Abstract
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.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.
Highlights
Phase-space methods represent a self-standing alternative to the conventional Hilbertspace formalism of quantum theory
Quantum mechanics appears as a statistical theory on phase space, which can make the corresponding classical limit emerge in a natural and intuitive manner
Quantum phenomena must be depicted in the proper phase space
Summary
Phase-space methods represent a self-standing alternative to the conventional Hilbertspace formalism of quantum theory. In this approach, observables are c-number functions instead of operators, with the same interpretation as their classical counterparts, composed in novel algebraic ways. We propose a similar way to deal with su(1, 1): starting with two orthonormal modes, and using the standard tools for continuous variables, we eliminate the spurious degrees of freedom and we get a description on the upper sheet of a two-sheeted hyperboloid, which is the natural arena to represent the physics associated to these systems. Our final upshot is that the Wigner function can be expressed as the average value of the displaced parity operator This is reassuring, for it is the case for continuous variables [56]. As this property has been employed for the direct sampling of the Wigner function for a quantum field [57,58,59], our result opens the way for the experimental determination of the Wigner function for SU(1,1)
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