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Pseudo-Boson Coherent and Fock States

D. A. Trifonov

Published Feb 21, 2009• 0 citations
OpenAlex

Abstract

Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.

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Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.
We introduce creation and annihilation operators of pseudo-Hermitian fermions for two-level systems described by a 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.
In order to construct time-dependent pseudo-bosonic coherent states, first, we extend the non-Hermitian integrals of motion method to cases where the quantum systems are described by time-dependent non-Hermitian Hamiltonians; second, we introduce a pseudo-bosonic annihilation operator defined as a time-dependent non-Hermitian linear invariant. The pseudo-bosons operators are a pseudo-Hermitian extension of usual boson operators. In fact, they are obtained from the modification of usual boson commutation relations where the annihilation and creation operators are related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. Thus, the pseudo-bosonic coherent states are just obtained as eigenstates of the pseudo-bosonic annihilation operator. As an illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian and we compare the obtained results with those in the literature (Swanson model but time-independent).
A new class of non-Hermitian Hamiltonians with real spectrum, which are written as a real linear combination of su(2) generators in the form H = ωJ 3 + αJ -+ βJ + , α = β, is analyzed.The metrics which allows the transition to the equivalent Hermitian Hamiltonian is established.A pseudo-Hermitian supersymmetic extension of such Hamiltonians is performed.They correspond to the pseudo-Hermitian supersymmetric systems of the boson-phermion oscillators.We extend the supercoherent states formalism to such supersymmetic systems via the pseudo-unitary supersymmetric displacement operator method.The constructed family of these supercoherent states consists of two dual subfamilies that form a bi-overcomplete and bi-normal system in the boson-phermion Fock space.The states of each subfamily are eigenvectors of the boson annihilation operator and of one of the two phermion lowering operators.
Recently an f -deformed Fock space which is spanned by | n 〉 λ was introduced. These bases are the eigenstates of a deformed non-Hermitian Hamiltonian. In this contribution, we will use rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by | n 〉 λ bases, to a new one, spanned by extended two-parameters bases | n 〉 λ 1 , λ 2 . These bases are now the eigenstates of a non-Hermitian Hamiltonian H λ 1 ,λ 2 = a † λ 1 ,λ 2 a +½, where a † λ 1 , λ 2 and a are, respectively, the deformed creation and ordinary bosonic annihilation operators. The bases | n 〉 λ 1 , λ 2 are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states in our two-parameter Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states numerically.
Nonlinear pseudo-fermions of degree n ( n -pseudo-fermions) are introduced as (pseudo) particles with creation and annihilation operators b and a , b ≠ a †, obeying the simple nonlinear anticommutation relation ab + b n a n = 1. The ( n + 1)-order nilpotency of these operators follows from the existence of unique (up to a bi-normalization factor) a -vacuum. Supposing appropriate ( n + 1)-order nilpotent para-Grassmann variables and integration rules the sets of n -pseudo-fermion number states, 'right' and 'left' ladder operator bi-overcomplete sets of coherent states are constructed. Explicit examples of n -pseudo-fermion ladder operators are provided, and the relation of pseudo-fermions to finite level pseudo-Hermitian systems is briefly considered. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.