Effect of classical noises on the coherent population trapping based on the Green’s function approach to the multiplicative stochastic processes

This paper introduces a mathematical framework for analyzing dynamical systems subjected to multiplicative classical stochastic noise, drawing inspiration from Green's function (GF) techniques used in quantum field theory and many-body physics. The researchers model the system's evolution as a set of stochastic differential equations (SDEs) where noise terms multiply the system variables. By expanding the state vector into an infinite perturbative series, they express the interacting (noisy) Green's function in terms of the noninteracting (noiseless) Green's function. This approach allows for the systematic inclusion of various broadening mechanisms, such as atomic collisions, Doppler effects, and laser-induced phase and amplitude fluctuations, into a single model.

The authors apply this formalism to CPT, a quantum interference phenomenon essential for the operation of chip-scale atomic clocks. In CPT, atoms are driven by two optical fields to form a 'dark state,' rendering the atomic medium transparent. The researchers demonstrate that the performance of CPT-based clocks is highly sensitive to the noise characteristics of the driving laser. Their model allows for the calculation of the CPT transmission resonance lineshape under the influence of both white and colored noise. For white noise, the perturbative series converges to an exact closed-form solution. For colored noise, they show that the formalism provides a reliable approximation when the noise bandwidth exceeds the system's damping rate.

As miniaturized atomic clocks become increasingly important for portable technology, understanding the impact of laser noise on clock stability is critical. This work provides a rigorous, versatile tool for researchers to quantify how different noise sources—specifically those arising from low-cost, miniaturized laser sources—degrade the precision of atomic frequency standards. Because the formalism is applicable to any dynamical system governed by multiplicative stochastic processes, it serves as a general-purpose method for analyzing decoherence in various quantum technological devices beyond just CPT-based clocks.