending through said unit cell.

4. An arrangement according to claim 2 or 3, wherein said plurality of unit cells are configured either as a 2D-confined waveguide or as a 3-D defined cavity.

5. An arrangement according to any one of the preceding claims, wherein said structure is configured for providing a desired mechanical mode with a wavevector and frequency that differs from the wavevectors and frequencies of any mechanical mode found in the continuum of mechanical modes associated with the substrate or cladding.

6. An arrangement according to any one of the preceding claims, wherein the transducer device is an optomechanical transducer (OMT).

7. An arrangement according to claim 6, wherein the optomechanical transducer (OMT) comprises a transducer device () which is manufactured from silicon and a substrate () which is manufactured from for example sapphire or silicon dioxide.

8. An arrangement according to any one of claims 1-5, wherein the transducer device () is an electromechanical transducer (EMT).

9. An arrangement according to claim 8, wherein the electromechanical transducer (EMC) comprises a transducer device () which is manufactured from a metal electrode layer such as NbTiN, a mechanically active crystalline layer such as silicon, and electromechanically active layer such as piezoelectric lithium niobate (LiNbOs) or DC-biased electrostrictive materials such as silicon and a substrate () which is manufactured from for example sapphire (AI2O3) or silicon dioxide (SiO2).

10. An arrangement according to any one of claims 1-5, wherein the transducer device is a microwave-to-optical transducer (hybrid EMT-OMT), using mechanical waves as an intermediary.11 . A transducer arrangement according to any one of the preceding claims, wherein said arrangement is configured for transduction at the quantum level with low noise, involving interaction between photons, corresponding to quantized units of electromagnetic radiation based on said microwave signals and optical signals; and phonons, corresponding to quantized units of mechanical vibrations based on said mechanical waves.

12. An arrangement according to any one of the preceding claims, wherein the OMT is configured for operating in a sideband-resolved regime where the mechanical frequency exceeds the optical linewidth.

13. An arrangement according to any one of the preceding claims, and being configured for either: transduction from mechanical waves to optical waves, and vice versa;transduction from microwaves to optical waves, via mechanical waves, and vice versa; wherein the arrangement is further configured for operating in a counter-propagating or inter-modal scattering regime where the high-wavevector mechanical mode interacts with two optical waves with opposite and / or sufficiently different wavevectors such that the optomechanical three-wave-mixing satisfies both frequency- and phase-matching conditions.

14. An arrangement according to any one of the preceding claims, wherein the transducer device (EMT; OMT; hybrid EMT-OMT) is non-suspended from the underlying substrate and clamped to the substrate in a manner so that said active region has at least one side in full contact with said substrate, thereby defining a thermal contact area between said active region and said substrate.

15. An arrangement according to any one of the preceding claims, wherein the transducer device (EMT; OMT; hybrid EMT-OMT) is based on a design in which the mechanical modes' wavevectors and frequencies are fully outside the continuum formed by acoustic excitations associated with the substrate, including surface and bulk acoustic waves.

16. Method for manufacturing a transducer arrangement for transducing signals according to at least one of the following alternative modes of operation: a) transduction from microwave waves to acoustic waves, and vice versa; b) transduction from acoustic waves to optical waves, and vice versa; c) transduction from microwave waves to optical waves, via acoustic waves, and vice versa; said method comprising:- providing a transducer device (EMT; OMT) having an active region which is either optomechanically or electromechanically active, or both, for said transduction; and- providing a substrate characterized in that the method further comprises:- a process for transfer of various combinations of thin-film layers of optomechanically and electromechanically active materials like silicon and lithium niobate so as to clamp said transducer (EMT ; OMT ; hybrid EMT-OMT) to the substrate in a manner so that said active region which is either optomechanically or electromechanically active, or both, has at least one side in full contact with said substrate, comprising- a process for assembling material layers, either the active EMT or OMT thin-films or composite substrates and claddings, through repeated micro-transfer printing.

17. Method for manufacturing a transducer arrangement, further comprising: - a process for assembling fully crystalline material stack of optomechanically and electromechanically active materials where the optomechanical stack is manufactured by selectively collapsing a pre-bonded silicon on sapphire wafer with SiO2 bonding layer.

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