Quantum Electrodynamics of graphene Landau levels in a deep-subwavelength hyperbolic phonon polariton cavity

Abstract

The confinement of electromagnetic radiation within extremely small volumes offers an effective means to significantly enhance light-matter interactions, to the extent that zero-point quantum vacuum fluctuations can influence and control the properties of materials. Here, we develop a theoretical framework for the quantum electrodynamics of graphene Landau levels embedded in a deep subwavelength hyperbolic cavity, where light is confined into ultrasmall mode volumes. By studying the spectrum, we discuss the emergence of polaritons, and disentangle the contributions of resonant quantum vacuum effects from those of purely electrostatic interactions. Finally, we study the hybridization between magnetoplasmons and the cavity's electromagnetic modes.

Figures (10)

• Figure 1: (Color online) A schematic representation of the system under study. The HPP cavity is formed by two metallic mirrors (depicted in gold) located at $|z| > L_z/2$. The region between the mirrors is filled with hBN, shown in light blue. A graphene sheet is positioned at $z=0$ and subjected to an external perpendicular magnetic field.
• Figure 2: (Color online) Different spectra are plotted as functions of $q_\parallel$ (expressed in ${\rm nm}^{-1}$) and $\hbar\omega$ (expressed in $\rm{eV}$). Panel (a) shows the imaginary part of the bare Green's function of the HPP cavity, i.e. $-e^2{\rm Im}~g^{(0)}(q_{\parallel},0,0, \omega)$, for $L_z=50~{\rm nm}$. All modes are taken into account via resummation of the series. Panel (b) displays the imaginary part of the bare density-density response function for LLs in graphene, $-{\rm Im}~\chi^{(0)}_{\rm nn} ({q}_{\parallel}, \omega)$, taking $B=5~{\rm T}$ and a cutoff for the LLs $N_{\rm c}=30$. In both panels we choose $\hbar \eta= 5\times 10^{-4}{\rm eV}$.
• Figure 3: (Color online) The RPA density-density response of the coupled system (HPP cavity plus graphene), $-{\rm Im}~{\chi}_{\rm nn} ({q}_{\parallel}, \omega)$. Panel (a) shows $-{\rm Im}~{\chi}_{\rm nn} ({q}_{\parallel}, \omega)$ as a function of the wavevector $q_\parallel$ for a given magnetic field, $B=5~{\rm T}$ (corresponding to $\ell_B\approx 11.5~{\rm nm}$). Panel (b) shows the same quantity as a function of $\hbar v_{\rm F}/\ell_B$, for fixed momentum $\bar{q}_\parallel=0.1~{\rm nm}^{-1}$. Other parameters are $L_z=50~{\rm nm}$, $N_{\rm c}=30$, and $\hbar \eta= 5\times 10^{-4}~{\rm eV}$. For $g^{(0)}( q_{\parallel}, z, z', \omega)$, the expression resummed over all modes was employed.
• Figure 4: (Color online) Panels (a,b) display the ratio $R_{q_\parallel,n_z,s,-,0,+, 1} = (\Omega_{q_\parallel,n_z,s,-,0,+, 1} / \omega_{q_\parallel,n_z,s})$, where $\Omega_{q_\parallel,n_z,s,-,0,+, 1}$ denotes the Rabi frequency for the Landau level (LL) transition $(0, +1)$, and $\omega_{q_\parallel,n_z,s}$ is the cavity mode frequency. This ratio is shown as a function of $L_z$ (in ${\rm nm}$) for different $n_z$ for a fixed $\bar{q}_\parallel=0.1 {\rm nm^{-1}}$ and $B=10 {\rm T}$. Panels (a) show these quantities for the upper Reststrahlen band using blue dotted curves, while panels (b) present the lower Reststrahlen band with red dashed curves. The line thickness decreases and the color of the curves darkens as the quantum number $n_z$ increases.
• Figure 5: (Color online) The RPA proper density-density response of the coupled system (HPP cavity plus graphene), $-{\rm Im}~{\chi}_{\rm nn} (\bar{q}_{\parallel}, \omega)$ as a function of $\hbar v_{\rm F}/\ell_B$ for a doped system ($N_{\rm F}=5$), for a fixed wavevector ($\bar{q}_{\parallel}=0.1{\rm nm}^{-1}$). Panel (a) illustrates this quantity $-{\rm Im}~{\chi}_{\rm nn} (\bar{q}_{\parallel}, \omega)$ in the case where quantum mechanical modes are not included, $g_{\rm q}^{(0)}( q_{\parallel}, z, z', \omega)=0$. Panel (b) shows the same quantity in the presence of these modes. Other parameters are $L_z=50{\rm nm}$, $N_{\rm c}=30$, $\hbar \eta= 5\times 10^{-4}{\rm eV}$. Panel (c) is the same of panel (b), with $L_z=10^3{\rm nm}$. For $g^{(0)}( q_{\parallel}, z, z', \omega)$, the expression resummed over all modes was employed.