Wigner function of coherent state of N components

In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms： the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius ｜β｜ = ｜α｜ with a separate angle of 2π/N, and the second term is composed of 1/2N（N - 1） Gaussian-cosine surfaces evenly centred in a circular region of radius ｜β｜ 〈 ｜α｜. Here, a is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.

Wigner function of coherent state of N components

In this paper, we study the Wigner function of coherent state of $N$ components, especially two components and three components. This function consists of two terms: the Gaussian term and the interference term with the negativity. The first term comprises $N$ Gaussian surfaces evenly centred on a circle of radius $|\beta|=|\alpha|$ with a separate angle of ${2\pi}/{N}$, and the second term is composed of $\frac{1}{2}N(N-1)$ Gaussian-cosine surfaces evenly centred in a circular region of radius $|\beta|<|\alpha|$. Here, $\alpha$ is the eigenvalue of the annihilation operator $a$, and $\beta$ is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.