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arXiv:quant-ph/0607074

Noncrossing normal ordering for functions of bosons

Toufik Mansour, Simone Severini B

Published Jul 11, 2006• 3 citations
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Abstract

Normally ordered forms of functions of boson operators are important in many contexts mainly concerning quantum field theory and quantum optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento 10(13):565--567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure [see P. Blasiak, quant-ph/0507206]. In this paper, we apply the linear representation of noncrossing partitions to contractions of normally ordered forms. In this way, we define the notion of noncrossing normal ordering. This appear to be a natural mathematical concept and it results indeed to be linked with well-known combinatorial objects. We explicitely give the noncrossing normally ordered form of the functions (a^{r}(a^{dagger})^{s})^n) and (a^{r}+(a^{dagger})^{s})^n, plus various special cases. We establish bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths. Physical applications of the noncrossing normal ordering are desiderata.

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Normally ordered forms of functions of boson operators are important in many contexts mainly concerning quantum field theory and quantum optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento 10(13):565--567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure [see P. Blasiak, quant-ph/0507206]. In this paper, we apply the linear representation of noncrossing partitions to contractions of normally ordered forms. In this way, we define the notion of noncrossing normal ordering. This appear to be a natural mathematical concept and it results indeed to be linked with well-known combinatorial objects. We explicitely give the noncrossing normally ordered form of the functions (a^{r}(a^{dagger})^{s})^n) and (a^{r}+(a^{dagger})^{s})^n, plus various special cases. We establish bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths. Physical applications of the noncrossing normal ordering are desiderata.
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.
We address a systematic combinatorial approach to the anti-normal ordering problem. In this way, we use the Stirling numbers and their generating function, the so-called Bell polynomials, together with the operational methods to anti-normal the operator eλa†a{e^{λ{a^†}a}}eλa†a. In fact, we exploit the new theorem given by Shähandeh et al. in a special case of anti-normal ordering.
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick’s theorem. The methodology can be straightforwardly generalized from the simple example we discuss to a wide class of operators.
We introduce the general method of converting a given function of boson operators into its s -ordered form. We state and prove a theorem representing the fact that any ordered expansion of some operator function might be considered as the combinatorial problem of counting the number of contractions. In this way, we reduce the general ordering problem of boson operators into a purely combinatorial one. We show the applications of the theorem through some generic examples from quantum optics, namely ordering of the Fock space projectors, together with some simple extensions of the previously given quantum optical lemmas.
We give a general expression for the normally ordered form of a function F[ŵ(a,a†)] where ŵ is a function of boson creation and annihilation operators satisfying [a,a†]=1. The expectation value of this expression in a coherent state becomes an exact generating function of Feynman-type graphs associated with the zero-dimensional quantum field theory defined by F(ŵ). This enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences. We give several examples of the use of this technique, including the applications to Kerr-type and superfluidity-type Hamiltonians.
We introduce the general method of converting a given operator function into its s -ordered form. We state and prove a theorem representing the fact that any ordered expansion of some operator function might be considered as the combinatorial problem of counting the number of contractions. This will also unify the two essentially distinct notions of 'taking an operator into some ordered form' and 'reordering, or formally, ordering an operator'. In this way, we reduce the general ordering problem into a purely combinatorial one. Finally, we show the application of the theorem through two generic examples from both quantum optics and field theory.