Antinormal Ordering Research Articles

Ordered product expansions of operators (AB)±m with arbitrary positive integer**Project supported by the National Natural Science Foundation of China (Grant No. 11804085) and the Natural Science Foundation of Shandong Province, China (Grant No. ZR2017MEM012).

We arrange quantum mechanical operators (a† a)m in their normally ordered product forms by using Touchard polynomials. Moreover, we derive the anti-normally ordered forms of (a† a)± m by using special functions as well as Stirling-like numbers together with the general mutual transformation rule between normal and anti-normal orderings of operators. Further, the ℚ- and ℙ-ordered forms of (QP)±m are also obtained by using an analogy method.

Recast combination functions of coordinate and momentum operators into their ordered product forms**Project supported by the National Natural Science Foundation of China (Grant No. 11347026), the Natural Science Foundation of Shandong Province, China (Grant Nos. ZR2016AM03 and ZR2017MA011), and the Natural Science Foundation of Heze University, China (Grant Nos. XY17KJ09 and XY18PY13).

By using the parameter differential method of operators, we recast the combination function of coordinate and momentum operators into its normal and anti-normal orderings, which is more ecumenical, simpler, and neater than the existing ways. These products are very useful in obtaining some new differential relations and useful mathematical integral formulas. Further, we derive the normally ordered form of the operator (fQ + gP)−n with n being an arbitrary positive integer by using the parameter tracing method of operators together with the intermediate coordinate–momentum representation. In addition, general mutual transformation rules of the normal and anti-normal orderings, which have good universality, are derived and hence the anti-normal ordering of (fQ + gP)−n is also obtained. Finally, the application of some new identities is given.

Photon enhanced chaotic light and its dissipation in an amplitude damping channel

In this paper, we investigate how a m-photon enhanced chaotic field (PACF), denoted by 1−e−λm+1a†me−λa†aam/m!, evolves in an amplitude damping channel with a damping constant κ. By using the technique of integration within ordered product of operators (IWOP), we find that it evolves into a Laguerre function-weighted chaotic field which is described by ⋮e1−eλe2κtaa†Lmeλ+2κtaa†⋮, which is in antinormal ordering. Time evolution of photon number average and fluctuation in this process are also derived. The corresponding evolution law of the Wigner function is also deduced analytically.

New approach for anti-normally and normally ordering bosonic-operator functions in quantum optics**Project supported by the Natural Science Foundation of Shandong Province, China (Grant No. ZR2015AM025) and the Natural Science Foundation of Heze University, China (Grant No. XY14PY02).

In this paper, we provide a new kind of operator formula for anti-normally and normally ordering bosonic-operator functions in quantum optics, which can help us arrange a bosonic-operator function f(λQ̂ + νP̂) in its anti-normal and normal ordering conveniently. Furthermore, mutual transformation formulas between anti-normal ordering and normal ordering, which have good universality, are derived too. Based on these operator formulas, some new differential relations and some useful mathematical integral formulas are easily derived without really performing these integrations.

Derivation of the Husimi symbols without antinormal ordering, scale transformation and uncertainty relations

We propose a new method for the derivation of Husimi symbols, for operators that are given in the form of products of an arbitrary number of coordinates, and momentum operators, in an arbitrary order. For such an operator, in the standard approach, one expresses coordinate and momentum operators as a linear combination of the creation and annihilation operators, and then uses the antinormal ordering to obtain the final form of the symbol. In our method, one obtains the Husimi symbol in a much more straightforward fashion, departing directly from operator explicit form without transforming it through creation and annihilation operators. With this method the mean values of some operators are found. It is shown how the Heisenberg and the Schrödinger–Robertson uncertainty relations, for position and momentum, are transformed under scale transformation . The physical sense of some states which can be constructed with this transformation is also discussed.

Star products on symplectic vector spaces: convergence, representations, and extensions

We briefly survey the general scheme of deformation quantization on symplectic vector spaces and analyze its functional analytic aspects. We treat different star products in a unified way by systematically using an appropriate space of analytic test functions for which the series expansions of the star products in powers of the deformation parameter converge absolutely. The star products are extendable by continuity to larger functional classes. The uniqueness of the extension is guaranteed by suitable density theorems. We show that the maximal star product algebra with the absolute convergence property, consisting of entire functions of an order at most 2 and minimal type, is nuclear. We obtain an integral representation for the star product corresponding to the Cahill-Glauber s-ordering, which connects the normal, symmetric, and antinormal orderings continuously as s varies from 1 to −1. We exactly characterize those extensions of the Wick and anti-Wick correspondences that are in line with the known extension of the Weyl correspondence to tempered distributions.

A generalized Weyl-Wigner quantization scheme unifying P—Q and Q—P ordering and Weyl ordering of operators

By extending the usual Wigner operator to the s-parameterized one aswith s being a real parameter, we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule. This rule recovers P-Q ordering, Q-P ordering, and Weyl ordering of operators in s = 1, - 1, 0 respectively. Hence it differs from the Cahill-Glaubers' ordering rule which unifies normal ordering, antinormal ordering, and Weyl ordering. We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P. The formula that can rearrange a given operator into its new s-parameterized ordering is presented.

Deriving new operator identities by alternately using normally, antinormally, and Weyl ordered integration technique

Dirac’s ket-bra formalism is the language of quantum mechanics. We have reviewed how to apply Newton-Leibniz integration rules to Dirac’s ket-bra projectors in previous work. 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 new integration formulas regarding Laguerre and Hermite polynomials. This may open a new route of directly deriving some complicated mathematical integration formulas by virtue of the quantum mechanical operator ordering technique, without really performing the integrations in the ordinary way.

Weyl Ordering Symbol Method for Studying Wigner Function of the Damping Field

The symbolic method (including normal ordering. antinormal ordering and Weyl ordering symbol) is usually utilized to tackle miscellaneous operators which have different commutative relations. Considering the Weyl ordering symbol’s remarkable properties, we have efficiently and conveniently derived the Wigner distribution function for field damping in a squeezed bath and a vacuum bath respectively, and then examined the decoherence processes from the plots of Wigner function and its contour in quantum phase space. Alternatively, we can employ a general Wigner operator under phase space transform to calculate distribution function and discuss the damping process.

Normal ordering and antinormal ordering of the operator (fQ + gP)n and some of their applications

In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator (fQ + gP)n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.

Coherent state path integrals in the Weyl representation

We construct a representation of the coherent state path integral using the Weyl symbol of the Hamiltonian operator. This representation is very different from the usual path integral forms suggested by Klauder and Skagerstan in \cite{Klau85}, which involve the normal or the antinormal ordering of the Hamiltonian. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. We show that the semiclassical limit of the coherent state propagator in Weyl representation is involves classical trajectories that are independent on the coherent states width. This propagator is also free from the phase corrections found in \cite{Bar01} for the two Klauder forms and provides an explicit connection between the Wigner and the Husimi representations of the evolution operator.

Observation of Antinormally Ordered Hanbury Brown–Twiss Correlations

We have measured antinormally ordered Hanbury Brown-Twiss correlations for coherent states of the electromagnetic field by using a stimulated parametric down-conversion process. Photons were detected by stimulated emission, rather than by absorption, so that the detection responded not only to actual photons but also to zero-point fluctuations via spontaneous emission. The observed correlations were distinct from normally ordered ones as they showed excess positive correlations, i.e., photon bunching effects, which arose from the thermal nature of zero-point fluctuations.

Lüders theorem for coherent-state POVMs

Lüders’ theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on Lie groups. In general, operators turn out to be invariant under the Lüders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lüders map generated by the coherent states of the Heisenberg–Weyl group, H3. The Lüders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.

Operator ordering in quantum optics theory and the development of Dirac s symbolic method

We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.

Generalized Weyl Ordering and Nonlinear Coherent States**The project supported by National Natural Science Foundation of China under Grant No. 10 175 057

In the context of the nonlinear coherent state (NLCS) theory we introduce the generalized Weyl ordering operator formulation. The corresponding generalized Wigner operator turns out to be the Weyl ordered Dirac δ-operator functions. The completeness relation of NLCS is recast into generalized Weyl ordering form. The relationship between normal ordering, antinormal ordering and the generalized Weyl ordering is established which constitute a self-consistent theory for NLCS.

Stochastic simulations: continuous approximation of quantum noise

The stochastic simulations of quantum evolution are introduced, alternatively, to the Ito-stochastic simulations, for the normal, symmetrical and antinormal ordering in the double-dimensional phase space. The nonclassical effects in the linearized parametric amplifier can be very well simulated in contrast to the collapses-revivals effect in a nonlinear version, which is stimulated by the discreteness of light. The reason is that the stochastic simulations use the continuous variables, whereas the quantum noise in nonlinear systems cannot generally be treated continuously.

Number-phase properties of correlated chaotic fields

We study the arrangement of two down-conversion crystals with aligned and partially aligned idler beams in the quasimonochromatic approximation. Treating the signal fields as correlated chaotic modes, we arrive at a detailed description and illustration of their mathematically well defined state in terms of number and phase properties. We show it to be a mixed partial phase-difference state. We derive the closed formulae for the phase-difference distributions in this state derived from the phase-space distributions related to the normal, symmetrical and antinormal orderings of field operators and show that such distributions depend exclusively on measures of correlation which are variants of the modulus of the familiar degree of coherence. Quantum correlation influences fluctuations of photon-number sum, photon-number difference and quantum phase difference. Starting from a quasidistribution which carries the same information as the statistical operator, we arrive at the joint distributions of number sum and phase difference and the quasidistribution of number difference and phase difference.

Various Re-ordering Expressions of General Exponential Quadratic Operators in Mutli-Mode Boson System**The project supported by National Natural Science Foundation of China

In this paper, we give various explicit expressions and transform equalities among usual, normal and anti-normal orderings for the exponential inhomogeneous quadratic (EIQ) operators and their inverse operators in multi-mode boson systems.

Ordered operator expansions and reconstruction from ordered moments

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.

Quasidistributions for noncommuting cosine and sine operators

A similarity between the photon annihilation and creation operators and the Susskind-Glogower exponential phase operators motivates the introduction and study of quasidistributions of the cosine and sine of phase. These quasidistributions are related to the standard and antistandard orderings of trigonometric phase operators and to the normal, antinormal, and symmetrical orderings of the exponential phase operators. The symmetrical ordering is connected to an optical ideal tomography of the appropriate quasidistribution from the rotated Susskind-Glogower cosine operator.