How Subradiance Enables Nonlinearity in Weakly Driven Quantum Arrays

Abstract

Harnessing the nonlinear response of a medium is essential for applications including frequency conversion and light amplification, as well as for the generation of quantum many-body correlations of light or matter. However, achieving these effects typically requires high drive intensities and thick samples, which induce undesired heating effects that typically suppress quantum correlations. In this work, we demonstrate that atom-thin arrays of quantum emitters exhibit a robust nonlinear response even at arbitrarily weak drive intensities. This discovery challenges the long-held assumption that weakly driven ensembles behave classically; instead, we reveal that subradiant states provide a dominant nonlinear contribution that persists in the low-intensity limit. Using a Dynamical Mean-Field Theory (DMFT) approach, we predict that these nonlinearities generate a quantum-correlated steady state composed of interacting pairs of subradiant excitations, characterized by long-range correlations and multi-mode squeezing. Our findings establish a new frontier for nonlinear quantum optics at minimal power, and provide a scalable protocol for preparing multimode squeezing, offering potential for applications in quantum metrology.

Figures (5)

• Figure 1: (a) A schematic of a 1D array of two-level emitters, with transition wavevector $k_0$ and lattice spacing $a$, under external driving. The driving electric field vector $\textbf{E}$ is oriented along the chain, inducing the orientation of the atomic dipoles indicated by an arrow. The emitters are subject to coherent dipole-dipole interactions $V_{ij}$ and collective decay processes with rates $\Gamma_{ij}$, both of which are long-ranged. (b) The dispersion relation $V_k$ (solid line) and decay rates $\Gamma_k$ (dash-dot line) of collective single-particle modes for $N\rightarrow \infty$ emitters, as functions of momentum $k$, for the subwavelength lattice spacing $k_0 a =2$: the modes for $\left\lvert k \right\rvert > k_0$ -- in the non-grayed area -- correspond to subradiant modes with zero decay rates (in the thermodynamic limit), that are linearly decoupled from the far-field; the illustration instead depicts a nonlinear process, in which two drive photons scatter resonantly into a pair of subradiant modes. (c) The number of steady state excitations $n_c(k) = \langle {\sigma_k^+ \sigma_k^-} \rangle_c$ calculated in DMFT/NCA for $N\rightarrow \infty$, as a function of momentum and drive detuning $\Delta$ (the dispersion relations is superimposed in yellow). Here the drive strength is $\Omega/\Gamma = 0.8$, the lattice spacing is the same as in (b). It shows that the subradiant modes are resonantly populated.
• Figure 2: Pair-correlations function of the steady state computed in DMFT/NCA, for the driving conditions illustrated in Fig. \ref{['fig:fig1']} and with a drive detuning of $\Delta/\Gamma=0.4$. The subscript "c"("d") indicates the connected (disconnected) component of such function, describing its nonlinear (linear) component. (a) The connected function as a function of momentum $k$ for different drive strengths $\Omega$: it shows multimode squeezing correlations of mode-pairs with momenta $(-k,k)$. Decreasing the drive strength it becomes more sharply peaked around the resonant modes. (b) The non-local component of the connected function in real space, where $i-j$ is the lattice-sites distance, showing long-range correlations. (a) and (b) also show that correlations increase in magnitude as the drive is decreased, as a non-trivial interplay of reduced occupations -- bounding correlations --, and reduced heating. (c) The ratio $W_{\rm subrad}$ (with a minus sign) between the connected component summed over the subradiant modes $c_{c,\rm subrad} = \underset{{|k|>k_0}}{\sum}\left\langle\sigma_k^{-} \sigma_{-k}^{-}\right\rangle_c/N$ and the disconnected one $c_{\rm d} = \left\langle\sigma_{k=0}^{-}\right\rangle\left\langle\sigma_{k=0}^{-}\right\rangle/N$, showing that the nonlinear contribution to the steady-state is far from a small perturbation.
• Figure S1: DMFT error as a function of number of DMFT iterations, starting close to the previously found solution at $k_0=2$ and $\Omega=0.3, \Delta=0$. Here $dt=0.1$ and $t_{\rm max}=200$.
• Figure S2: The local atomic Keldysh Green's function defined as in Fig. \ref{['fig:keldysh']} computed from a noninteracting theory for $\Omega\rightarrow 0$ (dashed line) and in DMFT (solid), for $k_0a =\pi$ and $\Delta=0$. This deviates from a Lorentzian shape for a single atom due the dipolar interactions between the emitters.
• Figure S3: The local atomic Keldysh Green's function ${\rm i} \chi_{11}^K(\tau) = \lim_{t\rightarrow\infty} \langle { \left\lbrace \sigma^-(t+\tau),\sigma^+(t) \right\rbrace } \rangle - \left\lbrace \langle { \sigma^-(t+\tau)} \rangle,\langle { \sigma^+(t) } \rangle \right\rbrace$ computed from a noninteracting theory for $\Omega\rightarrow 0$ and in DMFT in real time (a) and frequency (b), for the same parameters as in Fig. \ref{['fig:lowDrive']} (c).