Quantum Nonlinear Response of Emitter Lattices

TL;DR

The work addresses quantum nonlinearities in the optical response of a periodic lattice of two-level quantum emitters driven coherently by a laser. Using a mean-field approach, the emitter lattice is mapped to noninteracting units with a renormalized driving amplitude, revealing Bloch exciton states with parallel wavevectors $\mathbf{k}_{\parallel}$ that differ from the incident field, including states outside the light cone. Under strong driving, the lattice emits a broadband incoherent background across a wide range of frequencies and wavevectors, alongside a coherent Rayleigh component, due to the intrinsic quantum nonlinearity and resonance-fl fluorescence–like processes. Furthermore, as the lattice period approaches the driving wavelength, the effective driving rate $\Omega_{\text{eff}}$ exhibits bistability with hysteresis, enabling abrupt switching of the nonlinear optical response; this bistability is a purely quantum feature distinct from the classical bosonic case. These findings highlight the potential of quantum emitter lattices as tunable quantum metasurfaces for applications in single-photon storage and quantum information processing.

Abstract

We theoretically investigate the emergence of quantum nonlinearities in the optical response of lattices of two-level quantum emitters coherently driven by a laser. For subwavelength lattice periods, where the system behaves as a quantum metasurface, we find that a resonant incident plane wave can populate excitonic Bloch states with parallel wavevectors different from the incident field, including those lying outside the light cone. Closely related to resonance fluorescence, the far-field emission from the system in the strong-driving regime is dominated by a broadband background of photons spanning a wide range of frequencies and wavevectors. Moreover, we show that, for periods approaching the driving wavelength, the emitter lattice enters in a bistable regime due to the renormalization of the driving rate, in striking contrast with its classical (bosonic) analog. This bistable behavior enables the selective activation and deactivation of the optical quantum nonlinearities of the system.

Figures (3)

• Figure 1: (a) Schematics of a QE lattice with period $l$ lying in the $xy$ plane and excited with an $x$-polarized laser propagating in the $z$ axis. Orange photons indicate emission at $\omega_{\rm L}$ and $\bold{k}_{\rm L,\parallel}$, whereas blue photons highlight emission at other frequencies and wavevectors. The inset shows an schematics of an individual QE. (b) Population per emitter (green), along with its coherent (orange) and incoherent (blue) contributions, as a function of the driving rate for $\Delta=0$. For comparison, the black curve shows $\langle N\rangle_{\rm ss}$ for a lattice of CEs. The inset displays the population distribution within the 1BZ for $k_y=0$, $\Omega=2\gamma_0$, and $\Delta=0$, using the same color scheme.
• Figure 2: (a) Intensity integrated within the central peak of the Mollow-like spectrum, $\langle I_{\omega_{\rm L}} (\bold{k}_\parallel)\rangle_{\rm ss}$, evaluated along the path indicated in the inset for $\Omega=4\gamma_0$ (green). For comparison, the black curve corresponds to a CE lattice. The right inset displays $\langle I_{\omega_{\rm L}} (\bold{k}_\parallel)\rangle_{\rm ss}$ across the entire 1BZ. (b) Total intensity, $\langle I_\omega\rangle_{\rm ss}$, around $\omega_{\rm L}$ (green), and $\omega_{\pm}$ (purple). The black curve again corresponds to the CE lattice, while the orange and blue curves represent the coherent and incoherent contributions to $\langle I_{\omega_{\rm L}}\rangle_{\rm ss}$, respectively. The inset shows $\mathcal{S}^{\rm I}_{\rm QE, eff}(\omega)$ for $\Omega=4\gamma_0$. In all cases, we assume $\Delta=0$.
• Figure 3: Effective driving rates (a), central frequencies (b), and linewidths (c) of the incoherent emission spectrum $\mathcal{S}^{\rm I}_{\rm QE, eff}(\omega)$, as function of $\Omega$ for different values of $l$. (d) Population per emitter, along with its coherent and incoherent contributions, as a function of $\Omega$ for $l=0.9999\lambda_0$. (e) Total intensity emitted by the QE lattice for different frequency windows and $l=0.9999\lambda_0$. Arrows and dotted curves indicate the abrupt transitions and metastable states of the system, respectively ($\Delta =0$).