  Microwave pulses are used ubiquitously to control and measure qubits
fabricated on superconducting circuits. Due to continual environmental
coupling, the qubits undergo decoherence both when it is free and during its
interaction with the microwave pulse. As quantum logic gates are executed
through pulse-qubit interaction, we study theoretically the decoherence-induced
effects during the interaction, especially the variations of the pulse, under a
dissipative environment with linear spectral distribution. We find that a
transmissible pulse of finite width adopts an asymmetric multi-hump shape, due
to the imbalanced pumping and emitting rates of the qubit during inversion when
the environment is present. The pulse shape reduces to a solitonic pulse at
vanishing dissipation and a pulse train at strong dissipation. We give detailed
analysis of the environmental origin from both the perspectives of envelope and
phase of the propagating pulse.
