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Optica Acta International Journal of Optics

On the Calculation of Generating Functions in Quantum Optics

Jan Peřina

Published Jul 1, 1979• 6 citations
Publisher

Abstract

A relation between the normal generating and characteristic functions, recently obtained by Rockower and Abraham, is shown to be valid in a space of generalized functions if antinormal ordering is adopted. This is demonstrated for the quadratic non-linear interaction for which we derive the change of quantum statistics caused by the dynamics of the process as well as by the initial statistical state of the field. The possibilities that the uncertainty might increase and that anticorrelation might occur are discussed. The analyticity of the generating function is applied to demonstrate a smoothing process when calculating the normal generating function and to show that a recent statement of Timmermans and Zijlstra concerning the restricted validity of cumulants is incorrect.

DOI
10.1080/713820067

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A relation between the normal generating and characteristic functions, recently obtained by Rockower and Abraham, is shown to be valid in a space of generalized functions if antinormal ordering is adopted. This is demonstrated for the quadratic non-linear interaction for which we derive the change of quantum statistics caused by the dynamics of the process as well as by the initial statistical state of the field. The possibilities that the uncertainty might increase and that anticorrelation might occur are discussed. The analyticity of the generating function is applied to demonstrate a smoothing process when calculating the normal generating function and to show that a recent statement of Timmermans and Zijlstra concerning the restricted validity of cumulants is incorrect.
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.
The photon statistics of quadratic parametric two-mode processes are derived starting from the results of the first part of this paper. A general theory is presented and particular processes are discussed. Assuming the initial field to be coherent, fluctuations of the pumping field are taken into account. The effects of antibunching and anticorrelation of photons on the photon distribution, its factorial moments, fluctuations in modes and their correlations for the amplification process are demonstrated and their dependence on levels of pumping field fluctuations is shown.
In the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristic functionals that in order to determine the joint probability distributions of the complex field at N space-time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well-known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.
Starting from the characteristic function of an operator, we investigate cumulant expansions in quantum optics and apply them to two-dimensional distributions for the canonical variables of the phase space in the case of one degree of freedom (Wigner quasiprobability and its Fourier transform, uncertainty matrix) and to one-dimensional distributions (phase operator, time evolution operator to Hamiltonian). In the relations between cumulants and moments, we make emphasis on the central moments of an operator. It is shown that the determinant of the uncertainty matrix (modified uncertainty product) is invariant with respect to rotation and squeezing of the state in the phase space, whereas the uncertainty sum is only invariant with respect to rotations. We examine some problems for exponentials of the phase operator and show how mean values and variances are connected with the cumulants. The Hilbert–Schmidt distance of a state during time evolution to an initial state is discussed by cumulants.
We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and cumulants, quantities which characterize the fluctuations of an underlying probability distribution. In all of the cases we review, the fluctuations are those of a quantum system. We show that the original formalism for geometric phases, in which a quantum system is taken around an adiabatic cycle, can be extended to the case when degeneracy points are encountered along the cycle (quasiadiabatic cycles). The essential tool for this extension is a generalized Bargmann invariant which plays the role of a generating function. From the cumulants generated this way one can form ratios according to the Binder cumulant scheme in statistical mechanics. Such geometric Binder cumulants are sensitive to gap closure, as such, they are useful in identifying metal-insulator transitions, localization, and quantum phase transitions. We present example calculations on simple model systems, whose localization properties are well known, to validate to approach. We also complement our geometric Binder cumulant calculations with results for the fidelity susceptibility, a quantity directly related to the quantum geometry of the parameter space.