Fermionic coherent states for pseudo-Hermitian two-level systems

Abstract

We introduce creation and annihilation operators of pseudo-Hermitian fermions for two-level systems described by pseudo-Hermitian Hamiltonian with real eigenvalues. This allows the generalization of the fermionic coherent states approach to such systems. Pseudo-fermionic coherent states are constructed as eigenstates of two pseudo-fermion annihilation operators. These coherent states form a bi-normal and bi-overcomplete system, and their evolution governed by the pseudo-Hermitian Hamiltonian is temporally stable. In terms of the introduced pseudo-fermion operators the two-level system' Hamiltonian takes a factorized form similar to that of a harmonic oscillator.

A concise review of pseudobosons, pseudofermions, and their relatives

“…In 2007, in [20], an effective non self-adjoint hamiltonian describing a two level atom interacting with an electromagnetic field was analyzed in connection with pseudo-hermitian systems. We have shown in [19] that this model can be very naturally rewritten in terms of pseudo-fermionic operators, and that the structure previously described naturally arises.…”

Multi-mode entangled states represented as Grassmannian polynomials

“…(14) implies that any polynomial, which lives in the space of Grassmannian representatives, can be mapped to a state in the Hilbert space. Therefore, Assumimg X as an element of Grassmannian representative space, we can take X to its corresponding states by defining…”

“…This is due to the fact that these variables are useful mathematical tools for studying various problems in theoretical physics and quantum optics [7,[9][10][11][12][13][14]. Path integral over Grassmann variables plays an important role in modern field theory [15].…”

“…These variables, in the simplest case, describe the quantum mechanical system with two possible states; namely fermions. It is notable that, strictly within the framework of fermionic fields, the fermionic variables B Y. Maleki maleki@physics.tamu.edu which arise from Paulis exclusion principle anti-commute with each other, and hence, the physical nature of the fermions inevitably are described by the Grassmann variables [12,14,16]. Naturally, the generalization of these anti-commuting nilpotent variables is utilized in the study of multi-level quantum mechanical systems which are described in arbitrary but finite-dimensional Hilbert spaces [11,13,17,18].…”

Fermion Coherence Hamiltonians

“…The overcompleteness property of the set of fermion annihilation operator eigenstates has been proved in [14,15] using the Berezin integration rules for Grassmann variables. Extension of canonical CS to the case of pseudo-Hermitian fermions was performed in [4].…”