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arXiv:quant-ph/9608034 · Modern Physics Letters A

EIGENSTATES OF LINEAR COMBINATIONS OF TWO-BOSON CREATION AND ANNIHILATION OPERATORS: AN ALGEBRAIC APPROACH

P. SHANTA, S. Chaturvedi, V. SRINIVASAN

Published Sep 21, 1996• 6 citations
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Abstract

Eigenstates of the linear combinations a2+βa†2 and ab+βa†b†of two-boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett.72, 1447, (1994)] for constructing eigenstates of generalized annihilation operators. Expressions for the overlaps of these states with the number states, the coherent states and the squeezed states are given in a closed form.

DOI
10.1142/s021773239600237x

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Eigenstates of the linear combinations a2+βa†2 and ab+βa†b†of two-boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett.72, 1447, (1994)] for constructing eigenstates of generalized annihilation operators. Expressions for the overlaps of these states with the number states, the coherent states and the squeezed states are given in a closed form.
Eigenstates of the linear combinations a2+betaadagger2a^2+\\beta a^{\\dagger2}a2+betaadagger2 and $ab+\\beta a^\\dagger b^\\dagger$ of two boson creation and annihilation operators are presented. The algebraic procedure given here is based on the work of Shanta et al. [Phys. Rev. Lett. {\\bf 72}, 1447, 1994] for constructing eigenstates of generalized annihilation operators.
We introduce the inverse of the harmonic-oscillator creation and annihilation operators by their actions on the number states. We then show that three of the two-photon annihilation operators, a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^, a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$, and a^ 2^{2}2, possess eigenstates that are analogous to the often-used coherent and squeezed states. A family of the eigenstates of the operator a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^ is the customary squeezed vacuum S(\ensuremath{\sigma})\ensuremath{\Vert}0〉 while another family of the eigenstates of a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$ is the state S(\ensuremath{\sigma})\ensuremath{\Vert}n=1〉. We find some interesting properties related to squeezing, bunching, and antibunching exhibited by these states. An alternative method of summing some series by using these states is also considered. It is hoped that these states will find applications in quantum optics and quantum mechanics in general.
The k orthonormalized eigenstates of the k th power {( a q f ( N q ))} k ( k ⩾1) of the generalized q -boson annihilation operator a q f ( N q ) are obtained, and their properties are discussed. An alternative method to construct them is proposed, and it is shown that all of them can be expressed as a linear superposition of k q - f -coherent states that have the same amplitude but different phases. Physically, they can be generated by a linear superposition of the time-dependent q - f -coherent states at different instants.
In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (a†a^\daggera†) and annihilation (aaa) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator (a†x+ay)n(a^\dagger x+ay)^n(a†x+ay)n with nnn as the exponent and x, yx,\,yx,y are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.
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.