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InTech eBooks

The Husimi Distribution: Development and Applications

Sergio Curilef, F. Pennini

Published Apr 3, 2013• 3 citations
Publisher

Abstract

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.

DOI
10.5772/53846

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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.

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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.
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.
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.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
Abstract The Husimi function (Q-function) of a quantum state is the distribution function of the density operator in the coherent state representation. It is widely used in theoretical research, such as in quantum optics. The Wehrl entropy is the Shannon entropy of the Husimi function, and is non-zero even for pure states. This entropy has been extensively studied in mathematical physics. Recent research also suggests a significant connection between the Wehrl entropy and many-body quantum entanglement in spin systems. We investigate the statistical interpretation of the Husimi function and the Wehrl entropy, taking the system of N spin-1/2 particles as an example. Due to the completeness of coherent states, the Husimi function and Wehrl entropy can be explained via the positive operator-valued measurement (POVM) theory, although the coherent states are not a set of orthonormal basis. Here, with the help of the Bayes’ theorem, we provide an alternative probabilistic interpretation for the Husimi function and the Wehrl entropy. This interpretation is based on direct measurements of the system, and thus does not require the introduction of an ancillary system as in the POVM theory. Moreover, under this interpretation the classical correspondences of the Husimi function and the Wehrl entropy are just phase-space probability distribution function of N classical tops, and its associated entropy, respectively. Therefore, this explanation contributes to a better understanding of the relationship between the Husimi function, Wehrl entropy, and classical-quantum correspondence. The generalization of this statistical interpretation to continuous-variable systems is also discussed.
We introduce a new density for the representation of quantum states on phase space. It is constructed as a weighted difference of two smooth probability densities using the Husimi function and first order Hermite spectrograms. In contrast to the Wigner function, it is accessible by sampling strategies for positive densities. In the semiclassical regime, the new density allows one to approximate expectation values to second order with respect to the high frequency parameter and is thus more accurate than the uncorrected Husimi function. As an application, we combine the new phase space density with Egorov's theorem for the numerical simulation of time-evolved quantum expectations by an ensemble of classical trajectories. We present supporting numerical experiments in different settings and dimensions.
Husimi distributions [11] and Wigner distributions [20] are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems [15,16] and operational quantum physics [7], and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of Husimi distributions and mutual relation between Husimi distributions and other classes of quasi-probability distributions.