Light-sound interconversion in optomechanical Dirac materials

Analyzing the scattering and conversion process between photons and phonons coupled via radiation pressure in a circular quantum dot on a honeycomb array of optomechanical cells, we demonstrate the emergence of optomechanical Dirac physics. Specifically we prove the formation of polaritonic quasi-bound states inside the dot, and angle-dependent Klein tunneling of light and emission of sound, depending on the energy of the incident photon, the photon-phonon interaction strength, and the radius of the dot. We furthermore demonstrate that forward scattering of light or sound can almost switched off by an optically tuned Fano resonance; thereby the system may act as an optomechanical translator in a future photon-phonon based circuitry.


Theoretical modelling
To formulate the scattering problem we follow the standard approach of (i) linearizing the dynamics around the steady-state solution within the rotating-wave approximation in the red-detuned (Δ = ω L − ω cav <0) moderate-driving regime 16 and (ii) adapting the single-valley Dirac-Hamiltonian within the continuum approximation, valid for sufficiently low energies and barrier potentials that are smooth on the scale of the lattice constant a but sharp on the scale of the de Broglie wavelength 29 . Furthermore, focusing on the scattering by the barrier 2 Scientific REPORTs | 7: 9811 | DOI: 10.1038/s41598-017-09942-3 exclusively, we assume Δ = −Ω, and obtain (after the appropriate rescaling H → H/ħ − Ω) the optomechanical Dirac-Hamiltomian 24 , z x as a starting point (ħ = 1). Here, = with v o/m as the velocities of the optical/mechanical modes, τ and σ are vectors of Pauli matrices, k(r) gives the wavevector (position vector) of the Dirac wave, R is the quantum-dot radius, and g parametrizes the photon-phonon coupling strength, cf. Fig. 1. The low-energy dispersion follows as where τ = ±1 denote the two-fold degenerate, non-linear polariton branches with sublattice pseudospin σ = ±1. The eigenfunctions of (1) take the form 1/2 , ε τ,σ = v o σk − E τ,σ , and the bare (optical/mechanical) eigenstates o/m of τ z . For g = 0, the bandstructure simplifies to two independent photonic and phononic Dirac cones, and the scattering problem can be solved as for a graphene quantum dot [29][30][31] .
We expand the incident photonic wave (in x direction), the transmitted wave inside the dot (ψ  Figure 1. Setup considered in this work. Left part: Optomechanical graphene. Honeycomb array of optomechanical cells driven by a laser with frequency ω L . The co-localized cavity photon (ω cav ) and phonon (Ω) modes interact (linearly) via radiation pressure tunable by the laser power 16 waves appear (with wavevectors q ± and k o/m ), which-due to the symmetry of the problem-carry an angular momentum, i.e., their wavevectors have components in any planar direction 29,30 . Lower right part: Schematic bandstructure. Without photon-phonon coupling the photon (orange) and phonon (black) Dirac cones (obtained in low-energy approximation) simply intersect. In the quantum dot region with g > 0, weakly nonlinear (photon-phonon) polariton bands (green) emerge. Here, solid (dashed) lines correspond to pseudospin σ = 1 (−1). Connecting lines between q + and q − (k o and k m ) indicate that the corresponding states are superimposed. The dashed (solid) blue line gives the energy E (position-dependent profile of g). Model parameters: The continuum approximation is justified if  k a 1/ and  R a. Moreover, we have to avoid any 'phonon lasing' instabilities, i.e., the photon transfer element 2v o /3a has to be smaller than Ω/3 24 . If so, the effects discussed in this paper should be experimentally accessible for Ω  g/ 1 . With a lattice constant a ~ 50 μm 19   For E > 0, we can take σ = +1 and distinguish the branches of the incident and reflected waves by τ = ±1. For the transmitted wave, where ε ± = v o σ ± q ± − E,  E g is possible and we denote the two polaritonic branches by + and −. Here, for E > g (E < g) σ ± = 1 (τ ± = −1), and states with different τ ± = ±1 (σ ± = ±1) are superimposed, see Fig. 1. In eqs (3)(4)(5) the eigenfunctions of the Dirac-Weyl Hamiltonian σ · k are  are the Bessel [Hankel] function of the first kind (in the following we omit the upper index (1) of the Hankel functions). The continuity conditions at r = R give the reflection r o/m,l and transmission coefficients t ±,l : In eq. (7), Here, detA is obtained from eq. (10) when substituting Y l(+1) by H l(+1) and multiplying by g. Note that the scattering coefficients are invariant under the transformation (E, g, R −1 ) → (γE, γg, γR −1 ) with  γ ∈ . Furthermore, the reflection coefficients have upper bounds: |r o,l | ≤ 1 and From the current density of the reflected waves in the far field, we obtain the scattering efficiency, that is, the scattering cross section divided by the geometric cross section, as We note that in eqs (11), (12), and hereafter, l ≥ 0. The density ρ = ψ † ψ and the current j = ψ † sψ in-and outside the quantum dot region further specify the scattering.

Numerical results and discussion
Treating the scattering by the circular quantum dot region numerically, we adopt v m = 0.1v o and employ units such that v o = 1. Moreover, for the experimental reliable parameters quoted in the caption of Fig. 1, fixing g, 100a is a natural unit for the quantum dot radius R, where the number of cells (defects) enclosed in the quantum dot region is about 10 4 R 2 . Due to the scale invariance of the scattering coefficients, in what follows all physical quantities will be discussed in dependence on E/g and Rg. Figure 2 displays the complex pattern of both the photonic Q o and phononic Q m contributions to the scattering efficiency in the E/g-Rg plane. When the photon hits the quantum dot it stimulates mechanical vibrations (phonons) because of the optomechanical interaction. Then both scattered waves are inherently correlated. For energies of incident photon larger than the optomechanical coupling, Q o (Q m ) reveals a very broad (narrow) ripple structure with maxima of high (rather low) intensity. Above Ẽ g / 2 the phonon is hardly scattered, while the photon is still heavily influenced by the dot. This is because the phonon wave numbers take large values very quickly, compared to those of the photon, simply because v m is smaller than v o by an order of magnitude. If the dispersion of the phonon is unaffected by g, the wave numbers inside and outside are almost identical and scattering disappears. The same, in principle, happens to the photon, but at much larger energies. In this limit, photon scattering resembles the scattering of ultrarelativistic Dirac particles, which are massless outside the dot and carry an effective inside the quantum dot region (here, v o plays the role of vacuum ligth speed).
The situation becomes much more involved when the energy of the incident optical wave is smaller than the optomechanical coupling, see the right panels in Fig. 2 for E/g < 1. Let us first consider the case where the size-parameter ER is very small, i.e., the wavelengths 2π/k o/m are large compared to the quantum dot radius R. In Fig. 2 Fig. 3 gives a closer look at this limiting behavior and demonstrates that in each case two resonances occur, in fact, symmetrically around a point where the phonon scattering vanishes while the photon scattering is small but finite (see inset). These resonances, numbered by  ∈ n , belong to the lowest photonic/phononic partial waves with l = 0. Expanding the phononic reflection coefficients (8) with respect to the small size-parameter ER, the phonon-scattering depletion points result as , where j l,n are the n-th zero of the Bessel function J l . We note that here the phonon resonance peaks are larger than the photonic ones. Of course, such resonances also occur for the next higher partial wave with l = 1 at = Rg j v v n o m 1, , but are not visible in Fig. 3 left on account of their tiny linewidth/intensity.
In case that the size-parameter ER ~ 1, the wavelengths 2π/k o/m are in the order of the dot radius R. In this regime, only the lowest partial waves will be excited to any appreciable extent, and the photonic [phononic] resonances appear as bright spots [splitted stripes] at specific 'points' [lines] in the E/g-Rg plane, see , where the phononic scattering is perfectly suppressed. This is illustrated by the middle panel in Fig. 3: At .  Rg 1 7 [case (i)], the l = 1 photon mode is resonant and the scattering becomes purely photonic (i.e., the contribution of all phonon modes goes to zero). The phonon resonances of the l = 1 mode appear symmetrically about this photon resonance (at these points, on the other hand, the photonic contribution is significantly weakened). A similar scenario arises for the resonance of the l = 0 modes at .  Rg 1 24 and .  Rg 2 24. Vice versa, at certain radii the scattering becomes purely phononic, see, e.g., case (ii) where Rg = 1.566. This allows one to switch from entirely photon to phonon scattering just by varying the dot radius.  If the size-parameter increases further, the situation changes again. Now even higher partial waves will be excited. In this regime, the photon scattering efficiency is always a larger than the phononic one. Approximating the resonance points by the zeros of the Bessel function is no longer possible; as a result both Q o , Q m > 0, cf. Fig. 3 right. In the extreme limit  ER 1, however, phonon scattering is negligibly small and does not have to be considered.
Having discussed the global scattering efficiency of the quantum dot, let us now analyze the spatial resolution of the wave transmisson and reflection. We start by investigating the scattering characteristics, specified by the probability density ρ = ψ † ψ and current density ψ ψ = † j j , in the near field, see Fig. 4. In the quantum dot region polaritons (mixed photon-phonon states) are formed. For very small size-parameters  ER 1 and energies E/g < 1, the polariton density inside the dot becomes  The panels right in Fig. 4 show that the incident wave is fed into vortices which trap the polariton. For l = 0, two vortices arise for the n = 1 mode. Further vortices occur on the symmetry axis when n increases. In general, the vortex pattern of the l-th mode is dominated by 2(2l + 1) vortices which give rise to 2l + 1 preferred scattering directions in the far field for n = 1 (see below) 29 . We note that a very similar vortex pattern (scattering characteristics) arises for moderate size-parameters ER ~ 1, e.g., for the cases (i) and (ii) in the middle panel of Fig. 3.
The current density of the reflected waves in the far field given by eq. (11) exhibits the already mentioned cosinusoidal angle distribution with maxima at φ = l′π/(2l + 1) where l′ ∈ {0, …, ± l}. Consequently, if the l = 0 mode is resonant, only forward scattering takes place, whereas resonaces belonging to higher modes scatter the light respectively sound into different directions. This is illustrated by Fig. 5 (upper panels), for the far-field currents j o/m of a specific quantum dot system that preferably suppresses either the phonon [case(i)] or the photon [case(ii)] scattering [cf. Fig. 3, middle]. Accordingly, when the photonic partial wave with l = 1 becomes resonant, we observe three preferred scattering directions with equal intensity (left upper panel). Though a similar distribution results for the phononic resonance, now the forward scattering is somewhat enhanced as the lower l = 0 mode substantially contributes (right upper panel). Note that both waves will never be scattered in the angle range φ π ±  due to absence of backscattering. Most interestingly, the constructive and destructive interference between a resonant l mode and the off-resonant l = 0 mode can lead to a Fano resonance 32 that for its part may cause a depletion of Klein tunneling, i.e., a suppression of forward scattering 29 . In this way, the interference between the first two photonic and phononic partial waves depicted in the lower panels of Fig. 5 give rise to Fano resonances, which are reflected in the almost vanishing currents j o/m at certain ratios E/g(φ), even for φ = 0 (see upper panels). Varying the energy of the incident wave therefore allows to control the scattering into pure photon or phonon waves, having preferred directions of propagation, with or without forward scattering.
For larger size-parameters, ER > 1, where many partial waves may become resonant [e.g., case (iii) in Fig. 3  (right)], a rather complex structure of the far-field currents evolves. The two left panels in Fig. 6 display the ratio j o /j m in the Rg-φ plane and gives a polar plot of the light/sound emission. The figure corroborates the use of the considered setup as an optomechanical switch or light-sound translator. Finally, when  ER 1 and the extent of the quantum dot is much greater than the wavelengths, the scattering shows features known from ray optics [cf.  sound propagation is depleted [cf. Fig. 6, right]. The far-field currents strongly oscillate when φ becomes finite, whereby the phonon contribution is on average much smaller than those of the photon.
To sum up, we have demonstrated Dirac physics in an optomechanical setting. Solving-within Dirac-Weyl theory-the problem of light scattering by circular barriers in artificial graphene composed of tunable optomechanical cells, we show that large quantum dots enable photon lensing, while small dots trigger the formation of polariton (photon-phonon) states which cause a spatial and temporal trapping of the incident wave in vortex-like structures, and a subsequent direction-dependent re-emittance of light and sound. In the latter case (quantum regime), the quantum dot can be used to entangle photons and phonons and convert light to sound waves and vice versa. Equally important, the forward scattering and Klein tunneling of photons could switched off for small dots by optically tuning a Fano resonance arising from the interference between resonant scattering and the background partition. In this way optomechanical cells might be utilized to transfer, store, translate and process information in (quantum) optical communications, or simply to realize a coherent interface between photons and phonons.