Bosons-Bosons Oscillator Dynamics in a Quantum Control System
Roberto P. L. Caporali
Roberto P. L. Caporali, Department of Mathematics for Applied Physics, Roberto Caporali, Imola, BO, Italy.
Manuscript received on 10 January 2025 | First Revised Manuscript received on 14 January 2025 | Second Revised Manuscript received on 24 January 2025 | Manuscript Accepted on 15 February 2025 | Manuscript published on 28 February 2025 | PP: 25-31 | Volume-14 Issue-3, February 2025 | Retrieval Number: 100.1/ijitee.B104914020125 | DOI: 10.35940/ijitee.B1049.14030225
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© The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: 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 specifically addresses the interaction between a bosonic field, represented by the Cooper pairs, and another bosonic field, which is the System of vibrational modes of phonons in the crystal, in the context of applications related to Quantum Controllers. We consider an interaction of the boson-boson type, which refers explicitly to the possible coupling between the phonon field and the bosonic field constituted by Cooper pairs. We develop the Hamiltonian of the entire system, which comprises 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 KushnerStratonovich 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. Ultimately, we obtain the control law governing the temporal evolution of the bosonic system and demonstrate its representability through the logic gates of a quantum control system.
Keywords: Quantum Controller, Bosons-Bosons Interaction, Quantum Kalman Filter, Quantum Kushner-Stratonovich Equation, Pauli Matrices.
Scope of the Article: Computer Science and Applications