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Physica Scripta

Relations between scaling-transformed Husimi functions, Wigner functions and symplectic tomograms describing corresponding physical states

V. A. Andreev, Dragomir M Davidović, Ljubica Davidović, M. Davidović

Published Feb 1, 2011• 3 citations
Publisher

Abstract

Husimi Q -functions are the only functions from the class of Cohen quasi-distributions on phase space that after scaling transformation ( q , p )→(λ q , λ p ) remain in the same class when the modulus of the scaling parameter is smaller than unity and so, in this case, describe a physical state. We found the Wigner functions and symplectic tomograms of such states. We applied the obtained general results to the Fock states of the harmonic oscillator.

DOI
10.1088/0031-8949/2011/t143/014003

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Husimi Q -functions are the only functions from the class of Cohen quasi-distributions on phase space that after scaling transformation ( q , p )→(λ q , λ p ) remain in the same class when the modulus of the scaling parameter is smaller than unity and so, in this case, describe a physical state. We found the Wigner functions and symplectic tomograms of such states. We applied the obtained general results to the Fock states of the harmonic oscillator.
We consider scale transformations (q, p) → (λq, λp) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson–Schrödinger uncertainty relations for stretched states.
It is shown that dynamical equations for quantum tomograms retain the normalization conditions of their solutions during evolution only if the solutions satisfy a set of special conditions. These conditions are found explicitly. On the contrary, it is also shown that the classical Liouville equation, Moyal equation for Wigner function, and evolution equation for Husimi function retain normalization of any initially normalized and quickly decaying at infinity functions on the phase space. Other necessary and sufficient conditions for optical and symplectic tomogram-like functions to be tomograms of physical states are discussed.
The modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of Q or Husimi functions. A parameter λ specifies orderings in a mapping from the operator |ψ⟩⟨σ| to the corresponding phase space wavefunction, where σ is a given fiducial vector. The choice λ = 0 specifies the Weyl mapping and the Q -function so obtained is the usual one when |σ⟩ is the vacuum state. More generally, any choice of λ in the range (−1, 1) corresponds to orderings varying between standard and anti-standard. For all such orderings the generalized Q -functions are non-negative by construction. They are shown to be proportional to the expectation of the system state with respect to a generalized displaced squeezed state which depends on λ and position ( p , q ) in phase space. Thus, when a system has been prepared in the state , a generalized Q -function is proportional to the probability of finding it in the generalized squeezed state. Any such Q -function can also be written as the smoothing of the Wigner function for the system state by convolution with the Wigner function for the generalized squeezed state.
The phase space representation for a q -deformed model of the quantum harmonic oscillator is constructed. We have found explicit expressions for both the Wigner and Husimi distribution functions for the stationary states of the q -oscillator model under consideration. The Wigner function is expressed as a basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is shown that, in the limit case h → 0 ( q → 1), both the Wigner and Husimi distribution functions reduce correctly to their well-known non-relativistic analogues. Surprisingly, examination of both distribution functions in the q -deformed model shows that, when q ≪ 1, their behaviour in the phase space is similar to the ground state of the ordinary quantum oscillator, but with a displacement towards negative values of the momentum. We have also computed the mean values of the position and momentum using the Wigner function. Unlike the ordinary case, the mean value of the momentum is not zero and it depends on q and n . The ground-state-like behaviour of the distribution functions for excited states in the q -deformed model opens quite new perspectives for further experimental measurements of quantum systems in the phase space.
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.