Majorana stellar representation for mixed-spin $(s,\frac{1}{2})$ systems

TL;DR

This paper introduces a geometric method using Majorana stellar representation to visualize and analyze mixed-spin systems, decomposing them into multiple spins and representing their states with star trajectories on a Bloch sphere.

Contribution

The authors develop a practical approach to represent mixed $(s, 1/2)$ spin systems on a Bloch sphere, extending Majorana's stellar representation to high-spin coupled systems.

Findings

Effective visualization of mixed-spin states using Majorana stars.

Symmetric patterns observed in star trajectories on the Bloch sphere.

Unveiled properties of high-spin systems through geometric analysis.

Abstract

By describing the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, Majorana's stellar representation provides an intuitive geometric perspective to comprehend a quantum system with high-dimensional Hilbert space. However, the problem of the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, we present a practical method to resolve the problem for the mixed-spin $(s, 1/2)$ system. The system can be decomposed into two spins: spin-$(s+1/2)$ and spin-$(s-1/2)$ at the coupling bases, which can be regarded as independent spins. Besides, we may write any pure state as a superposition of two orthonormal states with one spin-$(s+1/2)$ state and the other spin-$(s-1/2)$ state. Thus, the whole state can be regarded as a state of a pseudo spin-$1/2$. In this way, the mixed spin decomposes into three spins. Therefore, we can represent the state by $(2s+1)+(2s-1)+1=4s+1$ sets of stars on a Bloch sphere. Finally, to demonstrate our theory, we give some examples that indeed show laconic and symmetric patterns on the Bloch sphere, and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on a Bloch sphere.

Equations

$$|\psi {⟩}_j$$

$$\sum _{n=0}^{2j}(-1{)}^n{\left(\genfrac{}{}{0ex}{}{2j}{n}\right)}^{1/2}{C}_j^{(n)}{z}^n=0,$$

$$z=\tan \frac{\theta }{2}{e}^{\text{i}\varphi },\theta \in [0,\pi ],\varphi \in [0,2\pi ],$$

$$|\psi {⟩}_{\frac{1}{2},\frac{1}{2}}=a|\uparrow \uparrow ⟩+b|\downarrow \uparrow ⟩+c|\uparrow \downarrow ⟩+d|\downarrow \downarrow ⟩.$$

$$|\psi {⟩}_{\frac{1}{2},\frac{1}{2}}=\sqrt{\frac{2-|b-c{|}^2}{2}}|\Uparrow ⟩+\frac{-b+c}{\sqrt{2}}|\Downarrow ⟩,$$

$$|\psi ⟩=C_{s+\frac{1}{2}}|\psi {⟩}_{s+\frac{1}{2}}+C_{s-\frac{1}{2}}|\psi {⟩}_{s-\frac{1}{2}},$$

$$z_{s,\frac{1}{2}}=\frac{C_{s-\frac{1}{2}}}{C_{s+\frac{1}{2}}},$$

$$z_1^{(\pm )}$$

$$z_{\frac{1}{2}}^{(1)}$$

$${\text{J}}^2={\text{S}}^2+\frac{1}{4}{\mathbf{\sigma }}^2+{\text{S}}_z{\sigma }_z+({\text{S}}_{+}{\sigma }_{-}+{\text{S}}_{-}{\sigma }_{+}),$$

$${\text{S}}_{+}|n{⟩}_s=\sqrt{(n+1)(2s-n)}|n+1{⟩}_s,$$

$${\text{S}}_{-}|n{⟩}_s=\sqrt{n(2s-n+1)}|n-1{⟩}_s,$$

$${\text{S}}_z|n{⟩}_s=(-s+n)|n{⟩}_s,$$

$$H={\text{J}}^2,$$

$$H={\oplus }_{n=0}^{2s+1}{A}_n,$$

$$A_n=(s^2+s+\frac{1}{4})I+(s-n+\frac{1}{2}){\sigma }_z+\sqrt{n(2s-n+1)}{\sigma }_x$$

$$|n{⟩}_{s+\frac{1}{2}}$$

$$|n{⟩}_{s-\frac{1}{2}}$$

$$|n{⟩}_1|0{⟩}_2$$

$$|n-1{⟩}_1|1{⟩}_2$$

$$|\psi ⟩$$

$$=D_{0,0}|0{⟩}_1|0{⟩}_2+D_{2s,1}|2s{⟩}_1|1{⟩}_2+\sum _{n=1}^{2s}\left(D_{n-1,1}|n-1{⟩}_1|1{⟩}_2+D_{n,0}|n{⟩}_1|0{⟩}_2\right)$$

$$=D_{0,0}|0{⟩}_1|0{⟩}_2+D_{2s,1}|2s{⟩}_1|1{⟩}_2+\sum _{n=1}^{2s}\left(E_{n,s+\frac{1}{2}}|n{⟩}_{s+\frac{1}{2}}+F_{n,s-\frac{1}{2}}|n{⟩}_{s-\frac{1}{2}}\right)$$

$$=\sum _{n=0}^{2s+1}{E}_{n,s+\frac{1}{2}}|n{⟩}_{s+\frac{1}{2}}+\sum _{n=1}^{2s}{F}_{n,s-\frac{1}{2}}|n{⟩}_{s-\frac{1}{2}},$$

$$E_{n,s+\frac{1}{2}}$$

$$F_{n,s-\frac{1}{2}}$$

$$|\psi {⟩}_{\frac{1}{2},\frac{1}{2}}=e^{-\text{i}{H}_{1}t}(\cos \phi |\uparrow ⟩+\sin \phi |\downarrow ⟩)\otimes (\sin \phi |\uparrow ⟩+\cos \phi |\downarrow ⟩),$$

$$z_1^2{e}^{-\text{i}\delta t}\sin (\phi )\cos (\phi )-z_1{e}^{\text{i}(\delta -2)t}+e^{-\text{i}\delta t}\sin (\phi )\cos (\phi )=0,$$

$${\theta }_{+}+{\theta }_{-}=\pi ,$$

$${\varphi }_{+}+{\varphi }_{-}=0,$$

$$\displaystyle\theta_{\pm}=2\arctan[\big{|}(e^{-2\text{i}t(\delta-1)}\pm\frac{\sqrt{2}}{2}\sqrt{\cos(4\varphi)+2e^{-4\text{i}t(\delta-1)}-1})\csc(2\varphi)\big{|}],$$

$$\displaystyle\phi_{\pm}=\arctan\big{(}\frac{\text{Im}[(e^{-2\text{i}t(\delta-1)}\pm\frac{\sqrt{2}}{2}\sqrt{\cos(4\varphi)+2e^{-4\text{i}t(\delta-1)}-1})\csc(2\varphi)]}{\text{Re}[(e^{-2\text{i}t(\delta-1)}\pm\frac{\sqrt{2}}{2}\sqrt{\cos(4\varphi)+2e^{-4\text{i}t(\delta-1)}-1})\csc(2\varphi)]}\big{)},$$

$$z_{\frac{1}{2},\frac{1}{2}}$$

$$\theta$$

$$\varphi$$

$$|\psi {⟩}_{1,\frac{1}{2}}$$

$$\otimes (\cos \phi |\uparrow ⟩+\sin \phi |\downarrow ⟩),$$

$$|\psi {⟩}_{1,\frac{1}{2}}$$

$$\displaystyle\quad+C_{3}^{(\frac{3}{2})}\left|\frac{3}{2},\frac{3}{2}\right\rangle\bigg{]}+\bigg{[}C_{0}^{(\frac{1}{2})}\left|\frac{1}{2},-\frac{1}{2}\right\rangle+C_{1}^{(\frac{1}{2})}\left|\frac{1}{2},\frac{1}{2}\right\rangle\bigg{]},$$

Full text

I Introduction

It is acknowledged that the evolution of an arbitrary two-level state can be exactly represented by the trajectory of a point on the Bloch sphere Bloch and Rabi (1945); Niu (2012); Schwinger (1965). Applied to a quantum state in a high-dimensional Hilbert space, this geometric interpretation seems difficult to imagine. Despite we can map the quantum pure state to a higher-dimensional geometric structure, this process is no more an intuitive and legible way to comprehend it. Fortunately, the Majorana’s stellar representation (MSR) builds us a wide bridge between the high dimensional projective Hilbert space and the two-dimensional Bloch sphere Majorana (1932). Employing the MSR, which represents a quantum pure state of spin-$J$ systems in terms of a symmetrized state of 2$J$ spin-$1/2$ systems, one can generalize this geometric approach to large spin systems or multilevel systems. Majorana’s perspective was that the evolution of a spin-$J$ state can be intuitively described by trajectories of 2$J$ points on the two-dimensional (2D) Bloch sphere, with these 2$J$ points generally coined as Majorana stars (MSs), rather than one point on an intricate high-dimensional geometric structure. Therefore, this representation spontaneously provides an intuitive way to study high spin systems from geometrical perspectives, which has made the MSR a useful tool in many different fields, e.g., classification of entanglement in symmetric quantum states Mäkelä and Messina (2010a, b); Ganczarek et al. (2012); Ribeiro and Mosseri (2011a); Markham (2011); Mandilara et al. (2014), analyzing the spectrum of the Lipkin-Meshkov-Glickmodel Ribeiro et al. (2007, 2008), studying Bose condensate with high spins Barnett et al. (2006, 2007); Yang et al. (2015); Stamper-Kurn and Ueda (2013); Kawaguchi and Ueda (2012); Lian et al. (2012); Cui et al. (2013), and calculating geometrical phases of large-spin systems Bruno (2012); Liu and Fu (2014).

Moreover, the MSR yields many useful insights for high dimensional quantum states. The Berry phase, which is a unique character of a quantum state Berry (1984) and has become a central unifying concept for quantum state Chruściński and Jamiołkowski (2004); Bohm et al. (2003), unveils the gauge structure associated with cyclic evolution in Hilbert space Simon (1983). When it comes to an arbitrary two-level state, the Berry phase is simply proportional to the solid angle subtended by the close trajectory of a point on the Bloch sphere, while every star in the MSR will trace out its own trajectory on the Bloch sphere for a cyclic evolution of a large spin state. For example, the Majorana stars can be driven moving periodically on the Bloch sphere, and making up the so-called Majorana spin helix Cui et al. (2013) by the spin-orbit coupling in high-spin condensates. Consequently, by asking what the explicit relation between the Berry phase and the Majorana stars helixes or loops is, it has turned into a significant topic in recent years Hannay (1998); Liu and Fu (2014); Bruno (2012); Tamate et al. (2011); Ogawa et al. (2015).

Except for the Berry phase, entanglement is another important unique character of a many-particle quantum state. Although the classification and measure are quite complex Coffman et al. (2000); Miyake (2003); Dür et al. (2000); Acín et al. (2001); Mandilara et al. (2014) for the multiqubit states, the MSR naturally provides an intuitive way to consider the multiqubit entanglement Liu and Fu (2016), since a spin-$J$ state is equivalent to a symmetric 2$J$-qubit state. The distribution of the Majorana stars not only disclose the relationship between the symmetry of the state and the multipartite entanglement measures, including geometric measures Aulbach et al. (2010); Markham (2011); Wang and Markham (2012, 2013) and Barycentric measures Ganczarek et al. (2012), but also can be employed to investigate entanglement classes Bastin et al. (2009); Mathonet et al. (2010), entanglement invariants Ribeiro and Mosseri (2011b), and so on. Hence, it is another challenge task to connect the quantum entanglement of the qubits to the distribution of the Majorana stars on the Bloch sphere.

Furthermore, it is interesting to apply this approach to study the multiband topological systems. For a two-band system, e.g., the Su-Schrieffer-Heeger (SSH) model Su et al. (1979, 1980); Su and Schrieffer (1981), the geometrical meaning of topologically different phases can be revealed by their distinct trajectories Ryu and Hatsugai (2002); Delplace et al. (2011),with mapping the Bloch state into a 2D Bloch sphere. As a paragon topological model Shen (2012), the SSH model supports either topologically trivial or nontrivial phase, characterized by the quantized Berry phase 0 or $\pi$ Berry (1984); Zak (1989); Xiao et al. (2010), verified in the recent cold atom experiment Atala et al. (2013).

From Refs. Majorana (1932); Liu et al. (2017); Yao et al. (2017), an arbitrary pure state for spin $J$ can be represented by

[TABLE]

where $j$ is the angular quantum number, $|n\rangle_{j}\equiv|-j+n\rangle_{j}$ is the basis state and $C_{j}^{(n)}$ is the corresponding coefficient. The star equation is given by Majorana (1932)

[TABLE]

where $z$ denotes the characteristic variable and we also have the normalization condition $\sum_{n=0}^{2j}|C^{(j)}_{n}|^{2}=1$. The root $z$ can be mapped to the stars on Bloch sphere via relation

[TABLE]

where $\theta$ and $\phi$ are the spherical coordinates.

The two-spin coupling system (a large spin coupling with a small spin) is significant in many physical fields, such as quantum criticality Quan et al. (2006) and quantum dynamics Ahmadi et al. (2010). However, the representation of a two-spin coupling system has been studied fragmentarily Fontana (1962); De Raedt and De Raedt (1983); Haaker et al. (2014), which did not generalize to an arbitrary spin-$s$ and was not visible enough. In this work, we chose mixed-spin $(s,1/2)$ systems as paragons to take advantage of the fact that it can be decomposed into two spins: spin-$(s+1/2)$ and spin-$(s-1/2)$. Benefiting from the MSR, we represent an arbitrary pure state on a Bloch sphere. Besides, we propose a practical method to decompose the arbitrary pure state that can be regarded as a state of a pseudo spin-$1/2$. For the arbitrary pure state of mixed spin, we always can decompose it into two spins: spin-$(s+1/2)$ and spin-$(s-1/2)$. Consequently, the arbitrary pure state of the mixed spin can be regarded as a state of a pseudo spin-1/2. In this way, the mixed spin decomposes into three spins and our task is resolving these star equations and obtaining $(2s+1)+(2s-1)+1=4s+1$ sets of stars.

Our study provides an intuitive perspective of a two-spin ($s$ and $1/2$) system, taking advantage of the MSR, and unveils the intrinsic property of the two-spin system on a Bloch sphere, which shall deepen our comprehension of the spin-$(s,1/2)$ system. The paper is organized as follows. In Sec. II, we study the the fundamental theory of the MSR, utilizing coupling bases to describe an arbitrary pure state that is spin-$(s,1/2)$. In Sec. III, we show concise examples for the MSR of the mixed-spin $(s,1/2)$ systems. A brief discussion and summary are given in Sec. IV.

II Theory of Majorana representation for mixed-spin $(s,\frac{1}{2})$ systems

In this section, we will study the fundamental theory of the MSR for the mixed-spin $(s,1/2)$ systems. Firstly, we illustrate with the mixed spin-$(1/2,1/2)$ system to give a legible view of our main logic. Besides, we introduce the coupling bases to present an arbitrary mixed spin- $(s,1/2)$ state as two independent spins. Finally, benefiting from the new form, we can construct a two-level system to describe the arbitrary pure state.

For two spin-$1/2$ particles, an arbitrary pure state can be represented by

[TABLE]

We can rewrite Eq. 4

[TABLE]

where $\left|\Uparrow\right\rangle=[a\left|\uparrow\uparrow\right\rangle+d\left|\downarrow\downarrow\right\rangle+(b+c)(\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle)/2]/\sqrt{1-|b-c|^{2}/2}$, $\left|\Downarrow\right\rangle=(\left|\uparrow\downarrow\right\rangle-\left|\downarrow\uparrow\right\rangle)/\sqrt{2}$. Then we notice that the terms of the bracket are the analogous triplet $|\psi\rangle^{j=1}$ and the last term is the analogous singlet $|\psi\rangle^{j=0}$.

For more general case, we can treat the system as a two-level energy system, which can be represented as two sets of stars on a Bloch sphere if it is a pure state. To normalize the states, we rewrite Eq. 5 as

[TABLE]

where $|\psi\rangle_{s+1/2}$ and $|\psi\rangle_{s-1/2}$ represent the normalized analogous triplet and the normalized analogous singlet. We also have the normalization condition $|C_{s+1/2}|^{2}+|C_{s-1/2}|^{2}=1$, so we only have one independent variable to describe the relation between them, which means that it needs one star on the Bloch sphere to be represented. Based on Eq. 6, we have the star (root) of the pseudo spin

[TABLE]

which reveals the relation between the analogous triplet and the analogous singlet. Therefore, we need $(2s+1)$ sets of stars to represent the analogous triplet, $(2s-1)$ sets of stars to represent the analogous singlet and one set of stars to combine the two parts on a Bloch sphere.

From Eqs. 1 and 2, we can get all stars

[TABLE]

where $z_{1}^{(\pm)}$ and $z_{1/2}^{(1)}$ are respective roots of the analogous triplet and the analogous state of the pseudo spin. Therefore, we need two sets of stars to represent the analogous triplet, no stars to represent the analogous singlet and one set of stars to combine the two parts on a Bloch sphere.

Now, we show a general method to gain the construct inside of the pseudo spin-1/2. For the mixed-spin $(s,1/2)$ systems, we have the total angular momentum $\textbf{J}=\textbf{S}+\bm{\sigma}/2$ (we choose $\hbar$=1), and S, $\bm{\sigma}/2$ are the angular momentums of the large spin-$s$ and small spin-$1/2$, respectively. Therefore, we have

[TABLE]

where the angular momentum $\textbf{S}=\textbf{S}_{x}+\textbf{S}_{y}+\textbf{S}_{z}$, the raising (lowering) operator $\textbf{S}_{+}=\textbf{S}_{x}\pm\text{i}\textbf{S}_{y}$ and $|n\rangle_{s}\equiv|-s+n\rangle_{s}$ is the basis state. Besides, we write the uncoupling basis state as $\left|n\right\rangle_{s}\bm{\otimes}|1\rangle_{1/2}\equiv|n\rangle_{1}|1\rangle_{2}$, $\left|n\right\rangle_{s}\bm{\otimes}|0\rangle_{1/2}\equiv|n\rangle_{1}|0\rangle_{2}$, and $n=0,1,\ldots,2s$.

Then we have the Hamiltonian

[TABLE]

and its vector space $W$. We find the space $V_{n}$ is the two-dimensional invariant subspace of the vector space $W=\bm{\oplus}^{2s}_{n=1}V_{n}\bm{\oplus}V_{0}\bm{\oplus}V_{2s+1}$, where $V_{0}$ and $V_{2s+1}$ are one-dimensional subspaces. The bases of the subspace $V_{n}$ $\left(1\leq n\leq 2s\right)$ are $|n\rangle_{1}|0\rangle_{2}$ and $\left|n-1\right\rangle_{1}|1\rangle_{2}$. The bases of the subspace $V_{0}$ and $V_{2s+1}$ are $|0\rangle_{1}|0\rangle_{2}$ and $|2s\rangle_{1}|1\rangle_{2}$, respectively.

From Eqs. 10, 11, 12, 13 and 14, We can rewrite the Hamiltonian as

[TABLE]

where the two-dimensional matrix

[TABLE]

for $n=1,\ldots,2s$, $A_{0}=A_{2s+1}=(s^{2}+2s+3/4)$. From Eq. 16, we diagonalize $A_{n}$ and get the eigenvalue $j_{\pm}(j_{\pm}+1)$ ($j_{\pm}=s\pm 1/2$) of the $A_{n}$, and the relationship between the coupling and the uncoupling representations

[TABLE]

where $n=1,\ldots,2s$, $|n\rangle_{s+1/2}$ and $|n\rangle_{s-1/2}$ denote the coupling basis states.

From Eqs. 17 and 18, we can get the uncoupling basis states

[TABLE]

where $n=1,\ldots,2s$.

For a mixed spin-$(s,1/2)$ system, using Eqs. 19 and 20, we can rewrite the arbitrary state

[TABLE]

where

[TABLE]

and $n=m+1$, $D_{m,1}=D_{n-1,1}$, $D_{n,0}$ are the coefficients of the eigenstates, respectively. We notice that the terms of the bracket are the analogous multiplet and the last term is the analogous singlet.

III Examples for Majorana representation for mixed-spin $(s,\frac{1}{2})$ systems

In this section, we will show concise examples for the MSR of the mixed-spin $(s,1/2)$ systems. Firstly, we will give an analytical example for the mixed-spin $(1/2,1/2)$ systems. In subsection. B, we will give brief examples with a real phase parameter $\varphi$, time fixed case, and with time evolution, fixed real phase parameter $\varphi$ case for the mixed- spin $(s,1/2)$ systems, respectively.

III.1 An Example for mixed-spin $(\frac{1}{2},\frac{1}{2})$ systems with a real phase parameter $\varphi$ and time evolution

In this subsection, we will show a concise example for the mixed-spin $(s,1/2)$ systems with a real phase parameter $\varphi$.

From Eqs. 8, 9 and 3, we use two sets of stars to represent the analogous triplet, no star to represent the analogous singlet and one set of stars to combine the two parts on a Bloch sphere.

To illustrate the idea above, we give a simple spin-$(1/2,1/2)$ system with time evolution:

[TABLE]

where the Hamiltonian $H_{1}=\sigma_{1x}\sigma_{2x}+\sigma_{1y}\sigma_{2y}+\delta\sigma_{1z}\sigma_{2z}$, $\delta\in[0,1]$ and the real phase parameter $\varphi\in(0,\pi/2)\cup(\pi/2,\pi)\cup(\pi,3\pi/2)\cup(3\pi/2,2\pi)$, considering the completeness condition.

Therefore, we gain triplet stars

[TABLE]

where $\theta_{\pm}$ and $\phi_{\pm}$ are the azimuth angle and zenith angle of the triplet stars. Obviously, we can find that the two sets of the stars are $180^{\circ}$ rotational symmetric around the $x$-axis (the intersecting line of the prime meridian plane and the equatorial plane).

However, singlet state have no star. The stars (roots) of the pseudo spin

[TABLE]

Hence, we know that the stars of the pseudo spin are always mapped to the north of the prime meridian and independent on time evolution. To show the idea above, we give a brief example of the two spin-$1/2$ particles with the real phase parameter $\varphi$, time evolution and $\delta=0$.

Fig. 1 shows the stars of the two spin-$1/2$ particles represented in Bloch sphere with real phase parameter $\varphi$, time evolution and $\delta=0$. Without time evolution, because the roots ($z_{1}$) in our example are always real, with the variety of the real phase parameter $\varphi$, each set of the triplet stars is mapped to the north or south prime meridian. Intuitively, the stars of the pseudo spin only cover the north prime meridian. Since star equation of the analogous singlet don’t have root in two spin-$1/2$ case, we don’t have singlet state star [Fig. 1]. Fixed the time at a special value, we can distinctly find that one set of the triplet stars and the others are $180^{\circ}$ rotational symmetric around the $x$-axis (the intersecting line of the prime meridian plane and the equatorial plane) [Figs. 1 and 1]. And the triplet stars are mapped to the prime meridian at $t=2k\pi/4$ , the $90^{\circ}W(E)$ meridian at $t=(2k+1)\pi/4$ ($k$ is integer). Moreover, the region that the two sets of stars are mapped to will shift from one hemisphere to the others [Fig. 1]. Fixed the phase parameter $\varphi$ at a special value, since stars of the pseudo spin are independent of time evolution, they are fixed at particular positions [Figs. 1 and 1]. Meanwhile, each set of the triplet stars is plane symmetry around the equatorial plane and the prime meridian plane at a whole period $T=\pi$, and each set still keeps the rotational symmetry at every single time and phase parameter value.

III.2 Examples for mixed-spin $(s,\frac{1}{2})$ systems with a real phase parameter $\varphi$ and time evolution

In subsection. B, we will give brief examples with a real phase parameter $\varphi$, time fixed case, and with time evolution, fixed real phase parameter $\varphi$ case for the mixed- spin $(s,1/2)$ systems, respectively. To illustrate the idea above, let’s consider the example spin-$1$ and spin-$1/2$ case as

[TABLE]

where the Hamiltonian $H_{2}=S_{1x}S_{2x}+S_{1y}S_{2y}+\delta S_{1z}S_{2z}$, $S_{1i}$ (respectively, $S_{2i}$, $i=x,y,z$) are the three direction components of the larger spin (respectively, spin-$1/2$), $|1\rangle,|0\rangle,|-1\rangle$ are the eigenstate of the spin-1 particle, and $\left|\uparrow\right\rangle,\left|\downarrow\right\rangle$ are the eigenstate of the spin-$1/2$ particle. So we have unitary transformation operator $U=\exp{(-\text{i}H_{2}t)}$.

Utilizing Eqs. 1 and 24, we have

[TABLE]

where $C_{n}^{(3/2)}$ and $C_{n}^{(1/2)}$ denote the coefficients of the $j=3/2$ and $j=1/2$ case, respectively.

From Eq. 2, we obtain

[TABLE]

where $z_{j}$, $z_{1,1/2}$ denote the characteristic variables of the spin-$j$ case and the pseudo spin case, respectively, and we also have the normalization condition $|C_{1+1/2}|^{2}+|C_{1-1/2}|^{2}=1$.

Solving Eqs. 34 and 35, we can easily gain the whole bloch representation of the spin-$s$ and spin-$1/2$ particles system with Eq. 3. To illustrate the idea above, we show the bloch representation of the spin-$s$ and spin-$1/2$ particles system with time evolution, the real phase parameter $\varphi$ and $\delta=0$.

Fig. 2 shows the stars of the spin-$s$ and spin-$1/2$ particles system with time evolution, fixed real phase parameter $\varphi$ and $\delta=0$. If the coefficients of the cubic equation are complex numbers, which means with time evolution, the roots are complex numbers generally. Obviously, if the roots are real numbers, the stars will be mapped to the prime meridian, or conversely. As shown in Figs. 2 and 2, we can use three stars, one star and one star to represent the analogous mutiplet stars, the analogous singlet star and the star of the pseudo spin, respectively. Moreover, focusing on one period of time, one set of the dots stars (mutiplet) will form a pattern that is plane symmetry around the prime meridian plane (red dots) and the others compose one pattern that have the same property together. Besides, all sets of the analogous mutiplet stars constitute closed curves together. While the blue stars (analogous singlet) only form one closed curves [Figs. 2 and 2]. However, unlike other sets of stars, the stars of the pseudo spin are mapped to the prime meridian and have the northernmost (the north pole) and southernmost position at $\varphi=\pi/4+2k\pi$ with $t=n\sqrt{2}\pi$ and $\varphi=5\pi/4+2k\pi$ with $t=(2n+1)\pi/\sqrt{2}$ ($k,n$ are integers), respectively [Figs. 2 and 2].

Fig. 3 shows the stars of the spin-$s$ and spin-$1/2$ particles system with the real phase parameter $\varphi$, fixed time and $\delta=0$. As the analysis above, the complex coefficients of the cubic equation generally lead to the complex roots, only if the roots are real numbers, the stars will be mapped to the prime meridian, or conversely. As shown in Fig. 3, we can use three stars, one star and one star to represent the analogous mutiplet stars, the analogous singlet star and the star of the pseudo spin, respectively. Without time evolution, two sets of the analogous mutiplet stars are mapped to the quarter of the prime meridian [Fig. 3]. Meanwhile, the other set of the analogous mutiplet stars and the analogous singlet stars are mapped to the half prime meridian [Figs. 3 and 3]. However, the stars of the pseudo spin can not form the half prime meridian, completely [Fig. 3]. Besides, the result is exactly the same with the result with $\delta=1$ and $t=0$ case, in other words, it is independent of time evolution when $\delta=1$ because of the symmetry of the three spin directions. As shown in Figs. 3 and 3, the stars of the pseudo spin have the northernmost (the north pole) and southernmost position at $\varphi=\pi/4+2k\pi$ with $t=n\sqrt{2}\pi$ and $\varphi=5\pi/4+2k\pi$ with $t=(2n+1)\pi/\sqrt{2}$ ($k,n$ are integers), respectively. Furthermore, we find that all sets of the analogous mutiplet stars form closed curves that are $180^{\circ}$ rotational symmetric around the $x$-axis (the intersecting line of the prime meridian plane and the equatorial plane) [Fig. 3]. Similarly, the analogous singlet stars compose four closed curves that start from the north pole and end with south pole, and the pattern which the curves form is $180^{\circ}$ rotational symmetric around the $x$-axis [Fig. 3].

IV Conclusions and discussions

Recently, the Majorana s stellar representation and relevant applications have demonstrated that the distributions and motions of the Majorana stars on the Bloch sphere have become a new and effective tool to study the symmetry-related questions in the high-dimensional or many-body system.Our study here shows that, utilizing the MSR, an arbitrary pure state always can be represented on a Bloch sphere with $4s+1$ stars in a two-spin ($s$ and $1/2$) system. We take the system described by coupling bases as a state of a pseudo spin-$1/2$. Furthermore, we propose a practical method to decompose the arbitrary pure state that can be regarded as a state of a pseudo spin-$1/2$.

As we know, a star on a Bloch sphere can represent a pure state, and a set of stars on a Bloch sphere can represent a high-dimensional pure states. However, this method can not be applied in arbitrary high-dimensional mixed states. Our result is not easy to be generalized to arbitrary high-dimensional mixed-spin systems, such as a mixed-spin $(s,1)$ system. Because we can not directly use an effective pseudo spin $1$ to describe it.

Taking advantage of the MSR, which provides an intuitive way to study high spin system from geometrical perspective, we can have a novel and holonomic view to investigate intricate system. Considering spin-$(1/2,1/2)$ and spin-$(1,1/2)$ systems, we find that the patterns on the Bloch sphere have legible and laconic symmetries. Since we have presented a general method to visualize the mixed-spin $(s,1/2)$ systems, one can further apply the method in many situations, such as quantum transitions and quantum tunneling in quantum spin baths and quantum dots.

V Acknowledgments

We thank Jing Cheng for useful discussions. This work was supported by the National Key Research and Development Program of China (No. 2017YFA0304202 and No. 2017YFA0205700), the NSFC through Grant No. 11875231, and the Fundamental Research Funds for the Central Universities through Grant No. 2018FZA3005.