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Physical Review

Ordered Expansions in Boson Amplitude Operators

Kevin E. Cahill, Roy J. Glauber

Published Jan 25, 1969• 1,117 citations
Publisher

Abstract

The expansion of operators as ordered power series in the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required $c$-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values $s=+1,0,\ensuremath{-}1$, respectively, of an order parameter $s$. In terms of this convention it is shown that for bounded operators the coefficients are finite when $s>0$, and the series are convergent when $s>\frac{1}{2}$. For each value of the order parameter $s$, a correspondence between operators and $c$-number functions is defined. Each correspondence is one-to-one and has the property that the function $f(\ensuremath{\alpha})$ associated with a given operator $F$ is the one which results when the operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for $F$ are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the $P$ representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter $s$.

DOI
10.1103/physrev.177.1857

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The expansion of operators as ordered power series in the annihilation and creation operators aaa and a\ifmmode†\else\textdagger\fi{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\ifmmode†\else\textdagger\fi is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required ccc-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values s=+1,0,\ensuremath−1s=+1,0,\ensuremath{-}1s=+1,0,\ensuremath−1, respectively, of an order parameter sss. In terms of this convention it is shown that for bounded operators the coefficients are finite when s>0s>0s>0, and the series are convergent when s>12s>\frac{1}{2}s>21​. For each value of the order parameter sss, a correspondence between operators and ccc-number functions is defined. Each correspondence is one-to-one and has the property that the function f(\ensuremathα)f(\ensuremath{\alpha})f(\ensuremathα) associated with a given operator FFF is the one which results when the operators aaa and a\ifmmode†\else\textdagger\fi{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\ifmmode†\else\textdagger\fi occurring in the ordered power series for FFF are replaced by their complex eigenvalues \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathα∗{\ensuremath{\alpha}}^{*}\ensuremathα∗. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of \ensuremathδ\ensuremath{\delta}\ensuremathδ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the PPP representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter sss.
The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the PPP representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the PPP representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators aaa and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the PPP representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the PPP representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter sss assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for Res<0\mathrm{Res}<0Res<0, has bounded eigenvalues for Res=0\mathrm{Res}=0Res=0, and has infinite eigenvalues for s=1s=1s=1. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter sss is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to s=+1s=+1s=+1, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the PPP representation is appropriate.
A new integral representation is obtained for the star product corresponding to the s -ordering of the creation and annihilation operators. This parametric ordering convention introduced by Cahill and Glauber enables one to vary the type of ordering in a continuous way from normal order to antinormal order. Our derivation of the corresponding integral representation is based on using reproducing formulas for analytic and antianalytic functions. We also discuss a different representation whose kernel is a generalized function and compare the properties of this kernel with those of the kernels of another family of star products which are intermediate between the - and pq -quantization.
The structure of ordered expansions in powers of boson operators and of canonical operators and the dual problem of operator reconstruction from ordered moments is derived and applied to the complete Gaussian class of ordering. In particular, the interpolation lines between normal and antinormal ordering and between standard and antistandard ordering with Weyl symmetrical ordering in their centre are dealt with in detail. The auxiliary operators for expansions in symmetrical ordering are explicitly found in the Fock-state representation and in other different representations. General and specialized formulae are derived for different ordering of powers of linear combinations of boson and of canonical operators which involve Hermite polynomials of operators. The link between symmetrical ordering of powers of boson operators and of canonical operators is expressed by means of Jacobi polynomials. Some basic formulae of operator ordering and operator expansion are collected for convenient use in the appendix.
From previous work on ordering expansions of the boson number operator exp(- mu a + a) formulae are derived for normally and antinormally ordered forms of arbitrary boson number operator functions f(a + a) in terms of the Fourier transform g( lambda ) of f(x). When the antinormally ordered form f (a) (a + a) is applied to states belonging to the domain of definition of f(a + a), the result differs in general from the action of f(a + a) on these states or is not even defined. For the number operator eigenstates mod m) two criteria are derived which give conditions on the Fourier transform g( lambda ) ensuring validity of f (a) (a + a) mod m)=f(m) mod m). The general considerations are demonstrated by two examples for functions f.
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.
This study derives the normal ordering expansion of degenerate integral powers of the number operator, (a†a)_{n,λ}, using recurrence relations for the coefficients and an operator action on number states. Here a† and a are respectively the boson creation and annihilation operators. We also determine the inverse of this normal ordering expansion. By analyzing diagonal coherent state elements of the degenerate integral powers of the number operator, we establish a combinatorial identity which yields a Dobinski-like formula for the degenerate Bell numbers at a specific value, connecting degenerate quantum operator calculus with combinatorics.