The paper reports measurements of birefringence in a single atom strongly coupled to a high-finesse optical resonator, observing nonlinear phase shifts even for intracavity photon numbers much less than one. The authors propose using these conditional phase shifts to implement quantum logic via a quantum-phase gate (QPG). Within a simple model for field transformation, they determine the parameters of the "truth table" for the QPG. The experiment demonstrates conditional dynamics at the single-photon level between two frequency-distinct fields in an optical resonator, with phase shifts conditioned on the intensity of a pump beam. The measured conditional phase shifts are around 16° per intracavity photon. The paper also discusses the potential of these capabilities for quantum nondemolition measurements and quantum cryptography, and explores the implementation of a QPG using single-photon pulses. The authors emphasize the significance of the measured parameter Δ, which represents the strength of the dispersive nonlinear interaction between intracavity fields, and propose operational strategies to evaluate the system's potential for quantum logic, including coherence and entanglement generation.The paper reports measurements of birefringence in a single atom strongly coupled to a high-finesse optical resonator, observing nonlinear phase shifts even for intracavity photon numbers much less than one. The authors propose using these conditional phase shifts to implement quantum logic via a quantum-phase gate (QPG). Within a simple model for field transformation, they determine the parameters of the "truth table" for the QPG. The experiment demonstrates conditional dynamics at the single-photon level between two frequency-distinct fields in an optical resonator, with phase shifts conditioned on the intensity of a pump beam. The measured conditional phase shifts are around 16° per intracavity photon. The paper also discusses the potential of these capabilities for quantum nondemolition measurements and quantum cryptography, and explores the implementation of a QPG using single-photon pulses. The authors emphasize the significance of the measured parameter Δ, which represents the strength of the dispersive nonlinear interaction between intracavity fields, and propose operational strategies to evaluate the system's potential for quantum logic, including coherence and entanglement generation.