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Physical Review Research

Symmetry-driven zero-frequency-pulling lasing via superradiant sidebands for active optical clocks

Mingyu Jeon, Jinuk Kim, Kyungwon An

Published Apr 1, 2026• 0 citations
Publisher

Abstract

Active optical clocks based on collective atomic emission promise narrow lasing linewidths as well as reduced sensitivity to cavity fluctuations, yet residual frequency pulling limits their ultimate precision. Here, we propose a laser scheme that simultaneously achieves vanishing frequency pulling and ultranarrow linewidths by exploiting symmetry in collective superradiant emission. Atoms in a dense beam traversing a bad cavity are prepared in controlled superposition states using a pump laser with a sinusoidally modulated Rabi frequency Ω m . The resulting emission spectrum consists of a central lasing peak at the atomic resonance and two superradiant sidebands at symmetric detunings ± Ω m . Owing to their symmetric interaction, the central peak is protected against cavity-induced frequency shifts over detunings comparable to the cavity linewidth. As the atom number increases, its linewidth narrows far beyond the conventional gain-narrowing limit, reaching values as small as one millionth of the cavity linewidth. Small pump detunings break the symmetry of the sideband intensities, providing a robust error signal to stabilize the pump frequency without an external reference cavity. These results establish a symmetry-driven route toward ultrastable active optical clocks and collective-radiation sources with exceptional frequency stability.

DOI
10.1103/q6lc-f9b8

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Active optical clocks based on collective atomic emission promise narrow lasing linewidths as well as reduced sensitivity to cavity fluctuations, yet residual frequency pulling limits their ultimate precision. Here, we propose a laser scheme that simultaneously achieves vanishing frequency pulling and ultranarrow linewidths by exploiting symmetry in collective superradiant emission. Atoms in a dense beam traversing a bad cavity are prepared in controlled superposition states using a pump laser with a sinusoidally modulated Rabi frequency Ω m . The resulting emission spectrum consists of a central lasing peak at the atomic resonance and two superradiant sidebands at symmetric detunings ± Ω m . Owing to their symmetric interaction, the central peak is protected against cavity-induced frequency shifts over detunings comparable to the cavity linewidth. As the atom number increases, its linewidth narrows far beyond the conventional gain-narrowing limit, reaching values as small as one millionth of the cavity linewidth. Small pump detunings break the symmetry of the sideband intensities, providing a robust error signal to stabilize the pump frequency without an external reference cavity. These results establish a symmetry-driven route toward ultrastable active optical clocks and collective-radiation sources with exceptional frequency stability.
Numerical simulations based on the quantum Langevin equations have been performed for a large number of two-level atoms in a beam interacting with a low-Q cavity with the atomic initial superposition states close to the north pole of the Bloch sphere. When the pump Rabi frequency was modulated at ΔpaΔ_{pa}Δpa​ with zero pump-atom detuning for various cavity-atom detunings, we obtained a lasing peak at the atomic resonance and superradiant lasing peaks at ±Δpa\pmΔ_{pa}±Δpa​ simultaneously while the central peak exhibiting a zero frequency pulling coefficient. The linewidth of the central peak was reduced beyond the gain narrowing as the mean number of atoms was increased, resulting in a minimum linewidth as small as a millionth of the atomic or cavity linewdith. A pump carrier detuning caused asymmetric heights for the side superradiant peaks, the height difference of which can be used to lock the pump laser to the atom within the linewidth of the central lasing peak. Our results may lead to development of a new type of ultra-stable active optical clocks for future frequency standards when applied to proper atomic systems.
Background: Theoretical studies of superradiant lasing on optical clock transitions predict a superb frequency accuracy and precision closely tied to the bare atomic linewidth. Such a superradiant laser is also robust against cavity fluctuations when the spectral width of the lasing mode is much larger than that of the atomic medium. Recent predictions suggest that this unique feature persists even for a hot and thus strongly broadened ensemble, provided the effective atom number is large enough. Methods: Here we use a second-order cumulant expansion approach to study the power, linewidth and lineshifts of such a superradiant laser as a function of the inhomogeneous width of the ensemble including variations of the spatial atom-field coupling within the resonator. Results: We present conditions on the atom numbers, the pump and coupling strengths required to reach the buildup of collective atomic coherence as well as scaling and limitations for the achievable laser linewidth. Conclusions: We show how sufficiently large numbers of atoms subject to strong optical pumping can induce synchronization of the atomic dipoles over a large bandwidth. This generates collective stimulated emission of light into the cavity mode leading to narrow-band laser emission at the average of the atomic frequency distribution. The linewidth is orders of magnitudes smaller than that of the cavity as well as the inhomogeneous gain broadening and exhibits reduced sensitivity to cavity frequency noise.
Superradiant lasers promise continuous, narrow-linewidth coherent emission at the bare atomic transition frequency, enabling frequency references of exceptional precision. Recent experiments have advanced the field, but achieving truly continuous operation remains technically challenging. Here we propose an alternative route to an active optical frequency reference with fewer emitters using all-to-all dipole-dipole interactions mediated by a nanophotonic waveguide. We show that selectively pumping only a sub-ensemble of emitters, rather than the full ensemble, substantially improves emission characteristics. The collective interactions with unpumped emitters provide narrowband frequency selection and establish an effective feedback mechanism analogous to the role of a macroscopic cavity. We find directional superradiant emission with strongly phase-synchronized emitter correlations and a narrow output spectrum close to the bare emitter resonance. Our results demonstrate a strong metrological gain from selective partial pumping of quantum emitters with the second-order intensity correlation g(2)(0)≃1g^{(2)}(0)\simeq 1g(2)(0)≃1, indicating reduced equal-time intensity fluctuations, and open a route to waveguide-based optical frequency references using small clock-atom ensembles for chip-scale precision metrology.
Abstract The implementation of a superradiant laser as an active frequency standard is predicted to provide better short-term stability and robustness to thermal and mechanical fluctuations when compared to standard passive optical clocks. However, despite significant recent progress, the experimental realization of continuous wave superradiant lasing still remains an open challenge as it requires continuous loading, cooling, and pumping of active atoms within an optical resonator. Here we propose a new scenario for creating continuous gain by using optical forces acting on the states of a two-level atom via bichromatic coherent pumping of a cold atomic gas trapped inside a single-mode cavity. Analogous to atomic maser setups, tailored state-dependent forces are used to gather and concentrate excited-state atoms in regions of strong atom-cavity coupling while ground-state atoms are repelled. To facilitate numerical simulations of a sufficiently large atomic ensemble, we rely on a second-order cumulant expansion and describe the atomic motion in a semi-classical point-particle approximation subject to position-dependent light shifts which induce optical gradient forces along the cavity axis. We study minimal conditions on pump laser intensities and detunings required for collective superradiant emission. Balancing Doppler cooling and gain-induced heating we identify a parameter regime of a continuous narrow-band laser operation close to the bare atomic frequency.
Active atomic clocks are predicted to provide far better short-term stability and robustness against thermal fluctuations than typical feedback-based optical atomic clocks. However, continuous laser operation using an ensemble of clock atoms still remains an experimentally challenging task. We study spatial self-organization in a transversely driven ensemble of clock atoms inside an optical resonator and coherent light emission from the cavity. We focus on the spectral properties of the emitted light in the narrow atomic linewidth regime, where the phase coherence providing frequency stability is stored in the atomic dipoles rather than the cavity field. The atoms are off-resonantly driven by a standing-wave coherent laser transversely to the cavity axis, allowing for atomic motion along the cavity axis as well as along the pump. In order to treat larger atom numbers we employ a second-order cumulant expansion which allows us to calculate the spectrum of the cavity light field. We identify the self-organization threshold where the atoms align themselves in a checkerboard pattern, thus maximizing light scattering into the cavity, which simultaneously induces cooling. For a larger driving intensity, more atoms are transferred to the excited state, reducing cooling but increasing light emission from the excited atoms. This can be enhanced via a second cavity mode at the atomic frequency spatially shifted by a quarter wavelength. For large enough atom numbers we observe laser-like emission close to the bare atomic transition frequency.
An ideal superradiant laser on an optical clock transition of noninteracting cold atoms is predicted to exhibit an extreme frequency stability and accuracy far below mHz-linewidth. In any concrete setup sufficiently many atoms have to be confined and pumped within a finite cavity mode volume. Using a magic wavelength lattice minimizes light shifts and allows for almost uniform coupling to the cavity mode. Nevertheless, the atoms are subject to dipole-dipole interaction and collective spontaneous decay which compromises the ultimate frequency stability. In the high density limit the Dicke superradiant linewidth enhancement will broaden the laser line and nearest neighbor couplings will induce shifts and fluctuations of the laser frequency. We estimate the magnitude and scaling of these effects by direct numerical simulations of few atom systems for different geometries and densities. For Strontium in a regularly filled magic wavelength configuration atomic interactions induce small laser frequency shifts only and collective spontaneous emission weakly broadens the laser. These interactions generally enhance the laser sensitivity to cavity length fluctuations but for optimally chosen operating conditions can lead to an improved synchronization of the atomic dipoles.