PHYSICAL REVIEW LETTERS 122, 043903 (2019) Anderson Photon-Phonon Colocalization in Certain Random Superlattices G. Arregui,1,2,* N. D. Lanzillotti-Kimura,3 C. M. Sotomayor-Torres,1,4 and P. D. García1,† 1 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 2 Dept. de Física, Universitat Autnònoma de Barcelona, 08193 Bellaterra, Spain 3 Centre de Nanosciences et de Nanotechnologies (C2N), CNRS, Univ. Paris-Sud, Université Paris-Saclay, 10 Boulevard Thomas Gobert, 91120 Palaiseau, France 4 Institució Catalana de Recerca i Estudis Avançats (ICREA), 08010 Barcelona, Spain (Received 1 June 2018; revised manuscript received 30 November 2018; published 1 February 2019) Fundamental observations in physics ranging from gravitational wave detection to laser cooling of a nanomechanical oscillator into its quantum ground state rely on the interaction between the optical and the mechanical degrees of freedom. A key parameter to engineer this interaction is the spatial overlap between the two fields, optimized in carefully designed resonators on a case-by-case basis. Disorder is an alternative strategy to confine light and sound at the nanoscale. However, it lacks an a priori mechanism guaranteeing a high degree of colocalization due to the inherently complex nature of the underlying interference processes. Here, we propose a way to address this challenge by using GaAs=AlAs vertical distributed Bragg reflectors with embedded geometrical disorder. Because of a remarkable coincidence in the physical parameters governing light and motion propagation in these two materials, the equations for both longitudinal acoustic waves and normal-incidence light become practically equivalent for excitations of the same wavelength. This guarantees spatial overlap between the electromagnetic and displacement fields of specific photon-phonon pairs, leading to strong light-matter interaction. In particular, a statistical enhancement in the vacuum optomechanical coupling rate, go , is found, making this system a promising candidate to explore Anderson localization of high frequency (∼20 GHz) phonons enabled by cavity optomechanics. The colocalization effect shown here unlocks the access to unexplored localization phenomena and the engineering of light-matter interactions mediated by Anderson-localized states. DOI: 10.1103/PhysRevLett.122.043903 The engineering of wave scattering across all the approach first proposed for microwave photons [11]. In frequency spectrum is at the heart of fundamental research optics, the far-field emission of excellent point sources— in scientific disciplines dealing with objects spanning i.e., quantum emitters—allows direct access to the statistics orders of magnitude in size. Scattering in the presence of Anderson localization and to spatially resolve the of disorder is pervasive in nature, with examples ranging localized field [12]. For elastic waves, experiments have from the diffusion of light as it passes through dust to the been so far restricted to ultrasound [13,14] while the direct dynamics of sea waves in the presence of obstacles. A observation of localization at higher frequencies with major breakthrough in wave scattering arises from the explicit inspection of the individual localized wave fields seemingly more inaccessible quantum-mechanical behav- simultaneously in the spatial and spectral domain remains ior of electrons in certain random atomic lattices [1]; elusive. The lack of practical phonon transitions in the solid diffusion halts when the scale of coherent multiple scatter- state and their limited far-field radiation complicate the ing is reduced to the wavelength itself, this being ascribed read-out and prevent the analogy to optics. to the (Anderson) localization of the eigenmodes’ wave The interaction between electromagnetic radiation and functions. This transition was found to be inherent to any mechanical motion in disordered periodic-on-average sil- system governed by a wave equation in a disordered icon nanobeams has been studied recently [15]. Since potential [2], prompting intense research activity [3–6]. disorder localizes both fields, the confined optical modes Not suffering from mutual interactions as electrons do, can interact with the localized mechanical motion as in electromagnetic radiation has concentrated much of the standard engineered cavity optomechanical systems [16]. effort. Experimental attempts to observe Anderson locali- These systems have proven to be suited to test fundamental zation of classical waves have often relied on the analysis of quantum physics with macroscopic objects [17,18] and are transport parameters [7–9], suffering from artifacts like promising candidates as transducers in quantum systems absorption [10]. Alternatively, one can look at the relative [19] or ultrahigh sensitivity mass and force sensors [20], size of the fluctuations in different transport parameters, an among many other applications [21–24]. When induced by 0031-9007=19=122(4)=043903(6) 043903-1 © 2019 American Physical Society PHYSICAL REVIEW LETTERS 122, 043903 (2019) disorder, they could be used as a fundamental local probe to (a) 0 0 observe Anderson localization of high frequency mechani- Transmission (dB) −20 −20 cal vibrations. Unfortunately, the electromagnetic and dis- −40 −40 placement fields interfere independently within a typical = 0·D = 0.05·D perturbed structure and hardly colocalize, thus hindering −60 = 0.1·D −60 the maximum achievable values of the vacuum optome- −80 = 0.15·D −80 chanical coupling rate go for such systems in the Anderson- 18.1 18.35 18.6 19.9 20.15 20.4 localization regime [15], generally making the read-out of Mechanical frequency (GHz) motion via the optics unlikely. We circumvent this issue by (b) 250 (c) using a certain type of epitaxially grown optical and acoustic = 0.05·D superlattices based on GaAs and AlAs. Engineered defects = 0.1·D Localization length /D 200 = 0.15·D in such a platform [25,26] or their three-dimensional version in the form of micropillars [27,28] are attracting increasing 150 interest for (very) high frequency cavity optomechanics. In particular, the GaAs=AlAs Fabry-Pérot resonators proposed 100 in Ref. [25] are designed to optimally confine photons and to 50 simultaneously behave as resonators for acoustic phonons with the same wavelength, quality factor, and field profile. 0 Here, we explore this idea in random structures and show 18.1 18.15 18.2 18.25 =0 = 0.15·D in our calculations that any multilayer composed by an Frequency (GHz) arbitrary combination of these two materials exhibits almost FIG. 1. Phonon localization in disordered GaAs=AlAs super- the same mechanical and optical interference pattern even lattices. (a) Transmission spectrum of perturbed GaAs=AlAs when induced by geometrical disorder. Subsequently, we superlattices (d1 ¼ 61.88 nm, d2 ¼ 73.48 nm, N ¼ 600), with propose GaAs=AlAs disordered multilayers as a cavity- Gaussian disorder N ðd1 ; σ 2 Þ in the position of the interface inside optomechanical platform to enable observation of Anderson the unit cell. (b) Localization length ξ normalized by the unit cell localization of mechanical excitations at frequencies so far thickness D for several standard deviations σ. (c) Schematic of a unexplored, as well as a test bed to explore field-localization perfect periodic superlattice (σ ¼ 0) and a perturbed one phenomena with coupled excitations. (σ ¼ 0.15D) on a thick GaAs substrate. To study the optical and acoustic properties of these superlattices, we use a transfer matrix formalism [29] (see Supplemental Material [30]). In this system, the solution to n1 =n2 ¼ 1.193–1.186 ¼ v2 =v1 , which approximately sati- the continuum-mechanics equations in the absence of sfies the ideal conditions (1). Despite this slight departure sources for longitudinal motion and to the Maxwell’s from the ideal case, any arbitrary GaAs=AlAs superlattice equations at normal incidence can be solved with exactly exhibiting a mechanical eigenmode with a field profile uðzÞ the same formalism [34], provided that the acoustic and angular frequency Ωm will also support an optical impedances Zi are replaced by refractive indexes ni and eigenmode with angular frequency ωo ∼ KΩm and field the longitudinal acoustic velocities vi by the speed of light, profile EðzÞ ∼ uðzÞ as shown in this Letter. To illustrate c=ni , in the different i layers. For a succession of two this, in our calculations we use a distributed Bragg reflector different materials (1) and (2), for which the ratios n1 =n2 ¼ (DBR) structure with a unit cell formed by a GaAs layer of Z1 =Z2 and n1 =n2 ¼ v2 =v1 then thickness d1 ¼ 61.88 nm and an AlAs layer of thickness d2 ¼ 73.48 (technically known as a λ=4, λ=4 DBR). This Mac ðΩm Þ ¼ Mop ðKΩm Þ; ð1Þ structure simultaneously opens a maximum-width first- order band gap with a center frequency of 344.6 THz where Mac ðΩm Þ and Mop ðωo Þ are the transfer matrices (λ ¼ 870 nm) for near infrared photons and 19 GHz for connecting the acoustic and optical coefficients in the microwave phonons. We introduce geometrical disorder in propagating-counterpropagating plane-wave basis between the position of the interface between the two materials. Its the incidence layer and the substrate and K ¼ ðc=n1 =v1 Þ natural position at z ¼ d1 is Gaussian distributed N ðd1 ; σ 2 Þ (see the Supplemental material [30]). Here, Ωm (ωo ) is used with varying standard deviation σ, while the period D ¼ for the mechanical (optical) angular frequency d1 þ d2 ¼ 135.36 nm is kept constant. Figure 1 shows the of a time-harmonic solution. This ideal condition implies acoustic transmission spectrum of a disordered superlattice the exact equivalence of the optical and acoustic with N ¼ 600 periods for different disorder levels, σ, problems in the system. Using standard values [35] for around the first-order odd band gap. the optical and mechanical coefficients for photons in the Noninteracting electronic Bloch modes in atomic crys- near-infrared and microwave phonons in GaAs(1) and tals undergo random multiple scattering in the presence of AlAs(2), we have n1 =n2 ¼ 1.193–1.199 ¼ Z1 =Z2 and disorder, eventually leading to the (Anderson) localization 043903-2 PHYSICAL REVIEW LETTERS 122, 043903 (2019) (a) m /2 (GHz) (b) (d) c (e) = 0.01· m /2 =18.38 GHz 1,79·104 vGaAs p p 18.2 18.3 18.4 18.5 18.6 330 10 nGaAs u(z) 2 /2 (THz) 328 4 0 1,82·10 p = 0.005· p 326 o PDF ( ) −10 30 35 40 45 log (T) 324 p = 0.002· p (c) /2 =327.68 THz −20 o 18.2 18.3 18.4 18.5 m/2 (GHz) 2 E(z) (f) −30 Si/Ge GaAs/AlAs = 0.0· PDF ( ) p p Optics Acoustics −40 324 326 328 330 332 30 35 40 45 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 o /2 (THz) Position (µm) Co-localization parameter FIG. 2. Photon-phonon colocalization. (a) Optical and acoustic transmission spectra of a disordered GaAs=AlAs superlattice with d1 ¼ 61.88 nm, d2 ¼ 73.48 nm, N ¼ 600, and σ ¼ 0.15D at the low-frequency optical and mechanical band-gap edge, respectively. The energy density of the displacement field uðzÞ and the electric field EðzÞ of the highlighted pair in (d) are plotted in (b) and (c), showing perfect spatial colocalization. The resonant frequencies of all the mechanical and optical eigenmodes are mapped onto each other and plotted in (d). (e) Histogram of Δ for 2 × 103 disordered GaAs=AlAs superlattices with their physical parameters fρ; v; ng normally distributed with a standard deviation σ p . (f) Histogram of the colocalization parameter Δ for a set of 2 × 103 disordered Si=Ge superlattices. of the wave function [1]. The electromagnetic and dis- C ¼ 1.82 × 104 − 1.79 × 104 ¼ K. To calculate the optical placement fields in artificial photonic and phononic crystals fωn ; En ðzÞg and the mechanical fΩm ; um ðzÞg eigenmodes, are also sensitive to disorder, especially at the band-gap we apply the transfer matrix method with outgoing boun- edges where the group velocity falls (ideally) to 0 [6], dary conditions and solve for the complex eigenfrequencies equally giving rise to disorder-induced or Anderson ω ¼ ωr − jωi of the quasinormal modes [39] (see details localization. In a one-dimensional structure such as a in Supplemental Material [30]). For the highlighted photon- GaAs=AlAs superlattice, the presence of Anderson-local- phonon pair of the structure simulated in Fig. 2(d), ized modes can be detected in the reflection or transmission Figs. 2(b) and 2(c) show how the displacement and electric spectrum through Lorentzian-shaped resonances centered field intensities respectively are also perfectly overlapping near the corresponding eigenfrequencies with a free spec- in real space. In order to quantify the degree of spatial tral range δω larger than the mean linewidth Δω [36]. colocalization we define the parameter Figure 1(a) shows these narrow mechanical resonances in R the transmission spectra visibly satisfying such criteria and jum ðzÞjjEn ðzÞjdz populating a spectral band that broadens with increasing Δnm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R R ; ð2Þ disorder level [37]. The ensemble-averaged decay of such jum ðzÞj2 dz jEn ðzÞj2 dz localized modes occurs with a localization length, ξ. It determines the minimum length of a finite sample for which where Δ ∈ [0, 1] with Δ ¼ 1 being fully colocalized. Here it is possible to statistically observe these modes as L ≥ ξ um ðzÞ is the mechanical displacement and En ðzÞ the electric [38], where L ¼ ND is the total length of the structure. field properly normalized by the amplitude of the field in Figure 1(b) shows ξ as a function of disorder level and the substrate, and the amplitude of the incident plane wave frequency, using the scaling of the ensemble-averaged from the air, respectively. We calculate Δ between all the logarithmic transmission hlogðTÞi ∝ −ðL=ξÞ and evidences photonic and phononic modes in a narrow frequency range that the system is deep in the localization regime (ξ ≪ L) around the lower band edge from a set of 2 × 103 near the band edges for the range of disorders and lengths GaAs=AlAs superlattices with the same structural param- (N ¼ 600) considered later in this Letter. eters as detailed above and the disorder level fixed at When condition (1) is fulfilled, the optical spectrum is σ ¼ 0.15D. The histogram of Δ, plotted in the lowest panel almost identical to the mechanical one when plotted with in Fig. 2(e), is strongly peaked for Δ > 0.95 for which the scaled frequency, as shown in Fig. 2(a). When the modes are perfectly colocalized and presents rather low mechanical and optical resonant frequencies are mapped values for the Δ < 0.95. To verify the robustness of this onto each other in spectral order, we recover [Fig. 2(d)] significant colocalization effect, we calculate the distribu- the predicted spectral behavior with ωo ≈ CΩm and tion of Δ while adding additional Gaussian disorder to the 043903-3 PHYSICAL REVIEW LETTERS 122, 043903 (2019) (a) material parameters fρ; v; ng of both GaAs and AlAs, with standard deviations σ p ¼ ½0.002; 0.005; 0.01 · μp , μp being the mean value for the given physical parameter. As plotted in the upper panel in Fig. 2(e), a disorder level larger than σ p ¼ 0.01μp is required to fully wipe out the effect, well above the layer-by-layer statistical fluctuations achievable in actual molecular-beam epitaxy-grown sam- ples [40] (see details in Supplemental Material [30]). For comparison, we calculate the distribution of Δ for a Si=Ge (b) disordered superlattice. The Si and Ge thicknesses (d1 ¼ 115.97 nm, d2 ¼ 93.88 nm, N ¼ 600) are chosen to tune the band edge of interest at approximately half the frequency of the GaAs=AlAs superlattice to avoid the effect of absorption. The disorder level is also shifted to σ ¼ 0.11D to satisfy the condition ξSi=Ge ∼ ξGaAs=AlAs . In this case, the distribution Δ [Fig. 2(f)] shows no colocalization at all. These well-confined spatially overlapping modes can interact with each other and are therefore candidates to explore their coupling. As expected from confined mechanical and light modes, the deformation profile associated with a normal mode um ðzÞ will locally change the optical properties of the structure. Thus, the electro- magnetic eigenmodes En ðzÞ will be affected, giving rise to FIG. 3. Vacuum optomechanical coupling rate go =2π in dis- both an optical frequency shift (dispersive interaction) and ordered multilayered systems. (a) Scatter plot of the mechanical a quality factor change (dissipative interaction). In and optical frequencies of the resonant modes found in a set of GaAs=AlAs superlattices and in the frequency range of 2 × 103 disordered GaAs=AlAs multilayers (σ ¼ 0.15D); the interest, mainly two acousto-optic interaction mechanisms color represents the coupling rate go =2π between the considered need to be considered [41,42]. First, the displacement of the pair in logarithmic scale. (b) The upper panel plots the histogram N þ 1 boundaries, or moving boundary effect, will change of the coupling rate go =2π considering poorly colocalized the interference pattern of multiple light paths. Second, the photon-phonon pairs (Δ < 0.95, blue bins) and only perfectly photoelastic effect will induce a change in the bulk colocalized pairs (Δ > 0.95, red bins). The bottom panel plots the histogram of the coupling rate gmod =2π defined in an inset. The permittivity tensor ϵ that can be written as the tensor corresponding dashed lines give the value of go =2π and gmod =2π product dðϵ−1 Þij ¼ Pijkl Ukl, where U is the second-order expected for a perfect Fabry-Pérot cavity based on two unper- strain tensor and P the fourth-order photoelastic tensor turbed DBR mirrors of 300 cells each. [43]. To account for the frequency shift induced by these two mechanisms, first-order perturbation theory applied to the area excited by a focused laser beam, in order to Maxwell’s equations [44] in the multilayered system leads calculate the effective mass meff of the mechanical reso- to the following expressions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nator and the zero point fluctuations xzpf ¼ ℏ=2meff Ωm . PNþ1 The product of the coupling integral by xzpf quantifies the ω um ðzi Þðϵi−1 − ϵi ÞjEn ðzi Þj2 gmb ¼− o i¼1 RL 2 xzpf ; ð3aÞ interaction between a single phonon and a single photon, 2 0 ϵðzÞjEn ðzÞj dz and is required since otherwise the parameter would depend R0 on the normalization chosen for um ðzÞ. ω P12 ðzÞ ∂um ðzÞϵðzÞ2 jEn ðzÞj2 We evaluate go between the optical and mechanical gpe ¼− o L R L ∂z 2 xzpf : ð3bÞ 2 0 ϵðzÞjEn ðzÞj dz modes of the GaAs=AlAs disordered superlattices simu- lated for Fig. 2(e). Because of the strongly dispersive nature The strength of the coupling parameter go ¼ gmb þ gpe of the localization length ξ [Fig. 1(b)], we consider only a between two (n-optical, m-mechanical) eigenmodes narrow frequency window, thus minimizing the effect of strongly depends on the overlap of the displacement varying the effective volume of the eigenmodes. The um ðzÞ or strain ½∂um =∂zðzÞ field with the electric field coupling values in the densely packed diagonal of intensity jEn ðzÞj2, which highlights the importance of Fig. 3(a), i.e., for perfectly colocalized photon-phonon colocalizing the displacement and the electric field profiles. pairs (Δ > 0.95), exhibit values higher than the poorly Since our model is by definition one dimensional, we have colocalized ones (Δ < 0.95). The probability density chosen an arbitrary size in the x-y plane of 2 × 2 μm2 , for function of the vacuum optomechanical coupling rate go 043903-4 PHYSICAL REVIEW LETTERS 122, 043903 (2019) is plotted in Fig. 3(b) either considering poorly colocalized This work was supported by the Spanish Ministerio modes (blue bars), or those perfectly colocalized (red bars), de Ciencia, Innovacion y Universidades (MICINN) via the pointing out a statistical evidence for this material Severo Ochoa Program (Grant No. SEV-2017-0706) and combination. The coupling mechanism considered in the project PHENTOM (Grant No. Fis 2015-70862-P) as (3b) is based on the product of the oscillating function well as by the Centres de Recerca de Catalunya (CERCA), ð∂um =∂zÞðzÞ by the always positive ϵðzÞ2 jEn ðzÞj2 . When and by the European Commission in the form of the H2020 two fields um ðzÞ and En ðzÞ are perfectly colocalized, the FET open project PHENOMEN (Grant No. 713450). G. A. function ð∂um =∂zÞðzÞ, π=2-shifted version of um ðzÞ, is null is supported by a Barcelona Institute of Science and whenever jEn ðzÞj2 attains a maximum, and vice versa for Technology PhD fellowship, N. D. L. K. by the ERC the maxima or minima of ð∂um =∂zÞðzÞ. Despite the perfect Starting Grant Nanophennec (Grant No. 715939) and coincidence of the fields, the integrand values stay low, P. D. G. by a Ramon y Cajal fellowship (Grant since ϵðzÞ2 and p12 ðzÞ have their own particular spatial No. RyC-2015-18124). dependence. 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