Ordering of "ladder" operators, the Wigner function for number and phase, and the enlarged Hilbert space

From the Weyl quantization of a particle on the circle to number–phase Wigner functions

“…This formalism enables us to define the Wigner function on any Lie group. In [63] the authors have found a Wigner function for the Euclidean group E (2). However, the relation between that Wigner function and our Wigner function on the cylinder given by Eq.…”

“…In fact, it has been motivated by a question raised by the referee of [1]. Namely, the referee pointed out that the Weyl quantization formalism developed in [1] should be closely related to the Wigner function and the Wigner representation of quantum phase investigated previously by the others [2][3][4]. So, here we follow this suggestion and we intend to display, how one can define the Wigner function which depends on the number and the phase.…”

Optimal waveform estimation for classical and quantum systems via time-symmetric smoothing. II. Applications to atomic magnetometry and Hardy’s paradox

“…An alternative definition in a 2N × 2N phase space, first suggested by Hannay and Berry [26], is 5) where the matrix elements with noninteger indices are assumed to be zero. One may also use either Wigner function to describe the energy level n and phase φ of a harmonic oscillator by letting n = q, φ = 2πp/N , and taking the N → ∞ limit at the end of a calculation [27,28]. Both definitions have a particularly desirable property for the purpose of smoothing, namely,…”

Quantum-Optical States in Finite-Dimensional Hilbert Space. I. General Formalism

“…Another generalization of discrete Wigner function for Schwinger's FD periodic Hilbert space was analyzed by, for instance, Hakioǧlu [60]. The Wigner function approach to FD systems can be developed from basic principles as was shown, for example, by Wootters [55], Leonhardt [37], Lukš and Peřinová [61], or Luis and Peřina [62]. Discrete Wigner function has successfully been applied to quantum-state tomography of FD systems [37] (for a review, see Ref.…”