In this article combinatorial aspects of normal ordering annihilation and creation operators of a multi-mode boson system are discussed. The modes are assumed to be coupled since otherwise the problem of normal ordering is reduced to the corresponding problem of the single-mode case. To describe the normal ordering in the multi-mode case for each mode a colour is introduced and coloured contractions are considered. A depiction for coloured contractions via coloured linear representations is given. In analogy to the single-mode case associated coloured Stirling numbers are defined as coefficients appearing in the process of normal ordering powers of the number operators. Several properties of these coloured Stirling numbers are discussed.
In this paper we define generalizations of boson normal ordering. These are based on the number of contractions whose vertices are next to each other in the linear representation of the boson operator function. Our main motivation is to shed further light onto the combinatorics arising from algebraic and Fock space properties of boson operators.
We present a combinatorial method of constructing solutions to the normal ordering of boson operators. Generalizations of standard combinatorial notions - the Stirling and Bell numbers, Bell polynomials and Dobinski relations - lead to calculational tools which allow to find explicitly normally ordered forms for a large class of operator functions.
We construct explicit representations of the Heisenberg-Weyl algebra [P,M]=1 in terms of ladder operators acting in the space of Sheffer-type polynomials. Thus we establish a link between the monomiality principle and the umbral calculus. We use certain operator identities which allow one to evaluate explicitly special boson matrix elements between the coherent states. This yields a general demonstration of boson normal ordering of operator functions linear in either creation or annihilation operators. We indicate possible applications of these methods in other fields.
We provide the solution to the normal ordering problem for powers and exponentials of two classes of operators. The first one consists of boson strings and more generally homogeneous polynomials, while the second one treats operators linear in one of the creation or annihilation operators. Both solutions generalize Bell and Stirling numbers arising in the number operator case. We use the advanced combinatorial analysis to provide closed form expressions, generating functions, recurrences, etc. The analysis is based on the Dobiński-type relations and the umbral calculus methods. As an illustrat…
We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasi-probabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-base…
We implement the normal ordering technique to study the quantum dissipation of a single mode harmonic oscillator system. The dynamic evolution of the system is investigated for a reasonable initial state by solving the Schrödinger equation directly through the normal ordering technique. The decoherence process of the system for the cases $T=0K$ and $T\neq0K$ is investigated as an application.
We solve the boson normal ordering problem for (q(a*)a + v(a*))^n with arbitrary functions q and v and integer n, where a and a* are boson annihilation and creation operators, satisfying [a,a*]=1. This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state representation to the exponential generating functions of Sheffer-type polynomials. This in turn opens a vast arena of combinatorial methodology which is applied to boson normal ordering and illustrat…
We solve the normal ordering problem for (A* A)^n where A* (resp. A) are one mode deformed bosonic creation (resp. annihilation) operators satisfying [A,A*]=[N+1]-[N]. The solution generalizes results known for canonical and q-bosons. It involves combinatorial polynomials in the number operator N for which the generating functions and explicit expressions are found. Simple deformations provide examples of the method.
We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobiski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a co…
We solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers, where a* and a are boson creation and annihilation operators satisfying [a,a*]=1. That is, we provide exact and explicit expressions for the normal form wherein all a's are to the right. The solution involves integer sequences of numbers which are generalizations of the conventional Bell and Stirling numbers whose values they assume for r=s=1. A comprehensive theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas)and generating function…
We construct and analyze a family of coherent states built on sequences of integers originating from the solution of the boson normal ordering problem. These sequences generalize the conventional combinatorial Bell numbers and are shown to be moments of positive functions. Consequently, the resulting coherent states automatically satisfy the resolution of unity condition. In addition they display such non-classical fluctuation properties as super-Poissonian statistics and squeezing.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators (which considers normal ordered, antinormally ordered and Weyl ordered product of operators as its special cases). The s-ordered operator expansion formula of density operators is derived, and the s-parameterized quantization scheme is thus completely established.
For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for F[(a*)^r a^s], with r,s positive integers,[a,a*]=1, i.e. we provide exact and explicit expressions for its normal form which has all a's to the right. The solution involves integer sequences of numbers which, for r,s >=1, are generalizations of the conventional Bell and Stirling numbers whose values they assume for r=s=1. A complete theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski - type formulas), recursion relations and generating functi…
In 1988, I translated a popular article 'Squeezed light' for the Hungarian mutation of Scientific American. How shall we say 'squeezed' in our language from now on? At the time, Janszky was already Hungary's top quantum optics theorist, he excelled in squeezed light research as well, so I did not want to decide without asking him. He voted for the word 'préselt' [prayshalt], and it worked,became standard in our talking and writing about squeezed light. To his memory, let me dedicate an easy route (two pages) to normal ordering the squeeze operator, based on my recent extension of the Wick theo…
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 given herein to a wide class of operators.
The normal ordering formulae for powers of the boson number operator $\hat{n}$ are extended to deformed bosons. It is found that for the `M-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = 1$, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the `P-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = q^{-\hat{n}}$, are found to depend on the operator $\hat{n}$. This distinction betwe…
Partial transpose is an important operation for quantifying the entanglement, here we study the (partial) transpose of any single (two-mode) operators. Using the Fock-basis expansion, it is found that the transposed operator of an arbitrary operator can be obtained by replacement of a^{†}(a) by a(a^{†}) instead of c-number within normal ordering form. The transpose of displacement operator and Wigner operator are studied, from which the relation of Wigner function, characteristics function and average values such as covariance matrix are constructed between density operator and transposed dens…
We solve the boson normal ordering problem for $(q(a^†)a+v(a^†))^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^†$ are boson annihilation and creation operators, satisfying $[a,a^†]=1$. This consequently provides the solution for the exponential $e^{λ(q(a^†)a+v(a^†))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illu…
We suggest and analyze a new scheme to adiabatically cool bosonic atoms to picokelvin temperatures which should allow the observation of magnetic ordering via superexchange in optical lattices. The starting point is a gapped phase called the spin Mott phase where each site is occupied by one spin-up and one spin-down atom. An adiabatic ramp leads to an xy-ferromagnetic phase. We show that the combination of time-dependent density matrix renormalization group methods with quantum trajectories can be used to fully address possible experimental limitations due to decoherence, and demonstrate that…
Dirac's ket-bra formalism is the "language" of quantum mechanics and quantum field theory. In Refs.(Fan et al, Ann. Phys. 321 (2006) 480; 323 (2008) 500) we have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors. In this work by alternately using the technique of integration within normal, antinormal, and Weyl ordering of operators we not only derive some new operator ordering identities, but also deduce some useful integration formulas regarding to Laguerre and Hermite polynomials. This opens a new route of deriving mathematical integration formulas by virtu…
Normally ordered forms of functions of boson operators are important in many contexts in particular 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, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in nonc…
In this paper, we show that the infinite generalised Stirling matrices associated with boson strings with one annihilation operator are projective limits of approximate substitutions, the latter being characterised by a finite set of algebraic equations.
We address the generalized measurement of the two-boson operator $Z_γ= a_1 + γa_2^†$ which, for $|γ|^2 \neq 1$, is not normal and cannot be detected by a joint measurement of quadratures on the two bosons. We explicitly construct the minimal Naimark extension, which involves a single additional bosonic system, and present its decomposition in terms of two-boson linear SU(2) interactions. The statistics of the measurement and the added noise are analyzed in details. Results are exploited to revisit the Caves-Shapiro concept of generalized phase observable based on heterodyne detection.
We analyze the spectrum and normal mode representation of general quadratic bosonic forms $H$ not necessarily hermitian. It is shown that in the one-dimensional case such forms exhibit either an harmonic regime where both $H$ and $H^\dagger$ have a discrete spectrum with biorthogonal eigenstates, and a coherent-like regime where either $H$ or $H^\dagger$ have a continuous complex two-fold degenerate spectrum, while its adjoint has no convergent eigenstates. These regimes reflect the nature of the pertinent normal boson operators. Non-diagonalizable cases as well critical boundary sectors separ…
In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated ``Feynman diagrams'' the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick's theorem in the present context.
We show how some Hamiltonians may be approximated using rotating wave approximation methods. In order to achieve this we use the algebra of boson ladder operators, and transformation formulas between normal and symmetric ordering of the operators. The method presented is studied in two special cases; the Morse and the Mathiue models. The connection with regular perturbation theory is given and the validity of the approximation is discussed.
A conventional context for supersymmetric problems arises when we consider systems containing both boson and fermion operators. In this note we consider the normal ordering problem for a string of such operators. In the general case, upon which we touch briefly, this problem leads to combinatorial numbers, the so-called Rook numbers. Since we assume that the two species, bosons and fermions, commute, we subsequently restrict ourselves to consideration of a single species, single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, specificall…
We formulate a bosonic dynamical mean-field theory (B-DMFT) which provides a comprehensive, thermodynamically consistent framework for the theoretical investigation of correlated lattice bosons. The B-DMFT is applicable for arbitrary values of the coupling parameters and temperature and becomes exact in the limit of high spatial dimensions d or coordination number Z of the lattice. In contrast to its fermionic counterpart the construction of the B-DMFT requires different scalings of the hopping amplitudes with Z depending on whether the bosons are in their normal state or in the Bose-Einstein …
It is shown that causality violations [M. de Haan, Physica 132A, 375, 397 (1985)], emerging when the conventional definition of the time-normal operator ordering [P.L.Kelley and W.H.Kleiner, Phys.Rev. 136, A316 (1964)] is taken outside the rotating wave approximation, disappear when the amended definition [L.P. and S.S., Annals of Physics, 323, 1989 (2008)] of this ordering is used.
We discuss how one calculates the coherent path integrals for locally interacting systems, where some inconsistencies with exact results have been reported previously. It is shown that the operator ordering subtlety that is hidden in the local interaction term modifies the Hubbard-Stratonovich transformation in the continuous time formulation, and it helps reproduce known results by the operator method. We also demonstrate that many-body effects in the strong interaction limit can be well characterized by the free-particle theory that is subject to annealed random potentials and dynamical gaug…
Wick's theorem, known for yielding normal ordered from time-ordered bosonic fields may be generalized for a simple relationship between any two orderings that we define over canonical variables, in a broader sense than before. In this broad class of orderings, the General Wick Theorem (GWT) follows from the Baker-Campbell-Hausdorff identity. We point out that, generally, the characteristic function does not induce an unambiguous scheme to order the multiple products of the canonical operators although the value of the ordered product is unique. We construct a manifold of different schemes for …
We investigate the role played by quantum operator ordering in the correlations that characterize two-photon polarization Bell measurements. The Clauser-Horne-Shimony-Holt (CHSH) criterion is investigated in the normal ordering imposed by the photodetection theory and in the symmetric ordering that constitutes the standard prescription for building Hermitian operators from products of noncommuting observables. The two approaches are obtained in a single theoretical framework, where operator ordering is directly associated with the representation used for the density matrix.Moreover, this discu…
The connection between real-time quantum field theory (RTQFT) [see, e.g., A.\ Kamenev and A.\ Levchenko, Advances in Physics {58} (2009) 197] and phase-space techniques [E.\ Wolf and L.\ Mandel, {\em Optical Coherence and Quantum Optics} (Cambridge, 1995)] is investigated. The Keldysh rotation that forms the basis of RTQFT is shown to be a phase-space mapping of the quantum system based on the symmetric (Weyl) ordering. Following this observation, we define generalised Keldysh rotations based on the class of operator orderings introduced by Cahill and Glauber [Phys.\ Rev.\ {177} (1969) 1882]. …
We derive a dynamical mean-field theory for mixtures of interacting bosons and fermions on a lattice (BF-DMFT). The BF-DMFT is a comprehensive, thermodynamically consistent framework for the theoretical investigation of Bose-Fermi mixtures and is applicable for arbitrary values of the coupling parameters and temperatures. It becomes exact in the limit of high spatial dimensions d or coordination number Z of the lattice. In particular, the BF-DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. Using the BF-DMFT we investigate …
