Ultranarrow Superradiant Lasing by Dark Atom-Photon Dressed States

We show that incoherent pumping of an optical lattice clock system with ultracold strontium-88 atoms produces laser light with a $\simeq 10\mathrm{Hz}$ linewidth when the atoms are exposed to a magnetic field. This linewidth is orders of magnitude smaller than both the cavity linewidth and the incoherent atomic decay and excitation rates. The narrow lasing is due to an interplay of multiatom superradiant effects and the coupling of bright and dark atom-light dressed states by the magnetic field.

Figure 1

Panel (a) illustrates thousands of ${}^{88}\mathrm{Sr}$ atoms trapped in an optical lattice inside an optical cavity. Panel (b) shows the ground state (${}^1{S}_0$) and the three excited states (${}^3{P}_1$) of the $k$th atom, which are split due to a magnetic field. The atom is pumped (black solid arrows) to the highly excited state ${}^3{S}_1$ via the $e_{0,k}$ excited state and decays rapidly (dashed arrows) to the ${}^3{P}_1$ excited states $e_{\pm ,k}$, and returns to the ground state by spontaneous emission (wavy arrows) or by the coherent coupling to the cavity mode (red thin double-head arrows). Panel (c) shows the coherent coupling to the cavity in the basis of bright and dark atomic excited states. Panel (d) shows the symmetric states of $N$ atoms and the cavity field sharing a single excitation, and the corresponding atom-photon dressed eigenstates, which explain the three peaks in the transmission spectrum and in the steady-state lasing spectrum.

Figure 2

Dependence of the steady state of $2.5\times {10}^5$ atoms on the atomic excitation rate $\eta$ (relative to the decay rate $\gamma$). Panel (a) shows the population $⟨A_{gg}^k⟩$ (black solid line), $⟨A_{bb}^k⟩$ (red dashed line), $⟨A_{dd}^k⟩$ (blue dotted line) of the ground state, the bright excited state and the dark excited state, respectively, as well as the imaginary part of the bright-dark state’s coherence $⟨A_{bd}^k⟩$ (green dot-dashed line). Panel (b) shows the correlations $⟨A_{gb}^k{A}_{bg}^{k^{\prime }}⟩$ (black solid line), $⟨A_{gd}^k{A}_{dg}^{k^{\prime }}⟩$ (red dashed line), and $\mathrm{Im}⟨A_{gd}^k{A}_{bg}^{k^{\prime }}⟩$ (blue dotted line) of the atoms ($k\ne k^{\prime }$) related to the dark and bright transitions. In panels (a) and (b) the Zeeman splitting is $\mathrm{\Delta }/2\pi =0.3\mathrm{MHz}$. Panel (c) shows a representation of the multiply excited atomic states by pseudo-Dicke collective quantum numbers $J_s$, $M_s$ of the bright $s=b$ (red dots) and dark $s=d$ (green dots) transition for increasing $\eta$ (arrows) for the system with $\mathrm{\Delta }=0$ (left) and $\mathrm{\Delta }=2\pi \times 0.3\mathrm{MHz}$ (right). The results are shifted for clarity and the dashed lines indicate the boundaries of each triangle of Dicke ladders. Panel (d) shows the intracavity photon number as a function of $\eta$ for different values of the Zeeman coupling $\mathrm{\Delta }$. Other parameters are specified in the text.

Figure 3

Magnetic field-controlled subradiance, superradiance, and superradiant lasing. Panels (a) and (b) show the normalized steady-state spectra for increasing pumping $\eta$ (relative to the decay rate $\gamma$), without ($\mathrm{\Delta }=0$) and with Zeeman coupling $\mathrm{\Delta }/2\pi =0.3\mathrm{MHz}$ for systems with $N=2.5\times {10}^5$ atoms [note that in the upper panel of (b) the frequency scale is in Hz]. (c) and (d) show the emission linewidth of the systems without and with Zeeman coupling as function of $\eta /\gamma$ ($N=2.5\times {10}^5$), and number of atoms $N$ ($\eta /\gamma =5$), respectively. In panel (c), the black dots and the red curves are the accurate results and those calculated with Eq. (2), respectively, and the black horizontal dotted line indicates the Purcell rate ${\mathrm{\Gamma }}_c$. Other parameters are specified in the text.