Generation of an optical Schr\"odinger-cat-like state in a non-ideal cavity by injecting opposite-phase atomic dipoles

We propose a method to generate an optical Schr\"odinger-cat-like state in a cavity in a substantial decoherence regime. Even when the cavity decay rate is considerably large, a cat-like state can be generated in a laser-like setting if the gain for the field is larger than the loss. Under the condition that opposite-phase atomic dipoles repeatedly traverse the cavity, the cavity field converges to a squeezed vacuum state in a steady state. A Schr\"odinger-cat-like state is then generated when a single photon decay occurs. The phase-space distribution of the cat state can be revealed in homodyne detection by using the decaying photon as a herald event. Quantum trajectory simulation was used to identify the conditions for the Schr\"odinger-cat-like state formation as well as to analyze the properties of those states. Based on these simulations, possible experiments are proposed within the reach of the current technology.