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Av. 99 7030, UMR CNRS Galil´ee, LIPN, Institut aoaor ePyiu herqed aMati`ere Condens´ la Th´eorique de Physique de Laboratoire aa obntrc n oo omlodrn:Agnl introduc gentle A ordering: normal Boson and Combinatorics edsusagnrlcmiaoilfaeokfroeao o operator for framework combinatorial general a discuss We .INTRODUCTION I. † NSUR70,Tu 4-2`m e. l use,F722Par 75252 F Jussieu, pl. 2i`eme ´et., 4 - 24 Tour 7600, UMR CNRS = a a † .Neoncank nttt fNcerPyis oihA Polish Physics, Nuclear of Niewodnicza´nski Institute H. and a 3,4,5,6,7 | .Teodrn rbe plays problem ordering The 1. + z a i † a h atr enda eigen- as deﬁned latter, The . † n annihilation and omi hc l annihila- all which in form ontcmuei proba- is commute not do n ntepaesaefor- space phase the in and l laz-azkwkeo12 L332Krak´ow, Poland 31342 PL 152, Eliasza-Radzikowskiego ul. 8 a lya important an play , .A Penson A. K. .Blasiak P. a operators .H .Duchamp E. H. G. order- ∗ ‡ n .Horzela A. and n .I Solomon I. A. and al h ocp fFc pc n nrdcstenormal the introduces and space Fock of concept the calls quantum of feature standard a yet undergraduate textbooks. as mechanics only not employs is and It algebra mechanics. context quantum the in of combinatorics and algebra between terplay oper- abstract associated the expressions. of problem ator meaning ordering the remark- the clarify this on light and that shed is will expectation interrelation The able set). a as of example, partitions (for can interpretations problem combinatorial the clear in have involved in objects especially the is because relation teresting This procedures. ordering op- and erator structures combinatorial between relation prising range wide a to problems. ordering applicable of straightforwardly is and tool hsietﬁain h ouinmyb on ihthe with found be using may the of solution it help The resolve then identiﬁcation. problem and this the identify structures to combinatorial is prob- with methodology ordering general the The to approach lem. systematic a illustrate to ipetcs fpwr rteepnnilo h num- the of exponential the or the operator powers ber In numbers of analysis. natural case combinatorial be in simplest nor- origin to the their have appear in ex- which emerging procedure It coeﬃcients ordering problems. the mal ordering that of fact class the ploits broad a to applicable to problems. desirable ordering thus lead normal is that of principles solutions It guiding to or expansion. formulas manageable series deﬁned ﬁnd inﬁnite operators consider an we when by tedious tractable be less and even complicated, can is is expression it the practice when cumbersome in and straightforward, and clear ordering. normal called is hc nmrt attoso set. a of partitions enumerate which h ae sognzda olw.ScinI reyre- brieﬂy II Section follows. as organized is paper The in- fertile a of example an is present we framework The sur- a obtain we methodology this of byproduct a As nti ae epeetagnrlfaeokta is that framework general a present we paper this In olm n obntra bet,and objects, combinatorial and roblems ¶ e nvri´ iree ai Curie, Marie et Universit´e Pierre ee, drn rbesb pligi othe to it applying by problems rdering s foperators. of ass tn attoso e.Ti frame- This set. a of partitions ating toooycnb straightforwardly be can ethodology Dobi´nski relation broeao.Teslto fthe of solution The operator. mber † N -33 iltnue France Villetaneuse, F-93430 , § = aeyo Sciences, of cademy a † a sCdx0,France 05, Cedex is hs r trigadBl numbers Bell and Stirling are these 4,5,6,7,8 , 10,11 lhuhtepoesis process the Although hc savr eﬀective very a is which 9 eueti example this use We tion - 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 ). These states take the explicit form h | h | F (a†,a) = :G(a†,a): . (7) ∞ n 1 2 z z = e− 2 |z| n . (6) The normally ordered form has the merit of enabling im- | i √ | i nX=0 n! mediate calculation of an operator’s coherent state ele- 3 ments which reduce, by virtue of Eq. (5), to substituting III. GENERIC EXAMPLE: STIRLING AND a z and a† z∗ in its functional representation, that BELL NUMBERS is,→ → A. Normal ordering: Combinatorial setting z :G(a†,a): z = G(z∗,z). (8) h | | i † Thus, by solving the normal ordering problem of Eq. (7), We consider the number operator N = a a and seek we readily obtain the normally ordered form of its nth power (n 1). We write the latter as ≥ † ∗ z F (a ,a) z = G(z ,z) . (9) n h | | i n a†a = S(n, k) (a†)kak, (11) This procedure may be illustrated in the example Xk=1 z aa†aaa†a z = z (a†)2a4 +4a†a3 +2a2 z h | | i h | | i which uniquely deﬁnes the integer sequences S(n, k) = (z∗)2z4 +4 z∗z3 +2 z2. for k = 1 ...n; these sequences are called the Stirling numbers.11,18 Information about this sequence for each n In brief, we have shown that the calculation of coher- may be captured in the Bell polynomials ent state matrix elements reduces to solving the normal n ordering problem. The converse statement is also true; B(n, x)= S(n, k) xk. (12) that is, if we know the coherent state expectation value Xk=1 of the operator, say Eq. (9), than the normally ordered form of the operator is given by Eq. (7).4,5 We also deﬁne the Bell numbers B(n)= B(n, 1) as A standard approach to the normal ordering problem n is to use Wick’s theorem.15 In our context, this theorem B(n)= S(n, k). (13) expresses the normal ordering of an operator by applying Xk=1 the double dot operation to the sum of all possible ex- pressions obtained by removing pairs of annihilation and Instead of operators a and a† we may equally well insert † creation operators where a precedes a , called contrac- into Eq. (11) the representation of Eq. (1) given by the tions in analogy to quantum ﬁeld theory, for example operator X deﬁned as multiplication by x, and by the derivative D = d 19 aa†aaa†a = : aa†aaa†a : dx no pair removed a† X, a D. (14) | {z } ←→ ←→ + : aa†aaa†a + aa†aaa†a + aa† aaa†a + aa†aaa†a : This substitution does not aﬀect the commutator of 6 6 6 6 6 6 6 6 1 pair removed Eq. (1), that is, [D,X] = 1, which is the only property | {z } relevant for the construction. Therefore in this represen- + : aa† aaa†a + aa†aaa†a : = (a†)2a4 +4 a†a3 +2 a2. tation Eq. (11) takes the form 6 6 6 6 6 6 6 6 2 pairs removed n | {z } (XD)n = S(n, k) XkDk. (15) This procedure may involve a large number of steps. For kX=1 polynomial expressions this diﬃculty may be overcome by using computer algebra, although this use does not provide an analytic structure. For nontrivial functions, B. Combinatorial analysis such as those having inﬁnite expansions, the problem still remains open. In the following we discuss the properties of the Stir- One approach to the problem relies on the disentan- ling and Bell numbers.20 For that purpose we use ele- gling properties of Lie algebraic operators and applica- mentary methods of combinatorial analysis based on a tion of the Baker-Campbell-Hausdorﬀ formula. Here is a versatile tool known as the Dobi´nski relation.10,11 The standard example: latter is obtained by acting with Eq. (15) on the expo- x ∞ xk † 2 † nential function e = yielding eλ(a+a ) = eλ /2 :eλ(a+a ): . (10) k=0 k! P ∞ xk n However, the use of this kind of disentangling property kn = ex S(n, k) xk. (16) k! of the exponential operators is restricted in practice to kX=0 Xk=1 quadratic expressions in boson operators.16 Another method exploits the recurrence relations and [xx all eqs. must be numbered xx] If we recall the deﬁni- solves the normal ordering problem by use of combina- tion of the Bell polynomials Eq. (12), we obtain 9,17 torial identities. This promising approach was the in- ∞ kn spiration for the systematic combinatorial methodology B(n, x)= e−x xk, (17) k! which is presented in this article. Xk=0 4 which is the celebrated Dobi´nski relation.10,11 It is usually An explicit expression for the Stirling numbers S(n, k) expressed in terms of the Bell numbers, which are given may be extracted from the Dobi´nski relation. Note that by in Eq. (17) the relevant series may be multiplied together using the Cauchy multiplication rule to yield ∞ n −1 k B(n)= e . (18) ∞ xl ∞ xk k! B(n, x)= ( 1)l kn (24a) Xk=0 − l! k! Xl=0 Xk=0 We observe that both series are convergent and express ∞ k k xk the integers B(n) or polynomials B(n, x) in a nontriv- = ( 1)k−j jn (24b) j − k! ial way. To mention one of the many applications notice Xk=0 Xj=1 that kn in Eq. (17) may be replaced by the integral repre- sentation kn = ∞ dλ λnδ(λ k). If we change the order By comparing the expansion coeﬃcients in Eq. (24b) with 0 − of the sum andR integral (allowable because both are con- Eq. (12), we obtain vergent), we obtain the solution to the Stieltjes moment k 1 k problem for the sequence of Bell polynomials S(n, k)= ( 1)k−j jn, (25) k! j − ∞ Xj=1 n B(n, x)= dλ Wx(λ)λ , (19) Z0 which yields an expression for S(n, k). If we use any of the above standard formulas Eqs. (25) where and (13) for the Stirling or Bell numbers, we can easily ∞ δ(λ k) calculate them explicitly. We remark that many other in- W (λ)= e−x − xk (20) teresting results may be derived by straightforward ma- x k! Xk=0 nipulation of the Dobi´nski relation Eq.(17) or the gener- ating function Eq.(23), see Appendix A.23 Some of these is a positive weight function located at integer points numbers are given in Table: and is called a Dirac comb.21 Note that Eq. (20) may be identiﬁed with the Poisson distribution with mean value S(n, k), 1 k n B(n) equal to x. ≤ ≤ A very elegant and eﬃcient way of storing and tackling n =1 1 1 information about sequences is attained through their n =2 1 1 2 11 generating functions. The exponential generating func- n =3 131 5 tion of the polynomials B(n, x) is deﬁned as n =4 1761 15 n =5 11525101 52 ∞ λn G(λ, x)= B(n, x) . (21) n =6 1319065 15 1 203 n! nX=0 n =7 1 63 301350 140 21 1 877 n =8 1 127 966 1701 1050 266 28 1 4140 It contains all the information about the Bell polynomi- ...... als. If we use the Dobi´nski relation, we may calculate G(λ, x) explicitly. We substitute Eq. (17) into Eq. (21), change the summation order,22 and then identify the ex- C. Normal ordering: Solution pansions of the exponential functions to obtain

∞ ∞ We return to normal ordering. By using the proper- xk λn G(λ, x)= e−x kn (22a) ties of coherent states in Eqs. (7)–(9), we conclude from k! n! nX=0 kX=0 Eqs. (11) and (12) that the diagonal coherent state ma- 17 ∞ k ∞ n trix elements generate the Bell polynomials −x x n λ = e k (22b) † n 2 k! n! z (a a) z = B(n, z ). (26) Xk=0 nX=0 h | | i | | † ∞ k λa a x λ If we expand the exponential e and take the diagonal = e−x eλk = e−xexe . (22c) k! coherent state matrix element, we ﬁnd Xk=0 ∞ n ∞ n † λ λ Thus, the exponential generating function G(λ, x) takes z eλa a z = z (a†a)n z = B(n, z 2) . h | | i h | | i n! | | n! the compact form nX=0 nX=0 (27) λ G(λ, x)= ex(e −1). (23) We observe that the diagonal coherent state matrix ele- † ments of eλa a yield the exponential generating function Note that in the context of the weight function of Eq. (20) of the Bell polynomials (see Eqs. (21) and (23)) G(λ, x) is the moment generating function of the Poisson † 2 λ distribution with the parameter x. z eλa a z = e|z| (e −1). (28) h | | i 5

Equations (26) and (28) allow us to read oﬀ the normally ordered forms

(a†a)n = :B(n,a†a): , (29) and † † λ eλa a = :ea a(e −1): . (30)

Notice that the normal ordering of the exponential of the number operator a†a amounts to a rescaling of the parameter λ eλ 1. We stress that this rescaling is characteristic→ for this− speciﬁc case only and in general the functional representation may change signiﬁcantly (see Sec. V). Just for illustration we give results that can be obtained by an analogous calculation24

† 2 † 2 λ(a ) a λ(a ) a FIG. 1: Illustration of Stirling numbers S(n, k) enumerating e = :exp 1−λa† : , (31) partitions of a set of n = 3 distinguishable marbles (white, 2 2 † n λ(a†) a −a†a ∞ λn(n−1) (a a) gray and black) into k = 1, 2, 3 su bsets. e = :e n=0 e n! : . (32) P

Equations (29) and (30) provide an explicit solution to IV. SOME APPLICATIONS the normal ordering problem for powers and the exponen- tial of the number operator. This solution was obtained A. Quantum phase space by identifying the combinatorial objects and resolving the problem on that basis. Furthermore, this surprising connection opens a promising approach to the ordering A curious application of the coherent state represen- problem through its combinatorial interpretation. tation is found in the phase space picture of quantum mechanics. For the conjugate pair of observablesq ˆ = (a† + a)/√2 andp ˆ = i(a† a)/√2, related to the posi- tion and momentum operators− of a particle or to quadra- D. Combinatorial interpretation: Bell and Stirling tures of the electromagnetic ﬁeld, coherent state expecta- numbers tion values have the simple form z qˆ z + i z pˆ z = √2z and minimize the uncertainty relation.h | | 27i Inh this| | sensei the coherent state z for z = (q +ip)/√2 may be interpreted We have deﬁned and investigated the Stirling and as the closest| quantumi approximation to the classical Bell numbers as solutions to the normal ordering prob- phase state (q,p). These properties are used to construct lem. These numbers are well known in combinatorics11,18 the quantum analog of phase space through the Husimi where the S(n, k) are called Stirling numbers of the sec- distribution, denoted by Q(q,p), which for the quantum ond kind. Their original deﬁnition is given in terms of state described by the density matrix ρ is deﬁned as2,8 partitions of a set; that is, the Stirling numbers S(n, k) count the number of ways of putting n diﬀerent objects 1 into k identical containers (none left empty), see Fig. 1. Q(q,p)= z ρ z . (33) 2π h | | i The Bell numbers B(n) count the number of ways of putting n diﬀerent objects into n identical containers Q(q,p) is interpreted as the probability density for the (some may be left empty). From these deﬁnitions the re- system to occupy a region in phase space of width currence relation of Eq. (A1) may be readily obtained and ∆ˆq = ∆ˆp = 1/2 centered at (q,p) which on experi- further investigated from a purely combinatorial view- mental groundsp refers to obtaining the result (q,p) from point. This formal correspondence establishes a direct an optimal simultaneous measurement ofq ˆ andp ˆ. Such link to the normal ordering problem of the number oper- measurements in quantum optics are obtained using the ator. As a result we obtain an interesting interpretation technique of heterodyne detection. of the ordering procedure in terms of combinatorial ob- This construction of the quantum phase space analog jects. We remark that other pictorial representations can raises the problem of eﬃciently calculating the coherent be also given, for example, in terms of graphs25 or rook state expectation values of an operator which, as we have numbers.26 seen in Sec. II B, is in practice equivalent to its normal In conclusion we point out that this method may be re- ordering. Hence ordering techniques are important for versed; that is, certain combinatorial families of numbers practical use. may be given an algebraic interpretation in the quantum As an illustration we observe that from Eq. (28) we mechanical context. can readily derive the explicit expression for the Husimi 6 distribution of the quantum harmonic oscillator in ther- in the same class are connected by contractions between mal equilibrium. For the Hamiltonian H = a†a+1/2 the their operator constituents. See the following examples † density matrix ρ of a thermal state is e−βa a/Z, where for illustration −βa†a −β Z = Tr e =1/(1 e ) and β =1/kBT . Thus from Eqs. (33) and (28) we− obtain a†a a†a a†a a†a a†a 1 2 3 4 5 1 2 3 4 5 ↔ ↔ 1 −β 2 2 † † † † † Q(q,p)= (1 e−β)e(e −1)(q +p )/2. (34) a a a a a a a a a a 1 2 3 4 5 1 2 3 4 5 2π − ↔ ↔ It is instructive to compare this quantum phase space a†a a†a a†a a†a a†a 1 2 3 4 5 1 3 2 4 5 distribution with its classical analog. The correspond- ↔ ↔ ing Hamiltonian for the classical harmonic oscillator is Observe that this construction may be reversed and thus 2 2 Hcl = (q + p )/2, and the probability distribution in provides a one-to-one correspondence between operator 2 2 −β(q +p )/2 † n the thermal state is Pcl(q,p) = e /Zcl, where contractions in (a a) and partitions of the set of n 2 2 −β(q +p )/2 28 Zcl = e dqdp =2π/β. Finally, we obtain objects. In this way the contractions of Wick’s theo- R rem may be seen as partitions of a set, providing the link 1 2 2 P (q,p)= βe−β(q +p )/2. (35) to the combinatorial framework presented in this paper. cl 2π The methodology of Sec. III oﬀers an alternative perspec- In both cases we obtain gaussians. However, observe that tive on the normal ordering problem and unlike Wick’s the quantum distribution of Eq. (34) is wider than its approach, exposes its analytical structure, thus yielding classical analog of Eq. (35). It is explained by additional practical calculational tools. ﬂuctuations due to the uncertainty relation that are in- herent in quantum mechanics. For β 0, that is, for large temperatures, the quantum distribution→ of Eq. (34) C. Operator identities correctly goes to the classical distribution in Eq. (35). An analogous analysis can be done for the whole spec- Manipulations of diﬀerently ordered operators often trum of models described by Hamiltonians constructed lead to interesting operator identities. For example, tak- in the second quantization formalism, provided the nor- ing the limit λ in Eq. (30), yields an interesting → −∞ 4,6 mally ordered form of the operators is known. Section III representation of the vacuum projection operator discusses the methodology which is applicable to a wide † set of problems, for example, the optical Kerr medium as 0 0 = :e−a a: . (36) in Eq. (32) and the open system described by Eq. (31). | ih | See Sec. V for a discussion of the range of applicability. Equation (36) leads to a coordinate representation of the ∗ 2 †2 squeezing transformation S(λ) = e(λ a −λa )/2, which is extensively used in quantum optics.29 It may be ob- B. Beyond the Wick theorem tained using the technique of integration within an or- dered product,30 yielding As mentioned in Sec. II B the standard approach to normal ordering through Wick’s theorem reduces the ∞ problem to ﬁnding all possible contractions in the op- S(λ)= e−λ/2 dq e−λq q , (37) Z | ih | erator expression. In practice, the process may be te- −∞ dious and cumbersome to perform, especially when a large number of operators are involved. Hence system- which oﬀers an interpretation of S(λ) as an explicit atic methods, like the one described in Sec. III, are of squeezing of the quadrature. importance in actual applications. To complete the picture we will show how to connect Wick’s approach to the combinatorial setting described V. SUMMARY AND OUTLOOK in this paper. The bridge is readily provided by the in- terpretation of Stirling numbers as partitions of a set, as We have presented a combinatorial framework for op- given in Sec. III D. To see the link we consider a string † † † † erator ordering problems by discussing a simple example a a a a ... a a consisting of n blocks a a which we la- of the powers and exponential of the number operator bel from 1 to n starting from the left, thus obtaining n a†a. We have provided a general method for relating distinguishable objects normally ordered operator expressions to combinatorial a†a a†a ... a†a 1 2 . . . n . objects and then solved the problem from that view- ←→ point. The solution involved using the representation of 1 2 3 the Heisenberg algebra in terms of operators on the space |{z} |{z} |{z} Then each choice of contraction in Wick’s theorem of polynomials and then applying the Dobi´nski relation, uniquely divides this set into classes such that objects which provides the exponential generating function and 7 explicit expressions. This approach provides eﬀective cal- function.23 culational tools and also exposes the analytic structure The recurrence relation for the Stirling numbers is behind the Wick theorem. One advantage of this methodology is that it can be S(n +1, k)= kS(n, k)+ S(n, k 1), (A1) − straightforwardly generalized to a wide class of opera- tor expressions.23 The simplest examples are provided with the initial conditions S(n, 0) = δn,0 and S(n, k)=0 by the powers and exponentials of (a†)ra and (a†)ras for k>n. with r and s integers.31 It may be further extended to The Bell polynomials may be shown to satisfy the fol- investigate the normal ordering of boson monomials25 in lowing recurrence relation † rn sn † r2 s2 † r1 s1 the form (a ) a . . . (a ) a (a ) a and more gen- n 32 n erally homogeneous boson polynomials, that is, linear B(n +1, x)= x B(k, x), (A2) combinations of boson expressions with the same excess k kX=0 of creation over annihilation operators.23 Further devel- opment of the method applies to the ordering of general with B(0, x) = 1. Consequently for Bell numbers we have n n operators linear only in the annihilation (or creation) op- B(n +1)= k=0 k B(k). † † erator, that is, q(a )a + v(a ), where q(x) and v(x) are The followingP exponential generating function of the arbitrary functions. The exponential of an operator of Stirling numbers S(n, k) is sometimes used in applica- this type constitutes a generalized shift operator and the tions 33 solution is in the class of Sheﬀer polynomials. In all of ∞ these cases the use of the Dobi´nski relation additionally λn (eλ 1)k S(n, k) = − . (A3) provides a solution of the moment problem,21 as well as a n! k! nX=k wealth of combinatorial identities for sequences involved In addition, the Stirling numbers S(n, k) may be inter- 34 in the result (including their deformations ). preted as the connection coeﬃcients between two sets xn Ordering problems are naturally inherent in the alge- and xn, n =1, 2,..., where xn = x (x 1) . . . (x n + 1) braic structure of quantum mechanics. It is remarkable is the falling factorial; that is, they· represent− a change− of that they may be described and investigated using ob- basis in the space of polynomials jects having a clear combinatorial interpretation. For the generic example considered here these are partitions of a n n k set. For more complicated expressions the interpretation x = S(n, k) x . (A4) can be provided by introducing correlations between el- kX=1 ements or by using a graph representation. We also note that the Bell polynomials belong to the class of Sheﬀer polynomials,36 which in particular share APPENDIX A: COMBINATORIAL IDENTITIES an interesting property called the Sheﬀer identity (note the resemblance to the binomial identity)

n We enumerate some properties of the Stirling and n Bell numbers deﬁned in Sec. III.35 The reader is invited B(n, x + y)= B(k,y) B(n k, x). (A5) k − to check these relations by straightforward manipula- kX=0 tion of the Dobi´nski relation or exponential generating

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12 Coherent states |zi are not orthogonal for diﬀerent z and 083511-1–8 (2005). ∗ ′ 1 2 1 ′ 2 26 the overlapping factor is hz|z′i = ez z − 2 |z| − 2 |z | . They A. I. Solomon, G. H. E. Duchamp, P. Blasiak, A. Horzela, constitute an overcomplete basis in the sense of resolution and K. A. Penson, “Normal order: Combinatorial graphs,” 1 2 in Proceedings of 3rd International Symposium on Quan- of the identity π RC d z |zihz| = 1. 13 The double dot notation is almost universal in quantum op- tum Theory and Symmetries, edited by P. C. Argyres, T. tics and quantum ﬁeld theory. Nevertheless some authors, J. Hodges, F. Mansouri, J. J. Scanio, P. Suranyi, and L. for example, Ref. 6, use an alternative notation. C. R. Wijewardhana (World Scientiﬁc, Singapore, 2004), 14 Careless use of the double dot notation may lead to incon- pp. 527–536, arXiv:quant-ph/0402082; A. Varvak, “Rook sistencies, for example if A = aa† and B = a†a+1, we have numbers and the normal ordering problem,” J. Combin. Theory Ser. A 112, 292–307 (2005). A = B, but : A : 6= : B : . Such problems are eliminated if 27 a rigorous deﬁnition, beyond this note is given (to be pub- The conjugate operatorsq ˆ andp ˆ, satisfying [ˆq, pˆ] = i, are lished by A. I. Solomon, P. Blasiak, G. H. E. Duchamp, A. subject to the Heisenberg uncertainty relation ∆ψqˆ∆ψpˆ ≥ Horzela, and K. A. Penson). 1/2. For the coherent state |zi the product of uncertain- 15 Wick’s theorem is usually formulated for the time ordered, ties exactly equals 1/2. These are the only states with this property that additionally have equal uncertainties also called chronologically ordered, products of ﬁeld oper- 5,7 ∆ψpˆ =∆ψqˆ (in general, we obtain the squeezed states. ators. See for example, J. D. Bjorken and S. D. Drell, Rel- 28 ativistic Quantum Fields (McGraw-Hill, New York, 1965). A detailed proof of this bijection should take into consider- ation the speciﬁc structure of contractions between blocks In our context operators do not depend on the space-time † coordinate and chronological ordering reduces to the nor- a a which make the order in such constructed class irrele- vant. mal ordering procedure discussed in this paper. See for 29 example, Ref. 2. D. Stoler, “Generalized coherent states,” Phys. Rev. D 16 R. M. Wilcox, “Exponential operators and parameter dif- 2, 2309–2312 (1971); H. P. Yuen, “Two-photon coherent ferentiation in quantum physics,” J. Math. Phys. 8, 962– states of the radiation ﬁeld,” Phys. Rev. A 13, 2226–2243 982 (1967); C. L. Mehta, “Ordering of the exponential of (1976). A useful review is V. V. Dodonov “‘Nonclassical’ a quadratic in boson operators. I. Single mode case,” J. states in quantum optics: A ‘squeezed’ review of the ﬁrst 75 years,” J. Opt. B 4, R1–R33 (2002). Math. Phys. 18, 404–407 (1977). See also A. 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Graham, shop on Contemporary Problems in Mathematical Physics, D. E. Knuth, and O. Patashnik, Concrete Mathematics edited by J. Govaerts, M. N. Hounkonnou, and A. Z. (Addison-Wesley, MA, 1994). Msezane (World Scientiﬁc, Singapore, 2004), pp. 439–449, 19 arXiv:quant-ph/0401126. The simplest representation acts on the space of polyno- 33 mials and is deﬁned by Xxn = xn+1 and Dxn = nxn−1. P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp, It may be naturally extended to the space of formal power and A. I. Solomon, “Boson normal ordering via substi- series, see Refs. 11,36. tutions and Sheﬀer-type polynomials,” Phys. Lett. A 37, 20 For convenience and to avoid inaccuracy, the deﬁnitions 108–116 (2005); P. Blasiak, G. Dattoli, A. Horzela, and of Stirling and Bell numbers are usually extended by the K. A. Penson, “Representations of monomiality principle following conventions: B(0,x) = B(0) = S(0, 0) = 1 and with Sheﬀer-type polynomials and boson normal order- ing,” Phys. Lett. A 352, 7–12 (2006). S(n, k) = 0 for k = 0 or k>n> 0. 34 21 P. Blasiak, A. Horzela, K. A. Penson, and A. I. Solomon, J. Katriel and M. Kibler, “Normal ordering for deformed “Dobi´nski-type relations: Some properties and physical boson operators and operator-valued deformed Stirling applications,” J. Phys. A 37, 4999–5006 (2006). numbers,” J. Phys. A 25, 2683–2691 (1992); J. Katriel and 22 Because the generating functions are formal series, the G. Duchamp, “Ordering relations for q-boson operators, question of convergence does not arise. continued fraction techniques and the q-BCH enigma,” J. 23 P. Blasiak, “Combinatorics of boson normal ordering and Phys. A 28, 7209–7225 (1995); M. Schork, “On the com- some applications,” Concepts of Physics 1, 177–278 (2004), binatorics of normal ordering bosonic operators and defor- arXiv:quant-ph/0510082. mations of it,” J. Phys. A 36, 4651–4665 (2003); See also P. 24 We suggest derivation of these formulas as a problem to Blasiak, A. Horzela, K. A. Penson, and A. I. Solomon, “De- be solved by students during classes on the Fock space formed bosons: Combinatorics of normal ordering,” Czech. J. Phys. 54, 1179–1184 (2004). methods. Detailed calculations may be found in Refs. 23, 35 31. M. Abramovitz and I. Stegun, Handbook of Mathematical 25 Functions (Dover, New York, 1972). M. A. M´endez, P. Blasiak, and K. A. Penson, “Combina- 36 torial approach to generalized Bell and Stirling numbers S. Roman, The Umbral Calculus (Academic Press, Or- and boson normal ordering problem,” J. Math. Phys. 46, lando, 1984).