A $$\mathcal{PT}$$ PT -symmetric simple harmonic oscillator

Nadjat Amaouche, Ishak Bouguerche, Rahma Zerimeche +1 more

Rahma Zerimeche, Rostom Moufok, Nadjat Amaouche +1 more

It is known that the standard and the inverted harmonic oscillator are different. Replacing thus ω by ±iω in the regular oscillator is necessary going to give the inverted oscillator H^{r}. This replacement would lead to anti- PT-symmetric harmonic oscillator Hamiltonian (∓iH^{os}). The pseudo-hermiticity relation has been used here to relate the anti-PT-symmetric harmonic Hamiltonian to the inverted oscillator. By using a simple algebra, we introduce the ladder operators describing the inverted harmonic oscillator to reproduce the analytical solutions.We construct the inverted coherent states which minimize the quantum mechanical uncertainty between the position and the momentum. This paper is dedicated to the memory of Omar Djemli and Nouredinne Mebarki who died due to covid 19.

Jonas F. G. Santos, Fabricio S Luiz

Abstract We present results concerning aspects of quantum thermodynamics under the background of non-Hermitian quantum mechanics for the dynamics of a quantum harmonic oscillator. Since a better control over the parameters in quantum thermodynamics processes is desired, we use concepts from collisional model to introduce a simple prototype of thermal reservoir based on PT -symmetric Hamiltonians and study its effects under the thermalization process of a single harmonic oscillator prepared in a displaced thermal state. We verify that controlling the PT -symmetric features of the reservoir allows to reverse the heat flow between system and reservoir, as well as to preserve the coherence over a longer period of time and reduce the entropy production. Furthermore, we considered a modified quantum Otto cycle in which the standard hot thermal reservoir is replaced by the thermal reservoir based on PT -symmetric Hamiltonians. By defining an effective temperature depending on the PT -symmetric parameter, it is possible to interchange the quantum Otto cycle configuration from engine to refrigerator by varying the PT -symmetric parameter. Our results indicate that PT -symmetric effects could be useful to achieve an improvement in quantum thermodynamics protocols such as coherence protection and entropy production reduction.

Lawrence R. Mead, Sungwook Lee, David Garfinkle

In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, C, for this system. All calculations are performed analytically and without approximation.