Slowing and stopping light using an optomechanical crystal array

One of the major advances needed to realize all-optical information processing of light is the ability to delay or coherently store and retrieve optical information in a rapidly tunable manner. In the classical domain, this optical buffering is expected to be a key ingredient to managing the flow of information over complex optical networks. Such a system also has profound implications for quantum information processing, serving as a long-term memory that can store the full quantum information contained in an optical pulse. Here we suggest a novel approach to light storage involving an optical waveguide coupled to an optomechanical crystal array, where light in the waveguide can be dynamically and coherently transferred into long-lived mechanical vibrations of the array. Under realistic conditions, this system is capable of achieving large bandwidths and storage/delay times in a compact, on-chip platform.


Introduction
Light is a natural candidate to transmit information across large networks due to its high speed and low propagation losses. A major obstacle to building more advanced optical networks is the lack of an all-optically controlled device that can robustly delay or store optical wave-packets over a tunable amount of time. In the classical domain, such a device would enable all-optical buffering and switching, bypassing the need to convert an optical pulse to an electronic signal. In the quantum realm, such a device could serve as a memory to store the full quantum information contained in a light pulse until it can be passed to a processing node at some later time.
A number of schemes to coherently delay and store optical information are being actively explored.
These range from tunable coupled resonator optical waveguide (CROW) structures [1,2], where the propagation of light is dynamically altered by modulating the refractive index of the system, to electromagnetically induced transparency (EIT) in atomic media [3,4], where the optical pulse is reversibly mapped into internal atomic degrees of freedom. While these schemes have been demonstrated in a number of remarkable experiments [5,6,7,8], they remain difficult to implement in a practical setting. Here, we present a novel approach to store or stop an optical pulse propagating through a waveguide, wherein coupling between the waveguide and a nearby nano-mechanical resonator array enables one to map the optical field into long-lived mechanical excitations. This process is completely quantum coherent and allows the delay and release of pulses to be rapidly and all-optically tuned. Our scheme combines many of the best attributes of previously proposed approaches, in that it simultaneously allows for large bandwidths of operation, on-chip integration, relatively long delay/storage times, and ease of external control. Beyond light storage, this work opens up the intriguing possibility of a platform for quantum or classical all-optical information processing using mechanical systems.

Description of system: an optomechanical crystal array
An optomechanical crystal [9] is a periodic structure that constitutes both a photonic [10] and a phononic [11] crystal. The ability to engineer optical and mechanical properties in the same structure should enable unprecedented control over light-matter interactions. Planar two-dimensional (2D) photonic crystals, formed from patterned thin dielectric films on the surface of a microchip, have been succesfully employed as nanoscale optical circuits capable of efficiently routing, diffracting, and trapping light. Fabrication techniques for such 2D photonic crystals have matured significantly over the last decade, with experiments on a Si chip [12] demonstrating excellent optical transmission through long (N ¡ 100) linear arrays of coupled photonic crystal cavities. In a similar Si chip platform it has recently been shown that suitably designed photonic crystal cavities also contain localized acoustic resonances which are strongly coupled to the optical field via radiation pressure [9]. These planar optomechanical crystals (OMCs) are thus a natural candidate for implementation of our proposed slow-light scheme.  Figure 1. (a) Illustration of a double optical cavity system forming the unit cell of the optomechanical array. A two-way optical waveguide is coupled to a pair of optical cavity modes a 1 and a 2 , whose resonance frequencies differ by the frequency of the mechanical mode b. Both optical modes leak energy into the waveguide at a rate κ ex and have an inherent decay rate κ in . The mechanical resonator optomechanically couples the two optical resonances with a cross coupling rate of h. (b)A simplified system diagram where the classically driven cavity mode a 2 is effectively eliminated to yield an optomechanical driving amplitude Ω m between the mechanical mode and cavity mode a 1 . (c) Frequency-dependent reflectance (black curve) and transmittance (red) of a single array element, in the case of no optomechanical driving amplitude Ω m 0 (dotted line) and an amplitude of Ω m κ ex {10 (solid line). The inherent cavity decay is chosen to be κ in 0.1κ ex . (inset) The optomechanical coupling creates a transparency window of width 4Ω 2 m {κ ex for a single element and enables perfect transmission on resonance, δ k 0. (d) Energy level structure of simplified system. The number of photons and phonons are denoted by n 1 and n m , respectively.
The optomechanical driving amplitude Ω m couples states |n m 1, n 1 y Ø |n m , n 1 1y while the light in the waveguide couples states |n m , n 1 y Ø |n m , n 1 1y. The two couplings create a set of Λ-type transitions analogous to that in EIT.
In the following we consider an optomechanical crystal containing a periodic array of such defect cavities (see figures 1(a),(b)). Each element of the array contains two optical cavity modes (denoted 1, 2) and a co-localized mechanical resonance. The Hamiltonian describing the dynamics of a single element is of the form Here ω 1,2 are the resonance frequencies of the two optical modes, ω m is the mechanical resonance frequency, andâ 1 ,â 2 ,b are annihilation operators for these modes. The optomechanical interaction cross-couples the cavity modes 1 and 2 with a strength characterized by h and that depends linearly on the mechanical displacementxWpb b X q.
While we formally treatâ 1 ,â 2 ,b as quantum mechanical operators, for the most part it also suffices to treat these terms as dimensionless classical quantities describing the positive-frequency components of the optical fields and mechanical position. In addition to the optomechanical interaction described by equation (1), the cavity modes 1 are coupled to a common two-way waveguide (described below). Each element is decoupled from the others except through the waveguide. The design considerations necessary to achieve such a system are discussed in detail in the section "Optomechanical crystal design." For now, we take as typical parameters ω 1 {2π 200 THz, ω m {2π 10 GHz, h{2π 0.35 MHz, and mechanical and (unloaded) optical quality factors of Q m ω m {γ m 10 3 (room temperature)-10 5 (low temperature) and Q 1 ω 1 {κ 1,in 3¢10 6 , where γ m is the mechanical decay rate and κ 1,in is the intrinsic optical cavity decay rate. Similar parameters have been experimentally observed in other OMC systems [9,13]. In practice, one can also over-couple cavity mode 1 to the waveguide, with a waveguide-induced optical decay rate κ ex that is much larger than κ in .
For the purpose of slowing light, the cavity modes 2 will be resonantly driven by an external laser, so that to good approximationâ 2 α 2 ptqe ¡iω 2 t can be replaced by its mean-field value. We furthermore consider the case where the frequencies are tuned such that ω 1 ω 2 ω m . Keeping only the resonant terms in the optomechanical interaction, we arrive at a simplified Hamiltonian for a single array element (see figure 1(b)), Here we have defined an effective optomechanical driving amplitude Ω m ptq hα 2 ptq and assume that α 2 ptq is real. Mode 2 thus serves as a "tuning" cavity that mediates population transfer (Rabi oscillations) between the "active" cavity mode 1 and the mechanical resonator at a controllable rate Ωptq, which is the key mechanism for our stopped-light protocol. In the following analysis, we will focus exclusively on the active cavity mode and drop the "1" subscript. A Hamiltonian of the form (2) also describes an optomechanical system with a single optical mode, when the cavity is driven off resonance at frequency ω 1 ¡ ω m and a 1 corresponds to the sidebands generated at frequencies¨ω m around the classical driving field. For a single system, this Hamiltonian leads to efficient optical cooling of the mechanical motion [14,15], a technique being used to cool nano-mechanical systems toward their quantum ground states [16,17,18,19]. While the majority of such work focuses on how optical fields affect the mechanical dynamics, here we show that the optomechanical interaction strongly modifies optical field propagation to yield the slow/stopped light phenomenon. Equation (2) is quite general and thus this phenomenon could in principle be observed in any array of optomechanical systems coupled to a waveguide. In practice, there are several considerations that make the 2D OMC "ideal." First, our system exhibits an extremely large optomechanical coupling h and contains a second optical tuning cavity that can be driven resonantly, which enables large driving amplitudes Ω m using reasonable input power [20]. Using two different cavities also potentially allows for greater versatility and addressability of our system. For instance, in our proposed design the photons in cavity 1 are spatially filtered from those in cavity 2 [20]. Second, the 2D OMC is an easily scalable and compact platform. Finally, as described below, the high mechanical frequency of our device compared to typical optomechanical systems allows for a good balance between long storage times and suppression of noise processes.

Static regime
We first analyze propagation in the waveguide when Ω m ptq Ω m is static during the transit interval of the signal pulse. As shown in the Appendix, the evolution equations in a rotating frame for a single element located at position z j along the waveguide are given by Equation (3) is a standard input relation characterizing the coupling of right-(â R,in ) and left-propagating (â L,in ) optical input fields in the waveguide with the cavity mode. Here κ κ ex κ in is the total optical cavity decay rate,â N pzq is quantum noise associated with the inherent optical cavity loss, and for simplicity we have assumed a linear dispersion relation ω k c|k| in the waveguide. Equation (4) describes the optically driven mechanical motion, which decays at a rate γ m and is subject to thermal noisê F N ptq. The cavity mode couples to the right-propagating field through the equation where k 0 ω 1 {c. We solve the above equations to find the reflection and transmission coefficients r, t of a single element for a right-propagating incoming field of frequency ω k (see Appendix). In the limit where γ m 0, and defining δ k ω k ¡ ω 1 , while t 1 r. Example reflectance and transmittance curves are plotted in figure 1(c).
For any non-zero Ω m , a single element is perfectly transmitting on resonance, whereas for Ω m 0 resonant transmission past the cavity is blocked. When Ω m $0, excitation of the cavity mode is inhibited through destructive interference between the incoming field and the optomechanical coupling. In EIT, a similar effect occurs via interference between two electronic transitions. This analogy is further elucidated by considering the level structure of our optomechanical system (figure 1(d)), where the interference pathways and the "Λ"-type transition reminiscent of EIT are clearly visible. The interference is accompanied by a steep phase variation in the transmitted field around resonance, which can result in a slow group velocity. These steep features and their similarity to EIT in a single optomechanical system have been theoretically [21,22] and experimentally studied [23,24], while interference effects between a single cavity mode and two mechanical modes have also been observed [25]. From r, t for a single element, the propagation characteristics through an infinite array (figure 2(a)) can be readily obtained via band structure calculations [10]. To maximize the propagation bandwidth of the system, we choose the spacing d between elements such that k 0 d p2n 1qπ{2 where n is a non-negative integer. With this choice of phasing the reflections from multiple elements destructively interfere under optomechanical driving. Typical band structures are illustrated in figures 2(b)-(f). The color coding of the dispersion curves (red for waveguide, green for optical cavity, blue for mechanical resonance) indicates the distribution of energy or fractional occupation in the various degrees of freedom of the system in steady-state. Far away from the cavity resonance, the dispersion relation is nearly linear and simply reflects the character of the input optical waveguide, while the propagation is strongly modified near resonance (ω ω 1 ω 2 ω m ). In the absence of optomechanical coupling (Ω m 0), a transmission band gap of width κ forms around the optical cavity resonance (reflections from the bare optical cavity elements constructively interfere). In the presence of optomechanical driving, the band gap splits in two (blue shaded regions) and a new propagation band centered around the cavity resonance appears in the middle of the band gap. For weak driving (Ω m Àκ) the width of this band is 4Ω 2 m {κ, while for strong driving (Ω m Áκ) one recovers the "normal mode splitting" of width 2Ω m [26]. This relatively flat polaritonic band yields the slow-light propagation of interest. Indeed, for small Ω m the steady-state energy in this band is almost completely mechanical in character, indicating the strong mixing and conversion of energy in the waveguide to mechanical excitations along the array.
It can be shown that the Bloch wavevector near resonance is given by (see Appendix) The group velocity on resonance, v g pdk eff {dδ k q ¡1 | δ k 0 2dΩ 2 m {κ ex can be dramatically slowed by an amount that is tunable through the optomechanical coupling strength Ω m . The quadratic and cubic terms in k eff characterize pulse absorption and group velocity dispersion, respectively. In the relevant regime where κ ex 4κ in , κ ex ÁΩ m ,   these effects are negligible within a bandwidth ∆ωmin The second term is the bandwidth over which certain frequency components of the pulse acquire a π-phase shift relative to others, leading to pulse distortion. This yields a bandwidth-delay product of for static Ω m and negligible mechanical losses. When intrinsic optical cavity losses are negligible, and if one is not concerned with pulse distortion, light can propagate over the full bandwidth 4Ω 2 m {κ of the slow-light polariton band and the bandwidth-delay product increases to ∆ωτ delay N (see Appendix). On the other hand, we note that if we had operated in a regime where k 0 d πn, constructive interference in reflection would limit the bandwidth-delay product to ∆ωτ delay 1, independent of system size.
In the static regime, the bandwidth-delay product obtained here is analogous to CROW systems [2]. In the case of EIT, a static bandwidth-delay product of ∆ωτ delay c OD results, where OD is the optical depth of the atomic medium. This product is limited by photon absorption and re-scattering into other directions, and is analogous to our result ∆ωτ delay N κ ex {κ in in the case of large intrinsic cavity linewidth. On the other hand, when κ in is negligible, photons are never lost and reflections can be suppressed by interference. This yields an improved scaling ∆ωτ delay N 2{3 or N, depending on whether one is concerned with group velocity dispersion. In atomic media, the weak atom-photon coupling makes achieving OD ¡ 100 very challenging [27]. In contrast, in our system as few as N 10 elements would be equivalently dense.

Storage of optical pulse
We now show that the group velocity v g ptq 2dΩ 2 m ptq{κ ex can in fact be adiabatically changed once a pulse is completely localized inside the system, leading to distortion-less propagation at a dynamically tunable speed. In particular, by tuning v g ptqÑ0, the pulse can be completely stopped and stored. This phenomenon can be understood in terms of the static band structure of the system (figure 2) and a "dynamic compression" of the pulse bandwidth. The same physics applies for CROW structures [1,28], and the argument is re-summarized here. First, under constant Ω m , an optical pulse within the bandwidth of the polariton band completely enters the medium. Once the pulse is inside, we consider the effect of a gradual reduction in Ω m ptq. Decomposing the pulse into Bloch wavevector components, it is clear that each Bloch wavevector is conserved under arbitrary changes of Ω m , as it is fixed by the system periodicity. Furthermore, transitions to other bands are negligible provided that the energy levels are varied adiabatically compared to the size of the gap, which translates into an adiabatic condition |pd{dtqpΩ 2 m {κq|Àκ 2 . Then, conservation of the Bloch wavevector implies that the bandwidth of the pulse is dynamically compressed, and the reduction in slope of the polariton band (figure 2) causes the pulse to propagate at an instantaneous group velocity v g ptq without any distortion. In the limit that Ω m Ñ0, the polaritonic band becomes flat and completely mechanical in character, indicating that the pulse has been reversibly and coherently mapped onto stationary mechanical excitations within the array. We note that since Ω m is itself set by the tuning cavities, its rate of change cannot exceed the optical linewidth and thus the adiabaticity condition is always satisfied in the weak-driving regime.
The maximum storage time is set by the mechanical decay rate, 1{γ m . For realistic system parameters ω m {2π 10 GHz and Q m 10 5 , this yields a storage time of 10 µs.
In CROW structures, light is stored as circulating fields in optical nano-cavities, where state of the art quality factors of Q10 6 limit the storage time to 1 ns. The key feature of our system is that we effectively "down-convert" the high-frequency optical fields to low-frequency mechanical excitations, which naturally decay over much longer time scales. While storage times of 10 ms are possible using atomic media [29], their bandwidths so far have been limited to 1 MHz [28]. In our system, bandwidths of 1 GHz are possible for realistic circulating powers in the tuning cavities.

Imperfections in storage
The major source of error in our device will be mechanical noise, which through the optomechanical coupling can be mapped into noise power in the optical waveguide output. In our system, mechanical noise emerges via thermal fluctuations and Stokes scattering (corresponding to the counter-rotating terms in the optomechanical interaction that we omitted from Equation (2)). To analyze these effects, it suffices to consider the case of static Ω m , and given the linearity of the system, no waveguide input (such that the output will be purely noise). For a single array element, the optomechanical driving Ω m results in optical cooling of the mechanical motion [14,15], with the mechanical energy E m evolving as (see Appendix) The first term on the right describes equilibration with the thermal surroundings, wheren th pe ωm{k B T b ¡ 1q ¡1 is the Bose occupation number at the mechanical frequency and T b is the bath temperature. The second (third) term corresponds to cooling (heating) through anti-Stokes (Stokes) scattering, with a rate proportional to Γ opt 4Ω 2 m {κ. The Stokes process is suppressed relative to the anti-Stokes in the limit of good sideband resolution κ{ω m 31. For an array of N elements, a simple upper bound for the output noise power at one end of the waveguide is given by where E ss is the steady-state solution of Equation (9). The factor of 1{2 accounts for the optical noise exiting equally from both output directions, Γ opt E ss is the optically-induced mechanical energy dissipation rate, and κ ex {κ describes the waveguide coupling efficiency. The term ω 1 {ω m represents the transduction of mechanical to optical energy and is essentially the price that one pays for down-converting optical excitations to mechanical to yield longer storage times -in turn, any mechanical noise gets "up-converted" to optical energy (whereas the probability of having a thermal optical photon is negligible). In the relevant regime where Γ opt 4γ m , This noise analysis is valid only in the weak-driving regime (Ω m Àκ) [14,15]. The strong driving regime, where the mechanical motion acquires non-thermal character [32] and can become entangled with the optical fields [33], will be treated in future work.
At room temperature,n th k B T b { ω m is large and thermal noise will dominate, yielding a noise power of 0.4 nW per element for previously given system parameters and κ ex {κ1. This is independent of Γ opt provided that Γ opt 4γ m , which reflects the fact that all of the thermal heating is removed through the optical channel.For high temperatures, the thermal noise scales inversely with ω m , and the use of high-frequency mechanical oscillators ensures that the noise remains easily tolerable even at room temperature.
Thermal noise in the high-frequency oscillator can essentially be eliminated in cryogenic environments, which then enables faithful storage of single photons. Intuitively, a single-photon pulse can be stored for a period only as long as the mechanical decay time γ ¡1 m , and as long as a noise-induced mechanical excitation is unlikely to be generated over a region covering the pulse length and over the transit time τ delay . The latter condition is equivalent to the statement that the power P ph ω 1 ∆ω in the single-photon pulse exceeds P noise . While we have focused on the static regime thus far, when thermal heating is negligible, realizing P ph {P noise Á1 in the static case in fact ensures that the inequality holds even when Γ opt ptq is time-varying. Physically, the rate of Stokes scattering scales linearly with Γ opt while the group velocity scales inversely, and thus the probability of a noise excitation being added on top of the single-photon pulse is fixed over a given transit length.
In a realistic setting, the optomechanical driving amplitude Ω m itself will be coupled to the bath temperature, as absorption of the pump photons in the tuning cavities leads to material heating. To understand the limitations as a quantum memory, we have numerically optimized the static bandwidth-delay product ∆ωτ delay for a train of single-photon pulses, subject to the constraints ∆ω min 2 and γ m τ delay 1. As a realistic model for the bath temperature, we take where T 0 is the base temperature and χ2 µK is a temperature coefficient that describes heating due to pump absorption (see Appendix). Using T 0 100 mK and Q m 10 5 , we find p∆ωτ delay q max 110, which is achieved for parameter values N 275, κ ex {2π1.1 GHz, and Ω m {2π130 MHz. and (c), respectively, for a single L2 defect cavity. (d) Bandstructure of the lineardefect waveguide (grey) and the zone folded superlattice of the entire coupled-resonator system (red). The cavity mode (green) crosses the superlattice band at mid-zone, and the waveguide-cavity interaction is shown in more detail in the insets (e) and (a) for the even (σ y 1) and odd (σ y ¡1) supermodes, respectively.

Optomechanical crystal design
A schematic showing a few periods of our proposed 2D OMC slow-light structure is given in figure 3. The structure is built around a "snowflake" crystal pattern of etched holes into a Silicon slab [20]. This pattern, when implemented with a physical lattice constant of a 400 nm, snowflake radius r 168 nm, and snowflake width w 60 nm (see In order to form the double-cavity system described in the slow-light scheme above, a pair of L2 cavities are placed in the near-field of each other as shown in the dashed box region of figure 3(a). Modes of the two degenerate L2 cavities mix, forming supermodes of the double-cavity system which are split in frequency. The frequency splitting between modes can be tuned via the number of snowflake periods between the cavities. As described in more detail in the Appendix, it is the optomechanical cross-coupling of the odd (E ¡ ) and even (E ) optical supermodes mediated by the motion of the odd parity mechanical supermode (Q ¡ ) of the double-cavity that drives the slow-light behaviour of the system. Since Q ¡ is a displacement field that is antisymmetric about the two cavities, there is no optomechanical self-coupling between the optical supermodes and this mechanical mode. On the other hand, the cross-coupling between the two different parity optical supermodes is large and given by h g{ c 2 2πp346 kHzq. By lettinĝ a 1 ,â 2 andb be the annihilation operators for the modes E ¡ , E and Q ¡ , we obtain the system Hamiltonian of equation (1). The different spatial symmetries of the optical cavity supermodes also allow them to be addressed independently. To achieve this we create a pair of linear defects in the snowflake lattice as shown in figure 3(a), each acting as a single-mode optical waveguide at the desired frequency of roughly 200 THz (see figure 3(d). Sending light down both waveguides, with the individual waveguide modes either in or out of phase with each other, will then excite the even or odd supermode of the double cavity, respectively. The waveguide width and proximity to the L2 cavities can be used to tune the cavity loading (see Appendix), which for the structure in figure 3(a) results in the desired κ ex {2π 2.4 GHz. It should be noted that these line-defect waveguides do not guide phonons at the frequency of Q ¡ , and thus no additional phonon leakage is induced in the localized mechanical resonance.
The full slow-light waveguide system consists of a periodic array of the doublecavity, double-waveguide structure. The numerically computed band diagram, for spacing d 15a periods of the snowflake lattice between cavity elements (the superlattice period), is shown in figure 3(d). This choice of superlattice period results in the folded superlattice band intersecting the E ¡ (â 1 ) cavity frequency ω 1 at roughly mid-zone, corresponding to the desired inter-cavity phase shift of kd π{2. A zoom-in of the bandstructure near the optical cavity resonances is shown in figures 3(e) and (f). In figure 3(e) the even parity supermode bandstructure is plotted (i.e., assuming the even supermode of the double-waveguide is excited), whereas in figure 3(f) it is the odd parity supermode bandstructure.
A subtlety in the optical pumping of the periodically arrayed waveguide system is that for the E (â 2 ) optical cavity resonance at ω 2 , there exists a transmission bandgap. In order to populate cavityâ 2 , then, and to create the polaritonic band  at ω 1 , the pump beam must be slightly off-resonant from ω 2 , but still at ω 1 ¡ ω m . We achieve this by choosing a double-cavity separation (14 periods) resulting in a cavity mode splitting (pω 1 ¡ ω 2 q{2π 9.7 GHz) slightly smaller than the mechanical frequency (ω m {2π 11.2 GHz), as shown in figures 3(e) and (f). By changing the detuning between ω 1 ¡ ω 2 and ω m , a trade-off can be made between the attenuation of the pump beam per unit cell, α expp¡ImtKudq, and total required input power, shown in figure 4. In Appendix D.3 we show that the total attenuation per unit cell, is given by α κ ex κ in {4δ 2 k . Interestingly, by using higher input powers such that α Ñ 0, it is in principle possible to eliminate completely effects due to absorption in the array, which may lead to inhomogeneous pump photon occupations.

Outlook
The possibility of using optomechanical systems to facilitate major tasks in classical optical networks has been suggested in several recent proposals [35,25,20]. This present work not only extends these prospects, but proposes a fundamentally new direction where optomechanical systems can be used to control and manipulate light at a quantum mechanical level. Such efforts would closely mirror the many proposals to perform similar tasks using EIT and atomic ensembles [4]. At the same time, the optomechanical array has a number of novel features compared to atoms, in that each element can be deterministically positioned, addressed, and manipulated, and a single element is already optically dense. Furthermore, the ability to freely convert between phonons and photons enables new possibilities for manipulating light through the manipulation of sound. Taken together, this raises the possibility that mechanical systems can provide a novel, highly configurable on-chip platform for realizing quantum optics and "atomic" physics.

Appendix A. Equations of motion for optomechanical crystal array
Here we derive the equations of motion for an array of optomechanical systems coupled to a two-way waveguide. Because each element in the array couples independently to the waveguide, it suffices here to only consider a single element, from which the result for an arbitrary number of elements is easily generalized. We model the interaction between the active cavity mode 1 and the waveguide with the following Hamiltonian, Hereâ R,k ,â L,k are annihilation operators for left-and right-going waveguide modes of wavevector k, and ω 1 is the frequency of the cavity mode. For convenience we have defined all optical energies relative to ω 1 , and assumed that the waveguide has a linear dispersion relation ωpkq c|k|. The last term on the right describes a point-like coupling between the cavity (at position z j ) and the left-and right-going waveguide modes, with a strength g. The operatorâ R pzq physically describes the annihilation of a rightgoing photon at position z and is related to the wavevector annihilation operators bŷ R,k (with a similar definition forâ L pzq). Eq. (A.1) resembles a standard Hamiltonian used to formulate quantum cavity input-output relations [36], properly generalized to the case when the cavity accepts an input from either direction. Note that we make the approximation that the left-and right-going waves can be treated as separate quantum fields, with modes in each direction running from ¡V k V.
This allows both the left-and right-going fields to separately satisfy canonical field commutation relations, râ R pzq,â R pz I qs râ L pzq,â L pz I qs δpz ¡ z I q, while commuting with each other. Thus each field contains some unphysical modes (e.g., wavevector components k 0 for the right-going field), but the approximation remains valid as long as there is no process in the system evolution that allows for the population of such modes. From the Hamiltonian above, one finds the following Heisenberg equation of motion for the right-going field, where k 0 ω 1 {c. A similar equation holds forâ L . The coupling of the cavity mode to a continuum of waveguide modes leads to irreversible decay of the cavity at a rate κ ex . Below, we will show that κ ex is related to the parameters in the Hamiltonian by κ ex 4πg 2 {c. With this identification, one recovers Eq. (5) in the main text.
The Heisenberg equation of motion for the cavity mode is given by d dtâ To cast this equation into a more useful form, we first integrate the field equation (A.2) across the discontinuity at z j , We can defineâ L,in pz j q â L pz j q andâ R,in pz j q â R pz ¡ j q as the input fields to the cavity.
It then follows that d dtâ and thus we indeed see that the waveguide induces a cavity decay rate κ ex {2 2πg 2 {c. In the case where the cavity has an additional intrinsic decay rate κ in , a similar derivation holds to connect the intrinsic decay with some corresponding noise input fieldâ in . From these considerations, and including the opto-mechanical coupling, one arrives at Eq. (3) in the main text, Finally, we consider the equation of motion for the mechanical mode given by Eq. (4) in the main text, The zero-mean noise operatorF N must accompany the decay term in the mechanical evolution in order to preserve canonical commutation relations ofb,b X at all times. In the case where the decay causes the mechanical motion to return to thermal equilibrium with some reservoir at temperature T b , the noise operator has a two-time correlation function given by xF N ptqF X N pt I qy γ m pn 1qδpt ¡ t I q [37], wheren e ωm{k B T b ¡ 1¨¡ 1 is the Bose occupation number at the mechanical frequency ω m .

Appendix B. Transfer matrix analysis of propagation
First we derive the reflection and transmission coefficients for a single element in the case of constant opto-mechanical driving amplitude Ω m . Given the linearity of the system, it suffices to treat Eqs. (A.2), (A.7), and (A.8) as classical equations for this purpose, and furthermore to set the noise termsF N 0,â in 0. For concreteness, we will consider the case of an incident right-going cw field in the waveguide. Upon interaction with the opto-mechanical system at z j , the total right-going field can be written in the form a R pzq e ikz¡iδ k t pΘp¡z z j q tpδ k qΘpz ¡ z j qq , while the left-going field is given by a L pzq e ¡ikz¡iδ k t rpδ k qΘp¡z z j q. Here Θpzq is the unit step function, δ k ck ¡ ω 1 is the detuning of the input field from the cavity resonance, and r, t are the reflection and transmission coefficients for the system. At the same time, we look for solutions of the cavity field and mechanical mode of the form To calculate propagation through an array of N elements, it is convenient to introduce a transfer matrix formalism. Specifically, the fields immediately to the right of the opto-mechanical element (at z z j ) can be related to those immediately to the left (at z z ¡ j ) in terms of a transfer matrix M om , On the other hand, free propagation in the waveguide is characterized by the matrix The transfer matrix for an entire system can then be obtained by successively multiplying the transfer matrices for a single element and for free propagation together.
In particular, the transfer matrix for a single "block", defined as interaction with a single opto-mechanical element followed by free propagation over a distance d to the next opto-mechanical element, is given by M block M f M om , and the propagation over N blocks is simply characterized by M N M N block .
Before studying the propagation through the entire array, we first focus on the propagation past two blocks, M 2 M 2 block . Because we want our device to be highly transmitting when the optomechanical coupling is turned on, we choose the spacing d between consecutive blocks to be such that k 0 d π 2 p2n 1q, where n is an integer.
Physically, this spaces consecutive elements by an odd multiple of λ{4, where λ is the resonant wavelength, such that the reflections from consecutive elements tend to destructively interfere with each other. This can be confirmed by examining the resulting reflection coefficient for the two-block system, where M 2 pi, jq denotes matrix elements of M 2 . Note that the reflection coefficient is now suppressed as a quadratic in the detuning, whereas for a single element riκ ex δ k {2Ω 2 m is linear. In the above equation, we have made the simplifying approximation that kdk 0 d, since in realistic systems the dispersion from free propagation will be negligible compared to that arising from interaction with an opto-mechanical element.
Now we can consider transmission past N {2 pairs of two blocks (i.e., N elements in total). Because the reflection r 2 is quadratic in the detuning, its effect on the total transmission is only of order δ 4 k (because the lowest order contribution is an event where the field is reflected twice before passing through the system). Thus, up to Opδ 3 particular, the opto-mechanical coupling termsâ X 1b h.c. andâ X 1b X h.c. induce anti-Stokes and Stokes scattering, respectively. These processes yield respective opticallyinduced cooling (Γ ¡ ) and heating (Γ ) rates In the case where δ L ¡ω m , the cooling process is resonantly enhanced by the cavity, yielding a cooling rate Γ opt Γ ¡ pδ L ¡ω m q 4Ω 2 m {κ as given in the main text. Also in this case, the optical heating rate is given by Γ Γ opt κ 2 . This leads to the net cooling dynamics given by Eq. (9) in the main text, Because the optical cooling process removes phonons from the mechanical system via optical photons that leak out of the cavity, one can identify pω 1 {ω m qΓ opt E m as the amount of optical power that is being leaked by the cavity in the anti-Stokes sideband during the cooling process. Similarly, the cavity leaks an amount of power pω 1 {ω m qΓ opt κ 2 κ 2 16ω 2 m pE m ω m q in the Stokes sideband. We have ignored this contribution in Eq. (10) in the main text, because its large frequency separation (2ω m ) from the signal allows it to be filtered out, but otherwise it approximately contributes an extra factor of 2 to the last term in Eq. (10). Finally, we remark that the expression for P noise given by Eq. (10) represents an upper bound in that it does not account for the possibility that the output spectrum from a single element may exceed the transparency bandwidth, which could cause some light to be absorbed within the system after multiple reflections and not make it to the end of the waveguide. (D.2) To simplify the notation, we define operators at the boundaries of the unit cells (immediately to the left of an optomechanical element) given by c j ¡i c ca R pz ¡ j q, d j ¡i c ca L pz ¡ j q. It is also convenient to re-write the transfer matrix M om in the form M om with the parameter βpδ k q given by βpδ k q ¡iκ ex δ k ¡iκ in δ k 2pΩ 2 m ¡ δ 2 k q . Physically, this diagonalization corresponds to finding the Bloch wavevectors Kpδ k q of the periodic system. The dispersion relation for the system can be readily obtained through the equation Writing k in terms of δ k , we arrive at kd ω 1 d{c δ k d{c. As described previously, the desirable operation regime of the system is such that the phase imparted in free propagation should be ω 1 d{c p2n 1qπ{2. For concreteness, we set here ω 1 d{c π{2, satisfying this condition. For the frequencies δ k of interest, which easily satisfy the condition |δ k | 3 d{c and ignoring the intrinsic loss κ in , the simple approximate dispersion can be found. This dispersion relation yields two bandgaps, which extend from¨κ ex {2 and¨2Ω 2 m {κ, in the weakly coupled EIT regime pΩ m À κ ex q. We therefore have three branches in the band structure, with the narrow central branch having a width of 4Ω 2 m {κ ex . This branch has an optically tunable width and yields the slow-light propagation.
The dispersive and lossy properties of the array can also be found by analyzing Eq. (D.6) perturbatively. Expanding Eq. (D.6) as a power series in δ k , we find which agrees with Eq. (7) in the main text.

Appendix D.2. Fractional Occupation Calculation
In our system, the Bloch functions are hybrid waves arising from the mixing of optical waveguide, optical cavity and mechanical cavity excitations. It is therefore of interest to calculate the hybrid or polaritonic properties of these waves, by studying the energy distribution of each Bloch mode. The number of photons n WG in the waveguide can be found by taking the sum of the left-and right-moving photons in a section of the device. Over one unit cell, one obtains The relation between this value and the amplitude of the hybrid Bloch wave may be found by considering the symmetry transformation used to diagonalize the unit-cell transmission matrix. Defining C j to be the amplitude of the Bloch mode of interest, one finds c j s 11 C j and d j s 21 C j , while from the properties of the symmetry matrix S, |C j | 2 |c j | 2 |d j | 2 . From here we can deduce the number of excitations in the waveguide n WG , the optical cavity n o and the mechanical cavity n m for a given Bloch wave amplitude: (D. 15) We then define the fractional occupation in the mechanical mode by n m {pn WG n 0 n m q (with analogous definitions for the other components). These relations were used to plot the fractional occupation and colored band diagrams shown in the main text.

Appendix D.3. Pump Input Power
The technicalities associated with pumping the optomechanical crystal array system are subtly distinct from those in its atomic system analogue. Due to the periodic nature of the structure and its strongly coupled property (κ ex 4 κ in ), a bandgap in the waveguide will arise at the frequency of the "tuning" or pump cavities. This prevents these cavities from being resonantly pumped via light propagating in the waveguide. This problem may be circumvented by making the pump frequency off-resonant from ω 2 (but still at ω 1 ¡ ω m ), and by for example changing the splitting ω 1 ¡ ω 2 to be less than the mechanical frequency. Interestingly, the periodic nature of the system also allows one to suppress attenuation of the pump beam through the waveguide (which would cause inhomogeneous driving of different elements along the array). Here we calculate the waveguide input powers required to drive the pump cavities. We then find the effect of the cavity dissipation rate κ in on the beam intensity, to provide estimates for the power drop-off as a function of distance propagated in the system. Finally we note that there is a trade-off between required pump intensity and pump power drop-off. In other words, by tuning the pump beam to a frequency closer to the pump cavity resonance, the required input power is reduced, but the attenuation per unit cell expp¡ImtKudq is increased.
To calculate the waveguide input power, we start by considering the net photon flux in the right moving direction, where The required input power P in ω o Φ R for the system parameters studied in the main text is shown in figure 4.
To find the attenuation per unit cell expp¡αq, with α ImtKpδ k qud, of this pump beam, we use a perturbative approach similar to that used to find the polaritonic band dispersion. By expanding the Bloch-vector as Kpδ k q k p0q k p¡1q {δ k k ¡2 {δ 2 k . . . and using equation (D.6), we find k p¡1q κ ex {2d, and k p¡2q iκ ex κ in {4d implying that α κ ex κ in {4δ 2 k . This approximate expression is shown along with the exact calculated values for the attenuation in figure 4.

Appendix E. Implementation in an Optomechanical Crystal
For the theoretical demonstration of our slow-light scheme, we confine ourselves to a simplified model of an optomechanical crystal where only the two-dimensional Maxwell equations for TE waves and the equation of elasticity for in-plane deformations of a thick slab are taken into account. These equations approximate fairly accurately the qualitative characteristics of in-plane optical and mechanical waves in thin slabs, and become exact as the slab thickness is increased. In this way, many of the intricacies of the design of high-Q photonic crystal cavities (which are treated elsewhere [34]) may be ignored, and the basic design principles can be demonstrated in a slightly simplified system.
The two-dimensional optomechanical crystal (2DOMC) system used here utilizes the "snowflake" design [34], which provides large simultaneous photonic and phononic bandgaps in frequency. Here we choose to use optical wavelengths in the telecom band, i.e., corresponding to a free-space wavelength of λ 1.5 µm. For this wavelength, we found that the crystal characterized by a lattice constant a 400 nm, snowflake radius r 168 nm, and width w 60 nm, shown in figure 2, should work well.
Appendix E.1. Optical and Mechanical Cavities Appendix E.1.1. Single Cavity System We begin our design by focusing on the creation of a single optomechanical cavity on the 2DOMC, with one relevant optical and mechanical mode. This cavity is formed by creating a point defect, consisting of two adjacent removed holes (a so-called "L2" cavity). We calculate the optical and mechanical spectra of this cavity using COMSOL, a commercial FEM package, and find a discrete set of confined modes. Of these, one optical and one mechanical mode were chosen, exhibiting the most promising value of the opto-mechanical coupling strength g (see below for calculation). These modes are shown in figure 2(b) and (c), and were found to have frequencies ν m 11.2 GHz and ν o 199 THz, respectively.
Appendix E.1.2. Double Cavity System From here we move to designing the nearlydegenerate double optical cavity system with large cross-coupling rates. As two separate L2 cavities are brought close to one another, their interaction causes the formation of even and odd optical and mechanical super-modes with splittings in the optical and mechanical frequencies. This splitting may be tuned by changing the spacing between the cavities. We take the even and odd optical modes of this two-cavity system as our optical resonances at ω 2 and ω 1 .
where now the cross-coupling rates can be calculated by the following expression: We denote this expression for convenience as g i,j xE i |Q|E j y.
For the modes of the L2 cavity shown in figures 2(b) and (c), the optomechanical coupling was calculated to be xE|Q|Ey{2π 489 kHz for silicon. When two cavities are brought in the vicinity of each other, super-modes form as shown in figure 2(a). We denote the symmetric ( ) and antisymmetric (¡) combinations by E¨and Q¨. These modes can be written in terms of the modes localized at cavity 1 and 2,  To simulate this coupling rate, we performed finite-element simulations using COMSOL where we placed the waveguide near our cavity, and placed absorbing boundaries at the ends of the waveguide away from the cavity. The resulting timeaveraged Poynting vector xSy t |E ¢ H ¦ |{2 is plotted in figure E2(a), showing how the power flows out of the system.

Appendix E.4. Estimates for Thermoelastic Damping
The achievable storage times of our system are determined by the lifetimes of the mechanical resonances. Since we use a phononic crystal, all clamping losses have been eliminated. However, other fundamental sources of mechanical dissipation remain, and here we provide estimates for one of these, the component due to thermoelastic damping (TED) [44,45].
Using the COMSOL finite-element solver [42], we solved the coupled thermal and mechanical equations for this system [46]. In these simulations the change in the thermal conductivity and heat capacity of silicon with temperature were taken into account. The TED-limited quality factors, Q m,TED are plotted in figure E2(d). In these simulations, we see that for the mode simulated, Q m,TED surpasses 10 6 at bath temperatures of T 0 5 K. To illustrate some representative results of these simulations, we have plotted the change in temperature field ∆T prq from the ambient temperature versus the phase of the mechanical oscillation in figure E2(d). At ω m t π{2, there are variations in temperature despite the displacement field Q being uniformly 0 at this time. This shows that at these frequencies, the temperature does not follow adiabatically the displacement.
Appendix E.5. Estimate for Optical Pump Heating As mentioned in the main text, in a realistic setting the optomechanical driving amplitude Ω m itself will be coupled to the bath temperature through absorption of optical pump photons in the tuning cavities. This optical pump heating of the structure is important in estimating the practical limits of the optomechanical system for quantum applications where thermal noise impacts system performance. As a realistic model for the bath temperature in our proposed Silicon optomechanical crystal array, we take T b T 0 χα 2 , where T 0 is the base temperature and χ is a temperature coefficient that describes the temperature rise in each cavity per stored cavity photon due to optical absorption. Our estimate of χ for a thin-film Silicon photonic crystal structure is as follows. The absorbed power for |α| 2 photons stored in a cavity is given simply by P loss ω o κ i |α| 2 , where ω o is the resonance frequency and κ i the optical (intrinsic) linewidth of the cavity. If we assume all of this power is being converted to heat, the change of temperature is ∆T P loss R th , where R th is the effective thermal resistance of the Silicon structure. There are a number of sources in the literature for R th in relevant photonic crystal geometries [47,48]. We choose here to use the value for a two-dimensional crystal system in Silicon, R th 2.7 ¢ 10 4 K/W, which yields a per photon temperature rise of χ2 µK assuming an intrinsic loss rate of κ i 4 ¢10 9 rad/s (Q i 3 ¢ 10 6 ).