Theoretical simulation of double resonance phenomenon in ladder-type Rydberg electromagnetically induced transparency

The double resonance phenomenon of electromagnetically induced transparency (EIT) in a ladder-type Rydberg atomic system is theoretically investigated. Based on the cesium <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>6</mml:mn> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mn>6</mml:mn> <mml:msub> <mml:mi>P</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>5</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> transition, we establish a density-matrix model incorporating the Doppler effect and simulate the absorption spectra under counter-propagating laser configurations. The numerical approach is adopted: a two-velocity-group approximation (Lorentzian line-shape superposition). The results show that when the probe detuning <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:math> is fixed and nonzero, scanning the coupling laser frequency yields two symmetric transparency peaks separated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:math> , in excellent agreement with recent experimental observations. The two-velocity-group approximation is computationally efficient and physically transparent, while the high-precision integration confirms its validity. We also reveal that conventional uniform velocity averaging fails to resolve the narrow EIT peaks due to insufficient sampling, and propose effective numerical strategies. This work provides a reliable theoretical tool for simulating Rydberg EIT spectra and offers insights into Doppler-induced double resonance, which may find applications in laser frequency stabilization and self-calibration of Rydberg atom-based sensors.