Antinormal Ordering Research Articles

Semiclassical theory of superfluorescence

The fluctuating properties of superradiant and superfluorescent systems are investigated by supplementing the equations of motion for the expectation values of quantum observables with random force terms. In the case of a small atomic system the time evolution of the radiated intensity as well as the intensity fluctuations are computed. We also find the time dependence of the average dipole moment in the presence of chirping. For a pencil-shaped geometry, a generalized system of Maxwell–Bloch equations is analysed. The inhomogeneous broadening is then included. We compare the semiclassical equation for the probability distribution of the atomic variables with the quantum equations based both on normal and antinormal ordering. With the exception of the region in the vicinity of the poles of the Bloch sphere, a close correspondence between the semiclassical and quantum theories is established. The oscillatory superfluorescence is treated similarly to quantum analysis by employing a new method of Lugiato for open systems.

Damped solutions for the photon statistics of radiation propagating through a random medium

The present paper uses results of the preceding paper (Czech. J. Phys.B 24 (1974), 374) to obtain the quantum characteristic functions, quasi-distributions and their moments for radiation passing through a random medium. It is shown that the quasi-distribution related to normal ordering does not exist as an ordinary function and the time behaviour of the quasi-distribution related to antinormal ordering is discussed. Exact photon counting distribution and its factorial moments are obtained and some useful approximations are derived. General theory is applied to the superposition of coherent and chaotic radiation. Finally some consequences are discussed such as self-radiation effects of the medium, the relation of the present active non-linear quantum description and the previous passive linear quantum description by Tatarski and passive linear quasi-classical description by Diament and Teich and the behaviour of the photon counting distribution for various levels of turbulence.

Ordering ofM-mode field operators in quantum optics

The general formalism of an arbitrary ordering ofM-mode field operators is developed and this formalism is applied to a model of the superposition of coherent and chaoticM-mode fields having importance for a description of laser light. Corresponding probability distributions and their moments as well as their interrelationships are derived. Physically important relations for the normal, symmetric and antinormal orderings are deduced as special cases and a criterion of the existence of the weighting function of the Sudarshan-Glauber diagonal representation of the density matrix inL2-space given.

Generalized phase-space distributions associated with a pseudo-oscillator

The problem of mapping the operator functions of $$\hat b$$ and $$\hat b^\dag $$ , which are the annihilation and creation operators associated with a pseudo-oscillator, onto thec-number functions is studied from a unified point of view. The approach is similar to the one recently developed by Agarwal and Wolf for the case of the ordinary oscillator. The results are illustrated by means of a number of examples. The phase-space distributions corresponding to Weyl’s, normal and antinormal orderings are studied in detail and the explicit representations for them in terms of coherentstate matrix elements are given. The connection with the so-called semiclassical description is also established.

Ordered Expansions in Boson Amplitude Operators

The expansion of operators as ordered power series in the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required $c$-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values $s=+1,0,\ensuremath{-}1$, respectively, of an order parameter $s$. In terms of this convention it is shown that for bounded operators the coefficients are finite when $s>0$, and the series are convergent when $s>\frac{1}{2}$. For each value of the order parameter $s$, a correspondence between operators and $c$-number functions is defined. Each correspondence is one-to-one and has the property that the function $f(\ensuremath{\alpha})$ associated with a given operator $F$ is the one which results when the operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for $F$ are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the $P$ representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter $s$.

Normal and antinormal ordering of field operators in quantum optics

The relations between the probability distributionp n vt of the number of photonsn vt in a volumeV at timet and the probability distributions PN(W) and PA(W) of the integrated intensityW related to the normal and antinormal ordering of field operators and the relations between corresponding moments are studied for a field containingM modes. As the normally ordered correlations are measured with photodetectors and the antinormally ordered correlations are measured with quantum counters the relations betweenn vt , PN(W) and PA(W) permit the statistical behaviour of light to be determined from measurements with photodetectors and quantum counters. The results obtained here are used for the superposition of coherent and thermal fields. It is also shown that the antinormal correlations depend explicitly on the number of modes and that in the classical limit, when the average photon occupation number per mode becomes large, the distributionsn vt , PN(W) and PA(W) become equal.

On the normal and antinormal ordering of field operators in quantum optics

The relations between the probability distribution P( n Vt ) of the number of photons n Vt and the probability distributions P N ( W) and P A ( W) of the integrated intensity W related to the normal and antinormal ordering of the field operators are studied.

A Continuous Representation of an Indefinite Metric Space

An overcomplete family of states (OFS) is constructed for a countably infinite linear vector space with an indefinite metric for the case that the metric is diagonal with eigenvalues (−1)n, where n is an integer. A continuous representation is indicated and the properties of a semiclassical description of a quantum mechanical system (the pseudo oscillator whose creation and destruction operators ā and ā+ satisfy [ā, ā+] = −1) defined in this vector space are studied. It is found that a consistent OFS |z> can be constructed if the operator G(z) which generates the state |z> from the vacuum is unitary. Furthermore, with the statistical state of this system specified by a bounded pseudo-Hermitian density matrix ρ̄, the related semiclassical complex function ρA(z) for antinormal ordering of operators in the indefinite metric space is found to be bounded, with ρA(z) and [ρA(z)]2 integrable, continuous, and a boundary value of an entire analytic function of two complex variables. The semiclassical function ρN(z) for normal ordering is associated with a sequence of functions ρN(ν)(z) whose square is integrable and related to a sequence of tempered distributions ρN(ν) such that the corresponding sequence of density matrices ρ̄(ν) converges to ρ̄ in the norm.

Relation between Quantum and Semiclassical Description of Optical Coherence

The problem of relating the semiclassical and quantum treatments of statistical states of an optical field is re-examined. The case where the rule of association between functions and operators is that of antinormal ordering is studied in detail. It is shown that the distribution function for each mode corresponding to this case is a continuous bounded function, and is also a boundary value of an entire analytic function of two variables. The nature of the distribution for the normal ordering rule of association and its relation to this entire function are discussed. It is shown that this distribution can be regarded as the limit of a sequence of tempered distributions in the following sense: One can find a sequence of density operators ${\stackrel{^}{\ensuremath{\rho}}}_{(\ensuremath{\nu})}$ which converges in the norm to the density operator $\stackrel{^}{\ensuremath{\rho}}$ of any given field (consisting of a single mode), such that each member of the sequence can be expressed in the form ${\stackrel{^}{\ensuremath{\rho}}}_{(\ensuremath{\nu})}=\ensuremath{\int}{\ensuremath{\varphi}}_{(\ensuremath{\nu})}(z)|z〉〈z|{d}^{2}z$, where ${\ensuremath{\varphi}}_{(\ensuremath{\nu})}$ is a tempered distribution.