Cavity-Free Δ-Type Coherent Population Trapping for Microwave Sensing

We investigated experimentally and theoretically a cavity-free microwave field that couples the two ground states of a Λ-type atomic system, thereby forming a closed Δ configuration. In this regime, the absence of cavity-imposed phase matching leads to a strong sensitivity of the ground-state coherence to the microwave field parameters. We observe that the coherent population trapping (CPT) resonance exhibits a pronounced dependence on the microwave power and detuning, resulting in measurable changes in resonance contrast, linewidth, and center frequency. To explain these effects, we develop a numerical density-matrix model in which the ground-state coherence explicitly incorporates the microwave coupling strength, capturing the essential physics of this no-phase-matching Δ system. The excellent agreement between theory and experiment establishes a simple and robust framework for microwave control of cavity-free Δ-type atomic systems, with direct implications for compact atomic clocks and quantum-enhanced quantum sensing platforms.