Local sampling of the SU(1,1) Wigner function

Abstract

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 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. I. INTRODUCTIONThe phase-space formulation of quantum theory 1-11 stands as a self-contained alternative to the conventional Hilbert-space formalism. In this approach, observables become 𝑐-number functions instead of operators and quantum mechanics appears as a statistical theory. Moreover, it is the most convenient construct for visualizing quantum states and processes.The foundations of the method were laid by Weyl 12 and Wigner. 13 Subsequently, Groenewold 14 and Moyal 15 developed all the indispensable tools that have evolved into an accomplished discipline with applications in many diverse fields.The phase-space picture of systems described by continuous variables, such as Cartesian position and momentum of a harmonic oscillator, gained popularity among the quantum optics community, mostly influenced by the authoritative work of Glauber. 16 In particular, the celebrated quasiprobability distributions, such as the 𝑃 (Glauber-Sudarshan), 17,18 𝑊 (Wigner), 13 and 𝑄 (Husimi) 19 representations, are nothing but the functions connected with the density operator.The formalism has been extended in a natural way to other dynamical symmetries (recall that a Lie group 𝐺, with Lie algebra 𝔤, is a dynamical symmetry if the Hamiltonian of the system under consideration can be expressed in terms of the generators of 𝐺; that is, the elements of 𝔤). Perhaps, the most significant example is that of SU(2), with the Bloch sphere as the underlying phase space, [20][21][22] which is of significance in dealing with two-level systems. [23][24][25][26][27][28] Entrhalling results have also been reported for the Euclidean group E(2), now with the cylinder as phase space: [29][30][31][32] this is of relevance in treating, e.g., the orbital angular momentum of twisted photons. 33,34 Additional developments for more general symmetries have appeared in the literature. [35][36][37][38] Moreover, the basic ideas have been adapted to discrete qudits, where the phase space becomes a finite grid. [39][40][41][42][43][44][45] It is apparent from the previous discussion that a satisfactory description of a physical phenomenon requires the use of a suitable geometric arena. Surprisingly, the phase-space description of systems with SU(1,1) dynamical symmetry has received comparatively little attention, 46,47 even if SU(1,1) plays a crucial role in connection with what has been br...