Tilted Material in an Optical Cavity: Light-Matter Moiré Effect and Coherent Frequency Conversion

Abstract

Exciton-polaritons formed inside optical cavities offer a highly tunable platform for exploring novel quantum phenomena. Here, we introduce and theoretically characterize a light-matter moiré effect (LMME) that arises when a 2D material is tilted inside a planar optical cavity, in contrast to stacking multiple layers at a twist angle as is done in forming 2D moiré hetero-structures. We show that this geometric tilt produces emergent periodicity in the light-matter coupling, yielding displaced replicas of the polariton dispersion and flat bands near the Brillouin-zone center. Through time-dependent quantum dynamical simulations, we demonstrate that LMME enables coherent frequency conversion and remains robust against phonon-induced decoherence. Our findings establish LMME as a new platform for engineering polariton band structures, the generation of flat bands and performing coherent frequency conversion relevant for developing polariton-based quantum devices.

Figures (4)

• Figure 1: Comparison of twisted graphene and tilted material in an optical cavity (a) Illustration of a graphene bilayer offset by a twist angle. (b) Illustration of a tilted material within a Fabry-Pérot cavity coupling to cavity radiation. (c) Schematic band structure of a graphene bilayer without (left) and with (right) a twist of $\theta$. (d) Schematic band structure of a tilted material without (left) and with (right) a tilt of $\theta$.
• Figure 2: Two-dimensional band structure of a flat and tilted material ($\theta = 5.98^\circ$). (a) Schematic representation of a flat material inside an optical cavity, coupled to cavity radiation. (b-c) One dimensional band structures along Y and X directions, respectively for a flat material inside a cavity. (d) Two dimensional band structure at E = $3$ eV of a flat material inside an optical cavity with Y (red) and X (orange) directional cuts, correlating with (b) and (c) respectively. (e) Schematic representation of a tilted material of angle $\theta$ in the X direction, inside an optical cavity, coupled to cavity radiation. (f-g) One dimensional band structures along Y and X directions, respectively for a tilted material inside a cavity. (h) Two dimensional band structure at E = $3$ eV of a tilted material inside an optical cavity with Y (red) and X (orange) directional cuts, correlating with (f) and (g) respectively.
• Figure 3: Tilt-induced separation of side bands and their dependence on material thickness. (a) Relation between tilt angle $\theta$ and $\Delta k_x$ for the model system. (b-d) Two dimensional band structures for tilts of $\theta = 3.58^{\circ}, 5.98^{\circ}, 8.38^{\circ}$ corresponding to $\Delta{k_x}$, where $\delta k_z = 2\pi/L_x$ and $p \in\{3,5,7\}$, respectively for a single layer. (e) Normalized spatially varying (bright-layer) light matter coupling at $n_z = 1, 10, 40$. (f-h) Two dimensional band structures of tilt $\theta = 5.98^{\circ}$ corresponding to $\Delta{k_x} = 5\delta k_x$ for $n_z = 5, 10, 40$, respectively. We use $N_x = N_y = 8001$.
• Figure 4: Coherent frequency conversion. (a) Band structure depicting an input coherent superposition $\frac{1}{\sqrt{2}} [ \hat{A}^{\dagger}_{\bf k_{\stretchrel*{\parallel}{\perp}}} + e^{i\phi}\hat{A}^{\dagger}_{\bf k'_{\stretchrel*{\parallel}{\perp}}}]|\bar{0}\rangle$, and an output coherent superposition of states $\frac{1}{\sqrt{2}} [ \hat{A}^{\dagger}_{\bf k_{\stretchrel*{\parallel}{\perp}} + 2\Delta k} + e^{i\phi}\hat{A}^{\dagger}_{\bf k'_{\stretchrel*{\parallel}{\perp}}+ 2\Delta k}]|\bar{0}\rangle$, for the material at angle of tilt $\theta = 5.98^{\circ}$ corresponding to $\mathbf{\Delta k} = 5\delta k_x$. (b) Real component of state $\hat{A}^{\dagger}_{\bf k_{\stretchrel*{\parallel}{\perp}}}|\bar{0}\rangle$ with $|\psi(t)\rangle$ over 48 fs, with phonon couplings equal to $\gamma_0, 2\gamma_0, 8\gamma_0$, where $\gamma_0 = 7.3067 \times 10^{-5}$ a.u. (c-d) The input and output superposition over 24 fs, without phonon coupling or an initial phase difference between states $\hat{A}^{\dagger}_{\bf k_{\stretchrel*{\parallel}{\perp}}}|\bar{0}\rangle$ and $\hat{A}^{\dagger}_{\bf k'_{\stretchrel*{\parallel}{\perp}}}|\bar{0}\rangle$. (e-f) The input and output superposition over 24 fs, without phonon coupling and an initial phase difference of $\phi = \pi$ between states $\hat{A}^{\dagger}_{\bf k_{\stretchrel*{\parallel}{\perp}}}|\bar{0}\rangle$ and $\hat{A}^{\dagger}_{\bf k'_{\stretchrel*{\parallel}{\perp}}}|\bar{0}\rangle$. (g-j) Mimic the structure of (c-f), differing as the phonon coupling $\gamma = 4\gamma_0$. (k-l) Polar plots displaying the initial phase difference (after $~.5$ fs) of the input and output in $45^{\circ}$ intervals, for $\gamma = 0$ (k) and $\gamma = 4\gamma_0$ (l).