Building the Quantum Mechanics of Time-Dependent Pseudo-Hermitian Hamiltonians.

• Grant number: 24/13689-1
• Support Opportunities: Regular Research Grants
• Start date: April 01, 2025
• End date: March 31, 2028
• Field of knowledge: Physical Sciences and Mathematics - Physics - General Physics
• Principal Investigator: Miled Hassan Youssef Moussa
• Grantee: Miled Hassan Youssef Moussa
• Host Institution: Instituto de Física de São Carlos (IFSC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

Our goal in this project is the construction of the quantum mechanics of time-dependent (TD) pseudo-Hermitian Hamiltonians and metric operators. i) A first step in this program was the extension to the TD scenario of the Mostafazadeh method, from 2002, which ensures the norm conservation of time-independent (TI) pseudo-Hermitian Hamiltonians and metric operators. ii) Another important step is to investigate the pseudo-hermiticity beyond the invariance of the Schrödinger equation for TD non-hermitian Hamiltonians which are PT-symmetric. Among other challenging steps to be implemented, we mention iii) the investigation of the processcalled PT-phase transition, well known in the context of the TI PT-symmetric Hamiltonians, bywhich the energy levels of the real spectrum of the Hamiltonian become degenerates into complex-conjugate pairs. This phase transition is a signature of the PT-symmetrical TI Hamiltonians, and it is essential to investigate this phase transition when considering the PT-symmetry breaking of TD pseudo-Hermitian Hamiltonians, or through item ii), when considering the breaking of more general symmetries than PT associated with the dynamics of the system.We also mention iv) the need for a comprehensive analysis of the energy cost for the constructionof TI or TD pseudo-Hermitian Hamiltonians generating dynamics which are invariant under PT-symmetry or others more general symmetries. We concluded in recent studies that the further awayfrom hermitality, the higher the energy associated with these Hamiltonians, which allows for intriguing phenomena such as the infinite squeezing degree of field states in finite times or the increment, in orders of magnitude, in the creation rate of photons via the pseudo-hermitian dynamical Casimir effect.v) The analysis of pseudo-hermiticity in the process of decoherence of quantum states, considering TD or TI Hamiltonians and metrics, is another topic of interest in recent literature, with which we can effectively contribute in view of the many works we have already presented on the theme.The vi) quantum simulacra is a completely new topic we are developing, which stems from theaction of non-unitary transformations to Hermitian Hamiltonians, which implies the redefinition ofthe metric of the problem, and consequently the redefinition of the observables and the measurement procedures, which demand the well-known weak measurements. Quantum simulacra result in the possibility of simulating, for example, interactions between effectively decoupled quantum systems, and consequently simulating the entanglement of states, which allows us to propose the development of simulacra based quantum information protocols.We must, finally, address vii) the phenomena of superradiance and superabsorption in dense atomicsamples, which are connected to the problem of convulsion in neuronal networks when modeled asconnected spin-bóson systems. The pseudo-Hermitian superradiance and superabsorption will also beaddressed. (AU)