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Quantum Optics Journal of the European Optical Society Part B

Displaced Fock states and their connection to quasiprobabilities

Alfred Wünsche

Published Dec 1, 1991• 110 citations
Publisher

Abstract

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.

DOI
10.1088/0954-8998/3/6/005

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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.
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 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.
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.
It is shown that the symmetrically ordered moments of boson operators for a single boson mode can be reconstructed from the corresponding moments of the Radon transform of the Wigner quasiprobability for discrete sets of equidistant inequivalent angles which solve the circle division problem. This reconstruction is sometimes simpler than the corresponding reconstruction of the normally ordered moments where one first has to multiply the Radon transform with Hermite polynomials in comparison to power functions for symmetrically ordered moments and then to integrate. The connection to the reconstruction for the general class of s-ordered moments is established. The transition from discrete sets of angles to integration over angles via averaging over the discrete angles is made. The results are applied to displaced squeezed thermal states. It is shown how the ordered moments for these states can be explicitly found from the calculated Radon transform of the Wigner quasiprobability. The obtained formulae for these moments possess independent interest since they contribute to the discussion of the properties of the most general class of states with quasiprobabilities of Gaussian form with many possible special cases as, for example, squeezed coherent states and squeezed thermal states.
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.