We solve the problem of the temporal evolution of one of two-modes embedded in a same dissipative environment and investigate the role of the losses after the preparation of a coherent state in only one of the two modes. Based on current cavity QED technology, we present a calculation of the fidelity of a superposition of coherent states engineered in a bimodal high-Q cavity. Our calculation demonstrates that the engineered superposition retains coherence for large times when the mean photon number of the prepared mode is on the order of unity.
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