Controlling Phonons and Photons at the Wavelength-Scale: Silicon Photonics Meets Silicon Phononics

Controlling phonons and photons at the wavelength-scale: silicon photonics meets silicon phononics

AMIR H.SAFAVI-NAEINI1,D RIES VAN THOURHOUT2,R OEL BAETS2, AND RAPHAËL VAN LAER1,2

1Department of Applied Physics and Ginzton Laboratory, Stanford University, USA 2Photonics Research Group (INTEC), Department of Information Technology, Ghent University–imec, Belgium .Center for Nano- and Biophotonics, Ghent University, Belgium

Compiled October 9, 2018

Radio-frequency communication systems have long used bulk- and surface-acoustic-wave devices supporting ultrasonic mechanical waves to manipulate and sense signals. These devices have greatly improved our ability to process microwaves by interfacing them to orders-of- magnitude slower and lower loss mechanical fields. In parallel, long-distance communications have been dominated by low-loss infrared optical photons. As electrical signal processing and transmission approaches physical limits imposed by energy dissipation, optical links are now being actively considered for mobile and cloud technologies. Thus there is a strong driver for wavelength-scale mechanical wave or “phononic” circuitry fabricated by scalable semiconduc- tor processes. With the advent of these circuits, new micro- and nanostructures that combine electrical, optical and mechanical elements have emerged. In these devices, such as optomechanical waveguides and resonators, optical photons and gigahertz phonons are ideally matched to one another as both have wavelengths on the order of micrometers. The development of phononic circuits has thus emerged as a vibrant field of research pursued for optical signal processing and sensing applications as well as emerging quantum technologies. In this review, we discuss the key physics and figures of merit underpinning this field. We also summarize the state of the art in nanoscale electro- and optomechanical systems with a focus on scalable platforms such as silicon. Finally, we give perspectives on what these new systems may bring and what challenges they face in the coming years. In particular, we believe hybrid electro- and optomechanical devices incorporating highly coherent and compact mechanical elements on a chip have significant untapped potential for electro-optic modulation, quantum microwave-to- optical photon conversion, sensing and microwave signal processing. arXiv:1810.03217v1 [physics.optics] 7 Oct 2018 CONTENTS 3 Photon-phonon interactions5 A Interactions between phonons and optical photons5 1 Introduction2 B Interactions between phonons and microwave photons ...... 7 2 Guiding and confining phonons3 4 State of the art8 A Total internal reflection ...... 3 B Impedance mismatch ...... 3 5 Perspectives9 C Geometric softening ...... 4 A Single-photon nonlinear optics ...... 9 D Phononic bandgaps ...... 4 B Efficient optical modulation ...... 10 E Other confinement mechanisms ...... 4 C Hybrid quantum systems ...... 11 F Material limits ...... 4 D Microwave signal processing ...... 12 2

E General challenges ...... 12 of length scales was used to demonstrate the first 2D- and 3D- confined systems in which both photons and phonons are con- 6 Conclusion 13 fined to the same area or volume [Fig.1]. These measurements have been enabled by advances in low-loss photonic circuits A Theoretical description 21 that couple light to material deformations through boundary A Cavities: 3D-confined ...... 21 and photoelastic perturbations. Direct capacitive or piezoelectric A.1 Linear detection of motion ...... 22 coupling to these types of resonances has been harder since the A.2 Back-action on the mechanical mode . . . 22 relatively low speed of sound in solid-state materials means that A.3 Understanding optomechanical coupling gigahertz-frequency phonons have very small volume, leading in a nanophotonic system ...... 23 to miniscule electrically-induced forces at reasonable voltages, B Waveguides: 2D-confined ...... 24 or in other words large motional resistances that are difficult to match to standard microwave circuits [7]. B Single-photon nonlinearity 25

Wavelength-scale Photons Phonons 1. INTRODUCTION con nement Microwave-frequency acoustic or mechanical wave devices have found numerous applications in radio-signal processing and sensing. They already form mature technologies with large mar- 0D kets. The vast majority of these devices are made of piezoelectric materials that are driven by electrical circuits [1–6]. A major technical challenge in such systems is obtaining the suitable 1D l matching conditions for efficient conversion between electrical and mechanical energy. Typically, this entails reducing the effective electrical impedance of the electromechanical component by increasing the capacitance of the driving element. This has 2D generally led to devices with large capacitors that drive mechan- τ τ ical modes with large mode volumes. Here, we describe a recent shift in research towards structures that are only about a wavelength, i.e. roughly one micron at gigahertz frequencies, across 3D in two or more dimensions. Greater confinement of mechanical waves in a device has both advantages and drawbacks depending on the application at hand. In the case of interactions with optical fields, higher Fig. 1. Confining photons and phonons to the wavelength- confinement increases the strength and speed of the interac- scale. Photonic and phononic systems can be classified action allowing faster switching and lower powers. A smaller cording to the number of dimensions in which they confine system demands less dissipated energy to achieve the same ef- photons and phonons to a wavelength. New structures have fects, simply because it focuses all of the optical and mechanical emerged in which both photons and phonons are confined energy into a smaller volume. High confinement also enables to the wavelength-scale in two or three spatial dimensions. scalable, less costly fabrication with more functionality packed Here we focus on such 2D- or 3D-confined wavelength-scale into a smaller space. Perhaps more importantly, in analogy to systems at gigahertz frequencies. Related reviews on inte- microwave and photonic circuits that become significantly easier grated opto- and electromechanical systems are [8–13]. The to engineer in the single- and few-moded limits, obtaining con- table gives as examples of 2D- and 3D-confined devices a sub- 2 3 trol over the full mode structure of the devices vastly simplifies µm silicon photonic-phononic waveguide [14] and a sub-µm designing and scaling systems to higher complexity. Confin- silicon optomechanical crystal [15]. The depicted 0D- and ing mechanical energy is not without its drawbacks; as we will 1D-confined structures are a long Fabry-Pérot cavity, a vertical- see below, focusing the mechanical energy into a small volume cavity surface-emitting laser [16] for the optical case and a also means that deleterious nonlinear effects manifest at lower thick quartz [17] and a thin aluminum nitride BAW resonator powers, and matching directly to microwave circuits becomes [18] for the mechanical case. significantly more difficult due to vanishing capacitances. We can classify confinement in terms of its dimensionality [Fig.1]. In this review, we primarily consider recent advances in The dimensionality refers to the number of dimensions where gigahertz-frequency phononic devices. These devices have been confinement is on the scale of the wavelength of the excitation demonstrated mainly in the context of photonic circuits, and in bulk. For example, surface acoustic wave (SAW) resonators share many commonalities with integrated photonic structures [1], much like thin-film bulk acoustic wave (BAW) resonators [2], in terms of their design and physics. They also have the poten- have wavelength-scale confinement in only one dimension – per- tial to realize important new functionalities in photonic circuits. pendicular to the chip surface – and are therefore 1D-confined. Despite recent demonstrations of confined phonon devices op- Until a few years ago, wavelength-scale phononic confinement erating at gigahertz frequencies and coupled to optical fields, at gigahertz frequencies beyond 1D remained out of reach. phononic circuits are still in their infancy, and applications be- Intriguingly, both near-infrared optical photons and giga- yond those of interest in integrated photonics remain largely hertz phonons have a wavelength of about one micron. This unexplored. Several attractive aspects of mechanical elements re- results from the five orders of magnitude difference in the speed main unrealized in chip-scale systems, especially in those based of light relative to the speed of sound. The fortuitous matching on non-piezoelectric materials. In this review, we first describe 3 the basic physics underpinning this field with specific attention cal [27] but not optical fields. Still, some structures find a sweet to the mechanical aspects of optomechanical devices. We dis- spot in this trade-off: the principle of total internal reflection is cuss common approaches used to guide and confine mechanical currently exploited to guide phonons in Ge-doped optical fibers waves in nanoscale structures in section2. Next, we describe the [28] and chalcogenide waveguides [9]. key mechanisms behind interactions between phonons and both Since silicon is “slower” than silicon dioxide optically, but optical and microwave photons in section3. These interactions “faster” acoustically, simple index guiding for co-confined optical allow us to efficiently generate and read-out mechanical waves and mechanical fields is not an option in the canonical platform on a chip. Section4 briefly summarizes the state of the art in of silicon photonics, silicon-on-insulator. Below we consider opto- and electromechanical devices. It also describes a few techniques that circumvent this limitation and enable strongly commonly used figures of merit in this field. Finally, we give co-localized optomechanical waves and interactions. our perspectives on the field in section5. In analogy to silicon photonics [19–24], the field may be termed “silicon phononics”. Ω Si While not strictly limited to the material silicon, its goal is to (a) Ω (b) develop a platform whose fabrication is in principle scalable to SiO τ τ 2 many densely integrated mechanical devices. K Γ K

SiO2 Si 2. GUIDING AND CONFINING PHONONS Si Si SiO2

SiO2 Phonons obey broadly similar physics as photons so they can be Si guided and conﬁned by comparable mechanisms, as detailed in Slow materials Impedance mismatch the following subsections.

A. Total internal reﬂection (c) Ω (d) Ω Ω In a system with continuous translational symmetry, waves incident on a medium totally reﬂect when they are not phase- Γ X K K matched to any excitations in that medium. This is called total K

Si internal reflection. The waves can be confined inside a slow Si medium sandwiched between two faster media by this mech- Si anism [Fig.2a]. This ensures that at fixed frequency Ω the SiO2 guided wave is not phase-matched to any leaky waves since its Geometric softening Bandgaps wavevector K(Ω) = Ω/vφ – with vφ its phase velocity – exceeds the largest wavevector among waves in the surrounding media at that frequency. In other words, the confined waves must have Fig. 2. Mechanisms for phononic confinement in micro- maximal slowness 1/vφ. This principle applies to both optical and nanostructures. We illustrate the main approaches with and mechanical fields [25, 26]. phononic dispersion diagrams Ω(K) and mark the operat- Still, there are important differences between the optical and ing point in black. (a) A waveguide core whose mechanical mechanical cases. For instance, a bulk material has only two excitations propagate slower than the slowest waves in the sur- transverse optical polarizations while it sustains two transverse rounding materials supports acoustic total internal reflection. mechanical polarizations with speed vt and a longitudinally po- Examples include chalcogenide rib waveguides on silica [9], larized mechanical wave with speed vl. Unlike in the optical silica waveguides cladded by silicon [27] and Ge-doped fibers case, these polarizations generally mix in a complex way at in- [28]. (b) Even when phonons are phase-matched to surface- or terfaces [25]. In addition, a boundary between a material and bulk-excitations, their leakage can be limited by impedance air leads to geometric softening (see next section), a situation mismatch such as in suspended silicon waveguides and disks in which interfaces reduce the speed of certain mechanical po- [14, 29–32] and silica microtoroids [33]. (c) In contrast to op- larizations. This generates slow SAW modes that are absent tical fields, mechanical waves can be trapped by surface per- in the optical case. So achieving mechanical confinement re- turbations that soften the elastic response such as in case of quires care in looking for the slowest waves in the surrounding Rayleigh surface waves [1], silicon fins on silica [34, 35] and structures. These are often surface instead of bulk excitations. all-silicon surface perturbations [36–39]. (d) Finally, phonons Among the bulk excitations, transversely polarized are slower can be trapped to line- or point-defects in periodic structures than longitudinally polarized phonons (vt < vl). with a phononic bandgap such as in silicon optomechanical Conflicting demands often arise when designing waveguides crystals [15, 40], line defects [41] and bullseye disks [42]. Many or cavities to confine photons and phonons in the same region: structures harness a combination of these mechanisms. photons can be confined easily in dense media with a high refractive index and thus small speed of light but phonons are naturally trapped in soft and light materials with a small speed B. Impedance mismatch of sound. In particular, the mechanical phase velocities scale as The generally conflicting demands between photonic and p vφ = E/ρ with E the stiffness or Young’s modulus and ρ the phononic confinement (see above) can be reconciled through mass density. For instance, a waveguide core made of silicon impedance mismatch [Fig.2c]. The characteristic acoustic

(refractive index nSi = 3.5) and embedded in silica (nSiO2 = 1.45) impedance of a medium is Zm = ρvφ with ρ the mass den- strongly confines photons by total internal reflection but cannot sity [25]. Interfaces between media with widely different easily trap phonons (for exceptions see next sections). On the impedances Zm, such as between solids and gases, strongly other hand, a waveguide core made of silica (vt = 5500 m/s) reflect phonons. In addition, gases have an acoustic cut-off fre- embedded in silicon (vt = 5843 m/s) can certainly trap mechani- quency – set by the molecular mean-free path – above which 4 they do not support acoustic excitations [43]. At atmospheric bandgap cavities and waveguides, does not lead to a full pressure this frequency is roughly Ωc/(2π) ≈ 0.5 GHz. Above phononic bandgap. Conversely, a rectangular array of cross- this frequency Ωc acoustic leakage and damping because of air shaped holes in a slab as has been used to demonstrate full are typically negligible. The cut-off frequency Ωc can be dras- phononic bandgaps in silicon and other materials, does not sup- tically reduced with vacuum chambers, an approach that has port a photonic bandgap. Nonetheless, both one-dimensional been pursued widely to confine low-frequency phonons [10]. [52] and two-dimensional crystals [40] with simultaneous pho- These ideas were harnessed in silicon-on-insulator waveguides tonic and phononic gaps have been proposed and demonstrated to confine both photons and phonons to silicon waveguide cores in technologically relevant material systems. [14, 29, 31] over milli- to centimeter propagation lengths. The Beyond enabling 3D-confined wavelength-scale phononic acoustic impedances of silicon and silica are quite similar, so cavities, phononic bandgaps also support waveguides or wires, in these systems the silica needs to be removed to realize low which are 2D-confined defect states. These have been realized in phonon leakage from the silicon core. In one approach [14], the silicon slabs with a pattern of cross-shaped holes supporting a silicon waveguide was partially under-etched to leave a small full phononic bandgap, with an incorporated line defect within silica pillar that supports the waveguide [Fig.2c]. In another, the the bandgap material [41, 53–55]. Robustness to scattering is silicon waveguide was fully suspended while leaving periodic particularly important to consider in such nano-confined guided silica or silicon anchors [29, 31]. wave structures, since as in photonics, intermodal scattering due to fabrication imperfections increases with decreasing cross- C. Geometric softening sectional area of the guided modes [56]. Single-mode phononic The guided wave structures considered above utilize full or par- wires are intrinsically more robust as they remove all intermodal tial under-etching of the oxide layer to prevent leakage of acous- scattering except backscattering. They have been demonstrated tic energy from the silicon into the oxide. Geometric softening is to allow robust and low-loss phonon propagation over millime- a technique that allows us to achieve simultaneous guiding of ter length scales [55]. Currently both multi- and single-mode light and sound in a material system without under-etching and phononic waveguides are actively being considered as a means regardless of the bulk wave velocities. Although phonons and of generating connectivity and functionality in chip-scale solid- photons behave similarly in bulk media, their interactions with state quantum emitter systems based on defects in diamond boundaries are markedly different. In particular, a solid-vacuum [57, 58]. boundary geometrically softens the structural response of the material below and thus lowers the effective mechanical phase E. Other confinement mechanisms velocity [Fig.2b]. This is the principle underpinning the 1D con- The above mechanisms for confinement cover many if not most finement of Rayleigh SAWs [25, 36]. This mechanism was used current systems. However, there are alternative mechanisms for in the 1970s in the megahertz range [36–38] to achieve 2D con- photonic and phononic confinement, including but not limited finement and was recently rediscovered for gigahertz phonons to: bound states in the continuum [59–61], Anderson localization where it was found that both light and motion can be guided [62, 63] and topological edge states [64, 65]. We do not cover in unreleased silicon-on-insulator structures [34]. More recently, these approaches here. fully 3D-confined acoustic waves have been demonstrated [35] with this approach on silicon-on-insulator where a narrow sil- F. Material limits icon fin, clamped to a silicon dioxide substrate, supports both localized photons and phonons. Phononic confinement, propagation losses and lifetimes are limited by various imperfections such as geometric disorder D. Phononic bandgaps [29, 40, 55, 66–68], thermo-elastic and Akhiezer damping [25, 69], Structures patterned periodically, such as a silicon slab with a two-level systems [70–72] and clamping losses [14, 73, 74]. grid of holes, with a period a close to half the phonons’ wave- Losses in 2D-confined waveguides are typically quantified by −1 length Λ = 2π/K result in strong mechanical reflections as in the a propagation length Lm = αm . In 3D-confined cavities one optical case. At this X-point – where K = π/a – in the dispersion usually quotes linewidths γ or quality factors Qm = ωm/γ. diagram forward- and backward-traveling phonons are strongly A cavity’s internal loss rate can be computed from the decay coupled, resulting in the formation of a phononic bandgap [Fig. length Lm through γ = vmαm in high-finesse cavities with negli- 2d] whose size scales with the strength of the periodic perturba- gible bending losses [75] with vm the mechanical group velocity. tion. The states just below and above the bandgap can be tuned Mechanical propagation lengths in bulk crystalline silicon are by locally and smoothly modifying geometric properties of the limited to Lm ≈ 1 cm at room-temperature and at a frequency lattice, resulting in the formation of line- or point-defects. This of ωm/(2π) = 1 GHz by thermo-elastic and Akhiezer damping. technique is pervasive in photonic crystals [44] and was adapted Equivalently, taking vm ≈ 5000 m/s one can expect material- to the mechanical case in the last decade [45–50]. This led to limited minimum linewidths of γ/2π ≈ 0.1 MHz and maximum 4 the demonstration of optomechanical crystals that 3D-confine quality factors of Qm ≈ 10 [25, 69] at ωm/(2π) = 1 GHz. both photons and gigahertz phonons to a wavelength-scale sus- These limits deteriorate rapidly at higher frequencies, typ- −2 −1 pended silicon nanobeams [40, 51, 52]. In these experiments, ically scaling as Lm ∝ ωm and Qm ∝ ωm [69, 72] or worse. confinement in one or two dimensions was obtained by periodic This makes the fm · Qm product a natural figure of merit for patterning of a bandgap structure, while in the remaining di- mechanical systems. For gigahertz frequency resonators at room mension confinement is due to the material being removed to temperature, the highest demonstrated values of fm · Qm are 13 obtain a suspended beam or film. on the order of 10 in several materials [76]. Intriguingly, the Conflicting demands similar to those discussed in sec- maximum length of time that a quantum state can persist inside tion2B complicate the design of simultaneous photonic-phononic a mechanical resonator with quality factor Qm at temperature hQ¯ m bandgap structures [49]. For example, a hexagonal lattice of T is given by tdecoherence = kT , and so requiring that the infor- circular holes in a silicon slab as is often used in photonic mation survive for more than a mechanical cycle is equivalent 5

−1 12 to the condition tdecoherence > ωm , or fm · Qm > 6 × 10 Hz as in Brillouin and Raman scattering and optomechanics, at room temperature [10]. Recently, new loss mitigation mech- where the latter includes capacitive electromechanics. anisms called “strain engineering” and “soft clamping” have been invented for megahertz mechanical resonators that enable • Direct coupling [Fig.3b]: one photon and one phonon inter- 8 mechanical quality factors and fm · Qm products beyond 10 act with each other directly as in piezoelectrics. and 1015 Hz respectively under high vacuum but without re- The parametric three-wave mixing takes place via two routes: frigeration [77–79]. This unlocks exciting new possibilities for quantum-coherent operations at room temperature. Finally, we • Difference-frequency driving (DFD): two photons with fre- note that many material loss processes, with the exception of quencies ω and ω0 drive the mechanical system through a 0 two-level systems [70], vanish rapidly at low temperatures (sec- beat note at frequency ω − ω = Ω ≈ ωm in the forces. tion4). • Sum-frequency driving (SFD): two photons with frequencies ω and ω0 drive the mechanical system through a beat (a) Parametric coupling 0 note at frequency ω + ω = Ω ≈ ωm in the forces. Photons Phonons

y ω ω’ Three-wave DFD is the only possible mechanism when the pho- ensi t ω tons and phonons have a large energy gap, as in interactions Ω al D

t r Ω ω’ Ω between phonons and optical photons. In contrast, microwave K photons can interact with phonons through any of the three- Spe c β’ β K wave and direct processes. Ω

(b) Direct coupling A. Interactions between phonons and optical photons Parametric DFD in a cavity is generally described by an interac-

y Ω tion Hamiltonian of the form (see Appendix) ensi t ω al D H = ( ) † t r ω Ω int h¯ ∂xωo a ax (1) Spe c β K with ∂xωo the sensitivity of the optical cavity frequency ωo to mechanical motion x and a the photonic annihilation operator. The terminology “parametric” refers to the parameter ω , Fig. 3. Generating and detecting phonons. (a) Interactions o essentially the photonic energy, being modulated by the me- between phonons and high-frequency photons occur through chanical motion [82–85], whereas the term “three-wave mix- parametric three-wave mixing: two high-frequency photons ing” points out that there are three operators in the Hamilto- couple to one phonon via third-order nonlinearities such as nian given by equation1. This does not restrict the interaction photoelasticity and the moving-boundary effect [80, 81]. In- to only three waves, as discussed further on. Describing the teractions between low-frequency photons and a phonon can Hamiltonian H in this manner is a concise way of captur- also occur through these mechanisms. Depending on which int ing all the consequences of the interaction between the electro- of the three waves is pumped, the interaction results in ei- magnetic field a and the mechanical motion x. The detailed ther down/up-conversion (δa δb + h.c.) or state-swapping dynamics can be studied via the Heisenberg equations of mo- (δa δb† + h.c.) events. The frequency-difference between the tion defined by a˙ = − i [a, H ] when making use of the har- two high-frequency photons ω − ω0 must approximately equal h¯ int † the phononic frequency Ω for efficient parametric interactions monic oscillator commutator [a, a ] = 1 [10]. Since by definition † to occur (left). In addition, in structures with translational x = xzp δb + δb with xzp the mechanical zero-point fluctua- symmetry the wavevector-difference between the two high- tions and δb the phonon annihilation operator, this is equivalent frequency photons β − β0 must also approximately equal the to † † phononic wavevector K for efficient coupling (right). Here we Hint = hg¯ 0a a(δb + δb ) (2) depict only DFD, SFD proceeds analogously but with minus with signs replaced by plus signs. (b) Direct conversion via second- g = (∂xωo)xzp (3) order nonlinearities such as piezoelectricity is possible when 0 the photonic and phononic energies match. Stronger mechan- the zero-point optomechanical coupling rate, which quantifies ical waves can typically be generated by direct conversion the shift in the optical cavity frequency ωo induced by the zero- than by indirect three-wave mixing of two optical waves. A point fluctuations xzp of the mechanical oscillator. Here we microwave photon can be converted into a phonon and sub- neglect the static mechanical motion [10, 86, 87]. Achieving sequently into an optical photon by cascading two of these large g0 thus generally requires small structures with large sen- 1/2 processes. sitivity ∂xωo and zero-point motion xzp = (h¯ /(2ωmmeff)) , where meff is the effective mass of the mechanical mode. This is brought about by ensuring a good overlap between the phononic 3. PHOTON-PHONON INTERACTIONS field and the photonic forces acting on the mechanical system [14, 80, 81] and by focusing the photonic and phononic energy In this section we describe the key mechanisms underpinning into a small volume to reduce meff. There are typically separate the coupling between photons and phonons. Photon-phonon bulk and boundary contributions to the overlap integral. The interactions occur via two main mechanisms: bulk contribution is associated with photoelasticity, while the • Parametric coupling [Fig.3a]: two photons and one phonon boundary contribution results from deformation of the interfaces interact with each other in a three-wave mixing process between materials [80, 81, 88, 89]. Achieving strong interactions 6 requires careful engineering of a constructive interference be- an infinitely long waveguide where phase-matching is strictly tween these contributions [14, 80, 81, 88]. Optimized nanoscale enforced. In contrast, a finite-length waveguide allows for in- silicon structures with mechanical modes at gigahertz frequen- teractions between a wider set of modes, although it suppresses cies typically have xzp ≈ 1 fm and g0/(2π) ≈ 1 MHz (section those with a large phase-mismatch (see Appendix). In essence, 4). The zero-point fluctuation amplitude increases with lower shorter waveguides permit larger violations of momentum con- frequency leading to an increase in g0: megahertz-frequency servation. The momentum selectivity can enable non-reciprocal mechanical systems with g0/(2π) ≈ 10 MHz have been demon- transport of both photons [95–98] and phonons [50, 99]. It is a strated [90]. continuum version of interference-based synthetic magnetism The dynamics generated by the Hamiltonian of equation2 schemes using discrete optomechanical elements [54, 100]. can lead to a feedback loop. The beat note between two photons Cavities can be realized by coiling up or terminating a 2D- with slightly different frequencies ω and ω0 generates a force that confined waveguide with mirrors. Then the cavity’s optome- 0 drives phonons at frequency ω − ω = Ω. Conversely, phonons chanical coupling rate g0 is connected to the waveguide’s cou- modulate, at frequency Ω, the optical field, scattering photons pling rate g0|β+K by into up- and downconverted sidebands. This feedback loop g0|β+K g0 = √ (8) can amplify light or sound, lead to electromagnetically induced L transparency, or cooling of mechanical modes. In principle, this with L the roundtrip length of the cavity (see Appendix). The interaction can even cause strong nonlinear interactions at the parameters g and g are directly related to the so-called few photon or phonon limit if g /κ > 1 [91], though current 0|β+K 0 0 Brillouin gain coefficient G that is often used to quantify photon- solid-state systems are more than two orders of magnitude away B phonon interactions in waveguides [11, 14, 31]. In particular from this regime (see Figure5 and section5). [75], Assuming g /κ 1, valid in nearly all systems, we linearize 0 4g2 the Hamiltonian of equation2 by setting a = α + δa with α a 0|β+K GB = (9) classical, coherent pump amplitude, yielding vpvs(h¯ ω)γ with v and v the group velocities of the interacting photons, H = hg¯ δa + δa† δb + δb† (4) p s int h¯ ω the photon energy and γ the phononic decay rate. Equations 8 and9 enable comparison of the photon-phonon interaction with g = g α the enhanced interaction rate – taking α real – and 0 strengths of waveguides and cavities. Since this gain coefficient δa and δb the annihilation operators representing photonic and depends on the mechanical quality factor Q via γ = ω /Q , phononic signals respectively. Often there are experimental con- m m m it is occasionally worth comparing waveguides in terms of the ditions that suppress a subset of interactions present in Hamil- ratio G /Q . The g /(2π) ≈ 1 MHz measured in silicon op- tonian4. For instance, in sideband-resolved optomechanical B m 0 tomechanical crystals [15] is via equation9 in correspondence cavities (ωm > κ) a blue-detuned pump α sets up an entangling −1 −1 with the GB/Qm ≈ 10 W m measured in silicon nanowires interaction † † at slightly higher frequencies [14, 29]. Both g0 and GB/Qm have Hint = hg¯ δaδb + δa δb (5) an important dependence on mechanical frequency ωm: lower- that creates or annihilates photon-phonon pairs. Similarly, a frequency structures are generally more flexible and thus gener- red-detuned pump α sets up a beam-splitter interaction ate larger interaction rates. The Hamiltonians given in equations5,6 and7 describe a † † Hint = hg¯ δaδb + δa δb (6) wide variety of effects. The detailed consequences of the three- wave mixing depend on the damping, intensity, dispersion and that converts photons into phonons or vice versa. This beam- momentum of the interacting fields. Next, we describe some splitter Hamiltonian can also be realized by pumping the of the potential dynamics. We quantify the dissipation experi- phononic instead of the photonic mode. In that case, α rep- enced by the photons and phonons with decay rates κ and γ resents the phononic pump amplitude, whereas both δa and δb respectively. The following regimes appear: are then photonic signals. • Weak coupling: g κ + γ. The phonons and photons In multi-mode systems, such as in 3D-confined cavities with can be seen as independent entities that interact weakly. A several modes or in 2D-confined continuum systems, the in- common figure of merit for the interaction is the coopera- teraction Hamiltonian is a summation or integration over each tivity C = 4g2/(κγ), which quantifies the strength of the of the possible interactions between the individual photonic feedback loop discussed above. In particular, for C 1 and phononic modes. For instance, linearized photon-phonon the optomechanical back-action dominates the dynamics. interactions in a 2D-confined waveguide with continuous trans- The pair-generation Hamiltonian5 generates amplification, lational symmetry are described by [92–94]: whereas the beam-splitter interaction6 generates cooling ZZ h¯ † and loss. Whether the phonons or the photons dominantly Hint = √ dβ dK gβ+K aβbK + gβ−K aβbK + h.c. (7) 2π experience this amplification and loss depends on the ratio κ/γ of their decay rates. The linewidth of the phonons is In this case the three-wave mixing interaction rate gβ+K = effectively (1 ∓ C)γ when κ γ where the minus-sign in ∓ ? g0|β+Kαβ+K is proportional to the amplitude αβ+K of the mode holds for the amplification case (Hamiltonian5). In contrast, with wavevector β + K, which is usually considered to be the linewidth of the photons is effectively (1 ∓ C)κ when pumped strongly. In contrast to the single-mode cavity de- γ κ. A lasing threshold is reached for the phonons or the scribed by equation4, in the waveguide case the symmetry photons when C = 1. In waveguide systems described by between the two-mode-squeezing δaδb and the beam-splitter equation7, C = 1 is equivalent to the transparency point † δa δb terms is broken by momentum selection from the onset as GBPp/α = 1 with Pp the pump power and α the waveguide generally gβ+K 6= gβ−K. The Hamiltonian of equation7 assumes propagation loss per meter. In fact, interactions between 7

0 photons and phonons in a waveguide can also be captured sum-frequency ω + ω = Ω ≈ ωm [3]. Such interactions can be in terms of a cooperativity which is identical to C under realized in capacitive electromechanics, where the capacitance of only weakly restrictive conditions [75]. an electrical circuit depends on mechanical motion. In particular, this sets up an interaction • Strong coupling: g κ + γ. The phonons and photons interact so strongly that they can no longer be considered (∂ C)V2x H = − x (11) independent entities. Instead, they form a photon-phonon int 2 polariton with an effective decay rate (κ + γ)/2. The beam- with ∂ C the sensitivity of the capacitance C(x) to the me- splitter interaction6 sets up Rabi oscillations between pho- x chanical motion x and V the voltage across the capacitor. In tons and phonons with a period of 2π/g [10, 33, 101]. This terms of ladder operators we have V = V (a + a†) and is a necessary requirement for broadband intra-cavity state zp = ( + †) swapping, but is not strictly required for narrowband itin- x xzp δb δb such that erant state conversion [102, 103]. aa + a†a† 1 H = hg¯ a†a + + δb + δb† (12) Neglecting dynamics and when the detuning from the mechani- int 0 2 2 cal resonance is large (∆Ω γ), the phonon ladder operator is This interaction contains three-wave DFD (Hamiltonian2) as a δb = (g /∆Ω)a†a such that Hamiltonian2 generates an effective 0 subset via the a†a term with an interaction rate g given by dispersive Kerr nonlinearity described by 0 hg¯ = −(∂ C)V2 x (13) g2 0 x zp zp H = h¯ 0 a†aa†a (10) Kerr ∆Ω In addition to three-wave DFD, it also contains three-wave SFD via the aa and a†a† terms. These little explored terms enable This effective Kerr nonlinearity [104–108] is often much stronger electromechanical interactions beyond the canonical three-wave than the intrinsic material nonlinearities. Thus a single optome- DFD optomechanical and Brillouin interactions. chanical system can mediate efficient and tunable interactions be- Further, by applying a strong bias voltage V the capacitive tween up to four photons in a four-wave mixing process that an- b interaction gets linearized: using V = V + δV and keeping only nihilates and creates two photons. The mechanics enhances the b the 2V δV term in V2 yields intrinsic optical material nonlinearities for applications such as b wavelength conversion [109] and photon-pair generation [110]. Hint = −(∂xC)VbδVx (14) Additional dynamical effects exist in the multi-mode case. † For instance, in a waveguide described by equation7 there is With δV = Vzp(δa + δa ) this generates an interaction a spatial variation of the photonic and phononic fields that is † † absent in the optomechanical systems described by equation4. Hint = hg¯ (δa + δa )(δb + δb ) (15) This includes: which is identical to the linearized optomechanics Hamiltonian • The steady-state spatial Brillouin amplification of an optical in expression4 with an interaction rate set by sideband. This has been the topic of recent research in chip- hg¯ = −(∂ C)V V x (16) scale photonic platforms. One can show that an optical x b zp zp Stokes sideband experiences a modified propagation loss that is enhanced with respect to g0 by g = g0α and α = Vb/Vzp (1 − C)α with C = GBPp/α the waveguide’s cooperativity the enhancement factor. The linearized Hamiltonian 15 realizes [75]. This Brillouin gain or loss is accompanied by slow or a tunable, effective piezoelectric interaction that can directly fast light [111]. Here we assumed an optical decay length convert microwave photons into phonons and vice versa. Piezo- exceeding the mechanical decay length, which is valid in electric structures are described by equation 15 as well with an nearly all systems. In the reverse case, the mechanical wave intrinsically fixed bias Vb determined by material properties. ( − C) 1/2 experiences a modified propagation loss 1 αm and Since Vzp = ((h¯ ωµ)/(2C)) with ωµ the microwave fre- there is slow and fast sound [75, 112, 113]. quency and C the total capacitance, the electromechanical coupling rate can be written as • Traveling photonic pulses can be converted into traveling phononic pulses in a bandwidth far exceeding the mechan- ∂ C g = − x ω x (17) ical linewidth. This is often called Brillouin light storage 0 2C µ zp [114–117]. The traveling optical pump and signal pulses or alternatively as g = (∂ ω )x – precisely as in section3A may counterpropagate or occupy different optical modes. 0 x µ zp but with the optical frequency√ ωo replaced by the microwave Several of these and other multi-mode effects have received lit- frequency ωµ with ωµ = 1/ LinC and Lin the circuit’s induc- tle attention so far. This may change with the advent of new tance. Typically the capacitance C = Cm(x) + Cp consists of a nanoscale systems realizing multi-mode and continuum Hamil- part that responds to mechanical motion Cm(x) and a part Cp tonians with strong coupling rates [14, 31, 92, 93, 118]. that is fixed and usually considered parasitic. This leads to ∂ C B. Interactions between phonons and microwave photons g = − x m η ω x (18) 0 2C p µ zp The above section3A on parametric three-wave DFD also applies m to interactions between phonons and microwave photons. How- with ηp = Cm/C the participation ratio that measures the frac- ever, microwave photons may interact with phonons via two tion of the capacitance responding to mechanical motion. For the additional routes: (1) three-wave SFD and (2) direct coupling. canonical parallel-plate capacitor with electrode separation s, we In three-wave SFD, two microwave photons with a frequency have ∂xCm = Cm/s such that g0 = −ηpxzpωµ/(2s). Similar to below the phonon frequency ωm excite mechanical motion at the the optomechanics case, this often drives research towards small 8 structures with large zero-point motion xzp and small electrode Comparably high quality factors are measured electrically in separation s. Contrary to the optomechanics case, however, in- quartz and sapphire at lower frequencies [159, 161]. It is an open creasing the participation ratio ηp motivates increasing the size question whether these extreme lifetimes have reached intrinsic and thus the motional capacitance Cm of the structures until material limits. The long lifetimes make mechanical systems ηp ≈ 1. In gigahertz-range microwave circuits with unity partic- attractive for delay lines and qubit storage [162] (section5). ipation and electrode separations on the order of s ≈ 100 nm, we Next, we look at the coupling strengths in these systems 5 have g0/(2π) ≈ −10 Hz, about a factor ωo/ωµ ≈ 10 smaller [Fig.5]. As discussed in section3, a few different figures of merit than the optomechanical g0/(2π) ≈ 1 MHz (section3A). De- are commonly used depending on the type of system. We believe spite the much smaller g0, it is still possible to achieve large the dimensionless ratios g0/κ and the cooperativity C are two 2 cooperativity C = 4g /(κγ) in electromechanics as the typical of the most powerful figures of merit (section5). The ratio g0/κ microwave linewidths are much smaller and the enhancement determines the single-photon nonlinearity, the energy-per-bit in factors α can be larger than in the optical case [119–121]. optical modulators as well as the energy-per-qubit in microwave- to-optical photon converters. The cooperativity C must be unity

1011 for efficient state conversion as well as for phonon and photon lasing. In the context of waveguides it measures the maximum Brillouin gain as C = GBPp/α [75]. 1010 optical cavity Thus we compute g0/κ for about fifty opto- and electrome- microwave cavity chanical cavities and waveguides [Fig.5a]. We convert the optical waveguide simulation waveguide Brillouin coefficients GB to g0 via expressions8 and 109 other 9 by estimating the minimum roundtrip length L a cavity made from the waveguide would have. In addition, we convert the

108 waveguide propagation loss α to the intrinsic loss rate κin = αvg with vg the group velocity. This brings a diverse set of systems together in single figure. No systems exceed g0/κ ≈ 0.01, with 107 the highest values obtained in silicon optomechanical crystals [15, 52], Brillouin-active waveguides [14, 29, 163] and Raman cavities [145]. There is no strong relation between g0/κ and C: 106 systems with low interactions rates g0 often have low decay rates κ and γ as well since they do not have quite as stringent fabrication requirements on the surface quality. 105 The absolute zero-point coupling rates g0 illustrate the power of moving to the nanoscale. We plot them as a function of the 104 maximum quantum cooperativity Cq = C/n¯th with n¯th the thermal phonon occupation [Fig.5b]. When Cq > 1, the state transfer between photons and phonons takes place more rapidly than 103 the mechanical thermal decoherence [10]. This is a requirement for hybrid quantum systems such as efficient microwave-to- optical photon converters (section5). There are several chip- 102 105 106 107 108 109 1010 1011 1012 1013 scale electro- and optomechanical systems that obtained Cq > 1, with promising values demonstrated in silicon photonic crystals. A main impediment to large quantum cooperativities in Fig. 4. Mechanical quality factors. The systems include opto- optomechanics is the heating of the mechanics caused by optical and electromechanical cavities and waveguides at room tem- absorption [128]. perature (red), at a few kelvin (blue) and at millikelvin temper- Further, we give an overview of the Brillouin coefficients G atures (blue). The highest quality factors are demonstrated in B found in 2D-confined waveguides [Fig.5c]. The current record millikelvin cavities. The data points correspond to references 4 −1 −1 [14, 15, 28–31, 33, 35, 40, 52, 67, 81, 90, 101, 120, 122–160]. GB = 10 W m in the gigahertz range was measured in a suspended series of silicon nanowires [29]. However, larger Bril- louin amplification was obtained with silicon and chalcogenide rib waveguides which have disproportionately lower optical 4. STATE OF THE ART propagation losses α and can handle larger optical pump pow- Here we give a concise overview of the current state of the art in ers Pp [163, 164]. We stress that the maximum Brillouin gain opto- and electromechanical systems by summarizing the param- is identical to the cooperativity [75]. They are both limited by eters obtained in about fifty opto- and electromechanical cavities the maximum power and electromagnetic energy density the and waveguides. First, we plot the mechanical quality factors system in question can withstand. At room temperature in sili- as a function of mechanical frequency [Fig.4] including room con, the upper limit is usually set by two-photon and free-carrier temperature (red) and cold (blue) systems. As discussed in sec- absorption [14, 31, 89]. Moving beyond the two-photon bandgap tion2F, cold systems usually reach much higher quality factors. of 2200 nm in silicon or switching to materials such as silicon ni- The current record is held by a 5 GHz silicon optomechanical tride, lithium niobate or chalcogenides can drastically improve 10 crystal with Qm > 10 , yielding a lifetime longer than a sec- the power handling [89, 144, 165, 166]. In cold systems, it is ond [122] at millikelvin temperatures. Measuring these quality instead set by the cooling power of the refrigerator and the heat- factors requires careful optically pulsed read-out techniques, as ing of the mechanical system [128]. Another challenge for 2D- the intrinsic dissipation of continuous-wave optical photons eas- confined waveguides is the inhomogeneous broadening of the ily heats up the mechanics thus destroying its coherence [128]. mechanical resonance. This arises from atomic-scale fluctuations 9

(a)10-2 optical cavity

10-3 microwave cavity optical waveguide simulation 10-4 other

10-5

10-6 10-2 10-1 100 101 102 103 104 105

(b) (c) 106

106

104

102

102 100

100 10-3 10-2 10-1 100 101 102 103 104 100 101 102 103

Fig. 5. Interaction rates in 3D-confined cavities and 2D-confined waveguides. (a) The dimensionless nonlinearity g0/κ versus the maximum cooperativity C. The largest interaction rates are achieved in nanoscale cavities and waveguides. Cooperativities are typically highest in cold systems (blue) and can be as high in less tightly confined systems since they often have lower photonic and phononic decay rates than the smallest systems. (b) The zero-point coupling rate g0 versus the maximum quantum cooperativity Cq = C/n¯th with n¯th the thermal phonon occupation for a selection of cold 3D-confined systems. It is significantly harder to reach Cq > 1 than C > 1. In particular, optical absorption easily heats up mechanical systems – effectively increasing n¯th [128]. (c) The Brillouin gain coefficient GB versus maximum net Brillouin gain (GBPp/α − 1) for a selection of 2D-confined waveguides. Up to extrinsic cavity losses, C − 1 = (GBPp/α − 1). Achieving this maximum Brillouin gain requires the waveguide to be have a length L = 1/α with α the optical propagation loss per meter. However, this is challenging as longer waveguides can suffer from an inhomogeneous broadening of the mechanical resonance due to atomic-scale disorder in the waveguide geometry. in the waveguide geometry along its length effectively smearing tuning from the mechanical resonance in the above expressions: out the mechanical response [14, 29, 31, 66]. Finally, compared ωm → 2∆Ω with the detuning ∆Ω = ∆ω − ωm. This enhances to gigahertz systems, flexible megahertz mechanical systems the shift per photon so that quantum nonlinearities are realized 6 −1 −1 give much higher efficiencies of GB ≈ 10 W m as measured at [170, 171] 9 −1 −1 g2 in dual-nanoweb [167] fibers and of GB ≈ 10 W m as pre- 2 0 ϑcav = ≈ π (19) dicted in silicon double-slot waveguides [168]. κ∆Ω with ∆Ω ωm. The photon blockade effect also requires > 5. PERSPECTIVES sideband-resolution (∆Ω κ) so g 0 > 1 (20) A. Single-photon nonlinear optics κ The three-wave mixing interactions discussed in section3 in is generally a necessary condition for single-photon nonlinear principle enable single-photon nonlinear optics in opto- and optics with opto- and electromechanical cavities [10, 172]. In the electromechanical systems [91, 169, 170]. For instance, in the case of 2D-confined waveguides, it can similarly be shown [118] photon blockade effect a single incoming photon excites the that a single photon drives a mechanically-mediated cross-Kerr motion of a mechanical system in a cavity, which then shifts phase shift 2 the cavity resonance and thus blocks the entrance of another g0|β+K photon. Realizing such quantum nonlinearities sets stringent ϑwg = (21) vg∆Ω requirements on the interaction strengths and decay rates. on another photon with v the optical group velocity (see Ap- For instance, in an optomechanical cavity the force exerted g † pendix). The cross-Kerr phase-shift ϑwg can be enhanced drasti- by a single photon is hFi = −h∂xHinti = −h¯ (g0/xzp)ha ai = cally by reducing the group velocity vg via Brillouin slow light −h¯ (g /xzp). To greatly affect the optical response seen by an- 0 [111, 118, 173]. If sufficiently large, the phase-shifts ϑ and ϑ other photon impinging on the cavity, this force must drive a cav wg can be used to realize controlled-phase gates between photonic mechanical displacement that shifts the optical resonance by qubits – an elementary building block for quantum information about a linewidth κ or xπ = κ/(∂xωo) = (κ/g0)xzp. In other 2 processors [118, 174–176]. Using equation8, we have words, we require F/(meffωm) = xπ which leads to ϑcav ≡ 4g2/(κω ) ≈ π where ϑ is the mechanically-mediated cross- F 0 m cav ϑcav = ϑwg (22) phase shift experienced by the other photon assuming critical π coupling to the cavity. This extremely challenging condition is with F = 2π/(κTrt) the cavity finesse and Trt the cavity relaxed when two photonic modes with a frequency difference roundtrip time. Therefore, cavities generally yield larger single- ∆ω roughly resonant with the mechanical frequency are used. photon cross-Kerr phase shifts than their corresponding optome- In this case, the mechanical frequency can be replaced by the de- chanical waveguides. 10

Currently state-of-the-art solid-state and sideband-resolved a microwave cavity [187, 188]. The required Vπ corresponds to 2 (ωm > κ) opto- and electromechanical systems yield at best an energy-per-bit Ebit = CVπ/2 which again can be rewritten as g0/κ ≈ 0.01 in any material [Fig.5]. Significant advances in g0 expression 23. Electro-optic materials such as lithium niobate may be made in e.g. nanoscale slotted structures [90, 153, 177], [189] may yield up to g0/(2π) ≈ 10 kHz, corresponding to an but it remains an open challenge to not only increase g0 but also energy-per-bit Ebit ≈ 10 fJ/bit keeping the optical linewidth κ g0/κ by a few orders of magnitude [178]. Beyond exploring constant – on the order of today’s world records [19]. novel structures, other potential approaches include effectively Although full system demonstrations using mechanics for boosting g0/κ by parametrically amplifying the mechanical mo- electro-optic modulation are lacking, based on estimates like tion [179], by employing delayed quantum feedback [180] or these we believe that mechanics will unlock highly efficient via collectively enhanced interactions in optomechanical arrays electro-optic systems. The expected much lower energy-per-bit [181, 182]. Although single-photon nonlinear optics may be out implies that future electro-opto-mechanical modulators could of reach for now, many-photon nonlinear optics can be enhanced achieve much higher bitrates at fixed power, or alternatively, very effectively with mechanics. Specifically, mechanics realizes much lower dissipated power at fixed bitrate than current direct Kerr nonlinearities orders of magnitude beyond those of typical electro-optic modulators. Although the mechanical linewidth intrinsic material effects. This is especially so for highly flexible, does not enter expression 23, bandwidths of a single device are low-frequency mechanical systems [104, 105, 183, 184] but has usually limited by the phononic quality factor or transit time been shown in gigahertz silicon optomechanical cavities and across the device. Interestingly, the mechanical displacement waveguides as well [14, 109]. corresponding to the estimated 1 aJ/bit is only xπ ≈ 10 pm. Here we highlighted the potential for optical modulation B. Efficient optical modulation based on mechanical motion at gigahertz-frequencies. However, Phonons provide a natural means for the spatiotemporal mod- similar arguments can be made for optical switching networks ulation of optical photons via electro- and optomechanical in- based on lower frequency mechanical structures. In particular, teractions. Hybrid circuits that marry photonic and phononic voltage-driven capacitive or piezoelectric optical phase-shifters excitations give us access to novel opto-electro-mechanical sys- exploiting mechanical motion do not draw static power and can tems. Two aspects of the physics make phononic circuits very generate large optical phase shifts in small devices [12, 190–196]. attractive for the modulation of optical fields. These “photonic MEMS” are thus an attractive elementary build- First, there is excellent spatial matching between light and ing block in reconfigurable and densely integrated photonic sound. As touched upon above, the wavelengths of microwave networks used for high-dimensional classical [19, 197–200] and phonons and telecom photons are both about a micron in tech- quantum [201–203] photonic information processors. They may nologically relevant materials such as silicon. The matching meet the challenging power- and space-constraints involved in follows from the four to five orders of magnitude difference running a complex programmable network. between the speed of sound and the speed of light. Momentum Demonstrating fully integrated acousto-optic systems re- conservation, i.e. phase-matching, between phonons and opti- quires that we properly confine, excite and route phonons on a cal photons (as discussed in section3) is key for non-reciprocal chip. Among the currently proposed and demonstrated systems nonlinear processes and modulation schemes with traveling are acousto-optic modulators [97] as well as optomechanical phonons [95, 97, 185, 186]. beam-steering systems [68, 204]. Besides showing the power of Second, the optomechanical nonlinearity is strong and essen- sound to process light with minuscule amounts of energy, these tially lossless. Small deformations can induce major changes phononic systems have features that are absent in competing on the optical response of a system. For instance, in an optome- approaches. For instance, gigahertz traveling mechanical waves chanical cavity (equation1) the mechanical motion required to with large momentum naturally enable non-reciprocal features encode a bit onto a light field has an amplitude of approximately in both modulators [97] and beam-steering systems [68]. This is essential for isolators and circulators based on indirect photonic xπ = (κ/g0)xzp. Generating this motion requires energy, and 2 2 transitions [96, 98, 205–208]. this corresponds to an energy-per-bit Ebit = meffωmxπ/2 which we rewrite as In order to realize these and other acousto-optic systems, it 2 is crucial to efficiently excite mechanical excitations on the sur- h¯ ωm κ E = (23) face of a chip. In this context, electrical excitation is especially bit 4 g 0 promising as it allows for stronger mechanical waves than opti- Thus the energy-per-bit also depends on the dimensionless cal excitation. With optical excitation of mechanical waves, the quantity g0/κ: a single phonon can switch a photon when this flux of phonons is upper-bounded by the flux of optical pho- quantity reaches unity, in agreement with section5A. For sil- tons injected into the structure. The ratio of photon to phonon −3 icon optomechanical crystals with g0/κ ≈ 10 , this yields energy limits the mechanical power to less than a microwatt, Ebit ≈ 1 aJ/bit: orders of magnitude more efficient than com- corresponding to 10-100 milliwatts of optical power. Neverthe- monly deployed electro-optic technologies [19]. less, proof-of-concept demonstrations [96, 98] have successfully The similarity between the fundamental interactions in op- generated non-reciprocity on a chip using optically generated tomechanics [10] and electro-optics [187, 188] allows to com- phonons. In contrast, microwave photons have a factor 105 pare the two types of modulation head-to-head. In particu- larger fluxes than optical photons for the same power. Therefore, lar, in an optical cavity made of an electro-optic material the microwave photons can drive milliwatt-level mechanical waves voltage drop across the electrodes required to encode a bit is in nanoscale cavities and waveguides. Such mechanical waves Vπ = κ/(∂V ωo) = (κ/g0)Vzp with g0 = (∂V ωo)Vzp the electro- can have displacements up to a nanometer and strains of a few optic interaction rate [187, 188], which is defined analogously to percent – close to material yield strengths. the optomechanical interaction rate. It is the parameter appear- Electrical generation of gigahertz phonons in nanoscale struc- † † ing in the interaction Hamiltonian Hint = hg¯ 0a a(b + b ), with tures has received little attention so far, especially in non- b + b† now proportional to the voltage across the capacitor of piezoelectric materials such as silicon and silicon nitride. As 11 discussed in section3B, this can be realized either via capacitive states from the microwave to the optical domain and vice versa. or via piezoelectric electromechanics. Capacitive approaches To realize a microwave-to-optical photon converter, one can work in any material [209] and have recently been demonstrated start from a classical electro-optic modulator and modify it to in a silicon photonic waveguide [158]. They require small capaci- protect quantum coherence. Several proposals aim to achieve tor gaps and large bias voltages to generate effects of magnitude this by coupling a superconducting microwave cavity to an comparable to piezoelectric approaches. More commonly, piezo- optical cavity made of an electro-optic material. For instance, electrics such as lithium niobate, aluminum nitride [210, 211] the beam-splitter Hamiltonian can be engineered by injecting a and lead-zirconate titanate can be used as the photonic plat- strong optical pump red-detuned from the cavity resonance in form, or be integrated with existing photonic platforms such as an electro-optic cavity. In order to suppress undesired Stokes silicon and silicon nitride in order to combine the best of both scattering events, the frequency of the microwave cavity needs worlds [68, 212–214]. Such hybrid integration typically comes to exceed the optical cavity linewidth, i.e. sideband-resolution with challenging incompatibilities in material properties [215], is necessary. In this scenario, continuous-wave state conver- especially when more than one material needs to be integrated sion with high fidelity requires an electro-optic cooperativity Ceo on a single chip. Efficient electrically-driven acoustic waves in close to unity: 2 2 photonic structures have the potential to enable isolation and 4g0|α| Ceo = = 1 (24) circulation with an optical bandwidth beyond 1 THz – limited κγµ only by optical walk-off [95, 98, 216–218]. with g0 the electro-optic interaction rate as defined in the pre- 2 C. Hybrid quantum systems vious section, |α| the number of optical pump photons in the cavity and γ the microwave cavity linewidth. The quantum Strain and displacement alter the properties of many different µ conversion is accompanied by an optical power dissipation systems and therefore provide excellent opportunities for con- P = h¯ ω |α|2κ with κ the intrinsic decay rate of the op- necting dissimilar degrees of freedom. In addition, mechanical diss o in in tical cavity. Operating the converter in a bandwidth of γ and systems can possess very long coherence times and can be used µ inserting condition 24, this leads to an energy-per-qubit of to store quantum information. In the field of hybrid quantum systems, researchers find ways to couple different degrees of ! h¯ ωo κκin freedom over which quantum control is possible to scale up and Eqbit = (25) 4 g2 extend the power of quantum systems. Realizing hybrid systems 0 by combining mechanical elements with other excitations is a which is the quantum version of the energy-per-bit 23. This widely pursued research goal. Studies on both static tuning of yields an interesting relation between the efficiency of classical quantum systems using nanomechanical forces [219–221] as well and quantum modulators as on quantum dynamics mediated by mechanical resonances Eqbit ω and waveguides [18, 57, 219, 222, 223] are being pursued. ≈ o (26) Among the emerging hybrid quantum systems, microwave- Ebit ωm to-optical photon converters utilizing mechanical degrees of We stress that Eqbit is the optical dissipated energy in a quan- freedom have attracted particular interest recently [123, 222, 224– tum converter, whereas Ebit is the microwave or mechanical 227]. In particular, one of the leading platforms to realize scal- energy necessary to switch an optical field in a classical mod- able, error-corrected quantum processors [228, 229] are supercon- ulator [238]. The quantum electro-optic modulator dissipates ducting microwave circuits in which qubits are realized using roughly five orders of magnitude more energy per converted Josephson junctions [230, 231] in a platform compatible with sili- qubit as it requires an optical pump field to drive the conver- con photonics [232]. To suppress decoherence, these microwave sion process. Strategies developed to minimize Ebit, as pursued circuits are operated at millikelvin temperatures inside dilution for decades by academic groups and the optical communica- refrigerators. Heat generation must be restricted in these cold tions industry, also tend to minimize Eqbit. Recently a coupling environments [233]. The most advanced prototypes currently rate of g0/(2π) = 310 Hz was demonstrated in an integrated consist of on the order of fifty qubits on which gates with at best aluminum nitride electro-optic resonator [239]. Switching to 0.1% error rates can be applied [233, 234]. Scaling up these sys- lithium niobate and harnessing improvements in the electro- tems to millions of qubits, as required for a fully error-corrected optic modal overlap may increase this to g /(2π) ≈ 10 kHz, quantum computer, is a formidable unresolved challenge [229]. 0 corresponding to Eqbit ≈ 1 nJ/qbit. Electro-optic polymers [240] Also, the flow of microwave quantum information is hindered may yield higher interaction rates g but bring along challenges outside of the dilution refrigerators by the microwave thermal 0 in optical and microwave losses κ and γµ. Cooling powers of noise present at room temperature [235, 236]. Optical photons roughly 10 µW at the low temperature stage of current dilution travel for kilometers at room temperature along today’s opti- refrigerators [233] imply that conversion rates with common cal fiber networks. Thus quantum interfaces that convert mi- electro-optic materials will likely not exceed about 10 kqbits/s. crowave to optical photons with high efficiency and low noise Considering that the g0/κ demonstrated optomechanical de- should help address the scaling and communication barriers vices is much larger than those found in electro-optic systems, hindering microwave quantum processors. They may pave the and following a reasoning similar to that presented in section way for distributed quantum computing systems or a “quantum 5B for classical modulators, it is likely that microwave-to-optical internet” [237]. Besides, such interfaces would give optical sys- photon converters based on mechanical elements as interme- tems access to the large nonlinearities generated by Josephson diaries will be able to achieve large efficiencies. It has been junctions, which enables a new approach for nonlinear optics. theoretically shown that electro-opto-mechanical cavities with The envisioned microwave-to-optical photon converters are dynamics described in section3 allow for efficient state trans- in essence electro-optic modulators that operate on single pho- duction between microwave and optical fields when tons and preserve entanglement [187]. They exploit the beam- splitter Hamiltonian discussed in section3 to swap quantum Cem ≈ Com 1 (27) 12 with Cem and Com the electro- and optomechanical coopera- tive level, as excessive fiber-to-chip losses and high modulator tivities. Noiseless conversion additionally requires negligible drive voltages associated with the discrete photonic devices cur- thermal microwave and mechanical occupations [102, 103, 224]. rently being used are the main origin of the low system efficiency. Since the dominant dissipation still arises from the optical pump, Also, the photonic-phononic emit-receive scheme proposed in the energy-per-qubit can still be expressed as in equation 25 for [254, 255] results in a lower RF-insertion loss. Although it gives a electro-opto-mechanical cavity. Given the large nonlinearity up tunability, additional advantages of this approach are that g0/κ enabled by nanoscale mechanical systems [Fig.5], we ex- its engineerable filter response [254, 256] and its cascadability pect conversion rates up to 100 Mqbits/s are feasible by operat- [255]. Exploiting the phase response of the SBS resonance also ing multiple electro-opto-mechanical photon converters in paral- phase control of RF signals has been demonstrated [250]. Both lel inside the refrigerator. State-of-the-art integrated electro- and pure phase shifters and relative time delay have been demon- opto-mechanical cavities have achieved Cem > 1 and Com > 1 in strated. Again interferometric approaches allow to amplify the separate systems [Fig.5]. It is an open challenge to achieve con- intrinsic phase delay of the system, which is limited by the avail- dition 27 in a single integrated electro-opto-mechanical device. able SBS gain. In the examples above, the filter is driven by a Finally, the long lifetimes and compact nature of mechanical single-frequency pump, resulting in a Lorentzian filter response. systems also makes them attractive for the storage of classi- More complex filter responses can be obtained by combining cal and quantum information [13, 157, 160, 162, 223, 241–244]. multiple pumps [111]. However, this comes at the cost of the Mechanical memories are currently pursued both with purely overall system response since the total power handling capacity electromechanical [18, 157, 160] and purely optomechanical of the system is typically limited. As such there is still a need for [114, 115] systems. Interfaces between mechanical systems waveguide platforms that can handle large optical powers and and superconducting qubits may lead to the generation of non- at the same time provide high SBS-gain. classical states of mesoscopic mechanical systems [10, 245, 246], Further, low-noise oscillators are also a key building block probing the boundary between quantum and classical behavior. in RF-systems. Two approaches, equivalent with the two dissipation hierarchies (γ κ and γ κ) identified in section D. Microwave signal processing 3, have been studied. In the first case, if the photon lifetime In particular in the context of wireless communications, com- exceeds the phonon lifetime (γ κ), optical line narrowing and pact and cost-effective solutions for radio-frequency (RF) signal eventually self-oscillation is obtained at the transparency con- processing are rapidly gaining importance. Compared to purely dition C = 1 – resulting in substantial narrowing of the Stokes electronic and MEMS-based approaches, RF processing in the wave and thus a purified laser beam [257–259]. Cascading this photonics domain – microwave photonics – promises compact- process leads to higher-order Stokes waves with increasingly ness and light weight, rapid tunability and integration density narrowed linewidths. Photomixing a pair of cascaded Brillouin [247, 248]. Currently demonstrated optical solutions however lines gives an RF carrier with phase noise determined by the still suffer from high RF-insertion loss and an unfavorable trade- lowest order Stokes wave. Using this approach in a very low- off between achieving sufficiently narrow bandwidth, high rejec- loss silica disk resonator a phase noise suppression of 110 dBc at tion ratio and linearity. Solutions mediated by phonons might 100 kHz offset from a 21.7 GHz carrier was demonstrated [260]. overcome this limit as they offer a narrow linewidth without suf- In the alternate case, with the phonon lifetime exceeding the fering from the power limits experienced in high-quality optical photon lifetime (γ κ), the Stokes wave is a frequency-shifted cavities [249, 250] copy of the pump wave apart from the phase noise added by the Given the high power requirements, 2D-confined waveg- mechanical oscillator. At the transparency condition C = 1 the uides lend themselves more naturally to many RF-applications. phonon noise goes down, eventually reaching the mechanical As such, stimulated Brillouin scattering (SBS) has been exten- Schawlow-Townes limit. Several such “phonon lasers” have sively exploited. Original work focused on phonon-photon in- been demonstrated already, relying on very different integration teractions in optical fibers, which allows for high SBS-gain and platforms [150, 156, 261–263]. Further work is needed to deter- high optical power but lacks compactness and integrability. Fol- mine if these devices can deliver the performance required to lowing the demonstration of SBS-gain in integrated waveguide compete with existing microwave oscillators. platforms [14, 31, 144], several groups now also demonstrated In the examples above, the mechanical mode is excited all- RF-signal processing using integrated photonics chips. In the optically via a strong pump beam. Both in terms of efficiency and most straightforward approach, the RF-signal is modulated on in terms of preventing the pump beam from propagating further a sideband of an optical carrier which is then overlayed with through the optical circuit this may be not the most appropriate the narrowband SBS loss-spectrum generated by a strong pump method. Recently, several authors have demonstrated electrical [251]. Tuning the carrier frequency allows rapid and straight- actuation of optomechanical circuits [53, 97, 123, 158, 264–266]. forward tuning of the notch filter over several GHz and a band- While this provides a more direct way to drive the acousto- width below 130 MHz was demonstrated. The suppression optic circuit, considerable efforts are still needed to improve the was only 20 dB however, limited by the SBS gain achievable overall efficiency of these systems and to develop a platform in the waveguide platform used, in this case a chalcogenide where all relevant building blocks including e.g. actuators and waveguide. This issue is further exacerbated in more CMOS- detectors, optomechanical oscillators and acoustic delay lines compatible platforms, where the SBS gain is typically limited can be co-integrated without loss in performance. to a few dB. This can be overcome by using interferometric approaches, which enable over 45 dB suppression with only 1 dB E. General challenges of SBS gain [252, 253]. 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THEORETICAL DESCRIPTION minum nitride piezo-acousto-photonic crystal nanocavity with high quality factors,” Applied Physics Letters 102 We first give a description of 3D-confined cavity optomechanics. (2013). Next, we connect to the description of 2D-confined waveguides. 273. H. Liang, R. Luo, Y. He, H. Jiang, and Q. Lin, “High-quality lithium niobate photonic crystal nanocavities,” Optica 4, A. Cavities: 3D-confined 1251 (2017). We focus on cavity optomechanics in this section although much 274. M. Mitchell, D. P. Lake, and P. E. Barclay, “Realizing of it applies to electromechanics as well. The dynamics of an op- Q>300000 in diamond microdisks for optomechanics via tomechanical system involves taking into account the interplay etch optimization,” arXiv:1808.09883 pp. 44–46 (2018). between the coupled acoustic and optical degrees of freedom 275. A. Krause, M. Winger, T. Blasius, Q. Lin, and O. Painter, “A in a system. The frequencies of these two coupled degrees of high-resolution microchip optomechanical accelerometer,” freedom are typically different by many orders of magnitude Nature Photonics 6 (2012). so that the only physical significant coupling arises paramet- 276. K. L. Ekinci, Y. T. Yang, and M. L. Roukes, “Ultimate limits rically in the form described below. As a first step, a modal to inertial mass sensing based upon nanoelectromechanical decomposition of time varying deformations in the elastic struc- systems,” Journal of Applied Physics 95, 2682–2689 (2004). ture is considered, so a set of parameters xj(t), each encoding 277. C. Baker, D. T. Nguyen, W. Hease, C. Gomez, A. Lemaître, the deformation due to a particular vibrational mode is consid- S. Ducci, G. Leo, and I. Favero, “High-frequency nano- ered. A similar decomposition of Maxwell’s equations leads optomechanical disk resonators in liquids,” Nature Nan- to a set of electromagnetic modes of the structure with ampli- otechnology 10, 810–816 (2015). tudes αj(t), which with the correct normalization would lead 2 2 278. P. Tiebot, R. Van Laer, and D. Van Thourout, “Thermal to Uj = h¯ ωo|αj| being the energy and |αj| the average photon tuning of Brillouin resonance in free standing silicon number in mode j. Each optical (mechanical) mode of the struc- nanowire,” in “Conference on Lasers and Electro-Optics,” ture has a frequency ωo (ωm) and their associated dynamics. At (OSA, Washington, D.C., 2018), p. JW2A.35. first we will only consider the interaction between two modes: 279. J.-C. Beugnot, T. Sylvestre, and H. Maillotte, “Guided a single optical and a single mechanical mode of the structure. acoustic wave Brillouin scattering in photonic crystal Optomechanical interactions give rise to coupling between these 22 modes in the following way: the deformation of the structure in The left-hand side of the above equation is simply the equation of a specific vibrational mode parametrized by x, causes a change motion for a damped harmonic oscillator and takes into account in the optical frequency given by ωo(x) = ωo(0) + Gx, where the dynamics of the modal degree of freedom being considered. the optomechanical coupling parameter G = ∂xωo has units The right-hand side of the equation are the forcing terms: FBA(t) −1 of Hz · m . The modal equation for the electromagnetic field, is the optical back-action, while Finput(t) is an input force which under laser excitation at frequency ωin = ωo(0) − ∆ with input we use to understand the linear response of the mechanical 2 photon flux given by |αin| is then expressed as system. The back-action force is given by radiation pressure described via the Maxwell stress tensor, which is quadratic in dα(t) κ √ 2 = − i(∆ + Gx) + α − κ α . (28) the field or proportional to |α| . By considering the total energy ex in 2 dt 2 of the system (see section A.3), we find that FBA(t) = hG¯ |α(t)| . The cavity decay rate κ represents the full-width half-maximum Equations 28 and 31 now describe the dynamics of the coupled (FWHM) of the optical mode excitation spectrum and contains system and can be solved to obtain the effects of back-action in all decay channels coupling to the photonic system. Typically, κ the classical domain. In particular, we are primarily interested in the modification of the linear response of the mechanical system consists of an engineered extrinsic decay rate κex as well as the to an input force, i.e. changes to its damping rate and frequency. intrinsic loss rate κin. These changes come about from the mechanical motion x modi- A.1. Linear detection of motion fying the intracavity field α(t) which then applies a force back First, we consider how motion is detected optically in such a onto x which can be proportional, lagging, or leading, leading setup, completely ignoring at first the effect of the light on the to a redefinition of the mechanical system’s complex frequency. mechanical system. We make a few approximations for this To calculate the laser power and frequency dependence of these particular analysis that are useful though not generally valid. modifications, we choose an operating point (α, x) and linearize First we assume that x(t) is slow compared to optical bandwidth the equations of motion by taking to account only the dynamics κ, or equivalently ωm κ. Also we assume that the laser field is of the fluctuations δx(t) and δα(t). This gives us a set of three driving the photonic cavity on resonance so ∆ = 0, and that the coupled linear differential equations optical decay rate is dominated by the out-coupling so κ = κ . √ ex δx¨(t) = −ω2 δx − γ δx˙ + The output field is then given by α = α + κα, which for m in out in ∗ oscillations that are small, i.e. when the laser-cavity detuning is hG¯ (α δα + c.c.)/meff + Finput(t)/meff (32) √ being modulated by the motion within the linear region of the δα˙ (t) = −(i∆ + κ/2)δα − iGαδx − κexδαin (33) cavity phase response such that Gx κ, can be solved to obtain ∗ ∗ ∗ √ ? δα˙ (t) = (i∆ − κ/2)δα + iGα δx − κ δα (34) a relation representing phase modulation of the output field: ex in αout = −αin(1 − i4Gx(t)/κ). This is a first order expansion which can be solved for input forces Finput(t) taking δαin = 0 of e−iφ(t) where φ(t) = Gx(t)/κ. An alternate way of writing for now. Solving these equations in the Fourier domain, we obtain an expression for the small-signal response of the mechan- this expression is in terms of the intracavity field α(t) which,√ neglecting the mechanical motion, is given by α = −2αin/ κ. ical system to the input force in terms of a dispersion relation, The output field is then δx(ω) ≡ χx(ω)Finput(ω), with p 1 αout = −α − i Γmeasx/xzp (29) χ (ω) = , (35) in x 2 2 meff(ωm − ω − iωγin + Σopt(ω)) with the measurement rate defined as where 2 2 2 2 2 4G xzp|α| 4g |α| 2 2 ∗ Γ ≡ = 0 (30) Σopt(ω) = −ihG¯ |α| (χα(ω) − χα(−ω)) /meff (36) meas κ κ −1 and χα(ω) = (i(∆ − ω) + κ/2) is the optical resonance re- being the rate at which photons are scattered from the laser beam sponse function. The expression in equation 35 represents the to sidebands due to motion of amplitude xzp and an intracavity response of a damped mechanical resonance that is modified by | |2 zero-point optical field intensity of α . The subscript represents , a “self-energy” term, Σopt(ω), due to interaction with optical res- assigned in anticipation of the quantum analysis below though onance. The real and imaginary parts of this self-energy cause an for a classical description xzp can be used to normalize the above effective modification of the mechanical frequency and linewidth expression without changing the physics. The measurement rate ωm and γin. This shift in the complex frequency, often referred as defined here is central to understanding the operation of to as the “optical spring” and “optical damping/amplification” optomechanical systems in the linear regime and will be used effects can be expressed succinctly in terms of Σopt(ω): throughout the text below often denoted alternatively as γOM ≡ ( ) ΓOM ≡ Γmeas. The zero-point coupling rate g0 = Gxzp defined ImΣopt ωm γOM ≈ − , and (37) here is consistent with section3 in the main text. ωm ReΣopt(ωm) A.2. Back-action on the mechanical mode δω ≈ . (38) m 2ω Now we consider how the motion of the mechanical system m is modified due to interaction with the optical field resonating These expression are good approximations in the weak-coupling in the structure. In addition to equation 28, to understand the regime (g κ + γ). In the strong-coupling regime (g κ + γ), back-action arising from the interplay between the optical field the full frequency-dependence of the self-energy Σopt(ω) should and mechanical motion, we must consider the dynamics of the be considered. motional degree of freedom: A common mode of operation of optomechanical systems that are sideband-resolved (ωm κ) is to tune the laser approx- 2 x¨(t) + γinx˙ + ωmx = (FBA(t) + Finput(t))/meff. (31) imately a mechanical frequency to the red side of the optical 23 resonance so ∆ = ωm. In this case, the above relations lead us in the equations above are then physically significant since, e.g., 2 2 2 2 2 † to γOM ≈ 2hG¯ |α| /(κmeffωm) = 4G xzp|α| /κ which is seen the expectation value of h¯ ωjaj aj represents the energy stored in to be equal to the measurement rate calculated in equation 30, the jth mode of the electromagnetic field. We can use the classical though that was for a different regime of operation. The equality expressions for energy in the fields to calculate the energy for of these two rates can be understood as such: with the red- a single photon/phonon above the vacuum state, |ψi = |1ij, detuned scheme of driving, all of the sideband scattering, which which we will then set equal toh ¯ ωj: occurs at rate Γmeas, causes up-shifting of the laser photons into the photonic mode, and thus effectively damps the mechanical |ψi Z h¯ ω ≡ U = hψ| dr Qˆ˙ (r)ρ(r)Qˆ˙ (r)|ψi resonator’s motion. In the above we focused on the effect of j mech the optomechanical interaction on the mechanical resonator’s re- Z −h | r Qˆ˙ (r) (r)Qˆ˙ (r)| i sponse function. However, there are equally important changes vac d ρ vac in the electromagnetic response. These effects, including Bril- Z = 2 ∗( ) ( ) ( ) louin gain/loss and slow/fast light, can be derived similarly 2ωj dr Qj r ρ r Qj r [75]. 2 2 = 2meffωj max[|Qj(r)| ]. (43) A.3. Understanding optomechanical coupling in a nanophotonic system where we’ve defined the effective mass and zero-point motion In the previous section we studied how optomechanical cou- of the mode to be pling can be used to detect mechanical motion and modify the R ∗ dr Qj (r)ρ(r)Qj(r) linear response of a mechanical resonator. Here we will see m = , and eff,j max[|Q (r)|2] how such an interaction comes about in a realistic nanophotonic j s system. Though a toy model with a one-dimensional scalar h¯ wave-equation and a simplified mass-spring system has long xzpf,j ≡ max[|Qj(r)|] = . (44) 2meff,jωj been used in studying optomechanical systems, obtaining a precise understanding of the coupling rates given nontrivial A similar consideration for the electromagnetic fields leads to wavelength-scale optical and elastic mode profiles requires care- Z ful consideration of the fields and calculation of the interactions. h¯ ωj ≡ Uem = hψ| dr Eˆ (r)e(r)Eˆ (r)|ψi − The goal of this section is to show how we can obtain equa- Z tions similar to equations 32-34 where x(t) and α(t) now repre- hvac| dr Eˆ (r)e(r)Eˆ (r)|vaci sent mode amplitudes for acoustic and optical excitations in a Z nanophotonic device. ∗ = 2 dr ej (r)e(r)ej(r) We start by solving separately the dynamical equations for ∗ electromagnetics and elastodynamics which can be expressed = 2Veffmax[ej (r)e(r)ej(r)]. (45) as eigenvalue equations for the magnetic field H(r) and elastic displacement field Q(r) respectively: with R ∗( ) ( ) ( ) e dr ej r e r ej r 2 2 0 V = Lhj = ωj hj, L(·) = c curl curl (·) . (39) eff,j ∗ ., and e(r) max[ej (r)e(r)ej(r)] s h¯ ωj ω2Q (r) = LQ (r) max[|ej(r)|] = . (46) j j j 2Veff,jediel λ + µ µ L(·) = − ∇(∇ · (·)) − ∇2(·). (40) ρ ρ Having defined the quantization of the fields and mode normalizations, we can now write a Hamiltonian for the optome- The set of solutions of these two equations are the normal elec- chanical system, tromagnetic and acoustic modes of the structure, and define the † † spectrum. Typically, a software package such as COMSOL is H = h¯ ∑ ωjaj aj + h¯ ∑ ωjbj bj +Hint (47) used to obtain these solutions in dielectric structures that don’t j j | {z } | {z } permit analytic analysis. Valid solutions of the electromagnetic Ho Hm and elastic field in the structure can then be expressed as normal mode expansions that can capture the quantum dynamics. The challenge remains calculation of the optomechanical interaction term Hint. We are ˆ ( ) = ( ) ( ) + † † E r, t ∑ ej r aj t h.c., (41) interested in interactions of the type aj ak(bl + bl ) which are the j simplest type that allow energy conservation, assuming that the photonic frequencies for mode j and k are many orders of ˆ ˆ magnitude larger than the mechanical frequency of mode l. For Q(r, t) = ∑ Qk(r)bk(t) + h.c. (42) k the case where j = k, we are considering a shift in the frequency of optical mode j due to a mechanical displacement in mode l. ˙ ˆ with a˙ j(t) = −iωjaj(t), and bk(t) = −iωkbk(t). In defining The relevant interaction energy or rate can be calculated using the quantum field theory, we assign to each mode j a Hilbert first-order cavity perturbation theory. In a dielectric structure space {|nij, n = 0, 1, 2, 3, ..} where |nij is the state representing characterized by e0(r), modifications due to deformations can n photons or phonons in the jth mode. Phonons and photons be taken into account with the expression in each of these Hilbert spaces are then annihilated with the operators aj and bj respectively. The normalizations of ej and Qk e(r) = e0(r) + δe(r). (48) 24

To first order, such a modification of the dielectric causes a shift B. Waveguides: 2D-confined in the optical resonance frequency of a mode with mode profile A theory for 2D-confined waveguides with translational sym- e(r) of metry can be developed similarly to the previous section on 3D-confined cavities. We refer to [89, 92, 93] for a thorough treatment from first principles. Here, we focus on connecting the (1) ω0 he|δe|ei ω = − . (49) waveguides’ dynamics to the cavities’ dynamics described in the 2 he|e|ei previous section. For a waveguide the translational symmetry implies that the photonic and phononic eigenproblems can be There are two ways that the dielectric constant changes due to expressed in terms of wavevectors β and K, yielding as solution this deformation. First, a deformation of the optical resonator dispersion relations ω(β) and Ω(K). As shown in [89, 92, 93], boundaries affects the dielectric tensor at the between different the Hamiltonian of the waveguide is H = H + H with the (r) free int materials. This is because the high-contrast step profile of e free Hamiltonian given by across a boundary is shifted by deformations of the structure. By Z Z relating a deformation to a change in the dielectric constant, we H = dβ h¯ ω a† a + dK h¯ ω b† b (54) can use equation 49 to calculate the optomechanical coupling. free β β β K K K Johnson has derived a useful expression [283] for this shift in and the interaction Hamiltonian given by frequency, which when adapted to optomechanics [48], gives a ZZ frequency shift per unit displacement of h¯ † Hint = √ dβ dK gβ+K aβbK + gβ−K aβbK + h.c. (55) 2π R k 2 −1 ⊥ 2 ω Qn(r)(∆e|e | − ∆(e )|d | )dA in the momentum-description where the three-wave mixing = − 0 gOM,B R 3 (50) † 2 max(|Q|) e(r)|e(r)|2d r interaction rate gβ±K = g0|β±Kαβ±K is proportional to the pump amplitude αβ±K of the mode with wavevector β ± K. Here we for a mechanical vector displacement field Q(r) with component assume phase-matching (∆K = −βp + β ± K = 0 with βp the pump wavevector) and usually consider α to represent a strong Qn(r) normal to the boundary. Secondly, a photoelastic contribution to the optomechanical coupling arises from local changes pump that can be treated classically. in the refractive index due to stress in the material induced by Previous work [75] linked the waveguide’s coupling rate the mechanical deformation. For a particular displacement vec- g0|β±K and Brillouin gain coefficient GB to an optomechanical cavity’s coupling rate g via a mean-field transition performed tor Q(r) corresponding strain tensor S = 1 ∂ Q + ∂ Q , the 0 ij 2 i j j i on the photonic and phononic equations of motion both in the dielectric perturbation is given by limit of large and small cavity finesse. Here, we derive the same connection but now via a mean-field transition in the large- p · S finesse limit performed directly on the waveguide’s Hamiltonian δe(r) = e · · e, (51) given by expression 55. We consider a cavity of roundtrip length e0 L constructed from a waveguide described by equation 55 and focus on a triplet of two photonic and one phononic mode(s). = − 4 which reduces to δeij e0n pijklSkl for an isotropic medium. The operator corresponding to the number of excitations in each The fourth-rank tensor p with components pijkl is called the pho- of the modes can be expressed as toelastic tensor and is a property of the material. The roto-optic Z k +π/L effect, which captures permittivity changes induced by rota- † c † c c = dk ck ck (56) tion, must be included as well in optically anisotropic materials kc −π/L [284–286]. Composite metamaterials may yield enhanced pho- 2π = c† c toelasticity [286]. Often, when considering the symmetries in L kc kc the atomic structure of the material, a reduced tensor is used with kc a relevant center wavevector. Therefore, the cavity and with elements pij. The perturbation integral can then be used to calculate the shift in frequency per unit displacement: waveguide operators are linked by r 2π R 3 c = ck (57) ω e · δe · e d r L c = − 0 gOM,PE R 3 . (52) 2 max(|Q|) e(r)|e(r)|2d r with c either a photonic or phononic ladder operator. Consider- ing the first term in Hint, we therefore have These expressions give the boundary and photoelastic compo- ZZ 2 nents for a shift in the optical cavity frequency per unit displace- 2π dβ dK gβ+K aβbK = gβ+K aβbK (58) ment of the maximum deflection point of a deformation profile L Q(r). A natural unit for displacement is the zero-point fluctua- 2π = g α δaδb (59) tion amplitude found by multiplying the expressions (50) and L 0|β+K β+K p (52) by the zero-point fluctuation length x = h¯ /(2m ω ) g0|β+K zpf eff m = √ αδaδb (60) (see equation 44). The coupling rate is g0 = g0,Bnd + g0,PE, and L the corresponding optomechanical interaction Hamiltonian can = g αδaδb (61) be written as 0 with δa and δb the cavity’s photonic and phononic ladder opera- † Hint = h¯ (gOM,PE + gOM,Bnd)xa a tors. Thus we obtain g0|β+K † † g0 = √ (62) = hg¯ 0(b + b)a a. (53) L 25 as in equation8 in the main text. This link between the coupling suppresses the interaction in the equations of motion via a rotat- rates holds both for forward and backward as well as for intra- ing term, for instance and inter-modal scattering [75]. (∂ + v ∂ )b = −iω b − ig αa† e−i∆Kcz (71) Next, we consider the waveguide’s dynamics in slightly more t m z m 0|βp detail. The dynamics of an optomechanical waveguide is usually The range of spatiotemporal effects described by these and ex- considered after transforming from momentum- to real-space tended versions of these equations of motion are considered in operators Z detail in amongst others [75, 89, 92, 94, 108, 146, 167, 287, 288]. dk −i(k−kc)z c(z) = e ck (63) 2π B. SINGLE-PHOTON NONLINEARITY We are usually concerned with narrow bandwidths relative to the group-velocity-dispersion so the frequencies can be ex- In this section we give a derivation for relationships 21 and 22. panded to first-order as We consider a 2D-confined waveguide of length L and inject a photon flux hΦi = vg/L with vg the optical group velocity that ωk ≈ ωc + vg(k − kc) (64) corresponds to one photon on average in the waveguide. This photon excites the mechanical system with a displacement x1, A few manipulations [92] starting from equation 54 lead to which in turn yields a phase-shift ϑwg on a second probe photon. Z The phase-shift can be expressed as h † † i Hfree = h¯ dz a (z)ωˆ aa(z) + b (z)ωˆ mb(z) (65) ϑwg = k0(∂xneff)x1L (72) with ωˆ k = ωc − ivg∂z the real-space operator corresponding with k0 the vacuum optical wavevector and ∂xneff the sensitivity to the momentum-space dispersion relation ωk. Higher-order of the effective optical index to mechanical motion x1. Assuming expansions of the dispersion relation in equation 64 yield higher- a static displacement, we have x1 = hFi/keff with hFi the force order spatial derivatives the operator ωˆ k. exerted by the first photon and keff the effective mechanical Further, dropping the second term in equation 55 the real- stiffness per unit length. In addition, from power-conservation 1 space interaction Hamiltonian becomes it can be shown that hFi = c ∂xneff(h¯ ω)hΦi [289]. Substitution Z leads to † h¯ ω2 1 2 Hint = h¯ dz g | + α (z)a(z)b(z) + h.c. (66) 0 β K ϑwg = ∂xneff vg (73) keff c Further, the Heisenberg equations of motion c˙(z) = Since the Brillouin gain coefficient can be expressed as [14] − i [ ( ) H ] h¯ c z , int together with the equal-time commutator 2 † 0 0 Qm 1 [c(z), c (z )] = δ(z − z ) yield GB = 2ω ∂xneff (74) keff c (∂t + vα∂z)α = −iωαα − ig0|β+K a b this yields † h¯ ω GB (∂t + va∂z)a = −iωaa − ig0|β+Kα b (67) ϑwg = vg (75) 2 Qm ( + ) = − − † ∂t vm∂z b iωmb ig0|β+Kα a Making use of equation9 gives These equations describe the spatiotemporal evolution of the 2 2g0|β+K three interacting fields in absence of dissipation and phase- ϑ = (76) wg v mismatch. Intrinsic propagation losses can be included via a gωm dissipative term, e.g. This is the single-photon cross-phase shift for a statically driven mechanical waveguide. When the mechanics is driven with † (∂t + vm∂z)b = −iωmb − γb − ig0|β+Kαa (68) a detuning ∆Ω > γ close to the resonance, one must replace x → (ωm/(2∆Ω))x so the cross-Kerr phase shift increases to −1 1 1 for the phononic field with γ = vmαm with Lm = αm the 2 phononic decay length. Similarly, a non-zero phase-mismatch g0|β+K ∆K 6= 0 effectively reduces the interaction rate. In particular, ϑwg = (77) vg∆Ω dropping the second term for simplicity the momentum-space interaction Hamiltonian 55 becomes which is in agreement with more rigorous analysis [118]. This phase-shift is independent of the waveguide length L since the ZZZ h¯ † ∆KL photon flux and the optical forces are inversely proportional Hint = √ dβ dK dβp g0|β αβ aβbK L sinc + h.c. 2π p p 2 to length L when there is on average a single photon in the (69) waveguide. L with the waveguide length. Thus the finite length weakens the Next, we link the waveguide single-photon phase-shift ϑwg wavevector-selectivity, generating interactions between a larger to the cavity single-photon phase shift set of modes. The strongest interactions are obtained between modes for which ∆K = 0. This corresponds to the real-space 2g2 ϑ = 0 (78) Hamiltonian cav κ∆Ω Z derived in section5A of the main paper. Making use of equation † i∆Kcz Hint = h¯ dz g α (z)a(z)b(z) e + h.c. (70) 0|β+K 8, the cavity phase-shift ϑcav can be expressed as 2 with ∆Kc = −βpc + βc + Kc the phase-mismatch between the 2g0|β+K center wavevectors of the photonic and phononic fields. This ϑcav = (79) Lrtκ∆Ω 26 with Lrt the cavity roundtrip length. The cavity finesse is F = 2π/(κTrt) with Trt = Lrt/vg the cavity roundtrip time so we obtain F ϑ = ϑ (80) cav π wg The expression for the cavity phase-shift assumed critical coupling and a small phase shift given by ϑcav = 2∆/κ with ∆ = ϑwg/Trt the mechanically-induced detuning from the cavity resonance.