Long-lifetime coherence in a quantum emitter induced by a metasurface

An anisotropic quantum vacuum (AQV) has been predicted to induce quantum interferences during the spontaneous emission process in an atomic $V$ transition [G. S. Agarwal, Phys. Rev. Lett. 84, 5500 (2000)]. Nevertheless, the finite lifetime of the excited states is expected to strongly limit the observability of this phenomenon. In this paper, we predict that an AQV can induce a long-lifetime coherence in an atomic $\mathrm{\Lambda }$ transition from the process of spontaneous emission, which has an additional advantage of removing the need for coherent laser excitation. We also carry out two metasurface designs and compare their respective efficiencies for creating an AQV over remote distances. The detection of this coherence induced by a metasurface, in addition to being yet another vindication of quantum electrodynamics, could pave the way towards the remote distance control of coherent coupling between quantum emitters, which is a key requirement to produce entanglement in quantum technology applications.

Figure 1

Three-level quantum emitter with a $\mathrm{\Lambda }$ structure. The upper level $|0\rangle$ can decay via two transitions: either to the state $|1\rangle$ with the emission of a right circularly polarized photon denoted ${\sigma }^{+}$, or to the state $|2\rangle$ with the emission of a left circularly polarized photon denoted ${\sigma }^{-}$. ${\rho }_{12}$ denotes the coherence between the two ground states $|1\rangle$ and $|2\rangle$.

Figure 2

Unit cell of a reflect-array metasurface made of a metallic mirror of thickness $h_1$, a dielectric spacer of thickness $h_2$, and a rectangular nanoantenna of dimensions $l_x\times l_y$ and of thickness $h_3$. The dimensions of the unit cell are ${\mathrm{\Lambda }}_x^{\mathrm{UC}}\times {\mathrm{\Lambda }}_y^{\mathrm{UC}}$.

Figure 3

Power reflectance (in purple) and phase shift $\phi$ (in green) of an incident $x$-polarized (respectively, $y$-polarized) wave as a function of the length $l_x$ of the nanoantennas, computed for a 2D grating (see main text). The symbols represent the simulated points, and the solid lines are guides to the eyes. The dotted black lines are spaced by $2\pi /5$. The number of Fourier modes used for the simulations is 30.

Figure 4

Illustration of the phase-mapping approach for the 1D design of the resonant-phase delay metasurface. (a) Phase profiles to be encoded by the metasurface: the wrapped (respectively, unwrapped) spherical phase profile ${\phi }_x$ of Eq. (17) (red full line) (respectively, red dashed line) desired for the $x$ -polarization, and the flat phase profile ${\phi }_y$ (blue line) desired for the $y$ -polarization, starting from the center of the metasurface at $r=0$. (b) Corresponding nanoantennas to encode the desired phase shifts ${\phi }_x$ and ${\phi }_y$. The unit cells of length ${\mathrm{\Lambda }}^{\text{UC}}$ (black dashed box) containing the nanoantennas are encompassed into supercells of length ${\mathrm{\Lambda }}^{\text{SC}}$ (red box), spanning the $2\pi$ phase space.

Figure 5

Diffraction performances of a linear-phase gradient metasurface. The inset shows a supercell of the gradient metasurface of size ${\mathrm{\Lambda }}_x^{\text{SC}}\times {\mathrm{\Lambda }}_y^{\text{SC}}$ (see main text). (a) Reflection angle ${\theta }_r$ in the diffracted order $m=-1$ of an incident plane wave polarized along $\vec{x}$ as a function of the incident angle ${\theta }_i$ (green circles). The generalized Snell's law of reflection [Eq. (18)] is also plotted (black dashed line). (b) Power reflectance in the diffracted orders $m=0,-1,-2$ (blue stars, green circles, and orange triangles, respectively) and total reflection efficiency (black squares) of an incident plane wave polarized along $\vec{x}$ as a function of the incident angle ${\theta }_i$. The number of Fourier modes used to compute the reflectance is 30.

Figure 6

Geometric phase $\phi$ as a function of the rotation angle $\varphi$ of the nanorods in the plane $(\vec{x},\vec{y})$ (see inset), computed for a 2D grating (see main text) (green circles). The analytical expression [Eq. (19)] is also plotted (black dashed line). The number of Fourier modes used to compute the geometric phase is 30.

Figure 7

Three-dimensional design of the geometric metasurface, made for a distance of the dipole source $d=10{\lambda }_0$ from the metasurface, and an emission wavelength of ${\lambda }_0=852\text{nm}$. The white box highlights the first supercell (starting from the center) of size ${\mathrm{\Lambda }}_1^{\text{SC}}=2.7\mu \text{m}$, made of nine nanorods. This computer aided design was drawn using the software SOLIDWORKS developed by Dassault Systèmes.

Figure 8

Cross-polarization (CP) power reflectance, which characterizes the conversion efficiency in energy between a light circularly polarized ${\sigma }^{+}$ and a light circularly polarized ${\sigma }^{-}$, as a function of the incident angle ${\theta }_i$, computed for a 2D grating (see main text). The number of Fourier modes used to compute the CP reflectance is 30.

Figure 9

Relative decay rate modifications ${\gamma }_x/{\gamma }_0$ (green circles) and absolute coherence $|{\rho }_{12}|$ (red triangles) as a function of the numerical aperture of the metasurface NA. For comparison, the relative decay rate (respectively, coherence) for an ideal spherical mirror of power reflectance $R_x=1$ for the $x$ polarization only is also shown [green (respectively, red) dashed line].