Quantum light by atomic arrays in optical resonators

Light scattering by a periodic atomic array is studied when the atoms couple with the mode of a high-finesse optical resonator and are driven by a laser. When the von-Laue condition is not satisfied, there is no coherent emission into the cavity mode, and the latter is pumped via inelastic scattering processes. We consider this situation and identify conditions for which different nonlinear optical processes can occur. We show that these processes can be controlled by suitably tuning the strength of laser and cavity coupling, the angle between laser and cavity axis, and the array periodicity. We characterize the coherence properties of the light when the system can either operate as degenerate parametric amplifier or as a source of antibunched light. Our study permits us to identify the individual multiphoton components of the nonlinear optical response of the atomic array and the corresponding parameter regimes, thereby in principle allowing one to control the nonlinear optical response of the medium.

Figure 1

(a) An array of atoms, with interparticle distance $d$, is confined along the axis of a standing-wave optical cavity at frequency ${\omega }_c$ and is transversally driven by a laser, whose wave vector forms the angle $\Theta$ with the cavity axis. The atomic internal transition and the relevant frequency scales are given in (b), with $|1\rangle$ and $|2\rangle$ ground and excited state of an optical transition with frequency ${\omega }_0$ and natural linewidth $\gamma$. The frequencies ${\omega }_z={\omega }_0-{\omega }_p$ and ${\delta }_c={\omega }_c-{\omega }_p$ denote the detunings between the laser frequency ${\omega }_p$ and the atomic and cavity frequency, respectively. The other parameters are the laser Rabi frequency $\Omega$, the atom-cavity coupling strength $g$, and the decay rate $\kappa$ of the optical cavity.

Figure 2

Schematic diagram of transitions which fulfill the phase-matching condition till fourth order. State $|\tilde{n}\rangle$ denote the number state of the polariton mode ${\gamma }_1$. The blue (light gray) arrows denote the laser-induced couplings and the red (dark gray) arrows denote the creation (annihilation) of polaritons due to the coupling with the cavity field. See text for a detailed discussion.

Figure 3

Squeezing spectrum for the maximum squeezed quadrature when $k=G/2$, $Q^{\prime }=G/2$, and $Q^{\prime }\ne Q$ for $\varphi ,{\phi }_L=0$. The parameters are $\Omega =200\kappa$, ${\omega }_z={10}^3\kappa$, and $N=10,50,100$ atoms (from top to bottom) for (a) $g=4\kappa$ and (b) $g=10\kappa$. The detuning ${\delta }_c$ is chosen such that $\delta {\omega }_1$ is zero. The curves are evaluated from Eq. (39) by numerically calculating the density matrix of the polariton field for a dissipative dynamics, whose coherent term is governed by the effective Hamiltonian in Eq. (28). The value $g=4\kappa$ is consistent with the experimental data of Ref. [29].

Figure 4

(a) Second order correlation function at zero time delay $g^{\left(2\right)}\left(0\right)$ vs ${\omega }_p$ (in units of $\kappa$) when the cavity is solely pumped by inelastic processes (here $k=G/2$, $Q^{\prime }\ne Q,3Q$ and $Q^{\prime }=G^{\prime }/2$ for $\varphi ,{\phi }_L=0$). The correlation function is evaluated numerically solving the master equation for the polaritons in presence of cavity decay, with the coherent dynamics given by Hamiltonian (A1) (solid line) and by Hamiltonian (28) (dashed line). (b) Corresponding average number of intracavity photons $\langle a^{†}a\rangle$ (blue upper line) and spin wave occupation $\langle b_Q^{†}{b}_Q\rangle$ (red lower line). (c) Ratios $|\chi /\delta {\omega }_1|$ (blue upper line) and $|\alpha /\delta {\omega }_1|$ (red lower line) vs ${\omega }_p$. The parameters are $g=80\kappa$, $\Omega =30\kappa$, ${\omega }_z-{\delta }_c=70\kappa$, and $N=2$ atoms. At the minimum of $g^{\left(2\right)}\left(0\right)$, ${\omega }_z\simeq 137\kappa$ and ${\delta }_c\simeq 67\kappa$.