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International Symposium on Quantum Theory and Symmetries (QTS)

ON THE SU (2) WIGNER FUNCTION DYNAMICS

A. B. Klimov, S. M. Chumakov

Published Jun 1, 2002• 2 citations
Publisher
DOI
10.1142/9789812777850_0052

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SE ESTUDIA LA DINAMICA CUANTICA DE LA FUNCION DE DISTRIBUCION DE CUASIPROBABILIDAD DEL GRUPO SU(2) (FUNCION DE WIGNER) PARA UN SIMPLE HAMILTONIANO NO LINEAL (EL ANALOGO FINITO DEL MEDIO DE KERR H = S2 Z ). SE CONSIDERAN LA APROXIMACION CUASICLASICA PARA LA FUNCION DE WIGNER Y LA EVOLUCION CORRESPONDIENTE DE LOS VALORES MEDIOS. SE COMPARAN LAS SOLUCIONES EXACTA, CLASICA Y CUASICLASICA. Abstract WE STUDY THE QUANTUM DYNAMICS OF THE SU(2) QUASIPROBABILITY DISTRIBUTION (”WIGNER FUNCTION”) FOR THE SIMPLE NONLINEAR HAMILTONIAN (FINITE ANALOG OF THE KERR MEDIUM, H = S2 Z ). THE QUASICLASSICAL APPROXIMATION FOR THE WIGNER FUNCTION AND THE CORRESPONDING EVOLUTION OF MEAN VALUES ARE CONSIDERED AND COMPARED WITH THE EXACT AND CLASSICAL SOLUTIONS.
Despite its indisputable merits, the Wigner phase-space formulation has not been widely explored for systems with SU(1,1) symmetry, as a simple operational definition of the Wigner function has proved elusive in this case. We capitalize on unique properties of the parity operator, to derive in a consistent way a bona fide SU(1,1) Wigner function that faithfully parallels the structure of its continuous-variable counterpart. We propose an optical scheme, involving a squeezer and photon-number-resolving detectors, that allows for direct point-by-point sampling of that Wigner function. This provides an adequate framework to represent SU(1,1) states satisfactorily.
Despite the indisputable merits of the Wigner phase-space formulation, it has not been widely explored for systems with SU(1,1) symmetry, as a simple operational definition of the Wigner function has proved elusive in this case. We capitalize on the unique properties of the parity operator, to derive in a consistent way a \emph{bona fide} SU(1,1) Wigner function that faithfully parallels the structure of its continuous-variable counterpart. We propose an optical scheme, involving a squeezer and photon-number-resolving detectors, that allows for direct point-by-point sampling of that Wigner function. This provides an adequate framework to represent SU(1,1) states satisfactorily.