Floquet scattering of light and sound in Dirac optomechanics

The inelastic scattering and conversion process between photons and phonons by laser-driven quantum dots is analyzed for a honeycomb array of optomechanical cells. Using Floquet theory for an effective two-level system, we solve the related time-dependent scattering problem, beyond the standard rotating-wave approximation approach, for a plane Dirac-photon wave hitting a cylindrical oscillating barrier that couples the radiation field to the vibrational degrees of freedom. We demonstrate different scattering regimes and discuss the formation of polaritonic quasiparticles. We show that sideband-scattering becomes important when the energies of the sidebands are located in the vicinity of avoided crossings of the quasienergy bands. The interference of Floquet states belonging to different sidebands causes a mixing of long-wavelength (quantum) and short-wavelength (quasiclassical) behavior, making it possible to use the oscillating quantum dot as a kind of transistor for light and sound. We comment under which conditions the setup can be utilized to observe zitterbewegung.

Figure 1

Sketch of the scattering setup. Injected by a probe laser, an incident optical wave with energy $E>0$ and wave vector ${\mathit{k}}^o=k^o{\mathit{e}}_x$ hits a laser-driven quantum dot of radius $R$. Inside the dot, the photon-phonon coupling is $g=g_0-2g_{1}cos\left(\mathrm{\Omega }t\right)$. As a result, reflected optical and mechanical waves appear with wave vectors ${\mathit{k}}^{o/m}$ (${\mathit{k}}_{n=0}\equiv \mathit{k}$), in the central band with energy $E$, and in the sidebands with energies $E_n=E+n\hslash \mathrm{\Omega }$, where $n=\pm 1,\pm 2,...$. The reflected waves are directed away from the dot and carry an angular momentum. Since the dot allows for the conversion between light and sound, mechanical waves appear outside the dot even though the coupling vanishes here. Note that the figure is not true to scale and, since $v_o>v_m$, the photon-phonon wave vectors $k_n^{o/m}=\left|E_n\right|/\hslash v_{o/m}$ are not equal in magnitude.

Figure 2

Different scattering regimes for the static quantum dot in dependence on the strength parameter $R{g}_0$ and the size parameter $ER$. The latter determines the maximum angular momentum $l_{\max }$ being possible in the scattering. The energy-coupling ratio $E/g_0$ switches between the optomechanical ($E/g_0\ll 1$) and the pure optical regime ($E/g_0\gg 1$), where the optomechanical (optical) regime is characterized by the interconversion rate $Q^m/Q^o\sim 1$ ($\ll 1$). Depending on these parameter ratios the static dot acts as a (i) resonant scatterer in the quantum regime, (ii) strong reflector, (iii) weak reflector, or (iv) weak scatterer. On the axis of abscissae the first resonance point derived from the resonance condition (15) with $l=0$ is marked.

Figure 3

Crossing energies (CE) according to Eq. (16). Note that in some cases CE with different $p$ coincide (such situation is marked by points). Symmetric Floquet-resonant scattering occurs at $g_0=\mathrm{\Omega }\sqrt{v_o{v}_m}/(v_o+v_m)\simeq 0.287\mathrm{\Omega }$. Circles designate particular energy-coupling ratios: (1) $E/g_0={10}^{-4}$ ($E\simeq 0$), (2) $E/g_0\simeq 3.48$ ($E={10}^{-3}{g}_0+1\mathrm{\Omega }\simeq \mathrm{\Omega }$), and (3) $E/g_0\simeq 0.31$ ($E=0.123\mathrm{\Omega }$ at $g_0=0.394\mathrm{\Omega }$). They correspond to different scattering regimes of the static dot in Fig. 2: (1) $\to$ (i), (2) $\to$ (iii), (3) $\to$ (ii).

Figure 4

Scattering efficiency at weak coupling, $|g_1|\ll \mathrm{\Omega }$. To realize symmetric Floquet resonance for $E\simeq 0$ [case (1) in Fig. 3] we set $E={10}^{-4}{g}_0\simeq 0$, $g_0=\mathrm{\Omega }\sqrt{v_o{v}_m}/(v_o+v_m)\simeq 0.287\mathrm{\Omega }$, and $|g_1|=0.02\mathrm{\Omega }$ [except for (c)]. (a) Time-averaged scattering efficiency of the photon (red/gray) and the phonon (black), with resonance points $i=0,1,...$ of the static quantum dot for $l=0$ according to Eq. (15) (blue numbers). (b) Different contributions to the scattering efficiency of (a). (c) Enlarged scattering efficiency close to $i=5$ for (i) $g_1=0$, (ii) $|g_1|=2\times {10}^{-3}\mathrm{\Omega }$, (iii) $|g_1|=5\times {10}^{-3}\mathrm{\Omega }$, and (iv) $|g_1|=0.02\mathrm{\Omega }$. (d) Time-averaged scattering efficiency (dashed) and time evolution of the scattering efficiency at $i=5$, corresponding to case (iv) in (c) (here $Q_m$ is multiplied by a factor of 100).

Figure 5

Time-retarded scattering characteristics for the same parameter values as used in Fig. 4. Shown are the optical (red, top left) and mechanical (black, top right) parts of the probability density $\rho =\langle \psi |\psi \rangle$ inside and outside the quantum dot (marked by the blue circle) at $t=0$, as well as the angle-resolved time evolution of the far-field current density $j$ according to Eq. (14) (bottom) at $r=R$. For reasons of symmetry the angle dependence of the optical (mechanical) mode is given only for $\phi \ge 0$ ($\phi \le 0$). Note that the ring structure occurring in the photon probability density also exists in the phonon density, but is hard to resolve due to the small wavelength of the phonon wave ($v_m=v_o/10$) (the additional structures in the phonon density arise due to undesirable aliasing effects).

Figure 6

Scattering contributions ${\overline{Q}}_{n,l=0}^o$ [colored (thicker) lines] at moderate couplings $|g_1|=0.05\mathrm{\Omega }$ (a) and $|g_1|=0.14\mathrm{\Omega }$ (b). Other parameter values are the same as in Fig. 4 for the symmetric Floquet-resonant situation with $E\simeq 0$ [case (1) in Fig. 3]. In addition, the time-averaged scattering efficiency of the phonon is depicted [black (thin) line].

Figure 7

Time-averaged scattering efficiency (top panel) of the photon [red (gray)] and the phonon (black) and optical scattering contributions of different partial waves (lower panels) at weak couplings, where $|g_1|=5\times {10}^{-3}\mathrm{\Omega }$ in (a) and $|g_1|=0.02\mathrm{\Omega }$ in (b). In the top panels the scattering efficiencies of the static dot are included [turquoise (thicker) line]. The scattering contribution of the phonon for the sideband $n=-1$ with $l=0$ is denoted by the blue line. To realize a symmetric Floquet resonance at $E\simeq \mathrm{\Omega }$ [case (2) in Fig. 3], we choose $E={10}^{-3}{g}_0+\mathrm{\Omega }$, $g_0=\mathrm{\Omega }\sqrt{v_o{v}_m}/(v_o+v_m)\simeq 0.287\mathrm{\Omega }$.

Figure 8

Time-averaged scattering efficiency of the photon [red (gray)] and the phonon (black) slightly away from the symmetric Floquet resonance at $E=0.928\mathrm{\Omega }$, $g_0=0.3\mathrm{\Omega }$, and $|g_1|=0.1\mathrm{\Omega }$. For comparison, the corresponding scattering efficiencies of the static dot are shown (turquoise).

Figure 9

Time-averaged scattering efficiency at Floquet resonance without symmetry [case (3) in Fig. 3]. Here $E\simeq 0.12\mathrm{\Omega }$ and $g_0\simeq 0.39\mathrm{\Omega }$.

Figure 10

(a) Scattering contributions to ${\overline{Q}}^o$ given in Fig. 9 for $|g_1|=0.025\mathrm{\Omega }$. (b) Enlarged area near $R{g}_0=2$.

Figure 11

Polar plot of the current density of the optical reflected wave in the far field according to Eq. (14) at the time $t/2\pi =0.61$ (a), $t/2\pi =1.2$ (b), and $t/2\pi =1.33$ (c) for $r=R$. Parameter values are the same as in Fig. 10 with $R{g}_0\simeq 2$ (dashed line).

Figure 12

Quasienergies ${\varepsilon }^{\left(\pm \right)}+n\mathrm{\Omega }$, $n\in \mathbb{Z}$, obtained as eigenvalues of the Floquet matrix (A5) for $|g_1|=0.14\mathrm{\Omega }$ as a function of the wave number $q$ (using $\sigma =\pm 1$ for positive and negative values of $q$). Inserted are also the energies $E_n=E+n\mathrm{\Omega }$ of the central band $n=0$ and the sidebands $n=\pm 1$ (blue horizontal lines) for $E=0$ and static coupling $g_0=0.287\mathrm{\Omega }$, corresponding to the case of symmetric Floquet resonance close by $E\simeq 0$ (see discussion in Sec. 3b1 of the main text). The proper pair of quasienergies ${\varepsilon }^{\left(\pm \right)}$ (solid lines) that has to be used for the scattering problem is that which coincides with the dispersion of the static case at $q\to 0$. The wave numbers used for the scattering problem are determined by the zeros of $E_n-{\varepsilon }^{\left(\pm \right)}\left(q\right)$ and are marked exemplary for the cases $n=0,1$ in the lower panels (i) and (ii). For comparison the polariton branches of the energy dispersion of the static case, $E^{\mp }\left(q\right)$ (solid) and $E^{\pm }\left(q\right)\mp \mathrm{\Omega }$ (dashed), are shown (brown thin lines). Since $E=0$, the wave numbers reveal the symmetry $q_{-n}^{\left(\pm \right)}=-q_n^{\left(\mp \right)}$.