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Journal of Optics B Quantum and Semiclassical Optics

About integration within ordered products in quantum optics

Alfred Wünsche

Published Jan 1, 1999• 210 citations
Publisher

Abstract

The fundamentals of Fan's method of integration within ordered products are investigated and it is shown that it is a reliable calculation method in quantum optics. Differentiation within ordered products is included into the considerations. The integrations are, in applications to single boson modes, mostly one-dimensional or two-dimensional integrations over phase-space variables. We examine some examples of integration within ordered products and compare them with calculations of results by other methods such as Lie group methods. A new representation of the main class of quasiprobabilities including the Wigner and the Husimi-Kano quasiprobability is derived by the method. Different representations of squeezing operators are related by the method and discussed. As a further new result, the position and momentum representation of the general unitary squeezing operators are derived and specialized to cases with squeezing in the direction of the coordinate axes which were earlier dealt with by the method under consideration.

DOI
10.1088/1464-4266/1/3/201

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The fundamentals of Fan's method of integration within ordered products are investigated and it is shown that it is a reliable calculation method in quantum optics. Differentiation within ordered products is included into the considerations. The integrations are, in applications to single boson modes, mostly one-dimensional or two-dimensional integrations over phase-space variables. We examine some examples of integration within ordered products and compare them with calculations of results by other methods such as Lie group methods. A new representation of the main class of quasiprobabilities including the Wigner and the Husimi-Kano quasiprobability is derived by the method. Different representations of squeezing operators are related by the method and discussed. As a further new result, the position and momentum representation of the general unitary squeezing operators are derived and specialized to cases with squeezing in the direction of the coordinate axes which were earlier dealt with by the method under consideration.
It is shown that the symmetrically ordered moments of boson operators for a single boson mode can be reconstructed from the corresponding moments of the Radon transform of the Wigner quasiprobability for discrete sets of equidistant inequivalent angles which solve the circle division problem. This reconstruction is sometimes simpler than the corresponding reconstruction of the normally ordered moments where one first has to multiply the Radon transform with Hermite polynomials in comparison to power functions for symmetrically ordered moments and then to integrate. The connection to the reconstruction for the general class of s-ordered moments is established. The transition from discrete sets of angles to integration over angles via averaging over the discrete angles is made. The results are applied to displaced squeezed thermal states. It is shown how the ordered moments for these states can be explicitly found from the calculated Radon transform of the Wigner quasiprobability. The obtained formulae for these moments possess independent interest since they contribute to the discussion of the properties of the most general class of states with quasiprobabilities of Gaussian form with many possible special cases as, for example, squeezed coherent states and squeezed thermal states.
By the newly developed technique of integration within an ordered product (IWOP) of operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We can not only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. We derive GFO's normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area to finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the IWOP technique are fully used. All these confirms Dirac's assertion: " ... for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory".
A new formalism of the squeeze operator has been found by integration within normal ordered products. Based on this formalism, the essence of the formation of the squeezed state can be revealed directly and clearly. Various representations and properties of the squeezed state can be derived easily. It has been proved that the squeezed state is a coherent one in the quasiparticle space. The transformation relationship between squeezed states and eigenstates of both coordinate and momentum operators are presented. Also presented is the Wigner distribution function of the squeezed state.
Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product" technique, and the mapping is manifestly apparent in the derivation. New generalized coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.