Normal product form of two-mode Wigner operator

Rui He,Xiangfei Wei,Congbing Wu,Xiangyuan Liu,Gang Zhang,Min Kong

doi:10.1038/s41598-022-06124-8

Abstract

In the context of normal product, we use the method of the integration within an ordered product (IWOP) of operators to derive three representations of the two-mode Wigner operator: SU(2) symmetric description, SU(1,1) symmetric description and polar coordinate form. We find that two-mode Wigner operator has multiple potential degrees of freedom. As the physical meaning of the selected integral variable changes, Wigner operator shows different symmetries. In particular, in the case of polar coordinates, we reveal the natural connection between the two-mode Wigner operator and the entangled state representation.

Highlights

In quantum theory, according to the Heisenberg uncertainty principle, one cannot accurately measure the position and momentum of a particle at the same time, that is, one cannot determine a phase point in the phase space
In the phase space of quantum optics, two orthogonal harmonic oscillators can be properly described as the product of the corresponding Wigner operators
The focus of this article is to explore the characteristics of the normal product form of the two-mode Wigner operator

Summary

Wigner operator

Rui He1*, Xiangyuan Liu[1,2], Xiangfei Wei[1], Congbing Wu1, Gang Zhang1 & Min Kong[1]. In Ref.[3–5], the normal product form of two-mode Wigner operator was given as. Similar to “Two-mode Wigner operator for SU(2)” section, we obtain the normally ordered form of the Wigner operator in this case by integrating over the unphysical parameters. × exp[cos θ (a†a − b†b) + sin θ (e−iφa†b + eiφb†a) − (a†a + b†b)] : we have derived polarization related Wigner operator (θ , φ) via suitable marginals of distributions for the field quadratures by removing the degrees of freedom irrelevant for the specification of polarization, which has at least two significant meanings according to Ref.[8]. The result given by Eq (63) shows that the entangled state representation |η is exactly the form of the twodimensional momentum eigenstate |px, py in polar coordinates. Since pr and l are not two independent variables, their scheme must not work

Concluding remarks

Author contributions

Additional information