Optical dressing of the electronic response of two-dimensional semiconductors in quantum and classical descriptions of cavity electrodynamics

We study quantum effects of the vacuum light-matter interaction in materials embedded in optical cavities. We focus on the electronic response of a two-dimensional semiconductor placed inside a planar cavity. By using a diagrammatic expansion of the electron-photon interaction, we describe signatures of light-matter hybridization characterized by large asymmetric shifts of the spectral weight at resonant frequencies. We follow the evolution of the light dressing from the cavity to the free-space limit. In the cavity limit, light-matter hybridization results in a modification of the optical gap with sizable spectral weight appearing below the bare gap edge. In the limit of large cavities, we find a residual redistribution of spectral weight which becomes independent of the distance between the two mirrors. We show that the photon dressing of the electronic response can be fully explained by using a classical description of light. The classical description is found to hold up to a strong coupling regime of the light-matter interaction highlighted by the large modification of the photon spectra with respect to the empty cavity. We show that, despite the strong coupling, quantum corrections are negligibly small and weakly dependent on the cavity confinement. As a consequence, in contrast to the optical gap, the single-particle electronic band gap is not sensibly modified by strong coupling. Our results show that quantum corrections are dominated by off-resonant photon modes at high energy. As such, cavity confinement can hardly be seen as a knob to control the quantum effects of the light-matter interaction in vacuum.

Figure 1

(Top left) Sketch of a two-dimensional material placed between two perfectly conducting mirrors separated by a distance $L$. (Top right) Electronic bands of the two-dimensional system as obtained by using the periodic potential defined by the parameters $v_0=5.0\mathrm{eV}$, $\lambda =0.2{\mathrm{nm}}^{-1}$, $\alpha =2,$ and $\xi =0.1\mathrm{nm},$ see text. (Bottom) Photon dispersions for two cavity lengths. Red dashed line indicate a photon cutoff used in the calculations. The massless mode drawn with a dashed line represents the mode with zero in-plane component.

Figure 2

Bare ${\mathcal{F}}_0\left(\omega \right)$ and dressed $\mathcal{F}\left(\omega \right)$ spectral densities for the $\mathbf{q}=0$ photons for cavity mirrors at a distance $L=0.2\mu \mathrm{m}$. Bare photons broadening is set to ${\mathrm{\Gamma }}_{\mathrm{ph}}=0.01\mathrm{eV}$. Shaded area represents the bare conductivity of the material. Throughout the paper the conductivity is measured with respect to the quantum of conductance $G_0\equiv \frac{2e^2}{h}.$ The conductivity is plotted in the linear scale, while photon spectra are plotted in the logarithmic scale (see arrows). Bottom panels show the photon spectral densities in linear scale around the single photon resonances.

Figure 3

(Top) Bare ${\sigma }_0$ (orange) and dressed $\sigma$ (blue) conductivities for cavity length $L=0.2\mu \mathrm{m}$. The inverted photon spectrum is reported in the top part of the panel. Photon broadening is ${\mathrm{\Gamma }}_{\mathrm{ph}}=0.01\mathrm{eV}$. (Bottom) Blow up of the asymmetric feature in the dressed conductivity around $3.1\mathrm{eV}$ for different values of the photon broadening.

Figure 4

(Top) Dressed conductivity at different cavity lengths. (Bottom) Conductivity at resonant lengths $L_n=(2n+1)L_0$ for $L_0=0.27$ (left) and $0.32\mu \mathrm{m}$ (right).

Figure 5

Spectral weight transfer below the gap edge ${\mathrm{\Omega }}_{\mathrm{gap}}=2.25\mathrm{eV}$ as a function of the distance between the mirrors for two values of the photon damping. The shaded area highlights the crossover from the cavity to the free-space limit.

Figure 6

(Top) Dressed conductivity in the large $L$ limit $L=15\mu \mathrm{m}$. (Left) Purely electronic ${\sigma }_{PP}$, purely diamagnetic ${\sigma }_{AA}$ and mixed ${\sigma }_{AP}$ contributions to the conductivity. (Right) Bare conductivity ${\sigma }_0$ (orange full) compared to the dressed conductivity $\sigma ={\sigma }_{PP}+{\sigma }_{AP}+{\sigma }_{AA}$ (blue dashed). (Inset) Zoom of the dressed conductivity below the gap showing the transferred spectral weight for two values of the photon damping ${\mathrm{\Gamma }}_{\mathrm{ph}}$. (Bottom) Dressing of conductivity in the static limit as a function of the cavity length, and for different cutoff in the number of electronic bands $N_b$.

Figure 7

(Top) Dressed conductivity computed using current dressing in Maxwell's equations (full lines) for different values of the charge renormalization parameter $Z=1$ (red), 10 (blue), and 50 (green). White dashed lines indicate the same quantities computed from the QED Hamiltonian in the Gaussian approximation for the same values of $Z$. For comparison, the conductivity curves have been scaled by the factor $Z$. The cavity length is $L=0.27\mu \mathrm{m}$ and the broadening ${\mathrm{\Gamma }}_{\mathrm{ph}}=0.02\mathrm{eV}$. Bottom: Photon spectral densities around the cavity fundamental mode computed using the QED Hamiltonian in the Gaussian approximation for the same values of $Z$ shown above. Insets: shift of the photon resonance of the fundamental cavity mode as a function of $Z$ (circles, right axis). $\mathrm{\Delta }{\omega }_0$ is defined as the energy difference between the peak of the bare ${\mathcal{F}}_0$ and the peak of the dressed $\mathcal{F}$. Relative shift of the spectral weight Eq. (22) as a function of $Z$ (squares, left axis).

Figure 8

Imaginary parts of the electronic self-energy computed at the $\mathrm{\Gamma }$ point for the valence and first two conduction bands and for two cavity lengths $L=0.27$ (left) and $1.89\mu \mathrm{m}$ (right). Top panels show the self-energies for $Z=1$. Bottom panels show self-energies divided by $Z$ for $Z=10$ and 50. In the bottom right panel we reported a sketch of the self-energy diagram considered. The full line indicates the electronic Green's function $G$, while the dashed line indicates the photon propagator $\mathcal{D}$. The dots indicate the couplings. For simplicity, in the sketch we omitted all the labels of quantum numbers as well as the energy and momentum conservation at the vertices. The noise in the self-energies is due to the discretization of the $\mathbf{q}$ grid for momentum integration of the photon propagator.

Figure 9

Real parts of the electronic self-energy computed at the $\mathrm{\Gamma }$ point for the valence (top) and first conduction (bottom) bands and for two cavity lengths $L=0.27$ (left) and $1.89\mu \mathrm{m}$ (right). Panels show self-energies for $Z=1$ and different cutoff energies ${\mathrm{\Omega }}_{\mathrm{ph}}=10\mathrm{eV}$, ${\mathrm{\Omega }}_{\mathrm{ph}}=15\mathrm{eV},$ and ${\mathrm{\Omega }}_{\mathrm{ph}}=20\mathrm{eV}$.