Normal product form of two-mode Wigner operator

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.

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. Therefore, people naturally think of defining the quasi-distribution function in phase space to study the quantum state and motion of microscopic particles. In 1932, Wigner introduced a quasi-classical distribution function W(q, p) corresponding to the density operator ρ 1 , the marginal distributions of which corresponds to the particle probability measured in the coordinate q space and momentum p space, respectively. This gave the phase space a new meaning and opened the front page of phase space quantum mechanics.
In order to calculate the Wigner function of different quantum states, it is necessary to introduce the Wigner operator. Generally, if we know the density matrix ρ of a system's quantum state, we can calculate the Wigner function W(q, p) of the system by tracing the product of the density operator ρ and the Wigner operator (Wigner kernel) �(q, p) That is, for any state, the Wigner function of the system is the expected value of its Wigner operator. For any pure state ρ = |ψ��ψ| , the Wigner function of the system is W(q, p) = �ψ|�(q, p)|ψ�.For any mixed state ρ = ψ p ψ |ψ��ψ| , the Wigner function of the system is W(q, p) = ψ p ψ �ψ|�(q, p)|ψ�.
In classical optics, two orthogonal harmonic oscillators generate an elliptical motion, which is the simplest Lissajous figure. In the phase space of quantum optics, two orthogonal harmonic oscillators can be properly described as the product of the corresponding Wigner operators �(α) and �(β) are the Wigner operators of single-mode harmonic oscillator in the x and y directions, respectively. �(α, β) is actually a two mode displaced parity operator, and its general form is where a † , a are creation and annihilation operators respectively, D(α) is the standard displacement operator and D(α) = exp(αa † − α * a) . In Ref. 2 , through observing the correlations described by the Wigner function of the Einstein-Podolsky-Rosen state in the joint measurement of the operator �(α, β) , they demonstrated the Wigner function provided direct evidence of the nonlocal character of this state.

Two-mode Wigner operator for SU(2)
In classical optics, a Lissajous figure needs only three independent quantities to be fully characterized: the amplitudes of each oscillator and relative phase between them. Without loss of generality, we introduce the parametrization where χ is a global phase, the radial variable represents the total intensity, and the parameters θ and ϕ can be interpreted as the polar and azimuthal angles, respectively, on the Poincare sphere: θ describes the relative amount of intensity carried by each mode and ϕ is the relative phase between them 9 . Substituting Eq. (5) into Eq. (4), �(α, β) can be recast as www.nature.com/scientificreports/ After writing �(r, χ, θ, ϕ) as the normal product form of Eq. (7), we can use the IWOP method 6,7 to perform the integral operation of the operator. Since all Bose operators are in : : internally, they can be treated as integration parameters, so that the integration can proceed smoothly. In order to eliminate the variables unrelated to the idea of polarization (i.e., total intensity r and global phase χ ), we try to integrate over the two variables r and χ in �(r, χ, θ, ϕ) . Considering the integral measure d 2 αd 2 β ≡ 1 4 r 3 sin θdrdχ dθ dϕ , the integration must be carried out in two steps. In the first step, we remove the physically irrelevant global phase χ by using δ(k − k ′ ) = 2π 0 dχe iχ(k−k ′ ) and have Next, we integrate over the radial variable r to get By using the integral formula Thus, 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 �(r, χ, θ, ϕ) ≡�(α, β) www.nature.com/scientificreports/ has at least two significant meanings according to Ref. 8 . On the one hand, �(θ, ϕ) have an exact correspondence with polarization in classical optics. On the other hand, polarization related Wigner functions provide a feasible approach to examine and measure diverse polarization properties by using diverse experimental procedures, such as homodyne and heterodyne detection, tomography, and atom-field interactions. When θ = 0 , from the Eq. (10), we get Now, we introduce the operator which is defined in terms of the two mode realization of the SU(2) algebra with commutation relations U(ζ ) is the unitary operator representing SU(2) transformation and its normally ordered form is with so that Our purpose is to prove that In fact, according to Eq. (10), we only need to show that The parity operator (−1) b † b can be written as Utilizing the bosonic operator realization of normally ordered product form of SU n group 11,12 , we have  (15), (20) and (21), we can get So, Eq. (19) is proved, which also shows that the two-mode Wigner operator that integrates over the variables r and χ is indeed SU(2) symmetric. For the coherent state |z 1 , z 2 � , we can get its Wigner function immediately For other pure states, as long as the inner product of them and the coherent state can be given, in principle, the Wigner function can be calculated by inserting the completeness relations of the coherent states. Furthermore, for any mixed state, as long as we calculate the expected value of the pure state of its subsystem, we can immediately get its Wigner function.

Two-mode Wigner operator in polar coordinates
Two-dimensional isotropic harmonic oscillator. The Hamiltonian of a two-dimensional isotropic har- g(a, a † ) : : (42) |n x , n y � = (n x !n y !) − 1 2 a †n x b †n y |0, 0�. www.nature.com/scientificreports/ we also treat a ± and a † ± as creation and annihilation operators, whose corresponding number operators are N ± = a † ± a ± . The two commutated observables N + and N − form a set of complete basis. For each group of eigenvalues (n + , n − ) , there exists an eigenvector, which is marked with |n + , n − � and which constitutes a complete set of orthogonal eigenvectors. One can find and which shows that the two commutated observables N and L form a set of complete basis.
Two-dimensional coordinate eigenstates in polar coordinates. We know that in the Cartesian coordinates, the two-dimensional coordinate eigenstate is descriped as Now we write the Eq. (49) as a form in polar coordinates 16  Eq. (51) is a very important result, because |ξ � is actually the entangled state representation created by Fan 17,18 . |ξ � is the continuous variable version of the EPR entangled state, which is introduced in the following way.
Since [X + + X − , P + − P − ] = 0 , where X ± and P ± are the position and momentum operators of the rotating reference frame, respectively, we can give the common eigenstate |ξ � of X + + X − and P + − P − , that is By using the method of IWOP, we can prove its orthogonality and completeness and In Ref. 16 , we have shown that where we have set The RHS of Eq. (55) is a standard Laguerre-Gaussian mode in quantum optics.
�(r, φ) �(r, φ) �(r, φ)  70) and (71) demonstrate that the marginal distribution of the two mode Wigner operator is exactly the pure state density matrix of the entangled state representation. This conclusion first appeared in the Ref. 19 , which tells us that the marginal distributions of the Wigner function for entangled system should be understood in the sense of entanglement. In Ref. 20 , the authors first separated φ from the Wigner operator, and then tried to use the integration method to integrate over the radial momentum p r in the Wigner operator. They wanted to use the above method to obtain the operator kernel function with the radius r and the orbital angular momentum l as variables, that is, �(r, p r , φ, l) → �(r, l) , but their efforts were not successful. We notice that and Since p r and l are not two independent variables, their scheme must not work.

Concluding remarks
Using the IWOP method, we give three approaches to analyze the normal product form of the two-mode Wigner operator, which are SU(2) symmetric representation, SU(1,1) symmetric representation, and polar coordinate form. The two-mode Wigner operator has a variety of intrinsic degrees of freedom, and the IWOP method is more conducive to explaining the intrinsic relationship of these potential degrees of freedom. Our next work may be to use this approach to study the phase or angular momentum properties of the two-mode quantum state in phase space.

Data availablity
The data that support the findings of this study are available from the corresponding author upon reasonable request.