On the Bandgap quantum coupler and the harmonic oscillator interacting with a reservoir: Defining the relative phase gate

In order to be able to study dissipation, the interaction between a single system and their environment was introduced in quantum mechanics. Master and quantum Langeving equations was derived and, also, decoherence was studied using this approach. One of the most used model in this field is a single harmonic oscillator interacting with a reservoir. In this work we solve analytically this problem in the resonance case with the evolution operator method. We use this result to study the conditional dynamics of a finite system of coupling, a bandgap quantum coupler. We study the conditional dynamics of the coupler on the computational basis by choosing a proper interaction time. This conditional dynamics provides a distinct realization of a quantum phase gate, which we name the relative phase gate.