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arXiv:1210.4115 · New Journal of Physics

Wigner function for the orientation state

Timo Fischer, Clemens Gneiting, Klaus Hornberger

Published Jun 6, 2013• 20 citations
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Abstract

We introduce a quantum phase space representation for the orientation state of extended quantum objects, using the Euler angles and their conjugate momenta as phase space coordinates. It exhibits the same properties as the standard Wigner function and thus provides an intuitive framework for discussing quantum effects and semiclassical approximations in the rotational motion. Examples illustrating the viability of this quasi-probability distribution include the phase space description of a molecular alignment effect.

DOI
10.1088/1367-2630/15/6/063004

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We introduce a quantum phase space representation for the orientation state of extended quantum objects, using the Euler angles and their conjugate momenta as phase space coordinates. It exhibits the same properties as the standard Wigner function and thus provides an intuitive framework for discussing quantum effects and semiclassical approximations in the rotational motion. Examples illustrating the viability of this quasi-probability distribution include the phase space description of a molecular alignment effect.
We extend the Wigner-Weyl-Moyal phase-space formulation of quantum mechanics to general curved configuration spaces. The underlying phase space is based on the chosen coordinates of the manifold and their canonically conjugate momenta. The resulting Wigner function displays the axioms of a quasiprobability distribution, and any Weyl-ordered operator gets associated with the corresponding phase-space function, even in the absence of continuous symmetries. The corresponding quantum Liouville equation reduces to the classical curved space Liouville equation in the semiclassical limit. We demonstrate the formalism for a point particle moving on two-dimensional manifolds, such as a paraboloid or the surface of a sphere. The latter clarifies the treatment of compact coordinate spaces, as well as the relation of the presented phase-space representation to symmetry groups of the configuration space.
The framework of Wigner functions for the canonical pair angle and orbital angular momentum, derived and analyzed in 2 recent papers [H. A. Kastrup, Phys.Rev. A 94, 062113(2016) and Phys.Rev. A 95, 052111(2017)] is applied to elementary concepts of quantum information like qubits and 2-qubits, e.g., entangled EPR/Bell states etc. Properties of the associated Wigner functions of such superposed states (pure and mixed) are discussed and illustrated. The Wigner functions of ERP/Bell states are distinguished by their topologically "twisted" domain on the configuration subspace S^1xS^1, a torus, of the total phase space. Like the applications of Wigner functions in quantum optics the results presented in the present paper may be useful for the description and analysis of quantum information experiments with orbital angular momenta of light beams or electron beams, respectively.
The Weyl-Wigner transformation enables one to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined bv the quasi-probability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix. (GE)
The spin system with Hamiltonian (ξ and σ are external and internal field parameters) is treated as an example of the phase space description of spin dynamics using a master equation for the quasiprobability distribution function of spin orientations in the representation (phase) space of the polar angles (analogous to the Wigner phase space distribution for translational motion). The master equation yields (via the Wigner–Stratonovich transformation of the density matrix) the solution as a Fourier series in the spherical harmonics with Fourier coefficients given by the statistical moments in a manner analogous to the classical distribution. The relaxation time of the longitudinal component of the spin can be estimated using a quantum generalization of the classical integral relaxation time via the stationary distribution and diffusion coefficient of the master equation.