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Acta Physica Polonica A

Time-Frequency Analysis of Nonstationary Optical Signals Using Husimi Type Function

M. Davidović, MilenaD. Davidović, Vuk Vojisavljević

Published Oct 1, 2009• 5 citations

Abstract

Time-frequency analysis is important to identify the localized information of a non-stationary signal in the time and frequency domains simultaneously.There are few different time-frequency analysis methods available with their own specialty and suitability.In quantum mechanics Husimi function of any quantum mechanical state arises naturally whenever the simultaneous measurement of both coordinate and momentum is performed on this state with maximal accuracy allowed by the Heisenberg uncertainty relations.The Husimi function is a probability distribution for the statistics of simultaneous unsharp measurement of both coordinate and momentum.In general, like the Wigner function, the Husimi function can be defined in the space of any pair of conjugate variables.In particular, in the studies of signal processing, this space is the time-frequency space.In the present work we consider the Husimi function in this space, and apply it to analyse the multicomponent signals.The time-frequency representations of the simulated signals by using the Husimi distribution clearly show the frequency features along the time axis.The results are encouraging and indicate that, like in corresponding analogous problems in quantum mechanics, the Husimi distribution approach in the time-frequency analysis for non-stationary optical signals may provide some insights which are not so easily obtained in other, more spread approaches.

DOI: 10.12693/aphyspola.116.675

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Time-frequency analysis is important to identify the localized information of a non-stationary signal in the time and frequency domains simultaneously.There are few different time-frequency analysis methods available with their own specialty and suitability.In quantum mechanics Husimi function of any quantum mechanical state arises naturally whenever the simultaneous measurement of both coordinate and momentum is performed on this state with maximal accuracy allowed by the Heisenberg uncertainty relations.The Husimi function is a probability distribution for the statistics of simultaneous unsharp measurement of both coordinate and momentum.In general, like the Wigner function, the Husimi function can be defined in the space of any pair of conjugate variables.In particular, in the studies of signal processing, this space is the time-frequency space.In the present work we consider the Husimi function in this space, and apply it to analyse the multicomponent signals.The time-frequency representations of the simulated signals by using the Husimi distribution clearly show the frequency features along the time axis.The results are encouraging and indicate that, like in corresponding analogous problems in quantum mechanics, the Husimi distribution approach in the time-frequency analysis for non-stationary optical signals may provide some insights which are not so easily obtained in other, more spread approaches.

The Husimi distribution, introduced by Kodi Husimi in 1940 [1], is a quasi-probability distribution commonly used to study the correspondence between quantum and classical dynamics [2]. Also, it is employed to describe systems in different areas of physics such as Quantum Mechanics, Quantum Optics, Information Theory [3–8]. Additionally, in nanotechnology it is possible to obtain a clear description of localization –which corresponds to classicality– and is crucial to determine correctly the size of systems when the particle dynamics takes into account mobility boundaries [9]. Among its properties, it is always positive definite and unique, conversely it cannot be considered as a true probability distribution over the quantum-mechanical phase space, reason why it is often considered as a quasi probability distribution. Although it possesses no correct marginal properties, its usefulness is to allow the assessment of the expectation values in quantum mechanics in a way similar to the classical case [10]. The semiclassical Husimi probability distribution refers to a special type of probability, this is for simultaneous but approximate location of position and momentum in phase space.

Abstract This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.

We study the efficiency of quantum algorithms which aim at obtaining phase-space distribution functions of quantum systems. Wigner and Husimi functions are considered. Different quantum algorithms are envisioned to build these functions, and compared with the classical computation. Different procedures to extract more efficiently information from the final wave function of these algorithms are studied, including coarse-grained measurements, amplitude amplification, and measure of wavelet-transformed wave function. The algorithms are analyzed and numerically tested on a complex quantum system showing different behavior depending on parameters: namely, the kicked rotator. The results for the Wigner function show in particular that the use of the quantum wavelet transform gives a polynomial gain over classical computation. For the Husimi distribution, the gain is much larger than for the Wigner function and is larger with the help of amplitude amplification and wavelet transforms. We discuss the generalization of these results to the simulation of other quantum systems. We also apply the same set of techniques to the analysis of real images. The results show that the use of the quantum wavelet transform allows one to lower dramatically the number of measurements needed, but at the cost of a large loss of information.

The quantum dynamics in Husimi representation is studied for the commutation relation, time development of distribution function and eigen distribution function. In the integrable system, the Husimi eigen distribution function is confined almost on the classical torus and can be approximated by Gaussian distribution with the variance

(\Delta\xi_{l})^{2}{\sim}\hbar/2

around the torus. Further, considering the correspondence between quantum and classical mechanics, the Husimi distribution function is a better representation than the Wigner distribution function, because coarse graining is usually involved in observational process.

The Husimi function provides a phase-space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate-density momentum-density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.

The correspondence between classical and quantum mechanics is an important subject for the better understandings of “quantum chaos”. In particular, it is very important to investigate the correspondence between distribution functions in classical mechanics and in phase space representation of quantum mechanics. This is the review of our recent progresses in the study of distribution functions in classical and quantum mechanics, namely distribution functions in classical mechanics and in coarse-grained classical mechanics as well as the Wigner function and the Husimi function. Topics dealt with include formulations of the Wigner representation, the Husimi representation and coarse-grained classical mechanics, and their applications to the analyses of the eigenstates and time developments of the distribution functions.