Phase diagram of lasing under correlated pump from GPU-accelerated Truncated Wigner dynamics

Problem Statement

Superradiant (SR) lasers offer a pathway to ultra-narrow linewidths and long-term phase stability by storing optical coherence in the atomic medium rather than the cavity field, thereby circumventing thermal fluctuations of cavity mirrors. However, current implementations face a fundamental trade-off regarding the incoherent pump required to maintain population inversion:

• Local Pumping: Requires a drive rate scaling linearly with the atom number $N$ to reach optimal lasing. This imposes severe photon recoil heating on the atomic medium, degrading performance.
• Fully Collective Pumping: Removes the $N$-scaling of the drive but restricts the system to a symmetry that limits the second-order coherence to $g^{(2)} \to 6/5$ (partial coherence) at the optimal operating point. Restoring full coherence ($g^{(2)} \to 1$) in this regime typically requires complex three-level atomic schemes.

Existing strategies, such as spatially selective pumping, rely on coherent dipole-dipole interactions to filter emission but still suffer from strong recoil heating due to the intense drive on the active subensemble. The central question addressed is whether an incoherent, spatially correlated pump can interpolate between these limits to simultaneously reduce the required drive strength (suppressing recoil) and maintain high optical coherence.

Methodology

To investigate systems beyond the reach of exact analytical solutions (which are limited to permutation-symmetric cases), the authors employ the Truncated Wigner Approximation (TWA).

• Model: A one-dimensional chain of $N$ two-level atoms coupled to a single "bad" cavity mode. The cavity field is integrated out, resulting in collective spin dissipation at rate $\Gamma = 4g^2/\kappa$.
• Correlated Pump: The atoms are driven by an incoherent pump with rates $w_{ij} = w / (|i-j| + 1)^\alpha$. The exponent $\alpha$ acts as a tunable parameter:
  • $\alpha = 0$: Fully collective pump ($w_{ij} = w$).
  • $\alpha \to \infty$: Local pump ($w_{ij} = w\delta_{ij}$).
  • $0 < \alpha < \infty$: Intermediate spatially correlated pumping.
• Numerical Approach: The TWA maps the quantum master equation onto stochastic trajectories. Because these trajectories are independent, the problem is ideally suited for GPU parallelism. The authors utilized NVIDIA GPUs (A16, V100, A100, H100, GH200) to evolve up to $N=10^4$ atoms with $2^{13} \approx 8,000$ trajectories, enabling a full scan of the steady-state phase diagram at a computational cost practical for large systems.

Key Contributions

1. GPU-Accelerated TWA for Non-Symmetric Systems: The paper establishes GPU-accelerated TWA as a practical tool for driven-dissipative spin systems that break permutation symmetry, allowing for the characterization of steady states in regimes where exact solutions are unavailable.
2. Decoupling of Laser Properties: The study demonstrates that the three defining properties of an SR laser—drive strength, optical coherence, and linewidth—can be decoupled and independently tuned via the pump correlation range $\alpha$.
3. Phase Diagram Construction: A comprehensive phase diagram is generated mapping the transition between thermal radiation, partially coherent light, and coherent lasing as a function of pump rate and correlation exponent $\alpha$.

Key Results

• Drive Strength Reduction: The drive strength required to reach the lasing threshold is determined by the largest eigenvalue of the pump-rate matrix. Relative to the standard local-pump case, the required rate is reduced by:
  • $\sim N^{1-\alpha}$ for $\alpha < 1$.
  • $\sim \log N$ for $\alpha = 1$.
  • $\sim \zeta(\alpha)$ (Riemann zeta function) for $\alpha > 1$.
    This reduction parametrically suppresses recoil heating. Notably, $\alpha=1$ corresponds to the far-field envelope of dissipative couplings in free space, suggesting that standard free-space emission already offers a $\log N$ reduction in required drive intensity compared to local pumping.
• Persistence of Ultra-Narrow Linewidth: The ultra-narrow linewidth characteristic of SR lasers (scaling as $\Gamma$, independent of $N$) persists for all values of $\alpha$. The linewidth is primarily inherited from the collective loss mechanism and shows minor sensitivity to the pump range.
• Coherence vs. Range Trade-off: While the linewidth is robust, coherence depends on the pump range.
  • At $\alpha = 0$ (collective), $g^{(2)} \to 6/5$.
  • As $\alpha$ increases, coherence improves. The paper finds that fully coherent emission ($g^{(2)} \to 1$) survives at least down to $\alpha \approx 1$.
  • Finite-size extrapolations for $N$ up to $10^4$ suggest that for $\alpha \ge 0.9$, the system approaches the coherent limit $g^{(2)}=1$ in the thermodynamic limit.
• Phase Diagram: The study identifies a regime where the system emits coherent light ($g^{(2)} \approx 1$) with reduced drive requirements. The transition from partially coherent to fully coherent light occurs roughly around $\alpha \lesssim 1$.

Significance and Claims

The paper claims that correlated pumping acts as a "knob" that allows experimentalists to trade drive intensity (and the associated recoil heating) against optical coherence without sacrificing the ultra-narrow linewidth.

• Theoretical Laboratory: The power-law pump is presented as a minimal theoretical laboratory to isolate the role of pump correlations, rather than a specific microscopic implementation.
• Practical Implication: The findings suggest that existing free-space emission schemes (which naturally correspond to $\alpha \approx 1$) can achieve a significant reduction in required pump intensity ($\sim \log N$) compared to local pumping, while maintaining high coherence, without the need for complex engineering of additional atomic levels or coherent interactions.
• Methodological Impact: The work validates GPU-accelerated TWA as a scalable method for studying large-scale, non-permutation-symmetric quantum systems, bridging the gap between small-scale exact simulations and macroscopic mean-field theories.

The authors remain modest regarding specific experimental realizations, noting that future work is required to map these idealized kernels to specific light-matter platforms, accounting for oscillatory phases, polarization dependence, and coherent dipole-dipole exchanges that may arise in realistic setups.