Quantum phase transition in $\Xi$-configuration Tavis-Cummings model driven by an electromagnetic field for a finite number of atoms

The quantum phase transitions of a three-level Tavis-Cummings model in the \Xi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi mathvariant="normal">Ξ</mml:mi> </mml:math> -configuration driven by a single-mode electromagnetic field in a QED cavity are studied for n_a=1,\,2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="0.167em"/> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , and 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>4</mml:mn> </mml:math> atoms. Although conventional observables like photon number and energy level populations suffice to locate quantum phase transitions, they often lack sensitivity to finite-size effects. Here, we show that the expectation values of the quadratic Casimir invariants associated with the atomic subsystems offer a more robust measure, exhibiting sharp signatures of criticality that are validated by fidelity calculations. Additionally, the von Neumann entropy is employed to map the entanglement landscape between matter and the quantized field. A key finding is that the external driving field acts as a control parameter that shrinks the normal phase region; as the driving intensity increases, the separable ground state vanishes, leading to the dominance of field-matter entanglement across the parameter space.