The Exact Curve Equation for Majorana Stars

Majorana stars are visual representation for a quantum pure state. For some states, the corresponding majorana stars are located on one curve on the Block sphere. However, it is lack of exact curve equations for them. To find the exact equations, we consider a superposition of two bosonic coherent states with an arbitrary relative phase. We analytically give the curve equation and find that the curve always goes through the North pole on the Block sphere. Furthermore, for the superpositions of SU(1,1) coherent states, we find the same curve equation.


Star equation for the STCS and its solution.
To find exact curve equation for Majorana stars, we consider the following state, i.e., the STCS, where ϑ 1 is the phase difference between two coherent states α and β , and the coherent state is given by k k 2 0 2 where α is a complex number and | 〉 k is Fock state. The star equation for the STCS is  . So, the Eqs (5) and (6) can be reduced to n From above two equations, we know that θ n and φ n has no relation with ϑ 1 , moreover φ n is a constant if the total value of θ α and θ β is fixed. In this case, the stars' distributions are independent of the relative phase ϑ 1 between the two component states. A more particular case is the situation of α β = . the Eqs (9) and (10) are further reduced to These two equations indicate that there is always exist a certain value for γ n , and this certain value always leads to θ = 0 n . It means that the curve of STCS must through the North Pole. This characteristic of the curve will be displayed in figures later. Otherwise, from Eq. (14), we do know the curve which is composed of stars will rotate θ α about Z-axis when θ ≠ α 0.
To further obtain the exact curve equation, we set π ϑ = 0, 2 . For ϑ = 0 2 , from the Eqs (5) and (6), we can obtain n Because Eqs (15) and (16) are the functions on γ n , we can obtain the relation between θ tan 2 2 n and φ  tan 2 by eliminating γ n . Finally, we obtain Equation (18) is the exact equation of the curve. It is only dependent on the polar and azimuth angles and from this equation, all the stars can be determined on the Bloch sphere.
The above we have given the curve equation for N , which takes any positive integer. For coherent states, → ∞ N , based on the Eq. (7), we obtain ≈ R 1, since α β | | − | | 2 2 is a finite value. In this case, the Eq. (18) becomes a simpler form where λ, θ α and α | | are the constants. This is a curve equation for the superposition of two coherent states. This curve includes infinite stars θ φ ( , ) n n , n = 1, 2, 3, …., ∞ on the Bloch sphere, and we will give the corresponding figures in the next section. Based on the Eq. (21), we know that the curve on the Bloch sphere has no relation with N and the phase difference ϑ 1 between α and β . It means that, when the value of N or ϑ 1 change, the curve's shape and location on the Bloch sphere are invariant. Furthermore, combining the Eq. (21)   where x, y are the same as Eqs (19) and (20). Moreover, the corresponding curve equation of → ∞ N can be given as By now, we have derived the curve equation in the case of π ϑ = 0, 2 for the STCS. But for other values of ϑ 2 , it is hard to deduce an exact expression of the curve's equation. Meanwhile, we have to emphasize that a random curve given on the bloch sphere can not be regarded as the expression of the STCS.

Numerical results of the curve equation.
We have presented analytical result for the STCS. In this section, we will further give the numerical results about the STCS in the case of no argument difference (ϑ = 0 2 ) between parameters α and β.
In Fig. 1, we give the two-dimensional images about θ and φ with different α | | and β | |. From the Fig. 1, we know that the curve of STCS through the Northern Pole. And the curve is only located on the northern hemisphere in case of α β + ≥ 4 2 , but for α β + < 4 2 , the curve through the northern and southern hemisphere. In addition, comparing Fig. 1-(b) with Fig. 1-(c), the shape and location of the curve is invariant as long as α β + 2 fixed. Hence the numerical results mentioned above are consistent with the theory in Eq. (22).
In Fig. 2, we find that the location of the curve rotate around the northern pole with the increase of θ α and θ β while the shape and size is invariant. This numerical result is consistent with the theoretical result in Eq. (22).
In this section, we've discussed the impacts on the curve from parameters α, β, θ α and θ β with the aid of numerical calculation. And the numerical result is exactly consistent with theoretical result.

Methods
For a single-mode pure bosonic state For the state given in Eq. (1), the expansion coefficients can be obtained as, Substituting the above equation into Eq. (28) leads to the following equation for locating stars, i and z is the root of this star equation. After simplifying, we find out the n-th root of z satisfying the relation is a real number, and ϑ 2 is the argument difference between two coherent parameters α and β. By now, we have given the roots of star equation for STCS with H-W symmetry. And these roots z n can be mapped to the stars on Block sphere via relation n n i n n n where θ n and φ n are the spherical coordinates. The spherical coordinates can be calculated as Eqs (5) and (6).

Discussions
In conclusion, for giving curve equation of Majorana stars, we have examined the STCS. For this state, we have obtained exact equations of curve for stars in the case of no argument difference between two parameters of coherent states α, β, or with augment difference π. These analytic results agree with numerical calculations. Meanwhile, we have shown that the curves through the North Pole. We have further examined the superposition states of two SU(1,1) coherent states, and found the same curve equations. The details are presented in the Appendix. In our investigations, two arbitrary coherent states are superimposed together with equal probability. For the STCS with different probability amplitudes, our method can directly apply.

Appendix: The curve equation for the STCS of SU(1,1) system
Based on ref. 32 , we know that for the state where Γ is the Gamma-function, and k is Bargmann index, k n the corresponding star equation is This is the curve equation for the STCS of SU(1,1) system. Comparing these two equations with the Eqs (22) and (24) for the STCS of bosonic system, we find that they have the same equation form.