Strong coupling of phononic cavity modes in one-dimensional corrugated nanobeam structures

The interaction of phononic cavities modes in one-dimensional nanobeam-based structures is investigated theoretically. The overlapping of two closely spaced phononic cavity modes leads to the formation of symmetric and antisymmetric modes, which can be used for the modulation of the frequency and the quality factors of the propagating cavity modes. In addition, the efficient coupling of cavity modes with the modes of a phononic waveguide connecting the two phononic cavities has been demonstrated. We show that such coupling can be used for routing, controlling, and modulating the phononic cavity modes over large distances between cavities. This demonstration paves the way to engineering complex on-chip phonon networks utilizing mechanical waves to connect quantum systems as phononic or optomechanical cavities.

Figure 1

(a) Representation of one unit cell of the corrugated nanobeam where $a=500$ nm is the lattice parameter, $e=0.44a$ is the thickness of the nanobeam, $r=0.3a$ is the radius of the hole, and $d=0.5a$ and $l=a$ are the width and the length of one rectangular stub, respectively. (b) Schematic representation in the $x-y$ plane of the 1D periodic nanobeam with a single defect localized in the middle, defined by the geometrical parameters $\delta$ and $\eta$. (c) Transmittance in a logarithmic scale at the fixed frequency of 2.053 GHz as a function of the period mismatch, $\delta$, and for three different values of $\eta$.

Figure 2

(a) Schematic presentation of a 1D periodic nanobeam with a single defect and the corresponding profile of transverse sound velocity for the equivalent multilayer structure (the numbers correspond to the different layers). (b) Transmittance of the propagating mode for $d_c=2.4\mu \mathrm{m}$ and for $v_{\text{eff}}^t=6488$ m/s ($v_1^t<v_{\text{eff}}^t<v_2^t$). (c) Quality factor and position of the resonant frequency as a function of the barrier thickness $d_b$.

Figure 3

Resonant frequency and $Q$-factor of a single cavity as a function of the thickness of the middle (defective) layer. Inset: spatial distribution of longitudinal displacement ($u_x$) for $d_c=2.4\mu \mathrm{m}$ (left) and $d_c=7\mu \mathrm{m}$ (right).

Figure 4

(a) Transmittance of the propagating longitudinal mode through the nanobeam with the defect represented in Fig. 1 using the FE method. The dimensions of the defect are $\delta =0.4a$, and $\eta =0.5a$, with $a=500$ nm. The transmission is presented according to the scheme “-${\mathrm{o}}^N\mathrm{x}{\mathrm{o}}^N$-” for different values of $N$. (b) Quality factor and resonant frequency as a function of $N$. (c,d) Spatial distribution of the longitudinal and transverse displacements at the resonant frequency 2.053 GHz (left) in the “- ${\mathrm{o}}^4\mathrm{x}{\mathrm{o}}^4$-” structure for $\delta =0.4a$ (c) and $\delta =3.95a$ (d).

Figure 5

Schematic presentation of the phononic nanobeam including two defects with the schematic representation (a) “- ${\mathrm{o}}^4\mathrm{x}{\mathrm{o}}^N\mathrm{x}{\mathrm{o}}^4$-” and (b) “-${\mathrm{o}}^4\mathrm{x}{\mathrm{o}}^4-{\mathrm{o}}^4\mathrm{x}{\mathrm{o}}^4$-”.

Figure 6

(a) Transmittance of the propagating mode with an effective transverse sound velocity of 6488 m/s. The inset presents the transverse sound velocity profile when the two cavities are directly connected. (b) Evolution of the resonance frequency and the quality factor as a function of the distance $d$ between the cavities. (c) Same as (a) when the two cavities are connected through a phononic waveguide. (d) Evolution of the resonance frequency and the quality factor as a function of the distance $D$ of the connected phononic waveguide . The dashed lines correspond to the resonance position from the analytical expression for three-layers system with transverse sound velocities (in m/s) “6986/5845/6986” corresponding to mass densities (in ${\mathrm{kg}/\mathrm{m}}^3$) “2330/1631/2330” [Eq. (13)].

Figure 7

Evolution of the resonant frequency and the quality factor as a function of the distance between two cavities for the 3D nanobeam structure when the two cavities are (a) directly connected and (b) connected through a straight phononic waveguide.

Figure 8

(a, b) Spatial distribution of the longitudinal $\left(u_x\right)$ and transverse $\left(u_y\right)$ displacement components at the frequencies $f_A=2.042$ GHz and $f_B=2.062$ GHz, when the two cavity modes are directly coupled. (c, d, e) Spatial distribution of the longitudinal $\left(u_x\right)$ and transverse $\left(u_y\right)$ displacement components at the frequencies $f_C=2.049$ GHz, $f_D=2.058$ GHz, and $f_E=2.052$ GHz when the two cavity modes are coupled through the straight phononic waveguide of length $D=1.04\mu \mathrm{m}$.