Atomically thin semiconductors as nonlinear mirrors

We show that a transition-metal dichalcogenide monolayer with a radiatively broadened exciton resonance would exhibit perfect extinction of the transmitted field. This result holds for $s$- or $p$-polarized weak resonant light fields at any incidence angle, due to the conservation of the in-plane momentum of excitons and photons in a flat defect-free two-dimensional crystal. In contrast to extinction experiments with single quantum emitters, exciton-exciton interactions lead to an enhancement of reflection with increasing power for incident fields that are blue detuned with respect to the exciton resonance. By taking into account quantum fluctuations, we show that the extinction of such a nonlinear mirror is limited; for light intensities leading to maximal reflection, the transmitted field is in a multimode squeezed state.

Figure 1

(a) Schematic of the experimental setup. A collimated coherent laser field is incident on a transition-metal dichalcogenide (TMD) monolayer. In addition to coherent transmitted and reflected fields, a two-mode squeezed state (TMSS) is generated due to exciton-exciton interactions. Unlike the coherent fields, the TMSS is emitted into a large solid angle. The electromagnetic field can be characterized as consisting of right- (left-) propagating input $r_{\mathrm{in}}$ ($l_{\mathrm{in}}$) and output $r_{\mathrm{out}}$ ($l_{\mathrm{out}}$) modes. (b) The reflection from the 2D TMD layer for a weak near-resonant pump. Destructive interference between the directly transmitted field and the field generated by the TMD excitons leads to a suppression of transmission and an enhancement of reflection. For perfect coupling efficiency $\eta =1$ between the input light and the excitons on the TMD layer, the system constitutes a perfect mirror. Imperfect coupling efficiency due to the nonradiative lifetime of the excitons reduces the reflection maxima to ${\eta }^2$.

Figure 2

(a) The reduction of reflection due to depletion of the $k=0$ coherent state for three different values of $g$ while keeping $g{\beta }^2=3.5$ constant. All parameters are plotted in units of $\gamma$. In the limit that $g\to 0$ (orange line), the depletion does not modify the reflection maximum, since in this limit the excitonic density at resonance is infinite. For $g=1.75$ ($g=0.35$) [green (blue) line], the depletion reduces the reflection maximum down to $\approx 0.9$ ($\approx 0.97$). The green shaded area indicates the region where the mean-field treatment is expected to be not reliable for this choice of $g$. Here, a depletion of the coherent exciton density exceeding $10\%$ is used as a benchmark. The black arrows indicate the hysteresis curve. The faint blue curve is the same as Fig. 1 for $\eta =1$. The (b) antisqueezing and (c) squeezing spectra at $\mathrm{\Delta }=g{\beta }^2=3.5$, in units of $\gamma$. For this detuning and coherent exciton density, the fluctuations on top of the coherent field have a Bogoliubov-like spectrum of an interacting bosonic system. The red and the green lines show the dispersion of the particle and hole excitations, respectively. These lines are determined by the real part of the poles of the dressed propagator $G(\omega ,k)$. The black line is the decay rate of the excitations, which in this case is equal to $-\gamma$, and is given by the imaginary part of the poles of the dressed propagator $G(\omega ,k)$. The bandwidth of squeezing is determined by the decay rate of the excitons.