Star product and ordered moments of photon creation and annihilation operators arXiv The corresponding phase space is a two-dimensional lattice with nodes m, n given by pairs of nonnegative integers. The star-product kernel of symbols on the lattice and intertwining kernels to other schemes are found in explicit form. Analysis of peculiar properties of the star-product kernel results in new sum relations for factorials. Advantages of the developed star-product scheme for describing dynamics of quantum systems are discussed and time evolution equations in terms of the ordered moments are derived.
PACS numbers: Tb, Ca, Wj, Ordering of creation and annihilation operators was studied extensively in the s in connection with the quasiprobability distributions see, e. Similarly, the Wigner W -quasidistribution  and the Sudarshan-Glauber P -quasidistribution [7, 8] are expressed through symmetrically and antinormally ordered representations of the density operator, respectively see, e. The quasidistributions on phase space do not limit to Q, W , and P -functions and are reviewed in several papers see, e.
The recent detailed review of the phase-space approach is presented by Vourdas . Moreover, in the phase-space formalism, quantum phenomena are known to be interpreted in the classical-like manner . Being a rather good alternative to the density operator, the quasidistribution functions Q, W , and P are all equivalent in the sense that they all contain the thorough physically meaningful information about the state .
Ladder Operators (Creation/Annihilation Operators)
As the quasidistributions are equivalent, the question arises itself: why do not we use only one of them for all the problems? The answer lies in actual applications of the quasidistributions. For example, using the classical interpretation of the Wigner W -function in collision problems, it is possible to make some reasonable approximations and obtain an appropriate but still accurate solution to the problem without requiring excessive computer time and expense. On the other hand, the nonnegativity and smoothness of the Husimi-Kano Q-function make it advantageous for the analysis of classically chaotic nonlinear systems .
Also, the Q-function of the radiation field can be measured at optical frequencies via an eight- port homodyne detection scheme see, e. We can draw a conclusion that although all the quasidistribution functions are equivalent, the different functions exhibit different properties and obey different dynamical equations. Those differences specify the most advantageous representation for a particular problem. An interest to the ordered moments rose in the s and was encouraged by the advances of measuring quantum states of light.
The matter is that at optical wavelengths there is no need to calculate the normally ordered moments and then use them to find the density operator. The normally ordered moments are easily expressed through the measured optical tomogram as well [29, 30]. For details we refer the reader to the paper . Thus, in the microwave domain, the lower-order moments are experimentally determined and contain the primary information about a quantum state. Formula 1 is nothing else but the evolution equation for the normally ordered moments.
Therefore, the time development of a quantum state is expressed through the measurable characteristics, which resembles the expectation value approach [43, 44] and the tomographic-probability approach [28, 45, 46] to quantum mechanics. According to the latter one, the measurable tomographic probability is a primary object to describe quantum states. The evolution equations for tomograms were found, e.
Let us note that a unitary evolution of any quasidistribution function can be written in the form of Eq. The aim of this paper is to develop the star-product scheme of symbols given by the normally ordered moments, to explore properties of the star-product kernel, and to derive unknown evolution equations for the moments via the star-product approach. The article is organized as follows.
- Annihilation operator.
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Wave Functions, Creation and Annihilation Operators of Quantum Physical System
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Generalized creation and annihilation operators: Physical interpretation and reordering properties.
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