A relation of the Wigner quasiprobability to the Bargmann representation of pure and mixed states by convolution is derived and generalized to the main class of quasiprobabilities and its inversion is given. The derivation uses a realization of the abstract group SU (1,1) by second-order differentiation and multiplication operators for a pair of complex conjugated variables and disentanglement of exponential functions of these operators by group-theoretical methods. Examples for the calculation of the Wigner quasiprobability via the Bargmann representation of states demonstrate the action of this relation. A short collection of different basic representations of the Wigner quasiprobability is given. An appendix presents results for the disentanglement of SU (1,1)-group operators by products of special operators in different ordering.
A relation of the Wigner quasiprobability to the Bargmann representation of pure and mixed states by convolution is derived and generalized to the main class of quasiprobabilities and its inversion is given. The derivation uses a realization of the abstract group SU (1,1) by second-order differentiation and multiplication operators for a pair of complex conjugated variables and disentanglement of exponential functions of these operators by group-theoretical methods. Examples for the calculation of the Wigner quasiprobability via the Bargmann representation of states demonstrate the action of this relation. A short collection of different basic representations of the Wigner quasiprobability is given. An appendix presents results for the disentanglement of SU (1,1)-group operators by products of special operators in different ordering.
The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.
The Wigner quasiprobability, along with some of its essentialproperties, is introduced and discussed in two versions, first covering real canonical variables such as W(q,p) and second a pair of complex conjugate coordinates such as W(α,α*). The reconstruction of the density operator ϱ of states is also given. Building upon the Susskind–Glogower concept of quantum phase operators, further aspects of phase operator algebras in the quantum optics of a harmonic oscillator are discussed in relation to the realization of the su(1,1) Lie algebra. Coherent phase states |ε⟩ are introduced in analogy to the common coherent states |α⟩ in two ways, as both eigenstates of certain operators and as states generated from a ground state |0⟩ by operators of the Lie group SU(1,1). The limiting transition to the non-normalizable Fritz London phase states |eiφ⟩ on the unit circle and an (over)-completeness relation for the coherent phase states are derived. The Wigner quasiprobability W(q,p) for the coherent phase states is calculated and graphically represented. From the Wigner quasiprobability, a phase distribution W(φ) is calculated by integrating over the radius, and its uncertainty is defined and presented. The Hilbert–Schmidt distance is discussed as a measure of the non-classicality of states, where most of our with Viktor Dodonov work was carried out.
The requirements on the general structure of quasiprobabilities for a single boson mode are investigated. The complete Gaussian class of quasiprobabilities, which can be obtained by convolutions of the Wigner quasiprobability with the complete class of normalized Gaussian functions, is represented by a three-dimensional complex vector parameter with the property of additivity when composing convolutions and meaning that the transition between two quasiprobabilities with the vector parameters r and s is given by the convolution with a Gaussian function belonging to the vector parameter r - s . The scalar product of r with itself is related to the determinant of the second-rank symmetric tensor of the quadratic form in the exponent of the Gaussian functions. This is obtained by a mapping with the two symmetric Pauli spin matrices and the unity matrix. The Wigner quasiprobability takes on the central position within this Gaussian class with the vector parameter r = (0,0,0). The class of s-ordered quasiprobabilities is described by the vector parameters with and its diagonalization is connected with displaced Fock states . The class of quasiprobabilities corresponding to the linear interpolation between standard and antistandard ordering belongs to the vector parameter with . Its diagonalization is connected with discrete series of excitations of the eigenstates of the canonical operators Q and P. The diagonalization of the complete Gaussian class of quasiprobabilities with real vector parameter and leads to dual systems of discrete excitations of squeezed coherent states with mutually orthogonal squeezing axes and properties of orthogonality and completeness and provides a possible generalization of the displaced Fock states. The diagonalization is basically different for real and complex vector parameters r . A generalization of the coherent-state quasiprobability, which uses squeezed coherent states instead of coherent states, belongs to a vector parameter with real and imaginary and and with . New representations of special classes of quasiprobabilities are derived.
Starting from Wigner's definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator.One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics.For the coherent states we show that their Fourier transforms are again coherent states.We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries.It is not well suited for the calculation of expectation values.A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new.We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult.The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values.By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics.Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.
The class of quasiprobabilities obtainable from the Wigner quasiprobability by convolutions with the general class of Gaussian functions is investigated. It can be described by a three-dimensional, in general, complex vector parameter with the property of additivity when composing convolutions. The diagonal representation of this class of quasiprobabilities is connected with a generalization of the displaced Fock states in direction of squeezing. The subclass with real vector parameter is considered more in detail. It is related to the most important kinds of boson operator ordering. The properties of a specific set of discrete excitations of squeezed coherent states are given.
The properties of the displaced Fock states mod a,n) identical to d(a,a*) mod n), (a complex numbers, D(a,a*) displacement operators, n=0,1,2,. . .) are systematically investigated with emphasis on the connections to the Heisenberg-Weyl group and to its irreducible representations. The displaced Fock states comprise the coherent states mod a) identical to mod a,0) as well as the Fock states mod n) identical to mod 0,n) as particular cases. An orthocompleteness relation for the displaced Fock states in the form of the area integral of the operators mod a,m)(a,n mod over the complex a-plane is derived. It generalizes for m=n the well known completeness relation for the coherent states and leads for m not=n to identities expressing the overcompleteness of the displaced Fock states. A basic formula is obtained for the convolution of the operators mod a,m)(a,n mod with the class of Gaussian functions that have exponents proportional to aa*. The connection of the displaced Fock states to the transition operators from the density operator for a single boson mode to quasiprobabilities is studied in general form and specified to the class of transition operators with the displaced Fock states as their eigenstates and connected by convolutions with Gaussian functions to the coherent-state quasiprobability, the Wigner quasiprobability, and the Glauber-Sudarshan quasiprobability.