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Home , Cavity optomechanics, Optomechanics

Unifying and cavity

Rapha¨elVan Laer,∗ Roel Baets and Dries Van Thourhout Research Group, Ghent University–imec, Belgium Center for Nano- and Biophotonics, Ghent University, Belgium So far, Brillouin scattering and were mostly disconnected branches of re- search – although both deal with - coupling. This begs for the development of a broader theory that contains both fields. Here, we derive the dynamics of optomechanical cavi- ties from that of Brillouin-active waveguides. This explicit transition elucidates the link between phenomena such as Brillouin amplification and electromagnetically induced transparency. It proves that effects familiar from cavity optomechanics all have traveling-wave partners, but not vice versa. We reveal a close connection between two parameters of central importance in these fields: the Bril- louin gain coefficient and the zero-point optomechanical coupling rate. This enables comparisons between systems as diverse as clouds, plasmonic Raman cavities and nanoscale sili- con waveguides. In addition, back-of-the-envelope calculations show that unobserved effects, such as photon-assisted amplification of traveling , are now accessible in existing systems. Finally, we formulate both circuit- and cavity-oriented optomechanics in terms of vacuum coupling rates, cooperativities and gain coefficients, thus reflecting the similarities in the underlying .

I. INTRODUCTION Circuit Cavity g˜ ˜ 1 L 0 L a(z, t)dz = 0 G 0 z Brillouin scattering [1] and cavity optomechanics [2] ˜ R C G have been intensively studied in recent years. Both con- C z g0 cern the interaction between light and sound, but they z z = L/2 were part of separate traditions. Already in the early 1920s, diffraction of light by sound was studied by L´eon FIG. 1. From circuit to cavity optomechanics. We ex- plicitly derive the physics of optomechanical cavities (right) Brillouin. Therefore, such inelastic scattering is called from that of Brillouin-active waveguides (left). Therefore, Brillouin scattering [3, 4]. The effect is known as stim- both traveling-wave and cavity-based optomechanics can be ulated Brillouin scattering (SBS) [5–7] when a strong cast in terms of vacuum coupling rates (˜g0 and g0), coopera- intensity-modulated light field generates the sound, of- tivities ( ˜ and ) and gain coefficients ( ˜ and ). ten with classical applications such as spectral purifica- C C G G tion [8] and signal processing [9] in mind. In contrast, cavity optomechanics arose from Braginsky’s light generates motion and the motion phase-modulates efforts to understand the limits of de- light. Next, this spatiotemporal phase-modulation cre- tectors in the 1970s – and greatly expanded since the ates motional sidebands – which interfere with those demonstration of phonon lasing in microtoroids [10]. By initially present. The research fields share this essen- and large, it aims to control both optical and mechanical tial feedback loop. Some connections have already been quantum states [11–13]. made. For instance, electrostrictive forces were exploited Historically, a number of important differences hin- for sideband cooling [17, 18] and induced transparency dered their merger. For instance, SBS generally dealt [19, 20] while contributed to SBS in with high-group-velocity and cavity optomechanics with dual-web fibers [21] and silicon waveguides [22–25]. low-group-velocity acoustic phonons. In addition, bulk In this work, we derive the dynamics of optomechan- electrostrictive forces usually dominated phonon gener- ical cavities from that of Brillouin-active waveguides ation in SBS – while radiation pressure at the bound- (fig.1). The transition holds for both co- and counter- aries took this role in cavity optomechanics. Further, propagating pump and Stokes waves, i.e. for “for- cavity optomechanics typically studied resonators with ward” and “backward” scattering, and for opto-acoustic much lower phonon than photon dissipation – whereas coupling between two different or two identical optical arXiv:1503.03044v4 [physics.] 22 May 2016 Brillouin [8, 14, 15] operate in the reversed regime modal fields, i.e. for “inter-” [18, 26–29] or “intra-” [16]. Finally, SBS is often studied not in cavities but [22, 23, 28, 30–35] modal scattering (fig.2). Hence, all in optically broadband waveguides [1]. Thus, particular flavours of photon-phonon interaction are treated on the physical systems used to be firmly placed in either one same footing. Moreover, this spatially averaged cav- or the other research paradigm. ity dynamics is found to be equivalent to the standard Lately, the idea that these are mostly superficial classi- Hamiltonian of cavity optomechanics [2] – even in the fications has been gaining traction. Indeed, in both cases case of low-finesse phonons. It turns out that this cav- ity dynamics can be mapped – by swapping space and time (z t) – on the steady-state spatial evolution of the opto-acoustic↔ fields in the waveguide. The treatment ∗ [email protected] describes Raman effects as well. 2

ω fast slow to this phononic damping. Hence, there are two to date unexplored regimes of guided-wave optomechanics: 1 (1) photon-assisted amplification of traveling phonons ωp 3 2 Ω and (2) strong coupling between traveling and ωs phonons (fig.3). The strong coupling regime produces either traveling entangled photon-phonon pairs or state k swapping between light and sound along the waveguide, FIG. 2. Phase-matching diagrams. The optical disper- depending on the details (e.g. probe detuning) of the sion relation ω(k) shows phonon-mediated coupling between experiment. Although currently unobserved, both ef- co- or counter-propagating photons and between two identical fects may be an asset in future quantum phononic net- (intra-) or two different (inter-modal) optical modes. There- works [13, 48–51]. For instance, in the strong coupling fore, there are generally four types of optomechanics, of which regime the flying phonon – entangled to its photonic three are indicated in this diagram. The fourth is counter- partner – could be detected piezo-electrically or optically coupling between two different optical modes. and thereby enable Bell tests [52–54] between two differ- ent particle species. Our back-of-the-envelope estimates Space Time show that these regimes can be achieved in existing sys- z t αs αm αm αs κs κm κm κs tems, such as dual-web fibers and silicon nanowaveguides. photon gain phonon gain photon phonon laser The transition (fig.1) assumes that the photonic and g˜ α + α g κs + κm phononic modes of the waveguide are not disturbed too s m  strong coupling strong coupling strongly by looping it into a cavity. This is justified in many cases since cavity designs aim to minimize the g˜0 ˜ ˜ α Φ g0 κ n C G C G losses (e.g. due to bending) induced by any modal per- turbations. Within this approximation, it permits trans- FIG. 3. Symmetry of circuit and cavity optomechan- lations between circuit- and cavity-oriented figures of ics. Each temporal optomechanical effect has a spatial sym- merit. For instance, we identify a connection between metry partner. Thus, the description of these effects can be the Brillouin or Raman gain coefficient ˜ and the zero- cast in terms of conceptually similar figures of merit. The point coupling rate g . The former ( ˜G) quantifies the scheme assumes a red-detuned optical probe; “gain” and 0 G “laser” should respectively be replaced by “loss” and “cool- pump power and waveguide length required to amplify a ing” for a blue-detuned optical probe. The meaning of the Stokes seed appreciably [3, 4]. The latter (g0) captures figures of merit is discussed in the main text. the interaction strength between a single photon and a single phonon in an optomechanical cavity [2]. The tran- sition proves that these figures of merit are inextricably This implies that the plethora of optomechanical ef- linked by fects, such as stimulated Brillouin scattering [3, 4, ˜ 36], slow light [37–39], optomechanically induced trans- (~ωp)Ωm v v = g2 (1) parency [39, 40], ground-state cooling [11, 41] etc., are p s G 0 4L Qm ! different aspects of the same feedback loop. The rig- orous transition decisively indicates that both fields are with vp and vs the group velocities of the pump and a subset of a larger theory of photon-phonon interac- Ωm Stokes waves, ~ωp the pump photon energy, 2π the tion, which may be built on a single Hamiltonian [42]. mechanical resonance frequency, L the cavity roundtrip This is not to say that they are identical: a Brillouin- length and Qm the waveguide’s mechanical quality fac- active waveguide supports complex spatiotemporal phe- tor. This link is independent of the type of driving op- nomena [43–45] and noise dynamics [46, 47] not present tical force and of the relative photon and phonon damp- in a high-finesse optomechanical cavity. Nevertheless, ing. Similarly, we derive connections between each of the in the resulting picture (fig.3), both traveling-wave and circuit- and cavity-oriented figures of merit: between the cavity-based photon-phonon interaction can be classified vacuum coupling rates (˜g0 and g0, see (21)), the cooper- according to (1) the damping hierarchy of the photons ativities ( ˜ and , see (34)) and the gain coefficients ( ˜ and phonons and (2) the strength of the photon-phonon and , seeC (36)).C G coupling with respect to the largest dissipation channel. Notably,G this treatment goes beyond cavity optome- For weak coupling, the long-lived particle species – ei- chanical systems that have a clear circuit equivalent ther photons or phonons – triggers the photon-phonon (as in fig.1). Indeed, the standard cavity Hamiltonian conversion. The short-lived particle species cannot truly ˆ = ω (ˆx)ˆa aˆ + Ω ˆb ˆb [2] also captures the tempo- build up and is thus slaved to its long-lived partner; it is ~ c † ~ m † ralH dynamics of cavity optomechanics based on Bose- merely created in short segments (of space or time) and Einstein condensates [55, 56] or plasmonic Raman cavi- immediately decays afterwards. ties [57]. The physics of all these diverse systems can be All Brillouin-active waveguides so far exhibited far understood in the scheme of fig.3. On top of the similar stronger phononic than photonic propagation losses; dynamics, this means that the photon-phonon interac- in addition, the coupling was always weak relative tion efficiency of a larger class of systems can now be 3

vm Ω bulk the following dynamical evolution [42, 47, 59] 3 1 1 3 v− ∂tap + ∂zap = ig˜0asb χ˜− ap bulk p − − p 2 1 1 1 vs− ∂tas ∂zas = ig˜0b†ap χ˜s− as (2) 2 vm = ∂qΩ ± − − Ωc 1 1 vm− ∂tb + ∂zb = ig˜0as†ap χ˜m− b q 1 q − − Its derivation proceeds from Maxwell’s and the elastic- a b ity equations on the assumption that the envelopes vary FIG. 4. Example of phonon dispersion relation. a, The slowly in space and time [42, 47, 59]. This starting point frequency Ω(q) of transversally trapped acoustic phonons has and the following treatment holds quantum mechanically a Raman-like cut-off Ωc for low q and approaches the bulk relation for large q. b, Thus, the phonon group velocity vm if one takes care to treat the envelopes in (2) as operators vanishes for low q and becomes the bulk speed for large q. [42, 47] obeying the equal-time commutator

a (z, t), a† (z0, t) = v v δ δ(z z0) (3) γ γ0 √ γ γ0 γγ0 compared in a single framework. For instance, the gain − h i coefficient of a silicon nanowire can be converted to the with “γ” an index running over the pump “p”, Stokes vacuum coupling rate of a hypothetical cavity (through “s” and mechanical wave “m”, vγ the group velocities, (1)); which can next be compared to that of any other δγγ0 the Kronecker delta, δ(z) the Dirac delta distribu- cavity optomechanical system. In reverse, the link en- tion and am = b for notational convenience. We flux- ables the conversion of a vacuum coupling rate of an normalized the field operators aγ such that Φp = ap† ap, actual cavity optomechanical system into a hypotheti- Φs = as†as and Φm = b†b correspond to the number cal guided-wave coupling rate (through (21)); which can of pump photons, Stokes photons and phonons passing next be compared to that of any other waveguide. We through a cross-section of the waveguide per second. We give examples of such conversions, which can be tested will treat highly occupied (i.e. large mean flux Φγ ) empirically in many cases, in section V. modes as classical in the remainder of the paper,h asi is The paper is organized as follows: in section II we de- standard [2, 60–62]. Further, we denoteg ˜0 the traveling- scribe a minimal model of traveling-wave optomechanics wave vacuum coupling rate (to be discussed further on), and frame it in terms of a guided-wave vacuum coupling αγ χ˜ 1 = i∆˜ the susceptibilities, α the propagation rateg ˜ and cooperativity ˜. Next, we make the mean- γ− 2 γ γ 0 −˜ field transition to a cavityC in section III. At that point, losses and ∆γ the wavevector offsets between externally we also discuss the limitations of the analysis. The re- applied fields and the intrinsic waveguide modes. sulting dynamical effects are treated in section IV. The In some systems, e.g. for the Raman-like low-group- prospects for observing new effects are considered in sec- velocity phonons (fig.4) associated with forward intra- modal scattering [22–24, 28, 30], the phonon wavelength tion V and we conclude in section VI. 2π 1 q can be substantially larger than its decay length αm− – so its slowly-varying amplitude treatment breaks down. II. CIRCUIT OPTOMECHANICS Then the acoustic excitation is better treated as a lo- calized series of mechanical oscillators [23, 30, 59], essen- In particular, we study the interaction between a pump tially dealing with each cross-sectional slice of the waveg- uide as an artificial Raman-active molecule. The above field with envelope ap(z, t) and a red-detuned Stokes field dynamics (2), however, contains these systems as well by envelope as(z, t) mediated by an acoustic field with enve- 1 lope b(z, t). The envelopes contain only the slowly vary- letting the phonon decay length αm− vanish. Further, the sign ( ) in the Stokes equation indicates the difference ing part of the photonic-phononic fields; rapidly oscil- ± i(kz ωt) between forward (+) and backward ( ) photon-phonon lating factors e − were removed in each case. The coupling. Cascaded scattering [30, 63]− and noise [46, 47] guided optical modes correspond to the points (ωp, kp) can and should be added to this model in some instances. and (ωs, ks) in the optical dispersion relation (fig.2). By energy and wave- [58] conservation, the ex- In fact, (2) can be regarded as the unique, minimal model for guided-wave Brillouin scattering [42, 47, 59]. We dis- cited phonon has an angular frequency Ω = ωp ωs and − cuss potential extensions in section V; in the following, wavevector q = kp ks. The nature of the optical modes (co/counter and fast/slow)∓ and the acoustic dispersion we only need the minimal model (2), future extended relation determine the wavevector q and group velocity versions can be dealt with similarly. The Manley-Rowe relations [3, 64] guarantee that a vm of the excited phonons (fig.2&4). Traveling-wave photon-phonon coupling is governed by single, unique figure of meritg ˜0 captures all conservative optical forces and scattering. Indeed, in the lossless case (αγ = 0), the rate of pump photon destruction must equal the rate of Stokes photon and phonon creation:

∂ Φ = ∂ Φ = ∂ Φ = g˜ ia†b†a + h.c. (4) − z p ± z s z m − 0 s p  4

Similar to g0 in a cavity [2],g ˜0 quantifies the inter- with action strength between a single photon and a single 4˜g2Φ 4˜g2 phonon, but in this case flying along a waveguide in- ˜ = 0 p = (10) stead of trapped in a cavity. We takeg ˜0 real and positive C αsαm αsαm without loss of generality. Briefly specializing to forward intra-modal scattering, the mean-field transition of sec- the guided-wave cooperativity andg ˜ =g ˜0 Φp the pump-enhanced spatial coupling rate. Therefore, ˜ = 1 tion III will show that (see Appendices) p is the threshold for net phonon-assisted gain onC flying ∂kp g˜0 = x˜ZPF (5) photons. Since Pp = ~ωpΦp is the pump power, we ob- − ∂x ˜P ωp tain ˜ = G p and αs C with 4˜g2 δL ~ ˜ = 0 (11) x˜ZPF = xZPF = (6) G ω α v 2m v Ω ~ p m r m r eff m m the guided-wave zero-point motion and meff the effec- the well-known Brillouin gain coefficient [3, 4], here tive mass of the mechanical mode per unit length. In- framed in terms of a spatial coupling rateg ˜0 and cooper- deed, a short waveguide section of length δL contains ativity ˜. It characterizes the spatial exponential build- C n = δL Φ phonons with Φ the mean phonon up of a Stokes seed in case of highly damped phonons m vm m m h i h i h i αs flux. As particle fluxes – instead of numbers – are funda- (αs αm). Since Φm = ˜ Φs Φs , there are  h i αm Ch i  h i mental in the traveling-wave Manley-Rowe relations (4), on average far fewer phonons than photons flying along the zero-point motion is rescaled by precisely this factor the waveguide in this case. The system enters the strong 1/2 ˜ αm (δL/vm) relative to the actual zero-point motion xZPF coupling regime as soon as (see section IV). C ∼ αs [2] of the δL-section Second, when the phononic damping is lowest (α m  αs), we similarly get a slaved Stokes wave (∂zas 0) ~ → x = (7) given by as = iχ˜sg˜0b†ap resulting in (∆˜ γ = 0) ZPF 2m δLΩ − r eff m αm Therefore, the traveling-wave vacuum coupling rateg ˜0 is ∂zb = (1 ˜) b (12) determined by the wavevector shift induced by mechan- − − C 2 ical motion at fixed frequency, while the cavity vacuum such that ˜ = 1 also yields the threshold for net coupling rate g0 is determined by the frequency shift in- C photon-assisted gain on flying phonons. Since Φs = duced by mechanical motion at fixed wavevector [2]. No- αm h i ˜ Φm Φm , there are far fewer photons than tably, the interpretation ofg ˜0 as the coupling strength be- αs Ch i  h i tween a single traveling photon and phonon holds also for phonons flying along the waveguide in this case. The sys- tem enters the strong coupling regime as soon as ˜ αs . inter-modal and backward scattering (see Appendices). αm By replacing the undepleted pump with an undepleted,C ∼ In steady-state (∂t 0) and for a constant, strong → strong Stokes mode (˜g =g ˜0√Φs), it follows that an anti- pump (Φp(z) = Φp(0)), the evolution (2) reduces to Stokes seed sees larger loss by a factor (1+ ˜) convention- 1 C ∂zas = ig˜0b†ap χ˜− as (8) ally (α α ) and that a guided-wave phonon channel ∓ ∓ s s  m 1 can be cooled by a factor (1 + ˜) when it has the low- ∂zb = ig˜0as†ap χ˜m− b C − − est propagation loss (αm αs). An undepleted, strong 1  The phonon decay length αm− is generally largest for phononic beam (˜g =g ˜0√Φm) yields similar coupling be- backward scattering. Even then, it typically does not tween the pump and Stokes wave. 1 exceed α− 100 µm [3, 65]. Therefore, the photon de- The coupling rateg ˜ and the cooperativity ˜ respect the m ∼ cay length massively exceeds the phonon decay length symmetry between flying photons and phonons,C whereas in Brillouin-active waveguides to date (αs αm). A the gain coefficient ˜ (11) is most relevant in case of  full solution of (8) exists but yields little intuitive in- stronger phonon damping.G Therefore, we regardg ˜ and sight (see Appendices). Therefore, we initially focus on ˜ as more natural and fundamental figures of merit. It C two subcases: the conventional (αs αm) and the re- is straightforward to extend the above discussion for ab-  αpz versed case (αm αs), both in the weak coupling regime sorptive decay of the pump flux, i.e. Φp(z) = Φp(0)e− (˜g Φ α +α ). These examples illustrate how one ˜ 0 p  s m and non-zero wavevector detunings ∆γ = 0. can formulate guided-wave optomechanics, including the So far, we discussed two subcases of guided-wave6 Bril- p classical stimulated Brillouin regime, in terms of the vac- louin scattering. We treat the strong coupling regime in ˜ uum coupling rateg ˜0 and cooperativity . section IV and the full solution in the Appendices. Next, First, strongly damped phonons (α C α ) act as a s  m we move on to cavity optomechanics via the mean-field localized slave wave (∂ b 0) given by b = iχ˜ g˜ a†a . z → − m 0 s p transition. On resonance (∆˜ γ = 0), we thus have α ∂ a = (1 ˜) s a (9) z s ∓ − C 2 s 5

III. BRIDGE TO CAVITY OPTOMECHANICS (ω) ωp D phonon ∆p In this section, we transition to an – Ω photon κ κm made from a Brillouin-active waveguide – of roundtrip Ω length L (fig.1). To do so, we introduce the mean-field ∆m ∆s envelope operators ... ωs ω Ω ωc 1 L a m a(t) = a(z, t)dz (13) L Z0 (ω) ω D p ∆p for both the optical (ap/s(t)) and acoustic (b(t)) fields. Such mean-field models have found early use in the treat- Ω κp κm κs ment of fluorescence [66] and recently also in the context ∆m of frequency combs [67]. During roundtrip propagation, ∆s Ω each field obeys dynamics of the form (see (2)) ... ω ω ωcp 1 1 Ωm cs v− ∂ta + ∂za = ζ χ˜− a (14) b − FIG. 5. Cavity description. The photonic and phononic with ζ the nonlinear interaction term. To describe the density of states (ω). The mean-field equations (20) de- D cavity feedback (fig.1), we add the boundary condition scribe coupling between one phononic and either one (a) or two (b) photonic resonances. The latter case (b) is most a(0, t) = √1 α 1 µ eiϕa(L, t) + √µ s(t) (15) power-efficient, although hard to achieve in practice [8]. − 0 − p with α0 the additional loss fraction along a roundtrip (on top of α, such as bending losses), µ the fraction of pho- (16) also reverses. Therefore, potential dynamical dif- tons or phonons coupled to an in- or output channel, ϕ ferences between forward and backward scattering dis- the roundtrip phase shift and s(t) the flux-normalized appear in a high-finesse traveling-wave cavity – at least envelope of injected photons or phonons. By Taylor- in the minimal model (2) of guided-wave optomechanics. Comparing (2) to (14), we see that ζ fg with f and g expansion of (15), we get ∝ equal to ap/s or b. In the mean-field approximation, we α0 + µ assume these envelopes vary little over a roundtrip such a(L, t) a(0, t) iϕ a(t) √µ s(t) (16) − ≈ 2 − − that fg = f g holds (see Appendices). Finally, we ap-   ply the mean-field (14)-to-(19) transition to (2). Hence, with higher-order terms negligible and a(L, t) a(t) in an optomechanical cavity – constructed from a Brillouin- the high-finesse limit. Low-finesse situations, particularly≈ active waveguide – is governed by relevant for phonons, are treated further on (see (24)). ˙ 1 We operate close to the cavity resonance, such that ϕ ap = ig0asb χp− ap + √κcpsp  − − 2π. Next, we let (13) operate on (14) and use ∂ta = a˙(t): 1 a˙ = ig b†a χ− a + √κ s (20) s − 0 p − s s cs s 1 ˙ 1 1 ˙ 1 v− a(t) + L− a(L, t) a(0, t) = ζ(t) χ˜− a(t) (17) b = ig a†a χ− b + √κ s { − } − − 0 s p − m cm m We insert (16) in (17) and find with

1 √µ a˙ = vζ χ− a + s (18) vpvsvm − T g˜0 = g0 (21) L r 1 κ with χ− = i∆ the cavity’s photonic or phononic 2 − response function, κ = κi + κc the total decay rate, κi = the well-known temporal zero-point coupling rate [2]. In- α0+αL µ deed, equations (20) are equivalent (see Appendices) to T the intrinsic decay rate, κc = T the coupling rate, ˜ the Heisenberg equations of motion resulting from the ∆ = ϕ+∆L the detuning and T = L the roundtrip time. T v well-known Hamiltonian ˆ = ω (ˆx)ˆa aˆ + Ω ˆb ˆb [2]. √ ~ c † ~ m † Next, we multiply (18) by T and switch from flux- Remarkably, the equivalenceH holds even for inter-modal to number-normalized fields (a √T a): 7→ and backward scattering. The connection (21) between the traveling-wave and the cavity-based vacuum coupling 1 a˙ = v√T ζ χ− a + √κ s (19) − c ratesg ˜0 and g0 is at the heart of this work: other links such as (1) are based on this result. Further, the mean- From here on, n = a†a represents the number of quanta field transition transforms the guided-wave commutator in the cavity, while s†s still corresponds to the injected photon or phonon flux. The transition from (14) to (19) still holds when we replace z z because condition 7→ − 6

(3) into phonon coupling or bending losses. In case of doubt, it is safe to alternatively write (1) as L L √vγ vγ0 aγ , a† = δγγ dzdz0δ(z z0) γ0 2 0 (~ωp) αm 2 L 0 0 − v v v ˜ = g (25) h i Z Z p s m 4L G 0 √vγ vγ0 = δγγ0 (22) L Both ˜ and g0 are well-established in the study of photon- phononG interaction, but they operate on different levels. and through rescaling a by T into γ γ The ~ enters (1) because ˜ is classical G p while g0 is inherently quantum mechanical. In addition, † aγ , aγ = δγγ0 (23) ˜ quantifies the combined action of forces and scatter- 0 G h i ing and contains the phonon loss – while g0 does not. thus correctly retrieving the standard Further, larger L yields a smaller g0 while ˜ is length- ˜ commutators [2]. 2 ~ G independent. Therefore, g0 G . This mean-field To derive (20), we made the same mean-field transition ∝ L Qm derivation is but one way to prove the ˜ g conver- for photons and phonons. In particular, this supposes a G ↔ 0 2π sion, other approaches yield the same result (see Appen- large phonon finesse m = 1. Often there is only F κmTm  dices). This proof captures all reversible photon-phonon intrinsic phonon loss such that κm = vmαm and thus this requires 2π 1. In many systems, the phonon decay coupling mechanisms. αmL 1  length αm− is much shorter than the roundtrip length L. Then this phonon high-finesse limit does not hold. IV. SYMMETRY BETWEEN CIRCUIT AND However, we can completely neglect phonon propagation CAVITY OPTOMECHANICAL EFFECTS (∂ b 0 in (2)) if α is sufficiently large. The phonons’ z → m envelope operator b then obeys In this section, we describe both guided-wave and 1 1 cavity-based regimes of photon-phonon coupling. To be- v− ∂ b = ig˜ a†a χ˜− b m t − 0 s p − m gin with, we recover and briefly review the known cavity- based regimes of photon lasing, phonon lasing and strong Applying (13), multiplying by √Tm and switching from flux- to number-normalized envelopes results in coupling. Next, we map these regimes on the guided- wave spatial evolution of the opto-acoustic fields. The ˙ 1 mapping unveils two unobserved regimes of guided-wave b = ig a†a v χ˜− b (24) − 0 s p − m m Brillouin scattering. We pay particular attention to the strong coupling regime (˜g α + α ). where we used (21). Hence, this localized low-phonon-  s m finesse approach yields the same result as the previous Here, we assume zero photon and phonon input flux high-finesse limit with sm = 0 (see (20)). Therefore, and an undepleted pump. Then (20) reduces to even low-finesse phonons produce the same dynamics as 1 is commonly studied in cavity optomechanics [2]. a˙ s = ig0b†ap χ− as (26) − − s Notably, the standard treatment of cavity optomechan- ˙ 1 b = ig a†a χ− b ics [2] does not consider an explicit space variable: the − 0 s p − m Hamiltonian ˆ performs an implicit spatial average by H These equations treat the photons and phonons identi- describing the entire object as single mechanical oscilla- cally. Therefore, every photonic phenomenon must have tor, in contrast to the explicit spatial average (13) per- a phononic counterpart and vice versa. Even more, the formed in this work. However, even the implicit average temporal cavity dynamics (26) can be mapped (t z) on in ˆ requires low-loss acoustic excitations to set up a 7→ H the spatial steady-state waveguide evolution (8). Each ef- global mechanical mode self-consistently, precisely as in fect known from cavities therefore has a waveguide coun- the high-finesse approximation leading to (20). In the lo- terpart (but not vice versa as we will see). This also calized, low-finesse phonon approach that generates (24), implies that the spatial figures of merit have a temporal the spatial averaging can still be performed and yields the symmetry partner and vice versa; we prepared for this at same classical dynamics – but its meaning changes. Now the end of section II by defining a guided-wave vacuum ( < 1) the acoustic wave is too lossy to set up a global ˜ Fm coupling rateg ˜0 and cooperativity . To clearly expose mechanical mode for the entire cavity. Instead, the cav- these symmetries, we solve (26); keepingC in mind that ity consists of an ensemble of independent Raman-like the very same discussion holds spatially for (8). First, mechanical oscillators. It is no longer possible to address we decouple (26) and get phonons circulating in the cavity.

Finally, we combine (21) and (11). Using αmvm = d ? d 1 2 + χ− + χ− b(t) = g b(t) (27) Ωm/Qm, we obtain result (1) immediately. Note that dt s dt m Qm is defined here as the waveguide’s intrinsic phonon     quality factor, which could be different from the cavity’s Here, we introduced the pump-enhanced coupling rate γt phonon quality factor if there were e.g. non-negligible g = g0√np. Next, we insert the ansatz b e in (27) ∝ 7 and find two roots γ system enters the strong coupling regime as soon as ± κm . C ∼ κs 1 ? 1 ? 1 2 2 Further, if the photon-phonon coupling rate is suffi- γ = χs− + χm− χs− χm− + 4g ± 2 − ± − ¨ 2  q  ciently strong, (27) simplifies to b = g b. An identical   (28) derivation yields ¨b = g2b if the Stokes wave is consid- In general, these roots strongly mix the photon and − phonon response: the photon-phonon pair forms a po- ered undepleted. Therefore, a red-detuned probe pro- duces entangled photon-phonon pair generation (b(t) lariton [12, 56, 61, 68, 69]. The guided-wave analog of gt ∝ e± ), whereas a blue-detuned probe produces Rabi flop- (27) is igt ping between photons and phonons (b(t) e± ) [2]. A ? 1 2 situation of equally strong optical and mechanical∝ damp- ∂z χ˜s− ∂z +χ ˜m− b(z) = g˜ b(z) (29) ± ± ing (κ κ ) invalidates the above weak coupling treat- s ≈ m and it can be treated identically. The full spatial and ment even for small g. However, this is not sufficient to temporal dynamics is governed by the general solution see strong coupling behavior. From the general solution (28) (see Appendices). However, it is more instructive to (see Appendices), this requires g κs + κm. Indeed, in  consider the limiting cases of weak and strong photon- the strong coupling regime the hybridized photon-phonon phonon interaction relative to the system’s damping. polariton sees half the optical and half the mechanical g First, if the photon-phonon interaction is sufficiently damping [12]. Therefore, the state-swap frequency π κs+κm weak, i.e. g κs κm , the two roots in (28) discon- must be high compared to the average decay rate 2 nect. Usually, the | photon− | and phonon decay rates differ to observe an actual Rabi swap before the population significantly. Then there are two scenario’s depending decreases by 1/e. on the relative photonic and phononic decay rates. Es- By comparing (26) and (8), we prove an analogy be- sentially, the dynamics of the short-lived particle can be tween spatial and temporal optomechanical effects (fig.3): adiabatically eliminated, although it may still strongly the above cavity-based discussion still largely holds for modify the response of its long-lived partner. guided-wave optomechanics with the mapping g g˜ , 0 7→ 0 In particular, when the phonons decay slowly (κm ˜, κ α , κ α and n Φ. In case of all  C 7→ C s 7→ s m 7→ m 7→ κs), the photonic response is barely modified: a˙ s 0 co-propagating waves, and in the absence of cascading → [30, 63] and noise [46, 47], the mapping of cavity optome- and therefore as = iχsg0b†ap to a good approximation. However, the phononic− response can then dramatically chanics onto a Brillouin-active waveguide in steady-state 1 2 ? is an exact equivalence. However, when for instance one change to χm− g χs . Hence, we recover the spring effect 2 ?− 2 ? of the particles counter-propagates, such as the Stokes (δΩm = g χs ) and phonon lasing (δκm = 2g χs ) [2]. = − < photons in backward scattering, important differences At the photon resonance (∆s = 0), the phonon linewidth 4g2 arise that have no equivalent in cavity optomechanics. equals κm + δκm = (1 ) κm with = the coop- κsκm Indeed, as proven in section III, information regarding erativity. Therefore, the− Cthreshold forC mechanical lasing the propagation direction of the waves disappears in the is = 1. This instability was first contemplated by Bra- mean-field transition. Instead comparing (27) and (29), ginskyC [70] and received further study in systems ranging much can still be learned by instead mapping g2 g˜2 from gram-scale mirrors [71] to optomechanical crystals 0 7→ − 0 κm and κs αs. Note that this particular difference dis- [72, 73]. Since ns = nm nm , there are far 7→ − κs appears if the counter-propagating particle species is un- fewer Stokes photonsh i thanCh phononsi  in h thei cavity in this depleted: then it vanishes from the dynamics and the situation. The system enters the strong coupling regime situation is identical to the co-propagating case. as soon as κs . C ∼ κm Thus, guided-wave weak coupling requiresg ˜ Similarly, when the photons decay slowly (κs κm), α α withg ˜ =g ˜ Φ the spatial coupling rate (see ˙  s m 0 p the phononic response is barely modified: b 0 and Appendices).| ∓ | Under weak coupling, there are two cases → p therefore b = iχmg0a†ap to a good approximation. depending on the relative photon and phonon propaga- − s However, the photonic response can then dramatically tion losses. We have touched upon these subcases at the 1 2 ? change to χs− g χm. Hence, we recover the cav- end of section II and briefly consider them again here to − 2 ? ity frequency pull (δωcs = g χm) and photon lasing show the similarity with cavity-optomechanical effects. 2 ? = (δκs = 2g χ ) [4, 74, 75]. At the phonon reso- First, when the phonons propagate far (α α ), the − < m m  s nance (∆m = 0), the Stokes linewidth equals κs + δκs = photonic loss αs barely changes. However, the phononic 1 2 ? (1 ) κs with the same temporal cooperativity as response can then drastically change toχ ˜ g˜ χ˜ , which − C C m− s above. Therefore, the threshold for Brillouin lasing is includes a shift in both the phononic propagation− loss also = 1. First realized in fibers [76], this case was re- (δα = 2˜g2 χ˜?) and group velocity ( χ˜?); i.e. C m − < s ∝ = s cently also studied in CaF2 resonators [14], silica disks [8] traveling-phonon amplification and light-induced slowing and chalcogenide waveguides [15]. Such lasers are known down of sound. In section V, we show that this unob- for their excellent spectral purity [75, 77] and received served regime can be achieved in existing systems. attention for quantum-limited amplification [16]. Since Second, when the Stokes photons propagate far (αs n = κs n n , there are far fewer phonons  m κm s s αm), the phononic loss αm barely changes. However, thanh i StokesCh photonsi  in h thei cavity in this situation. The 8 the photonic response can then drastically change to phonon losses, so effectively lower , in the cavity than 1 2 ? C χ˜s− g˜ χ˜m. Hence, we are back in the conventional in the waveguide. This directly shows that domain− of phonon-assisted amplification of traveling pho- tons (δα = 2˜g2 χ˜? ) and sound-induced slowing down ˜ (34) s m C ≥ C −? < ˜ of light ( χ˜m) [38]. At resonance (∆m = 0), the Stokes ˜ ∝ = Pp ˜ given (33), κγ vγ αγ and ˜ = G . Clearly, the guided- propagation loss is (1 )αs as in (9). ≥ C αs If the coupling is sufficiently− C strong compared to the wave cooperativity exceeds the cavity-based cooperativ- 2 ity since the cavity has additional dissipation (e.g. cou- propagation losses (˜g αs+αm), (29) simplifies to ∂z b = g˜2b (see Appendices). In the forward (+) case, and with pling and bending losses). Finally, we define a gain coef- ± ficient for a cavity in analogy to (11) boundary condition b(0) = 0, this yields G a (z) = a (0) coshgz ˜ (30) 4g2 s s = 0 (35) G ωpκm b(z) = ia†(0) sinhgz ˜ ~ − s which characterizes the temporal exponential build-up of such that Φ (z) = Φ (0) cosh2 gz˜ and Φ (z) = s s m the Stokes when the phonons are heavily damped. The Φ (0) sinh2 gz˜ . Therefore, Φ (z) Φ (z) = Φ (0) and s s − m s gain coefficients therefore obey ∂zΦs = ∂zΦm as required by the Manley-Rowe relations (4). In the backward ( ) case, with L the waveguide ˜ L length and boundary condition− b(0) = 0, the evolution (36) G ≥ vpvs G along the waveguide has no cavity-optomechanics coun- terpart. Specifically, we retrieve given κm vmαm and (21). Hence, the guided-wave and cavity-based≥ optomechanical figures of merit are now as(L) as(z) = cosgz ˜ (31) conceptually similar and the relations between each of cosgL ˜ them were given in (1), (21), (34) and (36). a (L) b(z) = i s† singz ˜ − cosgL ˜ V. PROSPECTS Φs(L) 2 such that Φs(z) = cos2 gL˜ cos gz˜ and Φm(z) = Φs(L) 2 Φs(L) In this section, we first give a couple of examples of how cos2 gL˜ sin gz˜ . Therefore, Φs(z) + Φm(z) = cos2 gL˜ and the ˜ g0 connection (1) can be implemented – includ- ∂zΦs = ∂zΦm as required by Manley-Rowe (4). The G ↔ − π ing several systems in which it can be tested empirically. system has an instability atgL ˜ = 2 , which is reached before a full state swap between light and sound can be Next, we move on to the prospects for observing new completed. This situation is called “contraflow Hermi- regimes of guided-wave optomechanics, simultaneously il- tian coupling” in classifications of coupled-mode interac- lustrating the application of our framework. Finally, we tions [78, 79]. In case of anti-Stokes (instead of Stokes) briefly discuss potential extensions to the minimal model seeding in the strong coupling regime, an identical deriva- (2) of traveling-wave Brillouin scattering. 2 2 Table I presents four implementations of the conver- tion leads to ∂z b = g˜ b – which produces the same ˜ Rabi oscillations for forward− and backward scattering. sion from the gain coefficient to the vacuum coupling rate g ( ˜ g ) and four in reverseG ( ˜ g ). The sys- Although familiar in resonators [2], these strong-coupling 0 G → 0 G ← 0 effects have not yet been observed in the field of guided- tems range from silicon nanowires and dual-web fibers wave Brillouin scattering; see section V for the prospects. to ultracold atom clouds and GaAs disks. In five cases, We conclude this section by analyzing the relation be- such as for silicon nanowires, the conversion can clearly be tested empirically by measuring ˜ and g through in- tween the guided-wave and cavity-based cooperativities G 0 ( ˜ and ) and by introducing a gain coefficient ( ) for dependent, established methods [2, 23]. In three cases, anC optomechanicalC cavity. Note that the temporalG coop- the conversion is hypothetical but still allows for com- erativity parison of the photon-phonon interaction strengths. For instance, an ultracold atom cloud in a Fabry-P´erotcav- 4g2 ity [55] has no obvious traveling-wave equivalent. Never- = (32) theless, its hypothetical waveguide partner would have a C κsκm 8 1 1 large gain coefficient of 10 W− m− – which compares is the ratio between the roundtrip gain and loss: inserting favorably to optomechanical∼ waveguides realized to date. 2 2 PpTp g = g np, np = and (21) in (32) indeed leads to So far, all Brillouin-active waveguides had far lower 0 ~ωp 1 1 phonon than photon propagation lengths (αm− αs− ). ˜P v α ˜P L v α Cavity-optomechanical systems, by contrast, more often = G p m m = G p m m (33) C κs κ κ T κ than not had far lower phonon than photon damping vs m s s m rates (κm κs) [2]. Only uniquely high-optical-quality with P the intracavity pump power and vmαm a natu- [8, 14, 15, 74, 83, 84] systems succeed at reversing the p κm rally appearing correction factor that allows for higher latter hierarchy (κ κ ). The common reversal of this m  s 9

1 1 g0 Ωm ˜ [W− m− ] [Hz] [Hz] Qm [ ] L [µm] ng [ ] λ [µm] G ←→(1) 2π 2π − − 4 ? 1.5 106 10 3 Silicon nanowire [23, 24] 10 · 10 10 4.6 1.55 −→ √L [µm] − Silica standard fiber [4] 1 ? 70 1010 500 1.45 1.55 −→ √L [cm] − 6 ? 3 103 6 4 Silica dual-web fiber [21] 4 10 · 6 10 4 10 1.7 1.55 · −→ √L [cm] · · − 2 ? 7 105 9 Chalcogenide rib [80, 81] 3 10 · 8 10 230 2.6 1.55 · −→ √L [µm] · − Silica microtoroid [12] 600 3 103 8 107 2 104 97 1.45 0.78 ←− · · · Silicon optomechanical crystal [73] 4 104 ? 6 105 6 109 2 103 5 5 1.55 · ←− · · · Rb ultracold atom cloud [55] 108 6 105 4 104 42 400 1 0.78 ←− · · GaAs optomechanical disk [82] 5 104 3 105 109 2 103 8 4 1.55 · ←− · · TABLE I. Translation between waveguides and cavities. The table contains four conversions from a gain coefficient to a vacuum coupling rate and four in reverse. The right five columns contain the parameters necessary for the conversion through formula (1). These are order-of-magnitude estimates. In some cases, indicated by a ?, the conversion can be empirically tested. In other situations, the conversion is hypothetical: e.g. an ultracold atom cloud in a Fabry-P´erotcavity has no obvious guided-wave equivalent. damping hierarchy (going from waveguides to cavities) In contrast, silicon chips can produce significantly stems from the small phonon group velocities (v v ). lower αm ratios and therefore ease the condition on ˜ m s αs The question naturally arises if waveguides with larger for strong coupling. One can expect phonon propaga-C 1 1 1 phonon than photon propagation length (αm− αs− ) tion distances up to αm− 1 mm, as these are readily can be made, while still keeping high cooperativities achieved in surface-acoustic-wave∼ devices [89]. Together 4˜g2 1 ˜ ˜ = 1. Currently, the largest phonon decay with αs− 1 cm [23], this yields > 10 as the strong αsαm ∼ C C ∼ 1 coupling condition, which requires a reasonable pump lengths are of the order α− 100 µm in backward Bril- m αs 4 1 1 ∼ power of P > 10 = 100 mW with ˜ 10 W− m− louin scattering [23, 65]. To realize larger phonon propa- p ˜ G ∼ gation lengths, one must look for waveguides with large [23, 24]. Indeed, currentG nanoscale silicon systems have acoustic group velocities vm and small linewidths κm. already demonstrated ˜ 2 [24, 63]. Hence, taking Thus, one promising approach uses low-frequency flex- into account the rapid progressC ≈ in state-of-the-art devices ural modes (Ω q2) in backward mode (large q) at low [1, 23, 24, 63], we expect demonstrations of traveling- ∝ temperatures (large Qm). Indeed, then we have both phonon amplification and guided-wave strong coupling 7 Ωm 10 large vm q and small κm = 4 Hz = 1 kHz in the coming years. Such observations would open up ∝ Qm ∼ 10 1 vm entirely new realms of optomechanics. [12, 85, 86]. Since α− = in a waveguide (where m κm Finally, this work can be extended on several fronts. there is only propagation loss), we find that decay lengths First, the mean-field transition can be applied to the up to α 1 10 m are feasible given v 104 m/s and m− m noise models of [46, 47]. Second, the regime of “nonlin- κ 1 kHz.∼ Such a small phonon propagation∼ loss would m ear quantum optomechanics” [2, 90–92] should be trans- strongly∼ boost the cooperativity ˜, which could compen- C ferred to waveguides. This requires that strong coupling sate for a potentially lower coupling rateg ˜ in backward 1 is reached for merely one pump photon (Φ 1 s− ): mode. Clearly, nothing intrinsically forbids amplification p 7→ g˜ =g ˜ Φ =g ˜ > α + α . As α α usu- of traveling phonons in systems such as the dual-web fiber 0 p 0 s m m  s 1 ally, traveling-wave nonlinear quantum optomechanics is [21] – where αs− 10 cm. Besides its scientific inter- p est, such a traveling-phonon∼ amplifier may be useful in achieved wheng ˜0 > αm. Third, the coupling between the phonon networks [13, 48, 49, 87, 88]. phononic mode and the thermal bath [46, 47] must be Next, we look into achieving the strong coupling regime treated carefully to obtain truly quantum-coherent [12] coupling. And fourth, we focused mainly on the dynam- in the typical situation of high acoustic loss (αm αs). To see traveling-wave Rabi flopping, entangled photon- ics that optomechanical waveguides and cavities have in phonon pair production or contraflow Hermitian coupling common, but wisdom may be found in the differences as (see section IV), one must obtaing ˜ > α or equivalently well. We gave the example of contraflow strong coupling m in section IV. In addition, the cavity has input fluxes ˜ > αm . In optical fibers, in backward mode and given αs that have no equivalent in a typical guided-wave set-up, C 1 1 8 α− 10 km and α− 100 µm, this requires ˜ > 10 . s ∼ m ∼ C while the waveguide can display spatiotemporal effects This necessitates an unrealistic continuous-wave pump (both ∂z and ∂t in (2)) that are absent in a cavity (only 8 α 1 1 s ˜ − power of Pp > 10 ˜ = 10 kW with 1 W m− . ∂t in (20)). On top of this, the cavity breaks the symme- G G ∼ 10 try between Stokes and anti-Stokes scattering, whereas The link between the Brillouin gain coefficient ˜ and the G this symmetry prevents exponential build-up of noise in zero-point coupling rate g0 was derived in a platform- low-dispersion forward intra-modal scattering [47]. It has independent way. As illustrated for silicon nanowires also yet to be determined whether different cavity dy- and ultracold atom clouds, it significantly expands the namics results in the medium-finesse case, as section III variety of systems whose photon-phonon interaction effi- was limited to a low or high finesse. ciency can be compared. Through the mean-field tran- With slight modifications, (2) also captures Raman ef- sition, we derived the dynamics of optomechanical cavi- fects [4, 6, 93]. For instance, the phonon frequency is ties from that of Brillouin-active waveguides. We framed much larger so an optical phase-mismatch can arise. Fur- the behavior of both systems in terms of cooperativi- ther, the thousandfold higher optical phonon frequency ties and vacuum coupling rates. Next, we showed that puts most Raman modes in the at room phenomena familiar from cavity optomechanics all have temperature. This breaks the symmetry between Stokes guided-wave partners, but not the other way around. and anti-Stokes scattering, such that thermally seeded The broader theory predicts that several novel regimes, exponential build-up may be seen even in the forward such as guided-wave strong coupling, will be accessible intra-modal case [47]. Some parameters introduced here in state-of-the-art systems in the coming years. Hence, – such as the spatial vacuum couplingg ˜0 – lose elegance we showed that Brillouin scattering and cavity optome- in the Raman case: the impossibility of significant opti- chanics are subsets of a larger realm of photon-phonon cal phonon transport undermines their symmetric defi- interaction. nitions. Mathematically, expressions such as (5) diverge 1 as vm and αm− vanish. Then one must resort to the broken-symmetry Raman gain coefficient, which never- theless obeys (1). Therefore, the core of this work also applies to guided-wave [4, 94–96], cavity-based [97–100] and surface-enhanced Raman scattering [57]. ACKNOWLEDGEMENT

VI. CONCLUSION R.V.L. acknowledges the Agency for Innovation by Science and Technology in Flanders (IWT) for a PhD In conclusion, we unveiled a connection between grant. This work was partially funded under the FP7- Brillouin-active waveguides and optomechanical cavities. ERC-InSpectra programme and the ITN-network cQOM. R.V.L. thanks T.J. Kippenberg for related discussions.

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Appendix A: Link to cavity Hamiltonian Appendix B: Manley-Rowe relations

With the mean-field transition derived in the main In this section, we prove that the Manley-Rowe rela- text, we take a step beyond the ˜ g link. As we show tions guarantee the existence of a single real, positive G ↔ 0 in this section, the mean-field equations are equivalent to photon-phonon coupling coefficient in waveguides (˜g0) the cavity Langevin equations in the resolved-sideband and in cavities (g0). In waveguides, the Manley-Rowe re- limit (κ Ωm). In the case of coupling between one lations are formulated at the level of photon and phonon mechanical and one optical resonance (fig.2a), the usual fluxes Φ. In cavities, they are written down in terms of theory [2] starts from the Hamiltonian the total photon and phonon numbers n. ˆ ˆ ˆ ˆ = ~ωcaˆ†aˆ + ~Ωmb†b + int H H with a. Manley-Rowe in waveguides ˆ ˆ ˆ int = ~g0aˆ†aˆ b + b† H A Brillouin-active waveguide in steady-state (∂t 0)   obeys (see (2)) → the interaction Hamiltonian,x ˆ = xZPF ˆb + ˆb† the me- 1 ∂ a = iκ˜ a b χ˜− a chanical oscillator’s position, xZPF the zero-point  motion, z p − mop s − p p ˆ 1 aˆ and b ladder operators for the optical and mechanical ∂zas = iκ˜mosb†ap χ˜s− as (B1) ∂ωc ± − − oscillator and g0 = xZPF ∂x the zero-point coupling rate. 1 ∂ b = iκ˜ a†a χ˜− b When the pump is undepleted, the interaction Hamil- z − om s p − m tonian can be linearized:a ˆ = α + δaˆ with δaˆ a small with arbitrary normalizations of the pump, Stokes and fluctuation. Then we have acoustic envelope such that generallyκ ˜ =κ ˜ = mop 6 mos 6 ˆ(lin) ˆ ˆ κ˜om are different complex numbers. Using ∂z a†a = int = ~g0α δaˆ + δaˆ† b + b† H (∂za†)a + a†(∂za), we find ˙   i  ˆ  Using the equation of motion aˆ = [ˆa, ] and the ? − ~ H ∂zΦp = αpΦp + iκ˜mopas†b†ap + h.c. commutator [ˆa, aˆ†] = 1 (the same for ˆb), this linearized − ∂ Φ = α Φ iκ˜ a†b†a + h.c. (B2) Hamiltonian leads straight to the coupled equations [2] ± z s − s s − mos s p  κ ∂zΦm = αmΦm iκ˜omas†b†ap + h.c. δaˆ˙ = i∆ δaˆ ig α ˆb + ˆb† − − − 2 − − 0 Suppose now that the envelopes are flux-normalized such ˆ˙ κm  ˆ   b = iΩm b ig0α δaˆ + δaˆ† that Φ = a a ,Φ = a a and Φ = b b correspond − 2 − − p p† p s s† s m † to the number of pump photons, Stokes photons and with ∆ = ω ω . Next, we consider a blue-detuned p − c phonons passing through a cross-section of the waveg- pump in the resolved-sideband regime (κ Ωm). Then  iΩt uide per second. Then we demand that, in the lossless we can write the ladder operators as δaˆ aˆse and case (α = 0), the rate of pump photon destruction equals iΩt → j ˆb ˆbe− , witha ˆ and ˆb now slowly-varying. We neglect → s the rate of Stokes photon and phonon creation the ˆb-term in the optical equation and the δaˆ-term in the mechanical equation because they are off-resonant. This ∂zΦp = ∂zΦs = ∂zΦm (B3) − ± is the rotating-wave approximation, which corresponds These are the Manley-Rowe relations [3, 79] for a Bril- to the classical slowly-varying envelope approximation [3, louin waveguide. We deduce from (B2) and (B3) that 4]. Hence, the above equations reduce to ? 1 κ˜ =κ ˜mos =κ ˜om (B4) aˆ˙ = ig αˆb† χ− aˆ (A1) mop s − 0 − s s ˙ 1 This proves the existence of a single coupling coefficient ˆb = ig αaˆ† χ− ˆb − 0 s − m that captures all reversible optical forces and scattering. and we find that equations (A1) are identical to equations Note that (B4) also guarantees power-conservation since (16) givena ˆs as and ˆb b. Remarkably, the equiva- 7→ 7→ ∂z (~ωpΦp ~ωsΦs + ~ΩΦm) = 0 lence holds even though the pump and Stokes could be ± counter-propagating or in different optical modes. In the leads with (B2) in the lossless case to unresolved-sideband limit (Ωm κ), anti-Stokes gener-  ? ation and cascading must be added for forward intra- ωpκ˜ + ωsκ˜mos + Ω˜κom = 0 (B5) − mop modal, but not necessarily for backward or inter-modal which is true given (B4) and ω = ω +Ω. Next, we show Brillouin scattering. Indeed, comb generation is usually p s that this coefficient (B4) can be taken real and positive not accessible by backward or inter-modal coupling be- without loss of generality. Renormalizing the envelopes cause of the phase-mismatch (fig.2). This assumption to c a , c a and c b yields new coupling coefficients can be violated in Fabry-P´erotcavities [101] or when the p p s s m first-order Stokes becomes sufficiently strong to pump a cp cs cm κ˜mop ? κ˜mos ? κ˜om (B6) second-order Stokes wave [9]. cscm cpcm cpcs 14

iϕ as can be seen from (B1). Suppose thatκ ˜om =g ˜0e is b. Manley-Rowe in cavities complex withg ˜0 real and positive. Then we take cp = iϕ cs = cm = e− . Using (B4) and (B6), it follows that the Here, we apply the discussion of the previous section to renormalized coupling coefficients are real and positive: the mean-field cavity equations. With arbitrary envelope

κ˜mop =κ ˜mos =κ ˜om =g ˜0 (B7) normalizations and without input, equations (13) are ˙ 1 This unique coupling coefficient quantifies the coupling ap = iκmopasb χp− ap strength between a single photon and a single phonon − − ˙ † 1 propagating along a waveguide. Indeed, suppose that as = iκmosb ap χs− as (B14) 1/2 1 − − ap = as = b 1 s− such that Φp = Φs = Φm 1 s− ˙ 1 b = iκoma†ap χ− b at a certain7→ point along the waveguide. In the7→ lossless − s − m case, (B2) then becomes d with generally κmop = κmos = κom. Applying dt a†a = d d 6 6 ∂ Φ = 2˜g ( a†)a + a†( a) to (B14), we find z p − 0 dt dt  ∂zΦs = 2˜g0 (B8) ± d ? n = κ n + iκ a†b†a + h.c. ∂zΦm = 2˜g0 dt p − p p mop s p   So 2˜g0 gives the rate (per meter) at which the pump flux d n = κ n iκ a†b†a + h.c. (B15) decreases and the Stokes and phonon flux increase at a dt s − s s − mos s p point along waveguide through which one pump photon, d   n = κ n iκ a†b†a + h.c. one Stokes photon and one phonon are passing. dt m − m m − om s p The waveguide coupling coefficientg ˜0 can also be in-   terpreted in terms of a zero-point motion. As shown in Suppose now that the envelopes are number-normalized † (14), we have such that np = ap† ap, ns = as†as and nm = b b corre- spond to the number of pump photons, Stokes photons L and phonons in the cavity. We demand that, in the loss- g˜0 = g0 (B9) svpvsvm less case (κj = 0), the rate of pump photon destruction equals the rate of Stokes photon and phonon creation For forward intra-modal scattering (vp = vs = vg) n˙ p =n ˙ s =n ˙ m (B16) ∂ωp − g0 = xZPF (B10) ∂x These are the Manley-Rowe equations for an optome- kp chanical cavity. We deduce from (B15) and (B16) that is defined in terms of the zero-point motion and the cavity frequency pull at fixed wavevector [2]. Combining (B9), ? κ = κmos = κom (B17) (B10) and (D9), we obtain mop ω ∂n ∂k This proves the existence of a single coupling coefficient g˜ = p x˜ eff = x˜ p (B11) that captures all conservative optical forces and scatter- 0 − c ZPF ∂x − ZPF ∂x ωp ωp ing. Note that (B17) also guarantees energy-conservation with since

L ~ d x˜ZPF = xZPF = (B12) (~ωpnp + ~ωsns + ~Ωnm) = 0 v 2m v Ω dt r m r eff m m the waveguide “zero-point motion” and meff the effective leads with (B15) in the lossless case to mass per unit length. Indeed, a waveguide section of L ? length L contains on average nm = Φm phonons ωpκmop + ωsκmos + Ωκom = 0 (B18) h i vm h i − with Φ the mean phonon flux. As fluxes – instead of m which holds given (B17) and ω = ω + Ω. As in the pre- numbersh –i are the fundamental quantities in waveguides, p s vious section, one can show that this coupling coefficient the zero-point motion is corrected by precisely a factor can be chosen real and positive. This unique coupling L in (B12). vm coefficient must then be the well-known g0. It quantifies qOften the optical envelopes are power-normalized and the interaction strength between a single photon and a the acoustic envelope displacement-normalized. Starting single phonon trapped in a cavity. Indeed, suppose that from flux-normalized envelopes, one can switch to such ap = as = b 1 such that np = ns = nm 1 at a normalizations through certain point7→ in time. In the lossless case, (B15)7→ then becomes 2~Ωm cp = ~ωp cs = ~ωs cm = = 2˜xZPF k v r eff m n˙ p = 2g0 p p (B13) − n˙ s = 2g0 (B19) with keff the effective stiffness per unit length and by applying (B6). n˙ m = 2g0 15

So 2g0 gives the rate (per second) at which the number d. From independent full-vectorial definitions of pump photons decreases and the number of Stokes photons and phonons increases when there is one pump Here, we derive equation (D1) from the full-vectorial photon, one Stokes photon and one phonon in the cavity. definitions of ˜ and g0 – specializing to intra-modal for- Often the optical envelopes are energy-normalized and ward scattering.G We focus on the moving boundary con- the acoustic envelope displacement-normalized. Start- tribution. From the perturbation theory of Maxwell’s ing from number-normalized envelopes, one can switch equations with respect to moving boundaries [102], the to such normalizations through ∂ωc cavity frequency shift ∂x can be expressed as 2 1 2 2~Ωm ∂ωc ωp dA (u ˆn) ∆ E ∆− D cp = ~ωp cs = ~ωs cm = = 2xZPF = · | k| − | ⊥| k L 2 r eff ∂x 2 H dV  E  p p (B20) | | with xZPF the zero-point motion and by applying (B6). with u the normalized (max(R u ) = 1) acoustic field, ˆn the unit normal pointing from| material| 1 to material 2, 1 1 1 ∆ =   and ∆− = − − . The upper integral is 1 − 2 1 − 2 Appendix C: Mean-field approximation over the entire surface area of the cavity, the lower inte- gral across the cavity volume. Further, E is the electric k c. Justification of fg = fg field parallel to the boundary and D the displacement field perpendicular to the boundary. For⊥ a longitudinally invariant cavity, the surface integral can be reduced to a We denote f(z, t) and g(z, t) two operators that vary line integral and the volume integral to a surface integral: slowly on a lengthscale L. The mean-field operators 1 L 2 1 2 are defined as f(t) = f(z, t)dz. Clearly, when ∂ωc ωp dl (u ˆn) ∆ E ∆− D L 0 = · | k| − | ⊥| (D2) f(z, t) = f(0, t) and g(z, t) = g(0, t) are constants then 2 ∂x 2 H dA E  fg(t) = f(0, t)g(0, t) = fR(t)g(t). Let us assume now | | Further, the gain coefficientR ˜ is given by [23, 25, 103] that the amplitudes vary slowly enough such that they G can be Taylor-expanded as f(z, t) = f(0, t) + f 0z with ˜ Qm 2 f 0 = ∂zf(0, t) and the same for g. Then we have = ωp f, u (D3) G 2keff |h i| 1 L2 with f the power-normalized optical force density and f = f(0)L + f 0(0) L 2 f, u = f ∗ u dA. Note that keff is the effective stiffness   perh uniti length· . In the case of radiation pressure forces 1 L2 R g = g(0)L + g0(0) frp we have [103] L 2   1 2 1 2 frp = ∆ e ∆− d ˆnδ(r rboundary) where we dropped the time-dependence. Thus, we have 2 | k| − | ⊥| − with δ(r r ) a spatial delta-distribution at the L L2 − boundary fg = f(0)g(0) + (f 0g(0) + f(0)g0) + f 0g0 waveguide boundaries. The fields e and d are power- 2 4 normalized. Here we already assumed that the Stokes Similarly, and pump field profiles are nearly identical, which holds for intra-modal SBS given the small frequency shifts. L Hence, we get 1 2 fg = f(0)g(0) + (f 0g(0) + f(0)g0) z + f 0g0z dz L 0 1 2 1 2 Z frp, u = dl (u ˆn) ∆ e ∆− d (D4) L L2  h i 2 · | k| − | ⊥| = f(0)g(0) + (f 0g(0) + f(0)g0) + f 0g0 I 2 3 Additionally, the guided optical power P is given by

2 L vg vg 2 Therefore fg fg = f 0g0 12 which can be neglected if L P = E, E = dA E (D5) − 2 h i 2 | | is sufficiently small compared to the lengthscale on which Z f(z, t) and g(z, t) vary. Combining equations (D2), (D4) and (D5), we find

∂ωc vgωp = frp, u Appendix D: Alternative proofs of the ˜ g0 link ∂x 2 h i G ↔ A similar derivation can be done for the strained bulk, In this section, we describe two other approaches to so we have derive the link ∂ω v ω c = g p f, u ∂x 2 ˜ h i (~ωp)Ωm g2 = v2 G (D1) 2 ∂ωc 0 g 4L Q = f, u = (D6) m ! ⇒ h i vgωp ∂x 16 with f = frp +fes and fes the electrostrictive force density. ~ Substitution of (D10) in (D11) with xZPF = 2m LΩ Substituting equation (D6) in (D3) yields eff m results in q 2 2Q ∂ω 4LQm ˜ m c ˜ = g2 = 2 (D7) 2 0 G ω v k ∂x G ~ωpvgΩm p g eff   Finally, we use the definition of the zero-point coupling or the other way around rate g = x ∂ωc and the zero-point motion x = 0 ZPF ∂x ZPF ˜ (~ωp)Ωm ~ g2 = v2 G (D12) 2m LΩ with meff the effective mass per unit length. 0 g eff m 4L Qm ! qInserting these in (D7) yields This proof only holds for forward intra-modal scattering ˜ 2Qm 2meffLΩm 2 – whereas the mean-field transition applies to backward = 2 g0 G ωpvgkeff ~ and inter-modal scattering as well. 4L g2 = Q 0 (D8) m ( ω )Ω v2 ~ p m g Appendix E: Full solution of guided-wave evolution and (D8) is identical to (D1). In this derivation, we started from full-vectorial definitions that are only valid In this section, we give the full solution of the traveling- for intra-modal forward scattering. In contrast, the wave spatial dynamics (8) in the constant pump ap- mean-field transition shows that this result remains true proximation (Φp(z) = Φp(0)) where the pump is strong with vg √vpvs for inter-modal coupling. enough to be treated classically. From this solution, one → can derive the regimes treated in section IV as limiting cases. In addition, this solution can directly be mapped e. From independent derivative definitions on the cavity-based temporal dynamics (26). Thus, we start from

The cavity resonance condition is kpL = 2πm with m 1 ωpneff ∂zas = ig˜0b†ap χ˜s− as (E1) an integer. Given kp = and c the speed of light, ∓ ∓ c 1 this implies that ∂zb = ig˜0a†ap χ˜− b − s − m ∂ω ω ∂n which immediately yields p = p eff ∂x −neff ∂x ? 1 2 kp kp ∂ +χ ˜− ∂ χ˜− a (z) = g˜ a (z) (E2) z m z ± s s ± s

This can be recast in terms of the index sensitivity at where stands for forward (+) and backward ( ) scat- fixed frequency by tering.± Inserting the ansatz a (z) eγz leads to− s ∝ 2 ? 1 ? 1 2 ∂neff neff ∂neff γ + χ˜− χ˜− γ χ˜− χ˜− g˜ = 0 (E3) = m ± s ± m s − ∂x ng ∂x kp ωp Its solution is   c c with vph = n the phase velocity and ng = v the group eff g 1 ? 1 ? 1 2 2 γ = χ˜− χ˜− + χ˜m− χ˜s− + 4˜g index. Thus we have 1 2 − m ± s ∓  q  ∂ω ω ∂n   (E4) p = p eff (D9) ∂x − ng ∂x 1 ? 1 ? 1 2 2 kp ωp γ = χ˜− χ˜− χ˜m− χ˜s− + 4˜g 2 2 − m ± s − ∓  q  The cavity frequency pull must be calculated at fixed   ∂ωp Given these two roots, one can determine the exact evo- wavevector (g0 = xZPF ∂x ), so this yields kp lution of the photon-phonon fields along the waveguide if the correct boundary conditions are known. The bound- 2 2 ary condition b(0) = 0 and fixed probe flux Φs(0) = ∂neff 2 ωp − = g0 xZPF (D10) as†(0)as(0) are appropriate for forward scattering such ∂x ng ωp !   that

Previously [23], we showed that as(0) 1 γ1z 1 γ2z as(z) = γ2 +χ ˜s− e γ1 +χ ˜s− e γ2 γ1 − 2 −    (E5) ˜ Qm 1 ∂neff = 2ωp (D11) 1 1 G keff c ∂x as(0) γ2 +χ ˜s− γ1 +χ ˜s− ωp ! b (z) = i eγ1z eγ2z † g˜ γ2  γ1  { − } − 17

The backward case (fixed Φs(L) with L the waveguide actual photon-phonon pair generation can be seen. The length) can be solved similarly. This full solution con- photons and phonons indeed equally share the total prop- tains the important regimes discussed in section IV. For agation loss αm + αs in this regime, as in cavity settings instance, in the strong coupling regime (˜g α + α ) [12]. The spatial evolution (E5) then becomes identical  s m and at resonance (∆˜ j = 0) one can show that to (30). The weak coupling regimes of stimulated photon (αs αm) and phonon (αm αs) emission can equally α + α largeg ˜   γ m s +g ˜ g˜ (E6) well be obtained from the full solution (E5). This solu- 1 ≈ − 4 −→ tion also contains acoustic (α α ) [65] and optical s  m αm + αs (αm αs) build-up effects. γ2 g˜ g˜  ≈ − 4 − −→ − Therefore, the spatial coupling rateg ˜ must overcome the average photonic and phononic propagation loss before