Wigner function for SU(1,1)

U. Seyfarth1, A. B. Klimov2, H. de Guise3, G. Leuchs1,4, and L. L. Sanchez-Soto1,5

1Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany
2Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico
3Department of Physics, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada
4Institute for Applied Physics, Russian Academy of Sciences, 630950 Nizhny Novgorod, Russia
5Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain

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In spite of their potential usefulness, Wigner functions for systems with SU(1,1) symmetry have not been explored thus far. We address this problem from a physically-motivated perspective, with an eye towards applications in modern metrology. Starting from two independent modes, and after getting rid of the irrelevant degrees of freedom, we derive in a consistent way a Wigner distribution for SU(1,1). This distribution appears as the expectation value of the displaced parity operator, which suggests a direct way to experimentally sample it. We show how this formalism works in some relevant examples.

$\textbf{Dedication}$: While this manuscript was under review, we learnt with great sadness of the untimely passing of our colleague and friend Jonathan Dowling. Through his outstanding scientific work, his kind attitude, and his inimitable humor, he leaves behind a rich legacy for all of us. Our work on SU(1,1) came as a result of long conversations during his frequent visits to Erlangen. We dedicate this paper to his memory.

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Cited by

[1] Andrei B Klimov, Ulrich Seyfarth, Hubert de Guise, and Luis L Sánchez-Soto, "SU(1, 1) covariant s-parametrized maps", Journal of Physics A: Mathematical and Theoretical 54 6, 065301 (2021).

[2] M. Riabinin, P. R. Sharapova, T. J. Bartley, and T. Meier, "Generating two-mode squeezing with multimode measurement-induced nonlinearity", arXiv:1912.09097.

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