A constrained Dekker oscillator as a quasi-qubit applied in Hadamard gate in non-equilibrium conditions

We consider a squeezed harmonic oscillator (SHO) dissipated as a Dekker oscillator and exposed to an environment under thermal non-equilibrium conditions (TNEC). The eigenvalues and eigenvectors of the system are obtained to determine the thermal distribution. Using Tsallis statistics under TNEC, the oscillator is constrained to its two lowest energy levels as a quasi-qubit and its thermal state is represented as a quitrit based on a simple harmonic oscillator basis. The Hadamard operation on the environmental entanglement with quasi-qubit is analyzed. This analysis examines the impact of temperature, the frequencies of the system and the environment. The fidelity of the Dekker state with the superposition state is compared with that of its environmental entanglement. We find that reducing the entanglement with the environment increases the fidelity up to 99%, and this increase is directly linked to a reduction in system-environment entanglement. These results may be beneficial for the control of single-qubit gates exposed to environmental effects.