Spontaneous Exciton Valley Coherence in Transition Metal Dichalcogenide Monolayers Interfaced with an Anisotropic Metasurface

The control of the exciton intervalley coherence renders transition metal dichalcogenides monolayers promising candidates for quantum information science. So far, generating intervalley coherence has the need for an external coherent field. Here, we theoretically demonstrate spontaneous generation (i.e., without any external field) of exciton intervalley coherence. We achieve this by manipulating the vacuum field in the vicinity of the monolayer with a designed polarization-dependent metasurface, inducing an anisotropic decay rate for in-plane excitonic dipoles. Harnessing quantum coherence and interference effects in two-dimensional materials may provide the route for novel quantum valleytronic devices.

Figure 1

Noise-induced valley coherence in transition metal dichalcogenide monolayers. We considered a monolayer of $M{X}_2$, positioned at a distant height from the metasurface. We designed a metasurface such that the spontaneous emission from the locally excited TMDC monolayer is efficiently focused back towards the source at the single photon level. Moreover, we engineered the polarization-dependent response of the metasurface to achieve an anisotropic scattered field. The interaction between the suspended TMDC monolayer and the custom-designed metasurface yields an anisotropic quantum vacuum. In an ordinary vacuum such as free space, the emission from the $K$ valley with ${\sigma }_{+}$ polarization cannot radiatively excite the orthogonal $K^{\prime }$ valley with ${\sigma }_{-}$ polarization. However, by breaking the isotropic nature of the quantum vacuum in the vicinity of the monolayer via a metasurface, one can remarkably excite the $K^{\prime }$ valley with an emission from the $K$ valley. Such interaction leads to spontaneous generation of valley coherence and yields quantum interference among their emissions.

Figure 2

Geometric phase-based metasurface for an anisotropic quantum vacuum. (a),(b) Phase profiles for molding the incident light with right circular polarization (RCP) and left circular polarization (LCP), respectively, at a free-space wavelength of $\lambda =670\mathrm{nm}$ and a height of the dipole from the metasurface of $d=10\lambda$. The polarization helicity-dependent phase profiles are presented by the heat map and the corresponding metasurface realization, i.e., a nanorod antenna array with space-variant orientations $\theta (x,y)$, is shown on top. The width ($x$ dimension) and length ($y$ dimension) of each nanorod are 200 and 80 nm, respectively, and the thickness is 30 nm. The individual antenna design relies on a gap plasmon resonator nanoantenna with a 30-nm-thick silver nanorod, a 110-nm-thick dielectric (${\mathrm{MgF}}_2$) spacer layer, and a 130-nm-thick silver layer acting as a back reflector. We used a set of 16 nanoantenna orientations mimicking phase shifters with a $\pi /8$ phase increment. (c) Wavelength-dependent reflection efficiency of the GPM. By optimizing the dimensions of the nanorod, we achieved a broadband high cross-polarization reflectivity (90%) and an extremely low copolarization reflectivity ($\sim 0.1\%$), yielding a highly efficient GPM. The dots highlight the reflectivities at 670 nm, which is the resonant wavelength for the ${\mathrm{MoS}}_2$ monolayer.

Figure 3

Metasurface-induced anisotropic decay rate. (a) Simulated scattered field intensity (${|E_x|}^2$) distribution for the excitonic-dipole source located at $(0,0,10\lambda )$ and oriented along the $x$ axis. The distribution is shown in the $x\text{−}z$ plane, where the metasurface lies in the $z=0$ plane. With an optimized design, we achieved $\sim 47\%$ reflection of the incident field focused back towards the on-demand location of the dipole (50% is the upper limit). The scattered field intensity (${|E_y|}^2$) distribution for a $y$-polarized dipole is identical. (b) Nondegenerate imaginary part of the scattered field. At the position of the dipole (i.e., $x=0$), for $x$- and $y$-polarized dipoles, the imaginary part of the scattered field is $\pi$-phase shifted, i.e., minimized for an $x$-polarized dipole while symmetrically maximized for a $y$-polarized dipole. The upper limit for the scattered field is $E_0$, which is the imaginary part of the field induced by the dipole at its position. (c) An anisotropic decay rate is enabled by the GPM, where the normalized decay rate for $x$- (${\gamma }_{xx}$) and $y$-polarized dipoles (${\gamma }_{yy}$) is suppressed (0.53) and symmetrically enhanced (1.47), respectively.

Figure 4

Noise-induced valley population-coherence coupling. (a) Temporal evolution of the population of mutually orthogonal valleys ($K$ and $K^{\prime }$) in the presence (solid lines) and absence (dashed line) of the metasurface. In the presence of the metasurface, the decay rate of the $K$ valley population is reduced along with a finite excitation of the $K^{\prime }$ valley. (b) Temporal evolution of intervalley coherence. A nonzero intervalley coherence, indicating coherent excitation of the $K^{\prime }$ valley with a photon emitted from the $K$ valley, is evident. This excitation is forbidden in free space (dashed line) due to an optical selection rule. (Inset) Dependence of the maximum intervalley coherence on the intervalley scattering rate, revealing that the generated coherence survives even when the intervalley scattering rate is 1000-fold stronger than the spontaneous decay rate.