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Combinatorics and Boson Normal Ordering: a Gentle Introduction

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Combinatorics and Boson Normal Ordering: a Gentle Introduction Combinatorics and Boson normal ordering: A gentle introduction P. Blasiak∗ and A. Horzela† H. Niewodnicza´nski Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL 31342 Krak´ow, Poland K. A. Penson‡ and A. I. Solomon§ Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, Universit´ePierre et Marie Curie, CNRS UMR 7600, Tour 24 - 2i`eme ´et., 4 pl. Jussieu, F 75252 Paris Cedex 05, France G. H. E. Duchamp¶ Institut Galil´ee, LIPN, CNRS UMR 7030, 99 Av. J.-B. Clement, F-93430 Villetaneuse, France 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 frame- work 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 given herein to a wide class of operators. I. INTRODUCTION is called normal ordering.4,5,6,7,8 Although the process is clear and straightforward, in practice it can be tedious and cumbersome when the expression is complicated, and Hilbert space constitutes the arena where quantum is even less tractable when we consider operators defined phenomena can be described. One common realization is by an infinite series expansion. It is thus desirable to Fock space, which is generated by the set of orthonormal find manageable formulas or guiding principles that lead vectors n representing states with a specified numbers of to solutions of normal ordering problems. particles| ori objects. A particular role in this description is played by the creation a† and annihilation a operators In this paper we present a general framework that is representing the process of increasing and decreasing the applicable to a broad class of ordering problems. It ex- number of particles in a system, respectively. We con- ploits the fact that the coefficients emerging in the nor- sider operators that satisfy the boson commutation rela- mal ordering procedure appear to be natural numbers tion [a,a†] = 1 describing objects obeying Bose-Einstein which have their origin in combinatorial analysis. In the statistics, for example, photons or phonons. The fact simplest case of powers or the exponential of the num- † that the operators a and a† do not commute is proba- ber operator N = a a these are Stirling and Bell numbers 9 bly the most prominent characteristic of quantum the- which enumerate partitions of a set. We use this example ory, and makes it so strange and successful at the same to illustrate a systematic approach to the ordering prob- time.1,2 lem. The general methodology is to identify the problem with combinatorial structures and then resolve it using In this paper we are concerned with the order- ing problem which is one of the consequences of non- this identification. The solution may be found with the Dobi´nski relation 10,11 commutativity. This problem derives from the fact that help of the , which is a very effective the order in which the operators occur is relevant, for ex- tool and is straightforwardly applicable to a wide range ample, a†a = aa† = a†a + 1. The ordering problem plays of ordering problems. 6 As a byproduct of this methodology we obtain a sur- arXiv:0704.3116v1 [quant-ph] 24 Apr 2007 an important role in the construction of quantum me- chanical operators. The physical properties of differently prising relation between combinatorial structures and op- ordered operators may be quite distinct, which we can erator ordering procedures. This relation is especially in- see by considering their expectation values. An analysis teresting because the objects involved in the problem can of operator matrix elements reveals their physical prop- have clear combinatorial interpretations (for example, as erties observed as probabilities. There are two sets of partitions of a set). The expectation is that this remark- states of primary interest in this context: number states able interrelation will shed light on the ordering problem n and coherent states z . The latter, defined as eigen- and clarify the meaning of the associated abstract oper- states| i of the annihilation| i operator a, play an important ator expressions. role in quantum optics3,4,5,6,7 and in the phase space for- The framework we present is an example of a fertile in- mulation of quantum mechanics.8 terplay between algebra and combinatorics in the context The calculation of the number or coherent state expec- of quantum mechanics. It employs only undergraduate tation values reduces to transforming the original expres- algebra and is not as yet a standard feature of quantum sion to the normally ordered form in which all annihila- mechanics textbooks. tion operators are to the right. In this form the evalua- The paper is organized as follows. Section II briefly re- tion of the matrix elements is immediate. The procedure calls the concept of Fock space and introduces the normal 2 ordering problem. The main part containing the connec- These states are normalized, z z = 1, but are not or- tion to combinatorics is given in Sec. III. It illustrates thogonal and constitute an overcompleteh | i basis in the the methodology by discussing in detail the solution of Hilbert space.12 Coherent states have many useful prop- a generic example. Some applications are provided in erties which are exploited in quantum optics and in other Sec. IV. In Sec. V we point out extensions of this ap- areas of physics.3,4,5,6,7,8 proach and suggest further reading. B. Normal ordering: Introduction II. OCCUPATION NUMBER REPRESENTATION The noncommutativity of the creation and annihilation operators causes serious ambiguities in defining operator A. States and operators functions in quantum mechanics. To solve this problem the order of the operators has to be fixed. An important We consider a pair of one mode boson annihilation practical example of operator ordering is the normally a and creation a† operators satisfying the conventional ordered form in which all annihilation operators a stand boson commutation relation to the right of the creation operators a†. We now define two procedures on boson expressions yielding a normally [a,a†]=1. (1) ordered form, namely, normal ordering and the double dot operation.4,5,6,7,8 † The operators a,a , and 1 generate the Heisenberg alge- By the normal ordering of a general expression F (a†,a) bra. we mean F (n)(a†,a) which is obtained by moving all the The occupation number representation arises from the annihilation operators a to the right using the commuta- † interpretation of a and a as operators annihilating and tion relation of Eq. (1). This procedure yields an opera- creating a particle in a system. From this point of view tor whose action is equivalent to the original one, that is, the Hilbert space of states is generated by the number (n) † † H F (a ,a)= F (a ,a) as operators, although the form of states n , where n = 0, 1, 2,... counts the number of † | i the expressions in terms of a and a may be completely particles, or objects in general. The states are assumed different. to be orthonormal, m n = δ , and constitute a basis † h | i m,n The double dot operation :F (a ,a): consists of apply- in . This representation is usually called Fock space. ing the same ordering procedure but without taking into H † The operators a and a satisfying Eq. (1) may be re- account the commutation relation of Eq. (1), that is, alized in Fock space as moving all annihilation operators a to the right as if they commuted with the creation operators a†. This nota- a n = √n n 1 , a† n = √n +1 n +1 . (2) | i | − i | i | i tion, although widely used, is not universal.13 We observe that in general this procedure yields a different operator The number operator N, which counts the number of † † 14 particles in a system, is defined by F (a ,a) = :F (a ,a): . In addition6 to the fact that these two procedures yield N n = n n , (3) different results (except for operators that are already in | i | i normally ordered form), there is also a practical differ- and is represented as N = a†a. It satisfies the commuta- ence in their use. That is, although the application of tion relations the double dot operation is almost immediate, for the normal ordering procedure some algebraic manipulation [a,N]= a, [a†,N]= a†. (4) † − of the non-commuting operators a and a is needed. Here is an example of both procedures in action: The algebra defined by Eqs. (1) and (4) describes objects normal ordering obeying Bose-Einstein statistics, for example, photons or aa†aaa†a (a†)2a4 +4 a†a3 +2 a2 phonons. It is sometimes called the Heisenberg-Weyl al- −−−−−−−−−−−→[a,a†]=1 gebra, and occupies a prominent role in quantum optics, a† to the left, a to the right condensed matter physics, and quantum field theory. | {z } The second set of states of interest in Fock space are aa†aaa†a double dot a†a†aaaa. the coherent states z . They are defined as the eigen- † | i −−−−−−−−−−→a, a commute z }| { states of the annihilation operator (like numbers) a z = z z , (5) In general we say that the normal ordering problem for | i | i F (a†,a) is solved if we can find an operator G(a†,a) for where z is a complex number (the dual relation is z a† = which the following equality is satisfied z∗ z ).
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