Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries

Abstract

We present a detailed discussion of a general theory of phase-space distributions, introduced recently by the authors [J. Phys. A 31, L9 (1998)]. This theory provides a unified phase-space formulation of quantum mechanics for physical systems possessing Lie-group symmetries. The concept of generalized coherent states and the method of harmonic analysis are used to construct explicitly a family of phase-space functions which are postulated to satisfy the Stratonovich-Weyl correspondence with a generalized traciality condition. The symbol calculus for the phase-space functions is given by means of the generalized twisted product. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we consider the reconstruction method based on measurements of displaced projectors, which comprises a number of recently proposed quantum-optical schemes and is also related to the standard methods of signal processing. A general group-theoretic description of this method is developed using the technique of harmonic expansions on the phase space.03.65.Bz, 03.65.Fd

The Pauli equation for probability distributions

“…Recently, we established [19] a quite general principle of constructing measurable probabilities, which determine completely the quantum state in the tomographic approach; more refined treatments then followed [20,21]. Here, we apply our general approach to derive the evolution equation for the tomographic probabilities that is alternative in some sense to the Schrödinger equation for the wave function (or the quantum Liouville equation for the density matrix).…”

Spin Relaxation in Phase Space

“…This function was further developed both for closed and open spin systems (e.g., [35][36][37][38][39][50][51][52][53][54][55][56][57][58][59][60]) and is entirely analogous to the translational Wigner distribution ( , , ) W q p t in phase space ( , ), q p which is the quasiprobability representation of the density operator except that certain differences arise because of the angular momentum commutation relations. The basic ideas may be summarized as follows [51].…”

“…By way of background to the discussion which follows, we recall that in providing a phase space description of spin systems, Stratonovich in 1956 [49] introduced the quasiprobability Alternative quasiprobability distribution functions for spins have also been proposed using the spin coherent-state representation of the density matrix [40,[50][51][52][53][54][57][58][59][60] introduced by Glauber and Sudarshan and commonly used in quantum optics (see, e.g., [45,46,51]). Moreover, Várilly…”

“…35-40, 49-51. Furthermore, phase-space representations of quantum mechanical evolution equations provide a formal means of treating quantum effects in dynamical systems transparently linking to the classical representations, facilitating the calculation of quantum corrections to classical distribution functions. These representations generally based on the coherent state representation of the density matrix introduced by Glauber and Sudarshan and widely used in quantum optics [45,46] when applied to spin systems (e.g., [35][36][37][38][39][40][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69]) allow one to analyze quantum spin 8 relaxation using [70] dynamics onto c-number quasiprobability density evolution equations clearly shows how these equations reduce to the Fokker-Planck equation in the classical limit [40,62]. The phase-space distribution function for spins was originally introduced by Stratonovich [49] for closed systems and further developed both for closed and open spin systems [35][36][37][38][39][40][52][53][54][55][56]…”

Coherent-state path-integral calculation of the Wigner function

“…There exists a family of representations of operators by functions on the manifold, of which the matrix element and diagonal representative are of relevance here [3,28]. The matrix element Q (also known as the anti-normal or upper symbol) of an operator is…”