Single-photon detection is an energy quantum limit detection (EQLD) of extremely weak electromagnetic waves. Given that the sensitivity of the conventional electromagnetic induction microwave receiver working at room temperature is very limited, as a result of the significantly strong thermal noise, here we analyze the possibility of approaching the EQLD of the weak microwave signal by using a current-biased Josephson junction (CBJJ) detector. By numerically simulating the dynamics for the phase particle of the CBJJ, we propose an approach to describe the discriminability between the probabilistically escaped events of the phase particle with and without the microwave current driving, by measuring the minimum
dKC index. Our predictions indicate that the experimentally demonstrated CBJJ detector possesses the ability to resolve about a dozen photons. Furthermore, with the minimum detectable signal strength and the saturation threshold, we discuss how to determine the dynamic range of the proposed CBJJ detector. The feasibility of using the proposed CBJJ detector to achieve the EQLD in a microwave band is also discussed.