Bosons-Bosons Oscillator Dynamics in a Quantum Control System

In this work, we developed a study of a set of Cooper pairs in a superconducting system and the control dynamics of the corresponding Quantum Control System. This paper deals specifically with the interaction in a crystal between a bosonic field given by the Cooper pairs and another bosonic field given by the System of vibrational modes of the phonons in the crystal, regarding applications related to Quantum Controllers. We consider an interaction of the bosons-bosons type, which specifically refers to the possible coupling between the phonon field and the bosonic field constituted by Cooper pairs. We develop the Hamiltonian of the whole system, giving the Hamiltonian of the Cooper Pairs, the Hamiltonian of the interacting phonon system, and the interaction Hamiltonian. In our study, we define, for the Quantum Control System relative to the bosons-bosons interaction, a Quantum Kalman Filter. We obtain, as a first result of this work, the expression that makes explicit the Quantum Kushner-Stratonovich equation in our case of interaction between Cooper pairs and phonons of the crystal lattice of a superconductor. It expresses the density of states ρ of the Cooper pairs as a function of time in the non-Markovian statistical system described by us. Furthermore, we obtain a second fundamental result of this work, expressing the Hamiltonian of the overall open system as a function of the Pauli matrix operator. This operator allows us to represent the same Hamiltonian of the open system described by us through the logic gates of a quantum controller. So, in the end, we obtain the control law of the temporal evolution of the bosonic system and the possibility of representing it through the logic gates of a quantum control system.